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\section{Introduction}
A combined measurement of the hyperfine structure (HFS) splittings in hydrogenlike and lithiumlike ions of $^{209}$Bi has been suggested as early as 2001 \cite{Shabaev:01a} to be a sensitive probe for bound-state strong-field QED in the strongest static magnetic fields available in the laboratory. Such fields exist in the surrounding of heavy nuclei with nuclear spin and a large nuclear magnetic moment. The electron in H-like $^{209}$Bi$^{82+}$, for example, experiences on average a magnetic field of about 30\,000\,T, more than 1000 times stronger than available with the strongest superconducting magnet.
According to \cite{Shabaev:01a}, a special combination of the ground-state HFS splittings in H-like and Li-like ions ($\Delta E^{(1s)}$ and $\Delta E^{(2s)}$, respectively) of the same nuclear species, called the specific difference
\begin{eqnarray}
\Delta 'E = \Delta E^{(2s)} - \xi \Delta E^{(1s)},
\end{eqnarray}
provides the best means to test bound-state strong-field QED in the magnetic regime.
Here, the parameter $\xi = 0.16886$ \cite{Shabaev:01a,Volotka:12} is chosen to cancel the contributions of the nuclear-magnetization distribution (Bohr-Weisskopf effect) to $\Delta E^{(1s)}$ and $\Delta E^{(2s)}$.
This is required since the uncertainties of these contributions to the HFS splittings are commonly larger than the complete QED contribution and have failed all previous attempts to perform a QED test solely based on the HFS splitting in H-like heavy ions. However, at the time of the proposal \cite{Shabaev:01a} the experimental uncertainty of the HFS splitting in the Li-like $^{209}$Bi$^{80+}$ extracted
from x-ray emission spectra \cite{Beiersdorfer:98} was far too high to verify the predictions
for $\Delta 'E$.
The first laser spectroscopic observation of the splitting reported in 2014 was orders of magnitude more precise but still limited by systematical uncertainties \cite{Lochmann:14}. Finally, further improvement in accuracy by more than an order of magnitude was recently reported \cite{Ullmann:15,Ullmann:17} but the result was surprisingly more than $7\sigma$ off from the latest theoretical prediction \cite{Volotka:12}.
Since the experimental nuclear magnetic moment $\mu_I$ of $^{209}$Bi enters the calculation of the specific difference, an incorrect value will lead to a proportional change in $\Delta^\prime E$, which could be responsible for the discrepancy \cite{Karr:17}.
We also note that in Ref. \cite{Urrutia:96} the discrepancy between theory and experiment on the HFS splitting in H-like Ho was ascribed to an inaccurate value of the nuclear magnetic moment of ${^{165}}$Ho.
We have reexamined the literature value $\mu_I$($^{209}$Bi)
obtained from nuclear magnetic resonance (NMR) experiments from a theoretical point of view. This has motivated new NMR measurements of bismuth ions in different chemical environments. Results of these experiments are reported and analyzed applying high-level
four-component relativistic coupled cluster theory for advanced chemical shift calculations. We show that our result can completely resolve the hyperfine puzzle established in \cite{Ullmann:17}.
The specific difference $\Delta ' E$ has, so far, always been calculated using
the magnetic moment $\mu_I(^{209}{\rm Bi})=4.1106(2)\mu_N$ tabulated in
\cite{Raghavan:89}.
This value was obtained using the uncorrected (for shielding effects) experimental value of the magnetic moment $\mu_I(^{209}{\rm Bi})=4.03910(19)\mu_N$ reported in an NMR study \cite{Ting:53} of bismuth nitrate, Bi(NO$_3$)$_3$, which was then combined with the shielding constant for the Bi$^{3+}$ cation calculated in \cite{Johnson:68}.
In \cite{Bastug:96} the self-consistent relativistic molecular Dirac-Fock-Slater calculation of the shielding constant of the Bi(NO$_3$)$_3$ molecule using the Lamb formula \cite{Lamb:41} was performed.
The final value, $\sigma=17290(60)$\,ppm, with very small uncertainty was obtained by combining relativistic random phase approximation calculation of the Bi$^{3+}$ cation (17270 ppm) with the molecular correction. The authors concluded that the molecular correction is very small and thus supported the value from \cite{Raghavan:89}.
However,
the authors of \cite{Bastug:96} have not taken into account chemical processes that occur in an aqueous solution of bismuth nitrate molecule Bi(NO$_3$)$_3 \cdot$5H$_2$O: the compound
dissociates and the Bi$^{3+}$ cation is surrounded by water molecules (hydration). Neither the completeness nor the exact form of hydration as a function of concentration, \textit{p}H or temperature is well understood.
While it was suggested in \cite{Fedorov:98} that in strongly acidic solutions Bi$^{3+}$ exists as hexaaquabismuth(III)-cation [Bi(H$_2$O)$_6$]$^{3+}$, more recent studies \cite{Naslund:00} expect that the hydrated form is rather [Bi(H$_2$O)$_8$]$^{3+}$. We found that in both cases the electronic structure of the $n$-coordinated complex significantly differs from the Bi(NO$_3$)$_3$ molecule considered in \cite{Bastug:96}, which is expected. The molecular environment in
Bi(III/V)-containing
complexes strongly contributes to the shielding constant and a considerable chemical shift is introduced.
Consequently, the value of the shielding constant obtained in Ref.\,\cite{Bastug:96} cannot be used for the precise extraction of the $^{209}$Bi magnetic moment from the experimental NMR data.
There is, however, additional NMR data for another Bi containing system: the
hexafluoridobismuthate(V) anion ($^{209}$BiF$_6^-$) \cite{Morgan:83}. It has seven atoms and high spatial symmetry.
According to Morgan \textit{et} al.\ \cite{Morgan:83}, a measurement of BiF$_6^-$ with reference to a saturated solution of bismuth nitrate in concentrated nitric acid gave a chemical shift of $-24$\,ppm. Unfortunately, there is an inconsistency in the reported experimental data of \cite{Morgan:83}, since the measured frequency ratio is given as $\nu(^{209} {\rm BiF}_6^-) / \nu(^1{\rm H})$ =0.16017649(10). The comparison of this ratio with the one reported in \cite{Ting:53} indicates a massive chemical shift of about $\delta \approx +3200$\,ppm instead of $-24$\,ppm. We have performed NMR measurements of both samples to clarify these discrepancies.
\section{Experiment}
Since a dependence of the chemical state of the Bi$^{3+}$ ions in an aqueous solution is expected but details on the sample preparation are missing in the original NMR measurements \cite{Ting:53}, we performed a systematic study using various bismuth nitrate solutions.
Samples of ``Bi(NO$_3$)$_3$'' solutions were prepared with concentrations of 2.5\%, 5\% and 10\% Bi$^{3+}$ (wt \%) in concentrated (65 wt \%) and diluted aqueous solutions (50, 30, 20, 10 wt \%) of nitric acid (HNO$_3$).
\begin{figure}[t]
\includegraphics[width=0.98\linewidth]{Fig1New.pdf}
\caption{\label{fig:Spectra} NMR spectra of Bi(NO$_3$)$_3$ solution (10\% Bi (wt \%)) in concentrated nitric acid (gray) and NMe$_4$BiF$_6$ diluted in acetonitrile (blue).}
\end{figure}
BiF$_6^-$ anions were obtained by dissolution of $\mathrm{(CH_3)_4N^+BiF_6^-}$ (NMe$_4$BiF$_6$)
in acetonitrile to a saturated solution
\cite{Note1}.
All NMR measurements were performed at an 8.4-T magnet using the same double resonance probe for $^{209}$Bi NMR and $^{1}$H NMR calibration with tetramethylsilane. The sample temperature was stabilized with an accuracy of 1\,K employing a constant gas flow tempered by an electric heater. Spectra were obtained from the free induction decay following a 90$^\circ$ pulse of 3.5\,$\upmu$s length for $^{209}$Bi.
Typical spectra of the $^{209}$Bi atoms in BiF$_6^-$ and in the nitrate solution are shown in Fig.\,\ref{fig:Spectra}. The advantage of BiF$_6^-$ is obvious. It exhibits a much narrower linewidth (200 Hz) and the septet arising from indirect spin coupling of $^{19}$F atoms directly bonded to the bismuth atom assures the chemical environment.
The observed ratio of the peak intensities is close to the expected ratio 1\,:\,6\,:\,15\,:\,20\,:\,15\,:\,6\,:\,1 and a spin-spin coupling of 3807(14)\,Hz was determined, in good agreement with \cite{Morgan:83}.
Note that a $^{19}$F spectrum of the sample was taken as well and a decet consistent with the coupling of an $I=9/2$ nucleus to an octahedral environment of six fluorine atoms was observed.
The signal from the nitrate solution is much wider. Even at the highest temperature of 360\,K, the width of the $^{209}$Bi spectra was 4.4\,kHz due to the short spin-lattice and spin-spin relaxation times of $\approx 70$\,$\upmu$s. This width limits the accuracy of the $^{209}$Bi resonance frequency in the solution of the nitrate to 1\,ppm. The chemical shift of Bi$^{3+}$ in the solution of the bismuth nitrate with respect to Bi$^{5+}$ in BiF$_6^-$ is $-106$\,ppm, larger than the $-24$\,ppm reported in \cite{Morgan:83}. Contrary to \cite{Flynn:59} we found that the variation of the bismuth concentration between mass fractions of 2\% and about 40\% (saturation) in nitric acid of 30\% had no appreciable effect on the measured Larmor frequency as long as temperature and nitric acid concentration were kept constant.
Variations of the Bi(NO$_3$)$_3$ sample temperature from 250 to 360\,K were performed with the sample of 10\% Bi in concentrated nitric acid (65\%). We observed a strong linear temperature dependence of the frequency ratio in this range (Fig.\,\ref{fig:TempDependence}) with a slope of $+4.69(13)\times 10^{-7}$\,K$^{-1}$, corresponding to about 3\,ppm/K,
which might be caused by the change of density.
For standard NMR conditions at 298.15\,K a frequency ratio of $\nu_\mathrm{^{209}Bi^{3+}}/\nu_\mathrm{H}=0.160699(1)$ was determined, where the given uncertainty is purely statistical. This value is in excellent agreement with 0.160696(6) reported in \cite{Ting:53}. The temperature dependency of BiF$_6^-$ is 2 orders of magnitude smaller ($\approx 20$\,ppb/K) and of opposite sign. At 298.15\,K the frequency ratio to the proton is 0.1607167(2) far off from the value provided in \cite{Morgan:83}. However, the latter matches our value if one simply flips two digits [$0.160{\bf 17}65(1) \to 0.160{\bf 71}65(1)$].
\begin{figure}[t]
\includegraphics[width=0.98\linewidth]{Fig2Test3.pdf}
\caption{\label{fig:TempDependence} Temperature and HNO$_3$-concentration dependency of the NMR Larmor-frequency ratios of bismuth and hydrogen. A strong temperature effect is observed for Bi(NO$_3$)$_3$ solutions, here exemplified for a
10\% Bi$^{3+}$ (wt \%) solution in concentrated nitric acid (black), whereas only a minor effect was measured for NMe$_4$BiF$_6$ dissolved in acetonitrile (blue).
Inset: Larmor-frequency ratios
measured by NMR in Bi(NO$_3$)$_3$ solutions with 2.5\% Bi$^{3+}$ (wt \%) in nitric acid (HNO$_3$) of various concentrations at 300\,K.
The $y$ axis is identical to the main graph and the gray band represents the total variation.}
\end{figure}
Finally, we have studied the resonance position of Bi(NO$_3$)$_3$ as a function of the nitric acid concentration
(inset in Fig.\,\ref{fig:TempDependence}).
A clear dependence on the acidity is observed for all Bi$^{3+}$ concentrations, covering a range of typically $\approx 60$\,ppm.
In summary, the results clearly demonstrate that a large uncertainty is connected with the extraction of the magnetic moment of $^{209}$Bi from NMR measurements in aqueous solutions of Bi(NO$_3$)$_3$. The influence of the chemical environment was strongly underestimated in theory since the calculations performed to extract the chemical shift do neither account for the temperature nor for the concentration or acidity of the sample. In this respect, BiF$_6^-$ is a much better candidate to obtain a reliable value of the magnetic moment which will be substantiated now also from a theoretical point of view.
\section{Theory}
In the presence of the external uniform magnetic field \textbf{B} and nuclear magnetic moment $\mu_j$ of $j-$th atom in a molecule the corresponding Dirac-Coulomb Hamiltonian includes the following terms:
\begin{equation}
\label{HB}
H_B={\rm \bf{B}}\cdot \frac{c}{2}(\bm{r}_G \times \bm{\alpha}),
\end{equation}
\begin{equation}
\label{HHFS}
H_{\rm hyp}=\frac{1}{c} \sum_j \bm{\mu}_j\cdot \frac{(\bm{r}_j \times \bm{\alpha})}{r_j^3},
\end{equation}
where $\bm{r}_G = \bm{r} - \bm{R_G}$, $\bm{R_G}$ is the gauge origin,
$\bm{r}_j=\bm{r} - \bm{R_j}$, $\bm{R_j}$ is the position of nucleus $j$, and $\bm{\alpha}$ are the Dirac matrices.
The chemical shielding tensor of the nucleus $j$ can be defined as a mixed derivative of the energy with respect to the nuclear magnetic moment and the strength of the magnetic field
\begin{equation}
\label{SHIELDINGDer}
\left.\sigma^j_{a,b}=\frac{\partial^2E}{\partial\mu_{j,a}\partial B_b} \right|_{\bm{\mu}_j=0,{\rm \bf{B}}=0}.
\end{equation}
We are interested in its isotropic part.
In the one-electron case the shielding tensor (\ref{SHIELDINGDer}) can be calculated by the sum-over-states method within the second-order perturbation theory with perturbations (\ref{HB}) and (\ref{HHFS}). In the relativistic four-component approach the summation should include both positive and negative energy spectra \cite{Aucar:99}.
The part associated with positive energy is called the ``paramagnetic'' term while the part associated with negative energy states is called ``diamagnetic term'' though only their sum is gauge invariant \cite{Aucar:99}.
To avoid an ambiguity in calculations utilizing finite basis sets due to the choice of the gauge origin $\bm{R_G}$ one can use the so-called London atomic orbitals (LAOs) method (see e.g.\ \cite{Olejniczak:12,Ilias:13} for details).
In Refs.\,\cite{DIRAC15,Olejniczak:12,Ilias:13} the four-component density functional theory (DFT) using response technique and LAOs has been developed to calculate the shielding constant (\ref{SHIELDINGDer}).
To construct the atomic basis sets for the unperturbed Dirac-Coulomb Hamiltonian calculations one often uses the restricted kinetic balance (RKB) method. However, in the presence of the external magnetic fields the usual relation between the large and small component changes. In Ref.\,\cite{Olejniczak:12} the scheme of magnetic balance (MB) in conjunction with LAOs was proposed to take into account the modified coupling which is utilised below.
Most of the chemical shift calculations for heavy atom compounds are performed within the (relativistic) DFT. The drawback of the theory is that it is hard to control the uncertainty of the results as there is no systematic way of improving it. Even combinations with high-level nonrelativistic \textit{ab initio} wave-function-based calculations are also questionable in the case of heavy atom compounds. In Refs.\,\cite{Skripnikov:16b,Skripnikov:15a,Petrov:17b,Skripnikov:17c} it was shown that for such properties as the hyperfine structure constant and the molecular $g$ factor, the relativistic coupled cluster method gives the most accurate results if there are no multireference effects. Therefore, this method has been adopted here to control the uncertainty of the DFT results.
\section{Electronic structure calculation details}
In the present study we have used atomic basis sets of different qualities.
The NZ (where N~$=$~Double, Triple, Quadruple) basis set corresponds to the uncontracted core-valence N-zeta \cite{Dyall:07,Dyall:12} Dyall's basis set for Bi and augmented correlation consistent polarized valence N-zeta, aug-cc-pVNZ \cite{Dunning:89,Kendall:92} basis set for light atoms.
In the DZC basis set the contracted version of the aug-cc-pVDZ \cite{Dunning:89,Kendall:92} basis sets were used for light atoms.
Based on the nonrelativistic estimates, the hybrid density functional PBE0 \cite{pbe0} has been chosen because it reproduces the nonrelativistic coupled cluster value rather well.
Geometry parameters of the BiF$_6^-$ anion have been obtained in the scalar-relativistic DFT calculation using the generalized relativistic pseudopotential method \cite{Mosyagin:16}.
The contribution of the Gaunt interaction to the shielding constant was estimated as the difference between the values calculated at the Dirac-Hartree-Fock-Gaunt and Dirac-Hartree-Fock level of theory within the uncoupled scheme.
Nonrelativistic and scalar-relativistic calculations were performed within the {\sc us-gamess} \cite{USGAMESS1} and {\sc cfour} \cite{CFOUR} codes. Relativistic four-component calculations were performed within the {\sc dirac15} \cite{DIRAC15} and {\sc mrcc} \cite{MRCC2013} codes. For calculation of the hyperfine-interaction matrix elements and $g$ factors the code developed in Refs.\,\cite{Skripnikov:16b,Skripnikov:15b,Skripnikov:15a} was used.
\section{Results and discussion}
Table \ref{BiF6} contains results of the calculation of the BiF$_6^-$ anion.
\begin{table}[h]
\centering
\caption{The values of $^{209}$Bi shielding constants in BiF$_6^-$ in ppm.}
\label{BiF6}
\begin{tabular}{lccc}
\hline
\hline
Basis set/method & Diamagnetic & Paramagnetic & Total \\
\hline
\hline
DZ-MB-LAO/DHF & 8\,618 & 5\,768 & 14\,386 \\
DZ-MB-LAO/DFT & 8\,621 & 3\,726 & 12\,347 \\
TZ-MB-LAO/DFT & 8\,639 & 3\,733 & 12\,372 \\
\hline
DZC-RKB/DFT & & 3\,848 & \\
DZC-RKB/CCSD & & 4\,403 & \\
DZC-RKB/CCSD(T) & & 4\,286 & \\
\hline
QZ-MB-LAO/DFT & 8\,628 & 3\,763 & 12\,391 \\
Correlation correction & & 437 & \\
Gaunt correction & & -37 & \\
\hline
Final & & & 12\,792 \\
\hline
\hline
\end{tabular}
\end{table}
Comparing Dirac-Hartree-Fock (DHF) and DFT results in Table \ref{BiF6} it can be seen that the diamagnetic contribution to $\sigma(^{209}{\rm Bi})$ depends only weakly on the correlation effects, while the paramagnetic contribution is strongly affected.
To check the accuracy of the latter DFT result we have performed a series of relativistic coupled cluster calculations of $\sigma(^{209}{\rm Bi})$ taking into account only the positive energy spectrum.
Comparing values obtained within the coupled cluster with single, double and noniterative triple-cluster amplitudes (CCSD(T)) with that of CCSD shows that the triple amplitudes only slightly contribute to $\sigma(^{209}{\rm Bi})$ demonstrating good convergence of the results with respect to the electron correlation treatment
\cite{Note2}.
In the final value of $\sigma(^{209}{\rm Bi})$ we include the correlation correction calculated as the difference between the CCSD(T) and PBE0 results.
To investigate the importance of systematic treatment of the molecular environment
we have also performed an additional DHF study of one of the possible hydrated forms of Bi$^{3+}$ in an acidic solution of Bi(NO$_3$)$_3$ -- [Bi(H$_2$O)$_8$]$^{3+}$ cation in comparison with the unsolvated Bi$^{3+}$ cation.
It was found that the shielding constant of the $^{209}$Bi$^{3+}$ is significantly larger (by about 20\% at the DHF level) than that in [$^{209}$Bi(H$_2$O)$_8$]$^{3+}$.
Therefore, the interpretation of the \textit{molecular} NMR experiment in terms of the nuclear magnetic moment using a shielding constant obtained for the corresponding ion (as was done in earlier studies) is associated with considerable uncertainties.
We now use the value obtained for $\nu_\mathrm{^{209}BiF_6^-}/\nu_\mathrm{H}= 0.1607167(2)$ from our NMR measurements and the shielding constant of $\sigma(^{209}{\rm BiF}_6^-)=12\,792$\,ppm calculated above to obtain $\mu_I(\mathrm{^{209}Bi}) = 4.092(2)\,\mu_\mathrm{N}$ with an uncertainty dominated by theory.
Table \ref{NewOld} compares the experimental values \cite{Ullmann:17} of the HFS splittings with the theoretical values calculated with the old [$\mu_I$(old)$=$4.1106(2)$\mu_N$] and the new [$\mu_I$(new)$=$4.092(2)$\mu_N$] values of the nuclear magnetic moment \cite{Volotka:12}. The theoretical results include the most elaborated calculation of the Bohr-Weisskopf effect \cite{Senkov:02}.
\begin{table}[]
\centering
\caption{Theoretical values of $\Delta E^{(1s)}$ and $\Delta E^{(2s)}$ (in meV) calculated with old and new nuclear magnetic moment of $^{209}$Bi in comparison with the experimental values \cite{Ullmann:17}.
For the Bohr-Weisskopf effect the most elaborated calculation by Sen'kov and Dmitriev \cite{Senkov:02} was employed.
}
\label{NewOld}
\begin{tabular}{llll}
\hline
\hline
& \multicolumn{2}{l}{Theory} & Experiment \\
& $\mu_I$(old) & $\mu_I$(new) & \\
\hline
$\Delta E^{(1s)}$ & 5112(-5/+20) & 5089(-5/+20)(2) & 5085.03(2)(9) \\
$\Delta E^{(2s)}$ & 801.9(-9/+34) & 798.3(-9/+34)(4) & 797.645(4)(14) \\
\hline
\hline
\end{tabular}
\end{table}
\begin{figure}[!h]
\includegraphics[width=0.98\linewidth]{deltaPrimeE_NMR_2.pdf}
\caption{\label{fig:CompExpTheory} Specific difference obtained in theory \cite{Shabaev:01a,Volotka:12} (red) and experiment \cite{Lochmann:14,Ullmann:17} (blue). The new nuclear magnetic moment established in this work yields a new value for the specific difference which matches the most recent experimental value within uncertainty.
}
\end{figure}
The new magnetic moment has been used to recalculate the specific difference and we obtain
$\Delta 'E_{\mathrm{theo}} = -61.043(5)(30)$\,meV, where the first uncertainty is due to uncalculated terms and remaining nuclear effects, while the second one is due to the uncertainty of the nuclear magnetic moment obtained in the present work. Revised value of $\Delta 'E_{\mathrm{theo}}$
is plotted in Fig.\,\ref{fig:CompExpTheory} combined with the previous theoretical and experimental data. Theory and experiment are now in excellent agreement and the $7\sigma$ discrepancy reported in \cite{Ullmann:17} disappears.
Unfortunately, the uncertainty of $\Delta 'E_{\mathrm{theo}}$ is now 14\% of the total QED contribution and about 1.5 times larger than the experimental uncertainty.
Hence, an improved value for the nuclear magnetic moment of $^{209}$Bi is urgently required, either from an atomic beam magnetic resonance experiment or from a measurement on trapped H-like ions. The latter will have the advantage that no shielding corrections have to be applied. Such an experiment is planned at the ARTEMIS trap \cite{Quint:2008} at the GSI Helmholtz Centre in Darmstadt. Only such a measurement combined with an improved determination of the HFS splitting in $^{209}$Bi$^{80+,82+}$ as it is foreseen at SPECTRAP \cite{Andelkovic:2013} can provide a QED test in the magnetic regime of strong-field QED. Our result also proves that a measurement of the specific difference can also be used to extract the nuclear magnetic moment.
Doing so results in $\mu_I(^{209}\mathrm{Bi})=4.0900(15)\,\mu_{\mathrm{N}}$ in excellent agreement with the NMR value obtained here.
\begin{acknowledgments}
\section*{Acknowledgments}
We thank Petra Th\"orle from the Institute of Nuclear Chemistry at the University of Mainz for the preparation of the NMR samples and Dmitry Korolev from Saint-Peterburg State University for valuable discussions.
The development of the code for the computation of the matrix elements of the considered operators as well as the performance of all-electron coupled cluster calculations were funded by RFBR, according to Research Project No.~16-32-60013 mol\_a\_dk; performance of DFT calculations was supported by the President of Russian Federation Grant No. MK-2230.2018.2. This work was also supported by SPSU (Grants No. 11.38.237.2015 and No. 11.40.538.2017) and by SPSU-DFG (Grants No. 11.65.41.2017 and No. STO 346/5-1). The experimental part was supported by the Federal Ministry of Education and Research of Germany under Contract No 05P15RDFAA and the Helmholtz
International Center for FAIR (HIC for FAIR).
\end{acknowledgments}
|
{
"timestamp": "2018-03-08T02:06:13",
"yymm": "1803",
"arxiv_id": "1803.02584",
"language": "en",
"url": "https://arxiv.org/abs/1803.02584"
}
|
\section{Introduction}
Large-scale distributed systems often resort to replication techniques to
achieve fault-tolerance and load distribution. These systems have to make a
choice between availability and low latency or strong consistency
\cite{abadi-cap,brewer-cap,gilbert-cap,golab-pacelc}, many times opting for the
first \cite{consistency1,consistency2}. A common approach is to allow
replicas of some data type to temporarily diverge, making sure these replicas
will eventually converge to the same state in a deterministic way.
\emph{Conflict-free Replicated Data Types} (CRDTs) \cite{crdts1,crdts2} can be
used to achieve this. They are key components in modern geo-replicated systems,
such as Riak~\cite{riak}, Redis~\cite{redis}, and Microsoft Azure Cosmos
DB~\cite{cosmosdb}.
CRDTs come mainly in two flavors: \emph{operation-based} and
\emph{state-based}. In both, queries and updates can be executed immediately
at each replica, which ensures availability (as it never needs to coordinate
beforehand with remote replicas to execute operations). In operation-based
CRDTs \cite{pure-op,crdts1}, operations are disseminated assuming a reliable
dissemination layer that ensures exactly-once causal delivery of operations.
State-based CRDTs need fewer guarantees from the communication channel:
messages can be dropped, duplicated, and reordered. When an update operation
occurs, the local state is updated through a mutator, and from time to time
(since we can disseminate the state at a lower rate than the rate of the
updates) the full (local) state is propagated to other replicas.
Although state-based CRDTs can be disseminated over unreliable communication
channels, as the state grows, sending the full state
becomes unacceptably costly. Delta-based CRDTs \cite{deltas1,deltas2} address
this issue by defining delta-mutators that return a delta ($\delta$),
typically much smaller than the full state of the replica, to be merged with
the local state. The same $\delta$ is also added to an outbound
$\delta$-buffer, to be periodically propagated to remote replicas.
Delta-based CRDTs have been adopted in industry as part of Akka Distributed
Data
framework~\cite{akka-data} and IPFS~\cite{ipfs, ipfs-deltas}.
However, and somewhat unexpectedly, we have observed (Figure \ref{fig:problem})
that current
delta-propagation algorithms can still disseminate much
redundant state between replicas,
performing worse than envisioned, and no better than the state-based approach.
This anomaly becomes noticeable when concurrent update
operations always occur between synchronization rounds,
and it is partially justified due to inefficient redundancy detection in
delta-propagation.
\begin{figure}[t]
\begin{center}
\begin{minipage}{0.33\textwidth}
\includegraphics[width=\textwidth,keepaspectratio]{first}
\end{minipage}
\begin{minipage}{0.105\textwidth}
\includegraphics[width=\textwidth,keepaspectratio]{second}
\end{minipage}
\end{center}
\caption{Experiment setup: 15 nodes in a partial mesh topology replicating
an always-growing set. The left plot depicts the
number of elements being sent throughout the experiment,
while the right plot shows the CPU processing time ratio
with respect to state-based. Not only does delta-based
synchronization not improve state-based in terms of state transmission,
it even incurs a substantial processing overhead.}
\label{fig:problem}
\end{figure}
In this paper we identify two sources of redundancy in current algorithms,
and introduce the concept of join decomposition of a state-based CRDT,
showing how it can be used to derive optimal deltas (``differences'') between states, as
well as optimal delta-mutators.
By exploiting these concepts, we also introduce an
improved synchronization algorithm, and experimentally
evaluate it, confirming that it outperforms current approaches
by reducing the amount of state transmission,
memory consumption, and processing time
required for delta-based synchronization.
\section{Background on State-based CRDTs}
A state-based CRDT can be defined as a triple
$(\mathcal{L}, \sqleq, \join)$ where $\mathcal{L}$ is a
join-semilattice (lattice for short, from now on),
$\sqleq$ is a partial order, and
$\join$ is a binary join operator that derives
the least upper bound for any two elements of $\mathcal{L}$.
State-based CRDTs are updated through a set of
\emph{mutators} designed to be inflations,
i.e. for mutator $\m$ and state $x \in \mathcal{L}$, we have $x \sqleq \m(x)$.
Synchronization of replicas is achieved by having
each replica periodically propagate
its local state to other
neighbour
replicas. When a remote state is received, a replica updates its state to reflect
the join of its local state and the received state.
As the local state grows, more state needs to be sent, which might affect the
usage of system resources (such as network) with a negative
impact on the overall system performance.
Ideally, each replica should only propagate the most
recent modifications executed over its local state.
Delta-based CRDTs can be used to achieve this,
by defining \emph{delta-mutators} that
return a smaller state which, when merged with the current state, generates the
same result as applying the standard mutators, i.e.
each mutator $\m$ has in
delta-CRDTs
a corresponding $\delta$-mutator
$\m^\delta$ such that:
\[
\m(x) = x \join \m^\delta(x)
\]
In this model, the deltas resulting from $\delta$-mutators
are added to a $\delta$-buffer,
in order to be propagated to neighbor replicas, as a $\delta$-group, at the next synchronization step.
When a $\delta$-group is received from a neighbor, it is also added
to the buffer for further propagation.
\subsection{CRDT examples}
\label{subsec:examples}
In Figure \ref{fig:crdt-spec} we present the specification
of two simple state-based CRDTs,
defining their lattice states, mutators, corresponding $\delta$-mutators,
and the binary join operator $\join$.
These lattices
are typically bounded and thus a bottom value
$\bot$ is also defined.
(Note that the specifications
do not define the partial order $\sqleq$ since it can
always be defined,
for any lattice $\mathcal{L}$, in terms of $\join$:
$x \sqleq y \iff x \join y = y$.)
\begin{figure}[!ht]
\begin{center}
\begin{subfigure}{0.48\textwidth}
\begin{align*}
\af{GCounter} & = \mathds{I} \mathrel{\hookrightarrow} \mathds{N} \\
\bot & = \varnothing \\
\af{inc}_i(p) & = p\{i \mapsto p(i) + 1\} \\
\daf{inc}_i(p) & = \{i \mapsto p(i) + 1\} \\
\af{value}(p) & = \sum \{ v | k \mapsto v \in p \} \\
p \join p' & = \{k \mapsto \max(p(k), p'(k)) | k \in l \} \\
& \textbf{where } l = \dom(p) \union \dom(p')
\end{align*}
\caption{Grow-only Counter.}
\label{fig:gcounter-spec}
\end{subfigure}
\end{center}
\hfill
\begin{center}
\begin{subfigure}{0.48\textwidth}
\begin{align*}
\af{GSet}\tup{E} & = \pow{E} \\
\bot & = \varnothing \\
\af{add}(e, s) & = s \union \{e\} \\
\daf{add}(e, s) & =
\begin{cases}
\{ e \} & \textbf{if}\enskip e \not \in s \\
\bot & \textbf{otherwise}
\end{cases} \\
\af{value}(s) & = s \\
s \join s' & = s \union s'
\end{align*}
\caption{Grow-only Set.}
\label{fig:gset-spec}
\end{subfigure}
\end{center}
\caption{Specifications of two data types, replica $i \in \mathds{I}$.}
\label{fig:crdt-spec}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\begin{subfigure}{0.48\textwidth}
\centerxy{
<0.5cm,0pt>:
(0,6)*+{\countab{2}{2}}="atwobtwo";
(-2,4.5)*+{\countab{2}{1}}="atwobone";
(2,4.5)*+{\countab{1}{2}}="aonebtwo";
(-4,3)*+{\counta{2}}="atwo";
(0,3)*+{\countab{1}{1}}="aonebone";
(4,3)*+{\countb{2}}="btwo";
(-2,1.5)*+{\counta{1}}="aone";
(2,1.5)*+{\countb{1}}="bone";
(0,0)*+{\bot}="empty";
"empty"; "aone" **\dir{-};
"empty"; "bone" **\dir{-};
"aone"; "atwo" **\dir{-};
"bone"; "btwo" **\dir{-};
"aone"; "aonebone" **\dir{-};
"bone"; "aonebone" **\dir{-};
"atwo"; "atwobone" **\dir{-};
"aonebone"; "atwobone" **\dir{-};
"btwo"; "aonebtwo" **\dir{-};
"aonebone"; "aonebtwo" **\dir{-};
"atwobone"; "atwobtwo" **\dir{-};
"aonebtwo"; "atwobtwo" **\dir{-};
}
\caption{$\af{GCounter}$, with two replicas $\mathds{I} = \{\A, \B\}$.}
\label{fig:gcounter-hasse}
\end{subfigure}
\end{center}
\hfill
\begin{center}
\begin{subfigure}{0.48\textwidth}
\centerxy{
<0.5cm,0pt>:
(0,6)*+{\{a, b, c\}}="abc";
(-3,4)*+{\{a, b\}}="ab";
(0,4)*+{\{a, c\}}="ac";
(3,4)*+{\{b, c\}}="bc";
(-3,2)*+{\{a\}}="a";
(0,2)*+{\{b\}}="b";
(3,2)*+{\{c\}}="c";
(0,0)*+{\bot}="empty";
"empty"; "a" **\dir{-};
"empty"; "b" **\dir{-};
"empty"; "c" **\dir{-};
"a"; "ab" **\dir{-};
"a"; "ac" **\dir{-};
"b"; "ab" **\dir{-};
"b"; "bc" **\dir{-};
"c"; "ac" **\dir{-};
"c"; "bc" **\dir{-};
"ab"; "abc" **\dir{-};
"ac"; "abc" **\dir{-};
"bc"; "abc" **\dir{-};
}
\caption{$\af{GSet}\tup{\{a, b, c\}}$.}
\label{fig:gset-hasse}
\end{subfigure}
\end{center}
\caption{Hasse diagram of two data types.}
\label{fig:hasse}
\end{figure}
\newcommand{\{a, \underline b\}}{\{a, \underline b\}}
\newcommand{\{\underline a, \underline b, c\}}{\{\underline a, \underline b, c\}}
\begin{figure*}[t]
\centerxy{
\xymatrix@C=1.5em @R2em{
\A &
\varnothing \ar@{->}[rr]^(0.45){\add_a} & & \{a\}
\ar@{.}[rr] & & \{a, b\}
\ar@{.}[r] & \bullet^2
\ar@{->}[rd]^(.6){\{a, \underline b\}}
\ar@{.}[rrr] & & & \{a, b, c\}
\\
\B &
\varnothing \ar@{->}[rr]^(0.45){\add_b} & & \{b\}
\ar@{.}[r] & \bullet^1
\ar@{->}[ru]_(.65){\{b\}}
\ar@{.}[r] & \ar@{->}[r]^(0.45){\add_c} & \{b, c\}
\ar@{.}[r] & \{a, b, c\}
\ar@{.}[r] & \bullet^3
\ar@{->}[ru]_(.65){\{\underline a, \underline b, c\}}
}
}
\caption{Delta-based synchronization of a $\af{GSet}$ with 2 replicas
$\A, \B \in \mathds{I}$. Underlined elements represent the $\BP$ optimization.}
\label{fig:delta-ex1}
\end{figure*}
\newcommand{\{a, \overline b\}}{\{a, \overline b\}}
\begin{figure*}[t]
\centerxy{
\xymatrix@C=1.5em @R2em{
\A &
\varnothing \ar@{->}[rr]^(0.45){\add_a} & & \{a\}
\ar@{.}[rr] & & \{a, b\}
\ar@{.}[r] & \bullet^6
\ar@{->}[rrdd]^(.78){\{a, b\}}
\\
\B &
\varnothing \ar@{->}[rr]^(0.45){\add_b} & & \{b\}
\ar@{.}[r] & \bullet^4
\ar@{->}[ru]_(.65){\{b\}} \ar@{->}[rd]^(.6){\{b\}}
\\
\C &
\varnothing
\ar@{.}[rrrr] & & & & \{b\}
\ar@{.}[r] & \bullet^5
\ar@{->}[rd]_(.6){\{b\}}
\ar@{.}[rr] & & \{a, b\}
\ar@{.}[r] & \bullet^7
\ar@{->}[rd]_(.6){\{a, \overline b\}}
\\
\D &
\varnothing
\ar@{.}[rrrrrr] & & & & & & \{b\}
\ar@{.}[rrr] & & & \{a, b\}
}
}
\caption{Delta-based synchronization of a $\af{GSet}$ with 4 replicas
$\A, \B, \C, \D \in \mathds{I}$. The overlined element represents the $\RR$ optimization.}
\label{fig:delta-ex2}
\end{figure*}
A CRDT counter that only allows increments is known as
a \emph{grow-only counter} (Figure \ref{fig:gcounter-spec}).
In this data type, the set of replica identifiers $\mathds{I}$
is mapped to the set of natural numbers $\mathds{N}$.
Increments are tracked per replica $i$, individually,
and stored in a map entry $p(i)$. The value of
the counter is the sum of each entry's value in the map.
Mutator $\af{inc}$ returns the updated map (the notation $p\{k \mapsto v\}$ indicates that only entry $k$ in the map $p$ is updated to a new value $v$, the remaining entries left unchanged), while
the $\delta$-mutator $\daf{inc}$ only returns the
updated entry. The join of two $\af{GCounter}$s computes,
for each key, the maximum of the associated values.
The lattice state evolution (either by mutation or
join of two states)
can also be understood by looking at
the corresponding Hasse diagram (Figure \ref{fig:hasse}).
For example,
state $\countab{1}{1}$ in Figure \ref{fig:gcounter-hasse}
(where $\A_1$ represents entry $\{\A \mapsto 1 \}$ in the map,
i.e. one increment registered by replica $\A$),
can result from an increment on $\counta{1}$ by $\B$,
from an increment on
$\countb{1}$
by $\A$, or from the join of these two states.
A \emph{grow-only set}, Figures \ref{fig:gset-spec}
and Figure \ref{fig:gset-hasse},
is a set data type that only allows element additions.
Mutator $\add$ returns the updated set, while $\daf{add}$
returns a singleton set with the added element
(in case it was not in the set already).
The join of two $\af{GSet}$s simply computes the set union.
Although we have chosen as running examples very simple CRDTs,
the results in this paper can be extended to more complex ones,
as we show in
Appendix \ref{app:compositions}.
For further coverage of delta-based CRDTs see \cite{deltas2}.
\subsection{Synchronization Cost Problem}
Figures \ref{fig:delta-ex1} and \ref{fig:delta-ex2} illustrate possible
distributed executions of the classic delta-based synchronization algorithm
\cite{deltas2}, with replicas of a \emph{grow-only-set}, all starting with a
bottom value $\bot = \varnothing$. (This classic algorithm is captured in
Algorithm \ref{alg:both}, covered in Section \ref{sec:revisited}.)
Synchronization with neighbors is represented by $\bullet$ and synchronization
arrows are labeled with the state sent, where we overline or underline
elements that are being redundantly sent and can be removed (thus improving
network bandwidth consumption) by employing two simple and novel optimizations
that we introduce next.
In Figure \ref{fig:delta-ex1}, we have two replicas $\A, \B \in \mathds{I}$
and each adds an element to the replicated set.
At $\bullet^1$, $\B$ propagates the content of the $\delta$-buffer,
i.e. $\{b\}$, to neighbour $\A$.
At $\bullet^2$, $\A$ sends to $B$ $\{a, b\}$,
i.e.
the join of $\{a\}$ from a local mutation,
and the received $\{b\}$ from $\B$,
even though $\{b\}$ came from $B$ itself.
By simply tracking the origin of each $\delta$-group in
the $\delta$-buffer, replicas can
\textbf{avoid back-propagation of $\delta$-groups} ($\BP$).
Before receiving $\{a, b\}$, $\B$ adds a new element $c$ to the set,
also adding $\{c\}$ to the $\delta$-buffer.
Upon receiving $\{a, b\}$, and since what was received produces changes in the local
state, $\B$ adds it to the $\delta$-buffer.
At $\bullet^3$, $\B$ propagates all new changes since last synchronization
with $\A$: $\{c\}$ from a local mutation, and $\{a, b\}$ from $\B$,
even though $\{a, b\}$ came from replica $\B$.
When $\A$ receives $\{a, b, c\}$, it will also add it to the buffer to be further propagated.
Note that as long as this pattern keeps repeating (i.e. there's always a state change between
synchronizations), delta-based synchronization will propagate the same amount of state
as state-based synchronization would, representing no improvement.
This is illustrated in Figure \ref{fig:delta-ex1},
and demonstrated empirically in Section \ref{sec:eval}.
In Figure \ref{fig:delta-ex2}, we have four replicas $\A, \B, \C, \D \in \mathds{I}$,
and replicas $\A, \B$ add an element to the set.
At $\bullet^4$, $\B$ propagates the content of the $\delta$-buffer
to neighbours $\A$ and $\C$.
At $\bullet^5$, $\C$ propagates the received $\{b\}$ to $\D$.
At $\bullet^6$, $\A$ sends
the join of $\{a\}$ from a local mutation and the received $\{b\}$
to $\C$.
Upon receiving the $\delta$-group $\{a, b\}$,
$\C$ adds it to the $\delta$-buffer
and sends it to $\D$ at $\bullet^7$.
However, part of this $\delta$-group has already been in the $\delta$-buffer
(namely $b$), and thus, has already been propagated.
This observation hints for another optimization:
\textbf{remove redundant state in received $\delta$-groups} ($\RR$),
before adding them to the $\delta$-buffer.
Both $\BP$ and $\RR$ optimizations are detailed in Section \ref{sec:revisited},
where we incorporate them into the delta-based synchronization algorithm
with few changes.
\section{Join Decompositions and Optimal Deltas}
\label{sec:efficient}
In this section we introduce state decomposition in state-based CRDTs, by
exploiting the mathematical concept of \emph{irredundant join decompositions}
in lattices. We then demonstrate how this concept can be used
to derive deltas and delta-mutators that are optimal,
in the sense that they produce the
smallest delta-state possible.
In Section \ref{sec:revisited} we show how this
same concept plays a key role in the $\af{RR}$ optimization
briefly described in the previous section.
\subsection{Join Decomposition of a State-based CRDT}
\label{subsec:jd}
\begin{definition}[Join-irreducible state]
State $x \in \mathcal{L}$ is join-irreducible if it cannot
result from the join of any finite set of states $F \subseteq \mathcal{L}$ not
containing $x$:
\[
x = \bigjoin F \implies x \in F
\]
\end{definition}
\begin{example}
Let the following $p_1$, $p_2$ and $p_3$ be $\af{GCounter}$ states,
and $s_1$, $s_2$ and $s_3$ be $\af{GSet}$ states.
\begin{center}
\begin{minipage}{.25\textwidth}
\begin{itemize}
\item[\ding{51}] $p_1 = \counta{5}$
\item[\ding{51}] $p_2 = \countb{6}$
\item[\ding{55}] $p_3 = \countab{5}{7}$
\end{itemize}
\end{minipage}
\begin{minipage}{.23\textwidth}
\begin{itemize}
\item[\ding{55}] $s_1 = \bot$
\item[\ding{51}] $s_2 = \{a\}$
\item[\ding{55}] $s_3 = \{a, b\}$
\end{itemize}
\end{minipage}
\end{center}
States $p_3$ and $s_3$ are not join-irreducible states,
since they can be decomposed into (i.e. result from the join of) two states
different from themselves: $\counta{5}$ and $\countb{7}$ for $p_3$,
$\{a\}$ and $\{b\}$ for $s_3$.
Bottom (e.g., $s_1$) is never join-irreducible, as it is the join over an empty
set $\bigjoin \varnothing$.
\end{example}
In a Hasse diagram of a finite lattice (e.g., in Figure \ref{fig:hasse})
the join-irreducibles are those elements with exactly one link below.
Given lattice $\mathcal{L}$, we use $\mathcal{J}(\mathcal{L})$ for the set of all
join-irreducible elements of $\mathcal{L}$.
\begin{definition}[Join Decomposition]
Given a lattice state $x \in \mathcal{L}$, a set of join-irreducibles $D$
is a join decomposition \cite{birkhoff1937} of $x$
if its join produces $x$:
\[
D \subseteq \mathcal{J}(\mathcal{L}) \mathrel{\wedge} \bigjoin D = x
\]
\end{definition}
\begin{definition}[Irredundant Join Decomposition]
A join decomposition D is irredundant if no element in it is redundant:
\[
D' \subset D \implies \bigjoin D' \ple \bigjoin D
\]
\end{definition}
\begin{example}
\label{ex:jd}
Let $p = \countab{5}{7}$ be a $\af{GCounter}$ state,
$s = \{a, b, c\}$ a $\af{GSet}$ state,
and consider the following sets of states as tentative decompositions of $p$
and $s$.
\begin{center}
\begin{minipage}{.25\textwidth}
\begin{itemize}
\item[\ding{55}] $P_1 = \{\counta{5}, \countb{6}\}$
\item[\ding{55}] $P_2 = \{\counta{5}, \countb{6}, \countb{7}\}$
\item[\ding{55}] $P_3 = \{\countab{5}{6}, \countb{7}\}$
\item[\ding{51}] $P_4 = \{\counta{5}, \countb{7}\}$
\end{itemize}
\end{minipage}
\begin{minipage}{.23\textwidth}
\begin{itemize}
\item[\ding{55}] $S_1 = \{\{b\}, \{c\}\}$
\item[\ding{55}] $S_2 = \{\{a, b\}, \{b\}, \{c\}\}$
\item[\ding{55}] $S_3 = \{\{a, b\}, \{c\}\}$
\item[\ding{51}] $S_4 = \{\{a\}, \{b\}, \{c\}\}$
\end{itemize}
\end{minipage}
\end{center}
Only $P_4$ and $S_4$ are irredundant join decompositions of $p$ and $s$.
$P_1$ and $S_1$ are not decompositions since their join does not result in $p$
and $s$, respectively;
$P_2$ and $S_2$ are decompositions but contain redundant elements,
$\countb{6}$ and $\{b\}$, respectively;
$P_3$ and $S_3$ do not have redundancy, but contain reducible elements
($S_2$ fails to be an irredundant join decomposition for the same reason,
since its element $\{a, b\}$ is also reducible).
\end{example}
As we show in Appendix \ref{app:unique-jds} and \ref{app:compositions},
these irredundant decompositions exist, are unique,
and can be obtained for CRDTs used in practice.
Let $\dec{x}$ denote the unique decomposition of element $x$.
From the Birkhoff's Representation Theorem~\cite{latticesAndOrder},
decomposition $\dec{x}$ is given by the maximals of the
join-irreducibles below $x$:
\[
\dec x = \max \{ r \in \mathcal J(\mathcal{L}) | r \pleq x \}
\]
As two examples,
given a $\af{GCounter}$ state $p$ and a $\af{GSet}$ state $s$,
their (quite trivial) irredundant decomposition is given by:
\[
\hfill
\dec{p} = \{ \{k \mapsto v\} | k \mapsto v \in p \}
\qquad
\dec{s} = \{ \{e\} | e \in s\}
\hfill
\]
We argue that these techniques can be applied to most (practical) implementations of CRDTs
found in industry. The interested reader can find generic decomposition rules in Appendix~\ref{app:decomposing}.
\subsection{Optimal deltas and \texorpdfstring{$\delta$}{}-mutators}
Having a unique irredundant join decomposition, we can define a function which
gives the minimum delta, or ``difference'' in analogy to set difference,
between two states $a, b \in \mathcal{L}$:
\[
\Delta(a, b) = \bigjoin \{ y \in \dec{a} | y \not \pleq b \}
\]
which when joined with $b$ gives $a \join b$, i.e. $\Delta(a, b) \join b = a \join b$.
It is minimum (and thus, optimal)
in the sense that it is smaller than any other $c$
which produces the same result:
$c \join b = a \join b \implies \Delta(a, b) \pleq c$.
If not carefully designed, $\delta$-mutators can be a source of redundancy
when the resulting $\delta$-state contains information that has already been incorporated
in the lattice state.
As an example, the original $\delta$-mutator $\daf{add}$ of $\af{GSet}$ presented in
\cite{deltas1} always returns a singleton set with the element to be added,
even if the element is already in the set
(in Figure \ref{fig:gset-spec} we have presented a definition of
$\daf{add}$ that is optimal). By resorting to function $\Delta$, minimum delta-mutators can be trivially derived
from a given mutator:
\[
\m^\delta(x) = \Delta(\m(x), x)
\]
\section{Revisiting Delta-based Synchronization}
\label{sec:revisited}
Algorithm \ref{alg:both} formally describes
delta-based synchronization at replica $i$.
The algorithm contains lines that belong to
\HLBase{classic} delta-based synchronization \cite{deltas1,deltas2},
and lines with \HLOpt{$\BP$} and \HLOpt{$\RR$} optimizations,
while non-highlighted lines belong to both.
In classic delta-based synchronization,
each replica $i$ maintains a lattice state $x_i \in \mathcal{L}$
(\qline{alg:state}),
and a $\delta$-buffer $B_i \in \pow\mathcal{L}$ as a set of lattice states
(\qline{alg:buffer}).
When an update operation occurs (\qline{alg:delta-operation}),
the resulting $\delta$ is merged with the local state $x_i$ (\qline{alg:merge})
and added to the buffer (\qline{alg:add-buffer}),
resorting to function $\store$. Periodically,
the whole content of the $\delta$-buffer (\qline{alg:buffer-collect})
is propagated to neighbors (\qline{alg:buffer-send}).
For simplicity of presentation, we assume that communication channels between
replicas cannot drop messages (reordering and duplication is considered), and
that is why the buffer is cleared after each synchronization step
(\qline{alg:buffer-clear}). This assumption can be removed by simply tagging
each entry in the $\delta$-buffer with a unique sequence number, and by
exchanging acks between replicas: once an entry has been acknowledged by every
neighbour, it is removed from the $\delta$-buffer, as originally
proposed in \cite{deltas1}.
When a $\delta$-group is received
(\qline{alg:delta-receive}),
then it is checked whether it will induce an inflation in the local state
(\qline{alg:delta-check}). If this is the case, the $\delta$-group is merged with the local state and added to the buffer
(for further propagation),
resorting to the same function $\store$.
The precondition in \qline{alg:delta-check} appears to be harmless,
but it is in fact, the source of most redundant state propagated
in this synchronization algorithm. Detecting an inflation
is not enough, since almost always there's something new to
incorporate. Instead, synchronization algorithms must extract
from the received $\delta$-group the lattice state responsible for the
inflation, as done by the $\RR$ optimization.
Few changes are required in order to incorporate
this and the $\BP$ optimization in the classic
algorithm, as we show next.
This happens because our approach encapsulates most of its complexity
in the computation of join decompositions and function $\Delta$.
The fact that few changes are required to the classic synchronization
algorithm is a benefit, that will minimize the efforts in incorporating
these techniques in existing implementations.
\algsinglecol{t}{
\newcommand\tab{\hspace{0.7em}}
\newcommand\block[3]{\leavevmode\rlap{\colorbox{#2}{\vphantom{$X_0^1$}\hspace{#1}}}\makebox[#1][l]{#3}}
\newcommand\leftright[2]{\block{0.4\hsize}{gray!15}{#1}\block{0.57\hsize}{gray!45}{ #2}}
\SetKw{kwif}{if }
\SetKw{kwfor}{for }
\inputs{
$n_i \in \pow\mathds{I}$, set of neighbors \;
}
\vspace{0.2cm}
\state{
$x_i \in \mathcal{L}$, $x_i^0 = \bot$ \; \label{alg:state}
\leftright
{$B_i \in \pow \mathcal{L}$, $B_i^0 = \varnothing$}
{$B_i \in \pow{\mathcal{L} \times \mathds{I}}$, $B_i^0 = \varnothing$}
\; \label{alg:buffer}
}
\vspace{0.2cm}
\on({$\operation_i(\m^\delta)$}){
\label{alg:delta-operation}
$\delta = \m^\delta(x_i)$ \;
$\store(\delta, i)$ \;
}
\vspace{0.2cm}
\periodically(// synchronize){
\label{alg:delta-ship}
$\kwfor j \in n_i$ \;
\leftright
{\tab $d = \bigjoin B_i$}
{$d = \bigjoin \{s | \tup{s, o} \in B_i \mathrel{\wedge} o \not = j \}$}
\; \label{alg:buffer-collect}
\tab $\send_{i,j}(\af{delta}, d)$ \; \label{alg:buffer-send}
$B_i' = \varnothing$ \; \label{alg:buffer-clear}
}
\vspace{0.2cm}
\on({$\receive_{j,i}(\af{delta}, d)$}){
\label{alg:delta-receive}
\leftright{}{$d = \Delta(d, x_i)$} \; \label{alg:delta-extract}
\leftright{$\kwif d \not \sqleq x_i$}{$\kwif d \not = \bot$} \; \label{alg:delta-check}
\tab $\store(d, j)$ \;
}
\vspace{0.2cm}
\fun({$\store(s, o)$}){
$x_i' = x_i \join s$ \; \label{alg:merge}
\leftright{$B_i' = B_i \union \{s\}$}{$B_i' = B_i \union \{\tup{s, o}\}$}
\label{alg:add-buffer}
}
\vspace{0.2cm}
}
{Delta-based synchronization algorithms at replica $i \in \mathds{I}$:
\HLBase{classic} version
and version with \HLOpt{$\BP$} and \HLOpt{$\RR$} optimizations.}
{alg:both}
\paragraph*{Avoiding back-propagation of $\delta$-groups}
For $\BP$, each entry in the $\delta$-buffer is tagged
with its origin (\qline{alg:buffer} and \qline{alg:add-buffer}),
and at each synchronization step with neighbour $j$,
entries tagged with $j$ are filtered out
(\qline{alg:buffer-collect}).
\paragraph*{Removing redundant state in received $\delta$-groups}
A received $\delta$-group can contain redundant state,
i.e. state that has already been propagated to neighbors,
or state that is in the $\delta$-buffer $B_i$ still to be propagated.
This occurs in topologies where the underlying graph has cycles,
and thus,
nodes can receive the same information through different
paths in the graph.
In order to detect if a $\delta$-group has redundant state,
nodes do not need to keep everything in the $\delta$-buffer
or even inspect the $\delta$-buffer: it is enough to compare the
received $\delta$-group with the local lattice state $x_i$.
In classic delta-based synchronization,
received $\delta$-groups were added to
$\delta$-buffer only if they would strictly inflate
the local state (\qline{alg:delta-check}).
For $\RR$,
we extract from the $\delta$-group what strictly inflates
the local state $x_i$ (\qline{alg:delta-extract}),
and $\store$ it if it is different from bottom
(\qline{alg:delta-check}).
This extraction is achieved by selecting
which irreducible states
from the decomposition of the received $\delta$-group
strictly inflate the local state,
resorting to function $\Delta$
presented in Section \ref{sec:efficient}.
\section{Evaluation}
\label{sec:eval}
In this Section we evaluate the proposed solutions and show the following:
\begin{itemize}
\item Classic delta-based synchronization
can be as inefficient as state-based synchronization in terms of
transmission bandwidth, while incurring an overhead in terms of memory usage
required for synchronization (Section \ref{sub:micro}).
\item In acyclic topologies, $\BP$ is enough to attain the best results,
while in topologies with cycles, only $\RR$ can greatly reduce the synchronization
cost (Section \ref{sub:micro}).
\item Alternative synchronization techniques (such as Scuttlebutt \cite{scuttlebutt} and operation-based synchronization \cite{crdts1,crdts2}) are metadata-heavy; this metadata represents a large fraction of all the data required for synchronization (over 75\%)
while for delta-based synchronization the metadata overhead can be as low as $7.7\%$
(Section \ref{sub:micro}).
\item In moderate-to-high contention workloads, $\BP$ + $\RR$
can reduce transmission bandwidth and memory consumption by several GBs;
when comparing with $\BP$ + $\RR$, classic delta-based synchronization has an unnecessary
CPU overhead of up-to 7.9$\af{x}$ (Section \ref{sub:retwis}).
\end{itemize}
Instructions on how to reproduce all experiments can be found in
our public repository\footnote{\url{https://github.com/vitorenesduarte/exp}}.
\subsection{Experimental Setup}
The evaluation was conducted in a Kubernetes cluster deployed in Emulab \cite{emulab}.
Each machine has a Quad Core Intel Xeon 2.4 GHz and 12GB of RAM.
The number of machines in the cluster is set such that two replicas
are never scheduled to run in the same machine, i.e. there is at least
one machine available for each replica in the experiment.
\paragraph*{Network Topologies}
Figure \ref{fig:topologies} depicts the two network topologies
employed in the experiments:
a partial-mesh, in which each node has 4 neighbors;
and a tree, with 3 neighbors per node, with the exception of the root
node (2 neighbors) and leaf nodes (1 neighbor).
The first topology exhibits redundancy in the links and tests the effect of cycles in the
synchronization, while the second represents an optimal propagation scenario
over a spanning tree.
\begin{figure}[t]
\begin{minipage}{.24\textwidth}
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\end{minipage}
\caption{Network topologies employed:
a 15-node partial-mesh (to the left)
and a 15-node tree (to the right).}
\label{fig:topologies}
\end{figure}
\subsection{Micro-Benchmarks}
\label{sub:micro}
We have designed a set of micro-benchmarks, in which
each node periodically
(every second)
synchronizes with neighbors and
executes an update operation over a CRDT. The update operation depends
on the CRDT type. In $\af{GSet}$, the update event is the addition of a globally
unique element to the set;
in $\af{GCounter}$, an increment on the counter;
and in $\af{GMap\ K\%}$ each node updates $\frac{\af{K}}{\af{N}}\%$ keys
($\af{N}$ being the number of nodes/replicas),
such that globally $\af{K\%}$ of all the keys in the \emph{grow-only map}
are modified within each synchronization interval.
Note how the $\af{GCounter}$ benchmark is a particular case of $\af{GMap\ K\%}$,
in which $\af{K} = 100$.
For $\af{GMap\ K\%}$ we set the total number of keys to 1000,
and for all benchmarks, the number of events per replica is set to 100.
\newcommand\mltext[2]{
\begin{minipage}{#1}
\vspace*{.02cm}
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#2
\end{flushleft}
\vspace*{-.42cm}
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}
\def1.5cm{1.5cm}
\def2cm{2.4cm}
\def1.7cm{2.6cm}
\begin{table}[]
\caption{Description of micro-benchmarks.}
\label{tab:micro}
\begin{center}
\begin{tabular}{c c p{1.7cm}}
\toprule
\mltext{1.5cm}{\textbf{Type}}
& \mltext{2cm}{\textbf{Periodic event}}
& \mltext{1.7cm}{\textbf{Measurement}} \\ \toprule
\mltext{1.5cm}{$\af{GCounter}$}
& \mltext{2cm}{single increment}
& \mltext{1.7cm}{number of entries in the map} \\ \midrule
\mltext{1.5cm}{$\af{GSet}$}
& \mltext{2cm}{addition of unique element}
& \mltext{1.7cm}{number of elements in the set} \\ \midrule
\mltext{1.5cm}{$\af{GMap\ K\%}$}
& \mltext{2cm}{change the value of $\frac{\af{K}}{\af{N}}\%$ keys}
& \mltext{1.7cm}{number of entries in the map} \\ \bottomrule
\end{tabular}
\end{center}
\end{table}
These micro-benchmarks are summarized in Table \ref{tab:micro},
along with the metric (to be used in transmission and memory measurements)
we have defined:
for $\af{GCounter}$ and $\af{GMap\ K\%}$ we count the number of map entries,
while for $\af{GSet}$, the number of set elements.
We setup this part of the evaluation with 15-node topologies
(as in Figure \ref{fig:topologies}).
As baselines, we have state-based synchronization,
classic delta-based synchronization,
Scuttlebutt, a variation of Scuttlebutt,
and operation-based synchronization.
\paragraph*{Scuttlebutt}
Scuttlebutt \cite{scuttlebutt} is an anti-entropy protocol
used to reconcile changes in values of a key-value store. Each value is uniquely identified
with a version $\tup{i, s} \in \mathds{I} \times \mathds{N}$, where the first component $i \in \mathds{I}$
is the identifier of the replica responsible for the new value,
and $s \in \mathds{N}$ a sequence number, incremented on each local update, thus being unique.
With this, the updates known locally can be summarized by a vector $\mathds{I} \mathrel{\hookrightarrow} \mathds{N}$,
mapping each replica to the highest sequence number it knows.
When a node wants to reconcile with a neighbor replica, it sends the summary vector,
and the neighbor replies with all the key-value pairs it has locally that have versions
not summarized in the received vector.
This strategy is performed in both directions, and in the end, both replicas
have the same key-value pairs in their local key-value store
(assuming no new updates occurred).
Scuttlebutt can be used to synchronize state-based CRDTs with few modifications.
Using as values the CRDT state would be inefficient, since changes to the CRDT
wouldn't be propagated incrementally, i.e. a small change in the CRDT would require
sending the whole new state, as in state-based synchronization.
Therefore, we use as values the optimal deltas resulting from $\delta$-mutators.
As keys, we can simply resort to the version pairs.
When reconciling two replicas, a replica receiving new key-delta pairs,
merges all the deltas with the local CRDT. If CRDT updates stop, eventually
all replicas converge to the same CRDT state. We label this approach \texttt{Scuttlebutt}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.46\textwidth,keepaspectratio]{gset_gcounter}
\end{center}
\caption{Transmission of $\af{GSet}$ and $\af{GCounter}$
with respect to delta-based $\BP + \RR$
-- tree and mesh topologies.}
\label{fig:gset-gcounter-transmission}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=.92\textwidth,keepaspectratio]{gmap}
\end{center}
\caption{Transmission of $\af{GMap\ 10\%}$,
$\af{30\%}$, $\af{60\%}$ and $\af{100\%}$
-- tree and mesh topologies.}
\label{fig:gmap-transmission}
\end{figure*}
This strategy is potentially inefficient in terms of memory: a replica has to keep in the
Scuttlebutt key-value store \emph{all} the deltas it has ever seen, since a neighbor
replica can at any point in time send a summary vector asking for \emph{any} delta.
Since the original Scuttlebutt algorithm does not support deleting keys from the key-value
store, we add support for \emph{safe} deletes of deltas,
in order to reduce its memory footprint.
If each node keeps track of what each node in the system has seen
(in a map $\mathds{I} \mathrel{\hookrightarrow} (\mathds{I} \mathrel{\hookrightarrow} \mathds{N})$ from replica identifiers to
the last seen summary vector),
once a delta has been seen by all nodes,
it can be safely deleted from the local Scuttlebutt store.
We compare with this improved Scuttlebutt variant (labeled \texttt{Scuttlebutt-GC})
that allows nodes to only be connected
to a subset of all nodes, not requiring all-to-all connectivity,
while supporting safe deletes.
For completeness, we also compare with the original Scuttlebutt design
that is unable to garbage-collect unnecessary key-delta pairs.
\paragraph*{Operation-based}
Operation-based CRDTs \cite{crdts1,crdts2} resort to a causal broadcast
middleware~\cite{causal-multicast-survey} that is used to disseminate CRDT operations.
This middleware tags each operation with a vector clock that
summarizes the causal past of the operation.
Such vector is then used by the recipient to ensure causal delivery of operations,
i.e. each operation is only delivered when every operation in its causal past
has been delivered as well.
In topologies with all-to-all connectivity, each node is only responsible for disseminating
its own operations. In order to relax this requirement, we have implemented a middleware
that \emph{stores-and-forwards} operations: when an operation is seen for the first time, it is
added to a transmission buffer to be further propagated in the next synchronization step;
if the same operation is received from different incoming neighbors, the middleware simply
updates which nodes have seen this operation so that unnecessary transmissions are avoided.
To the best of our knowledge, this is the best possible implementation of such a middleware.
We label this approach \texttt{Op-based}.
\subsubsection{Transmission bandwidth}
Figure \ref{fig:gset-gcounter-transmission}
shows, for $\af{GSet}$ and $\af{GCounter}$,
the transmission ratio (of all synchronization mechanisms previously mentioned)
with respect to delta-based synchronization with $\BP$ and $\RR$ optimizations enabled.
The first observation is that
classic delta-based synchronization presents almost
no improvement,
when compared to state-based synchronization.
In the tree topology, $\BP$ is enough to attain the best result,
because the underlying topology does not have cycles,
and thus, $\BP$ is sufficient to prevent redundant state to be propagated.
With a partial-mesh, $\BP$ has little effect,
and $\RR$ contributes most to the overall improvement.
Given that the underlying topology leads to redundant communication
(desired for fault-tolerance),
and classic delta-based can never extract that redundancy,
its transmission bandwidth is effectively similar to that of state-based synchronization.
Scuttlebutt and Scuttlebutt-GC are more efficient than classic delta-based for $\af{GSet}$
since both can precisely identify state changes between synchronization
rounds. However, the results for $\af{GCounter}$ reveal a limitation of this approach.
Since Scuttlebutt treats propagated values as opaque,
and does not understand that the changes in a $\af{GCounter}$ compress naturally
under lattice joins (only the highest sequence for each replica needs to be kept),
it effectively behaves worse than state-based and classic delta-based in this case.
Operation-based synchronization follows the \emph{same trend}
for the \emph{same reason}:
it improves state-based and classic delta-based for $\af{GSet}$
but not for $\af{GCounter}$ since the middleware is unable to compress
multiple operations into a single, equivalent, operation.
Supporting generic operation-compression at the middleware level
in operation-based CRDTs is an open research problem.
The difference between these three approaches is related with
the metadata cost associated to each,
as we show in Section~\ref{subsub:metadata}.
Even with the optimizations $\BP + \RR$ proposed,
the best result for $\af{GCounter}$ is not much
better than state-based.
This is expected since most entries of the underlying map are being updated
between each synchronization step:
each node has almost always something new from every other node
in the system to propagate
(thus being similar to state-based in some cases).
This pattern represents a special case of a map
in which $\af{100\%}$ of its keys are updated between
state synchronizations.
In Figure \ref{fig:gmap-transmission} we study
other update patterns, by measuring
the transmission of $\af{GMap\ 10\%}$, $\af{30\%}$, $\af{60\%}$,
and $\af{100\%}$.
These results are further evidence of what we have
observed
in the case of $\af{GSet}$:
$\BP$ suffices if the network graph is acyclic,
but $\RR$ is crucial in the more general case.
As seen previously, Scuttlebutt and Scuttlebutt-GC behave much better
than state-based synchronization, yielding a reduction in the transmission cost between
$46\%$ and $91\%$, and $20\%$ and $65\%$, respectively. This is
due to the underlying precise reconciliation mechanism of Scuttlebutt.
Operation-based synchronization leads to a transmission reduction between
$35\%$ and $80\%$ since it is able to represent
incremental changes to the CRDT as small operations.
Finally, delta-based $\BP + \RR$ is able reduce the transmission
costs by up-to $94\%$.
In the extreme case of $\af{GMap\ 100\%}$
(every key in the map is updated between synchronization rounds, which is a less likely workload in practical systems)
and considering a partial-mesh, delta-based $\BP + \RR$
provides a modest improvement in relation to state-based of about $18\%$
less transmission, and its performance is below Scuttlebutt variants and
operation-based synchronization.
Vector-based protocols (Scuttlebutt and operation-based) however, have an inherent scalability problem.
When increasing the number of nodes in the system, the transmission costs may
become
dominated by the size of metadata required for synchronization, as we show next.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.32\textwidth,keepaspectratio]{metadata}
\end{center}
\caption{Metadata required per node when synchronizing a $\af{GSet}$
in a mesh topology. Each node has 4 neighbours (as in
Figure \ref{fig:topologies}) and each node identifier has size 20B.}
\label{fig:metadata}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.46\textwidth,keepaspectratio]{memory}
\end{center}
\caption{Average memory ratio with respect to $\BP + \RR$
for $\af{GCounter}$, $\af{GSet}$,
$\af{GMap\ 10\%}$ and $\af{100\%}$
-- mesh topology}
\label{fig:memory}
\end{figure}
\subsubsection{Metadata Cost}\label{subsub:metadata}
Figure~\ref{fig:metadata} shows the size of metadata required
for synchronization per node while varying the
total number of replicas (i.e. nodes). The results show a linear and quadratic cost
(in terms of number of nodes) for Scuttlebutt and Scuttlebutt-GC (respectively),
and a
linear cost for operation-based synchronization
(in terms of both number of nodes and pending updates still to be propagated).
Given $N$ nodes, $P$ neighbors, and $U$ pending updates,
the metadata cost per node is:
\begin{itemize}
\item Scuttlebutt: $NP$ (a vector per neighbor)
\item Scuttlebutt-GC: $N^2 P$ (a map of vectors per neighbor)
\item Operation-based: $NPU$ (a vector per neighbor per pending update)
\item Delta-based: $P$ (a sequence number per neighbor)
\end{itemize}
This cost may represent a large fraction of all data propagated during
synchronization.
For example, in our measurements with 32 nodes, this metadata
represents $75\%$, $99\%$, and $97\%$ of the transmission costs
for Scuttlebutt, Scuttlebutt-GC and operation-based, respectively,
while the overhead of delta-based synchronization is only $7.7\%$.
\subsubsection{Memory footprint}
In delta-based synchronization,
the size of $\delta$-groups being propagated not only affects
the network bandwidth consumption, but also
the memory required to store them in the $\delta$-buffer
for further propagation.
During the experiments, we periodically measure
the amount of state (both CRDT state and metadata required
for synchronization) stored in memory for each node.
Figure~\ref{fig:memory} reports the average
memory ratio with respect to $\BP + \RR$.
State-based does not require synchronization metadata,
and thus it is optimal in terms of memory usage.
Classic delta-based and delta-based $\BP$ have an overhead
of 1.1$\af{x}$-3.9$\af{x}$ since the size of $\delta$-groups in the $\delta$-buffer
is larger for these techniques.
For $\af{GSet}$ and $\af{GMap\ 10\%}$, Scuttlebutt-GC is close to $\BP + \RR$
since deltas are removed from the key-value store
as soon as they are seen by all replicas.
Key-delta pairs are never pruned in the original Scuttlebutt, leading to an increasing memory usage.
As long as new updates exist, the memory consumption
for Scuttlebutt can only deteriorate, ultimately to a point where it will disrupt the system operation.
Operation-based has a higher memory cost than Scuttlebutt-GC, since each
operation in the transmission buffer is tagged with a vector,
while in Scuttlebutt and Scuttlebutt-GC each delta is simply tagged with a version pair.
Considering the results for $\af{GCounter}$, the three vector-based algorithms exhibit
the highest memory consumption. This is justified by
the same reason they perform poorly
in terms of transmission bandwidth
in this case (Figure~\ref{fig:gset-gcounter-transmission}):
these protocols are unable to compress incremental changes.
Overall, and ignoring state-based which doesn't present any metadata memory costs,
$\BP + \RR$ attains the best results.
\subsection{Retwis Application}
\label{sub:retwis}
\def1.5cm{1.5cm}
\def2cm{2cm}
\def1.7cm{1.7cm}
\begin{table}[]
\caption{Retwis workload characterization: for each operation, the number of
CRDT updates performed and its workload percentage.}
\label{tab:retwis}
\begin{center}
\begin{tabular}{c c p{1.7cm}}
\toprule
\mltext{1.5cm}{\textbf{Operation}}
& \mltext{2cm}{\textbf{\#Updates}}
& \mltext{1.7cm}{\textbf{Workload \%}} \\ \toprule
\mltext{1.5cm}{Follow}
& \mltext{2cm}{1}
& \mltext{1.7cm}{15\%} \\ \midrule
\mltext{1.5cm}{Post Tweet}
& \mltext{2cm}{1 + \#Followers}
& \mltext{1.7cm}{35\%} \\ \midrule
\mltext{1.5cm}{Timeline}
& \mltext{2cm}{0}
& \mltext{1.7cm}{50\%} \\ \bottomrule
\end{tabular}
\end{center}
\end{table}
We now compare classic delta-based with
delta-based $\BP + \RR$ using Retwis \cite{retwis},
a popular \cite{tapir,walter,tardis} open-source Twitter clone.
In Table \ref{tab:retwis} we describe the application workload,
similar to the one used in \cite{tapir}:
user $a$ can follow user $b$ by updating the set of followers of user $b$;
users can post a new tweet, by writing it in their wall and in the timeline
of all their followers;
and finally, users can read their timeline, fetching the 10 most recent tweets.
Each user has 3 objects associated with it:
1) a set of followers stored in a $\af{GSet}$;
2) a wall stored in a $\af{GMap}$ mapping tweet identifiers to their content; and
3) a timeline stored in a $\af{GMap}$ mapping tweet timestamps to tweet identifiers.
We run this benchmark with 10K users, and thus, 30K CRDT objects overall.
The size of tweet identifiers and content is 31B and 270B, respectively.
These sizes are representative of real workloads,
as shown in an analysis of Facebook's general-purpose key-value store
\cite{facebook-workload}. The topology is a partial-mesh, with 50 nodes,
each with 4 neighbors, as in Figure \ref{fig:topologies},
and updates on objects follow a Zipf distribution, with
coefficients
ranging from 0.5 (low contention) to 1.5 (high contention)
\cite{tapir}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.48\textwidth,keepaspectratio]{retwis}
\end{center}
\caption{Transmission bandwidth per node (top) and average memory per node (bottom) of
classic delta-based and $\BP + \RR$ for different Zipf coefficient values
(log scale).
The left and right side show these values for
the first and second half of the experiment (respectively).}
\label{fig:retwis}
\end{figure}
Figure~\ref{fig:retwis}
shows the transmission bandwidth and memory footprint of both algorithms,
for different Zipf coefficient values.
We can observe that in low contention workloads, classic delta-based
behaves almost optimally when compared to $\BP + \RR$.
Since updates are distributed almost evenly across all objects,
there are few concurrent updates to the same object between synchronization rounds,
and thus, the simple and naive inflation check in~\qline{alg:delta-check}
suffices.
This phenomena was not observed in the previous set of benchmarks,
since we had a single object, and thus, maximum contention.
As we increase contention, a more sophisticated approach like $\BP + \RR$ is required,
in order to avoid redundant state propagation.
For example, with a 1.25 coefficient, bandwidth is reduced from $1.46$GB/s to $0.06$GB/s per node,
and memory footprint per node drops from $1.58$GB to $0.62$GB (right side of the plots).
Also, as we increase the Zipf coefficient, we note that the bandwidth consumption continues to
rise, leading to an unsustainable situation in the case of classic delta-based,
as it can never reduce the size of $\delta$-groups being transmitted.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.34\textwidth,keepaspectratio]{retwis_processing}
\end{center}
\caption{CPU overhead of classic delta-based when compared to delta-based $\af{BP} + \af{RR}$.}
\label{fig:retwis-processing}
\end{figure}
During the experiment we also measured the CPU time spent in processing CRDT
updates, both producing and processing synchronization messages.
Figure \ref{fig:retwis-processing} reports the CPU overhead of classic delta-based,
when considering $\BP + \RR$ as baseline.
Since classic delta-based produces/processes larger messages
than $\BP + \RR$, this results in a higher CPU cost:
for the 1, 1.25 and 1.5 Zipf coefficients, classic delta-based incurs an overhead
of 0.4$\af{x}$, 5.5$\af{x}$, and 7.9$\af{x}$ respectively.
\section{Related Work}
In the context of remote file synchronization, \emph{rsync} \cite{rsync}
synchronizes two files placed on different machines,
by generating file block signatures,
and using these signatures to identify the missing blocks on the backup file.
In this strategy, there's a trade-off between the size of the blocks to be signed,
the number of signatures to be sent, and the size of the blocks to be received:
bigger blocks to be signed implies fewer signatures to be sent, but
the blocks received (deltas) can be bigger than necessary.
Inspired by \emph{rsync}, \emph{Xdelta}~\cite{xdelta} computes a difference between two files,
taking advantage of the fact that both files are present.
Consequently the cost of sending signatures can be ignored
and the produced deltas are optimized.
In \cite{join-decompositions}, we propose two techniques that can be used
to synchronize two state-based CRDTs after a network partition,
avoiding bidirectional full state transmission.
Let $\A$ and $\B$ be two replicas.
In \emph{state-driven} synchronization, $\A$
starts by sending its local lattice state to $\B$,
and given this state, $\B$ is able to compute a delta that reflects the updates missed by
$\A$.
In \emph{digest-driven} synchronization, $\A$ starts by sending a digest (signature)
of its local state (smaller than the local state), that still allows
$\B$ to compute the delta. $\B$ then sends the computed delta along with a digest
of its local state, allowing $\A$ to compute a delta for $\B$.
Convergence is achieved after 2 and 3 messages
in \emph{state-driven} and \emph{digest-driven}, respectively.
These two techniques also exploit the concept of join decomposition presented in this paper.
Similarly to \emph{digest-driven} synchronization,
$\Delta$-CRDTs \cite{making-deltas}
exchange metadata used to compute a delta that reflects missing updates.
In this approach, CRDTs need to be extended to maintain
additional metadata for delta derivation,
and if this metadata needs to be garbage collected,
the mechanism falls-back to standard bidirectional full state transmission.
In the context of anti-entropy gossip protocols,
\emph{Scuttlebutt} \cite{scuttlebutt}
proposes a \emph{push-pull} algorithm
to be used to synchronize a set of values between participants,
but considers each value as opaque, and does not try to represent
recent changes to these values as deltas. Other solutions try to minimize the communication overhead
of anti-entropy gossip-based protocols by exploiting either hash functions~\cite{clearhouse}
or a combination of Bloom filters, Merkle trees, and Patricia tries~\cite{Byers}. Still,
these solutions require a significant number of message exchanges to identify the source
of divergence between the state of two processes. Additionally, these solutions might incur
significant processing overhead due to the need of computing hash functions and manipulating
complex data structures, such as Merkle trees.
With the exception of \emph{Xdelta}, all these techniques do not assume knowledge prior
to synchronization, and thus delay reconciliation,
by always exchanging state digests in order to detect state divergence.
\section{Conclusion} \label{sec:conclusion}
Under geo-replication there is a significant availability and latency impact
\cite{abadi-cap} when aiming for strong consistency criteria such as linearizability
\cite{herlihy-linearizability}.
Strong consistency guarantees greatly simplify the programmers view of the system and are
still required for operations that do demand global synchronization.
However, several other system's components do not need that same level of coordination
and can reap the benefits of fast local operation and strong eventual consistency.
This requires capturing more information on each data type semantics, since a read/write
abstraction becomes limiting for the purpose of data reconciliation.
CRDTs can provide a sound approach to these highly available solutions and support the
existing industry solutions for geo-replication, which are still mostly grounded on state-based CRDTs.
State-based CRDT solutions quickly become prohibitive in practice, if there is no support for
treatment of small incremental state deltas. In this paper we advance the foundations of
state-based CRDTs by introducing minimal deltas that precisely track state changes.
We also present and micro-benchmark two optimizations,
\emph{avoid back-propagation of $\delta$-groups} and
\emph{remove redundant state in received $\delta$-groups},
that solve inefficiencies in classic delta-based synchronization algorithms.
Further evaluation shows the improvement our solution can bring to
a small scale Twitter clone deployed in a 50-node cluster,
a relevant application scenario.
\section*{Acknowledgments}
We would like to thank Ricardo Macedo, Georges Younes, Marc Shapiro and the anonymous reviewers for their valuable feedback on earlier drafts of this work.
Vitor Enes was supported by EU H2020 LightKone project (732505) and by a FCT - Funda{\c{c}}{\~{a}}o para a Ci{\^{e}}ncia e a Tecnologia - PhD Fellowship (PD/BD/142927/2018). Carlos Baquero was partially supported by SMILES within TEC4Growth project (NORTE-01-0145-FEDER-000020). Jo\~{a}o Leit\~{a}o was partially supported by project NG-STORAGE through FCT grant PTDC/CCI-INF/32038/2017,
and by NOVA LINCS through the FCT grant UID/CEC/04516/2013.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2019-03-12T01:25:49",
"yymm": "1803",
"arxiv_id": "1803.02750",
"language": "en",
"url": "https://arxiv.org/abs/1803.02750"
}
|
\section{Introduction}
Recovery of a signal from several measured intensity patterns, also known as the \emph{phase retrieval problem}, is of great interest in optics and imaging.
Recently it was shown in \cite{Antonello15} that the problem of estimating the wavefront aberration from measurements of the point spread functions can be formulated as a phase retrieval problem.
In this paper, we consider the general phase retrieval problem \cite{SheEld15}:
\begin{equation*}
\mbox{find}\quad \mathbf{a} \in \mathbb{C}^{n_a} \mbox{ such that }
{\mathbf{y}}_i = | {\mathbf{u}}_i^H {\mathbf{a}} |^2 \quad {\rm for}\;\; i=1,\ldots,n_y,
\end{equation*}
where ${\mathbf{y}}_i \in \mathbb{R}_+$ and ${\mathbf{u}}_i \in \mathbb{C}^{n_a}$ are known and $(\cdot)^H$ denotes the Hermitian transpose of a vector (matrix).
For brevity the following compact notation will be used in this paper to denote this general noise-free phase retrieval problem:
\begin{equation}\label{ProblemG}
\mbox{find}\quad \mathbf{a} \in \mathbb{C}^{n_a} \mbox{ such that }
{\mathbf{y}} = | U {\mathbf{a}} |^2,
\end{equation}
where ${\mathbf{y}} \in \Real_+^{n_y}$ are the measurements and $U \in \Complex^{n_y\times n_a}$ is the propagation matrix.
With noise on the measurements $y_i$, we consider the following related optimization problem:
\begin{equation}
\begin{aligned}
& \underset{{\mathbf{a}} \in \Complex^{n_a}}{\min}
& & \norm{{\mathbf{y}} - \abs{U{\mathbf{a}}}^2},
\end{aligned} \label{eq:problemG2}
\end{equation}
where $\norm{\cdot}$ denotes a vector norm of interest.
The sparse variant of the phase retrieval problem corresponds to the case that the unknown parameter $\mathbf{a}$ is a sparse vector.
A special case of this problem is when the measurements are the magnitude of the Fourier transform of multiples of $\mathbf{a}$ with certain phase diversity patterns.
A number of algorithms utilizing the Fourier transform have been proposed for solving this class of phase retrieval problems \cite{fienup1982phase,LukBurLyo02,Gespar}.
The fundamental nature of \eqref{ProblemG} has given rise to a wide variety of solution methods that have been developed for specific variants of this problem since the observation of Sayre in 1952 that phase information of a scattered wave may be recovered from the recorded intensity patterns at and between Bragg peaks of a diffracted wave \cite{Sayre52}.
Direct methods \cite{Hauptman86} usually use insights about the crystallographic structure and randomization to search for the missing phase information.
The requirement of such a-priori structural information and the expensive computational complexity often limit the application of these methods in practice.
A second class of methods first devised by Gerchberg and Saxton \cite{GSF72} and Fienup \cite{fienup1982phase} can be described as variants of the method of alternating projections on certain sets defined by the constraints. For an overview of these methods and latter refinements we refer the reader to \cite{Bauschke02,LukBurLyo02}.
In \cite{candes2015phase} \eqref{ProblemG} is relaxed to a convex optimization problem.
The inclusion of the sparsity constraint in the same framework of convex relaxations has been considered in \cite{Ohlsson}.
However, as reported in \cite{Gespar} the combination of matrix lifting and semidefinite programming (SDP) makes this method not suitable for large-scale problems.
To deal with large-scale problems, the authors of \cite{Gespar} have proposed an iterative solution method, called GESPAR, which appears to yield promising recovery of very sparse signals.
However, this method consists of a heuristic search for the support of ${\mathbf{a}}$ in combination with a variant of Gauss-Newton method, whose computational complexity is often expensive.
These algorithmic features are potential drawbacks of GESPAR.
In this paper, we propose a sequence of convex relaxations for the phase retrieval problem in \eqref{ProblemG}.
Contrary to existing convex relaxation schemes such as those proposed in \cite{candes2015phase,Ohlsson}, matrix lifting is not required in our strategy. The obtained convex problems are affine in the unknown parameter vector ${\mathbf{a}}$.
Contrary to \cite{candes2013phaselift}, our strategy does not require the tuning of regularization parameters when the measurements are corrupted by noise.
We then present an ADMM-based algorithm that can solve the resulting optimization problems effectively.
This potentially addresses the restriction of current SDP-based methods to only relatively small-scale problems.
In Section~\ref{sec:wavefrontestimation} we formulate the estimation problem of our interest for both zonal and modal forms.
In Section~\ref{sec:algorithm} we propose an algorithm for solving this problem.
Since this algorithm is based on minimizing a nuclear norm, a computationally heavy minimization problem, we suggest an ADMM-based algorithm in Section~\ref{sec:admm} that exploits the problem structure.
This ADMM algorithm features two minimization problems whose solutions can be computed exactly and with complexity $\bigO{n_y n_a}$, where $n_y$ is the number of measurements and $n_a$ is the number of unknown variables.
Analytic solutions for the ADMM algorithm update steps will be presented in Subsections~\ref{sec:aupdate} and \ref{sec:Xupdate}.
The convergence behaviour of the algorithm proposed in Section~\ref{sec:algorithm} is analysed in Section~\ref{sec:convergence}.
In Sections~\ref{sec:numericalexperiments} we describe and discuss the results of a number of numerical experiments that demonstrate the promising performances of our algorithms.
We end with concluding remarks in Section~\ref{sec:remarks}.
\section{Wavefront estimation from intensity measurements}\label{sec:wavefrontestimation}
The problem of phase retrieval from the point spread function images can be approached from 2 directions. We first describe the problem in zonal form, and then in modal form.
\subsection{Problem formulation in zonal form}
\label{subsec:zonal form}
In \cite{Antonello15} it was shown that reconstructing the wavefront from CCD recorded images of a point source may also be formulated as a phase retrieval problem.
These recorded images are called {\em point spread functions (PSFs)}.
As such approaches avoid the requirement of extra hardware to sense the wavefront, such as a Shack-Hartmann wavefront sensor, the problem is relevant and summarized here.
The PSF is derived from the magnitude of the Fourier transform of the generalized pupil function (GPF).
For an aberrated optical system the GPF is defined as the complex valued function \cite{goodman2008introduction}:
\begin{equation}
\label{eq:GPF}
P(\rho,\theta) = {\mathbf{A}}(\rho,\theta)\expp{j \phi(\rho,\theta)},
\end{equation}
where $\rho$ (radius) and $\theta$ (angle) specify the normalized polar coordinates in the exit pupil plane of the optical system.
In \eqref{eq:GPF}, $\mathbf{A}(\rho,\theta)$ is the amplitude apodisation function and $\phi(\rho,\theta)$ is the phase aberration function.
The aim of the wavefront reconstruction problem is to estimate $\phi(\rho,\theta)$.
Once this phase aberration of an optical system has been estimated, it can be corrected by using phase modulating devices such as deformable mirrors.
In order to estimate $\phi(\rho,\theta)$, a known phase diversity pattern $\phi_d(\rho,\theta)$ can be introduced (e.g., by using a deformable mirror) to transform the GPF in a controlled manner into the aberrated GPF:
\begin{equation}\label{eq:GPFd}
P_d(\rho,\theta)
= \mathbf{A}(\rho,\theta) \expp{j \phi(\rho,\theta)} \expp{j \phi_d(\rho,\theta)}.
\end{equation}
The noise-free intensity pattern of $P_d(\rho,\theta)$ measured at the image plane is denoted
\begin{equation}\label{eq:Intensity_d}
{\mathbf{y}}_d = \abs{\fourier{ {\mathbf{A}}(\rho,\theta) \expp{j \phi(\rho,\theta)} \expp{j \phi_d(\rho,\theta)} } }^2.
\end{equation}
If we sample the function $P_d(\rho,\theta)$ at points corresponding to a square grid of size $m \times m$ on the pupil plane, then ${\mathbf{A}}(\rho,\theta)$, $\phi_d(\rho,\theta)$ and $\phi(\rho,\theta)$ are square matrices of that size.
Let us define ${\ve}(\cdot)$ the vectorization operator such that ${\ve}(Z)$ yields the vector obtained by stacking the columns of matrix $Z$ into a column vector.
The inverse operator ${\ve}^{-1}(\cdot)$, which maps a column vector of size $m^2$ to a square matrix of size $m \times m$, is also well defined. Let in particular the matrix $Z$ and the vector ${\mathbf{a}}$ be defined as:
\begin{equation*}
Z = {\mathbf{A}}(\rho,\theta) e^{j\phi(\rho,\theta)} \in \mathbb{C}^{m \times m},\quad {\mathbf{a}} = {\ve}(Z) \in \mathbb{C}^{m^2}. \label{eq:unknownpupilF}
\end{equation*}
With the definition of the vector $\mathbf{p}_d$:
\begin{equation*}
\mathbf{p}_d = \vect{e^{j\phi_d(\rho,\theta)}}\in \mathbb{C}^{m^2},
\end{equation*}
and with $D_d = \dia{\mathbf{p}_d} \in \mathbb{C}^{m^2\times m^2} $ the diagonal matrix with diagonal entries taken from the vector $\mathbf{p}_d$, we can write the noise-free intensity measurements in \eqref{eq:Intensity_d} as
\begin{equation*}
{\mathbf{y}}_d = \abs{ \fourier{ \expp{j \phi_d(\rho,\theta)} Z} }^2
= \abs{ \fourier{ {\ve}^{-1}(D_d\mathbf{a})} }^2.
\end{equation*}
As the Fourier transform is a linear operator, we can write our noise-free intensity measurements in the form:
\begin{equation}\label{eq:Intensity_f}
{\mathbf{y}}_d = \left| U_d {\mathbf{a}} \right|^2,
\end{equation}
where in this case $U_d$ is a unitary matrix.
By stacking the vectors ${\mathbf{y}}_d$ and the matrices $U_d$, obtained from the $n_d$ images with $n_d$ different phase diversities, correspondingly into the vector ${\mathbf{y}}$ and the matrix $U$ (of size $n_d m^2 \times m^2$), the problem of finding ${\mathbf{a}}$ from noise-free intensity measurements can be formulated as in \eqref{ProblemG} and that from noisy measurements can be formulated as in \eqref{eq:problemG2} for $n_a=m^2$ and $n_y=n_dm^2$.
It is worth noting that the dimension of the unknown ${\mathbf{a}}$ with $m$ in the range of a couple of hundreds turns this problem into a non-convex large-scale optimization problem. For such a problem the implementation of PhaseLift \cite{candes2013phaselift} using standard semidefinite programming, using libraries like MOSEK \cite{mosek}, will not be tractable because of the large matrix dimensions of the unknown quantity.
If we assume that the computational complexity of semidefinite programming with matrix constraints of size $n \times n$ increases with $\bigO{n^6}$ \cite{vandenberghe2005interior}, then a naive implementation of the PhaseLift method applied to \eqref{eq:problemG2} involving a single image has worst-case computational complexity of $\bigO{m^{12}}$.
\subsection{Problem formulation in modal form}\label{sec:modal}
In general, only approximate solutions can be expected for a phase retrieval problem.
In the modal form of the phase retrieval problem, also considered in \cite{Antonello15} for extended Nijboer-Zernike (ENZ) basis functions, the GPF is assumed to be well approximated by a weighted sum of basis functions.
We make use of real-valued radial basis functions \cite{martinez2016computation} with complex coefficients to approximate the GPF. These are studied in the scope of wavefront estimation in \cite{Piet17} and an illustration of these basis function on a $4 \times 4$ grid in the pupil plane is given in Figure~\ref{fig:bf}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.5\columnwidth]{basis_functions.eps}
\caption{16 radial basis functions with centers in a $4 \times 4$ grid, with circular aperture support.}
\label{fig:bf}
\end{figure}
Switching from the polar coordinates $(\rho,\theta)$ to the Cartesian coordinates $(x,y)$ in the pupil plane, let us consider the radial basis functions and the approximate GPF given by
\begin{equation}
\begin{aligned}
G_i(x,y) &=
\chi(x,y)\expp{-\lambda_i \left((x-x_i)^2 + (y-y_i)^2\right) }, \\
P(x,y) &\approx \widetilde{P}(x,y,{\mathbf{a}}) = \sum_{i=1}^{n_a} a_i G_i(x,y),
\end{aligned} \label{eq:basis}
\end{equation}
where $(x_i,y_i)$ are the centers of basis functions $G_i(x,y)$, $a_i \in \Complex$, $\lambda_i \in \Real_+$ determines the spread of that function, $\chi(x,y)$ denotes the support of the aperture, and ${\mathbf{a}}$ is the coefficient parameter vector to be estimated. The parameters $\lambda_i$ are usually taken equal for all basis functions and for their tuning we refer to \cite{Piet17}.
The aberrated GPF corresponding to the introduction of phase diversity $\phi_d$ is
\begin{equation}\label{eq:aGPFd}
\widetilde{P}_d(x,y,{\mathbf{a}},\phi_d) = \sum_{i=1}^{n_a} a_i G_i(x,y) \expp{j\phi_d(x,y)}.
\end{equation}
The normalized complex PSF is the 2-dimensional Fourier transform of the GPF
\cite{janssen2002extended, braat2002assessment}.
The aberrated PSF corresponding to the aberrated GPF in \eqref{eq:aGPFd} is given as
\begin{equation}\label{eq:aPSF}
p_d({u},{v}) = \sum_{i=1}^{n_a} a_i \fourier{G_i(x,y) \expp{j\phi_d(x,y)} } = \sum_{i=1}^{n_a} a_i U_{d,i}({u},{v}),
\end{equation}
where $({u},{v})$ are the Cartesian coordinates in the image plane of the optical system.
We now drop the dependency on the coordinates and vectorize expression \eqref{eq:aPSF} for all $n_d$ diversities that have been applied to obtain the following compact form of a single matrix-vector multiplication,
\begin{equation}\label{p=Ua}
\mathbf{p} = U {\mathbf{a}}.
\end{equation}
The vector $\mathbf{p}$ is the obtained vectorization and combination over all the aberrated PSFs, and the matrix $U$ is the vectorized and concatenated version of the functions
$U_{d,i}$ sampled on a grid of size $m \times m$.
Let the intensity of the PSFs be recorded on the corresponding grid of pixels of size $m \times m$, and let the vectorization of this intensity pattern for different phase diversities be concatenated into the vector ${\mathbf{y}}$.
We can again formulate the problem of finding ${\mathbf{a}}$ from noise-free intensity measurements as in \eqref{ProblemG} and from noisy measurements as in \eqref{eq:problemG2}
for $n_y = m^2n_d$.
It is worth noting that the dimension of ${\mathbf{a}}$ is not dependent on the size of the sample grid (the size of the problem).
This is the fundamental advantage of the modal form formulation over the zonal form one, for which the size of ${\mathbf{a}}$ directly depends on the size of the problem, i.e. $n_a=m^2$.
In this paper two steps are combined to deal with the large-scale nature of optimization \eqref{eq:problemG2}:
\begin{enumerate}
\item The unknown pupil function $P(\rho,\theta)$ can be represented as a linear combination of a number of basis functions.
In \cite{Antonello15} use has been made of the ENZ basis functions, while in \cite{Piet17} use is made of radial basis functions instead of ENZ ones.
The radial basis functions are used here as \cite{Piet17} demonstrated their advantages over the ENZ type.
\item A new strategy is proposed for solving optimization \eqref{ProblemG} via a sequence of convex optimization problems.
Each of the subproblems can be solved effectively by an iterative ADMM algorithm that exploits the problem structure.
\end{enumerate}
In the following we assume that the problem is normalized such that all entries of ${\mathbf{y}}$ have values between 0 and 1.
\section{The COPR algorithm}\label{sec:algorithm}
Equation \ref{ProblemG} is equivalent to a rank constraint.
Define the matrix-valued function
\begin{equation}
M(A,B,C,X,Y) = \pmat{C+AY+XB+XY & A+X \\ B+Y & I}, \label{eq:M}
\end{equation}
where $I$ is the identity matrix of appropriate size.
Let ${\mathbf{b}} \in \Complex^{n_a}$ be a coefficient vector.
For notational convenience, we will denote
\begin{equation*}
\begin{aligned}
&M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}}) = \\
&M\left(\dia{{\mathbf{a}}^HU^H},\dia{U{\mathbf{a}}},\dia{{\mathbf{y}}},\dia{{\mathbf{b}}^HU^H},\dia{U{\mathbf{b}}}\right).
\end{aligned}
\end{equation*}
Our proposed algorithm in this paper relies on the following fundamental result.
\begin{lemma}\label{lem:rank}\cite{doelman2016sequential}
For any ${\mathbf{b}} \in \Complex^{n_a}$, the constraint ${\mathbf{y}} = \abs{U{\mathbf{a}}}^2$ is equivalent to the constraint
\begin{equation*}
\rank{M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})} = n_y.
\end{equation*}
\end{lemma}
For addressing problem \eqref{eq:problemG2}, Lemma~\ref{lem:rank} suggests a consideration of the following approximate problem, for a user-selected parameter vector ${\mathbf{b}}$,
\begin{equation} \label{min rank}
\min_{{\mathbf{a}} \in \Complex^{n_a}} \rank{M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})}.
\end{equation}
Since \eqref{min rank} is a non-convex problem and to anticipate the presence of measurement noise, we propose to solve the following convex optimization problem:
\begin{equation}
\min_{{\mathbf{a}} \in \Complex^{n_a}} f({\mathbf{a}}) := \norm{M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})}_*, \label{EQ:NN}
\end{equation}
where $\norm{\cdot}_*$ denotes the nuclear norm of a matrix, the sum of its singular values \cite{recht2010guaranteed}.
In the case that prior knowledge on the problem indicates that
${\mathbf{a}}$ is a sparse vector, the objective function in \eqref{EQ:NN} can easily be extended with an $\ell_1$-regularization to stimulate sparse solutions, since the vector ${\mathbf{a}}$ appears affinely in $M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})$:
\begin{equation}
\min_{{\mathbf{a}} \in \Complex^{n_a}} f({\mathbf{a}}) + \lambda \norm{{\mathbf{a}}}_1, \label{eq:sparse}
\end{equation}
for some regularization parameter $\lambda$.
Note that for ${\mathbf{b}} = -{\mathbf{a}}$,
\begin{equation}
\norm{M(U,{\mathbf{a}},-{\mathbf{a}},{\mathbf{y}})}_* = \norm{{\mathbf{y}} - \abs{U{\mathbf{a}}}^2}_1 + n_y.
\end{equation}
Since the result of optimization \ref{EQ:NN} might not produce a desired solution sufficiently fitting the measurements, we propose the iterative Convex Optimization-based Phase Retrieval (COPR) algorithm, outlined in Algorithm~\ref{ALG:COPR}.
\begin{algorithm}
\caption{Convex Optimization-based Phase Retrieval (COPR)}\label{ALG:COPR}
\begin{algorithmic}[1]
\Procedure{COPR}{${\mathbf{b}},\tau$}\Comment{Some guess for ${\mathbf{b}}$}
\While{$\norm{{\mathbf{y}} - \abs{U{\mathbf{a}}}}_1 > \tau$}\Comment{Termination criterion}
\State ${\mathbf{a}}_+ \in \argmin_{\mathbf{a}} \norm{M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})}_*$
\State ${\mathbf{b}}_+\gets -{\mathbf{a}}_+$
\EndWhile
\EndProcedure
\end{algorithmic}
\end{algorithm}
The nuclear norm minimization in Algorithm~\ref{ALG:COPR} is the main computational burden for an implementation.
Usual implementations of the nuclear norm involve semidefinite constraints, and require a semidefinite optimization solver.
If we assume that their computational complexity increases with $\bigO{n^6}$ \cite{vandenberghe2005interior} with constraint on matrices of size $n \times n$, then minimizing the nuclear norm of the matrix $M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})$ of size $2n_y \times 2n_y$ is computationally infeasible even for
relatively
small-scale problems.
Therefore, we propose a tailored ADMM algorithm of which the computational complexity of the iterations scales $\bigO{n_y n_a}$, and requires the inverse of a matrix of size $2n_a \times 2n_a$ for every iteration of Algorithm \ref{ALG:COPR}.
\section{Efficient computation of the solution to \eqref{EQ:NN}}\label{sec:admm}
The minimization problem \eqref{EQ:NN} can be reformulated as:
\begin{align}\label{eq:admmopt}
\underset{X,{\mathbf{a}}}{\min}\;
\norm{X}_*\quad \mbox{ subject to }\quad
X = M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}}).
\end{align}
Applying the ADMM optimization technique \cite{boyd2011distributed} to the constraint optimization problem \eqref{eq:admmopt}, we obtain the steps in Algorithm~\ref{alg:admm}.
\begin{algorithm}
\caption{An ADMM algorithm for solving \eqref{eq:admmopt}}\label{alg:admm}
\begin{algorithmic}[1]
\Procedure{NN-ADMM}{${\mathbf{b}},{\mathbf{y}},\rho,\tau$}
\State ${\mathbf{a}} \gets -{\mathbf{b}}$
\State $\pmb{X} \gets M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})$
\State $\pmb{Y} \gets 0 $
\While{$\abs{\norm{M(U,{\mathbf{a}}_{+},{\mathbf{b}},{\mathbf{y}})}_* - \norm{M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})}_*} > \tau$}
\State ${\mathbf{a}}_{+} \in $
\begin{equation}
\underset{{\mathbf{a}}}{\argmin} \norm{\pmb{X} - M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}}) + \frac{1}{\rho}\pmb{Y}}_F^2 \label{eq:aupdate}
\end{equation}
\State $\pmb{X}_{+} \in $
\begin{equation}
\underset{\pmb{X}}{\argmin} \norm{\pmb{X}}_* + \dfrac{\rho}{2}\norm{\pmb{X} - M(U,{\mathbf{a}}_{+},{\mathbf{b}},{\mathbf{y}}) + \dfrac{1}{\rho}\pmb{Y}}_F^2 \label{eq:Xupdate}
\end{equation}
\State $\pmb{Y}_{+} \gets \pmb{Y} + \rho\left(\pmb{X}_{+} - M(U,{\mathbf{a}}_{+},{\mathbf{b}},{\mathbf{y}})\right)$
\State update $\rho$ according to the rules in \cite{boyd2011distributed}
\EndWhile
\EndProcedure
\end{algorithmic}
\end{algorithm}
The advantage of using this ADMM formulation is that both of the update steps \eqref{eq:aupdate} and \eqref{eq:Xupdate} have solutions that can be computed analytically.
The efficient computation of the solutions are described in the following two subsections.
\subsection{Efficient computation of the solution to \eqref{eq:aupdate}}\label{sec:aupdate}
Upon inspection of \eqref{eq:aupdate}, we see that this is a complex-valued standard least squares problem since $M(U,{\mathbf{a}},{\mathbf{b}},{\mathbf{y}})$ is parameterized affinely in ${\mathbf{a}}$.
Let $\re{\cdot}$ and $\im{\cdot}$ respectively denote the real and the imaginary parts of a complex object.
Let the subscripts $(\cdot)_1$, $(\cdot)_2$ and $(\cdot)_3$ respectively denote the top-left, top-right and bottom-left submatrices according to \eqref{eq:M}.
Define
\begin{equation*}
\pmb{Z} = \pmb{X} + \dfrac{1}{\rho}\pmb{Y}, \quad X = \dia{b^HU^H}.
\end{equation*}
In the sequel, let $\adi{P}$ denote the vector with the diagonal entries of a square matrix $P$.
Reordering the elements in \eqref{eq:aupdate}, separating the real and the imaginary parts, removing all matrix elements in the argument of the Frobenius norm that do not depend on ${\mathbf{a}}$, and vectorizing the result, give the following least squares problem:
\begin{equation}
\min_{\mathbf{x}} \norm{{\mathbf{u}}_{ADMM} - {\mathbf{u}}_{COPR} - AB{\mathbf{x}}}_2^2. \label{eq:ls}
\end{equation}
The variables ${\mathbf{u}}_{ADMM},~{\mathbf{u}}_{COPR},~A,~B$ and ${\mathbf{x}}$ are given by
\begin{equation}
\begin{aligned}
&{\mathbf{u}}_{ADMM} = \pmat{\adi{\re{\pmb{Z}_{1}}} \\ \adi{\re{\pmb{Z}_{2}}} \\ \adi{\re{\pmb{Z}_{3}}} \\ \adi{\im{\pmb{Z}_{2}}} \\ \adi{\im{\pmb{Z}_{3}}} },
&&& &{\mathbf{u}}_{COPR} = \pmat{{\mathbf{y}}
+\adi{\abs{X}^2} \\ \adi{\re{X}} \\ \adi{\re{X}} \\ \adi{\im{X}} \\ -\adi{\im{X}}}, \\
&A = \pmat{2\re{X} & 2\im{X} \\ I & 0 \\ I & 0 \\ 0 & I \\ 0 & -I},
&&& &B = \pmat{\re{U} & - \im{U} \\ -\im{U} & -\re{U}},
\end{aligned}
\end{equation}
and ${\mathbf{x}} = \pmat{\re{{\mathbf{a}}}^T & \im{{\mathbf{a}}}^T}^T$.
This means that the optimal solution to \eqref{eq:ls} is given by
\begin{equation*}
{\mathbf{x}}^* = (B^TA^TAB)^{-1}B^TA^T({\mathbf{u}}_{ADMM} - {\mathbf{u}}_{COPR}).
\end{equation*}
During the ADMM iterations only ${\mathbf{u}}_{ADMM}$ changes. The inverse $ (B^TA^TAB)^{-1} $ has to be computed once for every iteration of Algorithm~\ref{ALG:COPR} (i.e. it remains constant throughout the ADMM iterations).
Since the complexity of computing an inverse is $\bigO{n^3}$ for matrices of size $n \times n$, the computational complexity of this inverse process scales cubically with the number of basis functions.
Once this inverse matrix is obtained, the optimal solution to the least squares problem in \eqref{eq:ls} can be computed by a simple matrix-vector multiplication, whose complexity scales with $\bigO{n_yn_a}$.
Note that in the case that the objective term includes regularization as in \eqref{eq:sparse}, the optimization \eqref{eq:ls} should be modified appropriately to include the additive regularization term $\lambda\norm{{\mathbf{a}}}_1$.
\subsection{Efficient computation of the solution to \eqref{eq:Xupdate}}\label{sec:Xupdate}
The optimization in \eqref{eq:Xupdate} is of the form
\begin{equation}
\underset{X}{\argmin} \norm{X}_* + \lambda\norm{X-C}_F^2. \label{eq:Xupdate_simple}
\end{equation}
Let $C= U_C\Sigma_CV_C^T$ be the singular value decomposition of $C \in \Complex^{2n_y \times 2n_a}$.
\begin{lemma}\label{lem:singular_vectors}
The solution $\pmb{X}$ to \eqref{eq:Xupdate_simple} has singular vectors $U_C$ and $V_C$.
\end{lemma}
\begin{proof}
Let
$X = U_X\Sigma_XV_X^T$ be a singular value decomposition of $X$.
Then
\begin{equation*}
\begin{aligned}
\norm{X}_* + \lambda\norm{X-C}_F^2 &= \trace{\Sigma_X} + \\
&\qquad \lambda\left(\inp{X}{X} + \inp{C}{C} -2\inp{X}{C}\right).
\end{aligned}
\end{equation*}
Using Von Neumann's trace inequality we get
\begin{equation*}
\begin{aligned}
& \min_{X} \paren{ \trace{\Sigma_X} + \lambda\left(\inp{X}{X} + \inp{C}{C} -2\inp{X}{C}\right)} \\
\geq\; &\min_{X} \paren{\trace{\Sigma_X} + \lambda\left(\inp{X}{X} + \inp{C}{C} -2\trace{\Sigma_X\Sigma_C}\right)}\\
\end{aligned}
\end{equation*}
with equality holds true when $C$ and $X$ are simultaneously unitarily diagonalizable.
The optimal solution $\pmb{X}$ to \eqref{eq:Xupdate_simple} therefore has the same singular vectors as $C$, i.e. $U_{\pmb{X}}=U_C,~ V_{\pmb{X}}=V_C$.
\end{proof}
Denote the singular values of $C$ in descending order as $\sigma_{C,1},\ldots,\sigma_{C,2n_y}$, and those of $X$ similarly.
Thanks to Lemma~\ref{lem:singular_vectors}, \eqref{eq:Xupdate_simple} can be simplified to
\begin{equation}\label{prob:sigma}
\underset{\sigma_{X,i}}{\argmin} \sum_{i=1}^{2n_y}
\paren{\sigma_{X,i} + \lambda\left(\sigma_{X,i} - \sigma_{C,i}\right)^2}.
\end{equation}
This problem is completely decoupled in $\sigma_{X,i}$ and the optimal solution to \eqref{prob:sigma} is computed with
\begin{equation*}
\sigma_{\pmb{X},i} = \max\left(0, \sigma_{C,i} - \frac{1}{2\lambda}\right),\quad i = 1,\ldots,2n_y.
\end{equation*}
By row and column permutations, the matrix $C$ is block-diagonal with blocks of size $2 \times 2$.
The SVD of this permuted matrix therefore involves block-diagonal matrices $U_C$, $\Sigma_C$ and $V_C$ and these blocks can be obtained separately and in parallel. Since the blocks are of size $2 \times 2$, the SVD can be obtained analytically.
This shows that a valid SVD can be computed very efficiently, in $\bigO{1}$.
That is, in theory, in a computation time independent of the number of pixels in the image, the number of images taken or of the number of basis functions.
\section{Convergence analysis of Algorithm \ref{ALG:COPR}}\label{sec:convergence}
Algorithm \ref{ALG:COPR} can be reformulated as a Picard iteration $\mathbf{a}_{k+1} \in T(\mathbf{a}_k)$, where the fixed point operator $T:\mathbb{C}^{n_a}\to \mathbb{C}^{n_a}$ is given by
\begin{equation}\label{T:operator}
T(\mathbf{a}) = \arg\min_{\substack{\mathbf{x}\in \mathbb{C}^{n_a}}}\;\norm{M(U,\mathbf{x},-\mathbf{a},\mathbf{y})}_*.
\end{equation}
Our subsequent analysis will show that the set of fixed points, $\Fix T$, of $T$ is in general nonconvex and as a result, iterations generated by $T$ can not be \emph{Fej\'er monotone} \cite[Definition 5.1 of]{BauCom11} with respect to $\Fix T$.
Therefore, the widely known convergence theory based on the properties of \emph{Fej\'er monotone operators} and \emph{averaging operators} is not applicable to the operator $T$ given at \eqref{T:operator}.
In this section, we make an attempt to prove convergence of Algorithm \ref{ALG:COPR}, which has been observed from our numerical experiments, via a relatively new developed convergence theory based on the theory of \emph{pointwise almost averaging operators} \cite{LukNguTam16}.
It is worth mentioning that we are not aware of any other analysis schemes addressing convergence of Picard iterations generated by general \emph{nonaveraging} fixed point operators.
Our discussion consists of two stages. Based on the convergence theory developed in \cite{LukNguTam16}, we first formulate a convergence criterion for Algorithm \ref{ALG:COPR} (Proposition \ref{p:convergence}) under rather abstract assumptions on the operator $T$.
Due to the highly complicated structure of the nuclear norm of a general complex matrix, we are unable to verify these mathematical conditions for general matrices $U$.
However, we will verify that they are well satisfied in the case that $U$ is a unitary matrix (Theorem \ref{T:MATRIX_GENERAL}).
From the latter result, we heuristically hope that Algorithm \ref{ALG:COPR} still enjoys the convergence result when the matrix $U$ is close to being unitary in a certain sense.
It is a common prerequisite for analyzing local convergence of a fixed point algorithm that the set of solutions to the original problem is nonempty.
That is, there exists $\mathbf{a}\in \mathbb{C}^{n_a}$ such that $\mathbf{y}=|U\mathbf{a}|^2$.
Before stating the convergence result, we need to verify that the fixed point set of $T$ is nonempty.
\begin{lemma}\label{LEM:FIX_NONEMPTY}
The fixed point operator $T$ defined at \eqref{T:operator} holds
\[
\parennn{\mathbf{a} \mid \mathbf{y} = |U\mathbf{a}|^2} \;\subseteq\; \Fix T := \left\{\mathbf{a}\in \mathbb{C}^{n_a}\mid \mathbf{a}\in T(\mathbf{a})\right\}.
\]
\end{lemma}
\begin{proof}
See Appendix~\ref{APP:FIX_NONEMPTY}
\end{proof}
The next proposition provides an abstract convergence result for Algorithm \ref{ALG:COPR}.
$\Fix T$ is supposed to be closed.
\begin{proposition}\label{p:convergence}\cite[simplified version of Theorem 2.2 of]{LukNguTam16}
Let $S\subset \Fix T$ be closed with $T(\mathbf{a}^*) \subset\Fix T$ for all $\mathbf{a}^* \in S$ and let $W$ be a neighborhood of $S$. Suppose that $T$ satisfies the following conditions.
\begin{enumerate}
\item[(i)]\label{t:subfirm convergence a} $T$ is \emph{pointwise averaging} at every point of $S$ with constant $\alpha\in (0,1)$ on $W$.
That is, for all $\mathbf{a}\in W$, $\mathbf{a}_+\in T(\mathbf{a})$, $\mathbf{a}^*\in P_S(\mathbf{a})$ and $\mathbf{a}^*_+\in T(\mathbf{a}^*)$,
\begin{align}\label{averaged of T}
\norm{\mathbf{a}_+ - \mathbf{a}^*_+}^2 \le \norm{\mathbf{a}-\mathbf{a}^*}^2 - \frac{1-\alpha}{\alpha}\norm{(\mathbf{a}_+ - \mathbf{a})-(\mathbf{a}^*_+ - \mathbf{a}^*)}^2.
\end{align}
\item[(ii)]\label{t:subfirm convergence b} The set-valued mapping $\psi:= T-\Id$ is \emph{metrically subregular} on $W$ for $0$ with constant $\gamma >0$, where $\Id$ is the Identity mapping.
That is,
\begin{equation}\label{met_subreg}
\gamma\dist(\mathbf{a},\psi^{-1}(0)) \le \dist(0,\psi(\mathbf{a})),\quad \forall \mathbf{a}\in W.
\end{equation}
\item[(iii)]\label{t:tech_assump} It holds $\dist(\mathbf{a},S) \le \dist(\mathbf{a},\Fix T)$ for all $\mathbf{a}\in W$.
\end{enumerate}
Then all Picard iterations $\mathbf{a}_{k+1}\in T(\mathbf{a}_k)$ starting in $W$ satisfy $\dist(\mathbf{a}_k,S)\to 0$ as $k\to \infty$ at least linearly.
\end{proposition}
Condition $(iii)$ in Proposition \ref{p:convergence} is, on one hand, a technical assumption and becomes redundant when $S=\Fix T$.
On the other hand, the set $S$ allows one to exclude from the analysis possible \emph{inhomogeneous} fixed points of $T$, at which the algorithm often exposes weird convergence behavior \cite[see Example 2.1 of]{LukNguTam16}.
The size of neighborhood $W$ appearing in Proposition \ref{p:convergence} indicates the robustness of the algorithm in terms of erroneous input (the distance from the starting point to a nearest solution).
We now apply the abstract result of Proposition \ref{p:convergence} to the following special, but important case.
\begin{thm}\label{T:MATRIX_GENERAL}
Let $U\in \mathbb{C}^{n_a\times n_a}$ be unitary and $\mathbf{a}^*\in \mathbb{C}^{n_a}$ be such that $|U\mathbf{a}^*|^2=\mathbf{y}$.
Then every Picard iteration generated by Algorithm~\ref{ALG:COPR} $\mathbf{a}_{k+1}\in T(\mathbf{a}_k)$ starting sufficiently close to $\mathbf{a}^*$ converges linearly to a point $\tilde{\mathbf{a}} \in \Fix T$ satisfying $|U\tilde{\mathbf{a}}|^2=\mathbf{y}$.
\end{thm}
\begin{proof}
See Appendix~\ref{APP:MATRIX GENERAL}.
\end{proof}
\section{Numerical experiments}\label{sec:numericalexperiments}
Three important numerical aspects of the CORP algorithm, including flexibility, complexity, and robustness, are tested on relevant problems.
First, we demonstrate the flexibility of the convex relaxation by comparing the COPR algorithm with an added $\ell_1$-regularization to the PhaseLift method \cite{candes2013phaselift} and to the CPRL method in \cite{Ohlsson} on an under-determined sparse estimation problem.
Second, we compare the practically observed computational complexity of COPR and a
naive implementation of PhaseLift \cite{candes2013phaselift}.
Finally, we investigate the robustness of CORP relative to noise
in a Monte-Carlo simulation for 25 and 100 basis functions. We compare four algorithms: COPR, PhaseLift \cite{candes2013phaselift}, a basic alternating projections method (Section 4.3 in \cite{candes2013phaselift}) and an averaged projections method based on \cite{Luke_Toolbox}.
We note that the latter method fundamentally employs the Fourier transform at every iteration and hence is, in generally, not applicable for phase retrieval in the modal form.
\subsection{Application of COPR to compressive sensing problems}
The first problem is to estimate 16 coefficients from 8 measurements, where the optimal vector is known to be sparse.
We generate a sparse coefficient vector ${\mathbf{a}}$ with two randomly generated non-zero complex elements. We generate two images ($n_d = 2,~m = 128$) by applying two different amounts of defocus with Zernike coefficients $-\frac{\pi}{8}$ and $\frac{\pi}{8}$, respectively. From each image we use the center $2 \times 2$ pixels, resulting in a total of $n_y = 8$ measurements.
The applied algorithms are the COPR algorithm, the COPR algorithm with an additional $\ell_1$-regularization, the PhaseLift algorithm \cite{candes2013phaselift} and the Compressive sensing Phase Retrieval (CPRL) algorithm of \cite{Ohlsson}. The results are displayed in Figure~\ref{fig:sparse}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\columnwidth]{sparse_solutions2.eps}
\caption{The absolute values of 16 estimated coefficients according to 4 different algorithms.}
\label{fig:sparse}
\end{figure}
As can be seen from the figure, COPR and PhaseLift fail to retrieve the correct solution. The CPRL method and the regularized COPR algorithm compute the correct solution.
\subsection{Computational complexity}
The second problem demonstrates the trends of the required computation time when the number of estimated coefficients increases.
The underlying estimation problem consists of 7 images with different amounts of defocus applied as phase diversity, where each image is of size 128 by 128 pixels. A subset of 20 by 20 pixels of each image is used in the estimation.
We compare the COPR algorithm to the PhaseLift algorithm, which is implemented according to optimization problem (2.5) in \cite{candes2013phaselift}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\columnwidth]{computation_time2.eps}
\caption{A computation time comparison between PhaseLift and COPR for different numbers of coefficients.}
\label{fig:trend}
\end{figure}
For PhaseLift, the reported time is the time it takes the MOSEK solver \cite{mosek} to solve the optimization problem. This does not include the time taken by YALMIP \cite{Lofberg2004} to convert the problem as given to the solver-specific form.
For COPR, the number of iterations is set beforehand according to convergence to the correct solution, and the total time is recorded.
By convergence we mean that the estimated vector $\hat{{\mathbf{a}}}$ satisfies the tolerance criterion:
\begin{equation}
\min_{c \in \Complex,~\abs{c} = 1} \norm{c\hat{{\mathbf{a}}} - {{\mathbf{a}}}^*}_2^2 \leq 10^{-5},
\end{equation}
where ${\mathbf{a}}^*$ is the exact solution.
The minimization over the parameter $c$ ensures that the (unobservable) piston mode in the phase is canceled.\footnote{Let $\pmat{\hat{{\mathbf{a}}} & {{\mathbf{a}}}^*} = QR$ be the QR decomposition. Then $\angle c^* = \angle \frac{R_{12}}{R_{11}}$. }
The computational complexity of PhaseLift is, as implemented, approximately $\bigO{n^4}$. The MOSEK solver ran into numerical issues for more than 25 estimated parameters.
The COPR algorithm's computational complexity is approximately $\bigO{n}$. The better complexity is offset by a longer computation time for very small problems.
\subsection{Robustness to noise}
When estimating of an unknown phase aberration, it is more logical to evaluate the performance of the algorithm on its ability to estimate the phase, and not the coefficients of basis functions.
We assume the phase is randomly generated with a deformable mirror. Let $H \in \Real^{m^2 \times n_u}$ be the mirror's influence matrix and ${\mathbf{u}} \in \Real^{n_u}$ be the input to the mirror's actuators, such that
\begin{equation}
\phi_{DM} = H {\mathbf{u}}. \label{eq:DM}
\end{equation}
The input values $u_i$ are drawn from the uniform distribution between 0 and 1.
The mirror has $n_u = 44$ actuators and the images have sides $m = 128$. The aperture radius is $0.4$.
Five different defocus diversities are applied with Zernike coefficients uniformly spaced between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.
Gaussian noise is added to the obtained images such that
\begin{equation}
{\mathbf{y}} = \max(0, \abs{\fourier{P_d(\rho,\theta)}}^2+ \varepsilon),~ \varepsilon \in N(0,\sigma I).
\end{equation}
and $\sigma$ is the noise variance.
No denoising methods were applied.
The signal-to-noise ratio (SNR) is computed according to
\begin{equation}
10 \log_{10} \frac{\norm{{\mathbf{y}} - \abs{\fourier{P_d(\rho,\theta)}}^2 }_2^2}{\norm{\abs{\fourier{P_d(\rho,\theta)}}^2}_2^2}.
\end{equation}
The phase is estimated from ${\mathbf{y}}$ using four different algorithms.
The first is the COPR algorithm.
The second is the averaged projections (AvP) algorithm \cite{Luke_Toolbox}.
The third is the alternating projections (AlP) method (\cite{candes2013phaselift}, section 4.3), and the fourth algorithm is the PhaseLift method \cite{candes2013phaselift}.
The COPR and the AlP methods are applied for two cases corresponding to using 25 and 100 basis functions.
The PhaseLift method is applied for only the case with 25 basis functions due to numerical problems in the solver for larger problems.
The AvP method is not based on the use of basis functions but on the Fourier transform.
Due to the sensitivity to noise of this method, 100 basis functions were fit to the estimated object plane field.
The phase generated by these weighted basis functions was used to report performance.
The use of basis functions improved the phase estimate.
We make use of the Strehl ratio as a measure of optical quality.
The Strehl ratio $S$ is the ratio of the maximum intensity of the aberrated PSF and that of the unaberrated one and can be approximated with the expression of Mahajan:
\begin{equation*}
S \approx \expp{-\delta^2},
\end{equation*}
where $\delta = \norm{\phi_{DM} - \hat{\phi}}_2$ and the mean residual phase has been removed \cite{roddier1999adaptive}.
For every noise level, 100 different phases were generated with the deformable mirror model \eqref{eq:DM}.
The results are presented in Figure~\ref{fig:noise}.
\begin{figure}[ht]
\centering
\includegraphics[width=1\columnwidth]{strehl5.eps}
\caption{The Strehl ratio of the estimated phase aberration as a function of SNR. The shaded areas indicate the 10\% and 90\% quantiles.}
\label{fig:noise}
\end{figure}
The resulting Strehl-ratio's are plotted with a trend line and shaded quantile lines at 10\% and 90\%.
In the case of PhaseLift, the tuning parameter that trades off measurement fit and the rank of the `lifted' matrix is tuned once and applied to all problems.
This has the effect that the reported performance is not as high as it could be with optimal tuning for individual problems.
This points to another advantage of COPR: the absence of tuning parameters aside from the choice of basis functions.
The figure shows that COPR appears to be robust to noise.
Also, the figure on the right shows that when the number of basis functions is high, the estimated phase is very close to the exact phase in low noise settings, something that cannot be done with 25 basis functions.
However, when the noise level is high, the choice for a smaller number of basis functions shows better performance.
We attribute this to overfitting in high noise level circumstances.
\section{Concluding Remarks}\label{sec:remarks}
The convex relaxations in solving the phase retrieval problem as proposed in \eqref{EQ:NN} have the advantage over current convex relaxation methods, such as PhaseLift, that our strategy is affine in the coefficients that are to be estimated.
This allows for easy extension of the proposed method to phase retrieval problems that incorporate prior knowledge on the coefficients by regularization of the objective function.
One such successful extension is the regularization with the $\ell_1$-norm to find sparse solutions, as demonstrated in Figure~\ref{fig:sparse}.
In Section~\ref{sec:admm} an ADMM algorithm was proposed for efficient computation of the solution to \eqref{EQ:NN}.
The result is that for the COPR algorithm a better computational complexity is observed compared to PhaseLift, see Figure~\ref{fig:trend}.
COPR is also able to solve phase estimation problems with larger numbers of parameters.
The required computations are favourable both in computation time and accuracy (they have simple analytic solutions) and in worst-case scaling behaviour $\bigO{ n_y n_a}$ for every ADMM iteration, where $n_y$ is the number of pixels and $n_a$ is the number of basis functions.
We discussed convergence properties of the COPR algorithm in Section~\ref{sec:convergence} and showed that for selected problems this convergence is linear or faster.
Finally, COPR has been shown to be robust against measurement noise, and outperform the two projection-based methods whose naive forms are often sensitive to noise as expected.
We are aware that in practice the performance of projection methods can be substantially better than what we have observed in this study provided that appropriate denoising techniques are also applied.
Keeping aside from the matter of using denoising techniques, we have chosen to compare the algorithms in their very definition forms.
\section{Funding Information}
The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 339681.
\bibliographystyle{ieeetr}
|
{
"timestamp": "2018-03-08T02:08:31",
"yymm": "1803",
"arxiv_id": "1803.02652",
"language": "en",
"url": "https://arxiv.org/abs/1803.02652"
}
|
\section{Introduction}
For years, medical informatics researchers have pursued data-driven methods to automate disease diagnosis procedures for early detection of many deadly diseases. Treatment of Alzheimer's disease, which has become the sixth leading cause of death in the United States \cite{xu2016mortality}, is one of the conditions that could benefit from computer-aided diagnostic techniques. A particular challenge of Alzheimer's disease is that it is difficult to detect in early stages before mental decline begins. But medical imaging holds promise for earlier diagnosis of Alzheimer's disease \cite{mckhann2011diagnosis}. Magnetic resonance imaging (MRI), computed tomography (CT), and positron emission tomography (PET) scans contain information about the effects of Alzheimer's disease on the brain’s structure and functioning. But analyzing such scans is very time consuming for doctors and researchers because each scan contains millions of voxels.
Deep learning systems are one potential solution for processing medical images automatically to make diagnosing Alzheimer's disease more efficient. 3D convolutional neural networks (3D-CNN), taking only MRI brain scans and disease labels as input and trained end-to-end, are reported to be on par with the performance of traditional diagnostic methods in Alzheimer's disease classification \cite{khvostikov20183d,korolev2017residual}. However, the process that 3D-CNNs use to arrive at their conclusions lacks transparency and cannot straightforwardly provide reasoning and explanations as human experts do in diagnosis. It is therefore difficult for human practitioners to trust such systems in evidence-centered areas like medical research.
The goal of this study is to break into the black box of 3D-CNNs for Alzheimer's disease classification. Particularly, we develop techniques to produce visual explanations that can indicate a 3D-CNN's spatial attention on MRI brain scans when making predictions. Our approaches give diagnosticians a better understanding of the behaviors of 3D-CNNs and provide greater confidence about integrating them into automated Alzheimer's disease diagnostic systems. In summary, the contributions of this study are as follows:
\begin{itemize}
\item
We propose a hierarchical MRI image segmentation based approach for sensitivity analysis of 3D-CNNs, which can discriminate the importances of homogeneous brain regions at different levels for Alzheimer's disease classification.
\item
We extend two state-of-the-art approaches for explaining CNNs in 2D natural image classification to 3D MRI images, which can track the spatial attention of 3D-CNNs when predicting Alzheimer's disease.
\item
We compare the developed approaches qualitatively by examining the visual explanations generated. We also conduct quantitative comparisons for their ability to localize important parts of the brain in diagnosing Alzheimer's disease.
\end{itemize}
The rest of the paper is organized as follows. Section \ref{sec2} surveys related work for this study. Section \ref{sec3} describes the methods development, data, and experimental setup. Section \ref{sec4} presents the qualitative and quantitative comparisons for proposed methods. Section \ref{sec5} presents study conclusions.
\section{Related Work}\label{sec2}
Works that are closely connected to this study are divided into three parts: 3D-CNNs for Alzheimer's disease classification, brain MRI segmentation, and visualizing and understanding CNNs for natural image classification.
\paragraph{3D-CNNs for Alzheimer's Disease Classification}
There are two major methods for using 3D convolutional neural networks for Alzheimer's disease classification from brain MRI scans. One uses 3D-CNNs to automatically extract generic features from MRIs and build other classifiers on top of them \cite{suk2014hierarchical,hosseini2016alzheimer}. The other trains the 3D-CNNs in an end-to-end manner that only takes MRI scans and labels as input \cite{korolev2017residual,khvostikov20183d}. Both approaches achieve comparable performance \cite{khvostikov20183d}. The user has more control over the first method and thus can understand it better. The latter needs little input from humans so that it is easier to use.
\paragraph{Brain MRI Segmentation}
As one of the fundamental problems in neuroimaging, brain segmentation is the building block for many Alzheimer's disease diagnosis methods. Semantic segmentation methods such as FreeSurfer \cite{fischl2012freesurfer} enable brain volume calculations from MRI scans of Alzheimer's disease subjects \cite{mulder2014hippocampal}. Unsupervised hierarchical segmentation methods detect homogeneous regions and separate them from coarse to finer levels, providing more flexibility for multilevel analysis than the one-level semantic segmentation \cite{corso2008efficient,yang2016supervoxel}.
\paragraph{Visualizing and Understanding CNNs for Natural Image Classification}
To explain the superior image classification performance for 2D-CNNs, researchers incorporate the spatial structure of the convolutional layer to visualize the discriminative object from activation maps \cite{zhou2016learning,selvaraju2016grad}. Sensitivity analysis by measuring the change of output class probability due to perturbed input is another popular method because it is not subject to the architectural constraints of CNNs. LIME, or local interpretable model-agnostic explanations \cite{ribeiro2016should}, is a regression-based sensitivity analysis approach that examines perturbed superpixels to make CNN results more interpretable. The perturbed superpixels could be further learned to be more semantically meaningful \cite{fong2017interpretable,yang2018global}. All these methods create a 2D spatial heatmap as a visual explanation that indicates where the CNN has focused to make its predictions. These can be extended to 3D for Alzheimer's disease classification.
\section{Method}\label{sec3}
In this section, we describe the methods that can produce visual explanations of predictions of Alzheimer's disease from brain MRI scans by deep 3D convolutional neural networks (3D-CNNs). First, we summarize the deep learning models we deploy for the Alzheimer's disease classification task. Then, we present the brain MRI data for the study and describe how we use the data in experiments. Finally, we introduce the three approaches that we develop for explaining the 3D-CNNs, which are sensitivity analysis by 3D ultrametric contour map (SA-3DUCM), 3D class activation mapping (3D-CAM), and 3D gradient-weighted class activation mapping (3D-Grad-CAM).
\subsection{Architecture of Deep 3D Convolutional Neural Networks}
The architecture of the deep 3D convolutional neural networks (3D-CNN) for Alzheimer's disease classification in this study are based on the network architectures proposed by Korolev et al.\cite{korolev2017residual}. Particularly, two types of 3D-CNNs are built for classifying brain MRI scans from an Alzheimer's disease cohort (AD) and a normal cohort (NC). The design ideas for both types of 3D-CNNs are rooted in successful 2D natural image classification models, specifically, VGGNet, the Very Deep Convolutional Networks \cite{simonyan2014very}, and ResNet, the Deep Residual Networks \cite{he2016deep}.
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=130mm]{resnet_gap}
\caption{\textbf{Left:} The architecture of 3D-VGGNet; \textbf{Middle:} The architecture of 3D-ResNet; \textbf{Right:} The modified architecture of 3D-ResNet with global average pooling layer, 3D-ResNet-GAP, to produce 3D class activation mapping (3D-CAM). The only difference is that a global average pooling layer directly outputs to the softmax output layer (yellow boxes), replacing the original max pooling and fully connected layers.}\label{arch}
\end{center}
\end{figure*}
\paragraph{3D Very Deep Convolutional Networks (3D-VGGNet)}
VGGNet stacks many layer blocks containing narrow convolutional layers followed by max pooling layers. The 3D very deep convolutional network (3D-VGGNet) \cite{korolev2017residual} for Alzheimer's disease classification is a direct application of this idea to 3D brain MRI scans. It contains four blocks of 3D convolutional layers and 3D max pooling layers, followed by a fully connected layer, a batch normalization layer \cite{ioffe2015batch}, a dropout layer \cite{srivastava2014dropout}, another fully connected layer, and the softmax output layer to produce the probabilities of disease in the Alzheimer's disease cohort (AD) and the normal cohort (NC). The full network architecture of 3D-VGGNet is visualized in Figure \ref{arch} (left). To optimize model parameters, the ADAM optimizer \cite{kingma2014adam} is used with a learning rate of 0.000027, a batch size of 5, and 150 training epochs. The two-class cross-entropy calculated from the probabilities output by the softmax layer and the ground-truth labels are used as loss functions.
\paragraph{3D Deep Residual Networks (3D-ResNet)}
Residual network is the most important building block of the state-of-the-art of 2D natural image classification \cite{he2016deep,xie2017aggregated}. 3D deep residual networks (3D-ResNet) \cite{korolev2017residual} for Alzheimer's disease classification prove their effectiveness in the 3D domain. We deploy this important type of 3D-CNN in this study and try to explain its predictions. Specifically, a six-residual-block architecture is built. Each residual block consists of two 3D convolutional layers with 3 $\times$ 3 $\times$ 3 filters that have a batch normalization layer and a rectified-linear-unit nonlinearity layer (ReLU) \cite{nair2010rectified} between them. Skip connections (identity mapping of a residual block) add a residual block element-by-element to the following residual block, explicitly enabling the following block to learn a residual mapping rather than a full mapping. This eases the learning process for deeper architectures and results in better performance. The full architecture of 3D-ResNet is depicted in Figure \ref{arch} (middle). For optimization, Nesterov accelerated stochastic gradient descent \cite{nesterov1983method} is used. Optimization parameters are set as 0.001 for learning rate, 3 for batch size, and 150 for training epochs. The same loss function as 3D-VGGNet, the two-class cross-entropy function, is used.
\subsection{Data and Experiment Setup}\label{cv}
Brain MRI scans from the Alzheimer's Disease Neuroimaging Initiative \footnote{Data used in preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: \url{http://adni.loni.usc.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf}} (ADNI) \cite{mueller2005alzheimer} are used for this study. Specifically, we used data from the "spatially normalized, masked, and N3-corrected T1 images" category to train the 3D-VGGNet and 3D-ResNet models to classify MRI scans from the Alzheimer's disease cohort (AD) and the normal cohort (NC). Each brain MRI scan is a 3D tensor of intensity values with size 110 $\times$ 110 $\times$ 110. As one subject could have more than one MRI scan in the database, to avoid potential information leak between the training and testing dataset, we only include the earliest MRI associated with each subject for this study. As a result, 47 MRI scans from the Alzheimer's disease cohort (AD) and 56 MRI scans from the normal cohort (NC) are selected for this study. We randomly set aside eight MRI scans (5 AD, 3 NC) for later visual explanation analysis. The rest of the dataset is used for training and testing the deep 3D convolutional neural networks (3D-CNNs).
For training and testing the 3D-VGGNet and 3D-ResNet models, we conduct five-fold cross-validation for five different splits of the dataset, totaling 25 training and testing rounds. As the batch size parameters are chosen as small numbers for both models (five for 3D-VGGNet and three for 3D-ResNet), we enforce that each batch in training contains samples from both the Alzheimer's disease cohort (AD) and normal cohort (NC) to stabilize the training process by avoiding biased loss.
\subsection{Explaining the 3D-CNNs}
In this section, we describe the methods that we develop for explaining the predictions of the 3D-CNNs in detail. We first revisit a baseline method using sensitivity analysis that can shed light on 3D-CNNs' attention \cite{korolev2017residual}. Then we show how we used an unsupervised 3D hierarchical volumetric image segmentation approach, the 3D ultrametric contour map (3D-UCM) \cite{yang2016supervoxel}, to improve the baseline, which we call sensitivity analysis by 3D ultrametric contour map (SA-3DUCM). Next, we describe how the successful 2D visual explanation method, class activation mapping (CAM) \cite{zhou2016learning} and its generalization, gradient-weighted class activation mapping (Grad-CAM) \cite{selvaraju2016grad}, are extended to 3D to explain predictions from 3D MRI scans. We call the two extended approaches 3D-CAM and 3D-Grad-CAM, respectively. As we mentioned, there are two major ways to explain the predictions of deep convolutional neural networks. One way applies perturbations to data and conducts sensitivity analysis. The baseline method and proposed SA-3DUCM approach belong to this category. The other way utilizes the architectural properties of CNNs to heuristically track the attention of neural networks. 3D-CAM and 3D-Grad-CAM fall into this category.
\paragraph{Baseline Approach}
A baseline approach is proposed alongside the work of 3D-VGGNet and 3D-ResNet \cite{korolev2017residual} to shed light on 3D-CNN's attention when classifying MRI scans. To be specific, for every voxel in the MRI scan, its 7 $\times$ 7 $\times$ 7 neighborhood is occluded from the image, and then the 3D-CNN re-evaluates the probability of Alzheimer's disease from the partially occuluded image. The change of probability is used as the importance of that voxel. More formally, for the brain MRI volume $V$ and each voxel of $V$ at $(x,y,z)$, we occlude the neighborhood $V_{x-3:x+3,y-3:y+3,z-3:z+3}$, resulting in a perturbed MRI volume occluded around $(x,y,z)$, denoted by $OV_{(x,y,z)}$. We want to measure the change of probability of Alzheimer's disease of $OV_{(x,y,z)}$, predicted by the 3D-CNN, compared to the original volume $V$. This change is assigned to the voxel at $(x,y,z)$. For a 3D heatmap, $C$, of the same size as $V$, to store these changes of probabilities as the importance score for all the voxels, the magnitude at $(x, y, z)$ of $C$ is calculated by
\begin{equation}
C_{x,y,z} = |P(OV_{(x,y,z)}) - P(V)|
\end{equation}
where $P(\cdot)$ is one forward pass of the 3D-CNN to evaluate the probability of Alzheimer's disease from the MRI volumes, and $|\cdot|$ is the absolute value function.
This approach is a direct application of the one-at-a-time sensitivity analysis at the single voxel level to test how the uncertainty of the output probability of the 3D-CNN could be assigned to different voxels of the MRI scan. This is straightforward to implement; however, this approach suffers from three important problems. First, the 7 $\times$ 7 $\times$ 7 cubical neighborhoods are not necessarily semantically meaningful and could be across different brain segments, e.g., half in cerebral cortex and half in white matter. Thus, occlusion of such an area results in an unaccountable change of output probability. Second, this approach could only capture the impact of the 7 $\times$ 7 $\times$ 7 local areas. The importances of larger or smaller areas are not tested. Third, as we evaluate a new output probability for each voxel, this approach is extremely computationally intensive. An MRI scan of size 110 $\times$ 110 $\times$ 110 has over 1 million voxels, requiring the same number of forward passes through the 3D-CNN, which could take hours even in GPU-assisted systems.
\paragraph{Sensitivity Analysis by 3D Ultrametric Contour Map (SA-3DUCM)}
We notice that the shortcomings of the baseline approach could be overcome by using a good segmentation of the brain volume instead of the 7 $\times$ 7 $\times$ 7 local neighborhood around each voxel. Particularly, we occlude each segment in the segmentation, instead of the cubical neighborhoods, before re-evaluating the probabilities. To resolve each of the three problems of the baseline approach, the segmentation method should be semantically meaningful, hierarchical, and compact. Most specifically, to be semantically meaningful, the segmentation should separate different homogeneous parts of the brain volume well, e.g., separating cerebral cortex and white matter, so that changes of probability could be ascribed to specific segments. To be hierarchical, the segmentation method should provide a hierarchy of segmentations that capture both coarse level parts, such as the whole white matter, as well as finer level parts. In this way, we can test the importances for both small and large areas. To be compact, the segmentation method should avoid over-segmentation and generate a manageable number of segments for analysis. Thus, we can reduce the number of forward passes needed through the 3D-CNN from the number of voxels to the number of segments, which is usually three to four orders of magnitude less.
3D Ultrametric Contour Map (3DUCM) \cite{yang2016supervoxel,huang2018supervoxel} is an effective approach for unsupervised hierarchical 3D volumetric image segmentation, which is the 3D extension of the 2D state-of-the-art, Ultrametric Contour Map for natural image segmentation \cite{arbelaez2011contour}. It provides compact hierarchical segmentation of high quality. For the brain MRI volume, $V$, it could generate a hierarchy of segmentation, $H=\{H_{1},H_{2},...,H_{N}\}$, where each level $H_{n}=S_{1}^{n}\cup S_{2}^{n}\cup ...\cup S_{K_{n}}^{n}$ is a full segmentation of the volume $V$. We occlude each segment $S_{k}^{n}$ , $k=1,2,...,K_{n}$, $n=1,2,...,N$, in $V$, denoting each resulting volume by $OV_{k}^{n}$ , and re-evaluate the probability of Alzheimer's disease through one forward pass of the 3D-CNN. The change of probabilities compared to what is obtained from the original volume, $|P(OV_{k}^{n})-P(V)|$, is assigned to every voxel in $S_{k}^{n}$. Since each voxel belongs to one segment at each level of the hierarchy, each voxel gets $N$ quantities from the calculation, where $N$ is the number of levels in the segmentation hierarchy. We compute the average quantity from the $N$ quantities as the importance score for each voxel and store it in a heatmap $C$. So for a voxel of $V$ at $(x, y, z)$, assuming that it belongs to $S_{k_{n}}^{n}$, for each level of hierarchy $H_{n}$, we calculate the importance score for it as
\begin{equation}
C_{x,y,z} = \frac{1}{N}\sum_{n=1}^{N}|P(OV_{k_{n}}^{n})-P(V)|
\end{equation}
Since the 3DUCM hierarchical segmentation usually provides homogeneous segments of the brain MRI, we expect the importance heatmap $C$ to distinguish important brain parts for Alzheimer's disease classification. In terms of computational burden, each level of the hierarchy contains at most hundreds of segments, and the hierarchy itself is no more than 20 levels. Thus, the number of forward passes needed to re-evaluate the probabilities is greatly reduced.
\paragraph{3D Class Activation Mapping (3D-CAM)}
One major problem with one-at-a-time sensitivity analysis based methods (baseline and SA-3DUCM) is that the correlations and interactions between segments of MRI volume are ignored. Although using the hierarchical segmentation method can cover most semantic segments from finer to coarser level, we cannot guarantee all combinations are tested. Therefore, we turn to methods based on the architectural properties of the 3D-CNN that directly visualize the activations of convolutional layers when predictions are made. Class activation mapping \cite{zhou2016learning} designs a global average pooling layer on top of convolutional layers in natural images classification, which enables remarkable localization performance on important objects in the images in spite of the fact that the CNN is trained on image-level labels. This fits our problem well. Our Alzheimer's disease labels (Alzheimer's disease cohort (AD) and normal cohort (NC)) are used at MRI scan level during the training of the 3D-CNNs. Our goal is to obtain visual explanations that can highlight brain parts important for Alzheimer's disease classification. Thus, extending class activation mapping to 3D provides a way to do this.
The idea of class activation mapping is that the last convolution layer of the CNN contains the spatial information indicating discriminative regions to make classifications. To visualize these discriminative parts, class activation mapping creates a spatial heatmap out of the activations from the last convolutional layer. Specifically, class activation mapping adopts a global average pooling layer between the final convolutional layer and output layer, which enables projection of class weights of the output layer onto the activation maps in the convolutional layer. The 3D extension of class activation mapping based on 3D-ResNet is shown in Figure \ref{arch} (right). Instead of using a max pooling layer and a fully connected layer before output, the modified 3D-ResNet only uses a global average pooling layer (3D-ResNet-GAP). To be specific, for a given MRI volume $V$ and a 3D-CNN, let $f_{u}(x,y,z)$ be the activation of unit $u$ in the last convolutional layer at location $(x, y, z)$. The global average pooling for unit $u$ is $F_{u}=\frac{1}{Z}\sum_{x,y,z}f_{u}(x,y,z)$, where $Z$ is the number of voxels in the corresponding convolutional layer. As the global average pooling layer is directly connected to the softmax output layer, by the definition of the softmax function, the probability of Alzheimer's disease, $P(V)$, given by
\begin{equation}
P(V) = \frac{\exp (\sum_{u}w_{u}^{AD}F_{u})}{\exp(\sum_{u}w_{u}^{AD}F_{u})+\exp(\sum_{u}w_{u}^{NC}F_{u})}
\end{equation}
where $w_{u}^{AD}$ and $w_{u}^{NC}$ are the class weights in the output layer for the Alzheimer's disease cohort (AD) and the normal cohort (NC), respectively. We ignore the bias term here because its impact is minimal on classification performance. Essentially, $\sum_{u}w_{u}^{AD}F_{u}$ and $\sum_{u}w_{u}^{NC}F_{u}$ are the class scores for AD and NC cohorts, respectively. By extending $F_{u}$ in the class score, we have
\begin{equation}
\textrm{Score}(AD) = \sum_{u}w_{u}^{AD}F_{u} = \sum_{u}w_{u}^{AD}\frac{1}{Z}\sum_{x,y,z}f_{u}(x,y,z) = \frac{1}{Z}\sum_{x,y,z}\sum_{u}w_{u}^{AD}f_{u}(x,y,z)
\end{equation}
The $\sum_{u}w_{u}^{AD}f_{u}(x,y,z)$ part of the quantity is defined for every spatial location $(x, y, z)$ and their sum is proportional to the class score for Alzheimer's disease. As areas significantly negatively contributing to the class score are also important, we adopt the absolute value and define the class activation mapping for the AD cohort as
\begin{equation}
\textrm{3D-CAM}_{x,y,z}(AD) = |\sum_{u}w_{u}^{AD}f_{u}(x,y,z)|
\end{equation}
which is essentially a heatmap of weighted sums of activations in every location $(x, y, z)$ and can be easily calculated by one forward pass when the volume $V$ is provided.
Though 3D-CAM is easy to obtain, and we expect it to highlight the important spatial areas for classification, there are two potential problems with this approach. First, as we modify the 3D-CNN architecture with the global average pooling layer, we need to re-train the model, possibly affecting the classification performance. Second, the resolution of the class activation mapping is of the same size as the last convolutional layer. We need to upsample it to the original MRI scan size to identify the discriminative regions, which means we would lose some details in the resulting heatmap. One solution could be to remove more layers and build the global average pooling layers on convolutional layers with higher resolution. But this could further decrease the classification performance.
\paragraph{3D Gradient-Weighted Class Activation Mapping (3D-Grad-CAM)}
To overcome class activation mapping's shortcoming of decreased classification performance, its generalization, gradient-weighted class activation mapping, is proposed in natural image classification \cite{selvaraju2016grad}. This approach does not need to modify the 3D-CNN's architecture and thus will do no harm to classification performance. Since no re-training is required, it is more efficient to deploy in deep learning systems. The core idea is still to identify the important activations from feature maps in convolutional layers. Using the same notation as the previous part, we first calculated the gradient of the $\textrm{Score}(AD)$ with respect to the activation of unit $u$ at location $(x, y, z)$, $f_{u}(x,y,z)$, in the last convolutional layer. Then, we use the global average pooling of the gradients, denoted by $a_{u}^{AD}$, as the importance weights for unit $u$ for the Alzheimer's disease cohort (AD). That is,
\begin{equation}
a_{u}^{AD}=\frac{1}{Z}\sum_{x,y,z}\frac{\partial \textrm{Score}(AD)}{\partial f_{u}(x,y,z)}
\end{equation}
where $Z$ is the number of voxels in the corresponding convolutional layer. Then, we combined the unit weights with the activations, $f_{u}(x,y,z)$, to get the heatmap of 3D gradient-weighted class activation mapping.
\begin{equation}
\textrm{3D-Grad-CAM}_{x,y,z}(AD) = |\sum_{u}a_{u}^{AD}f_{u}(x,y,z)|
\end{equation}
3D-Grad-CAM could be applied to a wider range of 3D-CNNs than 3D-CAM as long as the 3D-CNN has a fully convolutional layer. Also, it has been proven in 2D applications that CAM is a special case of Grad-CAM with the global average pooling layer \cite{selvaraju2016grad}. It does not require re-training so it quickly generates the 3D-Grad-CAM heatmap with just one forward pass. However, 3D-Grad-CAM still suffers from the low resolution problem because the 3D-Grad-CAM is a coarse heatmap of the same size as the last convolutional layer. We could have calculated it with gradients and activations from lower convolutional layers, but there is no guarantee that the spatial activations wouldn't change in the upper layers.
In summary, in this section, we introduce four approaches to obtain visual explanation heatmaps for predictions from 3D-CNNs. The baseline approach and sensitivity analysis by 3D ultrametric contour map (SA-3DUCM) are completely model-agnostic and can handle any type of 3D-CNNs, but they might have problems with correlations and interactions between different segments of the brain volume. 3D class activation mapping (3D-CAM) and 3D gradient-weighted class activation mapping (3D-Grad-CAM) are weighted visualizations of the activation maps in the convolutional layer, which avoids dealing with the correlations and interactions problem. However, they are limited by the low resolution of the convolutional layers. Upsampled heatmaps might not be able to provide enough detail to accurately identify important regions. For computational efficiency, the baseline approach is the slowest because it does a forward pass for every voxel. 3D-CAM only needs one forward pass to generate the heatmap, but it requires very time-consuming re-training. SA-3DUCM needs a few hundred forwarded passes. 3D-Grad-CAM is the best because it does not require re-training and only needs one forward pass when generating the heatmap. In the next section, we will compare the models' performances in identifying of discriminative brain parts for Alzheimer's disease classification from MRI scans.
\section{Results}\label{sec4}
In this section, we will present the classification performance of 3D-CNNs, visual comparisons of the heatmaps generated by the proposed visual explanation approaches, and a quantitative benchmark for the localization ability of the heatmaps in identifying important brain parts for Alzheimer's disease classification.
\subsection{Alzheimer's Disease Classification Performance}
We compare the classification performance of four different 3D-CNNs. These include 3D-VGGNet and 3D-ResNet as described. By implementing the 3D-CAM, we have a modified 3D-ResNet with global average pooling layer (GAP) as shown in Figure \ref{arch} (right), denoted as 3D-ResNet-GAP. The counterpart for 3D-VGGNet is not included because the classification performance drops too much, compared to 3D-VGGNet. Additionally, to obtain a higher resolution 3D-CAM, we remove the layers from {\tt conv4} to {\tt voxres9\_out}, resulting in a shallow version of 3D-ResNet-GAP, which we call 3D-ResNet-Shallow-GAP. All four 3D-CNNs are trained for classifying the Alzheimer's cohort (AD) in comparison to the normal cohort (NC). Classification performance is measured by the area under the ROC curve (AUC) and classification accuracy (ACC). Cross-validation as described in Section \ref{cv} is conducted. Average AUC and ACC and their standard deviations are reported. The results are presented in Table \ref{cls}. 3D-VGGNet and 3D-ResNet achieve good classification performances. However, there is a substantial drop in performance for 3DResNet-GAP and 3D-ResNet-Shallow-GAP, which means the global average pooling layer have a negative effect on classification performance.
\begin{table*}[t]
\begin{center}
\begin{tabular}{p{4cm}|p{3.5cm}p{3.5cm}}
\hline
\textbf{Method} & \textbf{AUC} & \textbf{ACC} \\
\hline
3D-VGGNet & 0.863$\pm$0.056 & 0.766$\pm$0.095 \\
3D-ResNet & 0.854$\pm$0.079 & 0.794$\pm$0.070 \\
3D-ResNet-GAP & 0.643$\pm$0.110 & 0.614$\pm$0.100 \\
3D-ResNet-Shallow-GAP & 0.751$\pm$0.083 & 0.585$\pm$0.122 \\
\hline
\end{tabular}
\end{center}
\caption{Classification performance of 3D-CNNs}
\label{cls}
\end{table*}
\subsection{Qualitative Comparison for Visual Explanations}
To visually check the quality of heatmaps generated by the introduced visual explanation methods, we take one MRI scan from the set-aside data for visual explanation analysis and present the heatmap from the horizontal, sagittal, and coronal sections. For comparison, we present the input brain MRI volume (Figure \ref{gt}) with highlighted areas of cerebral cortex, lateral ventricle, and hippocampus. These parts are believed to be important for Alzheimer's disease diagnosis by physicians \cite{juottonen1999comparative,mu1999quantitative}. The ground-truth cerebral cortex, lateral ventricle, and hippocampus regions are segmented by the FreeSurfer software \cite{fischl2012freesurfer}.
\paragraph{Baseline}The resulting heatmaps are labeled as VGG-Baseline and Res-Baseline and are presented in Figure \ref{vgg_baseline} and Figure \ref{resnet_baseline}, respectively. We can see from the figures that in both situations, the baseline method does not find the important areas. The heatmaps are irregularly shaped because heterogeneous regions are used for sensitivity analysis. Overall, the baseline method fails to identify discriminative regions.
\begin{figure*}[!ht]
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[Brain MRI with highlighted cerebral cortex, lateral ventricle, and hippocampus.]{\includegraphics[width=75mm]{gt_cortex.jpg}\label{gt}
}
\par\end{center}%
\end{minipage}
\\
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[VGG-Baseline]{\includegraphics[width=75mm]{vgg_baseline.jpg}\label{vgg_baseline}
}
\par\end{center}%
\end{minipage}
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[Res-Baseline]{\includegraphics[width=75mm]{resnet_baseline.jpg}\label{resnet_baseline}
}
\par\end{center}%
\end{minipage}
\\
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[VGG-SA-3DUCM]{\includegraphics[width=75mm]{vgg_3ducm.jpg}\label{vgg_3ducm}
}
\par\end{center}%
\end{minipage}
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[Res-SA-3DUCM]{\includegraphics[width=75mm]{resnet_3ducm.jpg}\label{resnet_3ducm}
}
\par\end{center}%
\end{minipage}
\\
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[Res-3D-CAM]{\includegraphics[width=75mm]{resnet_cam.jpg}\label{resnet_cam}
}
\par\end{center}%
\end{minipage}
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[Res-3D-CAM-Shallow]{\includegraphics[width=75mm]{resnet_shallow_cam.jpg}\label{resnet_shallow_cam}
}
\par\end{center}%
\end{minipage}
\\
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[VGG-3D-Grad-CAM]{\includegraphics[width=75mm]{vgg_grad_cam.jpg}\label{vgg_grad_cam}
}
\par\end{center}%
\end{minipage}
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[Res-3D-Grad-CAM]{\includegraphics[width=75mm]{resnet_grad_cam.jpg}\label{resnet_grad_cam}
}
\par\end{center}%
\end{minipage}
\\
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[VGG-3D-Grad-CAM-Shallow]{\includegraphics[width=75mm]{vgg_grad_cam_shallow.jpg}\label{vgg_grad_cam_shallow}
}
\par\end{center}%
\end{minipage}
\begin{minipage}[htp]{0.49\columnwidth}%
\begin{center}
\subfloat[Res-3D-Grad-CAM-Shallow]{\includegraphics[width=75mm]{resnet_grad_cam_shallow.jpg}\label{resnet_grad_cam_shallow}
}
\par\end{center}%
\end{minipage}
\caption{Horizontal, sagittal, and coronal view of the brain MRI and the visual explanation heatmaps.}\label{comp}
\end{figure*}
\paragraph{SA-3DUCM} After incorporating hierarchical segmentations into sensitivity analysis, we find that the results greatly improves, compared to baseline. Figure \ref{vgg_3ducm} presents the heatmap made by applying SA-3DUCM to 3D-VGGNet (VGG-SA-3DUCM), and the heatmap in Figure \ref{resnet_3ducm} is made by applying SA-3DUCM to 3D-ResNet (Res-SA-3DUCM). In both situations, the approach differentiates the importances of different homogeneous regions. There are clear boundaries separating the regions. The lateral ventricle area stands out as the most discriminative part. However, the cerebral cortex areas are not well identified. This is because cerebral cortex is widely and loosely distributed in the brain so the cerebral cortex is usually not segmented as one area in hierarchical segmentations. SA-3DUCM tested the importance of different segments one by one. Thus, it is not able to capture the correlations between all segments that belong to the cerebral cortex.
\paragraph{3D-CAM} We only apply 3D class activation mapping (3D-CAM) to 3D-ResNet because 3D-VGGNet loses too much classification performance after using the global average pooling layer. The class activation mapping heatmap of 3D-ResNet-GAP is labeled as Res-3D-CAM and is presented in Figure \ref{resnet_cam}. The heatmap is blurry because it is upsampled from a 14 $\times$ 14 $\times$ 14 coarse heatmap. To get a higher resolution 3D class activation mapping heatmap, Figure \ref{resnet_shallow_cam} (Res-3D-CAM-Shallow) is obtained from 3D-ResNet-Shallow-GAP with more convolutional layers removed. It is upsampled from a 55 $\times$ 55 $\times$ 55 heatmap and thus provides more detail. It identifies the lateral ventricle and most parts of the cortex as important areas, which matches the human experts' approach.
\paragraph{3D-Grad-CAM}The 3D gradient-weighted class activation mapping (3D-Grad-CAM) also has low resolution problems, especially when it is applied to 3D-VGGNet. Because the last convolutional layer of 3D-VGGNet is only of size 3 $\times$ 3 $\times$ 3, the resulting heatmap VGG-3D-Grad-CAM barely provides any information (Figure \ref{vgg_grad_cam}). When we apply the same approach to a lower convolutional layer, {\tt conv2b}, in 3D-VGGNet, the resulting heatmap, VGG-3D-Grad-CAM-Shallow (Figure \ref{vgg_grad_cam_shallow}), is able to highlight part of the lateral ventricle. 3D-ResNet has the same situation. Res-3D-Grad-CAM (Figure \ref{resnet_grad_cam}) and Res-3D-Grad-CAM-Shallow (Figure \ref{resnet_grad_cam_shallow}) are generated by the 3D-Grad-CAM approach applied to {\tt voxres9\_out} (last convolutional layer) and {\tt bn4} (an intermediate convolutional layer) of 3D-ResNet. They are of size 14 $\times$ 14 $\times$ 14 and 55 $\times$ 55 $\times$ 55, respectively. Though both of them identify most of the lateral ventricle and the cerebral cortex as discriminative, Res-3D-Grad-CAM-Shallow is of higher resolution and more accurate. However, as we stated, upper convolutional layers could change the activation maps from the lower convolutional layers. Thus sometimes, we may not trust the heatmap from lower layers as a good representation of spatial attention of the 3D-CNN.
To summarize the qualitative comparisons, SA-3UCM has the same resolution as the original MRI volume and differentiates homogeneous regions well. However, it fails to identify the correlations from the fragmented cerebral cortex segments because of the one-at-a-time process in sensitivity analysis. 3D-Grad-CAM and 3D-CAM both produce more blurry heatmaps than SA-3DUCM because of upsampling. But they are able to highlight the cerebral cortex that is loosely distributed in the brain.
\subsection{Quantitative Comparison for Localization}
Visual comparisons of the heatmap give us a general idea how well different visual explanation methods work. But we wonder how well these heatmaps could localize important regions such as cerebral cortex, lateral ventricle, and hippocampus. To quantitatively compare localization ability, we plot the precision-recall curve for the heatmaps that we have visualized in the previous section to identify cerebral cortex, lateral ventricle, and hippocampus regions from the 8 MRI scans that are set aside for visual explanation analysis. VGG-Baseline, Res-Baseline, and VGG-3D-Grad-CAM are not included because they do not generate usable heatmaps in the visual comparisons. The results are presented in Figure \ref{qf}.
\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=140mm]{pr_cortex_less.jpg}
\caption{Precision-recall curve to localize cerebral cortex, lateral ventricle, and hippocampus regions using heatmaps.}\label{qf}
\end{center}
\end{figure*}
From the results, we can see VGG-SA-3DUCM, Res-SA-3DUCM, and Res-3D-Grad-CAM-Shallow have high precision on the low recall end. This matches our visual comparisons as SA-3DUCM method puts the homogeneous lateral ventricle regions on top, and Res-3D-Grad-CAM-Shallow identifies cerebral cortex and lateral ventricle parts with high accuracy. However, the precision drops for all methods on the high recall end, implying no method is close to perfectly identifying all important regions. The reasons would be different. SA-3DUCM could not discriminate the cerebral cortex because of fragmented segments. 3D-CAM and 3D-Grad-CAM are limited by low resolution of the heatmaps.
Overall, both qualitative and quantitative comparisons indicate that all visual explanation methods have some limitations. The correct method may be chosen based on the specific goals. When the goal is to get the importance for a homogeneous region, SA-3DUCM is more suitable. If tracking the attention of the 3D-CNN is the goal, 3D-Grad-CAM is the preferred choice. Generally 3D-Grad-CAM is better than 3D-CAM because it does not modify the 3D-CNN architecture, requires less computation, and better localizes important regions.
\section{Conclusion and Discussion}\label{sec5}
In this study, we develop three approaches for producing visual explanations from 3D-CNNs for Alzheimer's disease classification. All approaches can highlight important brain parts for diagnosis. However, they have limitations in different aspects. The one-at-a-time sensitivity analysis procedure of SA-3DUCM is not able to handle correlated or interacting images segments, causing underestimation of attention in the loosely distributed area such as cerebral cortex in our case. 3D-CAM and 3D-Grad-CAM build heatmaps from convolutional layer activations that have lower resolution than the original MRI scan, resulting in loss of details and decreased localization accuracy. Therefore, we suggest users choose the right approach based on their use cases for MRI analysis.
Though all approaches are developed for Alzheimer's disease classification, they are generic enough for other type of 3D image analysis. SA-3DUCM is completely model agnostic and can adapt to any classifiers taking 3D volumetric images as input. 3D-CAM and 3D-Grad-CAM can work on any deep learning model that has a 3D convolutional layer. They could be applied to other types of 3D medical images or even video analysis.
One common limitation of these approaches is that the visual explanation is still one step away from fully understanding the 3D-CNN. Human experts measure cerebral cortex thickness as a biomarker for diagnosis \cite{fischl2000measuring}. In the generated visual explanations, there is no such explicit summarized representation on top of the visual attention from the cerebral cortex. This leads to our future work of explicit biomarker representation learning from medical imaging to fully interpret the 3D-CNNs.
\section*{\uppercase{Acknowledgments}}
This work is partially supported by NSF 1743050 to A.R. and S.R..
\makeatletter
\renewcommand{\@biblabel}[1]{\hfill #1.}
\makeatother
\bibliographystyle{unsrt}
|
{
"timestamp": "2018-07-09T02:03:14",
"yymm": "1803",
"arxiv_id": "1803.02544",
"language": "en",
"url": "https://arxiv.org/abs/1803.02544"
}
|
\section{Introduction}
\emph{Evolutionary game theory} has been established as a modeling tool for interactions between populations of strategic entities. In specific, evolutionary games describe the population dynamics resulting from pairwise interactions. It has found numerous applications in various areas of multi-agent systems such as in wireless networks~\cite{tembine1,tembine2}, swarm robotics~\cite{sun2015}, and dynamic routing protocols~\cite{tembine3}.
\emph{Evolutionary graphs} arise as an application of evolutionary game theory in modeling dynamic graph topologies. In such context, a population is organized as a network (graph) with the nodes (vertices) representing atoms (agents) and links (edges) representing interactions among them. This is captured by a weighted
graph with time-varying edge set determined by an evolutionary game. We will present a specific randomized decentralized mechanism for determining the graph topology based on the individual fitness functions of the agents. In our setting, each node maintains a local variable and computes its fitness function using its own value along with the values from its neighbors (both active and inactive, cf. Sec.~\ref{sec:problem}). Subsequently, it dynamically readjusts its neighbor set by randomly adding or deleting links with probabilities dictated by the resulting change of its fitness function.
\emph{Consensus} is a canonical example of in-network coordination in multi-agent systems. Each agent maintains a local value and the goal is for the entire network to reach agreement to a common value in a \emph{distributed} fashion, i.e., via local exchanges of messages between neighboring (adjacent) agents. An archetypal problem is \emph{average consensus}, where the goal is for each agent to asymptotically compute the average of all nodal values; cf.~\cite{linear_cons,xiao2004fast,boyd2006randomized,olfati}, for a largely non-exhaustive list of references.
This theme has proven a prevalent design tool for distributed multi-agent optimization~\cite{nedic2009distributed,nedich2016achieving}, signal processing~\cite{gossip}, numerical analysis~\cite{REK}, and estimation~\cite{RK,RK2}.
In this paper, we seek to bridge the gap between evolutionary game theory and distributed optimization by studying average consensus over evolutionary graphs. In our setting, each agent has a local variable that represents its `strategy': the strategies evolve following a consensus protocol over a time-varying graph capturing inter-agent cooperations. At each time instant, agents dynamically select the agents with which they cooperate (from a candidate set) based on their utility (i.e., fitness) that depends on their own strategy and the strategies of neighboring agents they intend to cooperate with. Specifically, agents create or drop links (i.e., cooperations) when they deem it beneficial for them, and they do so via a randomized decentralized mechanism, in a selfish manner. Examples enlist social networks~\cite{sn1} and coordination of robot swarms~\cite{sun2015}.
We consider the problem of average consensus over networks
of time-varying topology
captured by a Markov Decision Process (MDP).
Unlike prior work on the subject~\cite{olfati,nedich2016achieving}, the topology \emph{depends on the agents' values}, which renders previous analysis techniques inapplicable in our case. We proceed to establish average consensus a.s. (almost surely) and in m.s. (mean square) sense, using stochastic Lyapunov techniques. Additionally, we prove that the convergence is exponential in expectation, and provide a lower bound on the expected convergence rate. Finally, our method was empirically assessed via numerical simulations.
The remainder of the paper is organized as follows: Sec.~\ref{sec:preliminaries} exposes preliminaries on graph theory, consensus and evolutionary graph theory. In Sec.~\ref{sec:problem}, we present the problem formulation. The convergence analysis is presented in Sec.~\ref{sec:proof}. Sec.~\ref{sec:sim} illustrates simulation results, while Sec.~\ref{sec:conclusions} concludes the paper and discusses future research directions.
\section{Preliminaries}\label{sec:preliminaries}
In this section, we recap essential background on graph theory, consensus protocols and evolutionary games. In the remainder of the paper, we use boldface for vectors and capital letters for matrices. Vectors are meant as column vectors, and we use $\mathbf{0},\mathbf{1}$ to denote the vectors with all entries equal to zero, one, respectively, and $I$ to denote the identity matrix (with the dimension made clear from the context in all cases).
Last, we use the terms `agent', `vertex' and `node' interchangeably, and the same holds true for `edge' and `link,' in what follows.
\subsection{Graph theory}
Consider the case of $n$ interacting agents which aim to achieve consensus over a quantity of interest, for instance compute the average of their values.
Such problem is an instance of \emph{computing on graphs}, where each agent is modeled as a vertex of the graph and edges are drawn only between interacting nodes: we assume that two nodes can interact with each other (for instance, exchange private information) if and only if they are connected, i.e., there is an edge between them in the graph.
Formally, let a graph be denoted by $\mathcal{G}=(\mathcal{V}, \mathcal{E})$, where $\mathcal{V}$ is the non-empty set of vertices (nodes) and $\mathcal{E}$ is the set of edges.
In this paper, we restrict attention to \emph{undirected} graphs, that is to say the edge set $\mathcal{E}$ consists of unordered pairs: $(i,j)\in \mathcal{E}$ implies that agents $i,j$ can interact in a symmetric fashion, i.e., cooperate\footnote{The extension of our methods to directed graphs that model asymmetric interactions (e.g., asymmetric reward functions) will be the focal point of future work.}. We further assume that the graph does not contain self-loops (i.e., $(i,i)\notin \mathcal{E}$ for all $i\in \mathcal{V}$); this is without any loss in generality, since edges capture inter-agent interactions in our framework.
We set $n:=|\mathcal{V}|, m=|\mathcal{E}|$, to denote the number of nodes, edges respectively. We say that the graph $\mathcal{G}$ is \emph{connected} if there is a path between any two nodes $i,j\in\mathcal{V}$; otherwise, we say that the graph is disconnected.
The adjacency matrix $\mathsf{A}\in\mathbb{R}^{n\times n}$ of the graph captures connections between the nodes:
for any two nodes $i,j \in \mathcal{V}$, $a_{ij}$ is defined as:
$$a_{ij}=\left\lbrace \begin{array}{c l}
1, & \textrm{if } (i,j)\in\mathcal{E},\\%\textrm{if there is an edge between } i \textrm{ and }j\\
0, & \textrm{otherwise.
\end{array} \right.$$
The definition can be extended to \emph{weighted} graphs, in which case $a_{ij}$ can take an arbitrary
value if $(i,j)\in\mathcal{E}$. For an undirected graph, $a_{ij}=a_{ji}$, for all $i,j\in \mathcal{V}$, i.e., $\mathsf{A}$ is symmetric. Besides, $\mathsf{A}$ has a zero-diagonal ($a_{ii}=0$ for all $i$), since $\mathcal{G}$ is assumed to have no self-loops. The \emph{neighborhood} of a node $i$ contains all nodes that the node has a connection with, and is denoted by $\mathcal{N}_i := \{j:a_{ij}\ne 0\}$.
The \emph{degree} of node $i$ is defined as the
number of its neighbors, i.e., $d_i:= \abs{\mathcal{N}_i} = \sum_{j\in \mathcal{V}}a_{ij}$. Let $\mathsf{D}$ be the diagonal degree-matrix (i.e., its $(i,i)-$th entry equals $d_i$, and off-diagonal entries are zero). We define
$\L := \mathsf{D}-\mathsf{A},$
the \emph{Laplacian} of the graph. Clearly, $\L\in\mathbb{R}^{n\times n}$ is symmetric.
Additionally, it can be shown that $\L$ is positive semidefinite~\cite{graph_theory}.
It is well-known~\cite{graph_theory} that $\rank \L = n-k$, where $k$ is the number of connected components of the graph. In particular, a graph is connected if and only if $\rank \L = n-1.$
By its very definition, the Laplacian has a zero eigenvalue with corresponding eigenvector the all-one vector $\mathbf{1}$. In fact, the multiplicity of the zero eigenvalue equals the number of connected components of the graph.
The second smallest eigenvalue of the Laplacian is called the \emph{Fiedler value} or \emph{algebraic connectivity} of the graph, and is denoted as $\lambda_2(\mathcal{G}):=\lambda_2(\L)$. It is positive if and only if the graph is connected.
\subsection{Consensus}
Each agent $i\in \mathcal{V}$ maintains a local scalar value $x_i$. We call the \emph{state} of the network the vector $\mathbf{x}$ obtained by stacking all nodal values $\{x_i\}_{i\in\mathcal{V}}$. We use $x_{i,t}$ to denote the value of node $i$ at time $t$, and, correspondingly, $\mathbf{x}_t$ for the network state at time $t$.
A widely studied method which describes the evolution of $x_{i}$ is \emph{linear consensus}~\cite{linear_cons,olfati}.
The dynamics for $x_{i}$ can be written (up to a multiplicative constant) as
$$\dot{x}_{i,t} = \sum_{j\in \mathcal{N}_{i}} a_{i,j}(x_{j,t}-x_{i,t}),$$
which can further be written compactly in matrix form:
$$\dot{\mathbf{x}}_t = -\L\mathbf{x}_t.$$
For an undirected and connected graph, the network reaches \emph{average consensus}, i.e.,
$$\mathbf{x}_t \underset{t\to\infty}{\longrightarrow} Ave(\mathbf{x}_0)\mathbf{1},$$
where $Ave(\mathbf{x}_0):=\frac{1}{n}\mathbf{1}^\top\mathbf{x}_0$ is the average of the values at the initial time 0~\cite{olfati}. Besides, the convergence rate is exponential with rate lower-bounded by the Fiedler value~\cite{olfati}.
In this paper, we will study consensus over \emph{time-varying} graphs as abstracted by a time-varying Laplacian $\L_t$, dictated by agents' randomized decisions; that is to say, $\L_t = \L(\mathbf{x}_t,\mathcal{E}_{t^-})$ is a random matrix that depends on both the state at time $t$ and the topology `right before' time $t$; consequently, the analysis in existing literature~\cite{linear_cons,xiao2004fast,boyd2006randomized,olfati,nedich2016achieving} does not directly carry through.
\subsection{Evolutionary graphs}
An \emph{evolutionary graph} is a graph whose topology is specified by an evolutionary game~\cite {egraph1,egraph2} of a single population with finite number of players, $n$, placed on a graph $\mathcal{G}$. The interactions of players are captured by the edge set of $\mathcal{G}$. Each player $i$ has a set of actions $s_{i} \in S_{i}$ and receives a certain pay-off according to its utility (or fitness) function which is a mapping from the joint action space
$S:= S_1\times S_2\times \hdots S_n$
to the real numbers, $f_{i}:S\rightarrow \mathbb{R}$.
The evolution of graph topology follows a stochastic process administered through pairwise interactions. In particular, two players (that are allowed to interact) are randomly selected and a ``copy'' of the player with the higher fitness, takes the place of the player with the lower one. As a result, the strategy of the player with the smaller fitness is replaced by the strategy of the player with the higher one.
\section{Problem formulation}\label{sec:problem}
In a cyberphysical system, such as a wireless sensor network~\cite{mccabe2008controlled,freris2010fundamentals}, it is common that agents may opt to dynamically create new links with other agents or drop existing ones during the coordination process; this behavior results in time-varying graphs. In order to properly describe this process, it is necessary to formulate a dynamic graph, whose structure depends on time, topology and network state.
In what follows, we define a \emph{dynamic graph} as a graph with fixed predetermined vertex set $\mathcal{V}$ of agents,
in which the edge set $\mathcal{E}$ varies over time, in a \emph{state-dependent randomized} fashion. In particular, the edge set $\mathcal{E} = \mathcal{E}(\mathbf{x}_t,t,\omega) \subseteq \mathcal{V}\times\mathcal{V}$ is a state-/time-dependent random set in a probability space $(\Omega,\mathcal{F},\P)$ (where $\Omega$ is the sample space, $\mathcal{F}$ is the $\sigma-$algebra on $\Omega$, and $\P$ is the probability measure); correspondingly, we define the adjacency matrix $\mathsf{A}= \mathsf{A}(\mathbf{x}_t,t,\omega)$ and the Laplacian matrix $\L = \L(\mathbf{x}_t,t,\omega)$. In this paper, we focus on time-varying graphs with topology at time $t$ depending on both the state at time $t$ and the topology `right before' time $t$, i.e.,
$\L = \L(\mathbf{x}_t,\mathcal{E}_{t^-},\omega)$. We use the shorthand notation $\mathcal{E}_t,\mathsf{A}_t,\L_t$ and $\mathcal{G}_t = (\mathcal{V},\mathcal{E}_t)$ to emphasize the type-varying aspect.
For each node $i\in \mathcal{V}$, we denote by $\mathcal{N}_i$ the set of all \emph{feasible neighbors}, i.e., the set of all nodes that node $i$ can potentially create a connection with. Since we focus on undirected graphs, we assume that $ j\in \mathcal{N}_i$ implies that
$i \in \mathcal{N}_j$.
At each time $t$, for any given node $i\in \mathcal{V}$, we denote the set of \emph{active neighbors}, i.e., the set of nodes with which $i$ is connected, using
$\mathcal{N}_{i,t}^{(1)} \subseteq \mathcal{N}_i$. Furthermore, we let $\mathcal{N}_{i,t}^{(2)}:=\mathcal{N}_i \setminus \mathcal{N}_{i,t}^{(1)}$, the set of \emph{inactive neighbors}, i.e., the set of nodes that $i$ is not connected with, but may decide to connect with based on the evolutionary game.
The degree of a node $i$ at time $t$, $\abs{\mathcal{N}_{i,t}^{(1)}}$, is the total number of active neighbors of node $i$ at time $t$. For the subsequent treatment, we make the assumption that the graph obtained by taking the union of all feasible neighbor sets is connected:
\begin{Assumption}\label{ass:connectivity}
The graph $\mathcal{G}' = (\mathcal{V}, \mathcal{E}')$, with $\mathcal{E}':=\cup_{i\in\mathcal{V}}\cup_{j\in\mathcal{N}_i} \{(i,j)\}$ is connected.
\end{Assumption}
This condition is necessary for consensus to be achieved: otherwise, the graph will be disconnected at each time regardless of the agents' decisions, with no possible exchange of information across its connected components, which implies that consensus is infeasible. We will establish the sufficiency of the condition for the dynamic topology instructed by the evolutionary game we propose.
Let $\vecx{x}$ denote the \emph{coordination levels} of the agents, where we will restrict attention (without any loss in generality) to the case that $0 \leq x_{i,t} \leq 1, \forall i \in \mathcal{V},t\ge 0$. For instance, the evolution of the coordination levels may be considered as a resource allocation process, in which agents decide to share a percentage of a resource they own with their neighbors. Similarly, in an opinion dynamics setup, the coordination levels may reflect the beliefs of the agents, e.g., the information state of vehicles in a robot team.
The agents adjust their coordination levels based on interactions with their neighbors, according also to their tendency to create a link with other agents or drop an existing one. The evolution of an agent's coordination level may be described by the following dynamic consensus protocol:
\begin{equation}
\label{eq:wei_cons_pro}
\begin{array}{r l}
\dot{x}_{i,t}=
\sum_{j \in \mathcal{N}_{i,t^-}^{(1)}}\chi_{ij,t}^{m}(x_{j,t}-x_{i,t}) \\ &+ \sum_{k \in \mathcal{N}_{i,t^-}^{(2)}}\chi_{ik,t}^{c}(x_{k,t}-x_{i,t}),
\end{array}
\end{equation}
where $\chi_{ij,t}^{m},\chi_{ik,t}^{c}$ are $0-1$ variables that respectively indicate whether to: a) \emph{maintain} an existing link (i.e., a link that is active `right before' time $t$, equivalently $(i,j)$ with $j\in \mathcal{N}_{i,t^-}^{(1)}$), if $\chi_{ij,t}^{m}=1$ ($\chi_{ij,t}^{m}=0$ means that the link is dropped); and b) \emph{create} a new link $(i,k)$ with $k\in \mathcal{N}_{i,t^-}^{(2)}$, if $\chi_{ik,t}^{c}=1$.
Clearly, we set $\chi_{ji,t}^{m} \equiv \chi_{ij,t}^{m}, \chi_{ki,t}^{c} \equiv \chi_{ik,t}^{c}$.
The decisions are Bernoulli random variables with respective `success' probabilities (the probability of the value $1$) given by $0\leq w_{ij,t}^{m}\leq 1$, and $0 \leq w_{ik,t}^{c}\leq 1$.
In this paper, the decision rules are \emph{state-dependent} and time-invariant, i.e., $\chi_{ij,t}^{m},\chi_{ik,t}^{c}$ are independent Bernoulli random variables with success probabilities that depend on the coordination levels of the two neighbors:
$w_{ij,t}^{m} = w_{ij,t}^{m} (x_{i,t},x_{j,t})$, $w_{ik,t}^{c} = w_{ik,t}^{c} (x_{i,t},x_{k,t})$, cf.~\eqref{eq:weidrop},~\eqref{eq:weicreate} for their exact definition.
The following remark underlines the inapplicability of previous analysis~\cite{olfati} in our setting.
\begin{Remark}
The graph corresponding to the Laplacian matrix $\L_t$ may be disconnected.
\end{Remark}
Indeed, since decisions are probabilistic, there is a positive probability that the resulting graph is disconnected (even the event of an empty edge set has positive probability) at each given time instant.
We may stack the decisions $\{\chi_{ij,t}^{m},\chi_{ij,t}^{c}\}$ into a corresponding Laplacian matrix $\L = \L(\mathbf{x}_t,\mathcal{E}_{t^-},\omega)\equiv \L_t$ with entries $\{l_{ij}\}_{i,j\in \mathcal{V}}$ (dropping time dependency for notational simplicity) defined by:
\begin{eqnarray*}
l_{ij}=l_{ji} := &- \left(1_{\{j \in \mathcal{N}_i^{(1)}\}}\chi_{ij}^{m} + 1_{\{j \in \mathcal{N}_i^{(2)}\}}\chi_{ij}^{c}\right),\\
l_{ii} := &-\sum_{j\ne i} l_{ij},
\end{eqnarray*}
where $1_{\{\cdot\}}$ is the $0-1$ indicator function (1 if the event holds and 0 else). We proceed to write the update rule in matrix form as follows:
\begin{equation}\label{eq:state_evolution}
\dot{\mathbf{x}}_t = -\L_t\mathbf{x}_t.
\end{equation}
We call this the \emph{state evolution} equation; note that, by its very definition, it constitutes a continuous Markov Decision Process (MDP).
The following proposition shows that all coordination levels are guaranteed to remain in the interval $[0,1]$ if they are initialized in $[0,1]$, i.e., it establishes that the set $[0,1]^n$ is \emph{forward invariant}.
\begin{Proposition}\label{prop:invariance}
Suppose $\mathbf{x}_0 \in [0,1]^n$. Then, under state evolution~\eqref{eq:state_evolution}, $\mathbf{x}_t \in [0,1]^n$ for all $t>0$, i.e., $[0,1]^n$ is forward-invariant.
\end{Proposition}
\begin{proof}
The proof considers two cases: the first case considers a nodal value reaching the upper bound (1), and the second the lower bound (0).
Case 1: Assume that for some $t\ge 0$, there exists $i\in \mathcal{V}$ with $x_{i,t}=1$ and that $\mathbf{x}_s\in [0,1]^n$ for all $s\le t$. Then, given that $x_{j,t} \in [0,1]$ for all $j\ne i$ it follows that
$$\dot{x}_{i} = \sum_{j\in\mathcal{V}\setminus\{i\}} -l_{ij} (x_{j,t}-1) \le 0,$$
because $x_{j,t} \le 1$ and $-l_{ij}\ge 0$. Therefore $x_i$ can never exceed the value 1.
Case 2: Assume that for some $t\ge 0$, there exists $i\in \mathcal{V}$ with $x_{i,t}=0$ and that $\mathbf{x}_s\in [0,1]^n$ for all $s\le t$. Since $x_{j,t} \in [0,1]$ for all $j\ne i$ it follows that
$$\dot{x}_{i} = \sum_{j\in\mathcal{V}\setminus\{i\}} -l_{ij} x_{j,t} \ge 0,$$
therefore $x_i$ can never go below 0.
\end{proof}
\begin{Remark}
The forthcoming analysis applies irrespectively of the assumption that the values $\mathbf{x}_t \in [0,1]^n$, i.e., for arbitrary initial conditions $\mathbf{x}_0$. This assumption is adopted solely for the sake of interpretability in the context of evolutionary games.
\end{Remark}
\subsection{Evolutionary game}
In this section, we provide a rule for selecting the weights (i.e., probabilities) $w_{ij,t}^{m}, w_{ij,t}^{c}$ based on a particular evolutionary game. We use \emph{Continuous Actions Iterative Prisoner's Dilemma} (CAIPD)~\cite{caipd} to define the fitness function of a given node and illustrate how the weights are selected.
In CAIPD, there are $n$ agents that choose their coordination levels given their neighbors' decisions. Each agent $i$ has to pay a fee that is related to its coordination level and gains a reward related to the coordination levels of its neighbors: the higher the coordination level of agent $i$ and the coordination levels of its neighbors are, the higher the cost and reward are, respectively.
Formally, the reward of agent $i$ when it sets its coordination level to $x_{i}$ is defined using the following fitness function (where we drop dependency from time $t$ henceforth, since the definition of fitness in CAIPD is time-independent):
\begin{equation}
f_{i}(\vec{x})=b\sum_{j \in \mathcal{N}_{i}^{(1)}}{x_{j}}-c\abs{\mathcal{N}_{i}^{(1)}}x_{i},
\label{eq:fitness}
\end{equation}
where $b>c>0$ are constants (i.e., we assume that the gain--per unit of coordination--from cooperating with another agent $b$ is higher than the per-unit loss $c$).
The change in the fitness function of agent $i$ when it creates or drops a link, denoted by $\tilde{f}^{c}_{ij}$ (for $j\in \mathcal{N}_{i}^{(2)}$) and $\tilde{f}^{d}_{ij}$ (for $j\in \mathcal{N}_{i}^{(1)}$) respectively, is determined by:
\begin{eqnarray}
\label{eq:fcreate}
\tilde{f}^{c}_{ij}(\vec{x})&= & b\sum_{k \in \mathcal{N}^{(1)}_{i}\cup\{j\}} x_{k}-c\left(\abs{\mathcal{N}^{(1)}_{i}} +1 \right) x_{i}\nonumber\\
&-&\left(b\sum_{k \in \mathcal{N}_{i}^{(1)}}{x_{k}}-c\abs{\mathcal{N}_{i}^{(1)}}x_{i}\right)\nonumber\\
&=&bx_{j}-cx_{i},
\end{eqnarray}
\begin{eqnarray}
\label{eq:fdrop}
\tilde{f}^{d}_{ij}(\vec{x}) &= & b\sum_{k \in \mathcal{N}^{(1)}_{i} \setminus \{j\}} x_{k}-c\left(\abs{\mathcal{N}^{(1)}_{i}} -1 \right) x_{i} \nonumber\\
&-& \left(b\sum_{k \in \mathcal{N}_{i}^{(1)}}{x_{k}}-c\abs{\mathcal{N}_{i}^{(1)}}x_{i}\right) \nonumber\\
&=& cx_{i}-bx_{j}.
\end{eqnarray}
In the evolutionary game, a link may be created/dropped if both agents desire to coordinate or not based on the corresponding increment (or decrement) of their individual fitness functions. Essentially, if both agents benefit from maintaining/creating a link, the corresponding probability must be higher than the case where only one node benefits or when the fitness of both agents is decreased. In \cite{ecc} a sigmoid function was used to determine the weights $w_{ij,t}^{m}$ and $w_{ij,t}^{c}$. In our formulation, the weights correspond to the probabilities that agent $i$ maintains (one minus the probability that it drops) or creates link $(i,j)$, respectively.
Following a similar approach, the weights
are selected as (where we once again drop time dependency since the weight-rule is state-dependent but time-invariant):
\begin{equation}
\begin{array}{rl}
\label{eq:weidrop}
w_{ij}^{m}=&\frac{1}{2}-\frac{1}{2}\tanh\left( \tilde{f}_{ij}^{d}(\vec{x})+\tilde{f}_{ji}^{d}(\vec{x})\right )\\
=& \frac{1}{2}-\frac{1}{2}\tanh\left( (c-b)(x_{i}+x_{j})\right ),
\end{array}
\end{equation}
\begin{equation}
\begin{array}{rl}
\label{eq:weicreate}
w_{ij}^{c}=&\frac{1}{2}+\frac{1}{2}\tanh\left( \tilde{f}_{ij}^{c}(\vec{x})+\tilde{f}_{ji}^{c}(\vec{x})\right )\\
=& \frac{1}{2}+\frac{1}{2}\tanh\left((b-c)(x_{i}+x_{j})\right ).
\end{array}
\end{equation}
Note that, by definition, the two values are equal and lower-bounded by $\frac{1}{2}$ (in light of the fact that $\mathbf{x}_t\in[0,1]^n$, for all $t\ge 0$; cf. Proposition~\ref{prop:invariance}).
We define the \emph{weighted Laplacian} matrix $\mathsf{W}$ with entries (again dropping time dependency for notational simplicity) given by:
\begin{eqnarray}\label{eq:w_define}
w_{ij}=w_{ji} := &- \left(1_{\{j \in \mathcal{N}_i^{(1)}\}}w_{ij}^{m} + 1_{\{j \in \mathcal{N}_i^{(2)}\}}w_{ij}^{c}\right),\\
w_{ii} := &-\sum_{j\ne i} w_{ij},\nonumber
\end{eqnarray}
\section{Convergence analysis}\label{sec:proof}
Formally, for $t>0$, $\L_t$ is an $\mathcal{F}_{t^-}$-measurable random matrix where the $\sigma-$algebra $\mathcal{F}_{t^-}$ is defined by $\mathcal{F}_{t^-} := \sigma(\cup_{s<t} \sigma(\mathcal{E}_s),\mathbf{x}_t) = \sigma( \cup_{s<t}\sigma (\L_s),\mathbf{x}_t)$; $\sigma(\cdot)$ denotes the completion (i.e., adding all subsets of sets of zero measure~\cite{stroock}) of the $\sigma-$algebra generated by its argument (a random variable). Simply said, $\mathcal{F}_{t^-}$ is a formal way of describing the available information pertaining to the topology `right before' time $t$, along with the value $\mathbf{x}_t$
and it induces a \emph{filtration}~\cite{stroock}, i.e., $\mathcal{F}_{s^-} \subseteq \mathcal{F}_{t^-}$ for $s\le t$.
In the sequel, we use the notation $\mathbb{E}_t[\cdot]:=\mathbb{E}[\cdot|\mathcal{F}_{t^-}]$;
we assume the initial state $\mathbf{x}_0$ and topology $\mathcal{E}_0$ are deterministic and known.
It follows that
\begin{equation}\label{eq:cond_exp}
\mathsf{W}_t = \mathbb{E}_t[\L_t].
\end{equation}
Note that~\eqref{eq:state_evolution} is a `stochastic differential equation\footnote{We use this term in brackets as the differential equation is driven by the random Laplacian matrix and not a Brownian motion~\cite{stroock}.}' which is equivalent to the integral equation:
\begin{equation}\label{eq:integral_equation}
\mathbf{x}_t = \mathbf{x}_0 -\int_{0}^{t}\L_s\mathbf{x}_s ds.
\end{equation}
The following lemma characterizes the evolution of the
mean:
$$\bar{\mathbf{x}}_t:=\mathbb{E}[\mathbf{x}_t].$$
\begin{Lemma}[Evolution of the mean]\label{lemma:mean}
Under state evolution~\eqref{eq:state_evolution}, the mean value follows the differential equation:
\begin{equation}\label{eq:mean_evolution}
\dot{\bar{\mathbf{x}}}_t = -\mathbb{E}[\mathsf{W}_t \mathbf{x}_t].
\end{equation}
\end{Lemma}
\begin{proof}
Taking expectation in~\eqref{eq:integral_equation} yields
\begin{eqnarray*}
\bar{\mathbf{x}}_t & =& \bar{\mathbf{x}}_0 -\mathbb{E}[\int_{0}^{t}\L_s\mathbf{x}_s ds]\\
& = &\bar{\mathbf{x}}_0 - \int_{0}^{t}\mathbb{E}[\L_s\mathbf{x}_s] ds\\
& = & \bar{\mathbf{x}}_0 - \int_{0}^{t}\mathbb{E}[\mathbb{E}_s[\L_s\mathbf{x}_s]] ds\\
& = & \bar{\mathbf{x}}_0 - \int_{0}^{t}\mathbb{E}[\mathsf{W}_s\mathbf{x}_s] ds
\end{eqnarray*}
The first equality uses the definition of $\bar{x}_t$. The second one invokes Fubini's theorem~\cite{royden} (since $\mathbf{x}_t\in [0,1]^n$ and $\L$ is a finite dimensional matrix with bounded entries). The third equality uses the towering property of expectation~\cite{stroock}.
The fourth uses the fact that $\mathbf{x}_s$ is $\mathcal{F}_{s^-}-$measurable
along with~\eqref{eq:cond_exp}.
\end{proof}
The next theorem establishes the convergence of our scheme:
\begin{Theorem}[Average consensus]\label{thm:convergence}
Under Assumption~\ref{ass:connectivity} and state evolution~\eqref{eq:state_evolution} the system reaches average consensus:
$$\lim_{t\to \infty} \mathbf{x}_t = Ave(\mathbf{x}_0) \mathbf{1} \ \text{a.s. and in m.s.},$$
for any $\mathbf{x}_0\in[0,1]^n$, where $Ave(\mathbf{x}_0):=\frac{1}{n}\mathbf{1}^\top\mathbf{x}_0$ is the average of the initial nodal values.
\hide{
$$\lim_{t\to \infty} \mathbf{x}_t = Ave(\mathbf{x}_0) \mathbf{1} \ \text{a.s. and in } \ell_p(\Omega,\mathcal{F},\P),$$
for any $\mathbf{x}_0\in[0,1]^n$ and for any $p\in [1,\infty]$, where $Ave(\mathbf{x}_0):=\frac{1}{n}\mathbf{1}^\top\mathbf{x}_0$ is the average of the initial nodal values, and for a random vector $\mathbf{z}\in \mathbb{R}^n$
$$\|\mathbf{z}\|_{\ell_p(\Omega,\mathcal{F},\P)} := \mathbb{E}[\|\mathbf{z}\|_p].$$}
Furthermore, the m.s. convergence is exponential in expectation,
with the expected rate lower-bounded by $\lambda_2(\mathcal{G}')>0$.
\end{Theorem}
\begin{proof}
Since $\L$ is symmetric and $\L\mathbf{1} =\mathbf{0}$, pre-multiplying~\eqref{eq:integral_equation} by $\mathbf{1}^\top$ gives $\mathbf{1}^\top\mathbf{x}_t = \mathbf{1}^\top\mathbf{x}_0$ for all $t\ge 0$, i.e., the sum (and therefore the average) of entries is constant over time. We define the \emph{disagreement} vector $\mathbf{e}_t :=\mathbf{x}_t- Ave(\mathbf{x}_0)\mathbf{1}$: it follows that $\mathbf{e}_t\perp\mathbf{1}$, i.e., $\mathbf{1}^\top\mathbf{e}_t = \mathbf{0}$ for all $t\ge 0$. Consider the Lyapunov function $V(\mathbf{e}) = \frac{1}{2}\|\mathbf{e}\|_2^2 = \frac{1}{2}\mathbf{e}^\top\mathbf{e}$. Under~\eqref{eq:state_evolution} it follows that:
$$\dot{V}(\mathbf{e}_t) = -\mathbf{e}_t^\top\L_t\mathbf{e}_t,$$
where we have used the chain rule and the property that $\L_t\mathbf{1}=\mathbf{0}$. Note that the drift satisfies $-\mathbf{e}_t^\top\L_t\mathbf{e}_t\le 0$ since $\L_t$ is positive semidefinite.
Using the exact same line of analysis as in Lemma~\ref{lemma:mean} we get:
$$\mathbb{E}[V(\mathbf{e}_t)] = V(\mathbf{e}_0) - \int_{0}^{t}\mathbb{E}[\mathbf{e}_s^\top\mathsf{W}_s\mathbf{e}_s]ds,$$
or more generally:
$$\mathbb{E}_s[V(\mathbf{e}_t)] = V(\mathbf{e}_s) - \int_{s}^{t}\mathbb{E}_s[\mathbf{e}_\tau^\top\mathsf{W}_\tau\mathbf{e}_\tau]d\tau,$$
Therefore $V(\mathbf{e}_t)$ is a bounded (cf. Proposition~\ref{prop:invariance}) $(\Omega,\mathcal{F}_{t^-},\P)-$supermartingale, and therefore converges a.s. by the supermartingale convergence theorem~\cite{stroock}. Denote the limit by $\mathbf{e}_\infty(\omega)$; we will establish that $\mathbf{e}_\infty=\mathbf{0}$ a.s. Note that $\mathsf{W}(\mathbf{x})$ is a (state-dependent) weighted Laplacian on the graph $\mathcal{G}' = (\mathcal{V}, \mathcal{E}')$ which is connected (cf. Assumption~\ref{ass:connectivity}), therefore $\lambda_2(\mathcal{G}')>0$. Furthermore, the edge weights are positive and bounded away from zero uniformly over $\mathbf{x}$; to see this note that~\eqref{eq:weidrop},~\eqref{eq:weicreate} and the fact that $\mathbf{x}\in [0,1]^n$ imply that
$$\min (w_{ij}^m,w_{ij}^c) \ge \frac{1}{2}.$$
This also shows that $\lambda_2(\mathsf{W}) \ge \frac{1}{2}\lambda_2(\mathcal{G}')>0$. Since $\mathbf{e}_t\perp \mathbf{1}$ for all $t\ge 0$, and by the definition of $V(\cdot)$, we have:
$$\mathbb{E}[V(\mathbf{e}_t)] \le V(\mathbf{e}_0) - \lambda_2(\mathcal{G}')\int_{0}^{t}\mathbb{E}[V(\mathbf{e}_s)]ds,$$
consequently
$$\mathbb{E}[V(\mathbf{e}_t)] \le V(\mathbf{e}_0)e^{-\lambda_2(\mathcal{G}') t},$$
i.e.,
$$\lim_{t\to \infty}\mathbb{E}[V(\mathbf{e}_t)]=0.$$
This establishes m.s. convergence to $\mathbf{0}$, with expected exponential convergence with rate lower-bounded by $\lambda_2(\mathcal{G}')$;
a.s.-convergence follows by the supermartingale convergence theorem and Fatou's lemma~\cite{royden}.
\end{proof}
\begin{Corollary}[Convergence in expectation]
Under Assumption~\ref{ass:connectivity} and state evolution~\eqref{eq:state_evolution}:
$$\lim_{t\to \infty} \mathbb{E}[\mathbf{x}_t] = Ave(\mathbf{x}_0) \mathbf{1}.$$
\end{Corollary}
\begin{proof}
By Jensen's inequality, $\|E[\mathbf{e}_t]\|_2^2 \le \mathbb{E}[\|\mathbf{e}_t\|_2^2]$, therefore
$$\lim_{t\to \infty} \mathbb{E}[\mathbf{e}_t] = \mathbf{0},$$
and the result follows by the definition of $\mathbf{e}_t$.
\end{proof}
\section{Experiments}\label{sec:sim}
In this section, we present simulation studies that attest our convergence results.
We have employed the \emph{small world network} model~\cite{watts1999small}, i.e., a Bernoulli random graph in which any two agents are allowed to interact with a fixed probability $p_1$; this process generates the neighborhood sets $\{\mathcal{N}_i\}$ and therefore the graph $\mathcal{G}' = (\mathcal{V}, \mathcal{E}')$, cf. Assumption~\ref{ass:connectivity}. We took the network size $n=1000$ in our experiments and set $p_1=0.2$; we repeated the experiment until a connected graph $\mathcal{G}'$ was obtained as required by Assumption~\ref{ass:connectivity}. For initialization, we chose $\mathbf{x}_0$ uniformly distributed on $[0,1]^n$ and selected the active neighbor sets $\mathcal{N}_{i}^{(1)}$ as follows: for each $i$, neighbors in $\mathcal{N}_i$ were selected to be active with probability $p_2$ (independently from one another); we took $p_2=0.2$. Last, we set $b=5,c=4$ in~\eqref{eq:weidrop},~\eqref{eq:weicreate}.
For numerical simulation of the state evolution we have performed uniform discretization of~\eqref{eq:state_evolution} with a step-size $\Delta$, i.e., we set $t=\Delta k$ where $k$ is the discrete iterate counter and run:
$$\mathbf{x}_{k+1} = \mathbf{x}_k -\Delta \L_k \mathbf{x}_k.$$
We chose the step-size $\Delta=\frac{1}{n}$ which guarantees that the spectral radius of $(I - \Delta \L_k)$ is less than or equal to 1 (since the eigenvalues of the Laplacian are upper bounded by $n
~\cite{graph_theory}).
Figure \ref{fig:fig1} depicts the evolution of the coordination levels (for a single small world network and initialization of $\mathbf{x}_0$): it is evident that all coordination levels converge to the average value.
\begin{figure}
\includegraphics[scale=0.55]{cons_d}
\caption{Time evolution of coordination levels of 1000 agents.}
\label{fig:fig1}
\end{figure}
Figure \ref{fig:fig2} illustrates a logarithmic plot of the evolution of the normalized norm of the disagreement vector $\frac{\|\mathbf{e}_t\|_2}{{\|\mathbf{e}_0\|_2}}$, referred to as relative error, averaged over 1000 experiments (random topologies $\mathcal{G}'$ and initializations of $\mathbf{x}_0$); it is evident that the convergence is exponential, in full alliance with Theorem~\ref{thm:convergence}.
\begin{figure}
\includegraphics[scale=0.47]{norm}
\caption{Normalized disagreement vector norm over time.}
\label{fig:fig2}
\end{figure}
\section{Conclusions and future work}\label{sec:conclusions}
We have proposed and analyzed average consensus on evolutionary graphs. Linear consensus iterations are performed on a dynamic graph, where the topology is determined by an evolutionary game in which agents can randomly create new links or drop existing ones in a selfish manner based on their fitness function. We have established a.s. and m.s. average consensus with expected exponential rate regardless of the initial topology and agents' values.
Our future work will focus on devising distributed methods for multi-agent optimization over evolutionary graphs, as well as on extending the analysis to directed graphs and discrete-time iterations.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-03-08T02:05:41",
"yymm": "1803",
"arxiv_id": "1803.02564",
"language": "en",
"url": "https://arxiv.org/abs/1803.02564"
}
|
\section{Introduction}
Least squares approximations of the form
\[
\min_{{\mathbf{x}}\in\mathbb{R}^{d}}\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}-{\mathbf{b}}}
\]
are fundamental building blocks in computational science and statistical
data analysis, with applications ranging from statistical data analysis
to inverse problems. However, it is well appreciated, especially in
the aforementioned application areas, that regularization is often
the key to achieving the best results.
One of the basic methods for regularizing least squares approximations
is principal component regression (PCR)~\cite{hotelling1933analysis,kendall1957course,artemiou2009principal}.
Given a data matrix $\ensuremath{{\bm{\mathrm{A}}}}$, a right hand side ${\mathbf{b}}$ and a target
rank $k$, PCR is computed by first computing the coefficients $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$
corresponding to the top $k$ principal components of $\ensuremath{{\bm{\mathrm{A}}}}$ (i.e.,
to dominant right invariant subspace of $\ensuremath{{\bm{\mathrm{A}}}}$), then regressing
on $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$ and ${\mathbf{b}}$, and finally projecting the solution
back to the original space. In short, the PCR estimator is ${\mathbf{x}}_{k}=\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})^{\textsc{+}}{\mathbf{b}}$
and regularization is achieved via PCA based dimensionality reduction.
While there is some criticism of PCR in the statistical literature~\cite{artemiou2009principal,jolliffe1982note},
it is nevertheless a valuable tool in the toolbox of practitioners.
Up until recent breakthroughs on fast methods for least squares approximations,
there was little penalty in terms of computational complexity when
switching from ordinary least squares (OLS) to PCR. Indeed, the complexity
of SVD based computation of the dominant invariant subspace is $O(nd\min(n,d))$,
and this matches the asymptotic complexity of straightforward computation
of the OLS solution (i.e., via direct methods). However, recent progress
on fast sketching based algorithms for linear regression~\cite{DMMS11,RT08,MSM14,CW17,woodruff2014sketching}
has created a gap: exact computation of the principal components still
requires SVD so the overall complexity is still $O(nd\min(n,d))$,
even though the OLS stage is faster. The gap is not insubstantial:
when learning with large scale data (either large $n$, or large $d$),
$O(nd\min(n,d))$ is often infeasible, but modern sketching based
linear regression methods are.
\subsection{Contributions}
In this paper, we study the use of dimensionality reduction prior
to computing PCR (so we can compute PCR on a smaller input matrix).
In particular, for a data matrix $\ensuremath{{\bm{\mathrm{A}}}}$, we relate the PCR solution
of $\ensuremath{{\bm{\mathrm{A}}}}\mat R$, where $\mat R$ is any dimensionality reduction matrix,
to the PCR solution of $\ensuremath{{\bm{\mathrm{A}}}}$. To do so, we study the notion of
approximate PCR both from an optimization perspective and from a statistical
perspective, and provide conditions on $\mat R$ that guarantee that
after projecting the solution back to the full space (by multiplying
by $\mat R^{\textsc{T}}$) we have an approximate PCR solution with rigorous
statistical risk bounds. These results are described in Section~\ref{sec:dim-reduce-pcr}.
We then leverage the aforementioned results to design fast, sketching
based, algorithms for approximate PCR. We propose algorithms specialized
for the several cases (in the following, $n$ is number of data points,
$d$ is dimension of the data): large $n$ (using left sketching),
large $d$ (using right sketching), and both $n$ and $d$ large (using
two-sided sketching). Furthermore, we propose an input-sparsity time
algorithm for approximate PCR. These results are described in Section~\ref{sec:alg}.
We also consider computing approximate PCR in the streaming model,
providing the first algorithm for computing approximate PCR in a stream.
We also provide a fast algorithm for approximate Kernel PCR (polynomial
kernel only). These results are described in Section~\ref{sec:extensions}.
Finally, empirical results (Section \ref{sec:experiments}) clearly
demonstrate the ability of our proposed algorithms to compute approximate
PCR solution, the correctness of our theoretical analysis, and the
advantages of using our techniques instead of simpler techniques like
compressed least squares.
In general, unlike previous works on randomized methods for PCR (which
we discuss in the next subsection), we analyze the use of sketching
for PCR from a sketch-and-solve approach. We discuss the various advantages
and disadvantages of the sketch-and-solve approach in comparison to
iterative based approaches, in the next subsection.
\subsection{Related Work}
Recently matrix sketching, such as the use of random projections,
has emerged as a powerful technique for accelerating and scaling many
important statistical learning techniques. See recent surveys by Woodruff~\cite{woodruff2014sketching}
and Mahoney et al.~\cite{YMM15} for an extensive exposition on this
subject. So far, there has been limited research on the use of matrix
sketching in the context of principal component regression.
One natural strategy for leveraging sketching in the context of PCR
is to use approximate principal components. Approximate principal
components can be computed using fast sketching based algorithm for
approximate PCA (also known as 'randomized SVD')~\cite{halko2011finding,woodruff2014sketching}.
This was recently explored by Boutsidis and Magdon-Ismail~\cite{BM14}.
The authors show that the if the number of subspace iterations is
sufficiently large, one obtain a bound on the sub-optimality of the
approximate solution and on the error of the solution vector. We too
bound the sub-optimality of our solutions, but instead of bounding
the error of the solution vector, we bound their distance to the right
dominant subspace, or bound the distance of the projection to the
left dominant subspace.
Frostig et al. leverage fast randomized algorithms for ridge regression
to design iterative algorithms for principal component regression
and principal component projection~\cite{frostig2016principal}.
Forstig et al.'s results were later improved by Allen-Zhu and Li~\cite{AL17}.
Both of the aforementioned methods use iterations, while our work
explores the use of a sketch-and-solve approach. While it is true
that better accuracies can be achieved using iterative methods with
sketching based accelerators~\cite{RT08,AMT10,MSM14,GOS16,ACW17},
there are some advantages in using a sketch-and-solve approach. In
particular, sketch-and-solve algorithms are typically faster. However
this comes at the cost: sketch-and-solve algorithms typically provide
cruder approximations. Nevertheless, it is not uncommon for these
cruder approximations to be sufficient in machine learning applications.
Another advantage of the sketch-and-solve approach is that it is more
amenable to streaming and kernelization; we consider both in this
paper.
Closely related to our work is recent work on Compressed Least Squares
(CLS)~\cite{maillard2009compressed,Kaban14,slawski2017compressed,Slawski17,THM17}.
In particular, our statistical analysis (section~\ref{subsec:statistical})
is inspired by recent statistical analysis of CLS~\cite{slawski2017compressed,Slawski17,Kaban14}.
Additionally, CLS is sometimes considered as a computationally attractive
alternative to PCR~\cite{Slawski17,THM17}. While CLS certainly uses
matrix sketching to compress the matrix, it also uses the compression
to regularize the problem. The mix between compression for scalability
and compression for regularization reduces the ability to fine tune
the method to the needs at hand, and thus obtain the best possible
results. In contrast, our methods uses sketching primarily to approximate
the principal components and as such serves as a means for scalability
only. We propose methods that are computationally as attractive as
CLS, and are more faithful to the behavior of PCR (in fact, CLS is
a special case of one of our proposed algorithms). These advantages
over CLS are also evident in the experimental results reported in
Section~\ref{sec:experiments}.
Principal component regression is a form of least squares regression
with convex constraints (once the dominant subspace has been found).
Pilanchi and Wainwright recently explored the effect of regularization
on the sketch size for least squares regression~\cite{pilanci2015randomized,pilanci2016iterative}.
In the aforementioned papers, sketching is applied only to the objective,
while the constraint is enforced exactly. This is unsatisfactory in
the context of PCR since for PCR the constraints are gleaned from
the input, and enforcing them is as expensive as solving the problem
exactly. In contrast, our method uses sketching not only to compress
the objective function, but also to approximate the constraint set.
Ridge regression (also known as Tikhonov regularization) is another
popular and well studied method for regularizing least squares solutions.
It also closely related to PCR in the sense that the ridge term can
be viewed as a soft damping of the singular values. Recently several
sketching-based algorithms have been suggested to accelerate the solution
of ridge regression~\cite{ChenEtAl15,ACW17b,WGM17,CYD18}.
\section{Preliminaries}
\subsection{Notation and Basic Definitions}
We denote scalars using Greek letters or using $x,y,\dots$. Vectors
are denoted by ${\mathbf{x}},{\mathbf{y}},\dots$ and matrices by $\ensuremath{{\bm{\mathrm{A}}}},\mat B,\dots$.
The $s\times s$ identity matrix is denoted $\mat I_{s}$. We use the
convention that vectors are column-vectors. $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}$ denotes
the number of non-zeros in $\ensuremath{{\bm{\mathrm{A}}}}$. The notation $\alpha=(1\pm\gamma)\beta$
means that $(1-\gamma)\beta\leq\alpha\leq(1+\gamma)\beta$, and the
notation $\alpha=\beta\pm\gamma$ means that $|\alpha-\beta|\leq\gamma$.
Given a matrix $\ensuremath{{\bm{\mathrm{X}}}}\in\mathbb{R}^{m\times n}$, let $\ensuremath{{\bm{\mathrm{X}}}}=\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}\Sigma_{\ensuremath{{\bm{\mathrm{X}}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}^{\textsc{T}}$
be a \emph{thin SVD} of $\ensuremath{{\bm{\mathrm{X}}}}$, i.e. $\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}\in\mathbb{R}^{m\times\min(m,n)}$
is a matrix with orthonormal columns, $\Sigma_{\ensuremath{{\bm{\mathrm{X}}}}}\in\mathbb{R}^{\min(m,n)\times\min(m,n)}$
is a diagonal matrix with the non-negative singular values on the
diagonal, and $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}\in\mathbb{R}^{n\times\min(m,n)}$ is a matrix
with orthonormal columns. The thin SVD decomposition is not necessarily
unique, so when we use this notation we mean that the statement is
correct for any such decomposition. A thin SVD decomposition can be
computed in $O(mn\min(m,n))$. We denote the singular values of $\ensuremath{{\bm{\mathrm{X}}}}$
by $\sigma_{\max}(\ensuremath{{\bm{\mathrm{X}}}})=\sigma_{1}(\ensuremath{{\bm{\mathrm{X}}}})\geq\dots\geq\sigma_{\min(m,n)}(\ensuremath{{\bm{\mathrm{X}}}})=\sigma_{\min}(\ensuremath{{\bm{\mathrm{X}}}})$,
omitting the matrix from the notation if the relevant matrix is clear
from the context. For $k\leq\min(m,n)$, we use $\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k}$
(respectively $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k})$ to denote the matrix consisting
of the first $k$ columns of $\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}$ (respectively $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}$),
and use $\Sigma_{\ensuremath{{\bm{\mathrm{X}}}},k}$ to denote the leading $k\times k$ minor
of $\Sigma_{\ensuremath{{\bm{\mathrm{X}}}}}$. We use $\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k+}$ (respectively $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k+})$
to denote the matrix consisting of the last $\min(m,n)-k$ columns
of $\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}$ (respectively $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}$), and use $\Sigma_{\ensuremath{{\bm{\mathrm{X}}}},k+}$
to denote the lower-right $(\min(m,n)-k)\times(\min(m,n)-k)$ block
of $\Sigma_{\ensuremath{{\bm{\mathrm{X}}}}}$. In other words,
\[
\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}=\left[\begin{array}{cc}
\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k} & \ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k+}\end{array}\right]\quad\Sigma_{\ensuremath{{\bm{\mathrm{X}}}}}=\left[\begin{array}{cc}
\Sigma_{\ensuremath{{\bm{\mathrm{X}}}},k} & 0\\
0 & \Sigma_{\ensuremath{{\bm{\mathrm{X}}}},k+}
\end{array}\right]\quad\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}=\left[\begin{array}{cc}
\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k} & \ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}},k+}\end{array}\right].
\]
The \emph{Moore-Penrose pseudo-inverse} of $\ensuremath{{\bm{\mathrm{X}}}}$ is $\ensuremath{{\bm{\mathrm{X}}}}^{\textsc{+}}\coloneqq\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}\Sigma_{\ensuremath{{\bm{\mathrm{X}}}}}^{\textsc{+}}\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}^{\textsc{T}}$
where $\Sigma_{\ensuremath{{\bm{\mathrm{X}}}}}^{\textsc{+}}=\diag{\sigma_{1}(\ensuremath{{\bm{\mathrm{X}}}})^{\textsc{+}},\dots,\sigma_{\min(m,n)}(\ensuremath{{\bm{\mathrm{X}}}})^{\textsc{+}}}$
with $a^{+}=a^{-1}$ when $a\neq0$ and $0$ otherwise.
The \emph{stable rank }of a matrix $\ensuremath{{\bm{\mathrm{X}}}}$ is $\sr{\ensuremath{{\bm{\mathrm{X}}}}}\coloneqq\FNormS{\ensuremath{{\bm{\mathrm{X}}}}}/\TNormS{\ensuremath{{\bm{\mathrm{X}}}}}$.
The $k$-th \emph{relative gap} of a matrix $\ensuremath{{\bm{\mathrm{X}}}}$ is
\[
\gap{\ensuremath{{\bm{\mathrm{X}}}}}k=\frac{\sigma_{k}^{2}-\sigma_{k+1}^{2}}{\sigma_{1}^{2}}\,.
\]
For a subspace ${\cal U}$, we use $\ensuremath{{\bm{\mathrm{P}}}}_{{\cal U}}$ to denote the
\emph{orthogonal projection matrix} onto ${\cal U}$, and $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}$
for the projection matrix on the column space of $\ensuremath{{\bm{\mathrm{X}}}}$ (i.e. $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}=\ensuremath{{\bm{\mathrm{P}}}}_{\range{\ensuremath{{\bm{\mathrm{X}}}}}}$).
We have $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}=\ensuremath{{\bm{\mathrm{X}}}}\matX^{\textsc{+}}$. The \emph{complementary
projection matrix }is $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}^{\perp}=\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{X}}}}}$.
A useful property of projection matrices is that if ${\cal S}\subseteq{\cal T}$
then $\ensuremath{{\bm{\mathrm{P}}}}_{{\cal S}}\ensuremath{{\bm{\mathrm{P}}}}_{{\cal T}}=\ensuremath{{\bm{\mathrm{P}}}}_{{\cal T}}\ensuremath{{\bm{\mathrm{P}}}}_{{\cal S}}=\ensuremath{{\bm{\mathrm{P}}}}_{{\cal S}}$.
Furthermore, we note the following result.
\begin{thm}
[Theorem 2.3 in \cite{Stewart_perturbation}]\label{thm:projection_equality}
For any $\ensuremath{{\bm{\mathrm{A}}}}$ and $\ensuremath{{\bm{\mathrm{B}}}}$ with the same number of rows, the following
statements hold:
\begin{enumerate}
\item If $\rank{\ensuremath{{\bm{\mathrm{A}}}}}=\rank{\ensuremath{{\bm{\mathrm{B}}}}}$, then the singular values of $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}^{\perp}$
and $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}^{\perp}$ are the same, so
\[
\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}^{\perp}}=\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}^{\perp}}
\]
\item Moreover the nonzero singular values $\sigma$ of $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}^{\perp}$
correspond to pairs $\pm\sigma$ of eigenvalues of $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}$,
so
\[
\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}}=\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}^{\perp}}
\]
\item If $\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{B}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}}<1$, then $\rank{\ensuremath{{\bm{\mathrm{A}}}}}=\rank{\ensuremath{{\bm{\mathrm{B}}}}}$.
\end{enumerate}
\end{thm}
\subsection{Principal Component Regression and Principal Component Projection }
In the \emph{Principal Component Regression (PCR)} problem, we are
given an input $n\textrm{-by-}d$ data matrix $\ensuremath{{\bm{\mathrm{A}}}}$, a right hand
side ${\mathbf{b}}\in\mathbb{R}^{n}$, and a rank parameter $k$ which is smaller or
equal to the rank of $\ensuremath{{\bm{\mathrm{A}}}}$. Furthermore, we assume that there is
an non-zero eigengap at $k$: $\sigma_{k}>\sigma_{k+1}$. The goal
is to find the PCR solution, ${\mathbf{x}}_{k}$, defined as
\begin{equation}
{\mathbf{x}}_{k}\coloneqq\arg\min_{{\mathbf{x}}\in\range{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}}\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}-{\mathbf{b}}}.\label{eq:pcr-opt-problem}
\end{equation}
It is easy to verify that ${\mathbf{x}}_{k}=\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})^{\textsc{+}}{\mathbf{b}}=\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}\Sigma_{\ensuremath{{\bm{\mathrm{A}}}},k}^{-1}\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\textsc{T}}{\mathbf{b}}$.
The \emph{Principal Component Projection (PCP) }of ${\mathbf{b}}$ is ${\mathbf{b}}_{k}\coloneqq\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}=\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}{\mathbf{b}}$.
Straightforward computation of ${\mathbf{x}}_{k}$ and ${\mathbf{b}}_{k}$ via the SVD
takes $O(nd\min(n,d))$ operations\footnote{The complexity when using iterative algorithms (e.g. Lanczos) to compute
only the dominant invariant spaces depend on several additional facts
and in particular on spectral properties of the matrix and sparsity
level. Thus, to avoid overly complicating the discussion on computational
complexity, we refrain from further discussion of iterative methods
for computing dominant eigenspaces}. We are primarily interested in finding faster algorithms that compute
an approximate PCR or PCP solution (we formalize the terms 'approximate
PCP/PCR' in Section~\ref{sec:dim-reduce-pcr}). Throughout the paper,
we use $\ensuremath{{\bm{\mathrm{A}}}},{\mathbf{b}},$ and $k$ as the arguments of the PCR/PCP problem
to be solved.
\subsection{\label{subsec:Matrix-Perturbations}Matrix Perturbations and Distance
Between Subspaces}
Our analysis uses matrix perturbation theory extensively. We now describe
the basics of this theory and the results we use.
The \emph{principal angles} $\theta_{j}\in[0,\pi/2]$ between two
subspaces ${\cal U}$ and ${\cal W}$ are recursively defined by the
identity
\[
\cos(\theta_{j})=\max_{{\mathbf{u}}\in{\cal U}}\max_{{\mathbf{w}}\in{\cal W}}{\mathbf{u}}^{\textsc{T}}{\mathbf{w}}\,\text{s.t.}\,\TNorm{{\mathbf{u}}}=1,\TNorm{{\mathbf{w}}}=1,\forall i<j.{\mathbf{u}}_{i}^{\textsc{T}}{\mathbf{u}}=0,{\mathbf{w}}_{i}^{\textsc{T}}{\mathbf{w}}=0\,.
\]
We use ${\mathbf{u}}_{j}$ and ${\mathbf{w}}_{j}$ to denote the vectors for which $\cos(\theta_{j})={\mathbf{u}}_{j}^{\textsc{T}}{\mathbf{w}}_{j}$.
Let $\Theta({\cal {\cal U}},{\cal W})$ denote the $d\times d$ diagonal
matrix whose $j$th diagonal entry is the $j$th principal angle,
and as usual we allow writing matrices instead of subspaces as short-hand
for the column space of the matrix. Henceforth, when we write a function
on $\Theta(\cdot,\cdot)$, i.e. $\sin(\Theta(\ensuremath{{\bm{\mathrm{U}}}},\text{\ensuremath{\mat W}}$)),
we mean evaluating the function entrywise on the diagonal only. It
is well known~ \cite[section 6.4.3]{golub2012matrix} that if $\ensuremath{{\bm{\mathrm{U}}}}$
(respectively $\mat W$) is a matrix with orthonormal columns whose
column space is equal to ${\cal U}$ (respectively ${\cal W}$) then
\[
\sigma_{j}(\ensuremath{{\bm{\mathrm{U}}}}^{\textsc{T}}\mat W)=\cos(\theta_{j}).
\]
The following lemma connects the tangent of the principal angles to
the spectral norm of an appropriate matrix.
\begin{lem}
[Lemma 4.3 in \cite{drineas2016structural}] \label{lem:tan-to-spectral}Let
$\ensuremath{{\bm{\mathrm{Q}}}}\in\mathbb{R}^{n\times s}$ have orthonormal columns, and let $\mat W=(\begin{array}{cc}
\mat W_{k} & \mat W_{k+}\end{array})\in\mathbb{R}^{n\times n}$ be an orthogonal matrix where $\mat W_{k}\in\mathbb{R}^{n\times k}$ with
$k\leq s$. If $\rank{\mat W_{k}^{\textsc{T}}\ensuremath{{\bm{\mathrm{Q}}}}}=k$ then
\[
\TNorm{\tan\Theta(\ensuremath{{\bm{\mathrm{Q}}}},\mat W_{k})}=\TNorm{(\mat W_{k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{Q}}}})(\mat W_{k}^{\textsc{T}}\ensuremath{{\bm{\mathrm{Q}}}})}\,.
\]
\end{lem}
Matrix perturbation theory studies how a perturbation of a matrix
translate to perturbations of the matrix's eignevalues and eigenspaces.
In order to bound the perturbation of an eigenspace, one needs some
notion of distance between two subspaces. One common distance metric
between two subspaces is
\begin{equation}
d_{2}({\cal U},{\cal W})\coloneqq\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{{\cal U}}-\ensuremath{{\bm{\mathrm{P}}}}_{{\cal W}}}\,.\label{eq:distance_between_subspaces}
\end{equation}
If $\ensuremath{{\bm{\mathrm{U}}}}$ and $\ensuremath{{\bm{\mathrm{V}}}}$ have the same number of columns, and both
have orthonormal columns, then
\[
d_{2}(\ensuremath{{\bm{\mathrm{U}}}},\ensuremath{{\bm{\mathrm{V}}}})=\sqrt{1-\sigma_{\min}(\ensuremath{{\bm{\mathrm{U}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}})^{2}}=\sin(\theta_{\max})=\TNorm{\sin\Theta(\ensuremath{{\bm{\mathrm{U}}}},\ensuremath{{\bm{\mathrm{V}}}})}
\]
where $\theta_{\max}$ is the maximum principal angle between $\range{\ensuremath{{\bm{\mathrm{U}}}}}$
and $\range{\ensuremath{{\bm{\mathrm{V}}}}}$~\cite[section 6.4.3]{golub2012matrix}.
A classical result that bounds the distance between the dominant subspaces
of two symmetric matrices in terms of the spectral norm of difference
between the two matrices is the Davis-Kahan $\sin(\Theta)$ theorem~\cite[Section 2]{davis1970rotation}.
We need the following corollary of this theorem:
\begin{thm}
[Corollary of Davis-Kahan $\sin\Theta$ Theorem \cite{davis1970rotation}]\label{thm:sin-theta-1}
Let $\ensuremath{{\bm{\mathrm{A}}}},\tilde{\ensuremath{{\bm{\mathrm{A}}}}}\in\mathbb{R}^{n\times n}$ be two symmetric matrices,
both of rank at least $k$. Suppose that $\lambda_{k}>\tilde{\lambda}_{k+1}$
where $\lambda_{1}\geq\dots\geq\lambda_{n}$ and $\tilde{\lambda}_{1}\geq\dots\geq\tilde{\lambda}_{n}$
are the eigenvalues of $\ensuremath{{\bm{\mathrm{A}}}}$ and $\tilde{\ensuremath{{\bm{\mathrm{A}}}}}$. We have
\[
d_{2}(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k},\ensuremath{{\bm{\mathrm{V}}}}_{\tilde{\ensuremath{{\bm{\mathrm{A}}}}},k})\leq\frac{\TNorm{\ensuremath{{\bm{\mathrm{A}}}}-\tilde{\ensuremath{{\bm{\mathrm{A}}}}}}}{\lambda_{k}-\tilde{\lambda}_{k+1}}\,.
\]
\end{thm}
\begin{proof}
We use the following variant of the $\sin\Theta$ Theorem (see~\cite[Theorem 2.16]{stewart2001matrix}):
suppose a symmetric matrix $\ensuremath{{\bm{\mathrm{B}}}}$ has a spectral representation
\[
\ensuremath{{\bm{\mathrm{B}}}}=\ensuremath{{\bm{\mathrm{X}}}}\mat L\ensuremath{{\bm{\mathrm{X}}}}^{\textsc{T}}+\mat Y\ensuremath{{\bm{\mathrm{M}}}}\mat Y^{\textsc{T}}
\]
where $\left[\ensuremath{{\bm{\mathrm{X}}}}\,\mat Y\right]$ is square orthonormal. Let the
orthonormal matrix $\mat Z$ be of the same dimensions as $\ensuremath{{\bm{\mathrm{X}}}}$
and suppose that
\[
\mat R=\ensuremath{{\bm{\mathrm{B}}}}\mat Z-\mat Z\ensuremath{{\bm{\mathrm{N}}}}
\]
where $\ensuremath{{\bm{\mathrm{N}}}}$ is symmetric. Furthermore, suppose that the spectrum
of $\ensuremath{{\bm{\mathrm{N}}}}$ is contained in some interval $[\alpha,\beta]$ and that
for some $\delta>0$ the spectrum of $\ensuremath{{\bm{\mathrm{M}}}}$ lies outside of $[\alpha-\delta,\beta+\delta]$.
Then,
\[
\TNorm{\sin\Theta(\ensuremath{{\bm{\mathrm{X}}}},\mat Z)}\leq\frac{\TNorm{\mat R}}{\delta}\,.
\]
We prove Theorem~\ref{thm:sin-theta-1} by applying the aforementioned
variant of the $\sin\Theta$ Theorem with: $\ensuremath{{\bm{\mathrm{B}}}}=\tilde{\ensuremath{{\bm{\mathrm{A}}}}}$,
$\ensuremath{{\bm{\mathrm{X}}}}=\ensuremath{{\bm{\mathrm{V}}}}_{\tilde{\ensuremath{{\bm{\mathrm{A}}}}},k}$, $\mat Y=\ensuremath{{\bm{\mathrm{V}}}}_{\tilde{\ensuremath{{\bm{\mathrm{A}}}}},k+}$,
$\mat L=\diag{\tilde{\lambda}_{1},\dots,\tilde{\lambda}_{k}}$, $\ensuremath{{\bm{\mathrm{M}}}}=\diag{\tilde{\lambda}_{k+1},\dots,\tilde{\lambda}_{n}},$
$\mat Z=\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$, $\ensuremath{{\bm{\mathrm{N}}}}=\diag{\lambda_{1},\dots,\lambda_{k}}$,
and $\delta=\lambda_{k}-\tilde{\lambda}_{k+1}$ . It is easy to verify
that the conditions of the $\sin\Theta$ Theorem hold, so
\[
\TNorm{\sin\Theta(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k},\ensuremath{{\bm{\mathrm{V}}}}_{\tilde{\ensuremath{{\bm{\mathrm{A}}}}},k})}\leq\frac{\TNorm{\mat R}}{\lambda_{k}-\tilde{\lambda}_{k+1}}
\]
where $\mat R=\tilde{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}-\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}\ensuremath{{\bm{\mathrm{N}}}}$.
We have $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}=\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}\ensuremath{{\bm{\mathrm{N}}}}$ so $\TNorm{\mat R}=\TNorm{(\tilde{\ensuremath{{\bm{\mathrm{A}}}}}-\ensuremath{{\bm{\mathrm{A}}}})\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}\leq\TNorm{\tilde{\ensuremath{{\bm{\mathrm{A}}}}}-\ensuremath{{\bm{\mathrm{A}}}}}$.
Combining this inequality with the previous one and noting that $d_{2}(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k},\ensuremath{{\bm{\mathrm{V}}}}_{\tilde{\ensuremath{{\bm{\mathrm{A}}}}},k})=\TNorm{\sin\Theta(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k},\ensuremath{{\bm{\mathrm{V}}}}_{\tilde{\ensuremath{{\bm{\mathrm{A}}}}},k})}$
completes the proof.
\end{proof}
Under the conditions of Theorem~\ref{thm:sin-theta-1}, since $\ensuremath{{\bm{\mathrm{A}}}}$
and $\tilde{\ensuremath{{\bm{\mathrm{A}}}}}$ are symmetric matrices, Weyl's inequality implies
that
\[
d_{2}(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k},\ensuremath{{\bm{\mathrm{V}}}}_{\tilde{\ensuremath{{\bm{\mathrm{A}}}}},k})\leq\frac{\TNorm{\ensuremath{{\bm{\mathrm{A}}}}-\tilde{\ensuremath{{\bm{\mathrm{A}}}}}}}{\lambda_{k}-\lambda_{k+1}-\TNorm{\ensuremath{{\bm{\mathrm{A}}}}-\tilde{\ensuremath{{\bm{\mathrm{A}}}}}}}\,
\]
as long as $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}-\tilde{\ensuremath{{\bm{\mathrm{A}}}}}}<\lambda_{k}-\lambda_{k+1}$.
Thus, if $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}-\tilde{\ensuremath{{\bm{\mathrm{A}}}}}}\ll\lambda_{k}-\lambda_{k+1}$
then we can compute an approximation to the $k$-dimensional dominant
subspace of $\ensuremath{{\bm{\mathrm{A}}}}$ by computing the $k$-dimensional dominant subspace
of $\tilde{\ensuremath{{\bm{\mathrm{A}}}}}$.
\section{\label{sec:dim-reduce-pcr}PCR with Dimensionality Reduction}
Our goal is to design algorithms which compute an approximate solution
to the PCR or PCP problem. Our strategy for designing such algorithms
is to reduce the dimensions of $\ensuremath{{\bm{\mathrm{A}}}}$ prior to computing the PCR/PCP
solution. Specifically, let $\mat R\in\mathbb{R}^{d\times t}$ be some matrix
where $t\leq d$, and define
\begin{equation}
{\mathbf{x}}_{\mat R,k}:=\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}(\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}{\mathbf{b}}\,.\label{eq:apcr}
\end{equation}
The rationale in Eq.~(\ref{eq:apcr}) is as follows. First, $\ensuremath{{\bm{\mathrm{A}}}}$
is compressed by computing $\ensuremath{{\bm{\mathrm{A}}}}\mat R$ (this is the dimensionality
reduction step). Then we compute the rank $k$ PCR solution of $\ensuremath{{\bm{\mathrm{A}}}}\mat R$
and ${\mathbf{b}}$; this is $(\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}{\mathbf{b}}$. Finally,
the solution is projected back to the original space by multiplying
by $\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$. Obviously, given $\mat R$ we can
compute ${\mathbf{x}}_{\mat R,k}$ in $O(ndt)$ (and even faster, if $\ensuremath{{\bm{\mathrm{A}}}}$
is sparse), so if $t\ll\min(n,d)$ there is a potential for significant
gain in terms of computational complexity\emph{ }provided it is possible
to compute $\mat R$ efficiently as well. Furthermore, if we design
$\mat R$ to have some special structure that allows us to compute
$\ensuremath{{\bm{\mathrm{A}}}}\mat R$ in $O(nt^{2})$ time, the overall complexity would reduce
to $O(nt^{2})$.
Of course, ${\mathbf{x}}_{\mat R,k}$ is not the PCR solution ${\mathbf{x}}_{k}$ (unless
$\mat R=\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$). This suggests the following mathematical
question (which, in turn, leads to an algorithmic question): under
which conditions on $\mat R$ is ${\mathbf{x}}_{\mat R,k}$ a good approximation
to the PCR solution ${\mathbf{x}}_{k}$? In this section, we derive general
conditions on $\mat R$ that ensure deterministically that ${\mathbf{x}}_{\mat R,k}$
is in some sense (which we formalize later in this section) a good
approximation of ${\mathbf{x}}_{k}$. The results in this section are non algorithmic
and independent of the method in which $\mat R$ is computed. In the
next section we address the algorithmic question: how can we compute
such $\mat R$ matrices efficiently?
We approach the mathematical question from two different perspectives:
an optimization perspective and a statistical perspective. In the
optimization perspective, we consider PCR/PCP as an optimization problem
(Eq.~(\ref{eq:pcr-opt-problem})), and ask whether the value of the
objective function of ${\mathbf{x}}_{\mat R,k}$ is close to optimal value of
the objective function, while upholding the constraints approximately
(see Definition~\ref{def:apcr-apcp}). In the statistical perspective,
we treat ${\mathbf{x}}_{k}$ and ${\mathbf{x}}_{\mat R,k}$ as statistical estimators,
and compare their excess risk under a fixed-design model. Interestingly,
the conditions we derive for $\mat R$ are the same for both perspectives.
Before proceeding, we remark that an important special case of (\ref{eq:apcr})
is when $\mat R$ has exactly $k$ columns. In that case, for brevity,
we omit the subscript $k$ from ${\mathbf{x}}_{\mat R,k}$ and notice that
\begin{equation}
{\mathbf{x}}_{\mat R}=\mat R(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{+}{\mathbf{b}}\,.\label{eq:cls}
\end{equation}
Eq.~(\ref{eq:cls}) is valid even if $\mat R$ has more than $k$
columns and/or the columns are not orthonormal. Thus, an established
technique in the literature, frequently referred to as Compressed
Least Squares (CLS)~\cite{maillard2009compressed,Kaban14,slawski2017compressed,Slawski17,THM17},
is to generate a random $\mat R$ and compute ${\mathbf{x}}_{\mat R}$. To avoid
confusion, we stress the difference between (\ref{eq:apcr}) and (\ref{eq:cls}):
in (\ref{eq:apcr}) we compute a PCR solution on the compressed matrix
$\ensuremath{{\bm{\mathrm{A}}}}\mat R$, while in~(\ref{eq:cls}) ordinary least squares is
used. These two strategies coincide when $\mat R$ has $k$ columns.
In this paper, we focus on Eq.~(\ref{eq:apcr}) and consider Eq.~(\ref{eq:cls})
only when it is a special case of Eq.~(\ref{eq:apcr}) (when $\mat R$
has exactly $k$ columns). For an analysis of CLS from a statistical
perspective, see recent work by Slawski~\cite{Slawski17}.
\subsection{Optimization Perspective}
The PCR solution can be written as the solution of a constrained least
squares problem:
\[
{\mathbf{x}}_{k}=\arg\min_{\begin{array}{c}
\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}{\mathbf{x}}}=0\\
{\mathbf{x}}\in\range{\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}}
\end{array}}\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}-{\mathbf{b}}}.
\]
In order to analyze a candidate solution $\tilde{{\mathbf{x}}}$ from an optimization
perspective, we need to decide how to treat the constraints. One option
is to require a candidate $\tilde{{\mathbf{x}}}$ to be inside the feasible
set. Indeed, Pilanci and Wainwright recently considered sketching
based methods for constrained least squares regression~\cite{pilanci2015randomized}.
However, there is no evident way to impose $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\mat T}{\mathbf{x}}=0$
without actually computing $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}$, which is as expensive
as computing $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$. Thus, if we require an approximate
solution to be inside the feasible set, we might as well compute the
exact PCR solution. Thus, in our notion of approximate PCR, we relax
the constraints and require only that the approximate solution is
close to meeting the constraint, i.e. we seek a solution for which
$\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}-{\mathbf{b}}}$ is close to $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}-{\mathbf{b}}}$
\emph{and} $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}\tilde{{\mathbf{x}}}}$ is small.
Similarly, the if $\ensuremath{{\bm{\mathrm{A}}}}$ has full rank the PCP solution can written
as the solution of a constrained least squares problem:
\[
{\mathbf{b}}_{k}=\arg\min_{\begin{array}{c}
\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}\tilde{{\mathbf{b}}}}=0\\
\tilde{{\mathbf{b}}}\in\range{\ensuremath{{\bm{\mathrm{A}}}}}
\end{array}}\TNorm{\tilde{{\mathbf{b}}}-{\mathbf{b}}}
\]
Again, our notion of approximate PCP relaxes the constraint.
The discussion above motivates the following definition of approximate
PCR/PCP.
\begin{defn}
[Approximate PCR and PCP] \label{def:apcr-apcp}An estimator $\tilde{{\mathbf{x}}}$
is an \emph{$(\epsilon,\upsilon)$-approximate PCR} of rank $k$ if
\[
\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}-{\mathbf{b}}}=\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}-{\mathbf{b}}}\pm\epsilon\TNorm{{\mathbf{b}}}
\]
and $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}\tilde{{\mathbf{x}}}}\leq\upsilon\TNorm{{\mathbf{b}}}$.
An estimator $\tilde{{\mathbf{b}}}$ is an \emph{($\epsilon,\upsilon)$-approximate
PCP} of rank $k$ if
\[
\TNorm{\tilde{{\mathbf{b}}}-{\mathbf{b}}}=\TNorm{{\mathbf{b}}_{k}-{\mathbf{b}}}\pm\epsilon\TNorm{{\mathbf{b}}}
\]
and $\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}\tilde{{\mathbf{b}}}}\leq\upsilon\TNorm{{\mathbf{b}}}$.
\end{defn}
Before proceeding, a few remarks are in order.
\begin{enumerate}
\item Imposing no constraints on $\tilde{{\mathbf{x}}}$ (or $\tilde{{\mathbf{b}}}$) does not
make sense: we can always form or approximate the ordinary least squares
solution and it will demonstrate a smaller objective value. Indeed,
the main motivation for using PCR to impose some form of regularization,
so it is crucial the definition of approximate PCR/PCP have some form
of regularization built-in.
\item We require only additive error on the objective function, while relative
error bounds are usually viewed as more desirable. For approximate
PCR, requiring relative error bounds is likely unrealistic: since
it is possible that ${\mathbf{b}}=\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}$, any algorithm that provides
a relative error bound must search inside a space that contains $\range{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}$.
This is a strong restriction (and plausibly one that actually requires
computing $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$ ).
\item Approximate PCR implies approximate PCP: if $\tilde{{\mathbf{x}}}$ is an $(\epsilon,\nu)$-approximate
PCR then $\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}$ is an $(\epsilon,\sigma_{k+1}\nu)$-approximate
PCP.
\item Our notion of approximate PCP is somewhat similar to the notion of
approximate PCP proposed recently by Allen-Zhu and Li~\cite{AL17}.
\item Yet another notion of approximate PCR appears in~\cite[Theorem 5]{BM14}.
They too, consider an additive error on objective function, but instead
of considering the distance to the dominant subspace they bound the
distance of the approximate solution to the true solution. We remark
that a bound on $\TNorm{{\mathbf{x}}_{k}-\tilde{{\mathbf{x}}}}$ trivially implies a bound
on $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}\tilde{{\mathbf{x}}}}$.
\item Arguably, it would have been preferable to require the approximate
PCR solution $\tilde{{\mathbf{x}}}$ to be such that $\TNorm{{\mathbf{x}}_{k}-\tilde{{\mathbf{x}}}}$
is small (relative to $\TNorm{{\mathbf{x}}_{k}}$). However, be believe that
providing such guarantees with reasonable sketch sizes requires iterations.
In this paper, we focus predominately on algorithms that do not require
iterations (the only exception being the input sparsity algorithm
in subsection~\ref{subsec:input-sparsity}).
\end{enumerate}
We are now ready to state general conditions on $\mat R$ that ensure
deterministically that ${\mathbf{x}}_{\mat R}$ is an approximate PCR, and conditions
on $\mat R$ that ensure deterministically that $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}$
is an approximate PCP.
\begin{thm}
\label{thm:structural}Suppose that $\mat R\in\mathbb{R}^{d\times s}$ where
$s\ge k$. Assume that $\nu\in(0,1)$.
\begin{enumerate}
\item If $d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu$
then $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}$ is an $(\nu,\nu)$-approximate PCP.
\item \label{enu:structural_k_equal_s}If $s=k$, $\mat R$ has orthonormal
columns (i.e., $\mat R^{\textsc{T}}\mat R=\mat I_{k})$ and $d_{2}\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\right)\leq\nu(1+\nu^{2})^{-1/2}$
then ${\mathbf{x}}_{\mat R}$ is an $\left(\frac{\sigma_{k+1}}{\sigma_{k}}\nu,\frac{\nu}{\left(\sqrt{1-\nu^{2}}-\nu\right)\sigma_{k}}\right)$-approximate
PCR.
\end{enumerate}
\end{thm}
Before proving this theorem, we state a theorem which is a corollary
of a more general result proved recently by Drineas et al.~\cite{drineas2016structural},
and then proceed to proving a couple of auxiliary lemmas.
\begin{thm}
[Corollary of Theorem 2.1 in \cite{drineas2016structural}]\label{thm:krylov_srtuctural}
Let $\ensuremath{{\bm{\mathrm{A}}}}$ be an $m\times n$ matrix with singular value decomposition
$\ensuremath{{\bm{\mathrm{A}}}}=\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}^{\mat T}$ .
Let $k\ge0$ and let $\mat R\in\mathbb{R}^{d\times k}$ be any matrix such
that $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\mat T}\mat R$ has full rank. Then,
\end{thm}
\[
\TNorm{\sin\Theta(\ensuremath{{\bm{\mathrm{A}}}}\mat R,\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})}\leq\TNorm{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}},k+}}\cdot\TNorm{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{-1}}\cdot\TNorm{\tan\Theta(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})}
\]
\begin{lem}
\label{lem:11}Assume $\rank{\ensuremath{{\bm{\mathrm{A}}}}}\ge k$. If $\mat R\in\mathbb{R}^{d\times k}$
has orthonormal columns and $d_{2}\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\right)\leq\nu$
then the following bounds hold:
\begin{equation}
\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R}\leq\nu\label{eq:orho-to-minor}
\end{equation}
\begin{equation}
\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{A}}}}\mat R\right)\ge\sigma_{k}\left(\sqrt{1-\nu^{2}}-\nu\right)\label{eq:singular_values_bound}
\end{equation}
Furthermore, if $\nu<1$ then $\rank{\ensuremath{{\bm{\mathrm{A}}}}\mat R}=k$.
\end{lem}
\begin{proof}
Since both $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$ and $\mat R$ have orthonormal columns,
$d_{2}\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\right)\leq\nu$ implies that the
square of the singular values of $\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R$ lie
inside the interval $[1-\nu^{2},1]$. The eigenvalues of $\mat R^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R$
are exactly the square of the singular values of $\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R$,
so the eigenvalues of $\mat I_{k}-\mat R^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R$
lie in $[0,\nu^{2}]$. Let $\mat Z$ be any matrix with orthonormal
columns that completes $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}$ to a basis (i.e. $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}^{\textsc{T}}+\mat Z\matZ^{\textsc{T}}=\mat I_{d}$)
and is orthogonal to $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}$ (i.e., $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}^{\textsc{T}}\mat Z=0$).
Note that $\mat Z$ can be an empty matrix if $d\leq n$. Denote $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k\perp}=\left[\begin{array}{cc}
\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+} & \mat Z\end{array}\right]$. We have
\begin{eqnarray*}
\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k\perp}^{\textsc{T}}\mat R} & = & \sqrt{\TNorm{\mat R^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k\perp}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k\perp}^{\textsc{T}}\mat R}}\\
& = & \sqrt{\TNorm{\mat I_{k}-\mat R^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R}}\\
& \leq & \nu
\end{eqnarray*}
where we used the fact that $\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}+\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k\perp}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k\perp}^{\textsc{T}}=\mat I_{d}$.
We now note that $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}\mat R$ is a submatrix of $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k\perp}^{\textsc{T}}$
so $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R}\leq\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k\perp}^{\textsc{T}}\mat R}\leq\nu$.
This establishes the first part of the theorem.
As for the second part, recall the following identities: 1) for any
matrix $\ensuremath{{\bm{\mathrm{X}}}}$ and $\mat Y$ of the same size: $\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{X}}}}\pm\mat Y\right)\ge\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{X}}}}\right)-\sigma_{\max}\left(\mat Y\right)$
\cite[Theorem 3.3.19]{horn1990matrix}, 2) if the number of rows in
$\ensuremath{{\bm{\mathrm{X}}}}$ and $\mat Y$ is at least as large as the number of columns,
and $\ensuremath{{\bm{\mathrm{X}}}}\mat Y$ is defined, then $\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{X}}}}\mat Y\right)\ge\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{X}}}}\right)\sigma_{\min}\left(\mat Y\right)$.
We have
\begin{eqnarray*}
\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{A}}}}\mat R\right) & = & \sigma_{\min}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R+\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R)\\
& \geq & \sigma_{\min}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R)-\sigma_{\max}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R)\\
& \geq & \sigma_{\min}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k})\sigma_{\min}(\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R)-\sigma_{\max}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+})\sigma_{\max}(\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R)\\
& = & \sigma_{k}\sigma_{\min}(\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}\mat R)-\sigma_{k+1}\sigma_{\max}(\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R)\\
& \geq & \sigma_{k}\sqrt{1-\nu^{2}}-\sigma_{k+1}\nu\\
& \geq & \sigma_{k}(\sqrt{1-\nu^{2}}-\nu)
\end{eqnarray*}
where the first equality follows from the fact that $\ensuremath{{\bm{\mathrm{A}}}}(\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}^{\textsc{T}}+\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}})=\ensuremath{{\bm{\mathrm{A}}}}$.
When $\nu<1$ we have $\sigma_{\min}(\ensuremath{{\bm{\mathrm{A}}}}\mat R)>0$, so indeed the
rank of $\ensuremath{{\bm{\mathrm{A}}}}\mat R$ is $k$.
\end{proof}
\begin{lem}
\label{lem:14}Assume $\rank{\ensuremath{{\bm{\mathrm{A}}}}}\ge k$. Suppose that $\mat R\in\mathbb{R}^{d\times k}$
has orthonormal columns, and that $d_{2}\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\right)\leq\nu(1+\nu^{2})^{-1/2}<1$.
We have
\[
d_{2}(\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R},\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})\leq\frac{\sigma_{k+1}}{\sigma_{k}}\nu\,.
\]
\end{lem}
\begin{proof}
Since $\rank{\ensuremath{{\bm{\mathrm{A}}}}}\ge k$ and $\nu(1+\nu^{2})^{-1/2}<1$, according
to Lemma \ref{lem:11} the matrix $\ensuremath{{\bm{\mathrm{A}}}}\mat R$ has full rank. According
to Theorem~ \ref{thm:projection_equality} and the fact that $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R}$
and $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}$ are orthogonal projections we have
\begin{equation}
d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R},\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}\right)=d_{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\mat R,\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}\right)=\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}}=\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R}^{\perp}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}}\label{eq:projection_equality_k0}
\end{equation}
Combining Theorem \ref{thm:krylov_srtuctural} and Eq. (\ref{eq:projection_equality_k0}),
we bound:
\begin{eqnarray*}
d_{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\mat R,\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}\right) & = & \TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R}^{\perp}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}}\\
& = & \TNorm{(\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R})\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}\\
& = & \TNorm{\sin\Theta(\ensuremath{{\bm{\mathrm{A}}}}\mat R,\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})}\\
& \leq & \TNorm{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}.k+}}\cdot\TNorm{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}.k}^{-1}}\cdot\TNorm{\tan\Theta(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})}\\
& = & \frac{\sigma_{k+1}}{\sigma_{k}}\cdot\TNorm{\tan\Theta(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})}\\
& \leq & \frac{\sigma_{k+1}}{\sigma_{k}}\nu
\end{eqnarray*}
where the last inequality follows from the fact that $\Theta(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})$
is a diagonal matrix whose diagonal values are the inverse cosine
of the singular values of $\mat R^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$, and these,
in turn, are all larger than $\sqrt{1-\nu^{2}(1+\nu^{2})^{-1}}.$
\end{proof}
We are now ready to prove Theorem~\ref{thm:structural}.
\begin{proof}
[Proof of Theorem~\ref{thm:structural}]We need to show both the additive
error bounds on the objective function, and the error bound on the
constraints. We start with the additive error bounds on the objective
function, both for PCP (first part of the theorem) and for PCR (second
part of the theorem). We have
\[
\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}=\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\left(\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\right)^{+}{\mathbf{b}}=\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k}}{\mathbf{b}}
\]
and
\[
\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}=\mat A\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})^{+}{\mathbf{b}}=\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}}{\mathbf{b}}\,.
\]
Thus,
\begin{eqnarray*}
\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-{\mathbf{b}}} & = & \TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}-{\mathbf{b}}+\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}}\\
& = & \TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}-{\mathbf{b}}}\pm\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}}\\
& = & \TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}-{\mathbf{b}}}\pm\TNorm{(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}){\mathbf{b}}}\\
& = & \TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}-{\mathbf{b}}}\pm d_{2}(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k})\cdot\TNorm{{\mathbf{b}}}
\end{eqnarray*}
In the first part of the theorem, we have $d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu$,
while the second part of the theorem we have $\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}=\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R}$
(since $\ensuremath{{\bm{\mathrm{A}}}}\mat R$ has $k$ columns) and Lemma~\ref{lem:14} ensures
that $d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu\sigma_{k+1}/\sigma_{k}$.
Either way, the additive error bounds of the theorem are met.
We now bound the infeasibility of the approximate solution for the
PCP guarantee (first part of the theorem):
\begin{eqnarray*}
\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}} & = & \TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}+\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}}\\
& \leq & \TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\left(\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}\right)}+\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}}\\
& \leq & \TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}}\\
& \leq & d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\TNorm{{\mathbf{b}}}\\
& \leq & \nu\TNorm{{\mathbf{b}}}
\end{eqnarray*}
where we used the fact that $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}\in\range{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}$
so $\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}=0$.
We now bound the infeasibility of the approximate solution for the
PCR guarantee (second part of the theorem):
\begin{eqnarray*}
\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}{\mathbf{x}}_{\mat R}} & = & \TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{+}{\mathbf{b}}}\\
& \leq & \TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R}\cdot\TNorm{(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{+}}\cdot\TNorm{{\mathbf{b}}}\\
& \leq & \frac{\nu}{\left(\sqrt{1-\nu^{2}}-\nu\right)\sigma_{k}}\TNorm{{\mathbf{b}}}
\end{eqnarray*}
where we used Lemma~\ref{lem:11} to bound $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R}$
and $\TNorm{(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{+}}$.
\end{proof}
\subsection{\label{subsec:statistical}Statistical Perspective}
We now consider ${\mathbf{x}}_{\mat R,k}$ from a statistical perspective. We
use a similar framework to the one used in the literature to analyze
CLS~\cite{slawski2017compressed,Slawski17,THM17}. That is, we consider
a fixed design setting in which the rows of $\ensuremath{{\bm{\mathrm{A}}}}$, ${\mathbf{a}}_{1},\dots,{\mathbf{a}}_{n}\in\mathbb{R}^{d}$,
are considered as fixed, and ${\mathbf{b}}$'s entries, $b_{1},\dots,b_{n}\in\mathbb{R}$,
are
\[
b_{i}=f_{i}+\xi_{i}
\]
where $f_{1},\dots,f_{n}$ are fixed values and the noise terms $\xi_{1},\dots,\xi_{n}$
are assumed to be independent random values with zero mean and $\sigma^{2}$
variance. We denote by ${\mathbf{f}}\in\mathbb{R}^{n}$ the vector whose $i$th entry
is $f_{i}$ . The goal is to recover ${\mathbf{f}}$ from ${\mathbf{b}}$ (i.e., de-noise
${\mathbf{b}}$).
The optimal predictor $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}$ of ${\mathbf{f}}$ given $\ensuremath{{\bm{\mathrm{A}}}}$ is
a minimizer of
\[
\min_{{\mathbf{x}}\in\mathbb{R}^{d}}\Expect{\TNormS{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}-{\mathbf{b}}}/n}
\]
where here, and in subsequent expressions, the expectation is with
respect to the noise $\xi$ (if there are multiple minimizers, ${\mathbf{x}}^{\star}$
is the minimizer with minimum norm). It is easy to verify that $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}=\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}{\mathbf{f}}$.
Given an estimator $\theta=\theta(\ensuremath{{\bm{\mathrm{A}}}},{\mathbf{b}})$ of ${\mathbf{x}}^{\star}$ (which
we assume is a random variable since ${\mathbf{b}}$ is a random varaible),
its \emph{excess risk} is define as
\[
{\cal E}(\theta)\coloneqq\Expect{\TNormS{\ensuremath{{\bm{\mathrm{A}}}}\theta-\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}}/n}\,.
\]
The \emph{ordinary least square estimator }(OLS) $\hat{{\mathbf{x}}}$ is simply
a solution to $\min_{{\mathbf{x}}\in\mathbb{R}^{d}}\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}-{\mathbf{b}}}$: $\hat{{\mathbf{x}}}\coloneqq\ensuremath{{\bm{\mathrm{A}}}}^{+}{\mathbf{b}}$.
Simple calculations show that
\[
{\cal E}(\hat{{\mathbf{x}}})=\sigma^{2}\rank{\ensuremath{{\bm{\mathrm{A}}}}}/n\,.
\]
Thus, if the rank of $\ensuremath{{\bm{\mathrm{A}}}}$ is large, which is usually the case
when $d\gg n$, then the excess risk might be large (and it does not
asymptotically converge to $0$ if $\rank{\ensuremath{{\bm{\mathrm{A}}}}}=\Omega(n)$). This
motivates the use of regularization (e.g., PCR). Indeed, the excess
risk of the PCR estimator ${\mathbf{x}}_{k}$ can be bounded~\cite{Slawski17}:
\begin{equation}
{\cal E}({\mathbf{x}}_{k})\leq\frac{\InfNormS{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}^{\textsc{T}}{\mathbf{x}}^{\star}}\cdot\sum_{i=k+1}^{\min(n,d)}\sigma_{i}^{2}}{n}+\frac{\sigma^{2}k}{n}\,.\label{eq:pcr-old-risk-bound}
\end{equation}
In many scenarios, ${\mathbf{x}}_{k}$ has a significantly reduced excess risk
in comparison to the excess risk of $\hat{{\mathbf{x}}}$ (see~\cite{Slawski17}
for a discussion). This motivates the use of PCR when $d$ is large.
In this section, we analyze the excess risk of ${\mathbf{x}}_{\mat R,k}$ based
on properties of $\mat R$. The bounds are based on the following identity~\cite{Slawski17}\footnote{However, no proof of (\ref{eq:bias-var}) appears in~\cite{Slawski17},
so for completeness we include a proof in the appendix. }: for any $\ensuremath{{\bm{\mathrm{M}}}}$ of appropriate size
\begin{equation}
{\cal E}({\mathbf{x}}_{\ensuremath{{\bm{\mathrm{M}}}}})={\cal E}(\ensuremath{{\bm{\mathrm{M}}}}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{M}}}})^{\textsc{+}}{\mathbf{b}})=\underset{{\cal B}({\mathbf{x}}_{\ensuremath{{\bm{\mathrm{M}}}}})}{\underbrace{\frac{1}{n}\TNormS{(\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{M}}}}})\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}}}}+\underset{{\cal V}({\mathbf{x}}_{\ensuremath{{\bm{\mathrm{M}}}}})}{\underbrace{\sigma^{2}\frac{\rank{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{M}}}}}}{n}}}\,.\label{eq:bias-var}
\end{equation}
In the above, ${\cal B}({\mathbf{x}}_{\ensuremath{{\bm{\mathrm{M}}}}})$ can be viewed as a bias term,
and ${\cal V}({\mathbf{x}}_{\ensuremath{{\bm{\mathrm{M}}}}})$ can be viewed as a variance term. Eq.~(\ref{eq:pcr-old-risk-bound})
is obtained by bounding the bias term ${\cal B}({\mathbf{x}}_{k})$, although
our results lead to a bound on ${\cal E}({\mathbf{x}}_{k})$ that is tighter
in some cases (Corollary~\ref{cor:new-pcr-risk}). An immediate corollary
of~(\ref{eq:bias-var}) is the following bound for ${\mathbf{x}}_{\mat R,k}$:
\begin{equation}
{\cal E}({\mathbf{x}}_{\mat R,k})=\frac{1}{n}\TNormS{(\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}})\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}}+\frac{\sigma^{2}k}{n}\,.\label{eq:pcp_risk}
\end{equation}
The following results addresses the case where $\mat R$ has $k$ orthonormal
columns. The conditions are the same as the first part of Theorem~\ref{thm:structural}
(optimization perspective analysis).
\begin{thm}
\label{thm:stat-structural}Assume that $\rank{\ensuremath{{\bm{\mathrm{A}}}}}\geq k$. Suppose
that $\mat R\in\mathbb{R}^{d\times k}$ has orthonormal columns, and that $d_{2}(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})\leq\nu(1+\nu^{2})^{-1/2}<1$.
Then,
\[
{\cal E}({\mathbf{x}}_{\mat R})\leq\frac{\left(1+\nu\right)\cdot\TNormS{{\mathbf{x}}^{\star}}\cdot\sigma_{k+1}^{2}}{n}+\frac{\sigma^{2}k}{n}
\]
\end{thm}
For the proof, we need the following theorem due to Halko et al.~\cite{halko2011finding}.
\begin{thm}
[Theorem 9.1 in \cite{halko2011finding}]\label{thm:hmt_structural}
Let $\ensuremath{{\bm{\mathrm{A}}}}$ be an $m\times n$ matrix with singular value decomposition
$\ensuremath{{\bm{\mathrm{A}}}}=\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}^{\mat T}$ .Let
$k\ge0$. Let $\mat R$ be any matrix such that $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\mat T}\mat R$
has full row rank. Then we have
\[
\TNormS{(\mat I_{m}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R})\ensuremath{{\bm{\mathrm{A}}}}}\leq\TNormS{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}},k+}}+\TNormS{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}},k+}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\textsc{T}}\mat R\left(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\textsc{T}}\mat R\right)^{+}}
\]
\end{thm}
\begin{proof}
[Proof of Theorem~\ref{thm:stat-structural}]The condition that $d_{2}(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})\leq\nu(1+\nu^{2})^{-1/2}<1$
ensures that $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\textsc{T}}\mat R$ has full rank, and that
$\TNormS{\tan\Theta(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})}\leq\nu$ (since $\Theta(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})$
is a diagonal matrix whose diagonal values are the inverse cosine
of the singular values of $\mat R^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}$, and these, in turn,
are all larger than $\sqrt{1-\nu^{2}(1+\nu^{2})^{-1}}$). Thus we
have,
\begin{eqnarray*}
{\cal B}({\mathbf{x}}_{\mat R}) & = & \frac{1}{n}\TNormS{(\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R})\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}}\\
& \leq & \frac{1}{n}\TNormS{{\mathbf{x}}^{\star}}\cdot\left(\TNormS{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}},k+}}+\TNormS{\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}},k+}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\mat T}\mat R\left(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\mat T}\mat R\right)^{+}}\right)\\
& \leq & \frac{1}{n}\TNormS{{\mathbf{x}}^{\star}}\left(\sigma_{k+1}^{2}+\sigma_{k+1}^{2}\TNormS{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k+}^{\mat T}\mat R\left(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\mat T}\mat R\right)^{+}}\right)\\
& = & \frac{1}{n}\TNormS{{\mathbf{x}}^{\star}}\left(\sigma_{k+1}^{2}+\sigma_{k+1}^{2}\TNormS{\tan\Theta\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}\right)}\right)\\
& \leq & \frac{\left(1+\nu\right)\cdot\TNormS{{\mathbf{x}}^{\star}}\cdot\sigma_{k+1}^{2}}{n}
\end{eqnarray*}
where in the first inequality we used Theorem~\ref{thm:hmt_structural}
and for the second equality we used Lemma~\ref{lem:tan-to-spectral}.
The result now follows from the fact that $\rank{\ensuremath{{\bm{\mathrm{A}}}}\mat R}\leq k$.
\end{proof}
\begin{cor}
\label{cor:new-pcr-risk}For the PCR solution ${\mathbf{x}}_{k}$ we have
\[
{\cal E}({\mathbf{x}}_{k})\leq\frac{\TNormS{{\mathbf{x}}^{\star}}\cdot\sigma_{k+1}^{2}}{n}+\frac{\sigma^{2}k}{n}\,.
\]
\end{cor}
Next, we consider the general case where $\mat R$ does not necessarily
have orthonormal columns, and potentially has more than $k$ columns.
The conditions are the same as the second part of Theorem~\ref{thm:structural}
(optimization perspective).
\begin{thm}
\label{thm:struct-stat-pcp}Suppose that $\mat R\in\mathbb{R}^{d\times s}$
where $s\ge k$. Assume that $\rank{\ensuremath{{\bm{\mathrm{A}}}}\mat R}\geq k$. If $d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu<1$
then,
\[
{\cal E}({\mathbf{x}}_{\mat R,k})\leq{\cal E}({\mathbf{x}}_{k})+\frac{(2\nu+\nu^{2})\TNormS{{\mathbf{f}}}}{n}\,.
\]
\end{thm}
\begin{proof}
Since $\ensuremath{{\bm{\mathrm{A}}}}\mat R$ has rank at least $k$, we have $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}=\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}$.
From (\ref{eq:pcp_risk}), the fact that $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}=\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}{\mathbf{f}}$,
and $\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}=\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}$
(since the range of $\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$ is contained
in the range of $\ensuremath{{\bm{\mathrm{A}}}}$) we have
\begin{eqnarray*}
{\cal B}({\mathbf{x}}_{\mat R,k}) & = & \frac{1}{n}\TNormS{(\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}})\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}^{\star}}\\
& = & \frac{1}{n}\TNormS{(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}){\mathbf{f}}}\\
& = & \frac{1}{n}\TNormS{(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}+\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}){\mathbf{f}}}\\
& = & \frac{1}{n}\left(\TNormS{(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}){\mathbf{f}}}+\TNormS{(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}){\mathbf{f}}}+2(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}{\mathbf{f}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}{\mathbf{f}})^{\textsc{T}}(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}{\mathbf{f}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}{\mathbf{f}})\right)\\
& \leq & {\cal B}({\mathbf{x}}_{k})+d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)^{2}\frac{\TNormS{{\mathbf{f}}}}{n}+\frac{2}{n}\left|{\mathbf{f}}^{\textsc{T}}(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}})^{\textsc{T}}(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}){\mathbf{f}}\right|
\end{eqnarray*}
For the cross-terms, we bound
\begin{eqnarray*}
\left|{\mathbf{f}}^{\textsc{T}}(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}})^{\textsc{T}}(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}){\mathbf{f}}\right| & = & \left|{\mathbf{f}}^{\textsc{T}}\left(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}+\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}\right){\mathbf{f}}\right|\\
& = & \left|{\mathbf{f}}^{\textsc{T}}\left(\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}+\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}\right){\mathbf{f}}\right|\\
& = & {\mathbf{f}}^{\textsc{T}}\left(\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}\right)\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}{\mathbf{f}}\\
& = & {\mathbf{f}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}^{\perp}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}{\mathbf{f}}\\
& \leq & \TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}^{\perp}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}}\cdot\TNormS{{\mathbf{f}}}
\end{eqnarray*}
Since both $\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k}$ and $\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}$ are full
rank,\textcolor{red}{{} }we have (Theorem~\ref{thm:projection_equality})
\[
\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}^{\perp}\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}}=\TNorm{\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}}-\ensuremath{{\bm{\mathrm{P}}}}_{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}}=d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)
\]
Thus, we find that
\[
{\cal B}({\mathbf{x}}_{\mat R,k})\leq{\cal B}({\mathbf{x}}_{k})+(2\nu+\nu^{2})\frac{\TNormS{{\mathbf{f}}}}{n}\,.
\]
We reach the bound in the theorem statement by adding the variance
${\cal V}({\mathbf{x}}_{\mat R,k})$, which is equal to the variance of ${\mathbf{x}}_{k}$
because the ranks are equal.
\end{proof}
\paragraph{Discussion. }
Theorem~\ref{thm:stat-structural} shows that if $\mat R$ is a good
approximation to $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$, then there is a small relative
increase to the bias term, while the variance term does not change.
Since we are mainly interested in keeping the asymptotic behavior
of the excess risk (as $n$ goes to infinity), a fixed $\nu$ of modest
value suffices. However, for this result to hold, $\mat R$ has to
have exactly $k$ columns and those columns should be orthonormal.
Without these restrictions, we need to resort to Theorem~\ref{thm:struct-stat-pcp}.
In that theorem, we get (if the conditions are met) only an additive
increase in the bias term. Thus if, for example, $\TNormS{{\mathbf{f}}}/n\to c$
as $n\to\infty$ for some constant $c$, then $\nu$ should tend to
$0$ as $n$ goes to infinity, but a constant value should suffice
if $n$ is fixed.
\section{\label{sec:alg}Sketched PCR and PCP}
In the previous section, we considered general conditions on $\mat R$
which ensure that ${\mathbf{x}}_{\mat R,k}$ is an approximate solution to the
PCR/PCP problem. In this section, we propose algorithms to generate
$\mat R$ for which these conditions hold. The main technique we employ
is \emph{matrix sketching}. The idea is to first multiply the data
matrix $\ensuremath{{\bm{\mathrm{A}}}}$ by some random transformation (e.g., a random projection),
and extract an approximate subspace from the compressed matrix.
\subsection{Dimensionality Reduction using Sketching }
The compression (multiplication by a random matrix) alluded in the
previous paragraph can be applied either from the left side, or the
right side, or both. In left sketching, which is more appropriate
if the input matrix has many rows and a modest amount of columns,
we propose to use $\mat R=\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}},k}$ where $\mat S$ is
some sketching matrix (we discuss a couple of options shortly). In
right sketching, which is more appropriate if the input matrix has
many columns and a modest amount of rows, we propose to use $\mat R=\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$
where $\ensuremath{{\bm{\mathrm{G}}}}$ is some sketching matrix. Two sided sketching, $\mat R=\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k}$
, is aimed for the case that the number of columns and the number
of rows are large.
The sketching matrices, $\mat S$ and $\ensuremath{{\bm{\mathrm{G}}}}$, are randomized dimensionality
reduction transformations. Quite a few sketching transforms have been
proposed in the literature in recent years. For concreteness, we consider
two specific cases, though our results hold for other sketching transformations
as well (albeit some modifications in the bounds might be necessary).
The first, which we refer to as 'subgaussian map', is a random matrix
in which every entry of the matrix is sampled i.i.d from some subgaussian
distribution (e.g. $N(0,1)$) and the matrix is appropriately scaled
(however, scaling is not necessary in our case). The second transform
is a sparse embedding matrix, in which each column is sampled uniformly
and independently from the set of scaled identity vectors and multiplied
by a random sign. We refer to such a matrix as a \noun{CountSketch}
matrix~\cite{charikar2002finding,woodruff2014sketching}.
Both transformations described above, and a few other, have, provided
enough rows are used, with high probability the following property
which we refer to as \emph{approximate Gram property}.
\begin{defn}
Let $\ensuremath{{\bm{\mathrm{X}}}}\in\mathbb{R}^{m\times n}$ be a fixed matrix. For $\epsilon,\delta\in(0,1/2)$,
a distribution ${\cal D}$ on matrices with $m$ columns has the \emph{$(\epsilon,\delta)$-approximate
Gram matrix} property for $\ensuremath{{\bm{\mathrm{X}}}}$ if
\[
\Pr_{\mat S\sim{\cal D}}\left(\TNorm{\ensuremath{{\bm{\mathrm{X}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{X}}}}-\ensuremath{{\bm{\mathrm{X}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{X}}}}}\geq\epsilon\TNormS{\ensuremath{{\bm{\mathrm{X}}}}}\right)\leq\delta\,.
\]
\end{defn}
Recent results by Cohen et al.~\cite{cohen2015optimal}\footnote{Theorem 1 in \cite{cohen2015optimal} with $k=\sr{\ensuremath{{\bm{\mathrm{X}}}}}.$}
show that when $\mat S$ has independent subgaussian entries, then
as long as the number of rows in $\mat S$ is $\Omega((\sr{\ensuremath{{\bm{\mathrm{X}}}}}+\log(1/\delta))/\epsilon^{2})$
then we have $(\epsilon,\delta)$-approximate Gram property for $\ensuremath{{\bm{\mathrm{X}}}}$.
If $\mat S$ is a \noun{CountSketch} matrix, then as long as the number
of rows in $\mat S$ is $\Omega(\sr{\ensuremath{{\bm{\mathrm{X}}}}}^{2}/(\epsilon^{2}\delta))$
then we have $(\epsilon,\delta)$-approximate Gram property for $\ensuremath{{\bm{\mathrm{X}}}}$~\cite{cohen2015optimal}.
We first describe our results for the various modes of sketching,
and then discuss algorithmic issues and computational complexity.
\begin{thm}
[Left Sketching] Let $\nu,\delta\in(0,1/2)$ and denote
\[
\epsilon=\frac{\nu(1+\nu^{2})^{-1/2}}{1+\nu(1+\nu^{2})^{-1/2}}\cdot\gap{\ensuremath{{\bm{\mathrm{A}}}}}k.
\]
Suppose that $\mat S$ is sampled from a distribution that provides
a $(\epsilon,\delta)$-approximate Gram matrix for $\ensuremath{{\bm{\mathrm{A}}}}$. Then
for $\mat R=\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}},k}$, with probability $1-\delta$, the
approximate solution ${\mathbf{x}}_{\mat R}$ is a $\left(\frac{\sigma_{k+1}}{\sigma_{k}}\nu,\frac{\nu}{\left(\sqrt{1-\nu^{2}}-\nu\right)\sigma_{k}}\right)$-approximate
PCR and
\[
{\cal E}({\mathbf{x}}_{\mat R})\leq\frac{\left(1+\nu\right)\cdot\TNormS{{\mathbf{x}}^{\star}}\cdot\sigma_{k+1}^{2}}{n}+\frac{\sigma^{2}k}{n}
\]
Thus if, for example, $\mat S$ is a \noun{CountSketch} matrix, then
the conditions are met when the number of rows in $\mat S$ is
\[
\Omega\left(\frac{\sr{\ensuremath{{\bm{\mathrm{A}}}}}^{2}}{\gap{\ensuremath{{\bm{\mathrm{A}}}}}k^{2}\nu^{2}\delta}\right)
\]
rows. In another example, if $\mat S$ is a subgaussian map, then
the conditions are met when the number of rows in $\mat S$ is
\[
\Omega\left(\frac{\sr{\ensuremath{{\bm{\mathrm{A}}}}}+\log(1/\delta)}{\gap{\ensuremath{{\bm{\mathrm{A}}}}}k^{2}\nu^{2}}\right)\,.
\]
\end{thm}
\begin{proof}
Due to Theorems~\ref{thm:structural} and~\ref{thm:stat-structural},
it suffices to show that that $d_{2}\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\right)\leq\nu(1+\nu^{2})^{-1/2}$.
Under the conditions of the theorem, with probability of at least
$1-\delta$ we have $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}-\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}}\leq\epsilon\TNormS{\ensuremath{{\bm{\mathrm{A}}}}}$.
If that is indeed the case, $\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}$ has
rank at least $k$ since $\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}$ and $\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}$
are symmetric matrices and we know that $\sigma_{i}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\right)=\sigma_{i}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}\right)\pm\TNorm{\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}-\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}}$
(Weyl's Theorem and the fact that $\sigma_{k}^{2}>\epsilon\sigma_{1}^{2}$).
Furthermore, since $\nu>0$ we have $\epsilon<\gap{\ensuremath{{\bm{\mathrm{A}}}}}k$, and
Theorem~\ref{thm:sin-theta-1} implies that
\begin{eqnarray*}
d_{2}(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}) & \leq & \frac{\TNorm{\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}-\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}}}{(\sigma_{k}^{2}-\sigma_{k+1}^{2})-\TNorm{\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}-\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}}}\\
& \leq & \frac{\epsilon}{\gap{\ensuremath{{\bm{\mathrm{A}}}}}k-\epsilon}\\
& \leq & \nu(1+\nu^{2})^{-1/2}\,.
\end{eqnarray*}
Thus, we have shown that with probability $1-\delta$ we have $d_{2}\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\right)\leq\nu(1+\nu^{2})^{-1/2}$,
as required.
\end{proof}
\begin{thm}
[Right Sketching]\label{thm:right_sketching}Let $\nu,\delta\in(0,1/2)$
and denote
\[
\epsilon=\frac{\nu}{1+\nu}\cdot\gap{\ensuremath{{\bm{\mathrm{A}}}}}k.
\]
Suppose that $\ensuremath{{\bm{\mathrm{G}}}}$ is sampled from a distribution that provides
a $(\epsilon,\delta)$-approximate Gram matrix for $\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}$.
Then for $\mat R=\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$, with probability $1-\delta$, the approximate
solution $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}$ is an $(\nu,\nu)$-approximate PCP and
\[
{\cal E}({\mathbf{x}}_{\mat R,k})\leq{\cal E}({\mathbf{x}}_{k})+\frac{(2\nu+\nu^{2})\TNormS{{\mathbf{f}}}}{n}
\]
Thus if, for example, $\ensuremath{{\bm{\mathrm{G}}}}$ is a \noun{CountSketch} matrix, then
the conditions are met when the number of rows in $\ensuremath{{\bm{\mathrm{G}}}}$ is
\[
\Omega\left(\frac{\sr{\ensuremath{{\bm{\mathrm{A}}}}}^{2}}{\gap{\ensuremath{{\bm{\mathrm{A}}}}}k^{2}\nu^{2}\delta}\right)
\]
rows. In another example, if $\ensuremath{{\bm{\mathrm{G}}}}$ is a subgaussian map, then
the conditions are met when the number of rows in $\ensuremath{{\bm{\mathrm{G}}}}$ is
\[
\Omega\left(\frac{\sr{\ensuremath{{\bm{\mathrm{A}}}}}+\log(1/\delta)}{\gap{\ensuremath{{\bm{\mathrm{A}}}}}k^{2}\nu^{2}}\right)\,.
\]
\end{thm}
\begin{proof}
Due to Theorem~\ref{thm:structural} it suffices to show that that
$d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu$.
Under the conditions of the Theorem, with probability of at least
$1-\delta$ we have $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}}\leq\epsilon\TNormS{\ensuremath{{\bm{\mathrm{A}}}}}$.
If that is indeed the case, $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}$ has
rank at least $k$ since $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}$ and $\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}$
are symmetric matrices and we know that $\sigma_{i}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\right)=\sigma_{i}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}\right)\pm\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}}$
(Weyl's Theorem and the fact that $\sigma_{k}^{2}>\epsilon\sigma_{1}^{2}$).
Furthermore, since $\nu>0$ we have $\epsilon<\gap{\ensuremath{{\bm{\mathrm{A}}}}}k$, and
Theorem~\ref{thm:sin-theta-1} implies
\begin{eqnarray*}
d_{2}(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}) & \leq & \frac{\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}}}{(\sigma_{k}^{2}-\sigma_{k+1}^{2})-\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}}}\\
& \leq & \frac{\epsilon}{\gap{\ensuremath{{\bm{\mathrm{A}}}}}k-\epsilon}\\
& \leq & \nu\,.
\end{eqnarray*}
Thus, we have shown that with probability $1-\delta$ we have $d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu$,
as required.
\end{proof}
\begin{thm}
[Two Sided Sketching]\label{thm:left_right_sketching}Let $\nu,\delta\in(0,1/2)$
and denote
\[
\epsilon_{2}=\frac{\nu}{2(1+\nu/2)}\cdot\gap{\ensuremath{{\bm{\mathrm{A}}}}}k.
\]
Suppose that $\ensuremath{{\bm{\mathrm{G}}}}$ is sampled from a distribution that provides
a $(\epsilon_{2},\delta/2)$-approximate Gram matrix for $\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}$.
Denote
\[
\epsilon_{1}=\frac{\nu(1+\nu^{2}/4)^{-1/2}/2}{1+\nu(1+\nu^{2}/4)^{-1/2}/2}\cdot\gap{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}k
\]
and uppose that $\mat S$ is sampled from a distribution that provides
a $(\epsilon_{1},\delta/2)$-approximate Gram matrix for $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$.
Then for $\mat R=\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k}$ with probability
$1-\delta$ the approximate solution $\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}$ is an $(\nu,\nu)$-approximate
PCP and
\[
{\cal E}({\mathbf{x}}_{\mat R,k})\leq{\cal E}({\mathbf{x}}_{k})+\frac{(2\nu+\nu^{2})\TNormS{{\mathbf{f}}}}{n}
\]
Thus if, for example, $\mat S$ is a \noun{CountSketch} matrix and
$\ensuremath{{\bm{\mathrm{G}}}}$ is a subgaussian map, then the conditions hold when the numbers
of rows of $\mat S$ is
\[
\Omega\left(\frac{\sr{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}^{2}}{\gap{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}k^{2}\nu^{2}\delta}\right)
\]
and the number of rows in $\ensuremath{{\bm{\mathrm{G}}}}$ is
\[
\Omega\left(\frac{\sr{\ensuremath{{\bm{\mathrm{A}}}}}+\log(1/\delta)}{\gap{\ensuremath{{\bm{\mathrm{A}}}}}k^{2}\nu^{2}}\right)\,.
\]
\end{thm}
\begin{proof}
Due to Theorem~\ref{thm:structural} it suffices to show that that
$d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu$.
Under the conditions of the Theorem, with probability of at least
$1-\delta/2$ we have $\TNorm{(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}-(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}\leq\epsilon_{1}\TNormS{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}$,
and with probability of at least $1-\delta/2$ we have $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}}\leq\epsilon_{2}\TNormS{\ensuremath{{\bm{\mathrm{A}}}}}$.
Thus, both inequalities hold with probability of at least $1-\delta$.
If that is indeed the case, $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}$ has
rank at least $k$ since $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}$ and $\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}$
are symmetric matrices and we know that $\sigma_{i}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\right)=\sigma_{i}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}\right)\pm\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{A}}}}\matA^{\textsc{T}}}$
(Weyl's Theorem and the fact that $\sigma_{k}^{2}>\epsilon_{2}\sigma_{1}^{2}$).
Moreover, $(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$
has rank at least $k$ since $(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$
and $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}$ are symmetric matrices and we
know that $\sigma_{i}^{2}\left((\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\right)=\sigma_{i}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}\right)\pm\TNorm{(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{G}}}}\ensuremath{{\bm{\mathrm{A}}}}^{\textsc{T}}}$
(Weyl's Theorem and the fact that $\sigma_{k}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\right)>\epsilon_{1}\sigma_{1}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\right)$).
Since $\nu>0$ we have $\epsilon_{1}<\gap{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}k$ and
Theorem~\ref{thm:sin-theta-1} implies
\begin{eqnarray*}
d_{2}(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k}},\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k}) & \leq & \frac{\TNorm{(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}-(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}}{\sigma_{k}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\right)-\sigma_{k+1}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\right)-\TNorm{(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\mat S^{\textsc{T}}\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}-(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}}\\
& \leq & \frac{\epsilon_{1}\TNormS{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}}{\sigma_{k}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\right)-\sigma_{k+1}^{2}\left(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\right)-\epsilon_{1}\TNormS{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}}\\
& = & \frac{\epsilon_{1}}{\gap{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}k-\epsilon_{1}}\\
& \leq & \nu(1+\nu^{2}/4)^{-1/2}/2
\end{eqnarray*}
From Lemma~\ref{lem:14} (with $\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$) we get that $d_{2}(\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k}},\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k})\leq\frac{\sigma_{k+1}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})}{\sigma_{k}(\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})}(\nu/2)\leq\nu/2$.
We now bound
\begin{eqnarray*}
d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right) & \leq & d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k,k}\right)+d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\\
& \leq & \nu
\end{eqnarray*}
where we similarly use Theorem \ref{thm:right_sketching} to bound
$d_{2}(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k})\leq\nu/2.$
Thus, we have shown that with probability $1-\delta$ we have $d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu$,
as required.
\end{proof}
\subsection{Fast Approximate PCR/PCP}
A prototypical algorithm for approximate PCR/PCP is to compute ${\mathbf{x}}_{\mat R,k}$
with some choice of sketching-based $\mat R$. There are quite a few
design choices that need to be made in order to turn this prototypical
algorithm into a concrete algorithm, e.g. whether to use left, right
or two sided sketching to form $\mat R$, and which sketch transform
to use. There are various tradeoffs, e.g. using \noun{CountSketch}
results in faster sketching, but usually requires larger sketch sizes.
Furthermore, in computing ${\mathbf{x}}_{\mat R,k}$ there are also algorithmic
choices to be made with respect to choosing the order of matrix multiplications:
in computing $(\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{\textsc{+}}{\mathbf{b}}$ should
we first compute $\ensuremath{{\bm{\mathrm{A}}}}\mat R$ and then multiply by $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$,
or vice versa? Likely, there is no one size fit all algorithm, and
different profiles of the input matrix (in particular, the size and
sparsity level) call for a different variant of the prototypical algorithm.
\begin{table}
\begin{centering}
{\footnotesize{}}%
\begin{tabular}{|c|c|c|c|}
\hline
& & {\footnotesize{}${\mathbf{x}}_{\mat R,k}$} & {\footnotesize{}CLS}\tabularnewline
\hline
\hline
{\footnotesize{}Left sketching} & {\footnotesize{}subgaussian $\mat S$} & {\footnotesize{}$s_{1}\cdot\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+s_{1}d\min(s_{1},d)+nk^{2}$} & {\footnotesize{}N/A}\tabularnewline
\cline{2-4}
{\footnotesize{}$\mat R=\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}},k}$} & \noun{\footnotesize{}CountSketch}{\footnotesize{} $\mat S$} & {\footnotesize{}$k\cdot\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+s_{2}d\min(s_{2},d)+nk^{2}$} & {\footnotesize{}N/A}\tabularnewline
\hline
{\footnotesize{}Right sketching} & {\footnotesize{}subgaussian $\ensuremath{{\bm{\mathrm{G}}}}$} & {\footnotesize{}$t_{1}\cdot\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nt_{1}\min(n,t_{1})+t_{1}k\min(n,d)$} & {\footnotesize{}$t_{1}\cdot\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nt_{1}\min(n,t_{1})$}\tabularnewline
\cline{2-4}
{\footnotesize{}$\mat R=\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$} & \noun{\footnotesize{}CountSketch}{\footnotesize{} $\ensuremath{{\bm{\mathrm{G}}}}$} & {\footnotesize{}$\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nt_{2}\min(n,t_{2})+t_{2}k\min(n,d)$} & {\footnotesize{}$\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nt_{2}\min(n,t_{2})$}\tabularnewline
\hline
{\footnotesize{}Two sided } & \noun{\footnotesize{}CountSketch} & {\footnotesize{}$\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+s_{2}k^{2}+k\min(nt_{2},\nnz{\ensuremath{{\bm{\mathrm{A}}}}})+nk^{2}$} & {\footnotesize{}N/A}\tabularnewline
{\footnotesize{}$\mat R=\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k}$} & {\footnotesize{}$\ensuremath{{\bm{\mathrm{G}}}}$ and $\mat S$} & & \tabularnewline
\hline
\end{tabular}
\par\end{centering}{\footnotesize \par}
\caption{\label{tab:complex}Computational complexity of computing $\protect{\mathbf{x}}_{\protect\mat R,k}$
and CLS for various options $\protect\mat R$. For brevity, we omit
the $O$$()$ from the notation. }
\end{table}
Table~\ref{tab:complex} summarizes the running time complexity of
several design options. In order to better make sense between these
different choices, we first summarize the running time complexity
of various design choices using the optimal implementation (from an
asymptotic running-time complexity perspective). To make the discussion
manageable, we consider only subgaussian maps and \noun{CountSketch}.
Furthermore, for the sake of the analysis, we make some assumptions
and adopt some notational conventions. First, we assume that computing
$\ensuremath{{\bm{\mathrm{B}}}}^{\textsc{+}}{\mathbf{c}}$ for some $\ensuremath{{\bm{\mathrm{B}}}}\in\mathbb{R}^{m\times n}$ and ${\mathbf{c}}$ is
done via straightforward methods based on QR or SVD factorizations,
and as such takes $O(mn\min(m,n))$. We consider using fast sketch-based
approximate least squares algorithms in the next subsection. Next,
we let the sketch sizes be parameters in the complexity. In the discussion,
we use our theoretical results to deduce reasonable assumptions on
how these parameters are set, and thus to reason about the final complexity
of sketched PCR/PCP. We denote the number of rows in the left sketch
matrix $\mat S$ by $s_{1}$ for a subgaussian map, and $s_{2}$ for
\noun{CountSketch}. We denote the number of rows in the left sketch
matrix $\ensuremath{{\bm{\mathrm{G}}}}$ by $t_{1}$ for a subgaussian map, and $t_{2}$ for
\noun{CountSketch}. Finally, we assume $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}\geq\max(n,d)$,
and that all sketch sizes are greater than $k$.
Table~\ref{tab:complex} also lists, where relevant, the complexity
of the CLS solution ${\mathbf{x}}_{\mat R}$.
\paragraph{Discussion.}
We first compare the computational complexity of CLS to the computational
complexity of our proposed right sketching algorithm. For both choices
of $\ensuremath{{\bm{\mathrm{G}}}}$ we have for sketched PCP an additional term of $O(tk\min(n,d))$.
However, close inspection reveals that this term is dominated by the
term $O(nt\min(n,t))$. Thus our proposed algorithm has the same asymptotic
complexity as CLS for the same sketch size. However, our algorithm
does not mix regularization and compression and comes with stronger
theoretical guarantees.
Next, in order to compare subgaussian maps to \noun{CountSketch},
we first make some simplified assumptions on the required approximation
quality $\nu$, the relative eigengap $\gap{\ensuremath{{\bm{\mathrm{A}}}}}k$, and the rank
parameter $k$: $\nu$ is fixed, $\gap{\ensuremath{{\bm{\mathrm{A}}}}}k$ is bounded from below
by a constant, and we have $k=O(\sr{\ensuremath{{\bm{\mathrm{A}}}}})$. The first assumptions
is justified if we are satisfied with fixed sub-optimality in the
objective (optimization perspective), or a small constant multiplicative
increase in excess risk if left sketching is used, or $n$ is fixed
(statistical perspective). The first assumption is somewhat less justified
from a statistical point of view when $n\to\infty$ and right sketching
is used. The rationale behind the second assumption is that the PCR/PCP
problem is in a sense ill-posed if there is a tiny eigengap. The third
assumption is motivated by the fact that the stable rank is a measure
of the number of large singular values, which are typically singular
values that correspond to the signal rather than noise. With these
assumptions, our theoretical results establish that $s_{1},t_{1}=O(k)$
and $s_{2},t_{2}=O(k^{2})$ suffice. It is important to stress that
we make these assumptions only for the sake of comparing the different
sketching options, and we do not claim that these assumptions always
hold, or that our proposed algorithms work only when these assumptions
hold.
For left sketching, with these assumptions, we have complexity of
$O(k\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+k^{2}\max(n,d))$ for subgaussian maps and $O(k\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+dk^{2}\min(k^{2},d)+nk^{2})$
for \noun{CountSketch}. Clearly, better asymptotic complexity is achieved
with subgaussian maps. For right sketching, with these assumptions,
we have complexity of $O(k\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nk^{2})$ for subgaussian sketch
and $O(\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nk^{2}\min(n,k^{2}))$ for \noun{CountSketch}.
The complexity in terms of the input sparsity $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}$\noun{,}
which is arguably the dominant term, is better for \noun{CountSketch}.
For two sided sketching, we have complexity $O(\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+k^{4}+k\min(nk^{2},\nnz{\ensuremath{{\bm{\mathrm{A}}}}})+nk^{2}$).
If $n\gg d$ and $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}=O(n)$ (sparse input matrix, and constant
amount of non zero features per data point), left sketching gives
better asymptotic complexity. If $n\gg d$ and $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}=nd$ (full
data matrix), left sketching has better complexity unless $d\gg k^{3}$.
Furthermore, left sketching gives stronger theoretical guarantees.\emph{
Thus, for $n\gg d$} \emph{we advocate the use of left sketching}.
If $d\gg n$ and $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}=O(d)$ (sparse input matrix), right
sketching with a subgaussian maps always has better complexity than
left sketching, and potentially (but not always) right sketching with
\noun{CountSketch} has even better complexity. If $d\gg n$ and $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}=nd$
(full data matrix), right sketching with subgaussian maps has the
same complexity as left sketching, and potentially (but not always)
right sketching with \noun{CountSketch} has even better complexity
(if $d$ is sufficiently larger than $n$). \emph{Thus, for $d\gg n$
we advocate the use of right sketching}. If $n\approx d$ (both very
large), and $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}=n$ then it is possible to have $O(nk^{2})$
with all three options (left, right and two sided), as long as $k^{2}\leq n$.
A similar conclusion is achieved if $n\approx d$ and $\nnz{\ensuremath{{\bm{\mathrm{A}}}}}=nd$,
but if $k^{2}\ll n$ then two sided sketch is better.
\subsection{\label{subsec:input-sparsity}Input Sparsity Approximate PCP}
In this section, we propose an input-sparsity algorithm for approximate
PCP. In 'input-sparsity algorithm', we mean an algorithm whose running
time is $O(\nnz{\ensuremath{{\bm{\mathrm{A}}}}}\log(d/\epsilon)+\poly{k,s,t,\log(1/\epsilon})$,
where $\epsilon$ is some accuracy parameter (see formal theorem statement).
\begin{algorithm}
\begin{algorithmic}[1]
\STATE \textbf{Input: $\ensuremath{{\bm{\mathrm{A}}}}\in\mathbb{R}^{n\times d}$},\textbf{ ${\mathbf{b}}\in\mathbb{R}^{n}$},\textbf{
}$k\leq\min(n,d)$, $s,t\geq k$, $\epsilon\in(0,1)$
\STATE
\STATE Generate two \noun{CountSketch} matrices $\mat S\in\mathbb{R}^{s\times n}$
and $\ensuremath{{\bm{\mathrm{G}}}}\in\mathbb{R}^{t\times n}$.
\STATE Remove from $\ensuremath{{\bm{\mathrm{G}}}}$ any row that is zero.
\STATE $\ensuremath{{\bm{\mathrm{C}}}}\gets\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$.
\STATE $\ensuremath{{\bm{\mathrm{D}}}}\gets\mat S\ensuremath{{\bm{\mathrm{C}}}}$.
\STATE Compute $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{D}}}},k}$, the $k$ dominant right invariant
space of $\ensuremath{{\bm{\mathrm{G}}}}$ (via SVD).
\STATE For the analysis (no need to compute): $\mat R=\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{D}}}},k}$.
\STATE Solve $\min_{\gamma}\TNorm{\ensuremath{{\bm{\mathrm{C}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{D}}}},k}\gamma-{\mathbf{b}}}$
to $\epsilon/d$ accuracy using input-sparsity least squares regression
(see \cite[section 7.7]{CW17}). (Do not compute $\ensuremath{{\bm{\mathrm{C}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{D}}}},k}$.
In each iteration, multiplying a vector by $\ensuremath{{\bm{\mathrm{C}}}}\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{D}}}},k}$
is performed by first multiplying by $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{D}}}},k}$ and then by
$\ensuremath{{\bm{\mathrm{C}}}}$.)
\STATE Return ${\mathbf{y}}\gets\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}(\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{D}}}},k}\tilde{\gamma})$,
where $\tilde{\gamma}$ is the output of the previous step.
\end{algorithmic}
\caption{\label{alg:input_sparsity}Input Sparsity Approximate PCP}
\end{algorithm}
The basic idea is to use two sided sketching, with an additional modification
of using input sparsity algorithms to approximate $(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{\textsc{+}}{\mathbf{b}}=\arg\min_{\gamma}\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\mat R\gamma-{\mathbf{b}}}$.
Specifically, we propose to use the algorithm recently suggested by
Clarkson and Woodruff~\cite{CW17}. A pseudo-code description of
our input sparsity approximate PCP algorithm is listed in Algorithm~\ref{alg:input_sparsity}.
We have the following statement about the algorithm.
\begin{thm}
Run Algorithm \ref{alg:input_sparsity} with $\epsilon,s,t,k$ as
parameters. Under exact arithmetic\footnote{The results are likely too optimistic for inexact arithmetic. We leave
the numerical analysis to future work. }, after
\[
O(\nnz{\ensuremath{{\bm{\mathrm{A}}}}}\log(d/\epsilon)+\log(d/\epsilon)tk+sk^{2}+t^{3}k+k^{3}\log^{2}k)
\]
operations, with probability $2/3$, the algorithm will return a ${\mathbf{y}}$
such that
\[
\TNormS{{\mathbf{y}}-{\mathbf{x}}_{\mat R}}\leq\epsilon\TNormS{{\mathbf{x}}_{\mat R}}\,.
\]
\end{thm}
\begin{proof}
Denote $\ensuremath{{\bm{\mathrm{B}}}}=\ensuremath{{\bm{\mathrm{A}}}}\mat R$, and consider using the iterative method
described in~\cite[section 7.7]{CW17} to approximately solve $\min_{\gamma}\TNorm{\ensuremath{{\bm{\mathrm{B}}}}\gamma-{\mathbf{b}}}$.
Denote the optimal solution by $\gamma_{\mat R}$, and the solution
that our algorithm found by $\tilde{\gamma}$. Theorem 7.14 in \cite{CW17}
states that after the $O(\log(d/\epsilon))$ iterations the algorithm
would have returned $\tilde{\gamma}$ such that
\begin{equation}
\TNormS{\ensuremath{{\bm{\mathrm{B}}}}\mat Z(\tilde{\gamma}-\gamma_{\mat R})}\leq(\epsilon/d)\TNormS{\ensuremath{{\bm{\mathrm{B}}}}\mat Z\gamma_{\mat R}}\label{eq:gamma_err}
\end{equation}
for some invertible $\mat Z$ found by the algorithm. Furthermore,
$\kappa(\ensuremath{{\bm{\mathrm{B}}}}\mat Z)=O(1)$ where $\kappa(\cdot)$ is the condition
number (ratio between the largest singular value and smallest). Eq.~(\ref{eq:gamma_err})
implies that $\TNormS{\tilde{\gamma}-\gamma_{\mat R}}\leq\kappa(\ensuremath{{\bm{\mathrm{B}}}}\mat Z)^{2}(\epsilon/d)\TNormS{\gamma_{\mat R}}=O(\epsilon/d)\TNormS{\gamma_{\mat R}}$.
Now, noticing that ${\mathbf{x}}_{\mat R}=\mat R\gamma_{\mat R}$ and ${\mathbf{y}}=\mat R\tilde{\gamma}$,
we find that
\[
\TNormS{{\mathbf{y}}-{\mathbf{x}}_{\mat R}}\leq O(\epsilon/d)\kappa(\mat R)^{2}\TNormS{{\mathbf{x}}_{\mat R}}\,.
\]
We now need to bound $\kappa(\mat R)=\kappa(\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}}\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}},k})=\kappa(\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})$
where $\ensuremath{{\bm{\mathrm{G}}}}$ is a \noun{CountSketch} matrix. Since $\ensuremath{{\bm{\mathrm{G}}}}$ has
a single non zero in each column, then $\TNormS{\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}\leq\FNormS{\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}}\leq d$.
Furthermore, since we removed zero column from $\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}$, for
any ${\mathbf{x}}$ the vector $\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}}{\mathbf{x}}$ has in one of its coordinates
any coordinate of ${\mathbf{x}}$, so $\sigma_{\min}(\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})\geq1$. So
we found that $\kappa(\ensuremath{{\bm{\mathrm{G}}}}^{\textsc{T}})^{2}\leq d$. We conclude that
\[
\TNormS{{\mathbf{y}}-{\mathbf{x}}_{\mat R}}\leq O(\epsilon)\TNormS{{\mathbf{x}}_{\mat R}}\,.
\]
Adjusting $\epsilon$ to compensate for the constants completes the
proof.
\end{proof}
\section{\label{sec:extensions}Extensions}
\subsection{Streaming Algorithm}
We now consider computing an approximate PCR/PCP in the streaming
model. We consider a one-pass row-insertion streaming model, in which
the rows of $\ensuremath{{\bm{\mathrm{A}}}}$, ${\mathbf{a}}_{1},{\mathbf{a}}_{2},\dots,{\mathbf{a}}_{n}$, and the corresponding
entries in ${\mathbf{b}}$, $b_{1},b_{2},\dots,b_{n}$, are presented one-by-one
and once only (i.e., in a stream). The goal is to use $o(n)$ memory
(so $\ensuremath{{\bm{\mathrm{A}}}}$ cannot be stored in memory). The relevant resources to
be bounded for numerical linear algebra in the streaming model are
storage, update time (time spent per row), and final computation time
(at the end of the stream)~\cite{CW09}. Our goal is to bound these
by $O(\poly d)$ .
Our proposed streaming algorithm for approximate PCP uses left sketching.
It is easy to verify that if $\mat S$ is a subgaussian map or \noun{CountSketch,
}then $\mat R=\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}},k}$ can be computed in the streaming
model: one has to update $\mat S\ensuremath{{\bm{\mathrm{A}}}}$ as new rows are presented ($O(d)$
update for \noun{CountSketch}, and $O(sd)$ for subgaussian map),
and once the final row has been presented, factorizing $\mat S\ensuremath{{\bm{\mathrm{A}}}}$
and extracting $\mat R$ can be done in $O(sd\min(s,d))$ which is
polynomial in $d$ if $s$ is polynomial in $d$. However, to compute
${\mathbf{x}}_{\mat R}$ one has to compute $(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{\textsc{+}}{\mathbf{b}}$, and storing
$\ensuremath{{\bm{\mathrm{A}}}}\mat R$ in memory requires $\Omega(n)$ memory. To circumvent
this issue we propose to introduce another sketching matrix $\mat T$,
and approximate $(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{\textsc{+}}{\mathbf{b}}$ via $(\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{\textsc{+}}{\mathbf{b}}$
. Thus, for $\mat R=\ensuremath{{\bm{\mathrm{V}}}}_{\mat S\ensuremath{{\bm{\mathrm{A}}}},k}$ we approximate ${\mathbf{x}}_{\mat R}$
by $\tilde{{\mathbf{x}}}_{\mat R}=\mat R(\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{\textsc{+}}{\mathbf{b}}$. It is easy
to verify that $\tilde{{\mathbf{x}}}_{\mat R}$ can be computed in the streaming
model (by forming and updating $\mat T\ensuremath{{\bm{\mathrm{A}}}}$ while computing $\mat R$).
More generally, for \emph{any} $\mat R$ which can be computed in the
streaming model, we can also compute in the streaming model the following
approximation of ${\mathbf{x}}_{\mat R,k}$:
\[
\tilde{{\mathbf{x}}}_{\mat R,k}:=\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}(\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}\mat T{\mathbf{b}}\,.
\]
The next theorem, establishes conditions on $\mat T$ that guarantee
that $\tilde{{\mathbf{x}}}_{\mat R,k}$ is a an approximate PCR/PCP.
\begin{thm}
\label{thm:sketch-stream-structure-impl}Suppose that $\mat R\in\mathbb{R}^{d\times s}$
with $s\ge k$. Assume that $\nu\in(0,1)$. Suppose that $\mat T$
provides a $O(\nu)$-distortion subspace embedding for $\range{\left[\begin{array}{ccc}
\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k} & \ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k} & {\mathbf{b}}\end{array}\right]}$ that is
\[
\TNormS{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}{\mathbf{x}}_{1}+\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}{\mathbf{x}}_{2}+{\mathbf{b}} x_{3}}=(1\pm O(\nu))(\TNormS{{\mathbf{x}}_{1}}+\TNormS{{\mathbf{x}}_{2}}+x_{3}^{3})
\]
for every ${\mathbf{x}}_{1},{\mathbf{x}}_{2}\in\mathbb{R}^{k}$ and $x_{3}\in\mathbb{R}$. Then,
\begin{enumerate}
\item If $d_{2}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\mat A\mat R,k},\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\right)\leq\nu$
then $\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}_{\mat R,k}$ is an $(O(\nu),O(\nu))$-approximate
PCP.
\item If $s=k$ and $\mat R$ has orthonormal columns (i.e., $\mat R^{\textsc{T}}\mat R=\mat I_{k})$
and $d_{2}\left(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k}\right)\leq\nu(1+\nu^{2})^{-1/2}$
then $\tilde{{\mathbf{x}}}_{\mat R}$ is an $(O(\nu),\,O(\nu/\sigma_{k}))$-approximate
PCR.
\end{enumerate}
The subspace embedding conditions on $\mat T$ are met with probability
of at least $1-\delta$ if, for example, $\mat T$ is a \noun{CountSketch}
matrix with $O(k^{2}/\nu^{2}\delta)$ rows.
\end{thm}
\begin{proof}
We need to show both the additive error bounds on the objective function,
and the error bound on the constraints. We start with the additive
error bounds on the objective function for both for PCP (first part
of the theorem) and for PCR (second part of the theorem). The lower
bound on $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}_{\mat R,k}-{\mathbf{b}}}$ follows immediately
from the fact that $\tilde{{\mathbf{x}}}_{\mat R,k}\in\range{\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}$
and the fact that ${\mathbf{x}}_{\mat R,k}$ is a minimizer of $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}-{\mathbf{b}}}$
subject to ${\mathbf{x}}\in\range{\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}}$. For the upper
bound, we observe
\begin{eqnarray*}
\TNorm{\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}_{\mat R,k}-{\mathbf{b}}} & \leq & \left(1+O(\nu)\right)\TNorm{\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}(\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}\mat T{\mathbf{b}}-\mat T{\mathbf{b}}}\\
& \leq & \left(1+O(\nu)\right)\TNorm{\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}(\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}{\mathbf{b}}-\mat T{\mathbf{b}}}\\
& \leq & \left(1+O(\nu)\right)\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-{\mathbf{b}}}\\
& \leq & \TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-{\mathbf{b}}}+O(\nu)\TNorm{{\mathbf{b}}}
\end{eqnarray*}
where in the first and third inequality we used the fact that $\mat T$
provides a subspace embedding for $\range{[\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\,{\mathbf{b}}]}$
and in the second inequality we used the fact that $(\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}\mat T{\mathbf{b}}$
is a minimizer of $\TNorm{\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}{\mathbf{x}}-\mat T{\mathbf{b}}}$.
Bounds on $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{\mat R,k}-{\mathbf{b}}}$ (Theorem~\ref{thm:structural})
now imply the additive bound.
We now bound the constraint for the PCR guarantee (second part of
the theorem). Let
\[
\ensuremath{{\bm{\mathrm{C}}}}=(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}((\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{\textsc{T}})^{+}\,.
\]
Since $(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{\textsc{T}}$ and $(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}$
have the same row space, and $\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$ has more
rows than columns, $\ensuremath{{\bm{\mathrm{C}}}}$ is non-singular and we have $\ensuremath{{\bm{\mathrm{C}}}}(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{\textsc{T}}=(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}$.
Since $\mat T$ provides a subspace embedding for $\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$,
all the singular values of $\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$ belong to
the interval $[1-O(\nu),1+O(\nu)]$. We conclude that $\TNorm{\ensuremath{{\bm{\mathrm{C}}}}-\mat I_{k}}\leq O(\nu)$.
We also have $(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}=\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}^{-1}(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}$
since $\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$ has linearly independent columns
(since it provides a subspace embedding), and $\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}$
has all linearly independent rows. Thus,
\begin{eqnarray*}
\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}_{\mat R,k}} & = & \TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}_{\mat R,k}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k+}^{\textsc{T}}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}^{\textsc{T}}\mat T^{\textsc{T}}\mat T{\mathbf{b}}}\\
& \leq & \TNorm{\ensuremath{{\bm{\mathrm{A}}}}\tilde{{\mathbf{x}}}_{\mat R,k}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}^{\textsc{T}}\mat T^{\textsc{T}}\mat T{\mathbf{b}}}\\
& = & \TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}\mat T{\mathbf{b}}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}^{\textsc{T}}\mat T^{\textsc{T}}\mat T{\mathbf{b}}}\\
& = & \TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\ensuremath{{\bm{\mathrm{C}}}}(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{\textsc{T}}\mat T{\mathbf{b}}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}^{\textsc{T}}\mat T^{\textsc{T}}\mat T{\mathbf{b}}}\\
& \leq & \left(1+O(\nu)\right)\cdot\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\ensuremath{{\bm{\mathrm{C}}}}\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}^{\textsc{T}}\mat T^{\textsc{T}}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}^{\textsc{T}}\mat T^{\textsc{T}}}\cdot\TNorm{{\mathbf{b}}}\\
& \leq & \left(1+O(\nu)\right)^{2}\cdot\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\ensuremath{{\bm{\mathrm{C}}}}\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}^{\textsc{T}}}\TNorm{{\mathbf{b}}}\\
& \leq & \left(1+O(\nu)\right)\cdot\left(\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\left(\ensuremath{{\bm{\mathrm{C}}}}-\mat I_{k}\right)\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}^{\textsc{T}}}+\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}^{\textsc{T}}-\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}\ensuremath{{\bm{\mathrm{U}}}}_{\mat A,k}^{\textsc{T}}}\right)\cdot\TNorm{{\mathbf{b}}}\\
& \leq & \left(1+O(\nu)\right)\cdot\left(\TNorm{\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\left(\ensuremath{{\bm{\mathrm{C}}}}-\mat I_{k}\right)\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}^{\textsc{T}}}+\nu\right)\cdot\TNorm{{\mathbf{b}}}\\
& = & \left(1+O(\nu)\right)\cdot\left(\TNorm{\left(\ensuremath{{\bm{\mathrm{C}}}}-\mat I_{k}\right)}+\nu\right)\cdot\TNorm{{\mathbf{b}}}\\
& \leq & \left(1+O(\nu)\right)\cdot\left(O(\nu)+\nu\right)\cdot\TNorm{{\mathbf{b}}}\\
& = & O(\nu)\cdot\TNorm{{\mathbf{b}}}
\end{eqnarray*}
We now bound the constraint for the PCR guarantee (second part of
the theorem). To that end, and observe:
\begin{eqnarray*}
\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\tilde{{\mathbf{x}}}_{\mat R,k}} & \leq & \TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}(\mat T\ensuremath{{\bm{\mathrm{A}}}}\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}\mat T{\mathbf{b}}}\\
& \leq & \TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R}\cdot\TNorm{(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k})^{+}\mat T{\mathbf{b}}}\\
& \leq & \frac{\nu\cdot\left(1+O(\nu)\right)\cdot\TNorm{{\mathbf{b}}}}{\sigma_{\min}\left(\mat T\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\right)}\\
& \leq & \frac{\nu\cdot\left(1+O(\nu)\right)\TNorm{{\mathbf{b}}}}{\left(1-O(\nu\right))\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\mat{\Sigma}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k}\right)}\\
& \leq & \frac{O(\nu)\cdot\TNorm{{\mathbf{b}}}}{\sigma_{\min}\left(\ensuremath{{\bm{\mathrm{A}}}}\mat R\right)}\\
& \leq & \frac{O(\nu)}{\sigma_{k}}\cdot\TNorm{{\mathbf{b}}}
\end{eqnarray*}
where we used the fact that $\mat T$ provides a subspace embedding
for $\range{\left[\begin{array}{cc}
\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{A}}}}\mat R,k} & {\mathbf{b}}\end{array}\right]}$, and used Lemma~\ref{lem:11} to bound $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\mat A,k+}^{\textsc{T}}\mat R}$
and $\TNorm{(\ensuremath{{\bm{\mathrm{A}}}}\mat R)^{+}}$.
\end{proof}
\subsection{Approximate Kernel PCR}
For simplicity, we consider only the homogeneous polynomial kernel
${\cal K}({\mathbf{x}},{\mathbf{z}})=({\mathbf{x}}^{\textsc{T}}{\mathbf{z}})^{q}$. The results trivially extend to
the non-homogeneous polynomial kernel ${\cal K}_{n}({\mathbf{x}},{\mathbf{z}})=({\mathbf{x}}^{\textsc{T}}{\mathbf{z}}+c)^{q}$
by adding a single feature to each data point. We leave to future
work the development of similar techniques for other kernels (e.g.
Gaussian kernel).
Let $\phi:\mathbb{R}^{d}\to\mathbb{R}^{d^{q}}$ be the function that maps a vector
${\mathbf{z}}=(z_{1},\dots,z_{d})$ to the set of monomials formed by multiplying
$q$ entries of ${\mathbf{z}}$, i.e. $\phi({\mathbf{z}})=(z_{i_{1}}z_{i_{2}}\cdots z_{i_{q}})_{i_{1},\dots,i_{q}\in\{1,\dots,d\}}$.
For a data matrix $\ensuremath{{\bm{\mathrm{A}}}}\in\mathbb{R}^{d}$ and a response vector ${\mathbf{b}}\in\mathbb{R}^{n}$,
let $\Phi\in\mathbb{R}^{n\times d^{q}}$ be the matrix obtained by applying
$\phi$ to the rows of $\ensuremath{{\bm{\mathrm{A}}}}$, and consider computing the rank $k$
PCR solution $\Phi$ and ${\mathbf{b}}$, which we denote by ${\mathbf{x}}_{{\cal K},k}$.
The corresponding prediction function is $f_{{\cal K},k}({\mathbf{z}})\coloneqq\phi({\mathbf{z}})^{\textsc{T}}{\mathbf{x}}_{{\cal K},k}$.
While ${\mathbf{x}}_{{\cal K},k}$ is likely a huge vector (since ${\mathbf{x}}_{{\cal K},k}\in\mathbb{R}^{d^{q}}$),
and thus expensive to compute, in kernel PCR we are primarily interested
in having an efficient method to compute $f_{{\cal K},k}({\mathbf{z}})$ given
a 'new' ${\mathbf{z}}$. We can accomplish this via the kernel trick, as we
now show.
We assume that $\Phi$ has full row rank (this holds if all data points
are different). Let ${\mathbf{a}}_{1},\dots,{\mathbf{a}}_{n}$ be the rows of $\ensuremath{{\bm{\mathrm{A}}}}$.
As usual with PCR, we have ${\mathbf{x}}_{{\cal K},k}=\ensuremath{{\bm{\mathrm{V}}}}_{\Phi,k}\Sigma_{\Phi,k}^{-1}\ensuremath{{\bm{\mathrm{U}}}}_{\Phi,k}^{\textsc{T}}{\mathbf{b}}$.
Since $\ensuremath{{\bm{\mathrm{V}}}}_{\Phi,k}=\Phi^{\textsc{T}}\ensuremath{{\bm{\mathrm{U}}}}_{\Phi,k}\Sigma_{\Phi,k}^{-1}$
we have
\begin{equation}
f_{{\cal K},k}({\mathbf{z}})=\phi({\mathbf{z}})^{\textsc{T}}\Phi^{\textsc{T}}\ensuremath{{\bm{\mathrm{U}}}}_{\Phi,k}\Sigma_{\Phi,k}^{-2}\ensuremath{{\bm{\mathrm{U}}}}_{\Phi,k}{\mathbf{b}}=({\cal K}({\mathbf{z}},{\mathbf{a}}_{1})\,\cdots\,{\cal K}({\mathbf{z}},{\mathbf{a}}_{n}))\alpha_{{\cal K},k}\label{eq:kpcr}
\end{equation}
where $\alpha_{{\cal K},k}\coloneqq\ensuremath{{\bm{\mathrm{U}}}}_{\Phi,k}\Sigma_{\Phi,k}^{-2}\ensuremath{{\bm{\mathrm{U}}}}_{\Phi,k}^{\textsc{T}}{\mathbf{b}}$.
In the above, we used the fact that for any ${\mathbf{x}}$ and ${\mathbf{z}}$ we have
$\phi({\mathbf{x}})^{\textsc{T}}\phi({\mathbf{z}})=({\mathbf{x}}^{\textsc{T}}{\mathbf{z}})^{q}={\cal K}({\mathbf{x}},{\mathbf{z}})$. Let $\ensuremath{{\bm{\mathrm{K}}}}\in\mathbb{R}^{n\times n}$
be the\emph{ kernel matrix} (also called \emph{Gram matrix}) defined
by $\ensuremath{{\bm{\mathrm{K}}}}_{ij}={\cal K}({\mathbf{a}}_{i},{\mathbf{a}}_{j}$). It is easy to verify that
$\ensuremath{{\bm{\mathrm{K}}}}=\Phi\Phi^{\textsc{T}}$, so we can compute $\ensuremath{{\bm{\mathrm{K}}}}$ in $O(n^{2}(d+q))$
(and without forming $\Phi$, which is a huge matrix). We also have
$\ensuremath{{\bm{\mathrm{K}}}}=\ensuremath{{\bm{\mathrm{U}}}}_{\Phi}\Sigma_{\Phi}^{2}\ensuremath{{\bm{\mathrm{U}}}}_{\Phi}^{\textsc{T}}$ so $\alpha_{k}=\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{K}}}},k}\Sigma_{K,k}^{-1}\ensuremath{{\bm{\mathrm{U}}}}_{\ensuremath{{\bm{\mathrm{K}}}},k}^{\textsc{T}}{\mathbf{b}}$.
Thus, we can compute $\alpha_{{\cal K},k}$ in $O(n^{2}(d+q+n))$
time. Once we have computed $\alpha_{k}$, using~(\ref{eq:kpcr})
we can compute $f_{{\cal K},k}({\mathbf{z}})$ for any ${\mathbf{z}}$ in $O(ndq)$ time.
In order to compute an approximate kernel PCR, we introduce a right
sketching matrix $\mat R\in\mathbb{R}^{d^{q}\times t}$. Such a matrix $\mat R$
is frequently referred to, in the context of kernel learning, as a
randomized feature map. We use the \noun{TensorSketch} feature map~\cite{Pagh13,PP13}.
The feature map is defined as follows. We first randomly generate
$q$ $3$-wise independent hash functions $h_{1},\dots,h_{q}\in\{1,\dots,d\}\to\{1,\dots,t\}$
and $q$ $4$-wise independent sign functions $g_{1},\dots,g_{q}:\{1,\dots,d\}\to\{-1,+1\}$.
Next, we define $H:\{1,\dots,d\}^{q}\to\{1,\dots,t\}$ and $G:\{1,\dots,t\}^{q}\to\{-1,+1\}$:
\[
H(i_{1},\dots,i_{q})\coloneqq h_{1}(i_{1})+\dots+h_{q}(i_{q})\,\mod\,t
\]
\[
G(i_{1},\dots,i_{q})=g_{1}(i_{1})\cdot g_{2}(i_{2})\cdot\dots\cdot g_{q}(i_{q})
\]
To define $\mat R$, we index the rows of $\mat R$ by $\{1,\dots,d\}^{q}$
and set row $(i_{1},\dots,i_{q})$ to be equal to $G(i_{1},\dots,i_{q})\cdot{\mathbf{e}}_{H(i_{1},\dots,i_{q})}$,
where ${\mathbf{e}}_{j}$ denote the $j$th identity vector. A crucial observation
that makes \noun{TensorSketch} useful, is that via the representation
using $h_{1},\dots,h_{q}$ and $g_{1},\dots,g_{q}$ we can compute
$\mat R^{\textsc{T}}\phi({\mathbf{z}})$ in time $O(q(\nnz{{\mathbf{z}}}+t\log t))$ (see Pagh~\cite{Pagh13}
for details). Thus, we can compute $\Phi\mat R$ in time $O(q(\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nt\log t))$.
Consider right sketching PCR on $\Phi$ and $k$ with a \noun{TensorSketch}
$\mat R$ as the sketching matrix. The approximate solution is
\[
{\mathbf{x}}_{{\cal K},\mat R,k}\coloneqq\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\Phi\mat R,k}(\Phi\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\Phi\mat R,k})^{\textsc{+}}{\mathbf{b}}=\mat R\gamma_{{\cal K},\mat R,k}
\]
where $\gamma_{{\cal K},\mat R,k}\coloneqq\ensuremath{{\bm{\mathrm{V}}}}_{\Phi\mat R,k}(\Phi\mat R\ensuremath{{\bm{\mathrm{V}}}}_{\Phi\mat R,k})^{\textsc{+}}{\mathbf{b}}$.
We can compute $\gamma_{\mat R,k}$ in $O(q(\nnz{\ensuremath{{\bm{\mathrm{A}}}}}+nt\log t)+nt^{2})$
time. The predication function is
\[
f_{{\cal K},\mat R,k}({\mathbf{z}})\coloneqq\phi({\mathbf{z}})^{\textsc{T}}{\mathbf{x}}_{{\cal K},\mat R,k}=(\mat R^{\textsc{T}}\phi({\mathbf{z}}))^{\textsc{T}}\gamma_{{\cal K},\mat R,k}
\]
so once we have $\gamma_{{\cal K},\mat R,k}$ we can compute $f_{{\cal K},\mat R,k}({\mathbf{z}})$
in $O(q(\nnz{{\mathbf{z}}}+t\log t))$ time. Thus, the method is attractive
from a computational complexity point of view if $t\ll n$ or $d\gg n$
and $d\gg t$. The following theorem bound the excess risk of ${\mathbf{x}}_{{\cal K},\mat R,k}$.
\begin{thm}
Let $(\nu,\delta)\in(0,1/2)$. Let $\lambda_{1}\geq\dots\geq\lambda_{n}$
denote the eigenvalues of $\ensuremath{{\bm{\mathrm{K}}}}$. If $\mat R$ is a \noun{TensorSketch}
matrix with
\[
t=\Omega\left(\frac{3^{q}\Trace{\ensuremath{{\bm{\mathrm{K}}}}}^{2}}{(\lambda_{k}-\lambda_{k+1})^{2}\nu^{2}\delta}\right)
\]
columns, then with probability of at least $1-\delta$
\[
{\cal E}({\mathbf{x}}_{{\cal K},\mat R,k})\leq{\cal E}({\mathbf{x}}_{{\cal K},k})+\frac{(2\nu+\nu^{2})\TNormS{{\mathbf{f}}}}{n}
\]
where ${\mathbf{f}}$ is the expected value of ${\mathbf{b}}$ (recall the statistical
framework in section~\ref{subsec:statistical}).
\end{thm}
Before proving this theorem, we remark that the bound on the size
of the sketch is somewhat disappointing in the sense that it is useful
only if $d\gg n$ (since $\Trace{\ensuremath{{\bm{\mathrm{K}}}}}$ is likely to be large).
However, this is only a bound, and possibly a pessimistic one. Furthermore,
once the feature expanded data has been embedded in Euclidean space
(via \noun{TensorSketch}), it can be further compressed using standard
Euclidean space transforms like \noun{CountSketch} and subgaussian
maps (this is sometimes referred to as two-level sketching), or compression
can be applied from the left. We leave the task of improving the bound
and exploring additional compression techniques to future research.
\begin{proof}
The square singular values of $\Phi$ are exactly the eigenvalues
of $\ensuremath{{\bm{\mathrm{K}}}}$, so Theorem~\ref{thm:right_sketching} asserts that the
conclusions of the theorem hold if $\mat R^{\textsc{T}}$ provides $(\epsilon,\delta)$-approximate
Gram matrix for $\Phi$ where $\epsilon=O(\nu(\lambda_{k}-\lambda_{k+1})/\lambda_{1})$.
To that end, we combine the analysis of Avron et al.~\cite{ANW14}
of \noun{TensorSketch} with more recent results due to Cohen et al.~\cite{cohen2015optimal}.
Although not stated as a formal theorem, as part of a larger proof,
Avron et al. show that \noun{TensorSketch} has an OSE-moment property
that together with the results of Cohen et al.~\cite{cohen2015optimal}
imply that indeed the $(\epsilon,\delta)$-approximate Gram property
holds for the specified amounts of columns in $\mat R$.
\end{proof}
\section{\label{sec:experiments}Experiments}
In this section we report experimental results, on real data, that
illustrate and support the main results of the paper, and demonstrate
the ability of our algorithms to find appropriately regularized solutions.
\paragraph{Datasets. }
We experiment with three datasets, two regression datasets (\emph{Twitter
Buzz} and \emph{E2006-tfidf}) and one classification dataset (\emph{Gisette}).
\emph{Twitter Social Media Buzz~\cite{kawala2013predictions}} is
a regression dataset in which the goal is to predict the popularity
of topics as quantified by its mean number of active discussions given
77 predictor variables such as number of authors contributing to the
topic over time, average discussion lengths, number of interactions
between authors etc. We pre-process the data in a manner similar to
previous work \cite{lu2014fast,slawski2017compressed}. That is, several
of the original predictor variables, as well as the response variable
are log-transformed prior to analysis. We then center and scale to
unit norm. Finally, we add quadratic interactions which yielding a
total of $3080$ predictor variables (after preprocessing, the data
matrix is $583250\textrm{-by-}3080$). We used this dataset to explore
only sub-optimality of the objective and constraint satisfaction,
as we have found that the generalization error is very sensitive to
selection of the test set (when splitting a subset of the data to
training and testing).
\emph{E2006-tfidf}~\cite{kogan2009predicting} is regression dataset
where the features are extracted from SEC-mandated financial reports
published annually by a publicly traded company, and the quantity
to be predicted is volatility of stock returns, an empirical measure
of financial risk. We use the standard training-test split available
with the dataset\footnote{We downloaded the dataset from the LIBSVM website, \url{https://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/}.}.
We use this dataset only for testing generalization. The only pre-processing
we performed was subtracting the mean from the response variable,
and reintroducing it when issuing predictions.
The\emph{ Gisette} dataset is a binary classification dataset that
is a constructed from the MNIST dataset. The goal is to separate the
highly confusable digits '4' and '9'. The dataset has $6000$ data-points,
each having $5000$ features. We use the standard training-test split
available with the dataset (this dataset was downloaded from the same
website as the E2006-tfidf dataset). We convert the binary classification
problem to a regression problem using standard techniques (regularized
least squares classification). We use this dataset only for testing
generalization.
\paragraph{Baselines.}
A first reference are the performance of plain PCR. For small problems,
the dominant right subspace needed to compute the PCR solution can
be computed via MATLAB's dense SVD routine. For larger problems, we
compute the dominant right subspace using a PRIMME~\cite{SM10,WRS17},
a state-of-the-art iterative algorithm for SVD. As additional reference,
we also report results of two alternative algorithms: CLS and the
iterative algorithm of Frostig et al.~\cite{frostig2016principal}.
Both in the discussion, and in the graphs, we refer to the algorithm
Frostig et al.~ as ``Iterative-PCR''. We use the implementation
of Iterative-PCR supplied by the authors,\footnote{\url{https://github.com/cpmusco/fast-pcr}}
for which we used the default parameters, except for the ``tol''
parameter, which we set to $10^{-6}$ instead of the default $10^{-3}$.
We found that the use of tol=$10^{-3}$ produces results that generalize
poorly, while the use of tol=$10^{-6}$ produces much better results.
However, the running time of Iterative-PCR with tol=$10^{-6}$ is
considerably higher the the running time for tol=$10^{-3}$. Iterative-PCR
controls singular vector truncation via a cut-off parameter $\lambda$,
while in our experiments we set $k$ (the number of principal components
that are kept). We achieve this effect by setting $\lambda=(\sigma_{k}^{2}(\ensuremath{{\bm{\mathrm{A}}}})+\sigma_{k+1}^{2}(\ensuremath{{\bm{\mathrm{A}}}}))/2$
(when we report running times, we do not include the time to compute
the singular values). Finally, we remark that based of the documentation,
the algorithm analyzed by Frostig et al.~\cite{frostig2016principal}
does not completely correspond to the default parameters of the implementation
of Iterative-PCR supplied by the authors (e.g., the default parameter
for the ``method'' parameter is ``LANCZOS'', while ``EXPLICIT''
corresponds the algorithm analyzed in \cite{frostig2016principal}).
\paragraph{Sub-optimality of objective and constraint satisfaction.}
We explore the Twitter Buzz dataset from the optimization perspective,
namely measure the sub-optimality in the objective (vs. PCR) and constraint
satisfaction. Since $n\gg d$, we use left sketching with subgaussian
maps. We perform each experiment five times and report the median
value. Error bars, when present, represent the minimum and maximum
value of five runs. In the top panel, we use a fixed $k=60$ and vary
the sketch size (left sketching only), while in the bottom panel we
vary $k$ and set sketch size to be $s=4k$. The left panel explores
the value of the objective function, appropriately normalized (divided
by $\TNorm{\ensuremath{{\bm{\mathrm{A}}}}{\mathbf{x}}_{k}-{\mathbf{b}}}$ for fixed $k$, and divided by $\TNorm{{\mathbf{b}}}$
for varying $k$). The right panel explores the regularization effect
by examining the value of the constraints $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\textsc{T}}{\mathbf{x}}_{k}}/\TNorm{{\mathbf{b}}}$.
In the left panel, we see that value of the objective for the sketched
PCR solution follows the value of objective for the PCR solution.
In general, as the sketch size increases, the variance in the objective
value reduces (top left graph). The normalized value of the constraint
for sketched PCR is rather small (as a reference we note that $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\textsc{T}}{\mathbf{x}}_{OLS}}/\TNorm{{\mathbf{b}}}=0.4165$),
and generally decreases when the sketch size increases (top right
graph), but increases with $k$ for a fixed ratio between $s$ and
$k$ (bottom right graph). Furthermore, the results of sketched PCR
are very similar to the results of iterative PCR (bottom panel), while
running time is considerably shorter (see Table \ref{tab:time}).
The role of the constraints as a regularizer are illustrated by the
results for CLS (for fixed $k$ we use $t=4k$). As expected, CLS
achieves lower objective value at the price of larger constraint infeasibility.
The values of $\TNorm{\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}^{\textsc{T}}{\mathbf{x}}_{CLS}}/\TNorm{{\mathbf{b}}}$ are
much smaller than the OLS value, but much larger than the values for
sketched PCR. Furthermore, it is hard to control the regularization
effect for CLS: when sketch size increases the objective decreases
and the constraint increases (compare to PCR and sketched PCR, top
panel).
\begin{figure}
\noindent \begin{centering}
\begin{tabular}{ccc}
\includegraphics[width=0.45\textwidth]{twitter_objective_k_60} & ~ & \includegraphics[width=0.45\textwidth]{twitter_constraints_k_60}\tabularnewline
\includegraphics[width=0.45\textwidth]{twitter_objective_with_fpcr_slow} & & \includegraphics[width=0.45\textwidth]{twitter_constraints_with_fpcr_slow}\tabularnewline
\end{tabular}
\par\end{centering}
\caption{\label{fig:twitter}Sub-optimality of objective and constraint satisfaction
for the Twitter Buzz dataset. }
\end{figure}
\paragraph{Generalization results.}
We also explored the prediction error and the tradeoffs between compression
and regularization. We perform each experiment five times and report
the median value. Error bars, when present, represent the minimum
and maximum value of those five runs.
We report the Mean Square Error (MSE) of predictions for the E2006.tfidf
dataset in Figure~\ref{fig:E2006}. We compare CLS, iterative-PCR,
right sketching and two-sided sketching (the matrix is too large for
exact PCR, and $d\gg n$ so right sketching is more appropriate).
In the left panel we fix $k=600$ and vary the sketch size. The MSE
decreases as the sketch size increases for both sketching methods.
For CLS, initially the MSE decreases and is close to the MSE of the
two sketching methods, but for large sketch sizes the MSE starts to
go up, likely due to decreased level of regularization. We note that
the minimum MSE achieved by CLS is larger than achieved by both sketching
methods. A similar phenomenon is observed when we vary the value of
$k$ in the right panel.
\begin{figure}
\noindent \begin{centering}
\begin{tabular}{ccc}
\includegraphics[width=0.45\textwidth]{E2006_prediction_fixed_k_with_pcr} & ~ & \includegraphics[width=0.45\textwidth]{E2006_prediction_varing_k_with_fpcr_slow}\tabularnewline
\end{tabular}
\par\end{centering}
\caption{\label{fig:E2006}Mean squared error of predictions for the E2006.tfidf
dataset. }
\end{figure}
We report the classification error for the Giesette dataset in Figure~\ref{fig:gisette}.
In the left panel we fix $k=400$ and vary the sketch size. As a reference,
the error rate of exact PCR (with $k=400$) is $2.8\%$ and the error
rate for OLS is 9.3\%. Left sketching has error rate very close to
the error rate of exact PCR, especially when $s$ is large enough.
Right sketching does not perform as well as left sketching, but it
too achieves low error rate for large $s$. For both methods, the
error rate drops as the sketch sizes increase, and the variance reduces.
For CLS the error rate and variance initially drops as the sketch
size increases, but eventually, when sketch size is large enough,
the error rate and the variance increases. This is hardly surprising:
as the sketch size increase, CLS approaches OLS. This is due to the
fact that CLS uses the compression to regularize, and when the sketch
size is large there is little regularization. In the right panel,
we vary the value of $k$ and set $s=4k$ (left sketching) and $t=4k$
(right sketching and CLS). Left sketch and PCR consistently achieve
about the same error rate. For small values of $t$, CLS performs
well, but when $t$ is too large the error starts to increase. In
contrast, right sketching continues to perform well with large values
of $k$. Again, we see that CLS mixes compression and regularization,
and one cannot use a large sketch size and modest amount of regularization
with CLS.
\begin{figure}
\noindent \begin{centering}
\begin{tabular}{ccc}
\includegraphics[width=0.45\textwidth]{gisette_classification_k_400} & ~ & \includegraphics[width=0.45\textwidth]{gisette_classification_varying_vs_with_fpcr_slow}\tabularnewline
\end{tabular}
\par\end{centering}
\caption{\label{fig:gisette}Classification error for the Gisette dataset. }
\end{figure}
\paragraph{Running time. }
In Table~\ref{tab:time} we report a sample of the various running
times of the different algorithms. All experiments were conducted
using MATLAB, although the sketching routines were written in C. Running
times were measured on a machine with a 6-core Intel Xeon Processor
E5-1650 v4 CPU and 128 GB of main memory, running Ubuntu 16.04. For
plain PCR, we report running time using PRIMME, which we ran with
default parameters and no preconditioner. For PRIMME, we cap the number
of iterations at 100,000, and write ``FAIL'' in the table if the
PRIMME failed to convergence within that cap. For iterative PCR we
also report running times when we set tol to the default value, and
reduce the max number of iteration from 40 to 10. This results in
much faster running time, but much degraded generalization (not reported),
e.g. for E2006 the test MSE for Iter-PCR (tol=$10^{-3}$, iter=10)
is 0.32. With respect to running time, iterative-PCR is competitive
with sketched PCR only for E2006, but with worse classification error.
Using PRIMME for PCR is not competitive with sketched PCR. However,
we stress that we experimented with only three datasets, so the comparison
is not comprehensive.
\begin{table}
\centering{}\caption{\label{tab:time}Running times (in seconds). For sketched PCR, we
report in brackets the type of sketching used (left, right, or two-sided).}
{\footnotesize{}}%
\begin{tabular}{|c|c|c|>{\centering}p{2.4cm}|>{\centering}p{2.4cm}|c|}
\hline
& {\footnotesize{}CLS ($t=400)$} & {\footnotesize{}PRIMME-PCR} & {\footnotesize{}Iter-PCR }{\footnotesize \par}
{\footnotesize{}(tol=$10^{-3}$, iter=10)} & {\footnotesize{}Iter-PCR (tol=$10^{-6}$)} & {\footnotesize{}Sketched-PCR}\tabularnewline
\hline
\hline
{\footnotesize{}Twitter, $k=82$} & {\footnotesize{}21.2} & {\footnotesize{}1730} & {\footnotesize{}742} & {\footnotesize{}4067} & {\footnotesize{}4.7 (left)}\tabularnewline
\hline
{\footnotesize{}Twitter, $k=152$} & {\footnotesize{}48.9} & {\footnotesize{}5907} & {\footnotesize{}1278} & {\footnotesize{}7759} & {\footnotesize{}8.4 (left) }\tabularnewline
\hline
{\footnotesize{}E2006, $k=1000$} & {\footnotesize{}100} & {\footnotesize{}14694} & {\footnotesize{}140} & {\footnotesize{}601} & {\footnotesize{}150 (two sided)}\tabularnewline
\hline
{\footnotesize{}E2006, $k=2000$} & {\footnotesize{}270} & {\footnotesize{}FAIL} & {\footnotesize{}320} & {\footnotesize{}791} & {\footnotesize{}815 (two sided)}\tabularnewline
\hline
{\footnotesize{}Gisette, $k=400$} & {\footnotesize{}2.0} & {\footnotesize{}497} & {\footnotesize{}7.1} & {\footnotesize{}30.3} & {\footnotesize{}0.2 (left)}\tabularnewline
\hline
{\footnotesize{}Gisette, $k=1000$} & {\footnotesize{}9.3} & {\footnotesize{}FAIL} & {\footnotesize{}9.3} & {\footnotesize{}39.4} & {\footnotesize{}0.9 (left)}\tabularnewline
\hline
\end{tabular}
\end{table}
\section{Conclusions and Future work }
In this paper, we studied the use of sketching to accelerate the solution
of PCR and PCP. In particular, for a data matrix $\ensuremath{{\bm{\mathrm{A}}}}$, we relate
the PCR/PCP solution of $\ensuremath{{\bm{\mathrm{A}}}}\mat R$, where $\mat R$ is any dimensionality
reduction matrix, to the PCR/PCP solution of $\ensuremath{{\bm{\mathrm{A}}}}$. We presented
a notion of approximate PCR/PCP, motivated both from an optimization
perspective and from a statistical perspective, and provide conditions
on $\mat R$ that guarantee rigorous theoretical bounds. We then leverage
the aforementioned results to design fast, sketching based, algorithms
for approximate PCR/PCP, and demonstrate empirically the utility of
our proposed algorithms. Throughout, our focus in this paper has been
on algorithms that use the ``sketch-and-solve'' approach.
There are multiple ways in which the current work can be extended,
and the theoretical results improved. We have presented two notions
of approximation: approximate PCR and approximate PCP. Our results
for approximate PCR use only dimensionality reduction matrices $\mat R$
whose number of columns is equal to the target rank. It is natural
to conjecture that the use of dimensionality reduction matrices with
an higher number of columns will lead to stronger PCR bounds, but
we prove only PCP bounds. The underlying reason is that our bounds
for PCR are based on analyzing the distance between the column space
of $\mat R$ and the column space of $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$ . However,
once the number of columns in $\mat R$ is different from the number
of columns in $\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k}$, the definition of $d_{2}(\mat R,\ensuremath{{\bm{\mathrm{V}}}}_{\ensuremath{{\bm{\mathrm{A}}}},k})$
is no longer applicable. One possible strategy for analyzing PCR when
$\mat R$ has more than $k$ columns might be to use a generalization
of the distance between two subspaces that allows subspaces of different
size; see \cite{ye2016schubert} for such generalizations. Another
crucial component will then be to generalize the Davis-Kahan theorem
to bound such distances. We conjecture it is possible to derive algorithms
that depend on gaps between $\sigma_{k}$ and $\sigma_{k+l}$, where
$l$ is some oversampling parameter, as opposed to the smaller gap
between $\sigma_{k}$ and $\sigma_{k+1}$. We leave this for future
work.
Another interesting direction is in finding other ways to identify
a valid approximate dominant subspace. If we consider the statistical
perspective and inspect Eq.~\ref{eq:bias-var}, we see that all we
need is to find a subspace ${\cal S}\subseteq\range{\ensuremath{{\bm{\mathrm{A}}}}}$ of rank
$k$ such that $\FNorm{(\mat I-\ensuremath{{\bm{\mathrm{P}}}}_{{\cal S}})\ensuremath{{\bm{\mathrm{A}}}}}$ is small,
while our theoretical results try to achieve a stronger bound: having
the dominant subspaces align. One possible way for finding such a
${\cal S}$ is using so-called Projection-cost Preserving Sketches~\cite{CohenEMMP15}.
We leave this for future work.
\subsection*{Acknowledgments.}
This research was supported by the Israel Science Foundation (grant
no. 1272/17) and by an IBM Faculty Award.
\bibliographystyle{plain}
|
{
"timestamp": "2019-03-08T02:10:38",
"yymm": "1803",
"arxiv_id": "1803.02661",
"language": "en",
"url": "https://arxiv.org/abs/1803.02661"
}
|
\section{Introduction}
\label{intro}
Your text comes here. Separate text sections with
\section{Section title}
\label{sec:1}
Text with citations \cite{RefB} and \cite{RefJ}.
\subsection{Subsection title}
\label{sec:2}
as required. Don't forget to give each section
and subsection a unique label (see Sect.~\ref{sec:1}).
\paragraph{Paragraph headings} Use paragraph headings as needed.
\begin{equation}
a^2+b^2=c^2
\end{equation}
\begin{figure}
\includegraphics{example.eps}
\caption{Please write your figure caption here}
\label{fig:1}
\end{figure}
\begin{figure*}
\includegraphics[width=0.75\textwidth]{example.eps}
\caption{Please write your figure caption here}
\label{fig:2}
\end{figure*}
\begin{table}
\caption{Please write your table caption here}
\label{tab:1}
\begin{tabular}{lll}
\hline\noalign{\smallskip}
first & second & third \\
\noalign{\smallskip}\hline\noalign{\smallskip}
number & number & number \\
number & number & number \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\fi
\section{Introduction}
For a positive integer $i$, a set $S$ of vertices in a graph $G$ is {\em $\; i$-independent } if the distance in $G$ between any two
distinct vertices of $S$ is at least $i+1$. In particular, a $1$-independent set is simply an independent set.
A {\em packing $k$-coloring} of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$,
the set $V_i$ is $i$-independent. The {\em packing chromatic number}, $\chi_p(G)$, of a graph $G$, is the minimum $k$ such that $G$ has a packing $k$-coloring.
The notion of packing $k$-coloring was introduced in 2008 by
Goddard, Hedetniemi, Hedetniemi, Harris and Rall~\cite{GHHHR}
(under the name {\em broadcast coloring}) motivated by frequency assignment problems in broadcast networks.
The concept has attracted a considerable attention recently: there are around 30 papers on the topic
(see e.g.~\cite{ANT1,BF,BKR1,BKR2,BKRW1,BKRW2,BKRW3,CJ1,FG1,FKL1,G1,GT1,S1} and references in them). In particular,
Fiala and Golovach~\cite{FG1} proved
that finding the packing chromatic number of a graph
is NP-hard even in the class of trees. Sloper~\cite{S1} showed that there are graphs with maximum degree $4$ and arbitrarily large
packing chromatic number. In particular, coloring of {\em graph subdivisions} were considered.
For a graph $G$, let $D(G)$ denote the graph obtained from $G$ by subdividing every edge.
The questions on how large can $\chi_p(G)$ and $\chi_p(D(G))$ be if $G$ is a subcubic graph
(i.e., a graph with maximum degree at most $3$) were discussed in several papers (see~\cite{BKRW1,BKRW2,GT1,LBS1,S1}).
In particular, Gastineau and Togni~\cite{GT1} asked whether $\chi_p(D(G))\leq 5$ for every subcubic graph $G$.
Bre\v sar, Klav\v zar, Rall, and Wash~\cite{BKRW2} later conjectured this and proved the validity of their conjecture for some
special classes of subcubic graphs (e.g., the class of generalized Petersen graph).
However, no upper bounds for the whole class of (sub)cubic graphs
were proved in either case. Recently, the authors~\cite{BKL} showed that $\chi_p(G)$ is not bounded in
the class of cubic graphs and that `many' cubic graphs have
`high' packing chromatic number.
In contrast, in this paper we give the first upper bound on $\chi_p(D(G))$ for subcubic $G$: we
show that $\chi_p(D(G))$ is bounded by $8$ in
this class. We will prove the following slightly stronger result.
\begin{thm}\label{t1}
For every connected subcubic graph $G$, the graph $D(G)$ has a packing $8$-coloring such that color $8$ is used at most once.
\end{thm}
The theorem will be proved in the language of $S$-colorings introduced in~\cite{GX1} and used in~\cite{GT1,GX2}.
\begin{defn}
For a non-decreasing sequence $S=(s_1,s_2,\ldots,s_k)$ of positive integers, an $S${\em-coloring} of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1 \le i \le k$ the distance between any two distinct $x,y \in V_i$ is at least $s_i+1$.
\end{defn}
In particular, a $(1,\ldots,1)$-coloring is an ordinary coloring, and a $(1,2,\ldots,k)$-coloring is a packing $k$-coloring. For subcubic graphs, Gastineau and Togni~\cite{GT1} proved that they are $(1,1,2,2,2)$-colorable and $(1,2,2,2,2,2,2)$-colorable. We will use the
following observation of Gastineau and Togni~\cite{GT1}.
\begin{prop}[\cite{GT1} Proposition 1]\label{extension}
Let $G$ be a graph and $S
=(s_1, \ldots,s_k)$ be a non-decreasing sequence of integers. If $G$ is $S$-colorable then $D(G)$ is $(1,2s_1+1, \ldots,2s_k+1)$-colorable.
\end{prop}
In particular, if $G$ is $(1,1,2,2,3,3)$-colorable, then $D(G)$ has a packing $7$-coloring.
In view of this, by a
{\em feasible} coloring of $G$ we call a
coloring of $G$ with colors $1_a,1_b,2_a,2_b,3_a,3_b$ such that the distance between any two distinct vertices of color $i_x$
is at least $i+1$ for all $1\leq i\leq 3$ and $x\in \{a,b\}$.
\begin{defn}
A {\em k-degenerate graph} is a graph in which every subgraph has a vertex of degree at most $k$.
\end{defn}
In the next two sections we discuss feasible coloring
of $2$-degenerate subcubic graphs. In Section~2, we will
show that
if a $2$-degenerate subcubic graph $G$ has a feasible coloring $f$ and $v,u$ are vertices of $G$ with degree at most $2$, then we can
change $f$ to another feasible coloring with some control on the colors of $v$ and $u$. The long proof of one of the lemmas, Lemma~\ref{c2},
is postponed till the last section.
Based on the lemmas of Section~2, in Section~3 we prove
the following theorem (that gives a better bound than Theorem~\ref{t1} but for a more restricted class of graphs).
\begin{thm}\label{t2}
Every $2$-degenerate subcubic graph $G$ has a feasible coloring. In particular,
$D(G)$ has a packing $7$-coloring.
\end{thm}
In Section~4 we use Theorem~\ref{t2} and the lemmas in Section~2 to derive Theorem~\ref{t1}.
In the final section we present a proof of Lemma~\ref{c2}.
\section{Lemmas on feasible coloring}
\begin{defn} For a positive integer $s$ and a vertex $a$ in a graph $G$, the {\em ball $B_{G}(a,s)$ in $G$ of radius $s$ with center $a$}
is $\{v\in V(G)\,:\; d_G(v,a)\leq s\}$, where $d_G(v,a)$ denotes the distance in $G$ between $v$ and $a$. We abbreviate $B_G(a,s)$ to $B(a,s)$ when the graph $G$ is clear from the context.
\end{defn}
\begin{defn}
For a positive integer $k$, a {\em $k$-vertex} is a vertex of degree exactly $k$.
\end{defn}
For $A=\{a_1, \ldots, a_n\} \subseteq V(G)$ and a coloring $f$, by $f(A)$ we mean $\{f(a_1), \ldots, f(a_n)\}$.
\begin{lemma}\label{c1}
Let $G$ be a subcubic graph and $f$ be a feasible coloring of $G$. Suppose there are $2$-vertices $u,v\in V(G)$ with
$f(u)=f(v)=2_a$. Let $N(u)=\{u_1,u_2\}$ and $N(v)=\{v_1,v_2\}$. Then $G$ has a feasible coloring $g$ satisfying one of the following:\\
(a) $g(u)=2_a$ and $g(v)\in \{1_a,1_b\}$ or $g(v)=2_a$ and $g(u)\in \{1_a,1_b\}$;\\
(b) $\{g(u),g(v)\}=\{2_a,2_b\}$;\\
(c) $\{g(u_1),g(u_2)\}=\{g(v_1),g(v_2)\}=\{1_a,1_b\}$, and exactly one of $u,v$ has color $2_a$.
\end{lemma}
{\bf Proof.} If $\{f(u_1),f(u_2)\}\neq \{1_a,1_b\}$, then we recolor $u$ with a color $\alpha\in \{1_a,1_b\}-\{f(u_1),f(u_2)\}$,
and $(a)$ holds. Thus by the symmetry between $u$ and $v$ we may assume
\begin{equation}\label{1ab}
f(u_1)=f(v_1)=1_a\quad \mbox{and}\quad f(u_2)=f(v_2)=1_b.
\end{equation}
Since $f(u)=f(v)=2_a$, $N(u)\cap N(v)=\emptyset$. In other words,
\begin{equation}\label{distinct}
\mbox{\em all vertices $u_1,u_2,v_1$ and $v_2$ are distinct.}
\end{equation}
Let $G_1$ denote the subgraph of $G$ induced by the vertices of colors $1_a$ and $1_b$. If $u_1$ and $u_2$ are in distinct
components of $G_1$, then after switching the colors in the component of $G_1$ containing $u_2$, we obtain a coloring
contradicting~\eqref{1ab}. Thus we may assume
\begin{equation}\label{1ab'}
\mbox{\em $G$ has a $1_a,1_b$-colored $u_1,u_2$-path $P_u$ and a $1_a,1_b$-colored $v_1,v_2$-path $P_v$.}
\end{equation}
{\bf Case 1:} $u_1u_2\in E(G)$. If $|N(u_1)|=3$, then let $u_3\in N(u_1)-\{u,u_2\}$. Similarly, if
$|N(u_2)|=3$, then let $u_4\in N(u_2)-\{u,u_1\}$. If $2_b\notin f(N(u_1)\cup N(u_2))$, then after recoloring $u$ with $2_b$ we get a
coloring satisfying (b). Thus we may assume
\begin{equation}\label{1ab''}
\mbox{\em
$|N(u_1)|=3$ and $f(u_3)=2_b$.}
\end{equation}
Let $N(u_3)\subseteq \{u_1,u_5,u_6\}$. If $2_a\notin f(N(u_3))$, then since $f(u_4)\neq 2_a$ (because
$d(u,u_4)=2$) after switching the colors of $u$ and $u_1$
we obtain a coloring satisfying $(a)$. So we may assume $f(u_5)=2_a$.
\begin{figure}[ht]\label{f1}
\hspace{5mm}
\begin{minipage}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (1.15,0.60) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-1.88,-1.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (13) at (7.3,0.75) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (2.7,-1.4) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (15) at (4.2,-1.2) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (16) at (8.7,-1.2) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (17) at (-1.88,-3.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (0.5,-5.3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-2.8,-5.3) {$2_a$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0.5,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-1,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (2,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-1,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (6) at (-2,-6) {$u_5$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (7) at (0,-6) {$u_6$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (2) at (6.5,0) {$v$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (9) at (5,-2) {$v_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (10) at (8,-2) {$v_2$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\draw (2) edge (9);
\draw (2) edge (10);
\end{tikzpicture}
\caption{Case 1.1.}
\label{case 1.1.}
\end{minipage}
\hspace{15mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.4, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-4,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (15) at (11,1) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-7,-7.3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-5,-7.3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (40) at (-5,-8.2) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (8.5,-7.5) {$A$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-2,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (-3,-7.3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (13.5,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (8.5,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-1,-7.3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (41) at (-1,-8.2) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (-6,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (1,-7.3) {$1_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (6) at (-7,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (7) at (-5,-6) {$u_9$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (2) at (11,0) {$v$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (9) at (8.5,-2) {$v_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (10) at (13.5,-2) {$v_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (11) at (1,-6) {$u_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (12) at (7,-6) {$u_{14}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (13) at (5,-6) {$u_{12}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (14) at (3,-6) {$u_{10}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-2,-4) {$u_5$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (6,-4) {$u_6$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (v1) at (-3,-6) {$u_{11}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v2) at (-1,-6) {$u_{13}$};
\draw (2) edge (9);
\draw (2) edge (10);
\draw (1) edge (3);
\draw (1) edge (4);
\draw (4) edge (8);
\draw (4) edge (v7);
\draw (29) edge (3);
\draw (5) edge (3);
\draw (5) edge (6);
\draw (5) edge (7);
\draw (v1) edge (29);
\draw (29) edge (v2);
\draw (8) edge (11);
\draw (8) edge (14);
\draw (v7) edge (13);
\draw (v7) edge (12);
\draw (5.6,-5.5) ellipse (3.5 and 2.5);
\end{tikzpicture}
\caption{Case 2.1.}
\label{case 2.1.}
\end{minipage}
\end{figure}
{\bf Case 1.1:} $|N(u_2)|<3$ or $f(u_4)\neq 2_b$.
If $1_b\notin f(N(u_3))$, then we can recolor $u_3$ with $1_b$. By the case, we can recolor $u$ with $2_b$ to obtain a coloring satisfying $(b)$. So we may assume $f(u_6)=1_b$ (See Figure~\ref{case 1.1.}).
Then the coloring $g$
obtained from $f$ by recoloring $u$ and $u_3$ with $1_a$ and $u_1$ with $2_b$ satisfies $(a)$.
{\bf Case 1.2:} $|N(u_2)|=3$ and $f(u_4)= 2_b$. If $u_4=u_3$, then $N(u_3)=\{u_1,u_2,u_5\}$. Then $u$ has no vertices of
color $3_a$ at distance at most $3$, so after recoloring $u$ with $3_a$, we obtain a coloring $g$ satisfying (c).
Thus, $u_4\neq u_3$.
{\bf Case 1.2.1:} $1_b\notin f(N(u_3))$. We recolor $u_3$ with $1_b$. If $2_a \notin f(N(u_4)-u_2)$, then we recolor $u_2$ with $2_a$ and $u$ with $1_b$ to obtain a coloring satisfying $(a)$. If $1_a \notin f(N(u_4)-u_2)$, then we recolor $u_4$ with $1_a$, $u_2$ with $2_b$, and $u$ with $1_b$ to obtain a coloring satisfying $(a)$. Thus, we may assume
$$
f(N(u_4)-u_2)=\{1_a,2_a\}.
$$
Then recoloring $u_4$ with $1_b$, $u_2$ with $2_b$, and $u$ with $1_b$, we obtain a coloring satisfying $(a)$.
{\bf Case 1.2.2:} $1_b\in f(N(u_3))$. Since $f(u_5)=2_a$, this means $u_6$ exists and
$f(u_6)=1_b$. Then we recolor $u_3$ and $u_2$ with $1_a$ and $u_1$ with $1_b$. If $2_a \notin f(N(u_4)-u_2)$, then we recolor $u_2$ with $2_a$ and $u$ with $1_a$ to obtain a coloring satisfying $(a)$. If $1_b \notin f(N(u_4)-u_2)$, then we recolor $u_4$ with $1_b$ and $u$ with $2_b$ to obtain a coloring satisfying $(b)$. Thus, we may assume
$$
f(N(u_4)-u_2)=\{1_b,2_a\}.
$$
Then we recolor $u_4$ with $1_a$, $u_2$ with $2_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)$.
{\bf Case 2:} $u_1u_2\notin E(G)$. Then we may assume that $N(u_1)\subseteq \{u,u_3,u_5\}$,
$N(u_2)\subseteq \{u,u_4,u_6\}$ and
by~\eqref{1ab'}, $f(u_3)=1_b$ and $f(u_4)=1_a$. Furthermore, since by the case, $u_3\neq u_2$, we may assume
that $N(u_3)\subseteq \{u_1,u_7,u_9\}$ and $f(u_7)=1_a$. It is possible that $u_7=u_4$, but this will not affect the proof below.
Similarly, we will assume
that $N(u_4)\subseteq \{u_2,u_8,u_{10}\}$ and $f(u_8)=1_b$. As in Case 1, $2_b\in f(N(u_1)\cup N(u_2)),$ since otherwise we can recolor $u$ with $2_b$ and (b) will hold. In our notation,
this means $2_b\in \{f(u_5),f(u_6)\}$. By symmetry, we will assume $f(u_5)=2_b$. We also will assume
$N(u_5)\subseteq \{u_1,u_{11},u_{13}\}$ and $N(u_6)\subseteq \{u_2,u_{12},u_{14}\}$, where some vertices can coincide.
{\bf Case 2.1:} $|N(u_2)|<3$ or $f(u_{6})\neq 2_b$. If $1_b\notin f(N(u_5))$, then we can recolor $u_5$ with $1_b$,
and then $u$ with $2_b$. The resulting coloring satisfies (b). So we may assume
$f(u_{11})=1_b$. If $2_a\notin \{f(u_9),f(u_{13})\}$, then by switching
the colors of $u$ and $u_1$, we obtain a coloring satisfying $(a)$. Thus $2_a\in \{f(u_9),f(u_{13})\}$.
If $f(u_9)=2_a$ and $f(u_{13})\neq 1_a$ or if $f(u_{13})=2_a$ and $f(u_{9})\neq 2_b$, then after switching the colors of $u_1$ and $u_5$ and recoloring $u$ with $1_a$,
we again get a coloring satisfying $(a)$. So,
\begin{equation}\label{u9}
\mbox{\em either
$f(u_9)=2_a$ and $f(u_{13})= 1_a$ or $f(u_{13})=2_a$ and $f(u_{9})= 2_b$.}
\end{equation}
If $u_6$ does not exist, then by~\eqref{u9}, the only vertex in $B(u,3)-(N(u) \cup \{u\})$ that can be colored with $3_a$ or $3_b$
is $u_{10}$. Thus after recoloring $u$ with a color in $\{3_a,3_b\}-f(u_{10})$ we obtain a coloring satisfying (c). So suppose
$u_6$ exists.
Let
$A=\{u_6,u_{10},u_{12},u_{14}\}$. If $1_a\notin \{f(u_{12}),f(u_{14})\}$, then we can recolor $u_6$ with $1_a$ without changing color of any other vertex. Thus we may assume
\begin{equation}\label{u6}
1_a\in f(A).
\end{equation}
If a color $x\in \{2_a,2_b\}$ is not in $f(A)$, then after recoloring $u_2$ with $x$ and $u$ with $1_b$, we get a coloring satisfying $(a)$.
Thus
\begin{equation}\label{u6'}
2_a,2_b\in f(A).
\end{equation}
By the argument above, in particular, by~\eqref{u9}, colors $3_a$ and $3_b$ are not used on vertices in\\
$B=\{u_1,u_2,u_3,u_4,u_5,u_7,u_8,u_9,u_{11},u_{13}\}$. If at least one of them, say $3_a$, is also not used on $A$, then
after recoloring $u$ with $3_a$, we obtain a coloring satisfying (c). Thus
\begin{equation}\label{u6''}
3_a,3_b\in f(A) \text{ (See Figure~\ref{case 2.1.})}.
\end{equation}
Since $|f(A)| \le 4$, relations~\eqref{u6},~\eqref{u6'} and~\eqref{u6''} cannot hold at the same time, a contradiction.
{\bf Case 2.2:} $|N(u_2)|=3$ and $f(u_{6})= 2_b$. Suppose first that $u_{6}=u_{5}$ and that $N(u_5)=\{u_1,u_2,u_{11}\}$.
If $f(u_9)\neq 2_b$ and $f(u_{11})\neq 1_a$, then after switching the colors of $u_1$ and $u_5$ and recoloring $u$ with $1_a$,
we get a coloring satisfying $(a)$. So,
$f(u_9)=2_b$ or $f(u_{11})= 1_a$. Similarly, considering switching colors of $u_2$ and $u_5$, we obtain that $f(u_{10})=2_b$ or $f(u_{11})= 1_b$. Together, this means
\begin{equation}\label{u5}
\mbox{\em the colors of at least two vertices in $\{u_9,u_{10},u_{11}\}$ are in $\{1_a,1_b,2_b\}$.}
\end{equation}
By~\eqref{u5}, some color $y\in \{3_a,3_b\}$ is not used on $B(u,3)$. Then after recoloring
$u$ with $y$, we obtain a coloring satisfying (c).
Now we assume $u_{6}\neq u_{5}$. If $1_a\notin \{f(u_{12}), f(u_{14})\}$, then after recoloring $u_6$ with $1_a$, we
get Case 2.1. Thus below we assume $f(u_{12})=1_a$. If $2_a \notin \{f(u_{10}), f(u_{14})\}$, then we obtain a coloring satisfying $(a)$ by switching the colors of $u$ and $u_2$. Thus, $2_a\in \{f(u_{10}),f(u_{14})\}$. If $f(u_{14}) \neq 1_b$ and $f(u_{10}) \neq 2_b$, then after switching the colors of $u_2$ and $u_6$ and recoloring $u$ with $1_b$,
we again get a coloring satisfying $(a)$. So,
\begin{equation}\label{u10}
\mbox{\em either
$f(u_{10})=2_a$ and $f(u_{14})= 1_b$ or $f(u_{10})=2_b$ and $f(u_{14})= 2_a$.}
\end{equation}
Let $A=\{u_9,u_{11},u_{13}\}$. If $2_a \notin f(A)$, then we obtain a coloring satisfying $(a)$ by switching the colors of $u$ and $u_1$. Thus,
\begin{equation}\label{2afa}
2_a \in f(A).
\end{equation}
If $1_a \notin f(\{u_{11},u_{13}\})$ and $f(u_{9}) \neq 2_b$, then after switching the colors of $u_1$ and $u_5$ and recoloring $u$ with $1_a$,
we again get a coloring satisfying $(a)$. Therefore,
\begin{equation}\label{1a2bfa}
1_a \in f(\{u_{11},u_{13}\})\text{ or } f(u_{9}) = 2_b.
\end{equation}
By the argument above, in particular, by~\eqref{u10}, colors $3_a$ and $3_b$ are not used on vertices in\\
$B=\{u_1,u_2,u_3,u_4,u_5,u_7,u_8,u_{10},u_{12},u_{14}\}$. If at least one of them, say $3_a$, is also not used on $A$, then
after recoloring $u$ with $3_a$, we obtain a coloring satisfying (c). Thus,
\begin{equation}\label{3a3bfa}
3_a,3_b\in f(A).
\end{equation}
Since $|f(A)| \le 3$, relations~\eqref{2afa},~\eqref{1a2bfa}, and~\eqref{3a3bfa} cannot hold at the same time, a contradiction. \hfill\hfill \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi\\
Our second lemma is:
\begin{lemma}\label{c2}
Let $G$ be a subcubic graph and $f$ be a feasible coloring of $G$. Suppose there is a $2$-vertex $u \in V(G)$ with $N(u)=\{u_1,u_2\}$. If $f(u) \in \{3_a,3_b\}$, then we can recolor some vertices of $G$ so that the resulting coloring $g$ is feasible and satisfies the following:\\
(a) $g(u) \notin \{3_a,3_b\}$, and\\
(b) at most one vertex is recolored into $3_a$ or $3_b$, and this vertex (if there is such a vertex) is
at distance at most $3$ from $u$ and has degree $3$ in $G$, and at most one vertex of $f$-color $3_a$ or $3_b$ apart from $u$ is recolored into some other color, and this vertex (if there is such a vertex) has new color in $\{1_a, 1_b\}$.
\end{lemma}
The proof of this lemma is a long case analysis, so we postpone it to the last section.
\section{Proof of Theorem~\ref{t2}}
We prove the theorem by induction on the number $n$ of vertices. When $n \le 6,$ the claim holds obviously, since we have 6 colors. When $n>6$, we assume the argument holds for every graph with fewer than $n$ vertices. Let $G$ be any $2$-degenerate subcubic graph with $n$
vertices. We may assume $G$ is connected.
Since $G$ is $2$-degenerate, it has a vertex, say $w$, with degree at most 2.
{\bf Case 1:} $d(w)=1$. Let $N(w)=w'$. Since $G-w$ is an $(n-1)$-vertex connected subcubic graph with $d_{G-w}(w') \le 2$, by the induction hypothesis, $G-w$ has a $(1,1,2,2,3,3)$-coloring $f$. We color $w$ with a color $x \in \{1_a,1_b\}-f(w')$ to extend $f$ to $G$.
{\bf Case 2:}
$d(w)=2$. Let $N(w)=\{w_1,w_2\}$. Note that $G-w$ has at most two connected components and each connected component is a connected $2$-degenerate subcubic graph with less than $n$ vertices. By the induction hypothesis, $G-w$ has a feasible coloring $f$. We may assume that $|N_{G-w}(w_1)|=|N_{G-w}(w_2)|=2$. Otherwise we can first apply the induction hypothesis to obtain a $(1,1,2,2,3,3)$-coloring $f$ on $G-w$, then add leaves (vertices of degree one) to $w_1$ and $w_2$ to obtain a new graph $G'$ with $|N_{G'-w}(w_1)|=|N_{G'-w}(w_2)|=2$, then assign proper colors to those leaves we just added to obtain a $(1,1,2,2,3,3)$-coloring $f'$ on $G'-w$, then prove that $G'$ has a $(1,1,2,2,3,3)$-coloring, which can be used to get our desired coloring on $G$. So below we assume
$N(w_1)=\{w,w_3,w_4\}$ and $N(w_2)=\{w,w_5,w_6\}$.
By Lemma~\ref{c2}, $G-w$ has a feasible coloring $f_1$ such that $f_1(w_1)\notin \{3_a,3_b\}$. Then by Lemma~\ref{c2}
again, $G-w$ also has a feasible coloring $f_2$ such that $f_2(w_2)\notin \{3_a,3_b\}$ and no vertex of degree $2$
in $G-w$ changed its color to $3_a$ or $3_b$. Thus we also have $f_2(w_1)\notin \{3_a,3_b\}$. Therefore, $G-w$ has a feasible coloring $f_2$ such that $f_2(w_1) \notin \{3_a,3_b\}$ and $f_2(w_2) \notin \{3_a,3_b\}$.
{\bf Case 2.1:} Either $f_2(w_1)\neq f_2(w_2)$ or $f_2(w_1)= f_2(w_2)\in\{1_a,1_b\}$.
If $\{f_2(w_1),f_2(w_2)\} \neq \{1_a,1_b\}$, then we extend $f_2$ to $G$ by assigning $f_2(w)=\alpha \in \{1_a,1_b\}-\{f_2(w_1),f_2(w_2)\}$. By the case, if $f_2(w_1)=f_2(w_2)$, then $f_2(w_1)=f_2(w_2) \in \{1_a,1_b\}$. Therefore, the extension of $f_2$ to $G$ is feasible since we do not introduce new conflicts between $w_1$ and $w_2$ by adding $w$. Thus, we may assume
\begin{equation}\label{w1w2}
f_2(w_1)=1_a \quad \mbox{and}\quad f_2(w_2)=1_b.
\end{equation}
If $w_1$ and $w_2$ are in distinct
components of the subgraph $G_2$ of $G-w$ induced by the vertices of colors $1_a$ and $1_b$ in $f_2$,
then after switching the colors $1_a$ and $1_b$ with each other in the component of $G_2$ containing $w_2$, we obtain a coloring
contradicting~\eqref{w1w2}. Thus we may assume
\begin{equation}\label{1a1b'''}
\mbox{\em $G-w$ has a $1_a,1_b$-colored $w_1,w_2$-path $P_w$.}
\end{equation}
In particular, we may assume $f_2(w_3)=1_b$ and $f_2(w_5)=1_a$
(possibly, $w_3=w_2$ and then $w_5=w_1$).
If $\{2_a,2_b\} \nsubseteq f_2(N(w_1) \cup N(w_2)-\{w\})$, then we can extend $f_2$ to $G$ by assigning $f_2(w)=\beta \in \{2_a,2_b\}-f_2(N(w_1) \cup N(w_2)-\{w\})$. Thus, we may assume
\begin{equation}\label{w1w2'}
|N(w_1)|=|N(w_2)|=3, \{2_a,2_b\} \subseteq f_2(N(w_1) \cup N(w_2)-\{w\}), \text{ and by symmetry }
\end{equation}
\begin{equation}\label{w3w4}
f_2(w_4)=2_a \quad \mbox{and}\quad f_2(w_6)=2_b.
\end{equation}
If $1_b \notin f_2(N(w_4)-w_2)$, then we can extend $f_2$ to a feasible coloring of $G$ by recoloring $w_4$ with
$1_b$ and letting $f_2(w)=2_a$. By this and the symmetric statement for $w_6$ we can assume that
\begin{equation}\label{w3w4n}
\mbox{ \em $w_4$ has a neighbor $w_7$ with $f_2(w_7)=1_b$ and $w_6$ has a neighbor $w_8$ with $f_2(w_8)=1_a$.}
\end{equation}
{\bf Case 2.1.1:} $w_1w_2 \in E(G)$ (i.e.,
$w_3=w_2$ and $w_5=w_1$).
If $1_a \notin f_2(N(w_4)-w_1)$, then we obtain a feasible coloring on $G$ by switching colors of $w_1$ and $w_4$, assigning $1_a$ to $w$, and using $f_2$ on other vertices. Therefore, by \eqref{w3w4n}, we may assume $f_2(N(w_4)-w_1)=\{1_a,1_b\}$. Similarly, by \eqref{w3w4n}, we may assume $f_2(N(w_6)-w_2)=\{1_a,1_b\}$ (See Figure~\ref{case 2.1.1.}). With \eqref{w1w2}, \eqref{w3w4}, and the case, $3_a \notin f_2(B(w,3)-\{w\})$ and we can extend $f_2$ to $G$ by assigning $f_2(w)=3_a.$
{\bf Case 2.1.2:} $w_1w_2 \notin E(G).$ If $N(w_3)\cup N(w_4)$ does not contain a vertex $w_9$ of color $2_b$, then we can
recolor $w_1$ with $2_b$ and color $w$ with $1_a$. So we may assume that $N(w_3)\cup N(w_4)$ contains a vertex $w_9$ of color $2_b$ and symmetrically $N(w_5)\cup N(w_6)$ contains a vertex $w_{10}$ of color $2_a$. Furthermore, if
$1_a \notin f_2(N(w_4)-w_1)$ and $2_a \notin f_2(N(w_3)-w_1)$, then we can recolor $w_1$ with $2_a$
and color $w$ and $w_4$ with $1_a$. With \eqref{1a1b'''} and \eqref{w3w4n}, all vertices in $B(w_1,2)-w$ have colors in $\{1_a,1_b,2_a,2_b\}$.
Symmetrically, we can assume all vertices in $B(w_2,2)-w$ have colors in $\{1_a,1_b,2_a,2_b\}$ (See Figure~\ref{case 2.1.2.}). Then we can color $w$ with $3_a$.
\begin{figure}[ht]\label{f8}
\hspace{15mm}
\begin{minipage}[b]{0.25\textwidth}
\begin{tikzpicture}[scale=0.6, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0.5,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-1.9,-1.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (2.8,-1.4) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (16) at (2.5,-6) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (17) at (2.8,-3.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (-0.5,-6) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-1.9,-3.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (1.5,-6) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-1.5,-6) {$1_a$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0.5,0) {$w$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-1,-2) { $w_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (2,-2) {$w_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-1,-4) {$w_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2,-4) {$w_6$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\node (v1) at (-1.5,-5.5) {};
\node (v2) at (-0.5,-5.5) {};
\node (v3) at (1.5,-5.5) {};
\node (v4) at (2.5,-5.5) {};
\draw (5) edge (v1);
\draw (5) edge (v2);
\draw (v3) edge (8);
\draw (8) edge (v4);
\end{tikzpicture}
\caption{Case 2.1.1.}
\label{case 2.1.1.}
\end{minipage}
\hspace{15mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-4,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (15) at (6,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-6.5,-6.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-5.5,-6.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (40) at (-2.5,-6.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-2,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (-5.5,-7.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (1.5,-6.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (5.5,-6.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (42) at (2.5,-7.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (43) at (6.5,-7.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (44) at (6.5,-6.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-2.5,-7.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (41) at (2.5,-6.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (-6,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-1.5,-6.5) {$1_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$w$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $w_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$w_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$w_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2,-4) {$w_5$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-2,-4) {$w_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (6,-4) {$w_6$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (4) edge (8);
\draw (4) edge (v7);
\draw (29) edge (3);
\draw (5) edge (3);
\node (v1) at (-6.5,-6) {};
\node (v2) at (-5.5,-6) {};
\node (v3) at (-2.5,-6) {};
\node (v4) at (-1.5,-6) {};
\node (v5) at (1.5,-6) {};
\node (v6) at (2.5,-6) {};
\node (v8) at (5.5,-6) {};
\node (v9) at (6.5,-6) {};
\draw (5) edge (v1);
\draw (5) edge (v2);
\draw (29) edge (v3);
\draw (29) edge (v4);
\draw (8) edge (v5);
\draw (8) edge (v6);
\draw (v7) edge (v8);
\draw (v7) edge (v9);
\end{tikzpicture}
\caption{Case 2.1.2.}
\label{case 2.1.2.}
\end{minipage}
\end{figure}
\medskip
By the choice of $f_2$ and the symmetry of $2_a$ and $2_b$, the remaining case is:
{\bf Case 2.2:} $f_2(w_1)= f_2(w_2)=2_a$. In particular, this means $w_1w_2\notin E(G)$.
By Lemma~\ref{c1}, $G-w$ has a coloring $g$ satisfying one of the following:\\
(a) $g(w_1)=2_a$ and $g(w_2)\in \{1_a,1_b\}$ or $g(w_2)=2_a$ and $g(w_1)\in \{1_a,1_b\}$;\\
(b) $\{g(w_1),g(w_2)\}=\{2_a,2_b\}$;\\
(c) $\{g(w_3),g(w_4)\}=\{g(w_5),g(w_6)\}=\{1_a,1_b\}$, and exactly one of $w_1,w_2$ has color $2_a$.
If (a) or (b) occurs, then we again get Case 1. We do not get Case 1 only if (c) occurs and one of $w_1,w_2$ has
$g$-color in $\{3_a,3_b\}$. But then $2_b$ is not present in $B(w,2)$ and we can color $w$ with $2_b$.
\hfill\hfill \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi\\
\section{Cubic graphs}
\iffalse
\begin{lemma}\label{connected}
Let $G$ be a connected graph with at least two vertices, then there is at least two vertices $u,v$ such that they are both not cut vertices.
\end{lemma}
{\bf Proof.}
We pick a longest path $P$ in $G$ and let $u,v$ be its two endpoints. We claim that both $u$ and $v$ are not cut vertices. Since $u$ and $v$ can only have neighbors in $V(P)$, let $w \in V(G)-V(P)$, then every path from $w$ to $P$ do not go through $u$ or $v$. Therefore, if we delete $u$ or $v$ then the remaining graph is still connected.
\hfill \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi\\
\fi
A {\em good} coloring is a $(1,1,2,2,3,3,4)$-coloring with color $4$ used at most once. By Proposition~\ref{extension},
Theorem~\ref{t1} follows from the following fact.
\begin{thm}\label{1122334}
Every connected cubic graph has a good coloring.
\end{thm}
{\bf Proof.}
Let $G$ be a connected cubic graph with $n\geq 2$ vertices.
Since $G$ is connected, it has a non-cut vertex $w$ (simply take a leaf vertex of a spanning tree of $G$). Let $N(w)=\{w_1,w_2,w_3\}$.
{\bf Case 1:} $0 \le |E(G[\{w_1,w_2,w_3\}])| \le 1$. If $|E(G[\{w_1,w_2,w_3\}])|=0$, then let $G'=G-w+w_2w_3$. If $|E(G[\{w_1,w_2,w_3\}])|=1$, then by symmetry we may assume $w_2w_3 \in E(G)$. Let $G'=G-w$. Note that $G'$ is a connected subcubic graph with vertex $w_1$ of degree at most two. By Theorem~\ref{t2},
$G'$ has a feasible coloring. Hence by Lemma~\ref{c2}, $G'$ has a feasible coloring $f$ with
\begin{equation}\label{eeq0}
f(w_1)\notin \{3_a,3_b\}.
\end{equation}
Let $N_{G'}(w_1)=\{w_4,w_5\}$, $N_{G'}(w_2)=\{w_3,w_6,w_7\}$, and $N_{G'}(w_3)=\{w_2,w_8,w_9\}.$ It is possible that $|\{w_4,w_5,w_6,w_7,w_8,w_9\}|<6$, but this will not affect the proof below.
For $j\in\{1,2,3\}$ and $x,y\in V(G)-w$, a $(j,x,y)$-{\em conflict} in $(G,f)$ is the situation that
$f(x)=f(y)\in \{j_a,j_b\}$ and $d_G(x,y) \le j$. If $(G,f)$ has no $(j,x,y)$-conflicts for any $j\in\{1,2,3\}$ and $x,y\in V(G)-w$,
then we can extend $f$ to a good coloring of $G$ by letting $f(w)=4$.
Suppose now that $(G,f)$ has a $(j,x,y)$-conflict for some $j\in\{1,2,3\}$ and $x,y\in V(G)-w$ (there could be more than one conflict).
Then
\begin{equation}\label{eeq1}
\mbox{\em $d_{G}(x,y)\leq j<d_{G'}(x,y)$. This means $\{x,y\}\cap \{w_1,w_2,w_3\}\neq \emptyset$ and $j\geq 2$. }
\end{equation}
Since $w_2w_3\in E(G')$,~(\ref{eeq1}) yields that in each $(j,x,y)$-conflict, one of
$x$ and $y$ is in $ \{w_1,w_4,w_5\}$ and the other is in $ \{w_2,w_3,w_6,w_7,w_8,w_9\}$.
By~(\ref{eeq0}), we have the following two cases.
{\bf Case 1.1:} $f(w_1)\in \{1_a,1_b\}$, say $f(w_1)=1_a$. Then each conflict is a $(3,x,y)$-conflict.
{\bf Case 1.1.1:} There is only one conflict. We may assume it is a $(3,w_4,w_2)$-conflict,
where $f(w_4)=f(w_2)=3_a$. If $f(N_G(w_2)-w)\neq \{1_a,1_b\}$, then we can recolor $w_2$ with one of $1_a$ and $1_b$ and
eliminate the conflict.
If $f(w_3)\neq 1_b$, then we can recolor $w_4$ with $4$ and color $w$ with $1_b$.
So we may assume
\begin{equation}\label{eeq11}
\mbox{\em $f(N_G(w_2)-w)= \{1_a,1_b\}$ \quad and \quad $f(w_3)=1_b$. }
\end{equation}
Furthermore, if $f(w_5)\neq 1_b$ or $1_a \notin f(N_G(w_3)-w)$, then we can recolor $w_1$ and $w_3$
with the same color $\alpha\in \{1_a,1_b\}$, recolor $w_4$ with $4$ and color $w$ with $\beta\in \{1_a,1_b\}-\alpha$.
Otherwise, some $\gamma\in \{2_a,2_b\}$ is not present on $N(w_3)\cup \{w_5\}$, and by~(\ref{eeq11}) we can
recolor $w_4$ with $4$ and color $w$ with $\gamma$ (See Figure~\ref{case 1.1.1.}).
{\bf Case 1.1.2:} There are two conflicts. By the case and symmetry, we may assume
$f(w_4)=f(w_2)=3_a$ and $f(w_5)=f(w_3)=3_b$. Applying Lemma~\ref{c2} to vertex $w_2$ and coloring $f$ of
$G-w$, we obtain a feasible coloring $g$ of $G-w$ such that $g(w_2)=\gamma \notin \{3_a,3_b\}$ and at most one of
$w_3,w_4,w_5$ changed its color.
{\bf Case 1.1.2.1:} Neither $w_4$ nor $w_5$ changed its color. Then we color $w_3$ with color $4$, $w$ with a color $\beta \in \{1_a,1_b\}-\gamma$, $w_1$ with a color $\alpha \in \{1_a,1_b\}-\beta$, and use $g$ on other vertices.
{\bf Case 1.1.2.2:} One vertex of $\{w_4,w_5\}$ changed its color. We prove the case when $w_4$ changed its color, say $g(w_4)=\beta \in \{1_a,1_b\}$, the case $w_5$ changed its color is similar. We may assume that
\begin{equation}\label{gammabeta}
g(w_2)=\gamma \in \{1_a,1_b\}\quad \mbox{and}\quad \gamma = \beta,
\end{equation}
since otherwise we color $w_1$ with a color $\alpha \in \{1_a,1_b\}-\beta$, $w$ with a color $\mu \in \{1_a,1_b\}-\alpha$, $w_3$ with color $4$, and use $g$ on other vertices. We may also assume that some vertex, say $w_6 \in N(w_2)-w$, have color $\delta \in \{1_a,1_b\}-\gamma$, since otherwise we recolor $w_2$ with $\delta$ and it contradicts ~\eqref{gammabeta}. We may also assume that $g(\{w_8,w_9\})=\{1_a,1_b\}$, since otherwise we color $w_3$ with a color $\mu \in \{1_a,1_b\}-g(\{w_8,w_9\})$, $w$ with color $4$, and use $f$ on other vertices (See Figure~\ref{case 1.1.2.2.}). Note that $|g(N(w) \cup N(N(w))) \cap \{2_a,2_b\}| \le 1$. Then we color $w_1$ with a color $\alpha \in \{1_a,1_b\}-\beta$, $w_3$ with color $4$, $w$ with a color $\lambda \in \{2_a,2_b\}-g(N(w) \cup N(N(w)))$, and use $g$ on other vertices to obtain a good coloring.
\begin{figure}[ht]\label{f9}
\hspace{5mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$\gamma$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-5,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (5,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (15) at (6.5,-3) {$2_a /2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-1,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-6,-3) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (4,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (1,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (0.5,-1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-4,-3) {$1_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$w$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-5,-2) { $w_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (5,-2) {$w_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$w_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (4,-4) {$w_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-4,-4) {$w_5$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (6,-4) {$w_9$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (4) edge (8);
\draw (4) edge (v7);
\draw (29) edge (3);
\draw (5) edge (3);
\node (v1) at (-6.5,-6) {};
\node (v2) at (-5.5,-6) {};
\node (v3) at (-4.5,-6) {};
\node (v4) at (-3.5,-6) {};
\node (v5) at (3.5,-6) {};
\node (v6) at (4.5,-6) {};
\node (v8) at (5.5,-6) {};
\node (v9) at (6.5,-6) {};
\draw (5) edge (v1);
\draw (5) edge (v2);
\draw (29) edge (v3);
\draw (29) edge (v4);
\draw (8) edge (v5);
\draw (8) edge (v6);
\draw (v7) edge (v8);
\draw (v7) edge (v9);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v10) at (0,-2) {$w_2$};
\draw (1) edge (v10);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v11) at (-1,-4) {$w_6$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v12) at (1,-4) {$w_7$};
\draw (v10) edge (v11);
\draw (v10) edge (v12);
\draw (v10) edge (4);
\node (v13) at (-1.5,-6) {};
\node (v14) at (-0.5,-6) {};
\node (v15) at (0.5,-6) {};
\node (v16) at (1.5,-6) {};
\draw (v11) edge (v13);
\draw (v11) edge (v14);
\draw (v12) edge (v15);
\draw (v12) edge (v16);
\end{tikzpicture}
\caption{Case 1.1.1.}
\label{case 1.1.1.}
\end{minipage}
\hspace{15mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$\lambda$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-5,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (5,-1) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (15) at (6,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-1,-3) {$\delta$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-6,-3) {$\beta$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (4,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (1.5,-3) {$2_a /2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (0.5,-1) {$\gamma$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-4,-3) {$3_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$w$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-5,-2) { $w_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (5,-2) {$w_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$w_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (4,-4) {$w_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-4,-4) {$w_5$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (6,-4) {$w_9$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (4) edge (8);
\draw (4) edge (v7);
\draw (29) edge (3);
\draw (5) edge (3);
\node (v1) at (-6.5,-6) {};
\node (v2) at (-5.5,-6) {};
\node (v3) at (-4.5,-6) {};
\node (v4) at (-3.5,-6) {};
\node (v5) at (3.5,-6) {};
\node (v6) at (4.5,-6) {};
\node (v8) at (5.5,-6) {};
\node (v9) at (6.5,-6) {};
\draw (5) edge (v1);
\draw (5) edge (v2);
\draw (29) edge (v3);
\draw (29) edge (v4);
\draw (8) edge (v5);
\draw (8) edge (v6);
\draw (v7) edge (v8);
\draw (v7) edge (v9);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v10) at (0,-2) {$w_2$};
\draw (1) edge (v10);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v11) at (-1,-4) {$w_6$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v12) at (1,-4) {$w_7$};
\draw (v10) edge (v11);
\draw (v10) edge (v12);
\draw (v10) edge (4);
\node (v13) at (-1.5,-6) {};
\node (v14) at (-0.5,-6) {};
\node (v15) at (0.5,-6) {};
\node (v16) at (1.5,-6) {};
\draw (v11) edge (v13);
\draw (v11) edge (v14);
\draw (v12) edge (v15);
\draw (v12) edge (v16);
\end{tikzpicture}
\caption{Case 1.1.2.2.}
\label{case 1.1.2.2.}
\end{minipage}
\end{figure}
{\bf Case 1.2:} $f(w_1)\in \{2_a,2_b\}$, say $f(w_1)=2_a$. Since we cannot switch to Case 1.1, we need
$\{f(w_4),f(w_5)\}= \{1_a,1_b\}$. So the only possible conflict is a $(2,w_1,y)$-conflict, where $y\in \{w_2,w_3\}$.
We may assume $f(w_2)=2_a$. Then we recolor $w_1$ with $4$ and color $w$ with
$\alpha\in \{1_a,1_b\}-f(w_3)$.
{\bf Case 2:} $|E(G[\{w_1,w_2,w_3\}])|=2$, say $w_1w_2 \in E(G)$ and $w_2w_3 \in E(G)$. We obtain a good coloring $g$ of $G$ by using $f$ on $G-w$ and assigning color $4$ to $w$. Note that adding $w$ back will not create conflicts because the distance between any two vertices in $G-w$ remains the same.
{\bf Case 3:} $G[\{w_1,w_2,w_3\}]=K_3$. Then $G=K_4$, and $K_4$ has a good coloring. \hfill\hfill \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi\\
\section{Proof of Lemma~\ref{c2} }
Recall the claim of the lemma:
\medskip\noindent
{\bf Lemma~\ref{c2}.} {\em
Let $G$ be a subcubic graph and $f$ be a feasible coloring of $G$. Suppose there is a $2$-vertex $u \in V(G)$ with $N(u)=\{u_1,u_2\}$. If $f(u) \in \{3_a,3_b\}$, then we can recolor some vertices of $G$ so that the resulting coloring $g$ is feasible and satisfies the following:\\
(a) $g(u) \notin \{3_a,3_b\}$, and\\
(b) at most one vertex is recolored into $3_a$ or $3_b$, and this vertex (if there is such a vertex) is
at distance at most $3$ from $u$ and has degree $3$ in $G$, and at most one vertex of $f$-color $3_a$ or $3_b$ apart from $u$ is recolored into some other color, and this vertex (if there is such a vertex) has new color in $\{1_a, 1_b\}$. }
\bigskip
{\bf Proof.} Without loss of generality, we assume that $f(u)=3_a$. If $\{f(u_1),f(u_2)\} \neq \{1_a,1_b\}$, then we recolor $u$ with a color $x \in \{1_a,1_b\}-\{f(u_1),f(u_2)\}$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus we may assume
\begin{equation}\label{1a1b}
f(u_1)=1_a \quad \mbox{and}\quad f(u_2)=1_b.
\end{equation}
Let $G_1$ denote the subgraph of $G$ induced by the vertices of colors $1_a$ and $1_b$. If $u_1$ and $u_2$ are in distinct
components of $G_1$, then after switching the colors in the component of $G_1$ containing $u_2$, we obtain a coloring
contradicting~\eqref{1a1b}. Thus we may assume
\begin{equation}\label{1a1b'}
\mbox{\em $G$ has a $1_a,1_b$-colored $u_1,u_2$-path $P_u$.}
\end{equation}
{\bf Case 1:} $u_1u_2\in E(G)$. If $|N(u_1)|=3$, then let $u_3\in N(u_1)-\{u,u_2\}$. Similarly, if
$|N(u_2)|=3$, then let $u_4\in N(u_2)-\{u,u_1\}$. If $\{2_a, 2_b\} \nsubseteq f(N(u_1)\cup N(u_2))$, then after recoloring $u$ with a color $x \in \{2_a, 2_b\}-f(N(u_1)\cup N(u_2))$ we obtain a
coloring satisfying $(a)\text{ and }(b)$. By symmetry, we may assume
\begin{equation}\label{u3u41}
\mbox{\em
$|N(u_1)|=|N(u_2)|=3$, \quad $f(u_3)=2_a$ \quad and \quad $f(u_4) = 2_b$.}
\end{equation}
If $1_b \notin f(N(u_3))$, then we can recolor $u_3$ with $1_b$ and $u$ with $2_a$ to obtain a
coloring satisfying $(a)\text{ and }(b)$. So we may assume $1_b \in f(N(u_3))$. Similarly, we may assume $1_a \in f(N(u_4))$. If $|N(u_3)|=2$ or $1_a \notin f(N(u_3)-\{u_1\})$, then we can recolor $u_3$ with $1_a$, $u_1$ with $2_a$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. So we may assume
\begin{equation}\label{u5u6}
|N(u_3)|=3 \text{ and let }u_5,u_6 \in N(u_3)-\{u_1\} \text{ with } f(u_5)=1_a, f(u_6)=1_b.
\end{equation}
Similarly, we may assume
\begin{equation}\label{u7u8}
|N(u_4)|=3 \text{ and let } u_7,u_8 \in N(u_4)-\{u_2\} \text{ with } f(u_7)=1_a, f(u_8)=1_b.
\end{equation}
{\bf Case 1.1:} $u_5=u_7 \text{ and } u_6=u_8$. If $1_b \notin f(N(u_5))$, then we can recolor $u_5$ with $1_b$, $u_3$ with $1_a$, $u_1$ with $2_a$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. So we may assume $1_b \in f(N(u_5))$. Similarly, we may assume $1_a \in f(N(u_6))$. Then we can recolor $u_1$ with $3_a$ and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
{\bf Case 1.2:} $u_5=u_7$ or $u_6=u_8$, but not both. By symmetry, we may assume $u_6 = u_8$ and $u_5 \neq u_7$. It is possible that $u_5u_6 \in E(G)$ or $u_6u_7 \in E(G)$, but this will not affect the proof below.
Similarly to Case 1.1, we may assume
\begin{equation}\label{u5u6u7}
1_b \in f(N(u_5)), \quad 1_a \in f(N(u_6)) \quad \mbox{and}\quad 1_b \in f(N(u_7)).
\end{equation}
Since $3_a \notin f(N(u_6))$, we can also assume $3_a \in f(N(u_5))$, because otherwise we recolor $u_1$ with $3_a$ and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. With \eqref{u3u41} and \eqref{u5u6u7}, we have $f(N(u_5))=\{1_b,2_a,3_a\}$. However, we can recolor $u_1$ with $3_b$ and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
{\bf Case 1.3:} $u_5 \neq u_7$ and $u_6 \neq u_8$. Then $N(u_3) \cap N(u_4) = \emptyset$ and $d(u_3,u_4)\ge 3$. Similarly to Case 1.2, $\{1_a,1_b,3_a,3_b\} \subseteq f(N(u_5) \cup N(u_6) - \{u_3\})$ (See Figure~\ref{2-case 1.3.}). Therefore, we can recolor $u_3$ with $2_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
\begin{figure}[ht]\label{f2}
\centering
\begin{minipage}[b]{0.3\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (1.1,0.7) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-2,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (3,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (17) at (-2.85,-3.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (-0.45,-5.1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-3.5,-5.1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (3.8,-3.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (4.5,-5.1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (1.8,-5.1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-3.5,-8.5) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (-2.5,-8.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (27) at (-1.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (28) at (-0.5,-8.5) {$3_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0.5,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-2,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (3,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-2,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (6) at (-3,-6) {$u_5$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (7) at (-1,-6) {$u_6$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (3,-4) {$u_4$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v1) at (2,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v2) at (4,-6) {$u_8$};
\draw (8) edge (v1);
\draw (8) edge (v2);
\node (v3) at (-3.5,-8) {};
\node (v4) at (-2.5,-8) {};
\node (v5) at (-1.5,-8) {};
\node (v6) at (-0.5,-8) {};
\draw (6) edge (v3);
\draw (6) edge (v4);
\draw (7) edge (v5);
\draw (7) edge (v6);
\end{tikzpicture}
\caption{Case 1.3.}
\label{2-case 1.3.}
\end{minipage}
\hspace{15mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.4, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-4.15,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (17) at (-6,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (-2,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-3.05,-4.85) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-1,-4.85) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (5.5,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-7.05,-4.85) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (-5,-4.85) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2.5,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-4.5,-8.5) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (-6.5,-8.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (27) at (-7.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (28) at (-0.5,-8.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-5.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (31) at (-2.5,-8.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-1.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (33) at (-3.5,-8.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (34) at (-0.5,-9.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (35) at (-3.5,-9.5) {$3_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (6) at (-7,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (7) at (-5,-6) {$u_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2.5,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node (v3) at (-7.5,-8) {};
\node (v4) at (-6.5,-8) {};
\node (v5) at (-5.5,-8) {};
\node (v6) at (-4.5,-8) {};
\draw (6) edge (v3);
\draw (6) edge (v4);
\draw (7) edge (v5);
\draw (7) edge (v6);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-2,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (5.5,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (v1) at (-3,-6) {$u_9$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v2) at (-1,-6) {$u_{10}$};
\draw (29) edge (v1);
\draw (29) edge (v2);
\node (v8) at (-3.5,-8) {};
\node (v9) at (-2.5,-8) {};
\node (v10) at (-1.5,-8) {};
\node (v11) at (-0.5,-8) {};
\draw (v1) edge (v8);
\draw (v1) edge (v9);
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\end{tikzpicture}
\caption{Case 2.1.}
\label{2-case 2.1.}
\end{minipage}
\end{figure}
{\bf Case 2:} $u_1u_2\notin E(G)$. If $\{2_a, 2_b\} \nsubseteq f(N(u_1)\cup N(u_2))$, then after recoloring $u$ with a color $x \in \{2_a, 2_b\}-f(N(u_1)\cup N(u_2))$ we obtain a coloring satisfying $(a)\text{ and }(b)$. With \eqref{1a1b'}, we may assume that
\begin{equation}\label{u1u3u4}
N(u_1)=\{u,u_3,u_4\}, \quad f(u_3) = 2_a, \quad f(u_4) = 1_b,
\end{equation}
\begin{equation}\label{u2u5u6}
N(u_2)=\{u,u_5,u_6\}, \quad f(u_5) = 1_a \quad \mbox{and}\quad f(u_6) = 2_b.
\end{equation}
If $u_3u_4 \in E(G)$, then $1_a \in f(N(u_4)-\{u_1,u_3\})$ because of \eqref{1a1b'}. We also have $2_b \in f(N(u_3)-\{u_1,u_4\})$ because otherwise we can recolor $u_1$ with $2_b$ and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume $|N(u_3)|=|N(u_4)|=3$ and let $u_7 \in N(u_3)-\{u_1,u_4\}, u_8 \in N(u_4)-\{u_1,u_3\}$, $f(u_7)=2_b$, and $f(u_8) = 1_a$. Then, we can recolor $u_1$ with $2_a$, $u_3$ with $1_a$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Because of symmetry, we may assume
\begin{equation}\label{u3u4u5u6}
u_3u_4 \notin E(G)\quad \mbox{and}\quad u_5u_6 \notin E(G).
\end{equation}
If $1_b \notin f(N(u_3))$, then we recolor $u_3$ with $1_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. With \eqref{1a1b'}, we may assume that
\begin{equation}\label{1bu31au4}
1_b \in f(N(u_3))\quad \mbox{and}\quad 1_a \in f(N(u_4)).
\end{equation}
If $2_b \notin f(B(u_1,2))$, then we can recolor $u_1$ with $2_b$ and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{2bu3u4}
2_b \in f(N(u_3)) \cup f(N(u_4)).
\end{equation}
If $1_a \notin f(N(u_3)-\{u_1\})$ and $2_a \notin f(N(u_4))$, then we can recolor $u_3$ with $1_a$, $u_1$ with $2_a$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{u3u43}
|N(u_3)|=|N(u_4)|=3
\end{equation}
and
\begin{equation}\label{u3u4}
1_a \in f(N(u_3)-\{u_1\})\text{ or }2_a \in f(N(u_4)).
\end{equation}
Let $\{u_7,u_8\} \in N(u_3)$, $\{u_9,u_{10}\} \in N(u_4)$. By \eqref{1bu31au4}, we may assume
\begin{equation}\label{u8u9}
f(u_8)=1_b \quad \mbox{and}\quad f(u_9)=1_a.
\end{equation}
By \eqref{2bu3u4} and \eqref{u3u4}, we have
\begin{equation}\label{cases}
\mbox{\em either
$f(u_7)=2_b$ and $f(u_{10})=2_a$ or $f(u_7)=1_a$ and $f(u_{10})=2_b$.}
\end{equation}
If $3_a \notin f(B(u_1,3)-\{u\})$, then we can recolor $u_1$ with $3_a$ and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{3a}
3_a \in f(B(u_1,3)-\{u\}).
\end{equation}
Similarly, we may assume
\begin{equation}\label{3b}
3_b \in f(B(u_1,3)-\{u\}).
\end{equation}
{\bf Case 2.1:} $f(u_7)=2_b$ and $f(u_{10})=2_a$. By \eqref{u3u4u5u6} and $|N(u_2)|=3$, we have $$\{u_8,u_{10}\} \cap (\{u_i: i \in [6]\} \cup \{u\}) = \emptyset.$$ It is possible that $u_9=u_5$ or $u_7=u_6$, but this will not affect the proof below.
If $2_b \notin f(B(u_4,2))$, then we can recolor $u_4$ with $2_b$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{2bnnu4}
2_b \in f(B(u_4,2)).
\end{equation}
If $1_a \notin f(N(u_{10}))$, then we can recolor $u_{10}$ with $1_a$ and it contradicts \eqref{u3u4}. Thus, we may assume
\begin{equation}\label{1anu10}
1_a \in f(N(u_{10})).
\end{equation}
We may also assume
\begin{equation}\label{1a1bu7}
f(N(u_7)-\{u_3\}) = \{1_a,1_b\},
\end{equation}
because otherwise we can recolor $u_7$ with a color $x \in \{1_a,1_b\}-f(N(u_7)-\{u_1\})$ and it contradicts \eqref{cases}.
By \eqref{3a} and \eqref{3b}, we know that
\begin{equation}\label{3a3b78910}
\{3_a,3_b\} \subseteq f(N(u_7) \cup N(u_8) \cup N(u_9) \cup N(u_{10})).
\end{equation}
If $\{3_a,3_b\} \subseteq f(N(u_7) \cup N(u_8))$, then by \eqref{1a1bu7} we have $f(N(u_8))=\{2_a,3_a,3_b\}$. Then, we can recolor $u_8$ with $1_a$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. By symmetry, we may assume
\begin{equation}\label{3bu7u8}
3_b \notin f(N(u_7) \cup N(u_8)).
\end{equation}
By \eqref{3a3b78910} and \eqref{3bu7u8}, we know that $3_b \in f(N(u_9) \cup N(u_{10}))$. By \eqref{1a1b'}, $1_b \in f(N(u_9)-\{u_4\})$. With \eqref{2bnnu4}, \eqref{1anu10}, and $2_b \notin f(\{u,u_1,u_3,u_9,u_{10}\})$ we know that $$f(N(u_9) \cup N(u_{10})-\{u_4\}) = \{1_a,1_b,2_b,3_b\}\text{, hence }1_b \notin f(N(u_{10})-\{u_4\}) \text{ (See Figure~\ref{2-case 2.1.})}.$$ Therefore, we can recolor $u_{10}$ with $1_b$, $u_4$ with $2_a$, $u_3$ with $1_a$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
{\bf Case 2.2:} $f(u_7)=1_a$ and $f(u_{10})=2_b$. If $1_a \notin f(N(u_6))$, then we can recolor $u_6$ with $1_a$ and $u$ with $2_b$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{1au6}
1_a \in f(N(u_6)-\{u_2\}).
\end{equation}
Since some $u_i$ and $u_j$ may coincide, several cases are considered below.
{\bf Case 2.2.1}: $u_3u_5 \in E(G)$, i.e., $u_7=u_5$. It is possible that $u_4u_6 \in E(G)$, or $u_4u_5 \in E(G)$, or $\{u_4u_5, u_4u_6\} \subseteq E(G)$, but this will not affect the proof below. By \eqref{1a1b'},
\begin{equation}\label{1bu9}
1_b \in f(N(u_9)-\{u_4\}),
\end{equation}
and
\begin{equation}\label{1bu5}
1_b \in f(N(u_5)-\{u_2\}).
\end{equation}
If $1_a \notin f(N(u_{10})-\{u_4\})$, then we can recolor $u_{10}$ with $1_a$ and it contradicts \eqref{cases}. Thus, we may assume
\begin{equation}\label{1au10}
1_a \in f(N(u_{10})-\{u_4\}).
\end{equation}
If $1_a \notin f(N(u_8))$, then we can recolor $u_8$ with $1_a$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. If $2_b \notin f(N(u_8))$, then we can recolor $u_3$ with $2_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{u8}
f(N(u_8))=\{1_a,2_a,2_b\}.
\end{equation}
By \eqref{3a}, \eqref{3b}, \eqref{1bu9}, \eqref{1bu5}, \eqref{1au10}, and \eqref{u8}, we have
\begin{equation}\label{1a1b3a3b}
\{1_a,1_b,3_a,3_b\} \subseteq f(N(u_{9}) \cup N(u_{10})-\{u_4\}).
\end{equation}
By \eqref{1a1b3a3b}, $1_b \notin f(N(u_{10})-\{u_4\}),$ and $2_b \notin f(B(u_4,2)-\{u_{10}\})$ (See Figure~\ref{2-case 2.2.1.}). Then, we can recolor $u_{10}$ with $1_b$, $u_4$ with $2_b$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
With Case 2.2.1 handled, from now on by symmetry we may assume
\begin{equation}\label{u3u5u4u6}
u_3u_5 \notin E(G)\quad \mbox{and}\quad u_4u_6 \notin E(G).
\end{equation}
\begin{figure}[ht]\label{f3}
\hspace{5mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-4.15,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (2.5,-6) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (5.5,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-5,-5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2.5,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-7.5,-8.5) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (-6.5,-8.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-5.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (31) at (-6,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-2.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (34) at (-1.5,-8.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (35) at (-4.5,-8.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (38) at (-7,-5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-1.5,-5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (40) at (-2,-3) {$2_a$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (6) at (-7,-6) {$u_9$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (7) at (-5,-6) {$u_{10}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2.5,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node (v3) at (-7.5,-8) {};
\node (v4) at (-6.5,-8) {};
\node (v5) at (-5.5,-8) {};
\node (v6) at (-4.5,-8) {};
\draw (6) edge (v3);
\draw (6) edge (v4);
\draw (7) edge (v5);
\draw (7) edge (v6);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-2,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (5.5,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v2) at (-2,-6) {$u_{8}$};
\draw (29) edge (v2);
\node (v10) at (-2.5,-8) {};
\node (v11) at (-1.5,-8) {};
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\draw (29) edge (8);
\node (v12) at (2.5,-5.5) {};
\draw (8) edge (v12);
\node (v13) at (5,-6) {};
\node (v14) at (6,-6) {};
\draw (v7) edge (v13);
\draw (v7) edge (v14);
\end{tikzpicture}
\caption{Case 2.2.1.}
\label{2-case 2.2.1.}
\end{minipage}
\hspace{15mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-4.15,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (2.5,-6) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (5.5,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-1.5,-5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2.5,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-7.5,-8.5) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (-6.5,-8.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-5.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (31) at (-2,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-2.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (34) at (-4.5,-8.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (35) at (-1.5,-8.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (38) at (-7,-5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-5,-5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (40) at (-6,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (41) at (6,-6.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (42) at (5,-6.5) {$1_a$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (6) at (-7,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (7) at (-5,-6) {$u_{8}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2.5,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node (v3) at (-7.5,-8) {};
\node (v4) at (-6.5,-8) {};
\node (v5) at (-5.5,-8) {};
\node (v6) at (-4.5,-8) {};
\draw (6) edge (v3);
\draw (6) edge (v4);
\draw (7) edge (v5);
\draw (7) edge (v6);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-2,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (5.5,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v2) at (-2,-6) {$u_{10}$};
\draw (29) edge (v2);
\node (v10) at (-2.5,-8) {};
\node (v11) at (-1.5,-8) {};
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\draw (29) edge (8);
\node (v12) at (2.5,-5.5) {};
\draw (8) edge (v12);
\node (v13) at (5,-6) {};
\node (v14) at (6,-6) {};
\draw (v7) edge (v13);
\draw (v7) edge (v14);
\end{tikzpicture}
\caption{Case 2.2.2.}
\label{2-case 2.2.2.}
\end{minipage}
\end{figure}
{\bf Case 2.2.2:} $\{u_3u_5, u_4u_6\} \cap E(G) = \emptyset$ and $u_4u_5 \in E(G)$, i.e., $u_9=u_5$. If $2_a \notin f(N(u_5) \cup N(u_6))$, then we can recolor $u_2$ with $2_a$ and $u$ with $1_b$ to obtain a coloring satisfying $(a)\text{ and }(b)$. If $1_b \notin f(N(u_6)-\{u_2\})$ and $2_b \notin f(N(u_5)-\{u_2,u_4\})$, then we can recolor $u_6$ with $1_b$, $u_2$ with $2_b$, and $u$ with $1_b$ to obtain a coloring satisfying $(a)\text{ and }(b)$. With \eqref{1au6}, we know
$$f(N(u_5)-\{u_2,u_4\})=\{2_a\}\quad \mbox{and}\quad f(N(u_6)-\{u_2\})=\{1_a,1_b\}$$
$$\text{ or }f(N(u_5)-\{u_2,u_4\})=\{2_b\}\quad \mbox{and}\quad f(N(u_6)-\{u_2\})=\{1_a,2_a\}.$$ If $f(N(u_5)-\{u_2,u_4\})=\{2_b\}$ and $f(N(u_6)-\{u_2\})=\{1_a,2_a\}$, then we recolor $u_5$ with $2_a$, $u_2$ with $1_a$, and $u$ with $1_b$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we can assume that
\begin{equation}\label{u12u13}
f(N(u_5)-\{u_2,u_4\})=\{2_a\}\quad \mbox{and}\quad f(N(u_6)-\{u_2\})=\{1_a,1_b\}.
\end{equation}
If $1_b \notin f(N(u_7)-\{u_3\})$, then we can recolor $u_7$ with $1_b$ and it contradicts \eqref{cases}. Thus, we may assume
\begin{equation}\label{1bu7}
1_b \in f(N(u_7)-\{u_3\}).
\end{equation}
If $1_a \notin f(N(u_8)-\{u_3\})$, then we can recolor $u_8$ with $1_a$ and it contradicts \eqref{u8u9}. If $1_a \notin f(N(u_{10})-\{u_4\})$, then we can recolor $u_{10}$ with $1_a$ and it contradicts \eqref{cases}. Therefore, we may assume
\begin{equation}\label{u7u8u10}
1_a \in f(N(u_{10})-\{u_4\})\quad \mbox{and}\quad 1_a \in f(N(u_8)-\{u_3\}).
\end{equation}
If $2_b \notin f(N(u_7) \cup N(u_8) - \{u_3\})$, then we can recolor $u_3$ with $2_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{2bu7u8}
2_b \in f(N(u_7) \cup N(u_8) - \{u_3\}).
\end{equation}
By previous arguments, we know that $\{3_a,3_b\} \cap f(\{u_2,u_3,u_4,u_5,u_6,u_7,u_8,u_{10}\}) = \emptyset$.
With \eqref{3a}, \eqref{3b}, and \eqref{u12u13}, we know that $\{3_a,3_b\} \subseteq f(N(u_7) \cup N(u_8) \cup N(u_{10}) - \{u_3,u_4\})$. Moreover, by \eqref{1bu7}, \eqref{u7u8u10}, \eqref{2bu7u8}, and symmetry, we may assume that $$f(N(u_{10})-\{u_4\})=\{1_a,3_b\} \text{ (See Figure~\ref{2-case 2.2.2.})}.$$ But we can recolor $u_{10}$ with $1_b$, $u_4$ with $2_b$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
{\bf Case 2.2.3:} $\{u_3u_5, u_4u_6, u_4u_5\} \cap E(G) = \emptyset$ and $u_4u_7 \in E(G)$, i.e., $u_7=u_9$. If $1_a \notin f(N(u_8)-u_3),$ then we recolor $u_8$ with $1_a$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a) \text{ and } (b).$ Thus, we may assume $1_a \in f(N(u_8)-u_3)$. If $1_a \notin f(N(u_{10})-u_4),$ then we recolor $u_{10}$ with $1_a$, $u_1$ with $2_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a) \text{ and } (b).$ Thus, we may also assume $1_a \in f(N(u_{10})-u_4)$. If $2_b \notin f(N(u_7) \cup N(u_8)-\{u_3,u_4\})$, then we recolor $u_3$ with $2_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. With \eqref{3a}, \eqref{3b}, and symmetry, we may assume $f(N(u_7) \cup N(u_8) - \{u_3,u_4\}) = \{1_a,2_b,3_a\}$ and $f(N(u_{10})-u_4) = \{1_a,3_b\}$ (See Figure~\ref{2-case 2.2.3.}). We recolor $u_7$ with $1_b$, $u_4$ with $1_a$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
Thus, we may also assume $u_4u_7 \notin E(G)$.
\medskip
Below we have $\{u_3u_5, u_4u_6, u_4u_5, u_4u_7\} \cap E(G) = \emptyset$. Moreover, by the case (Case 2.2), $$\{u_3u_6,u_4u_8,u_3u_9,u_3u_{10}\} \cap E(G) = \emptyset.$$ Therefore, we also have $|\{u_i: i \in [10]\}|=10$.
\begin{figure}[ht]\label{f4}
\centering
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.5, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-4.15,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (17) at (-6,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (-2,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (5.5,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (-7.5,-5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2.5,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-7,-8.5) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (27) at (-8,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (28) at (-0.5,-8.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-4,-5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-1.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (33) at (-1,-5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (34) at (-4,-8.5) {$2_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (6) at (-4,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (7) at (-7.5,-6) {$u_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2.5,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node (v4) at (-4,-8) {};
\node (v5) at (-8,-8) {};
\node (v6) at (-7,-8) {};
\draw (6) edge (v4);
\draw (7) edge (v5);
\draw (7) edge (v6);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-2,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (5.5,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v2) at (-1,-6) {$u_{10}$};
\draw (29) edge (v2);
\node (v10) at (-1.5,-8) {};
\node (v11) at (-0.5,-8) {};
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\draw (6) edge (29);
\end{tikzpicture}
\caption{Case 2.2.3.}
\label{2-case 2.2.3.}
\end{minipage}
\hspace{15mm}
\begin{minipage}[b]{0.4\textwidth}
\begin{tikzpicture}[scale=0.45, transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-4,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (17) at (-6,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (-2,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-3.05,-4.85) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-1,-4.85) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (5.5,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-7,-5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (-5,-4.85) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2.5,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-4.5,-7) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (-4.5,-10.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (28) at (-0.5,-8.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-5.5,-10.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (31) at (-2.5,-8.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-1.5,-8.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (33) at (-3.5,-8.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (34) at (-7.5,-7) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (35) at (-3.5,-9.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (-0.5,-9.5) {$1_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (6) at (-7,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (7) at (-5,-6) {$u_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2.5,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-2,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (5.5,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v1) at (-3,-6) {$u_9$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v2) at (-1,-6) {$u_{10}$};
\draw (29) edge (v1);
\draw (29) edge (v2);
\node (v8) at (-3.5,-8) {};
\node (v9) at (-2.5,-8) {};
\node (v10) at (-1.5,-8) {};
\node (v11) at (-0.5,-8) {};
\draw (v1) edge (v8);
\draw (v1) edge (v9);
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\draw (6) edge (7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (45) at (-5,-8) {$u_{12}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (46) at (-7,-8) {$u_{11}$};
\draw (6) edge (46);
\draw (7) edge (45);
\node (v3) at (-5.5,-10) {};
\node (v4) at (-4.5,-10) {};
\draw (45) edge (v3);
\draw (45) edge (v4);
\end{tikzpicture}
\caption{Case 2.2.4.}
\label{2-case 2.2.4.}
\end{minipage}
\end{figure}
{\bf Case 2.2.4:} $u_7u_8 \in E(G)$. By \eqref{1a1b'}, $1_b \in f(N(u_9)-\{u_4\})$. If $1_a \notin f(N(u_{10})-\{u_4\})$, then we recolor $u_{10}$ with $1_a$, $u_1$ with $2_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume $1_a \notin f(N(u_{10})-\{u_4\})$. By \eqref{3a} and \eqref{3b}, $\{3_a,3_b\} \subseteq f(N(u_7) \cup N(u_8) \cup N(u_9) \cup N(u_{10}))$. If $\{3_a,3_b\} \subseteq f(N(u_9) \cup N(u_{10}))$, then $f(N(u_9) \cup N(u_{10})-\{u_4\})=\{1_a,1_b,3_a,3_b\}$, $1_b \notin f(N(u_{10})-\{u_4\})$ and $2_b \notin f(N(u_9)-\{u_4\})$. Then, we can recolor $u_{10}$ with $1_b$, $u_4$ with $2_b$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, by symmetry, we can assume
\begin{equation}\label{3au7u8}
3_a \in f(N(u_7) \cup N(u_8) - \{u_3\}) \quad \mbox{and} \quad 3_a \notin f(N(u_9) \cup N(u_{10})-u_4).
\end{equation}
If $2_b \notin f(N(u_7) \cup N(u_8) - \{u_3\})$, then we recolor $u_3$ with $2_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a) \text{ and } (b)$. Thus, we may assume $2_b \in f(N(u_7) \cup N(u_8) - \{u_3\})$. Let $u_{11} \in N(u_7)-\{u_3,u_8\}$ and $u_{12} \in N(u_8)-\{u_3,u_7\}$. We may assume
\begin{equation}\label{u11u12}
f(u_{11}) = 2_b\quad \mbox{and}\quad f(u_{12})=3_a,
\end{equation}
since, by symmetry, the proof for the case $f(u_{11}) = 3_a$ and $f(u_{12})=2_b$ is similar.
Note that $3_a \notin f(B(u_1,3)-u_{12})$. If $1_a \notin f(N(u_{12})-\{u_8\})$, then we recolor $u_{12}$ with $1_a$, $u_1$ with $3_a$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. If $1_b \notin f(N(u_{12})-\{u_8\})$, then we recolor $u_{12}$ with $1_b$, $u_8$ with $1_a$, $u_7$ with $1_b$, $u_1$ with $3_a$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{1a1bu12}
f(N(u_{12})-\{u_8\})=\{1_a,1_b\}.
\end{equation}
If $1_b \notin f(N(u_{11})-\{u_7\})$, then we can recolor $u_{11}$ with $1_b$, $u_3$ with $2_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}
1_b \in f(N(u_{11})-\{u_7\}) \text{ (See Figure~\ref{2-case 2.2.4.})}.
\end{equation}
Then, we can recolor $u_8$ with $2_a$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
{\bf Case 2.2.5:} $u_7u_8 \notin E(G), u_8u_9 \in E(G)$. Similarly to \eqref{1au10} and \eqref{1bu7}, we may assume
\begin{equation}\label{u7u10}
1_a \in f(N(u_{10})-\{u_4\})\quad \mbox{and}\quad 1_b \in f(N(u_7)-\{u_3\}).
\end{equation}
If $2_b \notin f(N(u_7) \cup N(u_8) - \{u_3\})$, then we recolor $u_3$ with $2_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume $2_b \in f(N(u_7) \cup N(u_8) - \{u_3\})$. If $1_b \notin f(N(u_{10})-\{u_4\})$ and $2_b \notin f(N(u_9)-\{u_4\})$, then we can recolor $u_{10}$ with $1_b$, $u_4$ with $2_b$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. From \eqref{3a} and \eqref{3b}, we know that $f(N(u_8) \cup N(u_9)-\{u_3,u_4\}) \subseteq \{2_b,3_a,3_b\}$ (See Figure~\ref{2-case 2.2.5.}). But it contradicts \eqref{1a1b'}.
Therefore, we may assume $u_8u_9 \notin E(G)$.
\begin{figure}[ht]\label{f5}
\begin{center}
\begin{tikzpicture}[scale=0.48,transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (0,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-4,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (4,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (17) at (-6.5,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-1.5,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-2.5,-5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-0.5,-5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (5.5,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-7.5,-5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (-5.5,-5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (2.5,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-8,-8.5) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (34) at (-5.5,-8.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (35) at (-2.5,-8.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (-7,-8.5) {$1_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (0,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-4,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (4,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-6.5,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (6) at (-7.5,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (7) at (-5.5,-6) {$u_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (2.5,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-1.5,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (5.5,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v1) at (-2.5,-6) {$u_9$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v2) at (-0.5,-6) {$u_{10}$};
\draw (29) edge (v1);
\draw (29) edge (v2);
\node (v8) at (-2.5,-8) {};
\node (v10) at (-1,-8) {};
\node (v11) at (0,-8) {};
\draw (v1) edge (v8);
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\node(45) at (-5.5,-8) {};
\node(46) at (-8,-8) {};
\draw (6) edge (46);
\draw (7) edge (45);
\draw (7) edge (v1);
\node (v3) at (-7,-8) {};
\draw (6) edge (v3);
\end{tikzpicture}
\caption{Case 2.2.5.}
\label{2-case 2.2.5.}
\end{center}
\end{figure}
{ \bf Case 2.2.6:} $u_7u_8 \notin E(G), u_8u_9 \notin E(G)$.
If $|N(u_7)|=|N(u_8)|=|N(u_9)|=|N(u_{10})|=3,$ then we let $$\{u_{11},u_{12}\} \subseteq N(u_7)-\{u_3\}, \quad \{u_{13},u_{14}\} \subseteq N(u_8)-\{u_3\}, \quad \{u_{15},u_{16}\} \subseteq N(u_9)-\{u_4\},$$$$\quad \mbox{and}\quad \{u_{17},u_{18}\} \subseteq N(u_{10})-\{u_4\}.$$
It is possible that $|\{u_i:i \in[18]-[10]\}| \neq 8$ or $\{u_5,u_6\} \cap \{u_i: i \in [18]-[10]\} \neq \emptyset$, but this will not affect the proof below.
Similarly to \eqref{1bu9}, \eqref{1bu5}, \eqref{1au10}, \eqref{u8}, we may assume
\begin{equation}\label{1a1b226}
f(u_{12})=f(u_{16})=1_b \quad \mbox{and}\quad f(u_{13})=f(u_{17})=1_a.
\end{equation}
Similarly to \eqref{2bu7u8} and \eqref{3au7u8}, we may assume
\begin{equation}\label{2b3au7u8}
\{2_b,3_a\} \subseteq f(N(u_7) \cup N(u_8) - \{u_3\}).
\end{equation}
If $1_b \notin f(N(u_{10})-\{u_4\})$ and $2_b \notin f(N(u_9)-\{u_4\})$, then we can recolor $u_{10}$ with $1_b$, $u_4$ with $2_b$, $u_1$ with $1_b$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. With \eqref{3b}, we may assume
\begin{equation}\label{u9u10}
\mbox{\em either
$f(u_{15})=3_b$ and $f(u_{18})=1_b$ or $f(u_{15})=2_b$ and $f(u_{18})=3_b$.}
\end{equation}
If $|N(u_{11})|=|N(u_{12})|=|N(u_{13})|=|N(u_{14})|=3$, then we let $\{u_{19},u_{20}\} \subseteq N(u_{11})$, $\{u_{21},u_{22}\} \subseteq N(u_{12})$, $\{u_{23},u_{24}\} \subseteq N(u_{13})$, $\{u_{25},u_{26}\} \subseteq N(u_{14})$.
By \eqref{2b3au7u8}, we have
\begin{equation}\label{cases26}
\mbox{\em either
$f(u_{11})=2_b$ and $f(u_{14})=3_a$ or $f(u_{11})=3_a$ and $f(u_{14})=2_b$.
}
\end{equation}
\begin{figure}[ht]\label{f6}
\begin{center}
\begin{tikzpicture}[scale=0.35,transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (51) at (-5.8,-12.35) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (52) at (-12.2,-12.35) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (54) at (-8.5,-17.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (58) at (3,-4.85) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (59) at (-11.5,-17.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (60) at (-7.5,-17.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (1,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (12) at (-12.5,-17.5) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (-18,-14.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (36) at (-10.5,-17.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (-0.5,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-4,-4.85) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-13,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (11,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-19,-8.35) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (23) at (-5.5,-17.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (24) at (-6.5,-17.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-7,-8.35) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (26) at (-9.8,-12.35) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (27) at (-8.2,-12.35) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (28) at (-9.5,-17.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-14.5,-14.65) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (61) at (-14.5,-15.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (31) at (4,-10.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (64) at (4,-11.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-16.5,-14.65) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (35) at (-5,-10.65) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (63) at (-5,-11.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (-15.5,-8.35) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (38) at (-20,-14.65) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (60) at (-20,-15.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-9,-4.85) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (40) at (6,-6.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (41) at (2,-10.65) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (42) at (-11,-8.35) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (43) at (-17,-4.85) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (44) at (10,-6.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (45) at (7,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (46) at (-7,-0.9) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (47) at (-3,-10.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (48) at (12,-6.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (49) at (8,-6.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (50) at (9,-0.9) {$1_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (1,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-7,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (9,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-13,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (6) at (-17,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (7) at (-9,-6) {$u_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (7,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v3) at (-19,-9.5) {$u_{11}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v4) at (-15.5,-9.5) {$u_{12}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v5) at (-11,-9.5) {$u_{13}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v6) at (-7,-9.5) {$u_{14}$};
\draw (6) edge (v3);
\draw (6) edge (v4);
\draw (7) edge (v5);
\draw (7) edge (v6);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-0.5,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (11,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (v1) at (-4,-6) {$u_9$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (v2) at (3,-6) {$u_{10}$};
\draw (29) edge (v1);
\draw (29) edge (v2);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v8) at (-5,-9.5) {$u_{15}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v9) at (-3,-9.5) {$u_{16}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v10) at (2,-9.5) {$u_{17}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v11) at (4,-9.5) {$u_{18}$};
\draw (v1) edge (v8);
\draw (v1) edge (v9);
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v12) at (-20,-13.5) {$u_{19}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v14) at (-18,-13.5) {$u_{20}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v13) at (-16.5,-13.5) {$u_{21}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v15) at (-14.5,-13.5) {$u_{22}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v16) at (-12,-13.5) {$u_{23}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v17) at (-10,-13.5) {$u_{24}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v18) at (-8,-13.5) {$u_{25}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v19) at (-6,-13.5) {$u_{26}$};
\node (v20) at (-12.5,-17) {};
\node (v22) at (-10.5,-17) {};
\node (v21) at (-11.5,-17) {};
\node (v23) at (-9.5,-17) {};
\node (v24) at (-8.5,-17) {};
\node (v25) at (-7.5,-17) {};
\node (v26) at (-6.5,-17) {};
\node (v27) at (-5.5,-17) {};
\draw (v3) edge (v12);
\draw (v4) edge (v13);
\draw (v3) edge (v14);
\draw (v4) edge (v15);
\draw (v5) edge (v16);
\draw (v5) edge (v17);
\draw (v6) edge (v18);
\draw (v6) edge (v19);
\draw (v16) edge (v20);
\draw (v16) edge (v21);
\draw (v17) edge (v22);
\draw (v17) edge (v23);
\draw (v18) edge (v24);
\draw (v18) edge (v25);
\draw (v19) edge (v26);
\draw (v19) edge (v27);
\node (v28) at (6,-6) {};
\node (v29) at (8,-6) {};
\node (v30) at (10,-6) {};
\node (v31) at (12,-6) {};
\draw (8) edge (v28);
\draw (8) edge (v29);
\draw (v7) edge (v30);
\draw (v7) edge (v31);
\end{tikzpicture}
\caption{Case 2.2.6.1.}
\label{2-case 2.2.6.1.}
\end{center}
\end{figure}
{\bf Case 2.2.6.1:} $f(u_{11})=2_b$ and $f(u_{14}) = 3_a$. If $1_b \notin f(N(u_{13})-\{u_8\})$, then we can recolor $u_{13}$ with $1_b$, $u_8$ with $1_a$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. If $2_b \notin f(N(u_{13}) \cup N(u_{14})-\{u_8\})$, then we can recolor $u_8$ with $2_b$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{2bu13u14}
2_b \in f(N(u_{13}) \cup N(u_{14})-\{u_8\}).
\end{equation}
If $2_a \notin f(N(u_{13}) \cup N(u_{14})-\{u_8\})$, then we can recolor $u_8$ with $2_a$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may also assume
\begin{equation}\label{2au13u14}
2_a \in f(N(u_{13}) \cup N(u_{14})-\{u_8\}).
\end{equation}
If $1_b \notin f(N(u_{11})-\{u_7\})$, then we can recolor $u_{11}$ with $1_b$ and it contradicts \eqref{cases26}. Similarly, $1_a \in f(N(u_{14})-\{u_8\})$. If $1_a \notin f(N(u_{12})-\{u_7\})$, then we can recolor $u_{12}$ with $1_a$, $u_7$ with $1_b$, and it contradicts \eqref{cases}. Similarly, $1_b \in f(N(u_{13})-\{u_8\})$. Thus, we may assume
\begin{equation}\label{u24u25}
|N(u_{13})|=|N(u_{14})|=3, f(u_{20})=f(u_{24})=1_b,\quad \mbox{and}\quad f(u_{21})=f(u_{25})=1_a.
\end{equation}
Furthermore, by \eqref{2bu13u14} and \eqref{2au13u14}, we assume
\begin{equation}\label{u23u26}
f(u_{23})=2_a\quad \mbox{and}\quad f(u_{26})=2_b,
\end{equation}
since the argument for $f(u_{23})=2_b$ and $f(u_{26})=2_a$ is similar.
If $\{1_a,1_b\} \neq f(N(u_{26})-\{u_{14}\})$, then we can recolor $u_{26}$ with a color $x \in f(N(u_{26})-\{u_{14}\})-\{1_a,1_b\}$, $u_8$ with $2_b$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{u26}
f(N(u_{26})-\{u_{14}\})=\{1_a,1_b\}.
\end{equation}
If $1_b \notin f(N(u_{25})-\{u_{14}\})$, then we can recolor $u_{25}$ with $1_b$, $u_{14}$ with $1_a$, and it contradicts \eqref{cases26}. Thus, we may assume
\begin{equation}
1_b \in f(N(u_{25})-\{u_{14}\}).
\end{equation}
If $f(u_{19}) \neq 1_a$ and $f(u_{22}) \neq 2_b$, then we can recolor $u_{11}$ with $1_a$, $u_7$ with $2_b$, $u_3$ with $1_a$, $u_1$ with $2_a$, and $u$ with $1_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. If $3_b \notin f(N(u_{11}) \cup N(u_{12})-\{u_7\})$, then we can recolor $u_3$ with $3_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we can assume
\begin{equation}\label{u19u22}
\mbox{\em either
$f(u_{19})=1_a$ and $f(u_{22})=3_b$ or $f(u_{19})=3_b$ and $f(u_{22})=2_b$.
}
\end{equation}
If $2_a \notin f(N(u_{25}) \cup N(u_{26}) - \{u_{14}\})$, then by \eqref{u19u22}, we can recolor $u_{14}$ with $2_a$, $u_3$ with $3_a$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. With \eqref{u26}, we may assume
\begin{equation}
2_a \in f(N(u_{25})-\{u_{14}\}).
\end{equation}
Similarly to \eqref{u24u25}, we may assume
\begin{equation}\label{u23u24}
1_a \in f(N(u_{24})-\{u_{13}\})\quad \mbox{and}\quad 1_b \in f(N(u_{23})-\{u_{13}\}).
\end{equation}
If $\{3_a,3_b\} \nsubseteq f(N(u_{23}) \cup N(u_{24}))$, then we can recolor $u_8$ with a color $x \in f(N(u_{23}) \cup N(u_{24})) - \{3_a,3_b\}$, $u_{14}$ with $1_b$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Therefore,
$$ f(N(u_{23}) \cup N(u_{24})-\{u_{13}\})=\{1_a,1_b,3_a,3_b\}\quad \mbox{and}\quad 2_b \notin f(B(u_{13})) \text{ (See Figure~\ref{2-case 2.2.6.1.})}.$$
We recolor $u_{13}$ with $2_b$, $u_8$ with $1_a$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$.
\begin{figure}[ht]\label{f7}
\begin{center}
\begin{tikzpicture}[scale=0.35,transform shape]
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (11) at (1,1) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (14) at (-18,-14.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (18) at (-0.5,-3) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (19) at (-4,-4.85) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (20) at (-13,-3) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (21) at (11,-3) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (22) at (-19,-8.35) {$3_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (25) at (-7,-8.35) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (30) at (-14.5,-14.65) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (71) at (-14.5,-15.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (31) at (4,-10.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (77) at (4,-11.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (32) at (-16.5,-14.65) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (35) at (-5,-10.65) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (75) at (-5,-11.5) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (37) at (-15.5,-8.35) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (38) at (-20,-14.65) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (70) at (-20,-15.5) {$3_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (39) at (-9,-4.85) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (40) at (6,-6.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (41) at (2,-10.65) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (42) at (-11,-8.35) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (43) at (-17,-4.85) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (44) at (10,-6.5) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (45) at (7,-3) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (46) at (-7,-1) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (47) at (-3,-10.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (48) at (12,-6.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (49) at (8,-6.5) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=25pt, font=\huge] (50) at (9,-1) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (58) at (3,-4.85) {$2_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (59) at (-6,-14.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (72) at (-6,-15.5) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (60) at (-8,-14.65) {$1_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (61) at (-10,-14.65) {$1_b$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (62) at (-12,-14.65) {$2_a$};
\node[circle, draw=white!0, inner sep=0pt, minimum size=15pt, font=\huge] (68) at (-12,-15.5) {$2_b$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (1) at (1,0) {$u$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (3) at (-7,-2) { $u_1$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (4) at (9,-2) {$u_2$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (5) at (-13,-4) {$u_3$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (6) at (-17,-6) {$u_7$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (7) at (-9,-6) {$u_8$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (8) at (7,-4) {$u_5$};
\draw (1) edge (3);
\draw (1) edge (4);
\draw (3) edge (5);
\draw (4) edge (8);
\draw (5) edge (6);
\draw (5) edge (7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v3) at (-19,-9.5) {$u_{11}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v4) at (-15.5,-9.5) {$u_{12}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v5) at (-11,-9.5) {$u_{13}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v6) at (-7,-9.5) {$u_{14}$};
\draw (6) edge (v3);
\draw (6) edge (v4);
\draw (7) edge (v5);
\draw (7) edge (v6);
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (29) at (-0.5,-4) {$u_4$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=25pt, font=\huge] (v7) at (11,-4) {$u_6$};
\draw (3) edge (29);
\draw (4) edge (v7);
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (v1) at (-4,-6) {$u_9$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=36pt, font=\huge] (v2) at (3,-6) {$u_{10}$};
\draw (29) edge (v1);
\draw (29) edge (v2);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v8) at (-5,-9.5) {$u_{15}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v9) at (-3,-9.5) {$u_{16}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v10) at (2,-9.5) {$u_{17}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v11) at (4,-9.5) {$u_{18}$};
\draw (v1) edge (v8);
\draw (v1) edge (v9);
\draw (v2) edge (v10);
\draw (v2) edge (v11);
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v12) at (-20,-13.5) {$u_{19}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v14) at (-18,-13.5) {$u_{20}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v13) at (-16.5,-13.5) {$u_{21}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v15) at (-14.5,-13.5) {$u_{22}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v16) at (-12,-13.5) {$u_{23}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v17) at (-10,-13.5) {$u_{24}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v18) at (-8,-13.5) {$u_{25}$};
\node[circle, draw=black!80, inner sep=0pt, minimum size=30pt, font=\huge] (v19) at (-6,-13.5) {$u_{26}$};
\draw (v3) edge (v12);
\draw (v4) edge (v13);
\draw (v3) edge (v14);
\draw (v4) edge (v15);
\draw (v5) edge (v16);
\draw (v5) edge (v17);
\draw (v6) edge (v18);
\draw (v6) edge (v19);
\node (v28) at (6,-6) {};
\node (v29) at (8,-6) {};
\node (v30) at (10,-6) {};
\node (v31) at (12,-6) {};
\draw (8) edge (v28);
\draw (8) edge (v29);
\draw (v7) edge (v30);
\draw (v7) edge (v31);
\end{tikzpicture}
\caption{Case 2.2.6.2.}
\label{2-case 2.2.6.2.}
\end{center}
\end{figure}
{\bf Case 2.2.6.2:} $f(u_{11})=3_a$ and $f(u_{14}) = 2_b$. Similarly to \eqref{u24u25}, we may assume
\begin{equation}\label{u20u21}
f(u_{20})=f(u_{24})=1_b \quad \mbox{and}\quad f(u_{21})=f(u_{25})=1_a.
\end{equation}
Similarly to \eqref{2au13u14}, we may assume
\begin{equation}\label{2au13u14'}
2_a \in f(N(u_{13}) \cup N(u_{14})-\{u_8\}).
\end{equation}
If $1_b \notin f(N(u_{14})-\{u_8\})$ and $2_b \notin f(N(u_{13}))$, then we can recolor $u_8$ with $2_b$, $u_{14}$ with $1_b$, $u_3$ with $1_b$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
$$
f(N(u_{13})-\{u_8\})=\{1_b,2_a\}\quad \mbox{and}\quad f(N(u_{14})-\{u_8\})=\{1_a,1_b\}
$$
\begin{equation}\label{u13u14}
\quad \mbox{ or } \quad f(N(u_{13})-\{u_8\})=\{1_b,2_b\}\quad \mbox{and}\quad f(N(u_{14})-\{u_8\})=\{1_a,2_a\}.
\end{equation}
If $2_b \notin f(N(u_{11}) \cup N(u_{12}))$, then we can recolor $u_7$ with $2_b$ and it contradicts \eqref{cases}. If $3_b \notin f(N(u_{11}) \cup N(u_{12}))$, then we can recolor $u_3$ with $3_b$ and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. Thus, we may assume
\begin{equation}\label{u11u12'}
f(N(u_{11}) \cup N(u_{12})-\{u_7\})=\{1_a,1_b,2_b,3_b\}.
\end{equation}
Specifically, we know that $1_a \notin f(N(u_{11})-\{u_7\})$ and $2_a \notin f(B(u_7,2) - \{u_3\})$ (See Figure~\ref{2-case 2.2.6.2.}). Therefore, we recolor $u_{11}$ with $1_a$, $u_7$ with $2_a$, $u_3$ with $3_a$, and $u$ with $2_a$ to obtain a coloring satisfying $(a)\text{ and }(b)$. \hfill \hfill \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi\\
\bigskip\noindent
{\bf Acknowledgment.} We thank Sandi Klav\v zar, Douglas West, and the referees for their helpful comments.
|
{
"timestamp": "2018-10-09T02:15:24",
"yymm": "1803",
"arxiv_id": "1803.02537",
"language": "en",
"url": "https://arxiv.org/abs/1803.02537"
}
|
\section{Introduction}\label{Sec:Introduction}
There is nowadays a huge amount of biological sequences available. The \textit{local score} for one sequence analysis, first defined by Karlin and Altchul in \cite{KAl90} (see Equation (\ref{Def:LocalScore}) below for definition) quantifies the highest level of a certain quantity of interest, e.g. hydrophobicity, polarity, etc..., that can be found locally inside a given sequence. This allows for example to detect atypical segments in biological sequences. In order to distinguish significantly interesting segments from the ones that could have appeared by chance alone, it is necessary to evaluate the $p$-value of a given local score. Different results have already been established using different probabilistic models for sequences: independent and identically distributed variables model (i.i.d.) \cite{MCC03,KAl90,KDe92,MDa01}, Markovian models \cite{KDe92, HMe07} or Hidden Markov Models \cite{DKM98}. In this article we will focus on the Markovian model.\par
An exact method was proposed by Hassenforder and Mercier \cite{HMe07} to calculate the distribution of the local score for a Markovian sequence, but this result is computationally time consuming for long sequences ($>10^3$).
Karlin and Dembo \cite{KDe92} established the limit distribution of the local score for a Markovian sequence and a random scoring scheme depending on the pairs of consecutive states in the sequence.
They proved that the distribution of the local score is asymptotically a Gumble distribution, as in the i.i.d. case. In spite of its importance, their result in the Markovian case is unforfunately very little cited or used in the literature. A possible explanation could be the fact that the random scoring scheme defined in \cite{KDe92} is more general than the ones classically used in practical approaches. In \cite{GRH06} and \cite{FBM17}, the authors verify by simulations that the local score in a certain dependence model follows a Gumble distribution, and use simulations to estimate the two parameters of this distribution.
In this article we study the Markovian case for a more classical scoring scheme. We propose a new approximation for the distribution of the local score of a Markovian sequence. We compare it to the one derived from the result of Karlin and Dembo \cite{KDe92} and illustrate the obtained improvement in a simple application case.
\subsubsection*{Mathematical framework}
Let $(A_i)_{i \geq 0}$ be an irreducible and aperiodic Markov chain taking its values in a finite set ${\cal A}$ containing $r$ states denoted $\alpha$, $\beta$, $\dots$ for simplicity. Let ${\bf P}=(p_{\alpha\beta})_{\alpha,\beta}$ be its transition probability matrix and $(\pi_{\alpha})_{\alpha}$ its stationary frequency vector. In order to simplify the presentation, we suppose that ${\bf P}$ is positive ($\forall \alpha,\beta,\ p_{\alpha\beta}>0$).
We also suppose that the Markov chain is stationary, i.e. with initial distribution of $A_0$ given by $\pi$.
$\mathbb P_{\alpha}$ will stand for the conditional probability given $\{A_0 = \alpha\}$.
We consider a lattice score function $f: {\cal A}\rightarrow d\mathbb Z$, with $d\in\mathbb N$ being the lattice step. Note that, since ${\cal A}$ is finite, we have a finite number of possible scores.
Since the Markov chain $(A_i)_{i \geq 0}$ is supposed to be stationary, the distribution of $A_i$ is $\pi$ for every $i \geq 0$. We will simply denote $\mathbb E[f(A)]$ the average score.
In this article we make the hypothesis that the average score is negative, i.e.
\begin{eqnarray}\label{Hyp:ScoreMoyNeg}
\mathbb E[f(A)]=\sum_{\alpha}f(\alpha)\pi_{\alpha}<0.
\end{eqnarray}
We will also suppose that for every $\alpha \in \mathcal{A}$ we have
\begin{eqnarray}\label{Hyp:ProbaScorePos}
\mathbb P_{\alpha}(f(A) > 0) > 0 \mbox{ and }\ \mathbb P_{\alpha}(f(A) < 0) > 0.
\end{eqnarray}
Let us introduce some definitions and notation. Let $S_{0}:=0$ and denote
$$S_{k}:=\sum_{i=1}^kf(A_i),$$
for $k\geq 1$ the successive partial sums. Let $S^+$ be the \textit{maximal non-negative partial sum}
$$S^+:=\max\{0,S_k : k \geq 0\}.$$
Further, let $\sigma^-:=\inf\{k\geqslant 1:S_k<0\}$ be the time of the first negative partial sum. Note that $\sigma^-$ is an a.s.-finite stopping time due to (\ref{Hyp:ScoreMoyNeg}).
Let $K_0 := 0$. For $i \geq 1$, we denote $K_i := \inf\{k > K_{i-1} : S_{k} - S_{K_{i-1}} < 0\}$
the successive decreasing ladder times of $(S_k)_{k\geq 0}$. Note that $K_1 = \sigma^-$.
Let us now consider the subsequence $(A_i)_{ 0 \leq i \leq n}$ for a given length $n \in \mathbb{N}\setminus \{0\}$.
Denote $m(n): = \max\{i \geq 0 : K_i \leq n\}$ the random variable corresponding to the number of decreasing ladder times arrived before $n$. For every $i=1,\dots,m(n)$, we call the sequence $(A_j)_{K_{i-1}< j\leq K_{i}}$ the $i$-th non-negative excursion.
Note that, due to the negative drift, we have $\mathbb E[K_1] < \infty$ (see Lemma \ref{lem:Esp_K1}) and $m(n) \to \infty$ \textit{a.s.} when $n \to \infty$.
To every non-negative excursions $i=1,\dots,m(n)$ we associate a \textit{maximal segmental score} (called also \textit{height}) $Q_{i}$ defined by
$$Q_i : = \max_{K_{i-1}\leq k < K_{i}} (S_k - S_{K_{i-1}}).$$
First introduced by Karlin and Altschul in \cite{KAl90}, the \textit{local score}, denoted $M_n$, is defined as the maximum segmental score for a sequence of length $n$:
\begin{equation}\label{Def:LocalScore}
M_{n}:=\max_{0\leqslant k\leqslant \ell\leqslant n}(S_{\ell}-S_{k}).
\end{equation}
Note that $M_{n}=\max(Q_1,\dots,Q_{m(n)},Q^*)$,
with $Q^*$ being the maximal segmental score of the last incomplete non-negative excursion $(A_j)_{K_{m(n)}< j\leq n}$.
Mercier and Daudin \cite{MDa01} give an alternative expression for $M_n$ using the Lindley process $(W_k)_{k \geq 0}$ describing the excursions above zero between the successive stopping times $(K_i)_{i\geq 0}$. With $W_0:=0$ and $W_{k+1}:=\max(W_{k}+f(A_{k+1}),0)$, we have
$M_{n}=\max_{0\leqslant k\leqslant n}W_k.$
\begin{rem}\label{KD}
Karlin and Dembo \cite{KDe92} consider a random score function defined on pairs of consecutive states of the Markov chain: they associate to each transition $(A_{i-1},A_{i})=(\alpha,\beta)$ a bounded random score $X_{\alpha\beta}$ whose distribution depends on the pair $(\alpha,\beta)$. Moreover, they suppose that, for $(A_{i-1},A_i)=(A_{j-1},A_j)=(\alpha,\beta)$, the random scores $X_{A_{i-1} A_i}$ and $X_{A_{j-1} A_j}$ are independent and identically distributed as $X_{\alpha\beta}$. The framework of this article corresponds to the case when the score function is deterministic, with $X_{A_{i-1}A_{i}}=f(A_i)$.\par
Note also that in our case the hypotheses (\ref{Hyp:ScoreMoyNeg}) and (\ref{Hyp:ProbaScorePos}) assure the so-called cycle positivity, i.e. the existence of some state $\alpha \in \mathcal{A}$
satisfying $$\mathbb P\left(\bigcap_{k=1}^{m-1} \{S_k>0\} \left |\ A_0=A_m=\alpha\right. \right) > 0.$$
In \cite{KDe92}, in order to simplify the presentation, the authors strengthen the assumption of cycle positivity by assuming that
$\mathbb P(X_{\alpha\beta} > 0) > 0 \text{ and } \mathbb P(X_{\alpha\beta} < 0) > 0$ for all $\alpha, \beta \in \mathcal{A}$ (see (1.19) of \cite{KDe92}), but precise that the cycle positivity is sufficient for their results to hold. Note that hypotheses (\ref{Hyp:ScoreMoyNeg}) and (\ref{Hyp:ProbaScorePos}) are usually verified in biological applications.
\end{rem}
In Section \ref{Sec:MainResults} we first introduce few more definitions and notation. Then we present the main results: a recursive result for the exact distribution of the maximal non-negative partial sum $S^+$ for an infinite sequence in Theorem \ref{res:exactS+}; based on the exact distribution of $S^+$, we further propose new approximations for the distribution of the height of the first non-negative excursion $Q_1$ in Theorem \ref{res:Q1} and for the distribution of the local score $M_n$ for a sequence of length $n$ in Theorem \ref{res:Mn}. Section \ref{Sec:Proofs} contains the proofs of the results of Section \ref{Sec:MainResults} and of some useful lemmas which use techniques of Markov renewal theory and large deviations. In Section \ref{sec:comp} we propose a computational method for deriving the quantities appearing in the main results. A simple scoring scheme is developed in Subsection \ref{subsec:simplecase}, for which we compare our approximations to the ones proposed by Karlin and Dembo \cite{KDe92} in the Markovian case.
\section{Statement of the main results}\label{Sec:MainResults}
\subsection{Definitions and notation}\label{subsec:notdef}
For every $\alpha, \beta \in \mathcal{A}$, we denote $q_{\alpha\beta}:=\mathbb P_{\alpha}(A_{K_1} = \beta)$ and
${\bf Q}:=(q_{\alpha\beta})_{\alpha,\beta}$.
Define $\mathcal{A}^-=\{\alpha\in\mathcal{A}:f(\alpha)<0\}$ and $\mathcal{A}^+=\{\alpha\in\mathcal{A}:f(\alpha)>0\}$. Note that the matrix ${\bf Q}$ is stochastic, with $q_{\alpha\beta}=0$ for $\beta \in \mathcal{A}\setminus \mathcal{A}^-$. Its restriction ${\bf \tilde Q}$ to $\mathcal{A}^-$ is stochastic and irreducible. The states $(A_{K_i})_{i \geq 1}$ of the Markov chain at the end of the successive non-negative excursions define a Markov chain on $\mathcal{A}^-$ with transition probability matrix ${\bf \tilde Q}$.
For every $i \geq 2$ we thus have
$\mathbb P(A_{K_i} = \beta \ | A_{K_{i-1}} = \alpha ) = q_{\alpha \beta}$
if $\alpha, \beta \in \mathcal{A}^-$ and 0 otherwise.
Denote $\tilde{z} > 0$ the stationary frequency vector of the irreducible stochastic matrix ${\bf \tilde Q}$ and let $z:=(z_{\alpha})_{\alpha\in\mathcal{A}}$ with $z_{\alpha}=\tilde{z}_{\alpha} > 0$ for $\alpha\in\mathcal{A}^-$ and $z_{\alpha}=0$ for $\alpha \in \mathcal{A} \setminus \mathcal{A}^-$. Note that $z$ is invariant for the matrix $Q$ i.e. $z{\bf Q}=z$.
\begin{rem}
Note that in Karlin and Dembo's Markovian model of \cite{KDe92} the matrix $Q$ is irreducible, thanks to their random scoring function and to their hypotheses recalled in Remark \ref{KD}.
\end{rem}
Using the strong Markov property, conditionally on $(A_{K_i})_{i \geq 1}$ the r.v. $(Q_i)_{i \geq 1}$ are independent, with the distribution of $Q_i$ depending only on $A_{K_{i-1}}$ and $A_{K_{i}}$.
For every $\alpha \in \mathcal{A}$, $\beta \in \mathcal{A}^-$ and $y \geq 0$, let
$$F_{\alpha \beta}(y):=\mathbb P_\alpha(Q_1 \leq y \ | A_{\sigma^-} = \beta)\
\mbox{ and }
\ F_\alpha(y): = \mathbb P_\alpha(Q_1 \leq y).$$
Note that for any $\alpha \in \mathcal{A}^-$ and $i \geq 1$, $F_{\alpha \beta}$ represents the cumulative distribution function (\textit{cdf}) of the height $Q_i$ of the $i$-th non-negative excursion given that it starts in state $\alpha$ and ends in state $\beta$, i.e.
$F_{\alpha \beta}(y)=\mathbb P(Q_i \leq y \ | A_{K_i} = \beta, A_{K_{i-1}} = \alpha)$, whereas $F_\alpha$ represents the \textit{cdf} of $Q_i$ conditionally on the $i$-th non-negative excursion starting in state $\alpha$, i.e. $F_{\alpha}(y)=\mathbb P(Q_i \leq y \ | A_{K_{i-1}} = \alpha)$.
We thus have
$$F_\alpha(y) = \sum_{\beta \in \mathcal{A}} F_{\alpha \beta}(y) q_{\alpha \beta}=\sum_{\beta \in \mathcal{A}^-} F_{\alpha \beta}(y) q_{\alpha \beta}. $$
We also introduce the stopping time $\sigma^+:=\inf\{k\geqslant 1:S_k>0\}$ with values in $\mathbb N \cup \{\infty\}$. Due to hypothesis $(\ref{Hyp:ScoreMoyNeg})$ we have $\mathbb P_\alpha(\sigma^+ < \infty) < 1$, for all $\alpha\in \mathcal{A}$.
For every $\alpha, \beta \in \mathcal{A}$ and $\xi > 0$, let
$$L_{\alpha\beta}(\xi):=\mathbb P_{\alpha}(S_{\sigma^+} \leqslant \xi, \sigma^+<\infty, A_{\sigma^+}=\beta).$$
Note that $L_{\alpha\beta}(\xi) = 0$ for $\beta \in \mathcal{A} \setminus \mathcal{A}^+$.
We have $L_{\alpha\beta}(\infty) \leq \mathbb P_\alpha(\sigma^+ < \infty) < 1$, and hence
\begin{equation}\label{L}
\int_{0}^\infty dL_{\alpha\beta}(\xi) = 1 - L_{\alpha\beta}(\infty) > 0.
\end{equation}
Let us also denote
$$L_{\alpha}(\xi):=\sum_{\beta \in \mathcal{A}^+}L_{\alpha \beta}(\xi) = \mathbb P_{\alpha}(S_{\sigma^+} \leqslant \xi, \sigma^+<\infty)$$
the conditional \textit{cdf} of the first positive partial sum when it exists, given that the Markov chain starts in state $\alpha$, and
$$L_{\alpha}(\infty) := \lim_{\xi \to \infty} L_{\alpha}(\xi) = \mathbb P_{\alpha}(\sigma^+<\infty).$$
For any $\theta \in \mathbb{R}$ we introduce the following matrix
$$\Phi(\theta):=\left(p_{\alpha\beta}\cdot\exp(\theta f(\beta))\right)_{\alpha,\beta \in \mathcal{A}}.$$
Since the transition matrix $\bf{P}$ was supposed to be positive, by the Perron-Frobenius Theorem, the spectral radius $\rho(\theta) > 0$ of the matrix $\Phi(\theta)$ coincides with its do\-mi\-nant eigenvalue, for which there exists a unique positive right eigen vector $u(\theta)=(u_i(\theta))_{1\leq i \leq r}$ (seen as a column vector) normalized so that $\sum_{i=1}^r u_i(\theta)=1$. Moreover, $\theta \mapsto \rho(\theta)$ is differentiable and strictly log convex (see \cite{Lancaster,DKa91b,KOs87}). In Lemma \ref{lem:rho'} we prove that $\rho'(0) = \mathbb E[f(A)]$, hence $\rho'(0) < 0$ by Hypothesis $(\ref{Hyp:ScoreMoyNeg})$. Together with the strict log convexity of $\rho$ and the fact that $\rho(0)= 1$,
this implies that there exists a unique $\theta^* > 0$ such that $\rho(\theta^*)=1$ (see \cite{DKa91b} for more details).
\\
\subsection{Main results. Improvements on the distribution of the local score}
Let $\alpha \in \mathcal{A}$. We start by giving a result which allows to compute recursively the \textit{cdf} of the maximal non-negative partial sum $S^+$.
We denote by $F_{S^+,\alpha}$ the \textit{cdf} of $S^+$ conditionally on starting in state $\alpha$:
$$F_{S^+,\alpha}(\ell d) := \mathbb P_{\alpha}(S^+\leq\ell d), \ \ \forall \ell \in \mathbb N $$
and for every $k \in \mathbb{N}\setminus \{0\}$ and $\beta \in \mathcal{A}$:
$$L^{(k)}_{\alpha \beta}:=\mathbb P_\alpha(S_{\sigma^+} = k d, \sigma^+ < \infty, A_{\sigma^+} = \beta).$$
Note that $L^{(k)}_{\alpha \beta}=0$ for $\beta \in \mathcal{A} \setminus \mathcal{A}^+$ and
$L_{\alpha}(\infty) = \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\infty} L^{(k)}_{\alpha \beta}.$
The following result gives a recurrence relation for the double sequence $(F_{S^+,\alpha}{(\ell d)})_{\alpha,\ell}$.
\begin{thm}[Exact result for the distribution of $S^+$]\label{res:exactS+}
For all $\alpha\in\mathcal{A}$ and $\ell \geq 1$:
\begin{align*}
F_{S^+,\alpha}(0) &=\mathbb P_\alpha(\sigma^+ = \infty) = 1 - L_{\alpha}(\infty),\\
F_{S^+,\alpha}{(\ell d)} &=1 - L_{\alpha}(\infty) + \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\ell} L^{(k)}_{\alpha \beta} \ F_{S^+,\beta}{((\ell-k)d)}.
\end{align*}
\end{thm}
The proof will be given in Section \ref{Sec:Proofs}.
In Theorem \ref{res:asymptS+} we obtain the asymptotic behavior of $S^+$ using Theorem \ref{res:exactS+} and ideas inspired from \cite{KDe92}
and adapted to our framework (see also the discussion in Remark \ref{KD}).
Before stating this result, we need to introduce few more notations.
For every $\alpha, \beta \in \mathcal{A}$ and $\ell \in \mathbb{N}$ we denote
$$G_{\alpha\beta}^{(\ell)} :=
\frac{u_{\beta}(\theta^*)}{u_{\alpha}(\theta^*)} e^{\theta^* \ell d}L_{\alpha\beta}^{(\ell)},\qquad
G_{\alpha\beta}{(\ell)} := \sum_{k=0}^{\ell} G_{\alpha\beta}^{(k)},\qquad
G_{\alpha\beta}{(\infty)} := \sum_{k=0}^{\infty} G_{\alpha\beta}^{(k)}.
$$
The matrix ${\bf G(\infty)}:=(G_{\alpha\beta}(\infty))_{\alpha, \beta}$ is stochastic, using Lemma \ref{EspeUsigma}; the subset $\mathcal{A}^+$ is a recurrent class, whereas the states in $\mathcal{A}\setminus \mathcal{A}^+$ are transient. The restriction of ${\bf G(\infty)}$ to $\mathcal{A}^+$ is stochastic and irreducible; let us denote $\tilde{w} > 0$ the corresponding stationary frequency vector. Define $w=(w_{\alpha})_{\alpha\in\mathcal{A}}$, with $w_{\alpha}=\tilde{w}_{\alpha} > 0$ for $\alpha\in\mathcal{A}^+$ and $w_{\alpha}=0$ for $\alpha \in \mathcal{A}\setminus \mathcal{A}^+$. The vector $w$ is invariant for $\bf G(\infty)$, i.e. $w{\bf G(\infty)}=w$.
\begin{rem}
Note that in Karlin and Dembo's Markovian model of \cite{KDe92} the matrix ${\bf G(\infty)}$ is positive, hence irreducible, thanks to their random scoring function and to their hypotheses recalled in Remark \ref{KD}.
\end{rem}
\begin{rem}
Note that the coefficients $L_{\alpha\beta}^{(k)}$ can be computed recursively (see Subsection \ref{Subsec:L}). In Subsection \ref{Subsec:FS+} we present in detail a recursive procedure for computing the \textit{cdf} $F_{S^+,\alpha}$, based on Theorem \ref{res:exactS+}. Note also that, for every $\alpha, \beta \in \mathcal{A}$, there are a finite number of $L_{\alpha\beta}^{(k)}$ different from zero. Therefore, there are a finite number of non-null terms in the sum defining $G_{\alpha \beta}(\infty)$.
\end{rem}
\begin{thm}[Asymptotic distribution of $S^+$]\label{res:asymptS+}
For every $\alpha\in {\cal A}$ we have
\begin{equation}\label{eq:cinf}
\lim_{k \rightarrow +\infty} \frac{e^{\theta^*{kd}}\mathbb P_{\alpha}(S^+>kd)}{u_{\alpha}(\theta^*)} = \frac{d}{c} \cdot \sum_{\gamma \in \mathcal{A}^+} \frac{w_{\gamma}}{u_{\gamma}(\theta^*)}\sum_{\ell \geq 0} (L_{\gamma}(\infty)-L_{\gamma}(\ell d)) e^{\theta^*\ell d}:= c(\infty),
\end{equation}
where
$w=(w_{\alpha})_{\alpha\in {\cal A}}$ is the stationary frequency vector of the matrix $\bf G(\infty)$
and
$$c:=\sum_{\gamma,\beta \in \mathcal{A}^+} \frac{w_{\gamma}}{u_{\gamma}(\theta^*)}u_{\beta}(\theta^*) \sum_{\ell \geq 0} \ell d \cdot e^{\theta^*{\ell d}} \ L^{(\ell)}_{\gamma \beta}.$$
\end{thm}
The proof is deferred to Section \ref{Sec:Proofs}.
\begin{rem}\label{rem:cinf}
Note that there are a finite number of non-null terms in the above sums over $\ell$.
We also have the following alternative expression for $c(\infty)$:
$$
c(\infty) = \frac{d}{c(e^{\theta^*d}-1)} \cdot \sum_{\gamma \in \mathcal{A}^+} \frac{w_{\gamma}}{u_{\gamma}(\theta^*)} \left\{
\mathbb E_\gamma\left[e^{\theta^* S_{\sigma^+}}; \sigma^+ < \infty\right] - L_\gamma(\infty)
\right\}.
$$
Indeed, by the summation by parts formula
$$
\sum_{\ell=m}^k f_\ell (g_{\ell+1}-g_\ell) = f_{k+1} g_{k+1} - f_m g_m - \sum_{\ell=m}^k (f_{\ell+1}-f_\ell)g_{\ell+1},
$$
we obtain
\begin{align*}
& \sum_{\ell = 0}^{\infty} (L_{\gamma}(\infty)-L_{\gamma}(\ell d)) e^{\theta^*\ell d}
= \frac{1}{e^{\theta^*d}-1} \sum_{\ell = 0}^{\infty} (L_{\gamma}(\infty)-L_{\gamma}(\ell d)) \left(e^{\theta^*(\ell+1) d} - e^{\theta^*\ell d} \right)\\
& = \frac{1}{e^{\theta^*d}-1}\\
&\quad\times \left\{ \lim_{k \to \infty} (L_{\gamma}(\infty)-L_{\gamma}(k d)) e^{\theta^*k d} - L_{\gamma}(\infty) - \sum_{\ell = 0}^{\infty} (L_{\gamma}(\ell d)-L_{\gamma}((\ell+1) d)) e^{\theta^*(\ell+1) d} \right\}\\
& = \frac{1}{e^{\theta^*d}-1} \left\{ - L_{\gamma}(\infty) + \sum_{\ell = 0}^{\infty} e^{\theta^*(\ell+1) d} \ \mathbb P_{\gamma}(S_{\sigma^+} = (\ell+1) d, \ \sigma^+<\infty) \right\}\\
& = \frac{1}{e^{\theta^*d}-1} \left\{
\mathbb E_\gamma\left[e^{\theta^* S_{\sigma^+}}; \sigma^+ < \infty\right] - L_\gamma(\infty)
\right\}.
\end{align*}
\end{rem}
Before stating the next results, let us denote for every integer $\ell < 0$ and $\alpha, \beta \in \mathcal{A}$,
$$Q_{\alpha\beta}^{(\ell)}:=\mathbb P_{\alpha}(S_{\sigma^-}=\ell d,A_{\sigma^-}=\beta).$$
Note that $Q_{\alpha\beta}^{(\ell)} = 0$ for $\beta \in \mathcal{A} \setminus \mathcal{A}^-$.
In Section \ref{sec:comp} we give a recursive computational method for obtaining these quantities.
Using Theorem \ref{res:asymptS+} we obtain the following
\begin{thm}[Asymptotic distribution of $Q_1$]\label{res:Q1}
We have the following asymptotic result on the distribution of the
height of the first non-negative excursion: for every $\alpha \in \mathcal{A}$ we have
\begin{equation}\label{eq:Q1}
\mathbb P_{\alpha}(Q_1>kd)\underset{k\rightarrow\infty}{\sim}\mathbb P_{\alpha}(S^+>kd)-\sum_{\ell<0}\sum_{\beta \in \mathcal{A}^-}\mathbb P_{\beta}\left (S^+>(k-\ell)d\right )\cdot Q_{\alpha\beta}^{(\ell)}.
\end{equation}
\end{thm}
The proof will be given in Section \ref{Sec:Proofs}.
Using now Theorems \ref{res:asymptS+} and \ref{res:Q1} we finally obtain the following result on the asymptotic distribution of the local score $M_n$ for a sequence of length $n$.
\begin{thm}[Asymptotic distribution of the local score $M_n$]\label{res:Mn}
For every $\alpha \in \mathcal{A}$:
\begin{align}\label{eq:Mn}
\mathbb P_\alpha &\left(M_n\leq \frac{\log (n)}{\theta^*}+x\right )
\underset{n\rightarrow\infty}{\sim} \exp\left \{-\frac{n}{A^*}\sum_{\beta \in \mathcal{A}^-} z_{\beta} \mathbb P_{\beta}\left (S^+> \left\lfloor\frac{\log(n)}{\theta^*}+x\right\rfloor\right )\right \} \nonumber \\
& \ \ \ \ \ \ \ \ \times\exp\left \{ \frac{n}{A^*}
\sum_{k < 0}\sum_{\gamma \in \mathcal{A}^-} \mathbb P_{\gamma}\left(S^+>\left\lfloor \frac{\log(n)}{\theta^*}+x \right \rfloor -kd\right) \cdot\sum_{\beta \in \mathcal{A}^-}z_{\beta}Q_{\beta\gamma}^{(k)}
\right \},
\end{align}
where $z=(z_{\alpha})_{\alpha\in {\cal A}}$ is the invariant probability measure of the matrix ${\bf Q}$ defined in Subsection \ref{subsec:notdef} and
$$A^* := \lim_{m\rightarrow +\infty} \frac{K_m}{m} = \frac{1}{\mathbb E(f(A))}\sum_{\beta \in \mathcal{A}^-} z_{\beta} \mathbb E_\beta(S_{\sigma^-})\mbox{ a.s.}$$
\end{thm}
\begin{rem}
\begin{itemize}
\item Note that the asymptotic equivalent in Equation (\ref{eq:Mn}) does not depend on the initial state $\alpha$.
\item We recall, for comparison, the asymptotic result of \cite{KDe92} (Equation (1.27)) for the distribution of $M_n$:
\begin{equation}\label{Res:AppxMnKDe}
\lim_{n\rightarrow +\infty}\mathbb P_\alpha \left(M_n\leq \frac{\log (n)}{\theta^*}+x\right )
=\exp\left ( -K^*\exp(-\theta^*)\right ),
\end{equation}
with
$K^*=v(\infty)\cdot c(\infty)$, where $c(\infty)$ given in Theorem \ref{res:asymptS+} is related to the defective distribution of the first positive partial sum $S_{\sigma^+}$ (see also Remark \ref{rem:cinf})
and $v(\infty)$ is related to the distribution of the first negative partial sum $S_{\sigma^-}$ (see Equations (5.1) and (5.2) of \cite{KDe92} for more details). A more explicit formula for $K^*$ is given in Subsection \ref{subsec:simplecase} for an application in a simple case.
\item Note that our asymptotic equivalent in Equation (\ref{eq:Mn}) keeps the dependence on $n$, whereas the approximation derived from Equation (\ref{Res:AppxMnKDe}) does not.
\end{itemize}
\end{rem}
\section{Proofs of the main results}\label{Sec:Proofs}
\subsection{Proof of Theorem \ref{res:exactS+}}
We have
\begin{align*}
F_{S^+,\alpha}(\ell d)&= \mathbb P_\alpha(\sigma^+ = \infty) + \mathbb P_\alpha(S^+ \leq \ell d, \sigma^+ < \infty)\\
&=1 - L_{\alpha}(\infty) + \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\ell} \mathbb P_\alpha(S^+ \leq \ell d, \sigma^+ < \infty, S_{\sigma^+} = k d, A_{\sigma^+} = \beta)\\
&= 1 - L_{\alpha}(\infty) + \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\ell} L^{(k)}_{\alpha \beta} \ \mathbb P_\alpha(S^+ \leq \ell d \ | \sigma^+ < \infty, S_{\sigma^+} = k d, A_{\sigma^+} = \beta).
\end{align*}
The last probability can further be written
$$
\mathbb P_\alpha(S^+ - S_{\sigma^+} \leq (\ell-k)d \ | \sigma^+ < \infty, S_{\sigma^+} = k d, A_{\sigma^+} = \beta)
= \mathbb P_\beta(S^+ \leq (\ell-k)d),
$$
by the strong Markov property applied to the stopping time $\sigma^+$.
The stated result easily follows. $\hfill\square$
\subsection{Proof of Theorem \ref{res:asymptS+}}
We first prove some preliminary lemmas.
\begin{lem}\label{Splus}
We have $\lim_{k \to \infty} \mathbb P_\alpha(S^+ > kd)= 0$ for every $\alpha \in \mathcal{A}$.
\end{lem}
\begin{proof}
With $F_{S^+,\alpha}$ defined in Theorem \ref{res:exactS+}, we introduce for every $\alpha$ and $\ell \geq 0$:
$$
b_{\alpha}(\ell d) := \frac{1-F_{S^+,\alpha}(\ell d)}{u_{\alpha}(\theta^*)}e^{\theta^*\ell d}, \ \ a_{\alpha}(\ell d) := \frac{L_{\alpha}(\infty) - L_{\alpha}(\ell d)}{u_{\alpha}(\theta^*)}e^{\theta^*\ell d}.
$$
Theorem \ref{res:exactS+} allows to obtain the following renewal system for the family $(b_{\alpha})_{\alpha\in\mathcal{A}}$:
$$\forall\ell > 0,\forall\alpha \in \mathcal{A},\quad
b_{\alpha}(\ell d)= a_{\alpha}(\ell d) + \sum_{\beta} \sum_{k = 0}^{\ell} b_{\beta}((\ell-k)d) G^{(k)}_{\alpha \beta}.
$$
Since the restriction of ${\bf \tilde G(\infty)}$ of ${\bf G(\infty)}$ to $\mathcal{A}^+$ is stochastic, its spectral radius equals 1 and a corresponding right eigenvector is the vector having all components equal to 1; a left eigenvector is the stationary frequency vector $\tilde w > 0$
\noindent \textit{Step 1}: For every $\alpha \in \mathcal{A}^+$, a direct application of Theorem 2.2 of Athreya and Murthy \cite{AMu75} gives the formula in Equation \ref{eq:cinf} for the limit $c(\infty)$ of $b_{\alpha}(\ell d)$ when $\ell \to \infty$, which implies the stated result.\\
\noindent \textit{Step 2}: Consider now $\alpha \notin \mathcal{A}^+$. By Theorem \ref{res:exactS+} we have
$$
\mathbb P_\alpha(S^+ > \ell d) = L_{\alpha}(\infty) - \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\ell} L^{(k)}_{\alpha \beta} \ \left\{1-\mathbb P_\beta(S^+ > (\ell-k) d)\right\}.
$$
Since $\mathbb P_\beta(S^+ > (\ell-k) d) = 1$ for $k > \ell$ and $L_{\alpha}(\infty) = \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\infty} L^{(k)}_{\alpha \beta}$, we deduce
\begin{equation}\label{eq:S+}
\mathbb P_\alpha(S^+ > \ell d) = \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\infty} L^{(k)}_{\alpha \beta} \ \mathbb P_\beta(S^+ > (\ell-k) d).
\end{equation}
Note that for fixed $\alpha$ and $\beta$, there are a finite number of non-null terms in the above sum over $k$. Using the fact that for fixed $\beta \in \mathcal{A}^+$ and $k \geq 1$ we have $\mathbb P_\beta(S^+ > (\ell-k) d) \longrightarrow 0$ when $\ell \to \infty$, as shown previously in Step 1, the stated result follows.\end{proof}
\begin{lem}\label{UmMartingale}
Let $\theta > 0$. With $u(\theta)$ defined in Subsection \ref{subsec:notdef}, the sequence of random variables $(U_m(\theta))_{m \geq 0}$ defined by $U_0(\theta):=1$ and
$$U_m(\theta):=\prod_{i=0}^{m-1}\left [ \frac{\exp(\theta f(A_{i+1}))}{u_{A_i}(\theta)}\cdot\frac{u_{A_{i+1}}(\theta)}{\rho(\theta)}\right ]=\frac{\exp(\theta S_m)u_{A_m}(\theta)}{\rho(\theta)^m u_{A_0}(\theta)} \ \ \text{, for } m \geq 1 $$
is a martingale with respect to the canonical filtration $\mathcal{F}_m = \sigma(A_0,\ldots, A_m).$
\end{lem}
\begin{proof}
We have
$$U_{m+1}(\theta)=U_m(\theta) \frac{\exp(\theta f(A_{m+1})) u_{A_{m+1}}(\theta)}{u_{A_m}(\theta)\rho(\theta)}.$$
Since $U_m(\theta)$ and $u_{A_m} (\theta)$ are measurable with respect to $\mathcal{F}_m$, we have
$$
\mathbb E[U_{m+1}(\theta) | \mathcal{F}_m] = U_m(\theta) \frac{\mathbb E[\exp(\theta f(A_{m+1})) u_{A_{m+1}}(\theta) | \mathcal{F}_m] }{u_{A_m} (\theta)\rho(\theta)}.
$$
By the Markov property we further have
$$
\mathbb E[\exp(\theta f(A_{m+1})) u_{A_{m+1}}(\theta) | \mathcal{F}_m]= \mathbb E[\exp(\theta f(A_{m+1})) u_{A_{m+1}}(\theta) | A_m]$$
and by definition of $u(\theta)$,
\begin{align*}
\mathbb E[\exp(\theta f(A_{m+1})) u_{A_{m+1}}(\theta) | A_m = \alpha] &= \sum_{\beta} \exp(\theta f(\beta)) u_{\beta}(\theta) p_{\alpha \beta}\\
& = (\Phi(\theta)u(\theta))_{\alpha}=u_{\alpha}(\theta)\rho(\theta).
\end{align*}
We deduce
$$\mathbb E[\exp(\theta f(A_{m+1})) u_{A_{m+1}}(\theta) | A_m] = u_{A_m}(\theta)\rho(\theta),$$
hence
$
\mathbb E[U_{m+1}(\theta) | \mathcal{F}_m]=U_{m}(\theta),
$
which finishes the proof.\end{proof}
\begin{lem
\label{EspeUsigma}
With $\theta^*$ defined at the end of Subsection \ref{subsec:notdef} we have
\begin{equation}
\forall\alpha \in \mathcal{A}: \ \quad \frac{1}{u_\alpha(\theta^*)} \sum_{\beta \in \mathcal{A}^+} \sum_{\ell = 1}^{\infty} L^{(\ell)}_{\alpha \beta} \ e^{\theta^* \ell d} \ u_\beta(\theta^*) = 1.
\end{equation}
\end{lem}
\begin{proof}
The proof uses Lemma \ref{Splus} and ideas inspired from \cite{KDe92} (Lemma 4.2). First note that the above equation is equivalent to
$$\mathbb E_{\alpha}[U_{\sigma^+}(\theta^*);\sigma^+<\infty]=1,$$
with $U_m(\theta)$ defined in Lemma \ref{UmMartingale}. By applying the optional sampling theorem to the bounded stopping time $\tau_n := \min(\sigma^+,n)$ and to the martingale $(U_m(\theta^*))_m$, we obtain
$$
1=\mathbb E_\alpha[U_0(\theta^*)]=\mathbb E_\alpha[U_{\tau_n}(\theta^*)] = \mathbb E_\alpha[U_{\sigma^+}(\theta^*); \sigma^+ \leq n] + \mathbb E_\alpha[U_{n}(\theta^*); \sigma^+ > n].
$$
We will show that $\mathbb E_\alpha[U_{n}(\theta^*); \sigma^+ > n] \longrightarrow 0$ when $n \to \infty$. Passing to the limit in the previous relation will then give the desired result. Since $\rho(\theta^*)=1$, we have
$$U_n(\theta^*) =\frac{\exp(\theta^* S_n)u_{A_n}(\theta^*)}{u_{A_0}(\theta^*)}$$
and it suffices to show that $\lim_{n \to \infty}\mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n] = 0$.\par
For a fixed $a > 0$ we can write
\begin{align}\label{esperance}
\mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n] &= \mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n, \ \exists k \leq n:S_k \leq -2a] \nonumber \\
&\ \ \ + \mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n, -2a \leq S_k \leq 0, \ \forall 0 \leq k \leq n].
\end{align}
The first expectation in the right-hand side of Equation (\ref{esperance}) can further be bounded as follows:
\begin{align}\label{esperance1}
\mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ & > n , \ \exists k \leq n:S_k \leq -2a] \leq \mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n, S_n \leq -a] \nonumber \\
& + \mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n, S_n > -a, \ \exists k < n:S_k \leq -2a].
\end{align}
We obviously have
\begin{equation}\label{esperance2}
\mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n, S_n \leq -a] \leq \exp(-\theta^* a).
\end{equation}
Let us further define the stopping time $T := \inf\{k \geq 1 : S_k \leq -2a\}$. Note that $T < \infty$ \textit{a.s.} since $S_n \longrightarrow -\infty$ \textit{a.s.} when $n \to \infty$. Indeed, by the ergodic theorem we have $S_n/n \longrightarrow \mathbb E[f(A)] < 0$ when $n \to \infty$. Therefore we have
\begin{align*}
\mathbb E_\alpha[\exp(\theta^* S_n)& ; \sigma^+ > n, S_n > -a, \ \exists k < n:S_k \leq -2a] \leq \mathbb P_\alpha(T \leq n, S_n > -a)\\
&= \sum_{\beta \in \mathcal{A}^-} \mathbb P_\alpha(T \leq n, S_n > -a \ | A_T = \beta)\mathbb P_\alpha(A_T = \beta)\\
& \leq \sum_{\beta \in \mathcal{A}^-} \mathbb P_\alpha(S_n - S_T > a \ | A_T = \beta)\mathbb P_\alpha(A_T = \beta)\\
& \leq \sum_{\beta \in \mathcal{A}^-} \mathbb P_\beta(S^+ > a) \mathbb P_\alpha(A_T = \beta),
\end{align*}
by the strong Markov property.
For every $a > 0$ we thus have
\begin{equation}\label{esperance3}
\limsup_{n \to \infty} \mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n, S_n > -a, \ \exists k < n:S_k \leq -2a] \leq \sum_{\beta \in \mathcal{A}^-} \mathbb P_\beta(S^+ > a).
\end{equation}
Considering the second expectation in the right-hand side of Equation (\ref{esperance}), we have
\begin{equation}\label{esperance4}
\lim_{n \to \infty} \mathbb P_\alpha(-2a \leq S_k \leq 0, \ \forall 0 \leq k \leq n) = \mathbb P_\alpha(-2a \leq S_k \leq 0, \ \forall k \geq 0) = 0,
\end{equation}
again since $S_n \longrightarrow -\infty$ \textit{a.s.} when $n \to \infty$.
Equations (\ref{esperance}),(\ref{esperance1}),(\ref{esperance2}),(\ref{esperance3}) and (\ref{esperance4}) imply that for every $a > 0$ we have
$$
\limsup_{n \to \infty} \mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n] \leq \exp(-\theta^*a) + \sum_{\beta \in \mathcal{A}^-} \mathbb P_\beta(S^+ > a).
$$
Using Lemma \ref{Splus} and taking $a \to \infty$ we obtain $\lim_{n \to \infty}\mathbb E_\alpha[\exp(\theta^* S_n); \sigma^+ > n] = 0.$
\end{proof}
We are now ready to prove the Theorem \ref{res:asymptS+}.
\noindent {\bf Proof of Theorem \ref{res:asymptS+}:}\\
For $\alpha \in \mathcal{A}^+$ the formula has been already shown in Step 1 of the proof of Lemma \ref{Splus}.
For $\alpha \notin \mathcal{A}^+$ we will prove the stated formula using Theorem \ref{res:exactS+}. From Equation (\ref{eq:S+}), we have
\begin{equation*}
\mathbb P_\alpha(S^+ > \ell d) = \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\infty} L^{(k)}_{\alpha \beta} \ \mathbb P_\beta(S^+ > (\ell-k) d),
\end{equation*}
hence
\begin{equation*}
\frac{e^{\theta^*{\ell d}}\mathbb P_{\alpha}(S^+>\ell d)}{u_{\alpha}(\theta^*)} = \sum_{\beta \in \mathcal{A}^+} \sum_{k=1}^{\infty} \frac{e^{\theta^*{(\ell-k)d}}\mathbb P_{\beta}(S^+>(\ell-k)d)}{u_{\beta}(\theta^*)} L^{(k)}_{\alpha \beta} e^{\theta^*{kd}} \frac{u_{\beta}(\theta^*)}{u_{\alpha}(\theta^*)}.
\end{equation*}
Note that for every $\alpha$ and $\beta$ there are a finite number of non-null terms in the above sum over $k$. Moreover, as shown in Lemma \ref{Splus}
$$\forall\beta \in \mathcal{A}^+,\ \forall k\geq 0:\ \quad
\frac{e^{\theta^*{(\ell-k)d}}\mathbb P_{\beta}(S^+>(\ell-k)d)}{u_{\beta}(\theta^*)} \underset{\ell \to \infty}{\longrightarrow} c(\infty).
$$
We finally obtain that
$$
\lim_{\ell \rightarrow +\infty} \frac{e^{\theta^*{\ell d}}\mathbb P_{\alpha}(S^+> \ell d)}{u_{\alpha}(\theta^*)} = \frac{c(\infty)}{u_\alpha(\theta^*)} \sum_{\beta \in \mathcal{A}^+} \sum_{k = 1}^{\infty} L^{(k)}_{\alpha \beta} \ e^{\theta^* k d} \ u_\beta(\theta^*),
$$
which equals $c(\infty)$ as desired, by Lemma \ref{EspeUsigma}.
\subsection{Proof of Theorem \ref{res:Q1}}
Since $S^+ \geq Q_1$, for every $\alpha \in \mathcal{A}$ we have
$$
\mathbb P_{\alpha}(S^+>kd) = \mathbb P_{\alpha}(Q_1>kd) + \mathbb P_{\alpha}(S^+>kd, Q_1 \leq kd).
$$
We will further decompose the last probability with respect to the values taken by $S_{\sigma^-}$ and $A_{\sigma^-}$, as follows:
\begin{align*}
& \mathbb P_{\alpha}(S^+>kd, Q_1 \leq kd) = \sum_{\ell<0}\sum_{\beta \in \mathcal{A}^-}\mathbb P_{\alpha}(S^+>kd, Q_1 \leq kd, S_{\sigma^-}=\ell d,A_{\sigma^-}=\beta)\\
&= \sum_{\ell<0}\sum_{\beta \in \mathcal{A}^-}\mathbb P_{\alpha}(S^+ -S_{\sigma^-} >(k-\ell)d \ | A_{\sigma^-}=\beta, Q_1 \leq kd, S_{\sigma^-}=\ell d ) \\
& \ \ \ \ \times \mathbb P_{\alpha}(Q_1 \leq kd, S_{\sigma^-}=\ell d, A_{\sigma^-}=\beta)\\
&= \sum_{\ell<0}\sum_{\beta \in \mathcal{A}^-}\mathbb P_{\beta}(S^+ >(k-\ell)d ) \cdot \left\{Q_{\alpha\beta}^{(\ell)}- \mathbb P_{\alpha}(Q_1 > kd, S_{\sigma^-}=\ell d, A_{\sigma^-}=\beta)\right \},
\end{align*}
by applying the strong Markov property to the stopping time $\sigma^-$. We thus obtain
\begin{align*}
& \mathbb P_{\alpha}(S^+>kd)-\sum_{\ell <0}\sum_{\beta \in \mathcal{A}^-}\mathbb P_{\beta}(S^+>(k-\ell )d)\cdot Q_{\alpha\beta}^{(\ell )} - \mathbb P_{\alpha}(Q_1>kd) \\
& = - \sum_{\ell <0}\sum_{\beta \in \mathcal{A}^-}\mathbb P_{\beta}(S^+ >(k-\ell )d ) \ \mathbb P_{\alpha}(Q_1 > kd, S_{\sigma^-}=\ell d, A_{\sigma^-}=\beta).
\end{align*}
By Theorem \ref{res:asymptS+} we have $\mathbb P_{\beta}(S^+ >kd ) = O(e^{-\theta^*kd})$ as $k \to \infty$, for every $\beta \in \mathcal{A}^-$, from which we deduce that the left-hand side of the previous equation is $o(\mathbb P_{\alpha}(Q_1 > kd))$ when $k \to \infty$.
The stated result then easily follows. $\hfill\square$
\subsection{Proof of Theorem \ref{res:Mn}}
We will first prove some useful lemmas.
\begin{lem}\label{lem:rho'}
We have
$\rho'(0)=\mathbb E[f(A)]<0.$
\end{lem}
\begin{proof}
By the fact that $\rho(\theta)$ is an eigenvalue of the matrix $\Phi(\theta)$ with corresponding eigenvector $u(\theta)$, we have
$$\rho(\theta)u_{\alpha}(\theta)=\left (\Phi(\theta)u(\theta)\right )_{\alpha}
=\sum_\beta p_{\alpha\beta}e^{\theta f(\beta)}u_\beta(\theta).$$
When derivating the previous relation with respect to $\theta$ we obtain
$$\frac{d}{d \theta}(\rho (\theta)u_\alpha(\theta))=
\sum_\beta p_{\alpha\beta} \left ( f(\beta)e^{\theta f(\beta)}u_\beta(\theta)+e^{\theta f(\beta)}u'_{\beta}(\theta)\right ).$$
We have $\rho(0)=1$ et $u(0)=^t(1/r,\dots,1/r)$. For $\theta=0$, we then get
\begin{equation}\label{eq:3}
\left. \sum_\alpha \pi_{\alpha} \frac{d}{d \theta}(\rho (\theta)u_\alpha(\theta)) \right |_{\theta=0} =\frac{1}{r}\mathbb E[f(A)]+\sum_{\alpha,\beta}\pi_{\alpha} p_{\alpha\beta}u'_{\beta}(0)
=\frac{1}{r}\mathbb E[f(A)]+\sum_{\beta}\pi_{\beta}u'_{\beta}(0).
\end{equation}
On the other hand,
$$ \sum_\alpha \pi_{\alpha} \frac{d}{d \theta}(\rho (\theta)u_\alpha(\theta))=
\frac{d}{d \theta}\left(\sum_\alpha \pi_{\alpha} \rho (\theta)u_\alpha(\theta)\right) =\rho'(\theta)\sum_\alpha \pi_{\alpha} u_\alpha(\theta)+\rho(\theta)\sum_\alpha \pi_{\alpha} u'_\alpha(\theta).$$
For $\theta=0$ we get
\begin{equation}\label{eq:4}
\left. \sum_\alpha \pi_{\alpha} \frac{d}{d \theta}(\rho (\theta)u_\alpha(\theta)) \right |_{\theta=0}=\frac{\rho'(0)}{r}+\rho(0)\cdot\sum_{\alpha}\pi_{\alpha}u'_{\alpha}(0).
\end{equation}
From Equations (\ref{eq:3}) and (\ref{eq:4}) we deduce
$$\frac{\rho'(0)}{r}+\sum_{\alpha}\pi_{\alpha}u'_{\alpha}(0)=\frac{1}{r}\mathbb E[f(A)]+\sum_\beta\pi_{\beta}u'_{\beta}(0),$$
from which the stated result easily follows.
\end{proof}
\begin{lem}\label{lem:DKa91}
There exist ${\cal I}>0$ and $n_0\geq 0$ such that $\forall n\geq n_0$, $\mathbb P_{\alpha}(S_n\geq 0)\leq \exp(-n{\cal I})$ for every $\alpha \in \mathcal{A}$.
\end{lem}
\begin{proof}
By a large deviation principle for additive functionals of Markov chains (see Theorem 3.1.2. in \cite{DKa91b}) we have
$$\limsup_{n\rightarrow +\infty}\frac{1}{n}\log\left ( \mathbb P_{\alpha}\left(\frac{S_n}{n} \in \Gamma\right)\right)
\leq -{\cal I},$$
with $\Gamma=[0,+\infty )$ and ${\cal I}=\inf_{x\in \bar{\Gamma}} \sup_{\theta\in\mathbb R} (\theta x-\log \rho(\theta))$.
Since $\cal A$ is finite, it remains to prove that ${\cal I}>0$.
For every $x \geq 0$, let us denote $g_x(\theta):=\theta x-\log \rho(\theta)$ and $I(x):=\sup_{\theta\in\mathbb R} g_x(\theta)$.
We will first show that $I(x)=\sup_{\theta\in\mathbb R^+} g_x(\theta)$.
Indeed, we have $g'_x(\theta)= x - \rho'(\theta)/\rho(\theta)$. By the strict convexity property of $\rho$ (see \cite{DKa91b, KOs87}) and the fact that $\rho '(0)=\mathbb E[f]<0$ (by Lemma \ref{lem:rho'}), we deduce that $\rho'(\theta)<0$ for every $\theta \leq 0$, implying that $g'_x(\theta)> x \geq 0$ for $\theta \leq 0$. The function $g'_x$ is therefore increasing on $\mathbb R^-$, and hence $I(x)=\sup_{\theta\in\mathbb R^+} g_x(\theta)$.
As a consequence, we deduce that $x \mapsto I(x)$ is non-decreasing on $\mathbb R^+$.
We thus obtain ${\cal I}=\inf_{x\in \mathbb R^+}I(x)=I(0)$.
Further, we have $I(0)=\sup_{\theta\in\mathbb R} \left(-\log \rho(\theta)\right)=-\inf_{\theta\in \mathbb R^+} \log(\rho(\theta))$. Using again the fact that $\rho'(0)<0$ (Lemma \ref{lem:rho'}), the strict convexity of $\rho$ and the fact that $\rho(0)=\rho(\theta^*)=1$, we finally obtain ${\cal I}=-\log\left(\inf_{\theta\in \mathbb R^+} \rho(\theta)\right) >-\log \rho(0)=0$. The statement then follows.
\end{proof}
\begin{lem}\label{lem:Esp_K1}
We have $\mathbb E_\alpha(K_1) < \infty$ for every $\alpha \in \mathcal{A}$.
\end{lem}
\begin{proof}
Note that $\mathbb P_\alpha(K_1>n)\leq\mathbb P_\alpha(S_n\geq 0)$. With $n_0 \in \mathbb N$ and ${\cal I}>0$ defined in Lemma \ref{lem:DKa91}, using a well-known formula for the expectation, we get
$$
\mathbb E_\alpha[K_1]=\sum_{n\geq 0}\mathbb P_\alpha(K_1>n)\leq\sum_{n\geq 0}\mathbb P(S_n\geq 0)\leq C + \sum_{n\geq n_0}\exp(-n{\cal I}),
$$
where $C >0$ is a constant. The statement easily follows.
\end{proof}
\begin{lem
\label{Km}
We have
$$\lim_{m\rightarrow +\infty} \frac{K_m}{m} = \sum_{\beta} z_\beta \mathbb E_\beta(K_1) \mbox{ a.s.}$$
\end{lem}
\begin{proof}
Recall that $K_1 = \sigma^-$. We can write
\begin{equation}\label{K}
\frac{K_m}{m} = \frac{K_1}{m} + \frac{1}{m}\sum_{i=2}^m (K_i - K_{i-1}) = \frac{K_1}{m} + \sum_\beta \frac{1}{m}\sum_{i=2}^m (K_i - K_{i-1}) \textbf{1}_{\{A_{K_{i-1}}=\beta\}}.
\end{equation}
First note that $\displaystyle \frac{K_1}{m} \to 0 \ a.s.$ when $m \to \infty$, since $K_1 < +\infty \ a.s.$ By the strong Markov property we have that, conditionally on $(A_{K_{i-1}})_{i \geq 2}$, the random variables $(K_i-K_{i-1})_{i \geq 2}$ are all independent and the distribution of $K_i-K_{i-1}$ depends only on $A_{K_{i-1}}$ and we have
$\mathbb P(K_i-K_{i-1}=\ell\ | A_{K_{i-1}} = \alpha) = \mathbb P_\alpha(K_1 = \ell)$.
Therefore, the couples $Y_i := (A_{K_{i-1}}, K_i-K_{i-1}), i \geq 2$ form a Markov chain on $\mathcal{A}^- \times \mathbb N$, with transition probabilities
$
\mathbb P(Y_i = (\beta,\ell) \ | Y_{i-1} = (\alpha,k)) = q_{\alpha \beta} \mathbb P_\beta(K_1 = \ell)$.
Recall that the restriction ${\bf \tilde Q}$ of the matrix ${\bf Q}$ to the subspace $\mathcal{A}^-$ is irreducible.
Therefore, the Markov chain $(Y_i)_i$ is also irreducible and we can show that
$\pi(\alpha,k):=z_\alpha \mathbb P_\alpha(K_1 = k)$ is its invariant distribution. Indeed, since $z$ is invariant for ${\bf Q}$, we easily deduce that
$$
\sum_{\alpha,k} \pi(\alpha,k) \cdot q_{\alpha \beta} \mathbb P_{\beta}(K_1=\ell) = \pi(\beta,\ell).
$$
For fixed $\beta$, when applying the ergodic theorem to the Markov chain $(Y_i)_i$ and the function $\varphi_\beta(\alpha, k) := k \textbf{1}_{\{\alpha=\beta\}}$, we deduce
$$
\frac{1}{m}\sum_{i=2}^m (K_i - K_{i-1}) \textbf{1}_{\{A_{K_{i-1}}=\beta\}} \longrightarrow \sum_{\alpha,k} \varphi_\beta(\alpha,k)\pi(\alpha,k) = z_\beta \mathbb E_\beta(K_1) \ \ a.s.
$$
when $m \to \infty$.
Taking the sum over $\beta$ and using the relation (\ref{K}) gives the desired result.
\end{proof}
\noindent {\bf Proof of Theorem \ref{res:Mn}:}\\
The proof is inspired from \cite{KDe92}.
Given $(A_{K_i})_{i \geq 0}$, the random variables $(Q_i)_{i \geq 1}$ are independent and the \textit{cdf} of $Q_i$ is $F_{A_{K_{i-1}}A_{K_i}}$. Therefore
\begin{eqnarray*}
\mathbb P_{\alpha}\left ( M_{K_m}\leq y \right)&=& \mathbb E_\alpha\left [ \prod_{i=1}^mF_{A_{K_{i-1}}A_{K_i}}(y)\right ]\\
&=& \mathbb E_\alpha\left [ \exp\left \{\sum_{\beta,\gamma \in \mathcal{A}} m \psi_{\beta\gamma}(m)\log(F_{\beta\gamma}(y))\right \}\right ],
\end{eqnarray*}
with $\psi_{\beta\gamma}(m):=\#\{i:1 \leq i\leqslant m,A_{K_{i-1}}=\beta,A_{K_{i}}=\gamma\}/m$. Given that $A_0 = \alpha \in \mathcal{A}^-$, the states $(A_{K_i})_{i \geq 0}$ form an irreducible Markov chain on $\mathcal{A}^-$ of transition matrix ${\bf \tilde Q}=(q_{ \beta \gamma})_{\beta, \gamma \in \mathcal{A}^-}$ and stationary frequency vector $\tilde z = (z_\beta)_{\beta \in \mathcal{A}^-} > 0$.
Consequently, for $\beta, \gamma \in \mathcal{A}^-$ the ergodic theorem implies that $\psi_{\beta\gamma}(m)\longrightarrow z_{\beta}q_{\beta\gamma}$ \textit{a.s.} when $m\rightarrow\infty$. On the other hand, for any $\alpha \in \mathcal{A}$, if $\beta \in \mathcal{A} \setminus \mathcal{A}^-$, then $\psi_{\beta\gamma}(m)\leqslant 1/m$ and thus $\psi_{\beta\gamma}(m)\longrightarrow 0$ \textit{a.s.} when $m\rightarrow\infty$, for any $\gamma \in \mathcal{A}$. With $z_{\beta} = 0$ for $\beta \in \mathcal{A} \setminus \mathcal{A}^-$, we thus have
$\psi_{\beta\gamma}(m)\longrightarrow z_{\beta}q_{\beta\gamma}$ \textit{a.s.} when $m\rightarrow\infty$, for every $\beta, \gamma \in \mathcal{A}$.
Denoting $d_{\beta\gamma}(m):=m\left[1-F_{\beta\gamma}\left(\frac{\log m}{\theta^*}+x\right)\right]$
and using the fact that $d_{\beta\gamma}(m)$ are uniformly bounded in $\beta$ and $\gamma$, we have
\begin{eqnarray*}
\underset{m\rightarrow\infty}{\lim}\mathbb P_{\alpha}\left(M_{K_m}\leq\frac{\log m}{\theta^*}+x\right)
&=&\underset{m\rightarrow\infty}{\lim}\mathbb E_{\alpha}\left [\exp\left(-\sum_{\beta,\gamma \in \mathcal{A}}\psi_{\beta\gamma}(m)d_{\beta\gamma}(m)\right )\right ] \\
&=&\underset{m\rightarrow\infty}{\lim}\exp\left(-\sum_{\beta,\gamma \in \mathcal{A}}z_{\beta}q_{\beta\gamma}d_{\beta\gamma}(m)\right ).
\end{eqnarray*}
Since
$$\sum_{\gamma \in \mathcal{A}}q_{\beta\gamma}d_{\beta\gamma}(m)=m\left[1-F_{\beta}\left(\frac{\log m}{\theta^*}+x\right)\right],$$
$$\underset{m\rightarrow\infty}{\lim}\mathbb P_{\alpha}\left(M_{K_m}\leq \frac{\log m}{\theta^*}+x\right)
=\underset{m\rightarrow\infty}{\lim}\exp \left(-m \sum_{\beta \in \mathcal{A}^-} z_{\beta}\left [1-F_{\beta}\left(\frac{\log m}{\theta^*}+x\right )\right]\right).$$
But $$1-F_{\beta}\left(\frac{\log m}{\theta^*}+x\right)
=\mathbb P_{\beta}\left(Q_1> \frac{\log(m)}{\theta^*}+x\right)
=\mathbb P_{\beta}\left(Q_1> \left \lfloor\frac{\log(m)}{\theta^*}+x\right\rfloor\right),$$
and hence, with $\displaystyle y=y(m):=\frac{\log(m)}{\theta^*}+x$ we get, using Theorem \ref{res:Q1}:
\begin{align*}
1-F_{\beta}\left (\frac{\log m}{\theta^*}+x\right )& \underset{m\rightarrow\infty}{\sim}
\mathbb P_{\beta}\left(S^+> \left \lfloor\frac{\log(m)}{\theta^*}+x\right \rfloor\right)\\
&\ \ \ \ \ \ \ -\sum_{k < 0}\sum_{\gamma \in \mathcal{A}^-} \mathbb P_{\gamma}(S^+>\lfloor y \rfloor -kd) \times \ \mathbb P_{\beta} (S_{\sigma^-}=kd, A_{\sigma^-}=\gamma)\ .
\end{align*}
This further leads to
\begin{align*}
& \underset{m\rightarrow\infty}{\lim}\mathbb P_{\alpha}\left(M_{K_m}\leq \frac{\log m}{\theta^*}+x\right)
=\underset{m\rightarrow\infty}{\lim}
\exp\left \{-\sum_{\beta \in \mathcal{A}^-} mz_{\beta} \mathbb P_{\beta}\left(S^+> \left \lfloor\frac{\log(m)}{\theta^*}+x\right\rfloor\right)\right \}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times\exp\left \{
\sum_{k < 0}\sum_{\gamma \in \mathcal{A}^-} \mathbb P_{\gamma}(S^+>\lfloor y \rfloor -kd) \cdot\sum_{\beta \in \mathcal{A}^-}z_{\beta}\mathbb P_{\beta}\left(S_{\sigma^-}=kd, A_{\sigma^-}=\gamma\right )
\right \}.
\end{align*}
Since $K_{m(n)} \leq n \leq K_{m(n)+1}$ and $m(n) \longrightarrow \infty$ \textit{a.s.}, Lemma \ref{Km} implies that $\displaystyle \frac{n}{m(n)} \longrightarrow A^*$ \textit{a.s.} Moreover, since $M_{K_{m(n)}} \leq M_n \leq M_{K_{m(n)+1}}$, we finally obtain
\begin{align*}
& \underset{n\rightarrow\infty}{\lim}\mathbb P_{\alpha}\left(M_n\leq \frac{\log n}{\theta^*}+x\right )
= \underset{n\rightarrow\infty}{\lim}\mathbb P_{\alpha}\left(M_{K_{\lfloor n/A^*\rfloor}}\leq \frac{\log n}{\theta^*}+x\right )\\
&=\underset{n\rightarrow\infty}{\lim}
\exp\left \{-\frac{n}{A^*}\sum_{\beta \in \mathcal{A}^-} z_{\beta} \mathbb P_{\beta}\left (S^+> \left \lfloor\frac{\log(n)}{\theta^*}+x\right \rfloor\right )\right \}\\
& \times\exp\left \{
\sum_{k < 0}\sum_{\gamma \in \mathcal{A}^-} \mathbb P_{\gamma}\left(S^+>\left \lfloor \frac{\log(n)}{\theta^*}+x \right\rfloor -kd\right) \cdot\sum_{\beta \in \mathcal{A}^-}z_{\beta}\mathbb P_{\beta} (S_{\sigma^-}=kd, A_{\sigma^-}=\gamma)
\right \}.
\end{align*}
It remains to prove the stated expression for $A^*:= \lim_{m\rightarrow +\infty} \frac{K_m}{m} \ a.s.$ in order to finish the proof.
Recall that $\sigma^- = K_1$. In Lemma \ref{Km} we proved that
$$A^* = \sum_{\alpha} z_{\alpha}\mathbb E_\alpha(\sigma^-).$$
Since $(U_m(\theta))_m$ is a martingale (see Lemma \ref{UmMartingale}) and $\sigma^-$ a stopping time, using the optional sampling theorem we get
$\mathbb E_{\alpha}\left [U_{\sigma^-}(\theta)\right ]=\mathbb E_{\alpha}\left [U_0(\theta)\right ]=1.$
Consequently,
\begin{eqnarray*}
1&=& \mathbb E_{\alpha}\left [\exp(\theta\cdot S_{\sigma^-})
\frac{u_{A_{\sigma^-}}(\theta)}{u_{A_{0}}(\theta)}\frac{1}{\rho(\theta)^{\sigma^-}}\right ]\\
&=&\mathbb E_{\alpha}\left [\exp(\theta\cdot S_{\sigma^-})
\frac{u_{A_{\sigma^-}}(\theta)}{u_{\alpha}(\theta)}\frac{1}{\rho(\theta)^{\sigma^-}}\right ]\\
&=& \sum_{\beta}\mathbb E_{\alpha}\left [\exp(\theta\cdot S_{\sigma^-})
\frac{u_{\beta}(\theta)}{u_{\alpha}(\theta)}\frac{1}{\rho(\theta)^{\sigma^-}}\big|A_{\sigma^-}=\beta\right ]\cdot \mathbb P_{\alpha}(A_{\sigma^-}=\beta)\\
&=&\sum_{\beta}\frac{u_{\beta}(\theta)}{u_{\alpha}(\theta)}\mathbb E_{\alpha}\left [\frac{\exp(\theta\cdot S_{\sigma^-})}{\rho(\theta)^{\sigma^-}} \big|A_{\sigma^-}=\beta\right ]\cdot q_{\alpha\beta}.
\end{eqnarray*}
We deduce
$$u_{\alpha}(\theta)=\sum_{\beta}\mathbb E_{\alpha}\left [\frac{\exp(\theta\cdot S_{\sigma^-})}{\rho(\theta)^{\sigma^-}}\big| A_{\sigma^-}=\beta \right ]\cdot u_{\beta}(\theta)q_{\alpha\beta} .$$
Derivating the above relation leads to
\begin{align*}
&u'_{\alpha}(\theta)=\\
& \sum_{\beta}q_{\alpha\beta} u_{\beta}(\theta)
\mathbb E_{\alpha}\left [ \frac{
S_{\sigma^{-}}\exp(\theta\cdot S_{\sigma^-})\rho(\theta)^{\sigma^-}
-\exp(\theta\cdot S_{\sigma^-})\sigma^-\rho(\theta)^{\sigma^--1}\rho'(\theta)}{\rho(\theta)^{2\sigma^-}} \big| A_{\sigma^-}=\beta\right ] \\
& \hfill \ \ + \sum_{\beta}q_{\alpha\beta} u'_{\beta}(\theta)\mathbb E_{\alpha}\left [\frac{\exp(\theta\cdot S_{\sigma^-})}{\rho(\theta)^{\sigma^-}}\big| A_{\sigma^-}=\beta \right ].
\end{align*}
Since $\rho(0)=1$, we obtain for $\theta = 0$:
$$u'_{\alpha}(0)=\sum_{\beta}q_{\alpha\beta}u_{\beta}(0)\left ( \mathbb E_{\alpha}\left [ S_{\sigma^-}\big| A_{\sigma^-}=\beta\right ]- \rho'(0)\mathbb E_{\alpha}\left [ \sigma^-\big| A_{\sigma^-}=\beta\right ]\right ) + \sum_{\beta}q_{\alpha\beta}u'_{\beta}(0).$$
By the fact that $u(0)=^t(1/r,\ldots,1/r)$, we further get
$$u'_{\alpha}(0)=\frac{1}{r}\mathbb E_{\alpha}[S_{\sigma^-}]-
\frac{\rho'(0)}{r} \mathbb E_\alpha(\sigma^-)+ \sum_{\beta}q_{\alpha\beta}u'_{\beta}(0).$$
From the last relation we deduce
\begin{equation}\label{eq1}
\sum_{\alpha}z_{\alpha}u'_{\alpha}(0)
=\frac{1}{r}\sum_{\alpha}z_{\alpha}\mathbb E_{\alpha}\left [ S_{\sigma^-}\right ]
-\frac{\rho'(0)}{r}\sum_{\alpha} z_{\alpha}\mathbb E_\alpha(\sigma^-) +\sum_{\alpha}\sum_{\beta}z_{\alpha} q_{\alpha\beta}u'_{\beta}(0).
\end{equation}
On the other hand, since $z$ is the stationnary frequency vector of the matrix ${\bf Q}$, we have $z=z \cdot {\bf Q}$
and thus
\begin{equation}\label{eq:2}
\sum_{\alpha}z_{\alpha}u'_{\alpha}(0)=^tz \cdot u'(0)=^t(z {\bf Q}) \cdot u'(0)=\sum_{\beta} {^t(z {\bf Q})_{\beta}}\cdot u'_{\beta}(0)
=\sum_{\beta}\sum_{\alpha} z_{\alpha} q_{\alpha\beta} u'_{\beta}(0).
\end{equation}
Equations (\ref{eq1}) and (\ref{eq:2}) imply that $\sum_{\alpha}z_{\alpha}\mathbb E_{\alpha}\left [ S_{\sigma^-}\right ]=\rho'(0)\cdot\sum_{\alpha} z_{\alpha} \mathbb E_\alpha(\sigma^-)$
and thus $A^*=\sum_{\alpha} z_{\alpha} \mathbb E_\alpha(\sigma^-)=\frac{1}{\rho'(0)}\sum_{\alpha}z_{\alpha}\mathbb E_{\alpha}\left [ S_{\sigma^-}\right ]$.
Using now the fact that $\rho'(0)=\mathbb E[f(A)]$ (see Lemma \ref{lem:rho'}) gives the stated expression for $A^*$. $\hfill\square$
\section{Applications and computational methods}\label{sec:comp}
In order to simplify the presentation, we suppose in this section that $d=1$. Let $-u,\dots,0,\dots,v$ be the possible scores, with $u, v \in \mathbb{N}$.
For $-u \leq j \leq v$, we introduce the matrix ${\bf P^{(j)}}$ with entries
$$P^{(j)}_{\alpha \beta} := \mathbb P_{\alpha}(A_1=\beta, f(A_{1})= j)$$
for $\alpha, \beta \in \mathcal{A}$.
Note that $ P^{(f(\beta))}_{\alpha\beta}=p_{\alpha\beta}$, $P^{(j)}_{\alpha\beta}=0$ if $j\neq f(\beta)$ and ${\bf P}=\sum_{j=-u}^{v}{\bf P^{(j)}}$, where ${\bf P}=(p_{\alpha \beta})_{\alpha, \beta}$ is the transition probability matrix of the Markov chain $(A_i)_i$.
In order to obtain the approximate distribution of $Q_1$ given in Theorem \ref{res:Q1}, we need to compute the quantities $Q^{(\ell)}_{\alpha \beta}$ for $-u \leq \ell \leq v, \alpha, \beta \in \mathcal{A}$ . This is the topic of the next subsection. We denote ${\bf Q^{(\ell)}}$ the matrix $(Q^{(\ell)}_{\alpha \beta})_{\alpha, \beta \in \mathcal{A}}$.
\subsection{Computation of ${\bf Q^{(\ell)}}$ for $-u \leq \ell \leq v$, and of ${\bf Q}$}\label{SubSec:Q}
Recall that $Q^{(\ell)}_{\alpha \beta}=\mathbb P_\alpha(S_{\sigma^-} = \ell, A_{\sigma^-} = \beta)$, and hence $Q^{(\ell)}_{\alpha \beta}=0$ for $\ell\geq 0$ or $\beta \in \mathcal{A} \setminus \mathcal{A}^-$. Note also that $\sigma^-=1$ if $f(A_1) < 0$.
Let $-u \leq \ell \leq -1$.
When decomposing with respect to the possible values $j$ of $f(A_{1})$, we obtain:
\begin{align*}
Q^{(\ell)}_{\alpha \beta}&= \mathbb P_\alpha( A_{1} = \beta, f(A_1) = \ell) + \mathbb P_\alpha(S_{\sigma^-} = \ell , A_{\sigma^-} = \beta, f(A_1) = 0) \\
& \ \ \ \ + \sum_{j = 1}^{v} \mathbb P_\alpha(S_{\sigma^-} = \ell , A_{\sigma^-} = \beta, f(A_1) = j).
\end{align*}
Note that the first term on the right hand side is exactly $P^{(\ell)}_{\alpha \beta}$ defined at the beginning of this section.
We further have, by the law of total probability and the Markov property:
\begin{align*}
\mathbb P_\alpha(S_{\sigma^-} = \ell , A_{\sigma^-} = \beta, f(A_1) = 0)
&= \sum_{\gamma} P^{(0)}_{\alpha \gamma} \ \mathbb P_\alpha(S_{\sigma^-} = \ell , A_{\sigma^-} = \beta \ | A_1 = \gamma, f(A_1) = 0) \\
&= \sum_{\gamma} P^{(0)}_{\alpha \gamma} \ \mathbb P_\gamma(S_{\sigma^-} = \ell , A_{\sigma^-} = \beta)=( {\bf P^{(0)}}{\bf Q^{(\ell)}})_{\alpha \beta}.
\end{align*}
Let $j \in \{1,\ldots,v\}$ be fixed. We have
\begin{align*}
\mathbb P_\alpha(S_{\sigma^-} &= \ell , A_{\sigma^-} = \beta, f(A_1) = j)= \sum_{\gamma} P^{(j)}_{\alpha \gamma} \ \mathbb P_\alpha(S_{\sigma^-} = \ell , A_{\sigma^-} = \beta \ | A_1 = \gamma, f(A_1) = j).
\end{align*}
For every possible $s \geq 1$, we denote $\mathcal{T}_s$ the set of all possible $s$-tuples $t=(t_1,\dots,t_s)$ verifying $-u\leq t_i\leq -1$ for $i=1,\dots, s$, $\ t_1+\dots+t_{s-1}\geq -j >0$ and $t_1+\dots+t_{s}=\ell-j >0$. Decomposing the possible paths from $-k$ to $\ell$ gives
$$
Q^{(\ell)}_{\alpha \beta}
= P^{(\ell)}_{\alpha \beta}+({\bf P^{(0)}} {\bf Q^{(\ell)}})_{\alpha \beta}+\sum_{j=1}^v\left ( {\bf P^{(j)}} \sum_s \sum_{t\in{\cal T}_s}\prod_{i=1}^s {\bf Q^{(t_i)}}\right)_{\alpha \beta},
$$
hence
\begin{equation}\label{Eq:RecurrenceQell}
{\bf Q^{(\ell)}}= {\bf P^{(\ell)}}+{\bf P^{(0)}} {\bf Q^{(\ell)}}+\sum_{j=1}^v {\bf P^{(j)}}\sum_s \sum_{t\in{\cal T}_s}\prod_{i=1}^s {\bf Q^{(t_i)}}.
\end{equation}
Recalling that ${\bf Q}=(q_{\alpha\beta})_{\alpha, \beta}$ with $q_{\alpha\beta}=\mathbb P_{\alpha}(A_{\sigma^-}=\beta)=\sum_{\ell<0}Q^{(\ell)}_{\alpha \beta}$,
we have
\begin{equation}\label{Eq:Q}
{\bf Q}=\sum_{\ell <0}{\bf Q^{(\ell)}}.
\end{equation}
\noindent \textbf{Example:} In the case where $u=v=1$, we only have the possible values $\ell=-1$, $j=1$, $s=2$ and $t_1=t_2=-1$, thus
\begin{equation}\label{Eq:RecQellExple}
{\bf Q^{(-1)}}={\bf P^{(-1)}}+{\bf P^{(0)}}\cdot {\bf Q^{(-1)}}+{\bf P^{(1)}}({\bf Q^{(-1)}})^2\mbox{ and } {\bf Q}={\bf Q^{(-1)}}.
\end{equation}
\subsection{Computation of $L_{\alpha \beta}^{(\ell)}$ for $0 \leq \ell \leq v$, and of $L_{\alpha}(\infty)$}\label{Subsec:L}
Recall that $L_{\alpha \beta}^{(\ell)}=\mathbb P_\alpha(S_{\sigma^+} = \ell ,\sigma^+ < \infty, A_{\sigma^+} = \beta)$. Denote ${\bf L^{(\ell)}}:=(L^{(\ell)}_{\alpha\beta})_{\alpha,\beta}$.
First note that $L_{\alpha \beta}^{(\ell)}=0$ for $\ell\leq0$ or $\beta \in \mathcal{A} \setminus \mathcal{A}^+ $. Using a similar method as the one used to obtain $Q^{(\ell)}_{\alpha \beta}$ in the previous subsection, we denote for every possible $s \geq 1$,
$\mathcal{T}'_s$ the set of all $s$-tuples $t=(t_1,\dots,t_s)$ verifying $1\leq t_i\leq v$ for $i=1,\dots, s$, $\ t_1+\dots+t_{s-1}\leq k$ and $t_1+\dots+t_{s}=\ell+k >0$.
For every $0<\ell\leq v$ we then have
\begin{equation}\label{Eq:RecurrenceLell}
{\bf L^{(\ell)}}= {\bf P^{(\ell)}}+{\bf P^{(0)}} {\bf L^{(\ell)}}+\sum_{k=1}^u {\bf P^{(-k)}}\sum_s \sum_{t\in{\cal T}'_s}\prod_{i=1}^s {\bf L^{(t_i)}}
\end{equation}
Since $L_{\alpha}(\infty)=\mathbb P_{\alpha}(\sigma^+<\infty)=\sum_{\beta}\sum_{\ell =1}^v L_{\alpha \beta}^{(\ell )}$, and denoting by ${\bf L}(\infty)$ the column vector containing all $L_{\alpha}(\infty)$ for $\alpha \in \mathcal{A}$, and by ${1\hspace{-0.2ex}\rule{0.12ex}{1.61ex}\hspace{0.5ex}}_r$ the column vector of size $r$ with all components equal to 1, we can write
\begin{equation}
{\bf L}(\infty)=\sum_{\ell=1}^v {\bf L^{(\ell)}}\cdot {1\hspace{-0.2ex}\rule{0.12ex}{1.61ex}\hspace{0.5ex}}_r.
\end{equation}
\noindent \textbf{Example:} In the case where $u=v=1$, equation (\ref{Eq:RecurrenceLell}) gives
\begin{equation}\label{Eq:LCassimple}
{\bf L^{(1)}}={\bf P^{(1)}}+{\bf P^{(0)}}\cdot {\bf L^{(1)}}+{\bf P^{(-1)}}\cdot ({\bf L^{(1)}})^2,
\end{equation}
\begin{equation}\label{Eq:LCassimple2}
{\bf L^{(\ell)}} = 0 \mbox{ for } \ell > 1, \mbox{ thus } {\bf L}(\infty)= {\bf L^{(1)}}\cdot {1\hspace{-0.2ex}\rule{0.12ex}{1.61ex}\hspace{0.5ex}}_r.
\end{equation}
\subsection{Computation of $F_{S^+,\alpha}{(\ell)}$ for $\ell \geq 0$}\label{Subsec:FS+}
For $\ell\geq 0$ let us denote ${\bf F}_{S^+,\cdot}{(\ell)}:=(F_{S^+,\alpha}{(\ell)})_{\alpha \in \mathcal{A}}$, seen as a column vector of size $r$. From Theorem \ref{res:exactS+} we deduce that for $\ell=0$ and every $\alpha\in{\cal A}$ we have
$$F_{S^+,\alpha}{(0)}=1 - L_{\alpha}(\infty).$$
For $\ell=1$ and every $\alpha \in \mathcal{A}$ we get
$$F_{S^+,\alpha}{(1)}=1 - L_{\alpha}(\infty) + \sum_{\beta \in \mathcal{A}} L^{(1)}_{\alpha \beta} \ F_{S^+,\beta}{(0)}.$$
With ${\bf L}(\infty)=(L_{\alpha}(\infty))_{\alpha \in \mathcal{A}}$, seen as a column vector, we can write
\begin{align*}
{\bf F}_{S^+,\cdot}{(1)}&=1 -{\bf L}(\infty) + {\bf L^{(1)}F}_{S^+,\cdot}{(0)},\\
{\bf F}_{S^+,\cdot}{(\ell)}&=1 - {\bf L}(\infty)+\sum_{k=1}^{\ell} {\bf L^{(k)} F}_{S^+,\cdot}{(\ell-k)}, \ \forall\ell\geq 1.
\end{align*}
See Subsection \ref{Subsec:L} for how to compute ${\bf L^{(k)}}$ for $k \geq 1$ and ${\bf L}(\infty)$.
\subsection{Application in a simple case}\label{subsec:simplecase}
Let us consider the simple case where the possible score values are $-1,0,1$, corres\-ponding to the case $u=v=1$.
We will use the results in the previous subsections (see Equations (\ref{Eq:RecQellExple}, \ref{Eq:LCassimple}, \ref{Eq:LCassimple2})) to derive the distribution of the maximal non-negative partial sum $S^+$. This distribution can be determined using the following matrix equalities:
\begin{equation}\label{eq:Linfini}
{\bf L}(\infty)=\left ( \sum_{\beta} L^{(1)}_{\alpha\beta}\right )_{\alpha}={\bf L^{(1)}}\cdot{1\hspace{-0.2ex}\rule{0.12ex}{1.61ex}\hspace{0.5ex}}_r,
\end{equation}
with ${\bf L^{(1)}}$ given in Equation (\ref{Eq:RecurrenceLell}) and
\begin{eqnarray}\label{Eq:LoiExacteS+CasSimple}
{\bf F}_{S^+,\cdot}{(0)}&=&1-{\bf L}(\infty),\\
{\bf F}_{S^+,\cdot}{(\ell)}&=& 1-{\bf L}(\infty)+{\bf L^{(1)} F}_{S^+,\cdot}{(\ell-1)}.
\end{eqnarray}
This allows to further derive the approximate distributions of $Q_1$ and $M_n$ given in Theorems \ref{res:Q1} and \ref{res:Mn}.
We present hereafter a numerical application for the local score of a DNA sequence. We suppose that we have a Markovian sequence whose possible letters are $\{A,C,G,T\}$ and whose transition probability matrix is given by
$${\bf P}=\left (
\begin{array}{cccc}
1/2&1/6&1/6&1/6\\
1/4&1/4&1/4&1/4\\
1/6&1/6&1/6&1/2\\
1/6&1/6&1/2&1/6\\
\end{array}
\right)\ .$$
We choose the respective scores $-1,-1,0,1$ for the letters $A, C,G,T$ for which Hypothesis (\ref{Hyp:ScoreMoyNeg}) and
(\ref{Hyp:ProbaScorePos}) are verified.
We use the successive iteration methodology described in Equation (5.12) of \cite{KDe92} in order to compute $\bf{L^{(1)}}$ and $\bf{Q^{(-1)}}$, solutions of Equations (\ref{Eq:RecQellExple}) and (\ref{Eq:LCassimple}), from which we derive the formulas proposed in our Theorems \ref{res:exactS+}, \ref{res:Q1} and \ref{res:Mn} for the approximate distributions of $S^+$, $Q_1$ and $M_n$ respectively. We also compute the different approximations proposed in Karlin and Dembo \cite{KDe92}.
We then compare these results with the corresponding empirical distributions computed using a Monte Carlo approach based on $10^5$ simulations. We can see in Figure \ref{Fig:Splus}, left panel, that for $n=300$ the empirical \textit{cdf} of $S^+$ and the one obtained using Theorem \ref{res:exactS+} match perfectly. We can also visualize the fact that Theorem \ref{res:exactS+} improves the approximation of Karlin and Dembo in Lemma 4.3 of \cite{KDe92} for the distribution of $S^+$. The right panel of Figure \ref{Fig:Splus} allows to compare, for different values of the sequence length $n$, the empirical \textit{cdf} of $S^+$ and the exact \textit{cdf} given in Theorem \ref{res:exactS+}: we can see that our formula performs very satisfactory even for sequence length $n=100$.
In this simple example the approximation of the distribution of $Q_1$ given in Theorem \ref{res:Q1} and the one given in Lemma 4.4 of \cite{KDe92} give quite similar numerical values.
In Figure \ref{Fig:Mn} we compare three approximations for the \textit{cdf} of $M_n$: the Karlin and Dembo's approximation given in Equation (1.27) of \cite{KDe92} (see also Equation (\ref{Res:AppxMnKDe})), our approximation proposed in Theorem \ref{res:Mn}, and a Monte Carlo approximation. For the simple scoring scheme of this application, the parameter $K^*$ of the Karlin and Dembo's approximation for $M_n$ is given by Equation (5.6) of \cite{KDe92}
$$K^*=(e^{-\theta^*}-e^{-2\theta^*})\cdot\mathbb E[-f(A)]\cdot\sum_{\gamma}z_{\gamma}u_{\gamma}(\theta^*)\cdot\sum_{\gamma}w_{\gamma}/u_{\gamma}(\theta ^*).$$
More precisely, in the left panel we plot the probability $p(n,x):=\mathbb P\left(M_n\leq \frac{\log(n)}{\theta^*}+x\right)$ as a function of $n$, for a fixed value $x=-8$. This illustrates the asymptotic behavior of this probability with growing $n$. We can also observe the fact that Karlin and Dembo's approximation does not depend on $n$.
In Figure \ref{Fig:Mn}, right panel, we compare the approximation of Karlin and Dembo \cite{KDe92} for the same probability $p(n,x)$ with our approximation, for varying $x$ and fixed $n=100$.
We observe that the improvement brought by our approximation is more significant for negative values of $x$. For fixed $n$ and extreme deviations (large $x$) the two approximations are quite similar and accurate.
\begin{flushleft}
\begin{figure
\centerline{
\begin{tabular}{cc}
\includegraphics[width=0.5\textwidth,trim=0 0 0 50,clip]{CdfS+compareEmpKDGMAn300remake.pdf}&
\includegraphics[width=0.5\textwidth,trim=0 0 0 50,clip]{Cdf_S+_compare_EmpGM_Adifferent_n.pdf}
\end{tabular}
}
\caption{Cumulative distribution function of $S^+$ for the simple scoring scheme $(-1,0,+1)$ and $A_0=$``$A$''. Left panel: Comparison between the approximation of Karlin and Dembo proposed in \cite{KDe92}, a Monte Carlo estimation with sequences of length $n=300$, and our exact formula proposed in Theorem \ref{res:exactS+}. Right panel: Comparison, for different values of $n$, of the Monte Carlo empirical cumulative distribution function and the exact one given in Theorem \ref{res:exactS+}.}
\label{Fig:Splus}
\end{figure}
\end{flushleft}
\begin{flushleft}
\begin{figure
\centerline{
\begin{tabular}{cc}
\includegraphics[width=0.5\textwidth,trim=0 0 0 50,clip]{CDFMnx-8nvariantRemake.pdf}&
\includegraphics[width=0.5\textwidth,trim=0 0 0 50,clip]{CDFMnn100xvariantRemake.pdf}
\end{tabular}
}
\caption{Comparison of the different approximations for $p(n,x)=\mathbb P\left(M_n\leq \frac{\log(n)}{\theta^*}+x\right)$ with the simple scoring scheme $(-1,0,+1)$: Karlin and Dembo's result \cite{KDe92} (see Equation (\ref{Res:AppxMnKDe})), our approximation proposed in Theorem \ref{res:Mn} and Monte Carlo estimation. Left panel: $p(n,x)$ as a function of $n$, for fixed $x=-8$. Right panel: $p(n,x)$ as a function of $x$, for fixed $n=100$.}
\label{Fig:Mn}
\end{figure}
\end{flushleft}
\vspace{-2.2cm}
|
{
"timestamp": "2018-03-08T02:11:21",
"yymm": "1803",
"arxiv_id": "1803.02769",
"language": "en",
"url": "https://arxiv.org/abs/1803.02769"
}
|
\section{Introduction}
Weyl semimetals (WSMs) represent materials allowing for a solid-state realization of three-dimensional massless fermions. After discovery of a nonvanishing neutrino mass, the Weyl fermions in condensed matter systems are unique particles with a massless linear energy dispersion and a definite chirality. This determines remarkable properties of WSMs, for a recent review see Ref.~\cite{RMP_2018}. A unique feature is the chiral anomaly consisting in a nonconservation of the particle number of a given chirality and leading to a negative magnetoresistance. Another important fact is that each Weyl node serves as a magnetic monopole in the reciprocal space with a nonzero Berry curvature. The important consequence is an existence of topologically induced Fermi arcs at a surface of a Weyl semimetal. Due to topological reasons, each Weyl node has its counterpart of opposite chirality. In contrast to neutrinos, the dispersion cones can be tilted or even overtilted which is realized, respectively, in type~I and type~II WSMs \cite{RMP_2018}.
The discovery of WSMs has been followed by theoretical and experimental studies of their transport and optical properties, both linear and nonlinear. In particular, it has been shown that the interaction of circularly-polarized light with chiral fermions is governed by the Berry curvature of the Weyl node. This allows a new look at the Circular PhotoGalvanic Effect (CPGE), an appearance of a helicity-dependent electric photocurrent upon illumination of the sample by circularly polarized light~\cite{EL_book}. The CPGE is allowed by the symmetry of gyrotropic (or optically active) media, and many WSMs belong to the family of gyrotropic crystals. It has been demonstrated that, in each Weyl node, the CPGE has universal features independent of details of the material. In particular, the generation rate of the CPGE current density is determined, except for the
intensity of the exciting light, by fundamental constants~\cite{Moore}. However, the corresponding CPGE current has the opposite polarity in the Weyl node of opposite chirality. Since the Weyl nodes exist in pairs of opposite chirality, the universality of CPGE is preserved only in those semimetals where such Weyl nodes have different energies. This is possible in the absence of any improper symmetry operations because such an operation transforms a given Weyl node to another one of the opposite chirality. In contrast, the point-group symmetry of available WSMs contains improper operations, and the universal contributions of individual nodes to the CPGE current mutually compensate each other. The net photocurrent can be generated in real Weyl semimetals owing to corrections to the Weyl Hamiltonian and, hence, to the carrier energy dispersion. It has been shown~\cite{Patrick} that the net CPGE current induced within two Weyl nodes of opposite chirality becomes nonvanishing in tilted WSMs where it, however, loses its universality and depends on the tilt. As we pointed out~\cite{JETP_Lett_2017}, the net CPGE current is sensitive to the sign of parameters describing corrections to the Weyl Hamiltonian rather than to chirality of the Weyl fermions. This opens a possibility to study the real Hamiltonians in WSMs with the help of CPGE. The theoretical study is also stimulated by the recent observations of the CPGE in the TaAs semimetal~\cite{Gedik,China}.
In the present paper we consider WSMs of the crystal classes containing improper symmetry operations and derive minimal models which allow for the CPGE. For this purpose we include into the electron effective Hamiltonian linear and nonlinear, spin-dependent and spin-independent terms leading to the net CPGE current under direct optical transitions between the valence and conduction band states as well as under intraband indirect transitions. It is demonstrated that the intraband CPGE current induced in an individual Weyl node has a universal value independent of the momentum relaxation details.
An additional effect specific to gyrotropic media is a Magneto-gyrotropic PhotoGalvanic Effect (MPGE) which is a generation of photocurrent odd in the magnetic field and independent of the light polarization. It has been investigated in bulk semiconductors~\cite{MPGE_bulk1,MPGE_bulk2} and quantum-well structures~\cite{MPGE_QWs}. Recently the MPGE has been theoretically studied in WSMs in weak magnetic fields and termed the helical magnetic effect~\cite{Kharzeev}. Here we consider gyrotropic WSMs in a quantizing magnetic field and investigate the MPGE current injected by direct optical transitions between magnetic subbands. In multi-valley semimetals of the C$_{2v}$ symmetry with Weyl nodes of opposite chiralities the net MPGE current appears with allowance for the
tilt.
The paper is organized as follows. In Sect.~\ref{interband}, we study the CPGE in semimetals of the C$_{2v}$ and C$_{4v}$ point-group symmetries at zero magnetic field. We take into account both the tilt and spin-dependent nonlinear terms in the effective Hamiltonian. The photocurrent generated under intraband indirect optical absorption is calculated in Section~\ref{intraband}. In Sect.~\ref{Sec_MPGE}, we consider the MPGE realized under the unpolarized photoexcitation in an external magnetic field. In Sect.~\ref{disc} we present and analyze a typical spectral dependence of the MPGE current and make some comments concerning non-stationary photocurrents and role of the electron-electron interaction. Section~\ref{concl} concludes the paper.
\section{Circular photocurrent at inter-band transitions}
\label{interband}
We start with the effective electron Hamiltonian
\begin{equation} \label{Hamilt3}
{\cal H} =\sigma_x d_x({\bm k}) + \sigma_y d_y({\bm k}) + \sigma_z d_z({\bm k}) + \sigma_0d_0({\bm k})\:,
\end{equation}
where $\sigma_{\alpha}$ ($\alpha = x, y, z$) are the Pauli spin matrices, $\sigma_0$ is the unit 2$\times$2 matrix, ${\bm k}$ is the electron wavevector referred to a particular Weyl node ${\bm k}_W$ and the functions $d_{\alpha}({\bm k}), d_0({\bm k})$ can be expanded in a Taylor series starting from the first-order terms. The energy dispersion consists of two branches $ E_{\pm, {\bm k}} = d_0({\bm k}) \pm d({\bm k})$, where $d = |{\bm d}|$.
The photocurrent generated under direct optical transitions in the vicinity of a Weyl node is given by
\begin{equation} \label{general}
{\bm j} = e \sum\limits_{\bm k} {\bm v}_{+-} ({\bm k}) \tau_p W_{+-}({\bm k})\:.
\end{equation}
Here $W_{+-}({\bm k})$ is the transition rate related with the optical matrix element $M_{+-}$ by the Fermi golden rule
\begin{equation} \label{W+-}
W_{+-} = \frac{2 \pi}{\hbar} \left\vert M_{+-} \right\vert^2 F({\bm k}) \delta \left[ 2d({\bm k}) - \hbar \omega \right]\:,
\end{equation}
$F({\bm k})$ is the difference $f_0(E_{-,{\bm k}}) - f_0(E_{+,{\bm k}})$ of the electron occupations of the initial and final states
with $f_0$ being the Fermi-Dirac distribution function, $\omega$ is the light wave frequency, $\tau_p$ is the electron and hole momentum relaxation times assumed to coincide, and $ {\bm v}_{+-} ({\bm k})$ is the difference of the group velocities
\[
{\bm v}_{+-} ({\bm k})= \frac{1}{\hbar} \frac{\partial }{\partial {\bm k}} \left( E_{+, {\bm k}} - E_{-, {\bm k}} \right)=
\frac{2}{\hbar} \frac{\partial d({\bm k})}{\partial {\bm k}}\:.
\]
A special property of WSMs is the following relation between the contribution to $\left\vert M_{+-} \right\vert^2$ dependent on the light circular polarization and the Berry curvature ${\bm \Omega_{\bm k}}$ \cite{JETP_Lett_2017}
\begin{equation} \label{MOmega}
\left\vert M_{+-} \right\vert^2_{\rm circ} = |{\bm E}|^2 \frac{2 e^2 d^2}{(\hbar \omega)^2} {\bm \varkappa}\cdot{\bm \Omega_{\bm k}} \:,
\end{equation}
where ${\bm E}$ is the amplitude of the light's electric field and ${\bm \varkappa} = i({\bm E} \times {\bm E}^*)/|{\bm E}|^2$ is the photon helicity. For the transverse electromagnetic wave, ${\bm \varkappa}$ is expressed via the light wave vector ${\bm q}$ and the degree of circular polarization $P_\text{circ}$ as $\bm \varkappa =P_\text{circ} {\bm q}/q$. The Berry curvature is given in terms of the vector ${\bm d}(\bm k)$ as follows
\begin{equation}
\label{Berrygeneral}
\Omega_{{\bm k},i} = \frac{\bm d }{2d^3} \cdot
\left( \frac{\partial \bm d }{\partial k_{i+1}} \times \frac{\partial \bm d}{\partial k_{i+2}} \right) \:,
\end{equation}
where the cyclic permutation of indices is assumed.
\subsection{Linear spin-orbit coupling}
\label{Linear_SOC}
If the spin-dependent part of the Hamiltonian (\ref{Hamilt3}) is linear in ${\bm k}$ and the tilt term vanishes then the circular photocurrent takes the universal form
\begin{equation} \label{Gamma00}
{\bm j} = {\cal C} \Gamma_0\tau_p i \left( {\bm E} \times {\bm E}^* \right)\:.
\end{equation}
Here $\Gamma_0 = \pi e^3 /3 h^2$, $e$ is the electron charge, $h$ is the
Planck constant, and ${\cal C} = \pm 1$ is the chirality (or topological charge) of the Weyl node
\begin{equation}
{\cal C} = {1\over 2\pi}\int\limits_\Sigma \bm \Omega_{\bm k} \cdot d\bm S_{\bm k},
\end{equation}
where the integration performed over a closed surface $\Sigma$ with the Weyl point inside.
The presence of crystal symmetry operations transforming the Weyl node ${\bm k}_W$ to another point of the Brillouin zone determines a multi-valley character of the WSM electron band structure. If the gyrotropic crystal class does not contain refection planes then the circular photocurrent related to the node ${\bm k}_W$ is given by Eq.~(\ref{Gamma00}) times the number of valleys and retains the universality. However, if the gyrotropic class contains a reflection plane $\sigma$ (or a rotoinversion axis $S_n$) then the nodes ${\bm k}_W$ and $\sigma{\bm k}_W$ (or $S_n\bm k_W$) are characterized by opposite chiralities, and their contributions to ${\bm j}$ can compensate each other, partly or completely. Let us analyze how the presence of
reflection planes in the point-symmetry group ${\cal F}$ of a gyrotropic crystal affects the CPGE
if the Hamiltonian has the form
\begin{equation} \label{lineartilt}
{\cal H} = {\cal C}\hbar v_0 \bm \sigma \cdot \bm k + \sigma_0 d_0(\bm k)\:,
\end{equation}
where $v_0>0$ is the electron effective velocity, and the tilt $d_0(\bm k)$ is an analytical function of ${\bm k}$ vanishing at the point ${\bm k}_W$. For this purpose it is useful to consider the one-dimensional representation ${\cal D}_v$ of the group ${\cal F}$ defined as follows: ${\cal D}_v(g) = {\cal C}_g/{\cal C}$, where ${\cal C}$ and ${\cal C}_g$ are the chiralities of the fixed node ${\bm k}_W$ and the node $g {\bm k}_W$. For the Hamiltonian (\ref{lineartilt}) one has ${\bm v}_{+-} ({\bm k}) = 2v_0{\bm k}/k$ and ${\bm \varkappa}\cdot{\bm \Omega_{\bm k}} \propto {\cal C} {\bm \varkappa}\cdot{\bm k}/2k^3$. Therefore the photocurrent $j_{\alpha}^{\beta} \propto {\varkappa}_{\beta}$ summed up over all the valleys $g {\bm k}_W$ can be presented as
\[
j_{\alpha}^{\beta} = {\cal C}\sum\limits_{\bm k} Q(k) k_{\alpha} k_{\beta} \sum_{g \in {\cal F}} {\cal D}_v(g) F\left( g^{-1} {\bm k}\right)\:,
\]
where $Q(k)$ is a function of the modulus $k$. By the variable transformation ${\bm k} \to g {\bm k}$ the sum is reduced to
\begin{equation} \label{kgk}
j_{\alpha}^{\beta} = {\cal C} \sum\limits_{\bm k} Q(k) F({\bm k}) \sum_{g \in {\cal F}} {\cal D}_v(g) \: g \left( k_{\alpha} k_{\beta} \right) \:.
\end{equation}
Thus, denoting by ${\cal D}^{(1)}$ the three-dimensional representation according to which the wavevector components $k_{\alpha}$ transform, we can formulate a theorem: among $j_{\alpha}^{\beta}$ there are nonzero values only if the symmetrized direct product $\left[{\cal D}^{(1)} \times {\cal D}^{(1)}\right]$ contains the representation ${\cal D}_v$.
For the point group C$_{2v}$ with the reflection planes $\sigma_v(xz)$ and $\sigma_v(yz)$, the representation ${\cal D}_v$ coincides with the representation $A_2$. Among linear combinations of the functions $k_{\alpha} k_{\beta}$ only the product $k_x k_y$ transforms according to $A_2$. It follows then that the sum in Eq.~(\ref{kgk}) is nonzero only for the pairs $\alpha = x, \beta = y$ and $\alpha = y, \beta = x$. With allowance for the fraction of $F({\bm k})$ of the $A_2$ symmetry the photocurrent off-diagonal components $j_x^y = j_y^x$ become nonzero. However, the antisymmetric contribution $(j_x^y - j_y^x)/2$
to the photocurrent also allowed by the C$_{2v}$ point group is absent in the model (\ref{lineartilt}). Note that in the axes $x',y'$ rotated around $z$ by 45$^{\circ}$ with respect to the $x,y$ axes the model (\ref{lineartilt}) allows the diagonal components $j_{x'}^{x'} = - j_{y'}^{y'}$.
For the point group C$_{4v}$, the representation ${\cal D}_v$ coincides with the representation denoted also by $A_2$ (or $\Gamma_2$ in the other notation). In this group, however, the symmetrized product $\left[{\cal D}^{(1)} \times {\cal D}^{(1)}\right]$ is decomposed into irreducible representations $2 A_1 + B_1 + B_2 + E$ (or $2 \Gamma_1 + \Gamma_3 + \Gamma_4 + \Gamma_5$) and does not contain the representation $A_2$. All the components $j_{\alpha}^{\beta}$ vanish for the linear spin-orbit coupling and an arbitrary tilt function $d_0({\bm k})$. Thus, we eventually come to conclusion that the CPGE in the C$_{4v}$ symmetry crystals, as well as the difference between $j_x^y$ and $j_y^x$ (or the components $j_{x'}^{y'} = - j_{y'}^{x'}$) allowed by the C$_{2v}$ symmetry, can be obtained only by adding to the spin-dependent part of ${\cal H}$ corrections of the higher order in ${\bm k}$. In the next section we demonstrate how this is done for the C$_{4v}$ point-group symmetry.
\subsection{Account for spin-dependent nonlinear terms}
In order to calculate the CPGE in WSMs of the C$_{4v}$ symmetry we present the functions $d_{\alpha}({\bm k})$ in a more general form
$${\cal C}\hbar v_0 k_x + P_x({\bm k}), \quad {\cal C}\hbar v_0 k_y + P_y({\bm k}), \quad {\cal C}\hbar v_0 k_z\:,$$ respectively, where the expansion of $P_{\alpha}({\bm k})$ in powers of ${\bm k}$ starts from the second or third order terms. We focus on the calculation of the symmetry allowed component $j_x^y$ given by a sum of the individual contributions
\begin{equation} \label{C4v}
j_x^y(Wi) = \frac{2\pi e^3 \tau_p }{\hbar^2} |{\bm E}|^2 \sum\limits_{\bm k} \frac{\partial d}{\partial k_x} \Omega_{{\bm k},y} F({\bm k})\ \delta ( 2d - \hbar \omega )\:,
\end{equation}
where the index $i$ runs from 1 to the number $N$ of the equivalent nodes. The component $j_y^x(Wi)$ differs from Eq.~(\ref{C4v}) only in sign.
We assign number 1 to one of the Weyl nodes ${\bm k}_{W1}$ lying, e.g., in the region of the Brillouin zone with the positive components ${\bm k}_{W1,\alpha} > 0$. Bearing in mind 8 elements of spatial symmetry of the C$_{4v}$ group and the time-inversion symmetry, we have 16 equivalent nodes. To analyze the net electric photocurrent, it is sufficient to consider only two nodes, the fixed one, ${\bm k}_{W1}$, and the node ${\bm k}_{W2}$ obtained by the reflection in the diagonal plane $\sigma_d$ which transforms $k_x, k_y, k_z$ into $k_y, k_x, k_z$ and $\sigma_x, \sigma_y, \sigma_z$ into $- \sigma_y, - \sigma_x, - \sigma_z$. This restriction to the two selected nodes is applicable because the eight nodes obtained from the node W1 by the $C_{2v}$ group operations $e, \sigma_v(xz), \sigma_v(yz), C_2$ and the time-inversion $T$ make coinciding contributions to $j_x^y$, and the remaining eight nodes also contribute equally. For example, we demonstrate this for the nodes ${\bm k}_{W3} = \sigma_v(xz) {\bm k}_{W1}$ and ${\bm k}_{W4} = T {\bm k}_{W1} = -{\bm k}_{W1}$. In these valleys the components $d^{(Wi)}_{\alpha}(k_x,k_y,k_z)$ ($i=3, 4$) have the form
\begin{eqnarray}
&&\mbox{}\hspace{1.3 cm}d^{(W3)}_x=- {\cal C}\hbar v_0 k_x - P_x(k_x,-k_y,k_z)\:, \nonumber\\
&&d^{(W3)}_y=- {\cal C}\hbar v_0 k_y + P_y(k_x,-k_y,k_z), d^{(W3)}_z= - {\cal C}\hbar v_0 k_z, \nonumber
\end{eqnarray}
and $d^{(W4)}_{\alpha}(k_x,k_y,k_z)=- d^{(W1)}_{\alpha}(-k_x,-k_y,-k_z)$.
One can check that the substitution of $d^{(Wi)}_{\alpha}({\bm k})$ into the sum~(\ref{C4v}) followed by the variable transformation $k_x, k_y, k_z$ to $k_x, -k_y, k_z$ or $-k_x, -k_y, -k_z$ completely restores the modulus $d({\bm k})$, and the product of functions to be summed over ${\bm k}$.
We take account for the additional terms $P_{\alpha}({\bm k})$ in the first order of the perturbation theory. There are several possible contributions to the photocurrent (\ref{general}) related to ${\bm P}({\bm k})$. They come from the corrections to (i)~the Berry curvature~(\ref{Berrygeneral}), (ii) the velocity ${\bm v}_{+-} ({\bm k})$ in Eq.~(\ref{general}), and (iii) the modulus $d({\bm k})$ in Eq.~(\ref{W+-}) as well as to the energy dependence of $\tau_p$ and $F({\bm k})$. Below we analyze these contributions, one after another. To avoid cumbersome formulas we simplify the functions $P_x({\bm k}), P_y({\bm k})$ assuming that they depend on $k_x, k_y$ but are independent of $k_z$.
\subsubsection{Correction to the Berry curvature}
In the linear in $\bm P(\bm k)$ approximation one has for the Berry curvature in the valley W1
\[
\Omega^{(W1)}_{y}({\bm k}) = \frac{(\hbar v_0)^2}{2d^3} \left( {\cal C}\hbar v_0 k_y + P_y + k_y \frac{\partial P_x}{\partial k_x} - k_x \frac{\partial P_y}{\partial k_x}\right).
\]
In the W2 valley one has $d^{(W2)}_x = -{\cal C}\hbar v_0 k_x - P_y(k_y, k_x)$, $d^{(W2)}_y = -{\cal C}\hbar v_0 k_y - P_x(k_y, k_x)$, $d^{(W2)}_z = - {\cal C}\hbar v_0 k_z$ and $d_0^{(W2)} = d_0(k_y,k_x,k_z)$. It follows from Eq.~(\ref{Berrygeneral}) that
\begin{eqnarray} \label{Omegay2}
&&\Omega^{(W2)}_{y}({\bm k}) = \frac{(\hbar v_0)^2}{2d^3} \left[ - {\cal C}\hbar v_0 k_y - P_x(k_y,k_x) \right.\\ &&\hspace{1 cm}\left. - k_y \frac{\partial P_y(k_y,k_x)}{\partial k_x} + k_x \frac{\partial P_x(k_y,k_x)}{\partial k_x}\right]. \nonumber
\end{eqnarray}
Substituting $\partial d/ \partial k_x = \hbar v_0 k_x /k$, $\Omega^{(W1)}_{y}({\bm k})$ or
$\Omega^{(W2)}_{y}({\bm k})$ into Eq.~(\ref{C4v}) and changing variables $k_x,k_y$ to $k_y,k_x$ in the sum for the W2 node contribution we obtain for the $N$-valley WSM
\begin{equation} \label{currentCURV}
j_x^y = \frac{4N \pi e^3 v_0^3 \tau_p }{\hbar^2 \omega^3} \left\vert {\bm E}\right\vert^2
\sum\limits_{\bm k} S_{\Omega} F({\bm k})\ \delta(2 \hbar v_0 k - \hbar \omega)\:,
\end{equation}
where
\begin{eqnarray}
S_{\Omega} &=& \frac{k_x}{k} \left( P_y({\bm k}) + k_y \frac{\partial P_x({\bm k})}{\partial k_x} - k_x \frac{\partial P_y({\bm k})}{\partial k_x} \right) \nonumber\\
&+& \frac{k_y}{k} \left( - P_x({\bm k}) - k_x \frac{\partial P_y({\bm k})}{\partial k_y} + k_y \frac{\partial P_x({\bm k})}{\partial k_y} \right)\:. \nonumber
\end{eqnarray}
\subsubsection{Correction to the velocity}
For the node W1, the corrections to the energy and the group velocity linear in ${\bm P}({\bm k})$ can be written as
\begin{equation} \label{Eapprox}
E_{\pm,{\bm k}} = d_0({\bm k}) \pm d({\bm k})\:,\: d({\bm k}) \approx \hbar v_0 k + {\cal C} \frac{ {\bm k} \cdot {\bm P}({\bm k})}{k}
\end{equation}
and
\begin{equation} \label{v+-1}
\delta v_{+-,x}({\bm k}) = \frac{2 {\cal C}}{\hbar k} \left( P_x - \frac{k_x}{k} \frac{ {\bm k}\cdot {\bm P}}{k} +{\bm k} \cdot \frac{\partial {\bm P}}{\partial k_x}\right)\:.
\end{equation}
The similar quantities for the node W2 are given by
\begin{eqnarray}
&&d^{(W2)}({\bm k}) \approx \hbar v_0 k + {\cal C} \frac{ \tilde{\bm k} \cdot \tilde{\bm P}}{k}\:, \nonumber\\
&&\delta v^{(W2)}_{+-,x} = \frac{2 {\cal C}}{\hbar k} \left( \tilde{P}_y - \frac{k_x}{k} \frac{ \tilde{\bm k}\cdot \tilde{\bm P}}{k} +\tilde{\bm k} \cdot \frac{\partial \tilde{\bm P}}{\partial k_x}\right), \label{v+-2}
\end{eqnarray}
where $\tilde{\bm k}= \sigma_d{\bm k}= (k_y,k_x)$, $\tilde {\bm P} = {\bm P}(\sigma_d{\bm k})= {\bm P}(k_y,k_x)$. Substituting Eq.~(\ref{v+-1}) or (\ref{v+-2}) into Eq.~(\ref{C4v}) we obtain, similarly to the previous subsubsection, the equation which differs from Eq.~(\ref{currentCURV}) by the replacement $S_{\Omega} \to S_v$ where
\begin{equation} \label{currentVELO}
S_v = \frac{k_y}{k} \left( P_x +{\bm k} \cdot \frac{\partial {\bm P}}{\partial k_x} \right) - \frac{k_x}{k} \left( P_y + {\bm k} \cdot \frac{\partial {\bm P}}{\partial k_y} \right). \nonumber
\end{equation}
A sum of the contributions due to the corrections to the Berry curvature and velocity reduces to
\begin{eqnarray}
\label{currentTOTAL}
&&j_x^y = \frac{4N \pi e^3 v_0^3 \tau_p }{\hbar^2 \omega^3} \left\vert {\bm E}\right\vert^2
\sum\limits_{\bm k} S \: F({\bm k}) \: \delta(2 \hbar v_0 k - \hbar \omega)\:,\\
&&S = 2 \frac{k_x k_y}{k} \left( \frac{\partial P_x}{\partial k_x} - \frac{\partial P_y}{\partial k_y} \right) - \frac{k_x^2 - k_y^2}{k} \left( \frac{\partial P_x}{\partial k_y} + \frac{\partial P_y}{\partial k_x}\right) \:.\nonumber
\end{eqnarray}
\subsubsection{Correction to the energy}
In this case the product $v_{+-}^x \Omega_y$ is proportional to $k_x k_y$ because of the linear dependence ${\bm d}({\bm k}) = {\cal C}\hbar v_0 {\bm k}$. This allows one to write the corresponding contribution to (\ref{C4v}) in the following general form
\[
{\cal C} \sum\limits_{\bm k} \frac{k_x k_y}{k} R[d({\bm k})] F({\bm k})\:,
\]
where $R$ is a function of the modulus $|{\bm d}({\bm k})|$. The same sum related to the node W2 differs by the sign of the Berry curvature ${\cal C}$ and the variable change in the product
\[
R[d({\bm k})] F({\bm k}) \to R[d(\sigma_d{\bm k})] F(\sigma_d{\bm k})\:.
\]
Changing the variable ${\bm k} \to \sigma_d {\bm k}$ in this sum and taking into account that the bilinear function $k_x k_y$ is invariant under the operation $\sigma_d$ we immediately find that the contributions from the nodes W1 and W2 differ in sign and cancel each other.
It follows then that Eq.~(\ref{currentTOTAL}) gives the total photocurrent linear in ${\bm P}$ and this equation can be used as the working formula for calculations in particular cases of the wavevector-dependence of ${\bm P}$.
\subsection{Particular cases}
Taking $P_x = D_x k_x^n k_y^m$ and $P_y = D_y k_x^{n'} k_y^{m'}$ we obtain
\begin{eqnarray} \label{kS}
k S &=& D_x k_x^n k_y^{m-1}\ \left[ (2 n +m) k_y^2 - m k_x^2 \right] \\&+& D_y k_x^{n'-1} k_{y'}^{m'} \left[ - (2 m' +n') k_x^2 + n' k_y^2 \right]\:. \nonumber
\end{eqnarray}
\subsubsection{Quadratic nonlinearity}
For a combination of linear-${\bm k}$ and double-Weyl Hamiltonians with $P_x = D_{x1} \left( k_x^2 - k_y^2\right) + 2D_{x2} k_x k_y$ and $P_y = D_{y1} \left( k_x^2 - k_y^2\right) + 2D_{y2} k_x k_y$ one obtains from Eq.~(\ref{kS})
\begin{eqnarray} \label{Sfinal}
S &=& 2 k^2 \sin^3{\theta} \left[\left( D_{x1} - D_{y2} \right)\sin{3 \varphi} \right.\\ &&\mbox{} \hspace{5 mm}\left.- \left( D_{x2} + D_{y1} \right) \cos{3 \varphi} \right], \nonumber
\end{eqnarray}
where $\theta, \varphi$ are the polar angles of the wavevector ${\bm k}$.
Therefore, the photocurrent becomes nonzero due to the angular harmonics of the third order contributing to the difference $F({\bm k})$ of the occupations of the initial and final states.
Then at $D_{x1} - D_{y2}=0$ we obtain from Eq.~\eqref{currentTOTAL}
\begin{equation}
j_x^y = -\left( D_{x2} + D_{y1} \right)\frac{Ne^3 \tau_p \omega}{8\pi\hbar^3 v_0^2} \left\vert {\bm E}\right\vert^2 \left< {k_x(k_x^2-3k_y^2)\over k^3} F\left(\bm k\right)\right>,
\end{equation}
where the angular brackets mean averaging over directions of the vector $\bm k$ at a fixed absolute value ${k=\omega/(2v_0)}$.
To move further we consider the particular case of zero temperature and linear-${\bm k}$ dependence of the tilt ${d_0({\bm k}) = {\bm a} \cdot {\bm k} = a_x k_x + a_y k_y }$. Then for the negative chemical potential $\mu = - |\mu|$ the function $F({\bm k})$ reduces to
\begin{equation} \label{omegarange}
\Theta\left(- |\mu| + \hbar \omega/2 - {\bm a} \cdot {\bm k}\right) - \Theta\left(- |\mu| - \hbar \omega/2 -{\bm a} \cdot {\bm k}\right)\:,
\end{equation}
where $\Theta(x)$ is the Heaviside step function.
For definiteness we set $a_y =0$ and $a_x > 0$, the general case of $a_x, a_y \neq 0$ is treated analogously.
We introduce the coordinate system $(x'y'z')$ which is obtained from $(xyz)$ by a rotation around the $y$ axis by $90^\circ$: $x'=-z$, $y'=y$, $z'=x$.
Then the photocurrent is given by
\begin{align}
\label{j_int_t}
j_x^y = & -\left( D_{x2} + D_{y1} \right)\frac{Ne^3 \tau_p \omega}{8\pi\hbar^3 v_0^2} \left\vert {\bm E}\right\vert^2 \\
& \times \left< {k_{z'}(k_{z'}^2-3k_{y'}^2)\over k^3} F\left({k_{z'}\over k}\right)\right>.
\nonumber
\end{align}
The average here can be presented as follows
\begin{equation}
\int\limits_{-1}^1 dt
{5t^3-3t\over 2} [\Theta\left(C- t \right) - \Theta\left(C'-t \right)],
\end{equation}
where
\begin{equation}
C={\hbar \omega - 2 |\mu| \over \hbar \omega b}, \qquad
C'={\hbar \omega + 2 |\mu| \over \hbar \omega b},
\end{equation}
and we use the dimensionless tilt parameter
\begin{equation}
\label{b_tilt_parameter}
b = {a_x \over \hbar v_0}.
\end{equation}
In type-I WSMs where $b<1$, $C'>1$ for any $\omega$ and $\mu$, and the photocurrent is nonzero at $|C|<1$ only.
This gives the lower and higher edges for the optical absorption \cite{TypeIboundaries}
\begin{equation} \label{range1}
\frac{1}{1 + b} < \frac{\hbar \omega}{2 |\mu|} < \frac{1}{1 - b}\:.
\end{equation}
Integration yields
for type-I WSM
\begin{multline}
\label{j_typeI}
j_x^y = -\left( D_{x2} + D_{y1} \right)\frac{Ne^3 \tau_p |\mu|}{64\pi\hbar^4v_0^2} \left\vert {\bm E}\right\vert^2 \\
\times {(1-C^2)(1 - 5C^2)\over 1-bC}\Theta(1-|C|).
\end{multline}
\begin{figure}[t]
\includegraphics[width=0.95\linewidth]{TypeI}\\
\includegraphics[width=0.95\linewidth]{TypeII}
\caption{Spectral dependence of the CPGE current calculated in type-I (a) and type-II (b) WSMs for two values of the tilt parameter $b$ defined by Eq.~\eqref{b_tilt_parameter}.
}
\label{fig:spectra}
\end{figure}
In type-II WSMs where $b> 1$, the photocurrent is nonzero at $C>-1$. This means that
the low-frequency range is cut by \cite{AbsCircRad}
\begin{equation} \label{range2}
\frac{1}{b + 1} < \frac{\hbar \omega}{2 |\mu|}.
\end{equation}
For $C'>1$ i.e. for ${2|\mu|/\hbar\omega>b-1}$ the photocurrent in type-II WSM is also given by Eq.~\eqref{j_typeI}. For $C'<1$, i.e. for $2|\mu|/\hbar\omega<b-1$ it is equal to
\begin{align}
&j_x^y =\left( D_{x2} + D_{y1} \right)\frac{Ne^3 \tau_p |\mu|}{8\pi\hbar^4v_0^2} \left\vert {\bm E}\right\vert^2 \\
&\times {1\over b^4} \left\{ 5\left[1+\left({2|\mu|\over\hbar\omega}\right)^2 \right] -3b^2 \right\} \Theta\left({\hbar\omega \over 2|\mu| } -{1\over b - 1}\right). \nonumber
\end{align}
Excitation spectra for type-I and type-II WSM are shown in Fig.~\ref{fig:spectra}. The spectra are identical in the low-frequency part. At high frequencies the photocurrent is zero for type-I WSM while it raises with frequency in type-II WSM.
\subsubsection{Cubic nonlinearity}
In general the cubic contribution to $P_x(k_x,k_y)$, $P_y(k_x,k_y)$ can be presented as
\begin{equation} \label{Pcubic}
P_x = \sum\limits_{n=0}^3 D_{xn} k_x^nk_y^{3 - n}\:,\quad P_y = \sum\limits_{n=0}^3 D_{yn} k_x^{3 - n}k_y^n\:.
\end{equation}
Substitution of these expressions into Eq.~(\ref{Sfinal}) and averaging over the solid angle $4\pi$, hereafter indicated by a bar, gives
\begin{equation} \label{Saverage}
\bar{S} = \frac{2 k^3}{15} \left[ 3\left( D_{x0} - D_{y0}\right) + D_{x2} - D_{y2} \right]\:.
\end{equation}
For illustration, in Ref.~\cite{JETP_Lett_2017}, we used the model with ${P_x = D k_y(k_x^2 + k_y^2)}$, $P_y = D k_x(k_x^2 + k_y^2)$. In the notation (\ref{Pcubic}) this corresponds to the choice $D_{x0} = D_{x2} = D_{y0} = D_{y2} = D$, $D_{x1} = D_{x3} =D_{y1} = D_{y3} =0$ in which case the average value $\bar{S}$ vanishes and the circular photocurrent becomes nonzero only with allowance for the tilt. However, if the combination of coefficients in square brackets of Eq.~(\ref{Saverage}) is nonzero then the photocurrent is induced even in the absence of tilt, particularly for $F({\bm k}) \equiv 1$ in Eq.~(\ref{currentTOTAL}).\\
\section{CPGE under indirect transitions}
\label{intraband}
Here we consider a WSM with a finite chemical potential $\mu>0$. At the photon energies $\hbar\omega<2\mu$, the light absorption is possible due to indirect transitions. We show that such indirect transitions are also accompanied by a photocurrent. For simplicity we assume a short-range scattering potential.
The photocurrent density is given by
\begin{equation}
j_z = e\sum_{\bm k \bm k'}W_{\bm k' \bm k} [v_z({\bm k'}) \tau_p(k')-v_z({\bm k}) \tau_p(k)].
\end{equation}
Here $W_{\bm k' \bm k}$ is the probability of the indirect optical absorption, and we introduced the momentum scattering time according to
\begin{equation}
{1\over \tau_p(k)} = {2\pi \over \hbar} \sum_{\bm k'} |U^{cc}_{\bm k' \bm k} |^2 (1-\cos{\Phi})\delta[\hbar v_0 (k - k')]
\end{equation}
with $\Phi$ being the angle between ${\bm k}'$ and ${\bm k}$. For the short-range scattering potential the scattering matrix element $U^{cc}_{\bm k' \bm k}$ is equal to $U_0 \left<c\bm k'|c\bm k \right>$ where the Fourier image $U_0$ of the potential is a constant. After the summation we obtain
\begin{equation}
{1\over \tau_p(k)}={{\cal N} |U_0|^2 k^2 \over 3\pi\hbar^2 v_0} ,
\end{equation}
where ${\cal N}$ is the concentration of the static scatterers.
\begin{figure*}[t]
\includegraphics[width=0.85\linewidth]{fig_Intraband}\caption{Quantum transitions resulting in the CPGE at ${\hbar\omega<2\mu}$. Dashed arrows indicate the direct optical transitions, and dotted arrows show the processes of impurity scattering.
The photon energy $\hbar\omega$ shown by the length of solid vertical arrows is smaller~(a) and larger~(b) than $\mu$.
Only transitions with the initial state in the conduction band (a) are allowed at $\hbar\omega <\mu$. At $\hbar\omega >\mu$ the transitions with the initial state in the valence band (b) also contribute to the CPGE.}
\label{fig:Intraband}
\end{figure*}
\subsection{$c \to c$ processes}
The probability of the intraband optical absorption is described by the Fermi golden rule
\begin{equation}
W_{\bm k' \bm k} = {2\pi \over \hbar} |M_{\bm k' \bm k}|^2 (f_k-f_{k'})\delta(\hbar v_0 k'- \hbar v_0 k-\hbar\omega),
\end{equation}
where $f_k=f_0(\hbar v_0 k)$.
The electron-photon interaction operator is taken in the form
\begin{equation}
\label{e-phot}
V = {\cal C}v_0 {ie\over \omega} \bm\sigma \cdot \bm E \text{e}^{-i\omega t} + {\rm h.c.}
\end{equation}
Then the compound matrix element of the indirect optical transition via the valence band, Fig.~\ref{fig:Intraband}(a), reads
\begin{equation} \label{compound}
M_{c\bm k' c\bm k} = {U^{cv}_{\bm k' \bm k}V^{vc}_{\bm k} \over E_v(k)-E_c(k)-\hbar\omega}
+{V^{cv}_{\bm k'}U^{vc}_{\bm k' \bm k} \over E_v(k')-E_c(k)},
\end{equation}
with the interband scattering matrix elements being
\begin{equation}
U^{cv}_{\bm k' \bm k} = U_0 \left<c\bm k'|v\bm k \right>, \qquad U^{vc}_{\bm k' \bm k} = U_0 \left<v\bm k'|c\bm k \right>,
\end{equation}
and the interband optical matrix elements, for the $\sigma^+$ and $\sigma^-$ circular polarizations, being
\begin{eqnarray}
&&V^{cv}_{\bm k}(\sigma^+) = - V^{vc,*}_{\bm k}(\sigma^-) = i{e |{\bm E}| v_0\over \omega} {{\cal C}\cos{\theta_{\bm k} + 1} \over \sqrt{2}}\text{e}^{i\varphi_{\bm k}},\nonumber\\
&&V^{vc}_{\bm k}(\sigma^+) = - V^{cv,*}_{\bm k}(\sigma^-) =i{e |{\bm E}| v_0\over \omega} {{\cal C}\cos{\theta_{\bm k} - 1} \over \sqrt{2}}\text{e}^{i\varphi_{\bm k}}.\nonumber
\end{eqnarray}
Note that the half difference of squared moduli of $V^{cv}_{\bm k}(\sigma^+)$ and $V^{cv}_{\bm k}(\sigma^-)$ yields Eq.~\eqref{MOmega}.
The energy conservation law and the electron-hole symmetry imply that
\begin{equation}
E_c(k') - E_c(k) = \hbar\omega, \quad E_v(k) = -E_c(k) = -\hbar v_0 k.
\end{equation}
Therefore the energy denominators in Eq.~(\ref{compound}) coincide, and this equation reduces to
\begin{equation}
M_{c\bm k' c\bm k} = -{U^{cv}_{\bm k' \bm k}V^{vc}_{\bm k} + V^{cv}_{\bm k'}U^{vc}_{\bm k' \bm k}\over 2\hbar v_0 k + \hbar\omega}.
\end{equation}
As a result, we obtain for the circular photocurrent density
\begin{eqnarray}
&&j_{z}^{c\to c}= {4e^3 v_0^3 \over 3\omega^2} P_\text{circ} {\cal C} |\bm E|^2 \\
&&\times \sum_{\bm k} {f_0(\hbar v_0 k)-f_0(\hbar v_0 k+\hbar\omega)\over (2\hbar v_0 k + \hbar\omega)^2} \left[1+\left(1+{\omega\over v_0 k}\right)^2 \right]\:. \nonumber
\end{eqnarray}
At low temperatures integration yields
\begin{eqnarray}
\label{j_c_c}
j_{z}^{c\to c} &=& {4 \Gamma_0\over \pi \omega} P_\text{circ} {\cal C} |\bm E|^2 \\&& \times \left\{
\begin{array}{l}
\frac{1}{1 - (\hbar\omega/2\mu)^2}, ~~ \mbox{if}~~ \hbar\omega < \mu
\\ {2\mu(\hbar \omega+\mu)\over \hbar\omega(2\mu+\hbar\omega)}, ~~\mbox{if}~~ \mu < \hbar\omega < 2\mu
\end{array}
\right. . \nonumber
\end{eqnarray}
\subsection{Photocurrent caused by $v\to c$ transitions}
At $\hbar\omega >\mu$ but still $\hbar\omega<2\mu$ the transitions from the valence-band states also contribute to both light absorption and the CPGE, Fig.~\ref{fig:Intraband}(b). The probability rate and matrix element of the indirect transitions $v\to c \to c$ and $v\to v \to c$ are given by
\begin{equation}
W_{c\bm k' v\bm k} = {2\pi \over \hbar} (f_k^v-f_{k'})\delta(\hbar v_0 k'+ \hbar v_0 k-\hbar\omega)
|M_{c\bm k' v\bm k}|^2,
\end{equation}
\begin{equation}
M_{c\bm k' v\bm k} = {U^{cc}_{\bm k' \bm k}V^{cv}_{\bm k} + V^{cv}_{\bm k'}U^{vv}_{\bm k' \bm k}\over 2\hbar v_0 k - \hbar\omega},
\end{equation}
where $f_k^v=f_0(-\hbar v_0 k)$.
As a result, we obtain for the photocurrent density
\begin{equation}
j_{z}^{v\to c} = e{\cal C}v_0\sum_{\bm k \bm k'} [\tau_1(k') \cos{\theta_{\bm k'}} + \tau_1(k) \cos{\theta_{\bm k}}] W_{c\bm k' v\bm k},
\end{equation}
where we took into account that the initial electron state has the velocity $-{\cal C}v_0$.
Summation over $\bm k'$ yields
\begin{align}
& j_{z}^{v\to c}=P_\text{circ}|\bm E|^2 \:{\cal C} {8\Gamma_0\over \pi\omega}
\\ & \times \int\limits_0^\infty dx {x^2+(1-x)^2\over (2x-1)^2} \bigl\{f_0(-x\hbar\omega)-f_0[(1-x)\hbar\omega]\bigr\}. \nonumber
\end{align}
At low temperatures we have
\begin{equation}
j_{z}^{v\to c} = P_\text{circ}|\bm E|^2 \:{\cal C} {4\Gamma_0\over \pi\omega}
{2\mu(\hbar \omega-\mu)\over \hbar\omega(2\mu-\hbar\omega)}.
\end{equation}
\subsection{Total photocurrent at intraband absorption}
Summing up the contributions from both the $c\to c$ and $v \to c$ transitions, we obtain for ${\hbar\omega > \mu}$
the dependence which is a continuation of the expression~\eqref{j_c_c} for $j_z^{c\to c}$ calculated at $\hbar\omega < \mu$.
Therefore, for the whole region $0 < \hbar\omega < 2\mu$ of intraband absorption, we have
\begin{equation}
j_z^\text{circ} = P_\text{circ}|\bm E|^2 \:{\cal C} {4\Gamma_0\over \pi \omega}{1\over 1 - (\hbar\omega/2\mu)^2}.
\end{equation}
We see that at small photon energies $\hbar\omega \ll \mu$ (but still $\omega\tau \gg 1$) the CPGE current in the given Weyl node has a universal form
\begin{equation}
j_z^\text{circ}(\hbar\omega \ll \mu) =
P_\text{circ}|\bm E|^2 \:{\cal C} {4\Gamma_0\over \pi \omega}
\end{equation}
determined by the fundamental constants and the frequency.
\section{Magnetogyrotropic photogalvanic effect}
\label{Sec_MPGE}
The MPGE electric current is induced under the unpolarized photoexcitation in the presence of a magnetic field and flows backwards with the field reversal. Here we consider strong magnetic fields where a current transverse to the magnetic field $\bm B$ is suppressed because of the quantized cyclotron motion. It follows then that, in WSMs of the $C_{2v}$ symmetry, the magnetic field can conveniently be applied along the $x'$ or $y'$ axis introduced in Sect.~\ref{Linear_SOC} and making the angle $45^\circ$ with the reflection planes $\sigma_v(xz), \sigma_v(yz)$. Phenomenologically the MPGE current density is described by~\cite{JETP_Lett_2017}
\begin{equation} \label{phenomquant}
j_{x'} = B_{x'} |\bm E|^2 \chi(B_{x'}^2), \quad
j_{y'} = -B_{y'} |\bm E|^2 \chi(B_{y'}^2),
\end{equation}
where $\chi$ is an even function of the magnetic field.
\subsection{Contribution of an individual Weyl node}
\label{MPGE_one_mode}
The magnetic field is included into the Weyl Hamiltonian by the Peierls substitution.
In accordance with Eq.~(\ref{phenomquant}) the field ${\bm B}$ is assumed to be directed along the $x'$ axis. The energy spectrum consists of one chiral subband
\begin{equation} \label{chiral}
E_0(p_{x'}) = -{\cal C}\text{sgn}(B_{x'}) \hbar v_0 k_{x'}
\end{equation}
and a series of the valence ($v$) and conduction ($c$) magnetic subbands enumerated by the positive integer ${n= 1,2\dots}$ and having the dispersion relations ${E_{n}^{(c,v)}(k_{x'}) = \pm E_n(k_{x'})}$, where
\begin{equation}
E_n(k_{x'}) = \hbar\sqrt{(v_0k_{x'})^2+n\omega_c^2}
\end{equation}
and
\begin{equation}
\omega_c = {v_0\sqrt{2}\over l_B}, \quad l_B=\sqrt{\hbar c\over |eB_{x'}|}.
\end{equation}
For $B_{x'}>0$, the wavefunctions are given by
\begin{equation} \label{wavefunc}
\Psi_0 = \left[
\begin{array}{c}
0\\
\Phi_0
\end{array}
\right], \qquad
\Psi_{n>0}^{(c,v)} =
\left[
\begin{array}{c}
a_{n,\pm}\Phi_{n-1}\\
\pm {\cal C} a_{n,\mp}\Phi_n
\end{array}
\right],
\end{equation}
while for the opposite direction, $B_{x'}<0$, they
are obtained from the above functions by the transformation
\begin{equation}
\Psi_{n}^{(c,v)}(B_{x'}<0) = -i{\cal C}\sigma_2 \Psi_{n}^{(v,c)}(B_{x'}>0).
\end{equation}
Here $\sigma_2$ is the second Pauli matrix,
\begin{equation}
\Phi_n = \text{e}^{i(k_{x'}x'+k_{y'}y')}\phi_n\left(z-{\hbar k_{y'} c\over eB_{x'}} \right),
\end{equation}
$\phi_n$ are the Landau-level oscillator functions, and the coefficients $a_{n,\pm}= \sqrt{(1\pm {\cal C} \hbar v_0 k_{x'}/ E_n)/2}$.
By using the wavefunctions (\ref{wavefunc}) and the perturbation (\ref{e-phot}) one can find the squared matrix element $| M_{c,n' \leftarrow v,n} |^2$ for the direct optical transitions of electrons between the $n$th magnetic subband in the valence band and the $n'$th conduction subband. For the linear polarization, $\bm E_\perp \perp \bm B$, the selection rules are $n' - n = \pm 1$. The direct calculation gives
\begin{align}
\label{M_sq_inter}
& \left| M_{c,n+1 \leftarrow v,n} (k_{x'})\right|^2 = \left({ev_0|\bm E_\perp|\over 2\omega}\right)^2 \\
& \times \left[ 1+{(\hbar v_0 k_{x'})^2\over E_nE_{n+1}} + {\cal C}\hbar v_0\text{sgn}(B_{x'})k_{x'} \left({1\over E_n} + {1\over E_{n+1} } \right)\right],\nonumber
\end{align}
\begin{equation}
\label{eh_time_inv}
\left| M_{c,n \leftarrow v,n+1}(k_{x'}) \right|^2 = \left| M_{c,n+1 \leftarrow v,n} (-k_{x'})\right|^2,
\end{equation}
for $n, n' \neq 0$, and
\begin{eqnarray} \label{10-01}
&&\left| M_{c1 \leftarrow 0}(k_{x'}) \right|^2 = \left| M_{0 \leftarrow v1}(-k_{x'}) \right|^2 \\
&&\hspace{1 cm}= {(ev_0|\bm E_\perp|)^2\over 2\omega^2}
\left[ 1+ { {\cal C}\hbar v_0\text{sgn}(B_{x'})k_{x'} \over E_1} \right]\nonumber
\end{eqnarray}
for $n= 0$ either $n'=0$.
Here the even in $k_{x'}$ terms are responsible for light absorption~\cite{magn_opt_cond}.
Equations (\ref{M_sq_inter}) and (\ref{eh_time_inv}) allow one to explain the origin of a photocurrent under the $v,n \rightarrow c, n+1$ transitions with $n> 0$.
Due to the terms odd both in $k_{x'}$ and $B_{x'}$, the probability rates are asymmetric with a predominance of states with ${\cal C}{\text{sgn}(B_{x'})k_{x'}>0}$ for the transitions {$v,n \rightarrow c, n+1$} and ${\cal C}{\text{sgn}(B_{x'})k_{x'}<0}$ for the transitions {$v, n+ 1 \to c,n$}. In these two kinds of transitions the electron energies of the initial (or final) states are different, Fig.~\ref{fig:MPGE_1}, and, therefore, their equilibrium occupation can be also different. This gives rise to the MPGE current in the polarization $\bm E \perp \bm B$.
For the light polarized along the magnetic field, the selection rules read $n'=n$, the squared matrix elements are even in $k_{x'}$ and the electron photoexcitation is not accompanied by the current generation.
\begin{figure}[t]
\includegraphics[width=0.8\linewidth]{MPGE_fig}
\caption{Scheme of direct optical transitions between the one-dimensional subbands in a quantized magnetic field in the case ${\cal C}B_{x'}>0$. The transitions $v,n \to c,n+1$ result in the MPGE current since the $v,n+1 \to c,n$ transitions are blocked (crosses) by the Pauli principle. For $n\neq0$ each transition occurs at two points $k_{x'}$ of opposite signs. The arrows illustrate the optical transitions, their starting points (open circles) are chosen at the side with the more probable transition rate.}
\label{fig:MPGE_1}
\end{figure}
It is instructive to divide the relevant light spectral area into three ranges as presented below.
\subsubsection{Range 1: $\omega/\omega_c>\sqrt{2}+1$}
The density of the interband photocurrent is given by
\begin{align}
\label{MPGE}
j_{x'}= {e \over \hbar l_B^2} \sum_{n,k_{x'}} & (v^{x'}_{n+1}\tau_{n+1} +v^{x'}_n \tau_n) \left| M_{c,n+1 \leftarrow v,n} \right|^2 \\
& \times F \: \delta \left[E_{n+1}(k_{x'}) + E_n(k_{x'}) -\hbar\omega\right]. \nonumber
\end{align}
Here $\tau_n$ is the momentum relaxation time in the $n$th magnetic subband, $v^{x'}_n$ is the velocity $\hbar v_0^2 k_{x'} / E_n$,
\begin{equation}
\label{F_MPGE}
F= (f^v_{n}-f^c_{n+1}) -(f^v_{n+1}-f^c_{n}),
\end{equation}
$f_n^{c,v}$ are the equilibrium occupations of the $n$th subbands in the conduction and valence bands. While deriving Eq.~(\ref{MPGE}) we took into account {that} the odd parts of the squared matrix elements (\ref{eh_time_inv}) are opposite in sign but equal in magnitude.
The interband transitions {$v,n \to c,n+1$}, Fig.~\ref{fig:MPGE_1},
contribute to the photocurrent with $n$ satisfying the cut-off frequency relation
\begin{equation}
\hbar\omega_c(\sqrt{n}+\sqrt{n+1}) \leq \hbar\omega, \hspace{1 mm}\mbox{or} \hspace{1 mm} n\leq \nu \equiv {(\omega^2-\omega_c^2)^2 \over 4\omega^2\omega_c^2}.
\end{equation}
For $n~\neq~0$ the direct transitions occur at the points $k_{x'} = \pm k_\omega^{(n)}$.
The quasi-momentum $\hbar k_\omega^{(n)}$ as well as the initial and final energies, $-E_n(\omega)$ and $E_{n+1}(\omega)$, are found from the energy conservation ${E_n (k_\omega^{(n)}) + E_{n+1}(k_\omega^{(n)}) = \hbar \omega}$ as follows
\begin{equation}
\label{En}
\hbar k_\omega^{(n)} =
{E_n (\omega)\over v_0}\sqrt{1-{n\over \nu}},
\: E_{n/n+1}(\omega) = \hbar {\omega^2\mp\omega_c^2\over 2\omega}.
\end{equation}
For $n=0$, i.e. for the process {$0 \to c1$}, the transition occurs in the point $k_{x'} = k_\omega^{(0)}$ if ${\cal C} B_{x'} > 0$ and the point $k_{x'} = - k_\omega^{(0)}$ if ${\cal C} B_{x'} < 0$. In the former case Eq.~(\ref{En}) is also valid if $E_n$ is replaced by $- E_0\left(k_\omega^{(0)}\right)$, where $E_0\left(k_\omega^{(0)}\right)$ is the initial energy in the chiral subband, Eq.~(\ref{chiral}). We draw attention to the fact that the energies of electrons involved in the transitions {$v,n \to c,n+1$} are independent of $n$ {and shifted} relative to those in the {$v,n+1 \rightarrow c,n$} transitions by $\hbar \omega^2_c/\omega$.
We start the analysis from zero temperature, the effect of temperature is considered in Sec.~\ref{T}.
In Range 1 the contributions of the transitions $v,n \to c, n+1$ and $v, n+1 \to c, n$ to the photocurrent do not compensate each other if the latter transition is suppressed by the Pauli principle, Fig.~\ref{fig:MPGE_1}. This takes place at values of the chemical potential $\mu>\hbar\omega_c$ where $f^c_{n+1}=0$ but $f^c_{n}=1$, i.e. at
\begin{equation}
\label{B_range}
E_n(\omega) < \mu < E_{n+1}(\omega), \hspace{1 mm}\mbox{or} \hspace{1 mm} |2\mu -\hbar\omega| < {\hbar\omega_c^2\over \omega},
\end{equation}
or, equivalently, in the frequency range ${\Omega^-(\mu)<\omega < \Omega^+(\mu)}$, where
\begin{equation}
\label{range_omega}
\hbar\Omega^\pm(\mu) = \mu + \sqrt{\mu^2\pm(\hbar\omega_c)^2}.
\end{equation}
In fact, the lower frequency edge is determined by the largest value between $(1 + \sqrt{2}) \hbar \omega_c$ and $\mu + \sqrt{\mu^2-(\hbar \omega_c)^2}$. The crossover between these two values occurs at the point
$\mu = \sqrt{2} \hbar \omega_c$.
Assuming the momentum relaxation times in all magnetic subbands to coincide and be equal to $\tau_p$, we obtain
\begin{equation}
\label{j_inter}
j_{x'} ={\cal C}\text{sgn}(B_{x'}) \Gamma_0 \tau_p |\bm E|^2
{2\omega_c^2 \over \omega^2 + \omega_c^2} \sum_{0\leq n<\nu} \sqrt{1- {n\over \nu}},
\end{equation}
where $\Gamma_0$ is introduced in Eq.~(\ref{Gamma00}) and we replaced $|{\bm E}_{\perp}|^2$ by $(2/3)|\bm E|^2$.
In the weaker (but still quantized) fields such as $\omega_c \ll \omega\approx 2\mu/\hbar$, we have $\nu \approx (\omega/2\omega_c)^2 \gg 1$ and, replacing the summation by integration, obtain a universal result
\begin{equation}
\label{MPGE_universal}
j_{x'} (\omega_c \ll \omega)= {\cal C}\text{sgn}(B_{x'}) {\Gamma_0 \over 3}\tau_p |\bm E|^2.
\end{equation}
\subsubsection{Range 2: $1<\omega/\omega_c<\sqrt{2}+1$}
The elastic scattering of electrons (or holes) in the chiral subband (\ref{chiral}) with the energies within the interval $(- \hbar \omega_c, \hbar \omega_c)$ radically differs from that in all other subbands. For energies outside this interval,
the backward scattering of the quasi-momentum within the same Weyl node is the main mechanism of the free-carrier momentum relaxation resulting in a short intravalley relaxation time $\tau_p$ comparable to the zero-field one.
However, the energies between $- \hbar \omega_c$ and $\hbar \omega_c$
are available only for the states in the chiral subband which has a linear dispersion and, therefore, forbids an intravalley backscattering. In this case the momentum relaxation occurs due to scattering to the chiral subbands in other Weyl points, Fig.~\ref{fig:rel_times}, with a characteristic time $\tilde{\tau}_0$ obviously by far exceeding $\tau_p$.
A photohole excited in the transition $0 \rightarrow c1$ in Range~2 has an energy $|E_0|$ below $ \hbar \omega_c$. The consideration of steady-state kinetics yields the following effective relaxation time which controls the photocurrent~\cite{JETP_Lett_2017}:
\begin{equation}
\label{tau_MPGE}
\tilde{\tau} = {\tilde{\tau}_0\tau_\varepsilon \over \tilde{\tau}_1} \gg \tau_p.
\end{equation}
Here $\tilde{\tau}_1$ is the time of elastic scattering of carriers between the 1st subbands in different monopoles, and $\tau_\varepsilon$ is the energy relaxation time describing phonon-involved transitions from the excited states in the 1st conduction subband to the 0th chiral subband, Fig.~\ref{fig:rel_times}. These scattering processes in WSMs and their effect on CPGE are discussed in Ref.~\cite{CPGE_scattering}. To summarize, the contribution to the photocurrent from the photoholes in the chiral subband dominates and one has for the photocurrent density
\begin{align}
\label{j01}
j_{x'} = -{e \tilde{\tau} \over \hbar l_B^2}\sum_{k_{x'}} & v^{x'}_0(k_{x'}) \left\vert M_{c1 \leftarrow 0}(k_{x'}) \right\vert^2 \\ & \times \delta \left[E_1(k_{x'}) - E_0(k_{x'}) -\hbar\omega\right]\:. \nonumber
\end{align}
Here the chemical potential lies between the Weyl-point energy and the energy $\sqrt{2} \hbar\omega_c$, and the temperature is set to zero. For $\hbar \omega_c > \mu > 0$, the frequency range narrows to $(\hbar \omega_c, \mu + \sqrt{\mu^2 + (\hbar \omega_c)^2})$, while for $\sqrt{2} \hbar \omega_c > \mu > \hbar \omega_c $ the photocurrent is generated in the range $[\mu + \sqrt{\mu^2 - (\hbar \omega_c)^2}, \hbar \omega_c(1 + \sqrt{2})]$.
\begin{figure}[t]
\includegraphics[width=0.8\linewidth]{rel_times}
\caption{Schematics of the magneto-gyrotropic photogalvanic effect and the related scattering times controlling the photocurrent formation. The current is governed by a combination of three relaxation times, namely, the intranode energy relaxation time $\tau_\varepsilon$ and the internode elastic scattering times between the chiral subbands ($\tilde{\tau}_0$) and between the first excited subbands ($\tilde{\tau}_1$), see Eq.~\eqref{tau_MPGE}. The position of chemical potential $\mu$ prevents the transitions from the valence subband $v1$ to the chiral subband with the linear dispersion.}
\label{fig:rel_times}
\end{figure}
The squared matrix element $\left| M_{c1 \leftarrow 0} \right|^2$ is even in magnetic field because, under the inversion of $B_{x'}$, the slope of dispersion $E_0(p_{x'})$ changes from negative to positive, and the direct transition $0 \to c1$ is replaced by the transition $v1 \to 0$ that takes place in the inverted value of $k_{x'}$. However the velocity in the chiral subband $v^{x'}_0 = - {\cal C}v_0\text{sgn}(B_{x'})$ is odd in $B_{x'}$, and we obtain~\cite{JETP_Lett_2017}:
\begin{equation}
\label{chircurrent}
j_{x'} = B_{x'}|\bm E|^2 \: {\cal C} \Gamma_0\tilde{\tau} {2v_0^2e\over \hbar c \omega^2}\:.
\end{equation}
For $B_{x'}=1$~T, $v_0=c/300$, and $\omega/(2\pi)=1$~THz a value of $j_{x'}$ is estimated as $j_0 = 80 \: \Gamma_0\tilde{\tau}|\bm E|^2$.
\subsubsection{Range 3: $\omega<\omega_c$. Direct intraband transitions}
If the chemical potential lies in the energy interval $(0,\hbar \omega_c)$ then the main contribution to the photocurrent comes from the $0 \to c1$ transition under the excitation in the window
\begin{equation} \label{chiral3}
\sqrt{\mu^2 + (\hbar\omega_c)^2} - \mu < \hbar \omega < \hbar\omega_c
\end{equation}
and is described by Eq.~(\ref{chircurrent}). Thus, we turn to the samples with chemical potential located above $\hbar \omega_c$, i.e. above the bottom of the 1st conduction subband.
At zero temperature the direct optical transitions $c,n \rightarrow c,n+1$ ($n > 0$) between the conduction subbands occur if the chemical potential lies above the bottom of the $c1$ subband and $n < \nu$~\cite{magn_opt_cond}. The intraband absorption is accompanied by a photocurrent generation because in this case the probability rate as well contains a part odd in $k_{x'}$. The corresponding photocurrent has the form
\begin{eqnarray}
j_\text{intra} &=& {e\tau_p\over \hbar l_B^2} \sum_{k_{x'}, n\leq \nu} \left| M^\text{odd}_{c,n+1 \leftarrow c,n}(k_{x'}) \right|^2 \\
&&\times (v^{x'}_{n+1}-v^{x'}_{n}) \delta[E_{n+1}(k_{x'})-E_{n}(k_{x'})-\hbar\omega]. \nonumber
\end{eqnarray}
Here the transition matrix element is given by Eq.~\eqref{M_sq_inter} where $E_{n+1}$ should be replaced by $-E_{n+1}$.
The electron energies of the initial and final states are equal to
\begin{equation}
E_n/E_{n+1}= \hbar {\omega_c^2 \mp \omega^2\over 2\omega}\:.
\end{equation}
The conditions $E_{n+1} - E_n = \hbar \omega$, $E_n < \mu < E_{n+1}$ and $n> 0$ restrict the frequency range to
\begin{equation}
\label{range_omega2}
\hbar\omega \in \left(\sqrt{\mu^2 + (\hbar\omega_c)^2} - \mu, \mu - \sqrt{\mu^2-(\hbar\omega_c)^2} \right).
\end{equation}
Analytically the photocurrent $j_{x'}$ is described by the same equation Eq.~\eqref{j_inter} derived
for the interband excitation bearing in mind that now $\omega < \omega_c$.
\subsubsection{Temperature dependence}
\label{T}
The discontinuities in the dependence of the MPGE current on the magnetic field are smeared by temperature $T$. At finite $T$, the populations of the initial and final states take values other than just 0 and 1. As a result, the ``complementary'' transitions ${v,n+1 \rightarrow c,n}$ forbidden in Fig.~\ref{fig:MPGE_1} by the Pauli exclusion principle become possible.
Using Eq.~\eqref{En} for the initial and final energies we obtain for the populations
\begin{eqnarray}
&&f^{c,v}_n=f_0\left(\pm{\hbar(\omega^2-\omega_c^2)\over2\omega}\right), \: \\
&&f^{c,v}_{n+1}=f_0\left(\pm{\hbar(\omega^2+\omega_c^2)\over2\omega}\right).\nonumber
\end{eqnarray}
Substitution of these functions into Eqs.~\eqref{MPGE} and~\eqref{F_MPGE} results in the following temperature dependence of the photocurrent for the interband transitions
\begin{equation}
\label{nonzero_T}
j_{x'} = j_{x'}(T=0) {2 \sinh{\left({\hbar\omega_c\over 2T}\right)}\sinh{\left({\hbar\omega\over 2T}\right)} \sinh{\left({\mu\over T}\right)} \over Z},
\end{equation}
where
\begin{eqnarray}
Z &=& \left[\cosh{\left({\mu\over T}\right)} + \cosh{\left({\hbar\omega_c\over 2T}\right)}\cosh{\left({\hbar\omega\over 2T}\right)} \right]^2 \nonumber \\&& \hspace{8 mm} - \left[ {\sinh{\left({\hbar\omega_c\over 2T}\right)}} \sinh{\left({\hbar\omega\over 2T}\right)} \right]^2. \nonumber
\end{eqnarray}
\subsection{Allowance for tilt}
The derived expressions for the MPGE current are different in sign for monopoles of opposite chirality. Here we demonstrate that account for the tilt terms in the Hamiltonian gives rise to the net MPGE current in gyrotropic WSMs.
In the linear-in-${\bm k}$ approximation the tilt term in Eq.~(\ref{Hamilt3}) can be written as $d_0({\bm k}) = {\bm a} \cdot {\bm k}$ and is characterized by the vector $\bm a$ which describes the magnitude of a spin-independent correction to the Hamiltonian. Let us use, instead of $x',y',z$, the axes $x',y'', z''$ with
$y''$ being the axis along the component ${\bm a}_{\perp}$ of the tilt vector $\bm a$ perpendicular to $\bm B$. Then the Hamiltonian~(\ref{lineartilt}) takes the form
\begin{equation}
{\cal H}={\cal C}\hbar v_0\bm \sigma \cdot \bm k + a_{x'} k_{x'} + a_{\perp} k_{y''}.
\end{equation}
With account for tilt, the energy dispersion in the magnetic subbands transforms to
\begin{align}
&E_0 = [-{\cal C}\hbar\text{sgn}(B_{x'})\tilde{v}_0 + a_{x'}] k_{x'}, \\
&E_{n}^{(c,v)} = a_{x'} k_{x'} \pm \tilde{E}_n. \nonumber
\end{align}
Here $\tilde{E}_n = \hbar\sqrt{\left(\tilde{v}_0 k_{x'}\right)^2 + n\tilde{\omega}_c^2}$,
the tilde marks the renormalized Fermi velocity, cyclotron frequency and magnetic length
\begin{equation}
\tilde{v}_0 = {v_0\over \gamma},\quad
\tilde{\omega}_c = {\tilde{v}_0 \sqrt{2}\over \tilde{l}_B}=\omega_c\gamma^{3/2}, \quad \tilde{l}_B= l_B\sqrt{\gamma},
\end{equation}
where
\begin{equation}
\gamma = {1\over \sqrt{1-\beta^2}}, \qquad \beta = {a_{\perp} \over \hbar v_0}.
\end{equation}
At $B_{x'} > 0$, the wavefunctions in the conduction and valence magnetic subbands are given by~\cite{PRL_2016_1,PRL_2016_2,PRL_2016_3}
\begin{equation}
\Psi_0 = {\text{e}^{i\alpha}\over \sqrt{\gamma}} \text{e}^{-\theta\sigma_1/2}\left[
\begin{array}{c}
0\\
\phi_0(z''-\zeta_0)
\end{array}
\right],
\end{equation}
\begin{equation}
\Psi_{n>0}^{(c,v)} =
{\text{e}^{i\alpha}\over \sqrt{\gamma}} \text{e}^{-\theta\sigma_1/2}\left[
\begin{array}{c}
\tilde{a}_{n,\pm}\phi_{n-1}\left(z''-\zeta_{n,\pm}\right)\\
\pm {\cal C} \tilde{a}_{n,\mp}\phi_n\left(z''-\zeta_{n,\pm}\right)
\end{array}
\right],
\end{equation}
where $\sigma_1$ is the first Pauli matrix, $\alpha=k_{x'}x'+k_{y''}y''$, $\tilde{a}_{n,\pm}= \sqrt{\left(1\pm {\cal C}\hbar{\tilde{v}_0 k_{x'}/ \tilde{E}_n}\right)/2}$, $\tanh{\theta} = \beta$, and the centers of the cyclotron orbits depend on the energies due to the tilt
\begin{eqnarray}
&&\zeta_0 = {c\hbar\over eB_{x'}}\left(k_{y''} - \beta\gamma{\cal C}k_{x'} \right),\\
&&\zeta_{n,\pm}={c\hbar\over eB_{x'}}\left(k_{y''} \pm {\beta\gamma\tilde{E}_n\over \hbar \tilde{v}_0} \right). \nonumber
\end{eqnarray}
We calculate the photocurrent due to the $0 \to c1$ transitions assuming that the chemical potential lies below the bottom of the $c1$ subband: $0<\mu<\hbar\omega_c$, Fig.~\ref{fig:j_10_tilt}.
At ${\omega/\tilde{\omega}_c<\sqrt{2}+1}$, the photocurrent is given by Eq.~\eqref{j01} where the substitutions $l_B \to \tilde{l}_B$, $E_{0,1} \to \tilde{E}_{0,1}$ are made and the velocity in the 0th subband is
$$v^{x'}_0= - {\cal C}\text{sgn}(B_{x'})\tilde{v}_0 + {a_{x'} \over \hbar}.$$
The electron-photon interaction operator can still be taken in the form of Eq.~\eqref{e-phot} because
the tilt term does not result in interband transitions.
Using the relations
\begin{equation}
\left< \phi_0(z''- \zeta_0) \big| \phi_1\left(z''-\zeta_{1,+}\right) \right> = \sqrt{u}\text{e}^{-u/2},
\end{equation}
\begin{equation}
\left< \phi_0(z''-\zeta_0) \big| \phi_0\left(z''-\zeta_{1,+}\right) \right> = \text{e}^{-u/2},
\end{equation}
where $u=\left[(\zeta_{1,+} - \zeta_0)/(\tilde{l}_B\sqrt{2})\right]^2$,
we obtain that the matrix element of the direct optical transition is given by
\begin{align}
& M_{c1 \leftarrow 0} = {\cal C}\tilde{v}_0 {ie\over \omega}\text{e}^{-u/2} \\
& \times \left[
\gamma E_{y''} (\tilde{a}_{1,+} - \beta \sqrt{u}\tilde{a}_{1,-}) - i\tilde{a}_{1,+}E_{z''} - \sqrt{u}\tilde{a}_{1,-}E_{x'} \right]. \nonumber
\end{align}
Energy conservation yields $u = (\beta \omega/\tilde{\omega}_c)^2$, and
we
obtain the MPGE current with allowance for tilt in the form
\begin{eqnarray}
\label{j_tilt}
j_{x'} &=& \left[{\cal C} \text{sgn}(B_{x'})-{a_{x'}\over \hbar \tilde{v}_0}\right] |\bm E|^2 \Gamma_0\tilde{\tau}\\ && \hspace{7 mm}\times \left( {\tilde{\omega}_c\over\omega}\right)^2 \exp{\left[-\left({\beta \omega\over\tilde{\omega}_c }\right)^2\right]} F. \nonumber
\end{eqnarray}
Here the current is averaged over polarization assuming an unpolarized light, and
\begin{eqnarray}
F&=&\sum_\pm \bigg[ f_0\left(\pm \tilde{E}_0+{\cal C} \text{sgn}(B_{x'}) a_{x'} k_\omega\right)\\
&&\mbox{}\hspace{3 mm} -f_0\left(\pm\tilde{E}_1+{\cal C} \text{sgn}(B_{x'}) a_{x'} k_\omega\right)\bigg],\nonumber
\end{eqnarray}
where
\begin{equation}
\tilde{E}_{0,1}={\hbar(\omega^2\mp\tilde{\omega}_c^2)\over2\omega},
\qquad
k_\omega={\tilde{E}_0\over \hbar\tilde{v}_0}.
\end{equation}
\subsubsection{Summation over monopoles in case of the $C_{2v}$ symmetry}
Equation~\eqref{j_tilt} demonstrates that, for $\bm B \parallel x'$, the MPGE current in each Weyl node depends on $a_{x'}$ and
\begin{equation}
a_{\perp}^2=a_{y'}^2+a_{z}^2.
\end{equation}
For the Weyl semimetals of $C_{2v}$ symmetry, the monopoles are located in four points of the Brillouine zone if they belong to the plane $z=0$ and in eight points if they are shifted from this plane. For simplicity we consider the former case. Let the first Weyl node be characterized by the chirality $\cal C$ and the tilt vector components be $a_{x'},a_{y'},a_{z}$.
The $C_2$ rotation makes a transition to another monopole:
\begin{equation}
{\cal C}, a_{x'},a_{y'}^2+a_{z}^2 \to {\cal C}, -a_{x'}, a_{y'}^2+a_{z}^2.
\end{equation}
The mirror reflections $\sigma_v$ yield two other monopoles:
\begin{equation}
{\cal C}, a_{x'},a_{y'}^2+a_{z}^2 \to -{\cal C}, a_{y'},a_{x'}^2+a_{z}^2 \hspace{4 mm}
\end{equation}
and
\begin{equation}
{\cal C}, a_{x'},a_{y'}^2+a_{z'}^2 \to -{\cal C}, -a_{y'},a_{x'}^2+a_{z}^2.
\end{equation}
\begin{figure}[t]
\includegraphics[width=0.8\linewidth]{j_10_tilt_1mech} \qquad
\caption{Microscopic mechanism of the MPGE in a tilted Weyl semimetal. The net current is caused by a disbalance between contributions of the Weyl nodes of opposite chiralities: due to the tilt-dependent factor $\Phi$ defined by Eq.~(\ref{Phia}) the direct transition rate in the Weyl node with ${\cal C}>0$ becomes larger than that in the node with ${\cal C}<0$.}
\label{fig:j_10_tilt}
\end{figure}
Therefore, at zero temperature, there are two sources of the MPGE. The first mechanism is realized at the chemical potential lying below the bottom of the $c1$ subband, Fig.~\ref{fig:j_10_tilt}. In the first mechanism ${F=1}$, the photocurrent is dependent on $a^2_{\perp}$ via the parameters $\beta$ and $\gamma$~\cite{JETP_Lett_2017}, and the summation over four monopoles yields
\begin{eqnarray}
j_{x'} &=& B_{x'}|\bm E|^2 {\cal C} \Gamma_0\tilde{\tau} {2v_0^2e\over \hbar c \omega^2}\\ &&\times\left[\Phi(a_{y'}^2+a_{z}^2)-\Phi(a_{x'}^2+a_{z}^2)\right],\nonumber
\end{eqnarray}
where
\begin{equation} \label{Phia}
\Phi(a_{\perp}^2) = 2\gamma^3 \exp{\left[-{a_{\perp}^2\over \hbar^2 v_0^2 \gamma^3}\left({\omega\over\omega_c }\right)^2\right]}.
\end{equation}
We remind that $\gamma=[1-(a_{\perp}/\hbar v_0)^2]^{-1/2}$. This equation demonstrates that the net MPGE current appears due to a difference in the direct optical transition rates in differently tilted Weyl nodes, Fig.~\ref{fig:j_10_tilt}.
The second mechanism is related with the interference of the $a_{x'}$-linear term in the velocity with the $a_{x'}$-dependent part of the distribution functions, Eq.~\eqref{j_tilt}. The photocurrent becomes nonzero due to a difference in the relaxation times $\tilde{\tau}$ and $\tau_p$. This mechanism working at quantized magnetic field is similar to that considered in Ref.~\cite{Kharzeev} for low magnetic fields.
\section{Discussion}
\label{disc}
The calculated frequency dependence of the MPGE current is shown in Fig.~\ref{fig:j_MPGE}. In agreement with considerations of Sect.~\ref{MPGE_one_mode}, at zero temperature the photocurrent is induced in a limited interval of frequencies dependent on the level of chemical potential. For $\mu = 1.2 \hbar \omega_c < \sqrt{2} \hbar \omega_c$, the photocurrent is controlled within the interval $\Omega_- < \omega < (\sqrt{2} + 1) \omega_c$ by the longer relaxation time $\tilde{\tau}$. On passing from Range~2 to Range~1 the photocurrent abruptly decreases because, although it is now contributed by the two transitions, $0 \to c1$ and $v1 \to c2$, the relaxation time shortens from $\tilde{\tau}$ to $\tau_p$. For $\mu = 1.6 \hbar \omega_c > \sqrt{2} \hbar \omega_c$, the interband photocurrent is generated under the transitions $0 \to c1$, $v1 \to c2$ and $v2 \to c3$ and controlled by the time $\tau_p$.
In Fig.~\ref{fig:j_MPGE} dashed lines illustrate the effect of temperature on the photocurrent spectral dependence. At $\omega>\omega_c(\sqrt{2}+1)$, the photocurrent density is close to the universal value \eqref{MPGE_universal}.
\begin{figure}[t]
\includegraphics[width=0.9\linewidth]{j_MPGE}
\caption{ Frequency dependence of the MPGE current related to $|\bm E|^2 \Gamma_0\tau_p$ and calculated for two values of the chemical potential $\mu$. The relaxation times $\tilde{\tau}$ and $\tau_p$ are taken at a ratio of 10 to 1. The vertical arrows show the onset of the corresponding interband transition. The frequencies $\Omega^-(\mu)$ and $\Omega^+(\mu)$ defined by Eq.~(\ref{range_omega}) indicate the range where the photocurrent is induced at $T=0$. Solid and dashed curves are calculated for zero temperature and $T=0.01\hbar\omega_c$, respectively. The horizontal arrow shows the universal photocurrent value, Eq.~\eqref{MPGE_universal}.
At $\omega < \omega_c$ the photocurrent is also contributed by the intraband transitions (not shown).}
\label{fig:j_MPGE}
\end{figure}
It should be noted that, in addition to the circular photogalvanic photocurrent (\ref{general}), a transient electric current can be generated under time-dependent optical excitation. Such a current appears due to a light-induced renormalization of the electron energy \cite{transient}. For the Weyl semimetal Hamiltonian (\ref{Hamilt3}) with a zero tilt term, the renormalized energy $\tilde{E}_{\pm, {\bm k}}$ differs from the unperturbed energy $E_{\pm, {\bm k}}$ by a correction $\delta E_{\pm, {\bm k}}$ which in the second-order approximation can be written as
\[
\delta E_{\pm, {\bm k}} = \pm \left\vert M_{+-}\right\vert^2 \frac{2 \hbar \omega}{(2 \hbar v_0 k)^2 - (\hbar \omega)^2}\:.
\]
The contribution (\ref{MOmega}) to the squared modulus of the matrix element $M_{+-}$ dependent on the circular polarization of the radiation is odd in the wavevector ${\bm k}$ and, therefore, the correction to the electron velocity ${\delta {\bm v}_{\pm, {\bm k}} = \hbar^{-1} \partial \delta E_{\pm, {\bm k}}/\partial {\bm k} }$ averaged over the direction of ${\bm k}$ does not vanish. As a result, at an abrupt switch-on of the light intensity at the moment $t=0$, a transient current
$\delta {\bm j}_{\rm tr} (t) ={\rm e}^{- t/\tau_p} {\bm j}^0_{\rm tr}$ does appear, where
\[
{\bm j}^0_{\rm tr} = \frac{e}{\hbar} \sum\limits_{\bm k} \frac{\partial \delta E_{+, {\bm k}} }{\partial {\bm k}}f_+(E_{+, {\bm k}})\:.
\]
The calculation performed at low temperature $T \approx 0$ for an $n$-doped sample with the Fermi wavevector satisfying the condition $2 v_0k_\text{F} < \omega$ results in
\[
{\bm j}^0_{\rm tr} = \frac{w}{\pi} \frac{ {\bm j}_{\rm CPGE}}{\omega \tau_p}\:,
\]
where the circular current ${\bm j}_{\rm CPGE}$ is given by Eq.~(\ref{Gamma00}) and
\[
w = {\frac{2 \omega^2}{(2 v_0 k_\text{F})^2 - \omega^2}}\:.
\]
It follows then that the non-stationary current $\delta {\bm j}_{\rm tr} (t)$, as compared with the photocurrent (\ref{Gamma00}), contains an additional small parameter $(\omega \tau_p)^{-1}$ and can be ignored, even in experiments with time-dependent light intensity.
Coulomb interaction effects on the direct optical absorption in systems with a linear dispersion has been investigated in graphene~\cite{Mishchenko,Schmalian_2009,JVH_2010,Teber_Kotikov_2014,Schmalian_2016}.
For three-dimensional WSMs, it has been shown that the electron-electron interaction yields a correction to the light absorption coefficient containing the factor $1/(N+1)$ which is small if $N \gg 1$, where $N$ is the number of Weyl points~\cite{WSM_opt_cond_2017,WSM_opt_cond_2018}. We remind that, for a Weyl semimetal of the $C_{4v}$ symmetry, $N=16$ for the nodes lying out of the plane $z=0$ and $N=8$ for the nodes on this plane.
The Coulomb correction to the CPGE current at interband transitions also contains the factor $1/(N+1)$. However, the analysis shows that the electron-electron effects are additionally suppressed in the CPGE current by a small factor $\ln^{-1}{(D/\omega)}$ where $D\gg\omega$ is the high-frequency cut-off.
\section{Conclusion}
\label{concl}
We have developed a theory of the circular and magneto-gyrotropic photogalvanic effects in Weyl semimetals with the point groups containing improper symmetry operations. In semimetals of the C$_{2v}$ symmetry with the linear energy dispersion, the net CPGE photocurrent becomes nonzero taking into account a spin-independent tilt term in the electron effective Hamiltonian. However, this is insufficient for the crystal class C$_{4v}$, like the TaAs Weyl semimetal. In this case one needs to add to the Hamiltonian not only the tilt but also spin-dependent terms of the second or third order in the electron quasi-momentum and take into account the nonlinear corrections, respectively, to the velocity and Berry curvature in the equation for the current density. Additionally, the theory has been extended by consideration of the indirect intraband optical transitions and their contribution to the CPGE. Here an important point to bear in mind is that the probability rate is contributed by the composite (two-quantum) processes with virtual states both in the conduction and valence bands.
Developing a theory of the polarization independent photocurrents in an external quantized magnetic field we have, in turn, analyzed three frequency ranges, namely, $\omega > \omega_c (\sqrt{2} + 1)$; $\omega_c (\sqrt{2} + 1) > \omega > \omega_c $ and
$\omega_c > \omega$ and found restrictions imposed by the level of chemical potential at zero temperature on the spectral intervals where the MPGE current is generated. The temperature smooths out the edges of these intervals. The calculation reveals that, for the C$_{2v}$ point-group symmetry, the net MPGE current is an even function of the tilt parameters $a_{\alpha}$.
\acknowledgements
We thank B.Z. Spivak for discussions. Financial support of the Russian Science Foundation (Project No. 17-12-01265) is acknowledged. Work of L.E.G. is supported by the Foundation for advancement of theoretical physics and mathematics ``BASIS''.
|
{
"timestamp": "2018-05-04T02:09:35",
"yymm": "1803",
"arxiv_id": "1803.02850",
"language": "en",
"url": "https://arxiv.org/abs/1803.02850"
}
|
\section{Introduction and the main result}
Let $H$ be a real separable Hilbert space and let $L=(L_t)_{t\geqslant 0}$ be an $H$-valued L\'evy process. We fix two bounded linear operators $A$, $\tilde{A}$ on $H$; should the space $H$ have finite dimension $d$, we think of $A$ and $\tilde{A}$ as $d\times d$ real matrices in some fixed basis. We consider the corresponding \emph{$H$-valued Ornstein--Uhlenbeck processes driven by a L\'evy process}, that is, solutions to the following stochastic differential equations:
\begin{equation}\label{equations}\left\{\begin{array}{lcl} {\rm d}X_t &=& A X_t\, {\rm d}t + {\rm d}L_t,\\ {\rm d}\tilde{X}_t &=& \tilde{A} \tilde{X}_t\, {\rm d}t + {\rm d}L_t\end{array}\right. \end{equation}
with the initial conditions $X_0 = \tilde{X}_0 = 0$. (The reader will find all the required definitions as well as proofs of the results to be presented in the subsequent sections.)\smallskip
It is customary to view sample paths of such processes (truncated to some initial interval $[0,T]$ for $T>0$) as elements of the Hilbert space $H_T:=L_2\big([0,T], H\big)$. On the other hand, we may be more restrictive and regard these processes as random variables assuming values in $\mathcal{D}_{H,T}$, the space of $H$-valued c\`adl\`ag functions on $[0,T]$ furnished with the Skorohod topology. This topology is induced by the so-called \emph{Skorohod metric}; the Borel $\sigma$-algebra of $\mathcal{D}_{H,T}$ coincides then with the cylindrical $\sigma$-algebra, that is, the smallest $\sigma$-algebra making the point evaluations $$p_t(f)=f(t)\quad (t\in [0,T], f\in \mathcal{D}_{H,T})$$ measurable--this is an important feature unavailable in the Hilbert space $H_T$. A natural question then arises. \begin{quote}\emph{In which circumstances such two processes have equivalent laws?}\end{quote} This line of research concerning the study of equivalence of laws was initiated by Kozlov (\cite{kozlov}) in the setting where $A,\tilde{A}$ are elliptic and self-adjoint operators on a smooth manifold without boundary and the equations are driven by a Brownian motion. This theory was developed further by Zabczyk (\cite{Zabczyk}) in much greater generality (see also the seminal monograph \cite{DZ}), Peszat (\cite{PeszatGoldys, Peszat2}), and other authors (\cite{BvN}, \cite{MvN}). \smallskip
The aim of this paper is to extend and complement already existing results for Ornstein--Uhlenbeck processes driven by a (cylindrical) Wiener processes that take values in a finite- or infinite-dimensional Hilbert space to Ornstein--Uhlenbeck processes driven by L\'evy processes that possibly have jumps. Our results appear to be new also in the case where $H = \mathbb R^d$ for some $d\in \mathbb N$. \smallskip
We fix a real separable Hilbert space $H$ and a~$H$-valued L\'evy process $L=(L_t)_{t\geqslant 0}$ on some probability space $(\Omega, \mathcal {F}, \mathsf P)$ that is expressed in the L\'evy--It\^o decomposition as $L_t = bt + W_t + Z_t$ ($t\geqslant 0$), where $b\in H$, $W=(W_t)_{t\geqslant 0}$ is a (possibly degenerate) Wiener process with the covariance operator $Q$, and $(Z_t)_{t\geqslant 0}$ is the jump part of $L$ (see Theorem~\ref{levyito} for more details).
\begin{theorema}Let $T>0$. Suppose that the eigenvalues of the covariance operator $Q$ corresponding to $W$ are strictly positive. Let $X,\tilde{X}\colon \Omega\to \mathcal{D}_{H,T}$ be the Ornstein--Uhlenbeck processes solving \begin{equation}\label{equations4}\left\{\begin{array}{rcl} {\rm d}X_t &=& A X_t\, {\rm d}t + {\rm d}L_t,\\ {\rm d}\tilde{X}_t &=& \tilde{A} \tilde{X}_t\, {\rm d}t + {\rm d}{L}_t, \\ X_0 & = & 0, \\ \tilde{X}_0 &=& 0.\end{array} \right. \end{equation}If $H$ is finite-dimensional or \begin{equation}\label{technical}\int\limits_0^t \, \big(AX_s - \tilde{A}X_s\big){\rm d}s \in {\rm im}\, Q^{1/2}\quad \big(t\in [0,T]\big), \end{equation} then the laws of $X$ and $\tilde{X}$ are equivalent.\end{theorema}
\noindent \emph{Note}. When the Hilbert space $H$ is finite-dimensional, the main hypothesis of Theorem A is equivalent to invertibility of the covariance matrix $Q$ corresponding to $W$, in which case \eqref{technical} holds vacuously as $Q^{1/2}$ is invertible. Otherwise, $Q^{1/2}$ is only a densely defined positive operator as $Q$ is trace-class.\medskip
In the case where the underlying L\'evy process $L$ is a purely jump process, solutions to \eqref{equations4} with respect to $L$ exhibit a remarkably rigid behaviour.
\begin{theoremb}Let $T>0$. Suppose that $L$ is a purely jump process, that is, $L = Z$ as written in the L\'evy--It\^o decomposition. Let $X_J,\tilde{X}_J\colon \Omega\to \mathcal{D}_{H,T}$ be the Ornstein--Uhlenbeck processes solving \eqref{equations4}. If the law of $X_J$ is absolutely continuous with respect to the law of $\tilde{X}_J$, then the processes $X_J$ and $\tilde{X}_J$ are equal to each other almost surely.\end{theoremb}
\section{Preliminaries} Let $(\Omega, \mathcal F, \mathsf P)$ be a probability space and let $(S, \mathcal S)$ be a measurable space. A function $X\colon \Omega \to S$ is an $S$-\emph{valued random variable} when $X^{-1}(E)\in \mathcal F$ for every $E\in \mathcal S$. The \emph{law} of $X$ is the pull-back measure $\mathsf P_X$ on $(S, \mathcal{S})$ given by $\mathsf P_X(E) = \mathsf P(X^{-1}(E))$ ($E\in \mathcal S$). When $S$ carries the structure of a metric space, by default we will take $\mathcal S= \mathsf{Bor}\, S$, the $\sigma$-algebra of Borel subsets of $S$. For two measures $\mu$ and $\nu$ we denote by $\mu \otimes \nu$ the product measure defined on the product $\sigma$-algebra, that is, the smallest $\sigma$-algebra containing all measurable rectangles from the respective measure spaces. For a separable metric space $S$, or more generally, a second-countable Hausdorff space, the product $\sigma$-algebra $\mathsf{Bor}\, S \otimes \mathsf{Bor}\, S$ coincides with $\mathsf{Bor}\, S\times S$ (see, \emph{e.g.}, \cite[Lemma 6.4.2]{Bogachev}).\smallskip
\subsection{The Skorohod metric} Let $(S, d)$ be a separable metric space and let $T>0$ be given. Let $\mathcal{D}_{S,T}$ denote the space of all $S$-valued c\`adl\`ag functions, that is, right-continuous functions $f\colon [0, T]\to S$ with the property that for each $t > 0$ the left limit at $t$, $f(t-)$, exists. Denote by $\Lambda_T$ the family of all strictly increasing functions $\phi$ from $[0,T]$ onto itself with $\phi(0)=0$ and $\phi(T)=T$. Then the formula
$$d_S(f,g)=\inf_{\phi\in \Lambda_T} \max\big\{\sup_{t\in [0,T]}|\phi(t)-t|, \sup_{t\in [0,T]} d\big(f(\phi(t), g(t) \big) \big\}\quad \big(f,g\in \mathcal{D}_{S,T}\big) $$
defines a metric on $\mathcal{D}_{S,T}$, called the \emph{Skorohod metric} (\cite[p.~265 \& Proposition 1.6]{Jakubowski}).
The Borel $\sigma$-algebra of the space of c\`adl\`ag functions with the Skorohod metric, $\mathsf{Bor}\, \mathcal{D}_{S,T}$, coincides with the $\sigma$-algebra of cylindrical sets--in other words, it is the smallest $\sigma$-algebra on $\mathcal{D}_{S,T}$ for which the point evaluations $p_t(f)=f(t)$ $(t\in [0,T], f\in \mathcal{D}_{S,T})$ are measurable (\cite[Corollary 2.4]{Jakubowski}). We shall frequently invoke the following consequence of this fact.
\begin{proposition}\label{usefulprojections}Let $(\Omega, \mathcal{F})$ be a measurable space. Then a function $X\colon \Omega\to \mathcal{D}_{S,T}$ is measurable if and only if, for every $t\in [0,T]$ the composite map $\pi_t\circ X$ is measurable. \end{proposition}
\subsection{Absolute continuity of measures} Let $\mu, \nu$ be measures on a measurable space $(S, \mathcal S)$. The measure $\mu$ is called \emph{absolutely continuous} with respect to $\nu$ (in short, $\mu \ll \nu$), when $\nu(E)=0$ ($E\in \mathcal S$) implies that $\mu(E)=0$. Two measures are \emph{equivalent} when they are mutually absolutely continuous. Let us record the following corollary to Fubini's theorem concerning absolute continuity of product measures (\cite[p.~92]{Folland}). Suppose that $\mu_i, \nu_i$ are $\sigma$-finite measures on measurable spaces $(S_i, \mathcal S_i)$ such that $\mu_i \ll \nu_i$ ($i=1,2$). Then $\mu_1\otimes \mu_2 \ll \nu_1 \otimes \nu_2$. We will make use of this fact stated in the following form.
\begin{lemma}\label{Lemma2new}Let $(\Omega, \mathcal F, \mathsf P)$ be a probability space and let $(S, \mathcal S)$ be a measurable space. Suppose that $X,Y_1,Y_2\colon \Omega\to S$ are random variables such that
\begin{romanenumerate}
\item $\mathsf P_{Y_1} \ll \mathsf P_{Y_2}$,
\item the variables $X, Y_i$ are independent $(i=1,2)$.\end{romanenumerate}
Then the law $\mathsf P_{(X, Y_1)}$ is absolutely continuous with respect to $\mathsf P_{(X, Y_2)}$.\end{lemma}
\begin{proof}Independence of $X$ and $Y_i$ is equivalent to $\mathsf P_{(X, Y_i)} = \mathsf P_X \otimes \mathsf P_{Y_i}$ ($i=1,2$) (see, \emph{e.g.}, \cite[Th\'eor\`eme IV.1.3]{metiver}).\end{proof}
\subsection{L\'evy processes} Let $B$ be a separable Banach space. A stochastically continuous, $B$-valued process $L = (L_t)_{t\geqslant 0}$, is \emph{L\'evy}, when $L_0 = 0$ almost surely, the increments of $L$ are independent and stationary, and almost every sample path $f(t):=L_t(\omega)$ ($\omega\in \Omega$) of $L$ is a~$B$-valued c\`adl\`ag function.\smallskip
Let $E\in \mathsf{Bor}\, B\setminus \{0\}$, $t\geqslant 0$ and $f\in \mathcal{D}_B$. We then define
\begin{equation}\label{Poissonmeasure}\pi_t(E, f) = {\rm card}\{s\leqslant t\colon \Delta f(s):=f(s) - f(s-) \in E \}.\end{equation}
For a L\'evy process $L = (L_t)_{t\geqslant 0}$ we may then set \begin{equation}\label{piL}\pi_t(E, L)(\omega)=\pi_t(E, f)\quad (\omega\in \Omega),\end{equation} where $f(t)=L_t(\omega)$ is a sample path of $L$. Put simply, $\pi_t(E, L)(\omega)$ counts the number of jumps in $E$ up to time $t$ that the sample path of $L$ at $\omega$ has in the set $E$. The family $\{\pi_t(\cdot, L)\colon t\geqslant 0\}$ is called \emph{the Poisson random measure} of $L$. The formula
$$\mu(E) = \mathsf E \,\left[\pi_t(E, L)\right] \quad (E\in \mathsf{Bor}\, B\setminus \{0\})$$
defines a Borel measure on $B\setminus \{0\}$, called the \emph{intensity measure} of $L$. For a \emph{bounded below} Borel set $E\subset B$, that is a set with ${\rm dist}(0,E) > 0$, and $f\in \mathcal{D}_B$ we set $$\hat{\pi}_t(E,f) = \pi_t(E, f) - t \cdot \mu(E)\quad (t\geqslant 0).$$ Whenever $E$ is bounded below, the expression
\begin{equation}\label{z1}Z^1_E(f,t):= \sum_{\begin{smallmatrix}0\leqslant s \leqslant t\\ \Delta f(s)\in E\end{smallmatrix}} \Delta f(s) = \int\limits_E u \,\pi_t({\rm d}u, f)\end{equation}
defines a function in $\mathcal{D}_{B,T}$. If $E$ is also bounded
\begin{equation}\label{z2}Z^2_E(f, t) : = Z^1_E(f, t) - \int\limits_E u\, \mu({\rm d}u)\end{equation}
defines an element in $\mathcal{D}_{B,T}$; we shall be primarily concerned with the $\mathcal{D}_{B,T}$-valued random variables of the form $Z^2_E(L, t)$.\bigskip
For every Borel set $E\subset B$ that is bounded below, the map $Z^1_E\colon \mathcal{D}_{B,T}\to \mathcal{D}_{B,T}$ is Borel. We are indebted to Mateusz Kwa\'snicki for sharing with us a direct proof of this fact (\cite{kwasnicki}); this argument replaces our previous, overly roundabout reasoning. Note that Borel measurability of $Z^2_E$ follows from Borel measurability of $Z^1_E$ as the former is a translation of the latter function. Let us then record these findings for the future reference.
\begin{lemma}\label{Z12Borel}Fix $T>0$ and $E\subset H$ be a non-empty Borel set that is bounded below. Then, the transformation $Z^1_E\colon \mathcal{D}_{B,T}\to \mathcal{D}_{B,T}$ given by \eqref{z1} is Borel. \smallskip
When $E$ is also bounded, the same is true for $Z^2_E\colon \mathcal{D}_{B,T}\to \mathcal{D}_{B,T}$ given by \eqref{z2} being a translation of $Z^1_E$.\end{lemma}
\begin{remark}\label{convergence}As observed by Applebaum (\cite[Section 4]{App2}), for any $t\geqslant 0$ and for every sequence $(E_n)_{n=1}^\infty$ of Borel sets in the unit ball $B_1$ of $B$ such that $E^c_n = B_1 \setminus E_n$ ($n\in \mathbb N$) is bounded below and the sets $E_n$ decrease to $\{0\}$, the random variables
$$Z^2_{E^c_n}(L, t) = \int\limits_{E^c_n} \hat{\pi}_t({\rm d}u, L),$$ converge almost surely as $n\to\infty$ to a random variable
$$Z^2_{B_1}(L,t):= \int\limits_{B_1} \hat{\pi}_t({\rm d}u, L)$$
(\cite[Section 2.3]{App1}; see also \cite[p.~80]{App2}). Moreover the above limit does not depend on the choice of $(E_n)_{n=1}^\infty$. \end{remark}
Under this framework, one recovers the L\'evy--It\^o decomposition for $B$-valued L\'evy processes (see \cite[Theorem 4.1]{App2}, \cite[Theorem 2.1]{Det}, and \cite[Theorem 6.3]{riedlevangaans}).
\begin{theorem}[L\'evy--It\^o decomposition]\label{levyito}Let $B$ be a separable Banach space and let $(L_t)_{t\geqslant 0}$ be a $B$-valued L\'evy process with the corresponding Poisson random measure $$\{\pi_t(\cdot, L)\colon t\geqslant 0\}$$ that has intensity measure $\mu$. Then there are $b\in B$ and a Wiener process $W_Q$ with a~(possibly degenerate) covariance operator $Q$ such that
$$L_t = bt + W_Q(t) + \int\limits_{B_1} \hat{\pi}_t({\rm d}u, L) + \int\limits_{B\setminus B_1} {\pi}_t({\rm d}u, L)\quad (t\geqslant 0). $$\end{theorem}
We term $$Z_t = \int\limits_{B_1} \hat{\pi}_t({\rm d}u, L) + \int\limits_{B\setminus B_1} {\pi}_t({\rm d}u, L) = Z^1_{B\setminus B_1}(L, t)+ Z^2_{B_1}(L, t) \quad (t\geqslant 0)$$ the \emph{jump part} of $L$. The Wiener process $W_Q$ and the jump part $Z$ are independent (\cite[Theorem 6.3]{riedlevangaans}).
\section{Proof of Theorem A}
We consider a~$H$-valued L\'evy process $L=(L_t)_{t\geqslant 0}$ on a probability space $(\Omega, \mathcal {F}, \mathsf P)$ that is expressed in the L\'evy--It\^o decomposition as $L_t = bt + W_t + Z_t$ ($t\in [0,T]$), where $b\in H$, $W=(W_t)_{t\geqslant 0}$ is a (possibly degenerate) Wiener process and $(Z_t)_{t\geqslant 0}$ is the jump part of $L$. Let $X,\tilde{X}\colon \Omega\to \mathcal{D}_{H,T}$ be the Ornstein--Uhlenbeck processes solving \eqref{equations}. Moreover, we consider solutions $X^J$ and $\tilde{X}^J$ to the auxiliary equations without the Wiener part:
\begin{equation}\label{equations2}\left\{\begin{array}{lcl}{\rm d}X^J_t &=& A X^J\, {\rm d}t + {\rm d}(L_t-W_t),\\ {\rm d}\tilde{X}^J_t &=& \tilde{A} \tilde{X}^J\, {\rm d}t + {\rm d}(L_t-W_t)\end{array}\right. \end{equation}
with the initial conditions $X^J(0)=\tilde{X}^J(0)=0$. Let us take a note that $X^J$ and $\tilde{X}^J$ (can be modified to) have c\`adl\`ag sample paths, which we will employ later.
\begin{proposition}The processes $X$ and $X^J + W$ have equivalent laws. \end{proposition}
\begin{proof}Let $(\mathcal{F}_t^W)_{t\geqslant 0}$ be the natural filtration of $W$ and let us consider the process
$$W^*_t = X_t - X^J_t = \int\limits_0^t A(X_s - X^J_s)\, {\rm d}s + W_t\quad \big(t\in [0,T]\big). $$
Then $W^*$ is the unique strong solution to ${\rm d}W^*_t = AW^*_t\, {\rm d}t + {\rm d}W_t$ with $W^*_0 = 0$ that is adapted to the filtration $(\mathcal{F}_t^W)_{t\geqslant 0}$. \smallskip
Since $Z$ and $W$ are independent processes, so are $Z$ and $W^*$. By Girsanov's theorem (see \cite[Theorem 1]{Loges} for a version of Girsanov's theorem for $H$-valued processes; this is where we apply the hypothesis that the eigenvalues of the covariance operator are strictly positive as well as \eqref{technical} in the case where $H$ is innfinite-dimensional), there is a probability measure $\tilde{\mathsf P}$ for which $W^*$ is a Wiener process on $(\Omega, \mathcal F, \tilde{\mathsf P})$ with the same covariance operator as $W$ and so the laws $\mathsf P_W$ and $\mathsf P_{W^*}$ are equivalent. The processes $X^J$ and $W$ are independent. Let us observe that the processes $X^J$ and $W^*$ are independent too. Indeed, $W^*$ being adapted to $(\mathcal{F}_t^W)_{t\geqslant 0}$ is $\mathcal{F}^W$-measurable, and thus independent from $(X^J_t)_{t\geqslant 0}$ (see also \cite[Theorem II.6.3]{IW}).\smallskip
We are now in a position to apply Lemma~\ref{Lemma2new} to conclude that the laws $\mathsf{P}_{(X^J, W)}$ and $\mathsf{P}_{(X^J, W^*)}$ are equivalent. Consequently, the laws $\mathsf{P}_{X^J+W}$ and $\mathsf{P}_{X^J+W^*} = \mathsf P_X$ are equivalent as well, which completes the proof. \end{proof}
Thus, in order to establish Theorem A, it is enough to prove the following proposition.
\begin{proposition}\label{prop1}The processes $X^J +W$ and $\tilde{X}^J + W$ have equivalent laws. \end{proposition}
\begin{proof}We will demonstrate that the law of $X^J +W$ is absolutely continuous with respect to the law of $\tilde{X}^J +W$ as the other direction would be completely analogous. \smallskip
For given $R > 0$ consider the set
$$\Omega_R:= \{\omega \in \Omega\colon \sup_{t\in [0,T]} \|AX_t(\omega) - \tilde{A}\tilde{X}_t(\omega)\| \leqslant R \} $$
and note that $\Omega_R\in \mathcal{F}$ (\emph{cf.}~Proposition~\ref{usefulprojections}). We may then consider the `truncated' process
$$W^R_t = W_t + \int\limits_0^t \big(AX_s - \tilde{A}\tilde{X}_s\big)\cdot \mathds{1}_{\Omega_R}\, {\rm d}s\quad (t\in [0,T]). $$
By hypothesis \eqref{technical}, we may apply Girsanov's theorem, so there is a probability measure $\mathsf P^R$ equivalent to $\mathsf P$, for which $W^R$ is a Wiener process on $(\Omega, \mathcal F,\mathsf P^R)$ with the same covariance operator as $W$; in particular the laws $\mathsf P_W$ and $\mathsf P_{W^R}$ are equivalent. Arguing as in the proof of Proposition~\ref{prop1}, we infer that the processes $X$ and $W^R$ are independent. By Lemma~\ref{Lemma2new} applied to $(\tilde{X}^J, W)$ and $(\tilde{X}^J, W^R$), we deduce that the processes $\tilde{X}^J+W$ and $\tilde{X}^J +W^R$ have equivalent laws.\smallskip
Since
$$X^J_t - \tilde{X}^J_t = \int\limits_0^t \big( AX_s - \tilde{A}\tilde{X}_s\big) \,{\rm d}s $$
we see that $\tilde{X}_t^J+W^R_t$ and $X^J_t + W_t$ agree on the set $\Omega_R$. It follows that for every set $E\in \mathsf{Bor}\, \mathcal{D}_{H,T}$ the condition $\mathsf P(\tilde{X}_t^J +W^R_t \in E)=0$ implies that for all numbers $R>0$ we have $\mathsf{P}\big( \Omega_R\cap (\tilde{X}^J_t + W)\big)=0$. The processes $X^J, \tilde{X}^J$ have c\`adl\`ag sample paths, which implies that they are bounded on bounded intervals. In particular, for any $\omega\in \Omega$ the function
$$t\mapsto AX_t(\omega) - \tilde{A}\tilde{X}_t(\omega) \quad (t\in [0,T]),$$
is bounded, which means that $\Omega = \bigcup_{R>0}\Omega_R$. Thus $\mathsf{P}_{{X}^J + W}\ll \mathsf{P}_{\tilde{X}^J + W}$. \end{proof}
\section{Proof of Theorem B}
This time we consider a purely jump L\'evy process $L$, that is $L = Z$ using the notation of Theorem~\ref{levyito}. Let $X^J, \tilde{X}^J\colon \Omega\to \mathcal{D}_{H,T}$ be solutions to \eqref{equations}, which now take the form
\begin{equation}\left\{\begin{array}{lcl}\label{equations3}{\rm d}X^J_t& =& A X^J_t\, {\rm d}t + {\rm d}Z_t,\\ {\rm d}\tilde{X}^J_t &= &\tilde{A} \tilde{X^J}_t\, {\rm d}t + {\rm d}{Z}_t\end{array}\right. \quad (t\in [0,T])\end{equation}
with the initial conditions $X^J_0 = \tilde{X}^J_0 = 0$.
\begin{lemma}For any $t\in [0,T]$ and $E\in \mathsf{Bor}\, H\setminus \{0\}$ we have
\begin{equation}\label{XJZ}\mathsf P\big(\pi_t(E,X_J ) = \pi_t(E,Z) \big) = 1.\end{equation} \end{lemma}
\begin{proof} Since
$$X^J_t = A \int\limits_0^t X^J_s\, {\rm d} s + Z_t,$$
we have
$$\begin{array}{lcl}\{s\leqslant t\colon X^J_s - X^J_{s-}\in E \} &=& \{s\leqslant t\colon A \int\limits_0^s X^J_u\, {\rm d} u + Z_s - A\int\limits_0^{s-} X^J_u\, {\rm d} u - Z_{s-}\in E\} \\
& = & \{s\leqslant t\colon Z_s - Z_{s-}\in E\} \end{array}$$
almost surely. \end{proof}
We may then derive the following conclusion.
\begin{corollary}\label{convergence2}
For every non-negative integer $k$, $t\in [0,T]$, and $E\in \mathsf{Bor}\, H\setminus \{0\}$ we have
$$\mathsf P_{X^J}\big(\{f\in \mathcal{D}_{H,T}\colon \pi_t(E,f) = k\} \big) = \mathsf P\big(\pi_t(E,X^J ) = k\big) = \mathsf P\big(\pi_t(E,Z ) = k\big).$$
In particular, for a sequence $(E_n)_{n=1}^\infty$ of Borel sets in the unit ball $H$, as in the statement of Remark~\ref{convergence}, the random variables $Z^2_{E^c_n}(X^J, t)$ converge almost surely to $Z^2_{H_1}(Z, t)$ as $n\to \infty$ $(t\in [0,T])$.\end{corollary}
For $t\in [0,T]$ and $f\in \mathcal{D}_{H,T}$ we set $$S(f,t) = \int\limits_0^t f(s)\,{\rm d}s.$$ Consequently, the assignment $f\mapsto S(f, \cdot)$ defines a function $\mathcal{D}_{H,T}\to \mathcal{D}_{H,T}$ as it takes continuous values.
\begin{lemma}\label{ASBorel}For a bounded linear operator $V\colon H\to H$, the assignment $\mathcal{D}_{H,T}\to \mathcal{D}_{H,T}$ given by $f\mapsto VS(f, \cdot)$ $(f\in \mathcal{D}_{H,T})$ is Borel. \end{lemma}
\begin{proof}Observe that if a sequence $(f_n)_{n=1}^\infty$ in $\mathcal{D}_{H,T}$ converges to some $f\in \mathcal{D}_{H,T}$, then for almost all $s\in [0,T]$ (with respect to the Lebesgue measure) we have $f_n(s)\to f(s)$ as $n\to\infty$.
Since the sequence $(f_n)_{n=1}^\infty$ is bounded with respect to the supremum norm, by the dominated convergence theorem (for Bochner-integrable functions) we conclude that for all $t\in [0,T]$
$$\int\limits_0^t f_n(s)\,{\rm d}s \to \int\limits_0^t f(s)\,{\rm d}s $$
as $n\to\infty$. Consequently, the map $\Phi(f)= VS(\cdot, f)$ ($f\in \mathcal{D}_{H,T}$) is Borel-measurable because it is continuous as a map from $\mathcal{D}_{H,T}$ with the Skorohod topology to $\mathcal{D}_{H,T}$ (actually even to $C([0,T], H)$) endowed with the topology of pointwise convergence. Indeed, by Proposition~\ref{usefulprojections}, $\Phi$ is Borel because for each $t\in [0,T]$, the map $\Phi(\cdot)(t)\colon \mathcal{D}_{H,T}\to H$ is continuous, hence Borel. \end{proof}
\begin{definition}For a bounded linear operator $V\colon H\to H$ we define
$$\Xi_V = \big\{f\in \mathcal{D}_{H,T}\colon \lim_{n\to\infty} \|f(t) - VS(f,t) -Z^1_{H\setminus H_1}(f,t) - Z^2_{E^c_n}(f, t)\|=0\text{ for all }t\in [0,T] \big\}.$$
\end{definition}
By Lemma~\ref{ASBorel}, the assignment $f\mapsto VS(f, \cdot)$ ($f\in \mathcal{D}_{H,T}$) is Borel measurable. Similarly, by Lemma~\ref{Z12Borel}, the assignments $f\mapsto Z^1_E(f, \cdot), Z^2_E(f, \cdot)$ ($f\in \mathcal{D}_{H,T}$) are Borel measurable for every bounded below set $E\subset H$ (in the latter case, $E$ is assumed to be additionally bounded). Let us invoke Proposition~\ref{usefulprojections} to see that in order to establish measurability of $\Xi_V$ it is enough to show measurability of $p_t[\Xi_V]$ for each $t\in [0,T]$. We have thus proved the following proposition.
\begin{proposition}For a bounded linear operator $V\colon H\to H$, the set $\Xi_V$ is Borel with respect to the Skorohod topology on $\mathcal{D}_{H,T}$.\end{proposition}
\begin{lemma}\label{lemma43}$\mathsf P_{X^J}(\Xi_A) = 1 = \mathsf{P}_{\tilde{X}^J}(\Xi_{\tilde{A}}).$ \end{lemma}
\begin{proof}By Corollary~\ref{convergence2}, $Z^1_{H\setminus H_1}(X^J,t) + Z^2_{E^c_n}(X^J, t)$ converges almost surely as $n\to\infty$ to $Z(L,t)$ ($t\in [0,T]$). Thus, $\mathsf P_{X^J}(\Xi_A)$ is equal to
$$\begin{array}{lcl}
\quad \mathsf P\big(X^J_t - AS(X^J, t) = Z(X^J, t)\; (t\in [0,T]) \big) = \mathsf P\big(X^J_t - A\int\limits_0^t X^J_s{\rm d}s = Z_t\; (t\in [0,T]) \big)= 1.\end{array} $$
The same proof applies for $\Xi_{\tilde{A}}$.\end{proof}
We are now ready to prove Theorem B.
\begin{proof}[Proof of Theorem B]Assume that $\mathsf P_{X^J} \ll \mathsf P_{\tilde{X}^J}$. By Lemma~\ref{lemma43}, $\mathsf P_{X^J}(\Xi_A) = 1 = \mathsf{P}_{\tilde{X}^J}(\Xi_{\tilde{A}})$. Consequently, by absolute continuity we must have $\mathsf P_{X^J}(\Xi_{\tilde{A}})=1$, which means that
$$X^J_t = \tilde{A} \int\limits_0^t X^J_s\,{\rm d}s + Z_t \quad \big(t\in [0,T]\big)$$
almost surely. We have thus proved that $X^J$ solves the stochastic differential equation ${\rm d}Y_t = AY_t {\rm d}t + {\rm d}Z_t$, so by the uniqueness of solutions, $X^J = \tilde{X}^J$ almost surely.\end{proof}
\subsection{Closing remarks}
In the case of stochastic processes in infinite dimensions it is customary to work with generators of infinitesimal semigroups rather than merely bounded linear operators. However, the proof methods employed in this paper required the operators $A$ and $\tilde{A}$ appearing in \eqref{equations} to be bounded (\emph{cf}.~the proofs of Proposition~\ref{prop1} and Lemma~\ref{ASBorel}). It is thus natural to ask whether Theorems A and B have their counterparts in the setting of generators of infinitesimal semigroups too. From this point of view, it is also desirable to investigate classes of those Feller processes for which analogous results can be established.
\subsection*{Acknowledgements} We wish to express our gratitude to Mateusz Kwa\'snicki (Wroc{\l}aw) for sharing with us a direct proof of Lemma~\ref{Z12Borel}. Furthermore, we are greatly indebted to the anonymous referee for spotting the need for the technical assumption \eqref{technical} that is indeed required when the underlying Hilbert space is infinite-dimensional.
|
{
"timestamp": "2019-05-14T02:39:52",
"yymm": "1803",
"arxiv_id": "1803.02655",
"language": "en",
"url": "https://arxiv.org/abs/1803.02655"
}
|
\section{Introduction}
The Kronecker product [1] generates fractals if it applies to the same matrix [1-3], i.e., it is the \emph{self Kronecker product}.
The origin, nature and generating technique related to the Kronecker product based (KPB) fractals are explained
in more details in [2,3]. A lot of well-known and new fractal samples are shown and KPB fractals generators
are offered both in JavaScript and in R [2,3].
A simple method of calculating the Hausdorff-Besicovitch dimension (HBD) of the Kronecker Product based
fractals was discovered and is presented together with a compact R script realizing it.
The proposed new formula is based on traditionally used values of the number of self-similar objects and
the scale factor that are now calculated using appropriate values of both the initial fractal matrix and the second order resultant matrix.
This method is reliable and producing dimensions equal to many already determined values of well-known canonical fractals.
It should be stressed in the beginning that this method only works and could be applied to any fractal matrix (defined below)
to determine KPB fractal dimension.
\section{Generating and plotting fractals}
Let's start with a few definitions that are related and applied only to matrices used for fractal images generation using
a \emph{self Kronecker product} (a.k.a. the \emph{Kronecker power}).
\begin{defn}
\textbf{The Kronecker power} $n$ of a matrix $M$ is defined as\\
$M \otimes M \otimes ... \otimes M$, i.e., n times self Kronecker product. In short: $M^{n\otimes}$.
\end{defn}
\begin{rem}
It is often said that matrix $M$ has an order (or level) $n$ if it has the Kronecker power $n$. Same order is used for the generated fractal image, related to matrix $M$.
\end{rem}
Note: a matrix declared an ``initial" has the order 1, even if it is, in fact, a resultant order $N$ matrix generated previously.\\
For clarity, a simple notation is used to stress the name of the fractal (using abbreviation) and its order (in the form `oN').
E.g., for the Sierpinski triangle fractal (STF) order 7 -- both generated matrix and plotted picture would be denoted as STFo7.
Here is another important definition:
\begin{defn}
A matrix containing zeros and ones will be called the \textbf{fractal matrix}
if it has at least one zero and one number one.
\end{defn}
Some initial matrices (even fractal ones) can't, produce fractals. So, here is a definition for them:
\begin{defn}
A matrix is called \textbf{degraded} if it contains all zeros or only one number one.
\end{defn}
To generate and plot KPB fractals R helper functions from [3] will be use. It should be explained that R helper
functions just simplifying and standardizing generating and plotting for fractal matrices. But, in fact, R
language has a built-in operator $\%x\%$ for the Kronecker product, i.e., for any matrix $M$ it could be used as $M\%x\%M$.
To start, two \emph{Sierpinski canonical fractals} -- triangle and carpet -- are shown in the figures below.
\begin{figure}[!ht]
\begin{minipage}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{STFo7.png}
\caption{STF, order 7}
\end{minipage}%
\begin{minipage}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{SCFo5.png}
\caption{SCF, order 5}
\end{minipage}%
\end{figure}
The initial fractal matrices for these two fractals and how to generate and plot them using
helper functions from [3] is shown in the R script \#1 below.
\begin{exm} R script \#1:\\
\begin{verbatim}
#### R SCRIPT BEGIN
## Initial matrix for Sierpinski triangle fractal (STF)
STFo1 <- matrix(c(1,1, 0,1), nrow=2, ncol=2, byrow=TRUE);
## Initial matrix for Sierpinski carpet fractal (SCF)
SCFo1 <- matrix(c(1,1,1, 1,0,1, (1,1,1), nrow=3, ncol=3, byrow=TRUE);
## Generate and plot fractals
gpKronFractal(STFo1, 7, "STFo7", "navy", "Sierpinski triangle fractal");
gpKronFractal(SCFo1, 5, "SCFo5", "maroon", "Sierpinski carpet fractal");
#### R SCRIPT END
\end{verbatim}
\end{exm}
Now, only matrices of the second order will be built (without plotting) in R.
\begin{exm} R script \#2 and its output:\\
\begin{verbatim}
#### R SCRIPT & OUTPUT BEGIN
## Generate 2-nd order fractal matrices
## Initial matrix for Sierpinski triangle fractal (STF)
STFo1 <- matrix(c(1,1, 0,1), nrow=2, ncol=2, byrow=TRUE);
STFo1
STFo2 = STFo
STFo2
## Initial matrix for Sierpinski carpet fractal (SCF)
SCFo1 <- matrix(c(1,1,1, 1,0,1, 1,1,1), nrow=3, ncol=3, byrow=TRUE);
SCFo1
SCFo2 = SCFo
SCFo2
## OUTPUT BEGIN:
> ## Generate 2-nd order fractal matrices
> ## Initial matrix for Sierpinski triangle fractal (STF)
> STFo1 <- matrix(c(1,1, 0,1), nrow=2, ncol=2, byrow=TRUE);
> STFo1
[,1] [,2]
[1,] 1 1
[2,] 0 1
> STFo2 = STFo
> STFo2
[,1] [,2] [,3] [,4]
[1,] 1 1 1 1
[2,] 0 1 0 1
[3,] 0 0 1 1
[4,] 0 0 0 1
> ## Initial matrix for Sierpinski carpet fractal (SCF)
> SCFo1 <- matrix(c(1,1,1, 1,0,1, 1,1,1), nrow=3, ncol=3, byrow=TRUE);
> SCFo1
[,1] [,2] [,3]
[1,] 1 1 1
[2,] 1 0 1
[3,] 1 1 1
> SCFo2 = SCFo
> SCFo2
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] 1 1 1 1 1 1 1 1 1
[2,] 1 0 1 1 0 1 1 0 1
[3,] 1 1 1 1 1 1 1 1 1
[4,] 1 1 1 0 0 0 1 1 1
[5,] 1 0 1 0 0 0 1 0 1
[6,] 1 1 1 0 0 0 1 1 1
[7,] 1 1 1 1 1 1 1 1 1
[8,] 1 0 1 1 0 1 1 0 1
[9,] 1 1 1 1 1 1 1 1 1
#### R SCRIPT & OUTPUT END
\end{verbatim}
\end{exm}
Although R script output is very clear, in the Fig. 3, 4 below find the initial, second and third order matrices
plotted using big square ``dots". This makes it even easier to determine the basic figure, counting and scaling.
\begin{figure}[!ht]
\centering
\begin{minipage}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{STo123.png}
\caption{Schemes of order 1, 2 and 3 matrices for STF}
\end{minipage}%
\begin{minipage}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{SCo12.png}
\caption{Schemes of order 1 and 2 matrices for SCF}
\end{minipage}%
\end{figure}
\section{Calculating Hausdorff-Besicovitch dimension}
A very simple formula to calculate Hausdorff-Besicovitch dimension of the fractal $F$ was introduced in [5] and further
explain in [6,7] and it fits our goal:
\[
\dim_{\mathrm{HB}} F = log N / log S,
\]
\noindent where $N$ - is the number of self-similar objects; $S$ - is the scale factor.
Let's apply the above formula to the Sierpinski triangle fractal.
Looking at Fig. 3 we can find that in the order 2 matrix the number of self-similar objects (i.e., objects equal to
the basic one in the initial matrix) $N=3$,
and the scale factor $S=2$. So, $\dim_{\mathrm{HB}} STF = log3/log2 = 1.584963$.
Applying the same formula to the Sierpinski carpet fractal (see Fig. 4) we have: the number of self-similar
objects $N=8$, the scale factor $S=3$ and\\ $\dim_{\mathrm{HB}} SCF = log8/log3 = 1.892789$.
As result of intensive testing, it was discovered that in case of KPB fractals:
\begin{itemize}
\item $N$ is the ratio $d2/d1$, where $d2$ is a number of dots in the resultant matrix of the second order
and $d1$ is a number of dots in the initial matrix.
\item $S$ is the ratio $m2/m1$, where $m2$ is a number of rows in the resultant matrix of the second order
and $m1$ is a number of rows in the initial matrix.
\end{itemize}
The new simple formula to calculate Hausdorff-Besicovitch dimension of the fractal $F$ is the following: \\
\[
\dim_{\mathrm{HB}} F = log(d2/d1) / log(m2/m1) ,
\]
\noindent where $d1, d2, m1, m2$ - are, accordingly, numbers of dots and rows in the initial and the second order matrices.
Such amazing simplicity could be explained by the fact that the Kronecker product is building resultant block-matrix in a uniform manner, so,
these two ratios are always describing correctly both the number of self-similar objects and the scale factor.
Moreover, these ratios are the same between other orders, e.g., between the second and third orders, etc.
Based on this discovered formula, -- a very simple R script was created. Just 7 lines of R code:
\emph{R script dimHB4kpf.R}
\begin{verbatim}
## dimHB4kpf.R.txt 3/11/17 aev
## Hausdorff-Besicovitch dimension for Kronecker product based
## fractals
## Note: only for Kronecker product based fractals created from
## the fractal matrix!
## dimHB4kpf(mat, ttl) -
## where: mat - initial matrix (filled with 0/1); ttl - title.
dimHB4kpf <- function(mat, ttl="") {
m1 = nrow(mat); dn1 = sum(mat!=0);
matr = ma
m2 = nrow(matr); dn2 = sum(matr!=0);
dimHB = log(dn2/dn1, m2/m1);
cat(" *** dimHB:", dimHB, ttl, "\n");
}
\end{verbatim}
Presented below testing R script \#3 is pretty simple. It includes two lines of code for
each of 7 selected well-known fractals.
\begin{exm} R script \#3:\\
\begin{verbatim}
## Testing dimHB for 7 well-known fractals:
STF <- matrix(c(1,1, 0,1), ncol=2, nrow=2, byrow=TRUE);
dimHB4kpf(STF, "'Sierpinski triangle'")
SCF <- matrix(c(1,1,1, 1,0,1, 1,1,1), ncol=3, nrow=3, byrow=TRUE);
dimHB4kpf(SCF, "'Sierpinski carpet'")
PTFm3 <- matrix(c(1,1,1, 0,1,1, 0,0,1), ncol=3, nrow=3, byrow=TRUE);
dimHB4kpf(PTFm3, "'Pascal triangle modulo 3'")
PTFm5 <- matrix(c(1,1,1,1,1, 0,1,1,1,1, 0,0,1,1,1, 0,0,0,1,1, 0,0,0,0,1),
ncol=5, nrow=5, byrow=TRUE);
dimHB4kpf(PTFm5, "'Pascal triangle modulo 5'")
VF <- matrix(c(0,1,0, 1,1,1, 0,1,0), ncol=3, nrow=3, byrow=TRUE);
dimHB4kpf(VF, "'Vicsek fractal'")
HGF <- matrix(c(1,1,0, 1,1,1, 0,1,1), ncol=3, nrow=3, byrow=TRUE);
dimHB4kpf(HGF, "'Hexagon/Hexaflake fractal'")
BF <- matrix(c(1,0,1, 0,1,0, 1,0,1), ncol=3, nrow=3, byrow=TRUE);
dimHB4kpf(BF, "'Box fractal'")
\end{verbatim}
\end{exm}
Note: last 5 fractals are shown in the Fig. 5-9 below.
\newpage
\begin{figure}[!ht]
\begin{minipage}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{PTFm3o5.png}
\caption{PTFm3o5}
\end{minipage}%
\begin{minipage}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{PTFm5o4.png}
\caption{PTFm5o4}
\end{minipage}%
\end{figure}
\begin{figure}[!ht]
\begin{minipage}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{VFo5.png}
\caption{VFo5}
\end{minipage}%
\begin{minipage}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{HGFo5.png}
\caption{HGFo5}
\end{minipage}%
\begin{minipage}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{BFo5.png}
\caption{BFo5}
\end{minipage}%
\end{figure}
Results of the testing script are presented below, and they are equal to results shown in [8] for
the same fractals.
\begin{exm} R script \#3 output:\\
\begin{verbatim}
## OUTPUT BEGIN:
> ## Testing dimHB for 7 well-known fractals:
> STF <- matrix(c(1,1, 0,1), ncol=2, nrow=2, byrow=TRUE);
> dimHB4kpf(STF, "'Sierpinski triangle'")
*** dimHB: 1.584963 'Sierpinski triangle'
> SCF <- matrix(c(1,1,1, 1,0,1, 1,1,1), ncol=3, nrow=3, byrow=TRUE);
> dimHB4kpf(SCF, "'Sierpinski carpet'")
*** dimHB: 1.892789 'Sierpinski carpet'
> PTFm3 <- matrix(c(1,1,1, 0,1,1, 0,0,1), ncol=3, nrow=3, byrow=TRUE);
> dimHB4kpf(PTFm3, "'Pascal triangle modulo 3'")
*** dimHB: 1.63093 'Pascal triangle modulo 3'
> PTFm5 <- matrix(c(1,1,1,1,1, 0,1,1,1,1, 0,0,1,1,1, 0,0,0,1,1,
+ 0,0,0,0,1), ncol=5, nrow=5, byrow=TRUE);
> dimHB4kpf(PTFm5, "'Pascal triangle modulo 5'")
*** dimHB: 1.682606 'Pascal triangle modulo 5'
> VF <- matrix(c(0,1,0, 1,1,1, 0,1,0), ncol=3, nrow=3, byrow=TRUE);
> dimHB4kpf(VF, "'Vicsek fractal'")
*** dimHB: 1.464974 'Vicsek fractal'
> HGF <- matrix(c(1,1,0, 1,1,1, 0,1,1), ncol=3, nrow=3, byrow=TRUE);
> dimHB4kpf(HGF, "'Hexagon/Hexaflake fractal'")
*** dimHB: 1.771244 'Hexagon/Hexaflake fractal'
> BF <- matrix(c(1,0,1, 0,1,0, 1,0,1), ncol=3, nrow=3, byrow=TRUE);
> dimHB4kpf(BF, "'Box fractal'")
*** dimHB: 1.464974 'Box fractal'
## OUTPUT END
\end{verbatim}
\end{exm}
\section{Conclusion}
The discovered new simple formula to calculate the Hausdorff-Besicovitch dimension together with
offered compact R script (dimHB4kpf.R) can help studies related to KPB fractals and even to any fractals in general.
For example, if any fractal can be simulated using matrix presentation and the Kronecker power,
then its Hausdorff-Besicovitch dimension can be calculated instantly, - helping understand many peculiarities of
this fractal.
Actually, it was done already here for Hexagon fractal (see Fig. 8). Our plotted Hexagon fractal is very similar to
the Hexaflake in [8], but it looks rather as a distorted Hexaflake. Anyway, they both have the equal dimension.
Here is another appropriate and interesting example related to the Rauzy gasket (RG) described in [9]. The authors of [9]
proved ``that the Rauzy gasket is homeomorphic to the usual Sierpinski gasket". If compare the Fig. 1 in the mentioned
article (which is presented the Rauzy gasket) and the Fig. 1 above (presented Sierpinski triangle fractal, aka
Sierpinski gasket), - it is very clear that the RG is looking like a distorted version of STF, and it is actually copying
the structure of STF.
It was already calculated that $\dim_{\mathrm{HB}} STF = 1.584963$, so $\dim_{\mathrm{HB}} RG$
is the same (at least, expected to be almost the same).
This value is in accordance with the fact proved in [10]: ``the Hausdorff dimension of the Rauzy gasket is less than 2".
One can look at this value as a reasonable approximation, until the precise value would be found, or it would be proved
that dimensions of both gaskets are equal.
|
{
"timestamp": "2018-03-08T02:11:13",
"yymm": "1803",
"arxiv_id": "1803.02766",
"language": "en",
"url": "https://arxiv.org/abs/1803.02766"
}
|
\section{Introduction}
We are here concerned with the problem of understanding the genericity of irreducible smooth representations of a general linear group over a $p$-adic field.
Let $G$ be a reductive $p$-adic group. Recall that a smooth irreducible representation $\pi$ of $G$ is called generic if $\pi$ appears in $\mathrm{Ind}_U^G \psi$ (i.e. admits a Whittaker model), where $\mathrm{Ind}$ denotes induction and $\psi$ is a nondegenerate character of a maximal unipotent subgroup $U$ of $G$.
We will start by recalling a few facts about the category of smooth representations. Let $C$ be an algebraically closed field of characteristic zero. Let $\mathcal{R}(G)$ be the category of all smooth $C$-representations of $G$. The Bernstein decomposition (\cite{MR771671}) expresses the category of smooth $C$-valued representations of $G$ as the product of certain indecomposable full subcategories, called Bernstein components. Those components are parametrized by the inertial classes, whose definition we now recall. Consider the set of pairs $(M, \rho)$, with $M$ a Levi subgroup of $G$ and $\rho$ an irreducible supercuspidal representation of $M$. We say that two pairs $(M_1, \rho_1)$ and $(M_2, \rho_2)$ are inertially equivalent if and only if there are $g \in G$ and an unramified character $\chi$ of $M_2$ such that $M_2=M_1^{g}$ and $\rho_2 \simeq \rho_1^g \otimes\chi$, where $M_1^g:=g^{-1}M_1g$ and $\rho_1^g(x) = \rho_1(gxg^{-1})$, for $x \in M_1^g$. The equivalence class of $(M, \rho)$ will be denoted by $[M,\rho]_{G}$, and is called \textit{inertial class}. The set of inertial classes will be denoted by $\mathcal{B}(G)$.
We denote by $i_{P}^{G} : \mathcal{R}(M) \longrightarrow \mathcal{R}(G)$ the normalized parabolic induction functor, where $P=MN$ is a parabolic subgroup of $G$ with Levi subgroup $M$. Let $\Omega:=[M,\rho]_{G}$ be an inertial equivalence class, where $\rho$ is a supercuspidal representation of $M$. To $\Omega$ we may associate a full subcategory $\mathcal{R}^{\Omega}(G)$ of $\mathcal{R}(G)$, such that the representation $(\pi,V)$ is an object of $\mathcal{R}^{\Omega}(G)$ if and only if every irreducible $G$-subquotient $\pi_{0}$ of $\pi$ appears as a composition factor of $i_{P}^{G}(\rho \otimes \omega)$ for $\omega$ some unramified character of $M$ and $P$ some parabolic subgroup of $G$ with Levi factor $M$. The category $\mathcal{R}^{\Omega}(G)$ is called a Bernstein component of $\mathcal{R}(G)$. According to \cite{MR771671}, the Bernstein decomposition is written as, $\mathcal{R}(G) = \prod_{\Omega \in \mathcal{B}(G)} \mathcal{R}^{\Omega}(G)$. It follows that if we want to understand the category $\mathcal{R}(G)$, it is enough to restrict our attention to the Bernstein components. This can be done via the theory of types. This theory allows us to parametrize all the irreducible representations of $G$ up to inertial equivalence using irreducible representations of compact open subgroups of $G$. Let $J$ be a compact open subgroup of $G$ and let $\lambda$ be an irreducible representation of $J$. We say that $(J, \lambda)$ is an $\Omega$-type, if for $(\pi,V)$ a representation of $G$, the representation $(\pi,V)$ is an object of $\mathcal{R}^{\Omega}(G))$ if and only if $V$ is generated by its $\lambda$-isotypical space $V^{\lambda}$ as a $G$-representation.
Let $F$ be a local non-archimedean field. For $G=GL_n(F)$, types can be constructed (cf. \cite{MR1204652}, \cite{MR1643417} and \cite{MR1711578}) for every Bernstein component. The simplest example of a type is $(I, 1)$, where $I$ is the standard Iwahori subgroup of $G$ and $1$ is the trivial representation. In this case $\Omega = [T,1]_G$, where $T$ is the subgroup of diagonal matrices and $1$ denotes the trivial representation of $T$. We will refer to this example as the Iwahori case.
Fix $K$ a maximal compact subgroup of $G=GL_n(F)$. Given a Bushnell--Kutzko type $(J, \lambda)$ with $J$ contained in $K$, in \cite[section 6]{MR1728541} (just above Proposition~2) the authors define irreducible $K$-representations $\sigma_{\mathcal{P}}(\lambda)$, where $\mathcal{P}$ belongs to some partially ordered set (cf. \cite[section 2]{MR1728541}). One has the decomposition :
\begin{equation}\label{decomp}
\mathrm{Ind}_{J}^{K} \lambda = \bigoplus_{\mathcal{P}} \sigma_{\mathcal{P}}(\lambda)^{\oplus m_{\mathcal{P},\lambda}},
\end{equation}
\noindent where the summation runs over the same partially ordered set as above. The integers $m_{\mathcal{P},\lambda}$ are finite and we call $m_{\mathcal{P},\lambda}$ the multiplicity of $\sigma_{\mathcal{P}}(\lambda)$. Let $\mathcal{P}_{max}$ be the maximal elements and let $\mathcal{P}_{min}$ the minimal one. Define $\sigma_{max}(\lambda):=\sigma_{\mathcal{P}_{max}}(\lambda)$ and $\sigma_{min}(\lambda):=\sigma_{\mathcal{P}_{min}}(\lambda)$. Both $K$-representations $\sigma_{max}(\lambda)$ and $\sigma_{min}(\lambda)$ occur in $\mathrm{Ind}_{J}^{K} \lambda$ with multiplicity 1. In the Iwahori case those representations have a very simple description. Indeed, $\sigma_{min}(\lambda)$ is the inflation of the Steinberg representation of $GL_n(k_F)$ to $K$ and $\sigma_{max}(\lambda)$ is the trivial representation.
Having introduced the main notation of this paper we may now state our main theorem:
\begin{thm}\label{p}
Let $\pi$ be an absolutely irreducible representation in the Bernstein component $\Omega$ and let $(J,\lambda)$ an $\Omega$-type. Then
$$
\dim_C \mathrm{Hom}(\sigma_{min}(\lambda), \pi) =
\begin{cases}
1\ \text{if}\ \pi \text{is a generic object of }\ \mathcal{R}^{\Omega}(G),\\
0\ \text{otherwise.}
\end{cases}
$$
\end{thm}
Theorem \ref{p} shows that the representation $\sigma_{min}(\lambda)$ has a very special role. One can wonder about other $\sigma_{\mathcal{P}}(\lambda)$'s. There is a recent result by Jack Shotton in that direction. He proves \cite[Thm.3.7]{MR3769675} that by modifying the proof of \cite[Proposition 2 Section 6]{MR1728541} and \cite[Proposition 6.5.3]{MR2656025} in the tempered case, one gets the same result in the generic case. In the author's thesis the result \cite[Thm.3.7]{MR3769675} was proven independently but with a different method. First using the theory of types of Bushnell--Kutzko, we reduce the statement to the Iwahori case. Then, in the Iwahori case, we use the results of Rogawski \cite{MR782228} on modules over Iwahori--Hecke algebra. In this case the proof relies on some easy combinatorics on partitions.
The multiplicity one statement can fail for other $\sigma_{\mathcal{P}}(\lambda)$'s. For example, consider the Iwahori case with $n=3$, i.e. $G=GL_3(F)$. Take $\pi=i_B^G(1 \otimes \chi_1 \otimes \chi_2)$, where $B$ is the subgroup of $G$ of upper triangular matrices, $1$ the trivial character and $\chi_1$, $\chi_2$ unramified characters such that $\chi_1.\chi_2^{-1} \neq |.|^{\pm 1}$. Then, writing $\sigma_{2,1}$ for the summand of $\mathrm{Ind}_I^K 1$ corresponding to the partition $(2,1)$ (see section 2), one can easily verify that $\dim \mathrm{Hom}_K(\sigma_{2,1}, \pi)=2$.
Let us say a few words about the proof of Theorem \ref{p}. First we use one of the main results of \cite{MR1711578}, which asserts that the Hecke algebra $\mathcal{H}(G,\lambda)$ is naturally isomorphic to a tensor product of affine Hecke algebras of type A. Moreover it is shown in \cite{MR1204652} that any Hecke algebra of a simple type is isomorphic to an affine Hecke algebra of type A. In this manner we can reduce the statement about irreducible representations of general type to the Iwahori case. It was pointed out to us, recently, by Peter Schneider that the Iwahori case was already treated by \cite{MR1915088}. This allowed us to simplify a little the original proof in the author's thesis.
Finally let us observe that to the best of our knowledge Theorem \ref{p} and \cite[Thm.3.7]{MR3769675} do not have an analogue for all reductive groups, because the crucial ingredient in the proofs is the tensor product decomposition of the Hecke algebra $\mathcal{H}(G,\lambda)$ and the existence of types, proven by Bushnell--Kutzko in \cite{MR1711578}. Indeed results of \cite{MR1204652}, \cite{MR1643417} and \cite{MR1711578} allow us to transfer the general situation to the Iwahori case, where the proofs are simpler. However we believe that those results should generalize easily to reductive groups with $A_n$ root system. It would be interesting to investigate the case of other reductive groups.
\subsection*{Notation}
For an arbitrary local non-archimedean field $L$, let $\mathcal{O}_L$ be its ring of integers and $k_{L}$ the residue field. We also choose a uniformizer $\varpi_L \in \mathcal{O}_L$. From now on fix $F$ a local non-archimedean field and $G=GL_n(F)$.
Recall that all the representations have their coefficients in an algebraically closed field $C$ of characteristic zero. Assume that $C$ has the same cardinality as the complex numbers ${\mathbb C}$. Fix an isomorphism $\iota : C \rightarrow {\mathbb C}$. Let $\tilde{G}$ be some $p$-adic group. A character $\chi: \tilde{G} \rightarrow C$ is defined by $\chi = \iota^{-1}(\iota \circ \chi)$, where $\iota \circ \chi$ is a character in a usual sense.
We are given an inertial class $\Omega=[M,\rho]_{G}$, where $\rho$ is a supercuspidal representation of $M$ and an $\Omega$-type $(J, \lambda)$ with $J \subset K$ a compact open subgroup of $G$. Write $\mathfrak{Z}_{\Omega}$ for the centre of the category $\mathcal{R}^{\Omega}(G)$. Recall that the centre of a category is the ring of endomorphisms of the identity functor. For example the centre of the category $\mathcal{H}(G,\lambda)\text{-Mod}$ is $Z(\mathcal{H}(G,\lambda))$, where $Z(\mathcal{H}(G,\lambda))$ is the centre of the ring $\mathcal{H}(G,\lambda)$.
The representations of a Bernstein component can be seen as modules over Hecke algebra. Let $\mathcal{R}_{\lambda}(G)$ be a full subcategory of $\mathcal{R}(G)$ such that $(\pi,V)$ is an object of $\mathcal{R}_{\lambda}(G)$ if and only if $V$ is generated by $V^{\lambda}$ (the $\lambda$-isotypical component of $V$) as $G$-representation. Define $\mathcal{H}(G,\lambda):= \mathcal{H}(G,J,\lambda):=\mathrm{End}_{G}({\mathrm{c\text{--} Ind}}_J^G \lambda)$, the Hecke algebra of the type $(J,\lambda)$. Then for any $\Omega$-type $(J,\lambda)$, by \cite[Theorem 4.2 (ii)]{MR1643417}, the functor:
$$\begin{array}{ccccc}
\mathfrak{M}_{\lambda} & : & \mathcal{R}_{\lambda}(G) & \to & \mathcal{H}(G,\lambda)\text{-Mod} \\
& & \pi & \mapsto & \mathrm{Hom}_{J}(\lambda, \pi) = \mathrm{Hom}_{G}({\mathrm{c\text{--} Ind}}_J^G \lambda, \pi)\\
\end{array}$$
\noindent is an equivalence of categories. Since $(J,\lambda)$ is an $\Omega$-type, we have $\mathcal{R}^{\Omega}(G)=\mathcal{R}_{\lambda}(G)$.
As in \cite{MR1728541} a \textit{partition} is a function $P:\Z_{\geq 1} \rightarrow \Z_{\geq 0}$ with finite support; we say that $P$ is a partition of an integer $k:=\sum_{n=1}^{+\infty} P(n).n$. Usually a partition $P$ of $k$ is represented by a sequence $(m_1,\ldots,m_k)$, with $m_1 \geq \ldots \geq m_k \geq 0$ and $m_1+\ldots+m_k=k$, where one omits the zeroes from that list. The integers $m_i$ are related to $P$ as follows, $m_k=P(k)$, $m_{k-1}=P(k)+P(k-1)$,..., $m_1=P(k)+\ldots+P(1)$. We define a partial ordering on the set $\mathbb{P}$ of partitions as follows. We write $\lambda =(\lambda_{1},\ldots, \lambda_{k}) \geqq \mu =(\mu_{1},\ldots, \mu_{k})$ if and only if $\sum_{i=1}^{j} \lambda_{i} \leq \sum_{i=1}^{j} \mu_{i}$ for all integers $j$. The smallest partition for this partial order is $(k)$ and the biggest is $(1,\ldots,1)$ ($k$ times $1$). This is the opposite of the usual order on partitions (\cite[Chapter 5, Section 5.1.4]{MR3077154}).
As in \cite[section 2]{MR1728541}, let $\mathcal{C}$ be a system of representatives for the irreducible supercuspidal representations of any $GL_k(F)$ ($ k \in \Z_{\geq 1}$) up to unramified twist. A partition-valued function is a function $\mathcal{P}: \mathcal{C} \rightarrow \mathbb{P}$ with finite support. The set of partition-valued functions is partially ordered with respect to the partial ordering on partitions defined in the paragraph above by setting $\mathcal{P}\leq \mathcal{P}'$ if and only if $\mathcal{P}(\tau) \leq \mathcal{P}'(\tau)$, $\forall \tau \in \mathcal{C}$. Choose a partition-valued function $\mathcal{P}^{min}$ which is minimal for this partial ordering as in \cite{MR1728541}. Recall the decomposition (\ref{decomp}) from the introduction:
\[\mathrm{Ind}_{J}^{K} \lambda = \bigoplus_{\mathcal{P}} \sigma_{\mathcal{P}}(\lambda)^{\oplus m_{\mathcal{P},\lambda}},\]
\noindent where the summation runs over partition-valued functions.
From now on let $\sigma_{min}(\lambda) := \sigma_{\mathcal{P}^{min}}(\lambda)$ with the notations of section 6 in \cite{MR1728541}.
Let me introduce some further notation. Denote by $W$ the vector space on which the representation $\lambda$ is realized. Next, let $(\check{\lambda},W^{\vee})$ denote the contragradient of $(\lambda,W)$. Then by \cite[(2.6)]{MR1711578}, the Hecke algebra $\mathcal{H}(G, \lambda)$ can be identified with the space of compactly supported functions $f : G \longrightarrow \mathrm{End}_{C}(W^{\vee})$ such that $f(j_1.g.j_2) = \check{\lambda}(j_1)\circ f(g) \circ \check{\lambda}(j_2)$, with $j_1$, $j_2 \in J$ and $g \in G$ and the multiplication of two elements $f_1$ and $f_2$ is given by convolution:
$$f_1*f_2(g) = \int_G f_1(x)\circ f_2(x^{-1}g) dx.$$
For $u \in \mathrm{End}_C(W^{\vee})$, we write $u^{\vee} \in \mathrm{End}_{C}(W)$ for the transpose of $u$ with respect of the canonical pairing between $W$ and $W^{\vee}$. This gives $(\check{\lambda}(j))^{\vee} = \lambda(j)$, for $j \in J$. For $f \in \mathcal{H}(G,\lambda)$, define $\check{f} \in \mathcal{H}(G, \check{\lambda})$, by $\check{f}(g) = f(g^{-1})^{\vee}$, for all $g \in G$.
\section{Simple types}\label{M.8}
Let $E=F[\beta]$ be a finite field extension of $F$. Define $R=n/[E:F]$. Let $(J, \lambda)$ a simple type in $G$, where $J$ is a compact open subgroup in $G$ and $\lambda= \kappa \otimes \sigma$ with $\kappa$ a $\beta$-extension and $\sigma$ the inflation of $\tau \otimes\ldots\otimes \tau$ ($e$-times), where $\tau$ a cuspidal representation of $GL_{f}(k_{E})$, and we have $R=ef$.
Let $W=S_{e}$ a symmetric group in $e$ elements, and let $S$ be the subset of $W$ of all the transpositions $(i,i+1)$. Then $(W,S)$ be a Coxeter group. The Hecke algebra of $(W,S)$, denoted $\mathcal{H}_{W}$, is spanned by elements $T_{w}$, $w\in W$, subject to relations:
$$T_{x}T_{y}=T_{xy} \mbox{ if } l(xy)=l(x)+l(y)$$
$$T_{s}^{2} = (q-1)T_{s}+q \mbox{ for all } s\in S,$$
\noindent where $l$ denotes the length of reduced decomposition of an elements in $W$.
In this section $\overline{B}:=B(k_{E})$ is the Borel subgroup of $\overline{G}_{e}=GL_{e}(k_{E})$ and let $\overline{G}=GL_{R}(k_{E})$. We will always identify $w\in W$ with a matrix in $\overline{G}_{e}$ or with a matrix in $\overline{G}$, depending on the context.
Let $\overline{P}$ be a subgroup of $\overline{G}$ consisting of is upper triangular matrices by blocs with bloc sizes $f\times f$. Let $\phi_{w} \in \mathcal{H}(\overline{G},\sigma)$ is null outside $\overline{P}w\overline{P}$ such that $\phi_{w}(p_{1}wp_{2})=\sigma(p_{1})\circ \phi_{w}(w) \circ \sigma(p_{2})$ and $\phi_{w}(w)(y_{1}\otimes\ldots\otimes y_{e})= y_{w(1)}\otimes\ldots\otimes y_{w(e)}$. The homomorphism of Hecke algebras, as in (5.6.1) \cite{MR1204652}:
$$\begin{array}{ccccc}
\Psi & : & \mathcal{H}_{W} & \to & \mathcal{H}(\overline{G},\sigma) \\
& & T_{w} & \mapsto & \phi_{w} \\
\end{array}$$
\noindent is actually an isomorphism according to Theorem 5.1 in Chapter 1 \cite{MR821216}. In fact one can carry out a calculation to prove that $\phi_{w}$ are generators of $\mathcal{H}(\overline{G},\sigma)$ and they satisfy the same relations as $T_{w}$ in $\mathcal{H}_{W}$.
We have the following isomorphisms of Hecke algebras:
$$\mathcal{H}_{W} \simeq \mathrm{End}_{\overline{G}_{e}}(\mathrm{Ind}_{B}^{\overline{G}_{e}}1)$$
\noindent and
$$\mathcal{H}(\overline{G},\sigma) \simeq \mathrm{End}_{\overline{G}}(\mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma)$$
Let $\mathscr{M}(\overline{G}_{e})$ be the category of $\overline{G}_{e}$-representations and $\mathscr{M}_{\nu}(\overline{G}_{e})$ the full subcategory of $\mathscr{M}(\overline{G}_{e})$ of all $\overline{G}_{e}$-representations whose irreducible constituents all have cuspidal support $\nu=\tau \otimes\ldots\otimes\tau$. Define:
$$\begin{array}{ccccc}
& & \mathscr{M}_{1}(\overline{G}_{e}) & \to & \mathcal{H}_{W} \\
& & \pi & \mapsto & \mathrm{Hom}_{\overline{G}_{e}}(\mathrm{Ind}_{B}^{\overline{G}_{e}}1, \pi) \\
\end{array}$$
$$\begin{array}{ccccc}
\Psi' & : & \mathcal{H}_{W}-Mod & \to & \mathcal{H}(\overline{G},\sigma)-Mod \\
& & M & \mapsto & M \otimes_{\mathcal{H}_{W}} \mathcal{H}(\overline{G},\sigma) \\
\end{array}$$
$$\begin{array}{ccccc}
& & \mathcal{H}(\overline{G},\sigma)-Mod & \to & \mathscr{M}_{\nu}(\overline{G}) \\
& & M & \mapsto & M \otimes_{\mathcal{H}(\overline{G},\sigma)} \mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma \\
\end{array}$$
Let $\overline{H}_e : \mathscr{M}_{1}(\overline{G}_{e}) \to \mathscr{M}_{\nu}(\overline{G})$ the composition of these 3 functors.
First notice that $\overline{H}_e(\mathrm{Ind}_{B}^{\overline{G}_{e}}1) = \mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma$. Let $\overline{Q}$ be any standard parabolic of $\overline{G}_{e}$. We obtain a standard parabolic $\widehat{Q}$ of $\overline{G}$ from $\overline{Q}$ by enlarging each entry of $\overline{Q}$ to a bloc of size $f\times f$. We use the same convention for Levi subroups.
\begin{lemma}\label{1.13}
Let $Q$ be a standard parabolic of $\overline{G}_{e}$ and $\tilde{Q}$ as above, a standard parabolic of $\overline{G}$. Then:
$$\overline{H}_e(\mathrm{Ind}_{\overline{Q}}^{\overline{G}_{e}}1)= \mathrm{Ind}_{\widehat{Q}}^{\overline{G}}\sigma$$
\end{lemma}
\begin{proof} Let $W_{\overline{Q}}$ be the parabolic subgroup of $W$ associated to $\overline{Q}$. We have
$$\overline{H}_e(\mathrm{Ind}_{\overline{Q}}^{\overline{G}_{e}}1) = \mathrm{Hom}_{\overline{G}_{e}}(\mathrm{Ind}_{\overline{B}}^{}1, \mathrm{Ind}_{\overline{Q}}^{\overline{G}_{e}}1) \otimes_{\mathcal{H}(\overline{G},\sigma)} \mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma$$
$$=\bigoplus_{w \in W/W_{\overline{Q}}} \mathrm{Hom}_{\overline{B}^{w}\cap \overline{Q}}(1, 1^{w})\otimes_{\mathcal{H}(\overline{G},\sigma)} \mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma$$
We identify, as usual, $\mathrm{Hom}_{B^{w}\cap Q}(1, 1^{w})$ with the set of functions in $\mathcal{H}_{W}$ supported on $\overline{B}w\overline{Q}$. Via the isomorphism $\Psi$ of Hecke algebras, the set functions in $\mathcal{H}_{W}$ supported on $\overline{B}w\overline{Q}$ is in bijection with the set of functions in $\mathcal{H}(\overline{G},\sigma)$ supported on $\overline{P}w\widehat{Q}$ and this set is indetified with the intertwining set $\mathrm{Hom}_{\overline{P}^{w}\cap \widehat{Q}}(\sigma, \sigma^{w})$. It follows, that:
$$\overline{H}_e(\mathrm{Ind}_{Q}^{\overline{G}_{e}}1)\simeq \bigoplus_{w \in W/W_{\overline{Q}}} \mathrm{Hom}_{\overline{P}^{w}\cap \tilde{Q}}(\sigma, \sigma^{w}) \otimes_{\mathcal{H}(\overline{G},\sigma)} \mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma$$
$$= \mathrm{Hom}_{\overline{G}}(\mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma, \mathrm{Ind}_{\tilde{Q}}^{\overline{G}}\sigma)\otimes_{\mathcal{H}(\overline{G},\sigma)} \mathrm{Ind}_{\overline{P}}^{\overline{G}}\sigma$$
\noindent The result follows.
\end{proof}
Let $st(\tau,e)$ be a representation of $\overline{G}_{fe}$, defined as the unique nondegenerate irreducible representation with cuspidal support $\tau \otimes\ldots\otimes \tau$ ($e$-times). Since $\overline{H}_e$ is exact, $\overline{H}_e(st(1,e))=\frac{\overline{H}_e(\mathrm{Ind}_{B}^{\overline{G}_{e}}1)}{\sum_{Q \varsupsetneq B}\overline{H}_e(\mathrm{Ind}_{Q}^{\overline{G}_{e}}1)}$ and by previous lemma, $\overline{H}_e(st(1,e))=st(\tau,e)$.
\begin{lemma}\label{1.14}
Let $\overline{M'}$ be the Levi subgroup of $\overline{Q}$. The following diagram commutes:
$$\xymatrixcolsep{5pc}\xymatrix{
\mathscr{M}_{1}(\overline{G}_{e}) \ar[r]^{\overline{H}_e} &\mathscr{M}_{\nu}(\overline{G}) \\
\mathscr{M}_{1}(\overline{M'}) \ar[u]^{\mathrm{Ind}_{\overline{Q}}^{\overline{G}_{e}}} \ar[r]^{\overline{H}_{\overline{M'}}} &\mathscr{M}_{\nu_{\widehat{M'}}}(\widehat{M'}) \ar[u]^{\mathrm{Ind}_{\widehat{Q}}^{\overline{G}}}
}$$
\noindent where the horizontal arrows are an equivalence of categories, and $\nu_{\widehat{M'}}$ denotes the restriction of the cuspidal support $\nu$ to $\widehat{M'}$.
\end{lemma}
\begin{proof} It is enough to check that this diagram commutes for every irreducible representation of $\mathscr{M}_{1}(\overline{M'})$. By definition, the irreducible representations of $\mathscr{M}_{1}(\overline{M'})$ are just unramified characters of $L$. Moreover, by Lemma \ref{1.13} we have that $\overline{H}_e(\mathrm{Ind}_{\overline{Q}}^{\overline{G}_{e}}1)= \mathrm{Ind}_{\widehat{Q}}^{\overline{G}}\overline{H}_{\overline{M'}}(1)$. The same identity holds if $1$ is replaced any unramified character of $L$, by the same argument as in Lemma \ref{1.13}.
\end{proof}
We identify the partitions and partition valued functions. Let $\tilde{\mathcal{P}}$ be the partition $(e_{1}f,\ldots,e_{k}f)$ of $R$ associated to parabolic subgroup $\tilde{Q}$ and $\mathcal{P}$ the partition $(e_{1},\ldots,e_{k})$ of $e$ associated to parabolic subgroup $Q$. Define $\pi(\tau,\tilde{\mathcal{P}})=\mathrm{Ind}_{\tilde{Q}}^{\overline{G}} st(\tau,e_{1}) \otimes \ldots\otimes st(\tau,e_{k})$ and $\sigma(\tau,\tilde{\mathcal{P}})$ the representation of $\overline{G}$ that occurs in $\pi(\tau,\tilde{\mathcal{P}})$ with multiplicity 1 and not in $\pi(\tau,\mathcal{Q})$ if $\mathcal{Q}> \tilde{\mathcal{P}}$.
\begin{lemma}\label{1.15}
Let $Q$ be a standard parabolic of $\overline{G}_{e}$ and $\tilde{Q}$ as above, a standard parabolic of $\overline{G}$. Then:
$$\overline{H}_e(\sigma(1,\mathcal{P}))= \sigma(\tau,\tilde{\mathcal{P}})$$
\end{lemma}
\begin{proof} By previous lemma we have that:
$$\overline{H}_e(\pi(1,\mathcal{P})) = \mathrm{Ind}_{\tilde{Q}}^{\overline{G}} F_{L}( st(1,e_{1}) \otimes \ldots\otimes st(1,e_{k}) )$$
$$ = \mathrm{Ind}_{\tilde{Q}}^{\overline{G}} F_{e_{1}}( st(1,e_{1})) \otimes \ldots\otimes F_{e_{k}}(st(1,e_{k})) = \pi(\tau,\tilde{\mathcal{P}})$$
\noindent Since $\overline{H}_e$ is exact:
$$\overline{H}_e(\sigma(1,\mathcal{P})) = \frac{\overline{H}_e(\pi(1,\mathcal{P})}{\sum_{\mathcal{Q} < \mathcal{P}}\overline{H}_e(\pi(1,\mathcal{Q})} = \frac{\pi(\tau,\tilde{\mathcal{P}})}{\sum_{\mathcal{Q} < \mathcal{P}}\pi(\tau,\tilde{\mathcal{Q}})}=\sigma(\tau,\tilde{\mathcal{P}})$$
\end{proof}
Let $\pi$ be an irreducible representation containing a simple type $(J,\lambda)$. In this case $\Omega = [GL_{r}(F)^{e}, \omega \otimes \ldots \otimes \omega]_{G}$ where the tensor product $\rho : = \omega \otimes \ldots \otimes \omega$ is taken $e$ times and $\omega$ is a supercuspidal representation of $GL_{r}(F)$. According to the description of Hecke algebras in section (5.6) of \cite{MR1204652} there is a support preserving isomorphism of Hecke algebras $\mathcal{H}(G_{L}, I_{L}, 1) \simeq \mathcal{H}(G, J, \lambda)$, where $L$ is an extension of $F$ (denoted by $K$ in \cite{MR1204652}), $G_{L}=GL_{e}(L)$ with $I_{L}$ the standard Iwahori subgroup of $G_{L}$. We denote by $K_{L}$ a maximal compact subgroup of $G_{L}$ containing $I_L$.
We will recall now the results on supercuspidal representations from chapter~6 of \cite{MR1204652} and describe the general form of the supercuspidal representation $\omega$ of $G_0=GL_{r}(F)$. The representation $\omega$ contains a maximal simple type $(J_0, \lambda_0)$. This means that there are a finite extension $E$ of $F$ and a uniquely determined representation $\Lambda_0$ of $E^{\times}J_0$ such that $\omega = {\mathrm{c\text{--} Ind}}_{E^{\times}J_0}^{G_0} \Lambda_0$ and $\Lambda_0|J_0 = \lambda_0$. Let $f=\frac{n}{e[E:F]}$. According to \cite[Proposition 5.5.14]{MR1204652}, the extension $L$ considered in the previous is unramified extension of degree $f$ of $E$. A special case of the support preserving isomorphism in the previous paragraph is the support preserving isomorphism $\Phi_{1}: \mathcal{H}(G_0, J_0, \lambda_0) \simeq \mathcal{H}(L^{\times}, \mathcal{O}_{L}^{\times}, 1)$ sending a function supported on $J_0\varpi_{E}J_0=\varpi_{E}J_0$ to a function supported on $\varpi_{E} \mathcal{O}_{L}^{\times}$, where $\varpi_{E}$ a uniformizer of both $E$ and $L$. Further we observe that the unramified characters of $G_0$ are determined by the image of $\varpi_{E}$, as are unramified characters of $L^{\times}$. Therefore we may and we will identify the unramified characters of $G_0$ with the unramified characters of $L^{\times}$.
The representation $\pi$ is a Langlands quotient of the form $Q(\Delta_{1},\ldots,\Delta_{s})$ (cf. \cite[Section 1.2 Theorem 1.2.5]{MR1265559}) such that for $i < j$ the segment $\Delta_{i}$ does not precede $\Delta_{j}$ (cf. \cite[Section 1.2 Definition 1.2.4]{MR1265559}). After twisting $\pi$ by some unramified character we may assume that all the segments are of the form $\Delta_{i}=[\omega(\alpha_{i}),\omega(\alpha_{i}+e_{i}-1)]$, where $\alpha_{i}\in C$ and $e_{i}$ an integer such that $\sum_{i=1}^{s} e_{i} = e$. Here the notation $\omega(\alpha_{i})$ means that $\omega(\alpha_{i}):=\omega \otimes \iota^{-1}(|\det|^{\iota(\alpha_{i})})$, the norm $|.|$ is viewed as taking values in $q^{\Z}\subset{\mathbb C}$.
Let $P$ be a standard parabolic of $G$ containing $M=GL_r(F)^e$ and let $B_L$ be a Borel subgroup of $G_L$ containing $T_L = (L^{\times})^{e}$.
According to \cite[Theorem 7.6.20]{MR1204652}, the diagram
\begin{equation}\tag{D1}\label{D1}
\xymatrix{
\mathcal{H}(G, J, \lambda) \ar[r]^{\Phi} &\mathcal{H}(G_{L}, I_{L}, 1)\\
\mathcal{H}(M, J_{M}, \lambda_{M}) \ar[u]^{t_P} \ar[r]^{\Phi_{1}^{\otimes e}} &\mathcal{H}(T_{L}, T_{L}^{\circ}, 1) \ar[u]^{t_{B_L}}}
\end{equation}
\noindent is commutative, where the horizontal arrows are support preserving isomorphisms and $\lambda_{M} = \lambda_{0}\otimes \ldots \otimes \lambda_{0}$ ($e$ times), $J_{M} = (J_{0})^{e}$, $T_{L}=(L^{\times})^{e}$ and $T_{L}^{\circ} = (\mathcal{O}_L^{\times})^{e}$. In \cite{MR1204652}, the horizontal isomorphisms in the commutative diagram above are given in the other direction. For $t:A\rightarrow A'$ a morphism $C$-algebras we write $t_{\ast} : A'-\mathrm{Mod}\longrightarrow A-\mathrm{Mod}$ for the induced functor given $\mathrm{Hom}_{A'}(A,\bullet)$. The diagram above produces the following commutative diagram:
\begin{equation}\tag{D2}\label{D2}\xymatrix{
\mathcal{R}_{\lambda}(G) \ar[r]^-{M_{\lambda}} &\mathcal{H}(G, J, \lambda)\text{-Mod} \ar[r]^{\Phi_{\ast}} &\mathcal{H}(G_{L}, I_{L}, 1)\text{-Mod}\ar[r]^-{T_{\lambda}} &\mathcal{R}_{1}(G_{L})\\
\mathcal{R}_{\lambda_{M}}(M) \ar[u]^-{i_{P}^{G}} \ar[r]_-{M_{\lambda_M}} &\mathcal{H}(M, J_{M}, \lambda_{M})\text{-Mod} \ar[u]^{(t_{P})_{\ast}} \ar[r]^{(\Phi_{1}^{\otimes e})_{\ast}} &\mathcal{H}(T_{L}, T_{L}^{\circ}, 1)\text{-Mod} \ar[u]^{(t_{B_L})_{\ast}} \ar[r]^-{T_1} &\mathcal{R}_{1}(T_{L})\ar[u]_-{i_{B_{L}}^{G_{L}}}}
\end{equation}
\noindent where the horizontal arrows are equivalences of categories, $T_{\lambda} = \bullet \otimes_{\mathcal{H}(G_{L}, I_{L}, 1)}\mathrm{c\text{--} Ind}_{I_{L}}^{G_{L}} 1$, $T_1=\bullet \otimes_{\mathcal{H}(T_{L}, T_{L}^{\circ}, 1)}\mathrm{c\text{--} Ind}_{T_{L}^{\circ}}^{T_{L}} 1$, $M_{\lambda}=\mathrm{Hom}_{J}(\lambda,\bullet)$ and $M_{\lambda_M}=\mathrm{Hom}_{J_{M}}(\lambda_{M},\bullet)$. The first and second squares are commutative as a consequence of \cite[Corollary 8.4]{MR1643417}, the commutativity of the middle square follows from diagram (\ref{D1}). Let $H$ be the composition of all the top horizontal arrows. Hence the functor $H:\mathcal{R}_{\lambda}(G) \longrightarrow \mathcal{R}_{1}(G_{L})$ from above is an equivalence of categories.
\begin{lemma}\label{1.20}
We have $H(i_{P}^{G} \rho)=i_{B_{L}}^{G_{L}} 1$.
\end{lemma}
\begin{proof}It follow from the commutative diagram above that:
$$\Phi_{\ast} (\mathrm{Hom}_{J}(\lambda,i_{P}^{G}(\rho)))\otimes_{\mathcal{H}(G_{L}, I_{L}, 1)}\mathrm{c\text{--} Ind}_{I_{L}}^{G_{L}} 1$$
$$= i_{B_{L}}^{G_{L}}((\Phi_{1}^{\otimes e})_{\ast}(\mathrm{Hom}_{J_{M}}(\lambda_{M},\rho))\otimes_{\mathcal{H}(T_{L}, T_{L}^{\circ}, 1)}\mathrm{c\text{--} Ind}_{T_{L}^{\circ}}^{T_{L}} 1)$$
Observe that the representation $\mathrm{c\text{--} Ind}_{T_{L}^{\circ}}^{T_{L}} 1$ is canonically a rank 1 free $\mathcal{H}(T_{L}, T_{L}^{\circ}, 1)$-module. This observation allows us to simplify the right hand side. Recall that $\rho : = \omega \otimes \ldots \otimes \omega$.
Since $(J, \lambda)$ is a simple type, $\lambda_{M} = \lambda_{0}\otimes \ldots \otimes \lambda_{0}$ ($e$ times), $J_{M} = (J_{0})^{e}$ and $(J_{0}, \lambda_{0})$ is a maximal simple type for the supercuspidal representation $\omega$, we have:
$$\mathrm{Hom}_{J_{M}}(\lambda_{M},\rho) \simeq \mathrm{Hom}_{J_{M}}(\lambda_{0}\otimes \ldots \otimes \lambda_{0},\omega | J_{0} \otimes \ldots \otimes \omega| J_{0})$$
$$\simeq \mathrm{Hom}_{J_{0}^{e}}(\lambda_{0}\otimes \ldots \otimes \lambda_{0},\lambda_{0}\otimes \ldots \otimes \lambda_{0})\simeq \mathrm{Hom}_{J_M}(\lambda_M, \lambda_M),$$
\noindent where have used that $\lambda_{0}$ occurs in $\omega$ with multiplicity one. Now notice that $\mathrm{Hom}_{J_M}(\lambda_M, \lambda_M)$ is the subspace of functions in $\mathcal{H}(M, J_{M}, \lambda_{M})$ supported on $J_M$. By our choice of $\lambda_0$, $\omega$ and $\Phi_{1}$, the support preserving isomorphism $\Phi_{1}^{\otimes e}$ maps this space isomorphically onto the space of functions in $\mathcal{H}(T_{L}, T_{L}^{\circ}, 1)$ supported on $T_{L}^{\circ}$. It follows that $\Phi_{1}^{\otimes e}(\mathrm{Hom}_{J_{M}}(\lambda_{M},\rho)) = \mathrm{Hom}_{T_{L}^{\circ}}(1, 1)$. Thus, the representation $\Phi_{1}^{\otimes e}(\mathrm{Hom}_{J_{M}}(\lambda_{M},\rho))\otimes_{\mathcal{H}(T_{L}, T_{L}^{\circ}, 1)}\mathrm{c\text{--} Ind}_{T_{L}^{\circ}}^{T_{L}} 1$ is a trivial character of $T_{L}$. Then an object $i_{P}^{G} \rho$ in $\mathcal{R}_{\lambda}(G)$ corresponds to an object $i_{B_{L}}^{G_{L}} 1$ in $\mathcal{R}_{1}(G_{L})$.
\end{proof}
\begin{lemma}\label{1.21}
$H:\mathcal{R}_{\lambda}(G) \longrightarrow \mathcal{R}_{1}(G_{L})$ is compatible with twisting by characters
\end{lemma}
\begin{proof}
Consider the representation $i_{P}^{G}((\omega \otimes \chi_1)\otimes \ldots (\omega \otimes \chi_e))$, where $\chi_1,\ldots,\chi_e$ are some unramified characters of $G_0$. Let $\rho' = (\omega \otimes \chi_1)\otimes \ldots (\omega \otimes \chi_e) = \rho \otimes \chi$, where $\chi$ is an unramified character of $M$. According to \cite[page 591]{MR1643417} the action of $\mathcal{H}(M, J_{M}, \lambda_{M})$ on $\mathrm{Hom}_{J_M}(\lambda_M, \rho')$ is given by
$$f\cdot\phi(w) = \int\limits_{M} \rho'(g)\phi(\check{f}(g^{-1})w)dg$$
\noindent where $f \in \mathcal{H}(M, J_{M}, \lambda_{M})$, $\phi \in \mathrm{Hom}_{J_M}(\lambda_M, \rho')$ and $w$ is a vector in the underlying vector space of $\lambda_M$. We want to understand the compatibility of this action with twisting and support preserving isomorphisms of Hecke algebras. Since $f$ has compact support, without loss of generality we may assume that $f$ is supported on $J_M m J_M$, for some $m \in M$. The element $m$ is a block-diagonal matrix with $e$ blocks. Without loss of generality we may assume that each block is some power of the uniformizer $\varpi_{E}$. We normalize measures on $G_0$ and on $L^{\times}$ such that $J_0$ and $\mathcal{O}^{\times}$ have both volume 1. Then taking the induced product measures on $M$ and $T_L$, we see that $\int_{J_M}dj=1$ and $\int_{T_{L}^{\circ}}dt=1$. Each block in the matrix $m$ normalizes $J_0$, hence $m$ normalizes $J_M$ and $J_M mJ_M =mJ_M$. It follows that
$$f \cdot\phi(w) = \int\limits_{j \in J_M} \rho'(mj)\phi(f(mj)^{\vee}w) dj.$$
By definition we have:
$$f(mj)^{\vee} = (f(m)\check{\lambda}_M(j))^{\vee} = \lambda_M(j^{-1}).f(m)^{\vee}.$$
Moreover $\phi$ is $J_M$-equivariant, thus
$$\phi(\lambda_M(j^{-1})f(m)^{\vee}) = \rho'(j^{-1}).\phi(f(m)^{\vee}).$$
This simplifies the integral above:
$$\int\limits_{j \in J_M} \rho'(m).\phi(f(m)^{\vee} w) dj=\rho'(m).\phi(f(m)^{\vee}w)=\chi(m).\rho(m).\phi(f(m)^{\vee}w).$$
The expression above is compatible with the support preserving isomorphism $\Phi_{1}^{\otimes e}$, in a sense that
$$\Phi_{1}^{\otimes e} (\chi(m).\rho(m).\phi(f(m)^{\vee}\bullet) )= \chi(m). \Phi_{1}^{\otimes e}(\phi)(\Phi_{1}^{\otimes e}(f)(m)^{\vee}\bullet),$$
\noindent where $m$ is naturally seen as an element of $T_L$ because its diagonal blocks are some powers of the uniformizer $\varpi_{E}$ and $\chi$ is seen as unramified character of $T_L$. This is, of course, compatible with the same computation of the integral replacing $ \mathcal{H}(M, J_{M}, \lambda_{M})$ by $\mathcal{H}(T_{L}, T_{L}^{\circ}, 1)$ and $\mathrm{Hom}_{J_M}(\lambda_M, \rho')$ by $\mathrm{Hom}_{T_{L}^{\circ}}(1, \chi)$.
\end{proof}
\begin{lemma}\label{1.22}
The $H:\mathcal{R}_{\lambda}(G) \longrightarrow \mathcal{R}_{1}(G_{L})$ preserves the segments.
\end{lemma}
\begin{proof}
We know that $\pi= Q(\Delta_{1},\ldots,\Delta_{s})$ is an irreducible subquotient of $i_{P}^{G}((\omega \otimes \chi_1)\otimes \ldots (\omega \otimes \chi_e))$, where $\chi_1,\ldots,\chi_e$ are some unramified characters of $G_0$.
Then by the equivalence of categories described in the diagram (\ref{D2}), $H(\pi)$ is an irreducible subquotient of $H(i_{P}^{G}((\omega \otimes \chi_1)\otimes \ldots \otimes(\omega \otimes \chi_e)))$. However by Lemma \ref{1.21}, we have $H(\pi)$ is an irreducible subquotient of $H(i_{P}^{G}((\omega \otimes \chi_1)\otimes \ldots \otimes(\omega \otimes \chi_e)))=i_{B_{L}}^{G_{L}} (\chi_1 \otimes \ldots \otimes \chi_e)$. Let now $\Delta = [\omega(\alpha),\omega(\alpha+e-1)]$, a segment in $G$, where $\alpha$ is some scalar. Then the commutative diagram above shows that the $G$-representation $i_{P}^{G}(\Delta)$ corresponds to the $G_{L}$-representation $H(i_{P}^{G}(\Delta))=i_{B_{L}}^{G_{L}}(\Delta_{L})$, where $\Delta_{L} = [1(\alpha),1(\alpha+e-1)]$ is a segment in $G_{L}$ and $1$ is the trivial character of $L^{\times}$. We know that $i_{P}^{G}(\Delta)$ and $i_{B_L}^{G_L}(\Delta_L)$, admit unique irreducible quotients $Q(\Delta)$ and $Q(\Delta_{L})$ respectively, so $H(Q(\Delta))= Q(\Delta_{L})$. Let $\tilde{P}=\tilde{M}\tilde{N}$, with Levi subgroup $\tilde{M}=GL_{re_1}(F) \times \ldots \times GL_{re_s}(F)$ and unipotent radical $\tilde{N}$, be a standard parabolic containing $P$ which is adapted to the segment decomposition of $\pi=Q(\Delta_{1},\ldots,\Delta_{s})$, so that $\pi$ is a quotient of $i_{\tilde{P}}^G(Q(\Delta_{1})\otimes\ldots\otimes Q(\Delta_s))$. Define $\tilde{M}_L = GL_{e_1}(L) \times \ldots \times GL_{e_s}(L)$ a Levi subgroup of a standard parabolic $\tilde{P}_L$ such that $B_L \subset\tilde{P}_L \subset G_L$. In the same way as for diagram (\ref{D2}), we get a commutative diagram:
$$\xymatrix{
\mathcal{R}_{\lambda}(G) \ar[r]^-{H} &\mathcal{R}_{1}(G_{L})\\
\mathcal{R}_{\lambda_{M}}(\tilde{M}) \ar[u]^{i_{\tilde{P}}^G} \ar[r]_-{H_{\tilde{M}}} &\mathcal{R}_{1}(\tilde{M}_{L}),\ar[u]_-{i_{\tilde{P}_L}^{G_L}} } $$
where the horizontal arrows are equivalences of categories, constructed in a similar fashion to the diagram (\ref{D2}) replacing $P$ by $\tilde{P}$, $B_L$ by $\tilde{P}_L$ and so on... Moreover by construction the functor $H_{\tilde{M}}=H_1 \times \ldots \times H_s$ is a product of functors $H_i$, where each individual functor $H_i : \mathcal{R}_{\lambda}(GL_{re_i}(F)) \longrightarrow \mathcal{R}_{1}(GL_{e_i}(L))$ is constructed in the same way as $H$, with $G$ replaced by $GL_{re_i}(F)$. Gathering all the results above we may write:
$$H\circ i_{\tilde{P}}^G(Q(\Delta_{1})\otimes\ldots\otimes Q(\Delta_s))= i_{\tilde{P}_L}^{G_L}(H_{\tilde{M}}(Q(\Delta_{1})\otimes\ldots\otimes Q(\Delta_s)))$$
$$=i_{\tilde{P}_L}^{G_L}(Q(\Delta'_{1})\otimes\ldots\otimes Q(\Delta'_s)),$$
where $\Delta'_{i}=[1(\alpha_{i}),1(\alpha_{i}+e_{i}-1)]$ for all $i$. Hence we have an equality $H(\pi)=Q(\Delta'_{1},\ldots,\Delta'_{s})$, since both representations are the Langlands quotient of $H\circ i_{\tilde{P}}^G(Q(\Delta_{1})\otimes\ldots\otimes Q(\Delta_s))=i_{\tilde{P}_L}^{G_L}(Q(\Delta'_{1})\otimes\ldots\otimes Q(\Delta'_s))$.
\end{proof}
According to the description of Hecke algebras in section (5.6) of \cite{MR1204652} the isomorphism of Hecke algebras $\Phi: \mathcal{H}(G, J, \lambda) \simeq \mathcal{H}(G_{L}, I_{L}, 1) $ is support preserving, in the sense that $\mathrm{supp}(\Phi(f))=I_L.\mathrm{supp}(f).I_L$. We have also a natural isomorphism between $\mathcal{H}(K_{L}, I_{L}, 1)= \left\lbrace f \in \mathcal{H}(G_{L}, I_{L}, 1) \mid \mathrm{supp}(f) \subset K_{L}\right \rbrace$ and $\mathcal{H}(K, J, \lambda)= \left\lbrace f \in \mathcal{H}(G, J, \lambda) \mid \mathrm{supp}(f) \subset K\right \rbrace$. We have then the following commutative diagram:
$$\xymatrix{
\mathcal{H}(G, J, \lambda) \ar[r]^{\Phi} &\mathcal{H}(G_{L}, I_{L}, 1)\\
\mathcal{H}(K, J, \lambda) \ar[u] \ar[r]^{\Phi} &\mathcal{H}(K_{L}, I_{L}, 1) \ar[u]}$$
As for diagram (\ref{D2}), the diagram above induces:
$$\xymatrix{
\mathcal{R}_{\lambda}(G) \ar[r]^-{M_{\lambda}} &\mathcal{H}(G, J, \lambda)\text{-Mod} \ar[r]^{\Phi_{\ast}} &\mathcal{H}(G_{L}, I_{L}, 1)\text{-Mod}\ar[r]^-{T_{\lambda}} &\mathcal{R}_{1}(G_{L})\\
\mathcal{R}_{\lambda}(K) \ar[u]^{\mathrm{c\text{--} Ind}_{K}^{G}} \ar[r]_-{M_{\lambda}} &\mathcal{H}(K, J, \lambda)\text{-Mod} \ar[r]^{\Phi_{\ast}} &\mathcal{H}(K_{L}, I_{L}, 1)\text{-Mod} \ar[r]_-{T_{K_L}} &\mathcal{R}_{1}(K_{L})\ar[u]_-{\mathrm{c\text{--} Ind}_{K_{L}}^{G_{L}}}} $$
\noindent where $T_{K_L}=\bullet \otimes_{\mathcal{H}(K_{L}, I_{L}, 1)}\mathrm{c\text{--} Ind}_{I_{L}}^{K_{L}} 1$.
If we denote the composition of all the top horizontal arrow by $H$ and the composition of all the bottom horizontal arrow by $H_{K}$, then $H(\mathrm{c\text{--} Ind}_{K}^{G} \sigma) = \mathrm{c\text{--} Ind}_{K_{L}}^{G_{L}} H_{K}(\sigma)$.
The functor $\mathrm{Ind}_{J_{max}}^{K}(\kappa_{max} \otimes \cdot) : \mathscr{M}_{\nu}(\overline{G}) \to \mathcal{R}_{\lambda}(K) $ is an equivalence of categories according the discussion above Proposition 11 in Section 5 \cite{MR1728541}. We will denote this functor by $"\kappa_{max}"$. Let $\sigma_{\tilde{\mathcal{P}}}(\lambda) = \mathrm{Ind}_{J_{max}}^{K}(\kappa_{max} \otimes \sigma(\tau,\tilde{\mathcal{P}}))$.
\begin{lemma}\label{1.23}
We have $H_{K}(\sigma_{min}(\lambda))=\mathrm{St}$, where $\mathrm{St}$ denotes the inflation of Steinberg representation of $GL_{n}$ over a finite field.
\end{lemma}
\begin{proof} By Lemma \ref{1.15} we have that $\overline{H}_e(st(1,e)) = \sigma(\tau,\tilde{\mathcal{P}}_{min})$. To conclude, use following commutative diagram:$$\xymatrix{
\mathscr{M}_1(\overline{G}_e) \ar[d]^{"\kappa_{max}"} \ar[r]^{\overline{H}_e} &\mathscr{M}_{\nu}(\overline{G}) \ar[d]^{"\kappa_{max}"}\\
\mathcal{R}_1(K_L) \ar[r]^{H_K^{-1}} &\mathcal{R}_{\lambda}(K) \quad,}$$
\noindent where every arrow is an equivalence of categories.
\end{proof}
\section{Generic representations}\label{M.11}
In this section we will use the results proven above to deduce our main theorem.
\begin{thm}\label{1.18}
Let $\pi$ be an absolutely irreducible representation in the Berstein component $\Omega$, then $\mathrm{Hom}_{K}(\sigma_{min}(\lambda), \pi) \neq 0$ if and only if $\pi$ is generic.
\end{thm}
\begin{proof} Let us first deal with a particular case before the general case.
\noindent \textbf{1. Simple type case.} Assume that $\pi$ contains a simple type $(J,\lambda)$. In this case $\Omega = [GL_{r}(F)^{e}, \omega \otimes \ldots \otimes \omega]_{G}$ where the tensor product $\rho : = \omega \otimes \ldots \otimes \omega$ is taken $e$ times and $\omega$ is a supercuspidal representation of $GL_{r}(F)$.
Recall that representation $\pi=Q(\Delta_{1},\ldots,\Delta_{s})$ such that for $i < j$ the segment $\Delta_{i}$ does not precede $\Delta_{j}$. If $s=1$ then $\pi$ is generic and contains $\sigma_{min}(\lambda)$. Assume that $s > 1$.
The functor $H:\mathcal{R}_{\lambda}(G) \longrightarrow \mathcal{R}_{1}(G_{L})$ from above is an equivalence of categories. To avoid notational overload let $\sigma : = \sigma_{min}(\lambda)$, then
$$\mathrm{Hom}_{G}(\mathrm{c\text{--} Ind}_{K}^{G} \sigma,\pi)= \mathrm{Hom}_{G_{L}}(H(\mathrm{c\text{--} Ind}_{K}^{G} \sigma),H(\pi))$$
Recall the following commutative diagram:
$$\xymatrix{
\mathcal{R}_{\lambda}(G) \ar[r]^-{M_{\lambda}} &\mathcal{H}(G, J, \lambda)\text{-Mod} \ar[r]^{\Phi_{\ast}} &\mathcal{H}(G_{L}, I_{L}, 1)\text{-Mod}\ar[r]^-{T_{\lambda}} &\mathcal{R}_{1}(G_{L})\\
\mathcal{R}_{\lambda}(K) \ar[u]^{\mathrm{c\text{--} Ind}_{K}^{G}} \ar[r]_-{M_{\lambda}} &\mathcal{H}(K, J, \lambda)\text{-Mod} \ar[r]^{\Phi_{\ast}} &\mathcal{H}(K_{L}, I_{L}, 1)\text{-Mod} \ar[r]_-{T_{K_L}} &\mathcal{R}_{1}(K_{L})\ar[u]_-{\mathrm{c\text{--} Ind}_{K_{L}}^{G_{L}}}} $$
\noindent where $T_{K_L}=\bullet \otimes_{\mathcal{H}(K_{L}, I_{L}, 1)}\mathrm{c\text{--} Ind}_{I_{L}}^{K_{L}} 1$.
Recall that $H$ is the composition of all the top horizontal arrows and $H_K$ is the composition of all the bottom horizontal arrows, then $H(\mathrm{c\text{--} Ind}_{K}^{G} \sigma) = \mathrm{c\text{--} Ind}_{K_{L}}^{G_{L}} H_{K}(\sigma)$. By Lemma \ref{1.23} we have $H_{K}(\sigma)=\mathrm{St}$, where $\mathrm{St}$ denotes the inflation of Steinberg representation of $GL_{n}$ over a finite field. Moreover by Lemma \ref{1.22}, we have $H(\pi)=Q(\Delta'_{1},\ldots,\Delta'_{s})$. Therefore:
$$\mathrm{Hom}_{K}(\sigma,\pi|K)=\mathrm{Hom}_{G}(\mathrm{c\text{--} Ind}_{K}^{G} \sigma,\pi)= \mathrm{Hom}_{G_{L}}(H(\mathrm{c\text{--} Ind}_{K}^{G} \sigma),H(\pi))$$
$$=\mathrm{Hom}_{G_{L}}(\mathrm{c\text{--} Ind}_{K_{L}}^{G_{L}} H_{K}(\sigma), Q(\Delta'_{1},\ldots,\Delta'_{s}))$$ $$=\mathrm{Hom}_{K_{L}}( \mathrm{St},Q(\Delta'_{1},\ldots,\Delta'_{s})|K_{L}).$$
According to \cite[Theorem 9.7]{MR584084} $\pi = Q(\Delta_{1},\ldots,\Delta_{s})$ is generic if and only if no two segments $\Delta_i$ are linked. By construction the relative positions of the segments $\Delta_i$ are the same as of the segments $\Delta'_i$. Therefore no two segments $\Delta_i$ are linked if and only if no two segments $\Delta'_i$ are linked. It follows that $Q(\Delta'_{1},\ldots,\Delta'_{s})$ is generic if and only if $\pi$ is generic and $\mathrm{Hom}_{K_{L}}(\mathrm{St},Q(\Delta'_{1},\ldots,\Delta'_{s})|K_{L}) \neq 0$ if and only if $\mathrm{Hom}_{K}(\sigma,\pi|K) \neq 0$ by the equality above. So we are reduced to consider the case when $(J,\lambda)=(I,1)$. However this was proven in \cite[section 7.2]{MR1915088}.
\noindent \textbf{2. Semi-simple type case (general case).} Let now $\lambda$ be some general semi-simple type. The second part of the Main Theorem of section 8 in \cite{MR1643417} gives a support preserving Hecke algebra isomorphism $j :\mathcal{H}(\overline{M}, \lambda_{M}) \rightarrow \mathcal{H}(G, \lambda)$ (here $\overline{M}$ is the unique Levi subgroup of $G$ which contains the $N_{G}(M)$-stabilizer of the inertia class $D=[M,\rho]_M$ and is minimal for this property), and section 1.5 of \textit{op. cit.} gives a tensor product decomposition $\mathcal{H}(\overline{M}, \lambda_{M}) = \mathcal{H}_{1} \otimes_{C}\ldots\otimes_{C}\mathcal{H}_{s}$, where $\mathcal{H}_{i} = \mathcal{H}(G_{i}, J_{i}, \lambda_{i})$ is an affine Hecke algebras of type A and $(J_{i}, \lambda_{i})$ is some simple type with $G_{i}$ some general linear group over a $p$-adic field.
Let $M=\prod_{i=1}^{s} GL_{n_i}(F)$ be a standard block-diagonal Levi subgroup of a standard parabolic $P=MN$, such that $K \cap M = \prod_{i=1}^{s} K_{i}$, where $K_{i}$ is a maximal compact subgroup of $GL_{n_i}(F)$. By definition, see the end of section 6 in \cite{MR1728541}, the restriction of the $K$-representation $\sigma:=\sigma_{min}(\lambda)$ to $K \cap N$ is trivial, and $\sigma|K\cap M \simeq \sigma_{1}\otimes\ldots\otimes \sigma_{s}$ where $\sigma_{i} := \sigma_{\mathcal{P}_{i}^{min}}(\lambda_{i})$ with obvious notations.
According to \cite[Theorem (8.5.1)]{MR1204652} the irreducible representation $\pi$ is of the form
$$\pi \simeq i_P^G(\pi_{1} \otimes \ldots \otimes \pi_{s}),$$
\noindent such that $\pi_{i}$ is an irreducible representation of $G_{i}$ and contains the simple type $(J_{i}, \lambda_{i})$. Moreover the supercuspidal support of $\pi_{i}$ is disjoint from the supercuspidal support of $\pi_{j}$ for $i \neq j$. Then
$$\mathrm{Hom}_{K}(\sigma,\pi) = \mathrm{Hom}_{K}(\sigma,\mathrm{Ind}_{K\cap P}^K(\pi_{1}|K_{1}\otimes\ldots\otimes \pi_{s}|K_{s}))$$ $$=\mathrm{Hom}_{K \cap P}(\sigma| K\cap P,\pi_{1}|K_{1}\otimes\ldots\otimes \pi_{s}|K_{s})$$
$$ = \mathrm{Hom}_{K \cap M}(\sigma_{1}\otimes\ldots\otimes \sigma_{s},\pi_{1}|K_{1}\otimes\ldots\otimes \pi_{s}|K_{s}),$$
where the first equality is obtained from the Mackey formula and Iwasawa decomposition, the second equality follows from Frobenius reciprocity, where $\pi_{1}|K_{1}\otimes\ldots\otimes \pi_{s}|K_{s}$ denotes the inflation of the representation of $K\cap M$ to $K\cap P$, and the last equality is obtained by taking the coinvariants of $\sigma|K\cap P$ with respect to $K \cap N$. Hence $\mathrm{Hom}_{K}(\sigma,\pi)$ is non zero if and only $\mathrm{Hom}_{K_{i}}(\sigma_{i},\pi_{i}|K_{i})$ are non zero for all $i$. However, $\mathrm{Hom}_{K_{i}}(\sigma_{i},\pi_{i}|K_{i})$ are non zero for all $i$ if and only if $\pi_i$ are generic for all $i$ (by the simple type case for each $i$). Finally $\pi_i$ are generic for all $i$ if and only if $\pi$ is generic, because the supercuspidal supports of $\pi_i$ are pairwise disjoint and all segment are pairwise disjoint. This finishes the proof.
\end{proof}
We may now deduce the multiplicity one statement:
\begin{lemma}\label{4.34}
We have $\dim \mathrm{Hom}_K(\sigma_{min}(\lambda),\pi)=1$, for $\pi$ an irreducible generic representation of $G$ in $\Omega$.
\end{lemma}
\begin{proof} Let $x:=\mathfrak{m}_x \in \Spm \mathfrak{Z}_{\Omega}$ the maximal ideal defined by $\pi$ and $\kappa(x):= \mathfrak{Z}_{\Omega}/\mathfrak{m}_x$. Since $\pi$ is generic we have that $\mathrm{Hom}_K(\sigma_{min}(\lambda),\pi)\neq 0$ by Theorem \ref{1.18}. It follows that we have ${\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(x) \twoheadrightarrow \pi$. Since the functor $\mathrm{Hom}_K(\sigma_{min}(\lambda),.)$ is exact, we have $\mathrm{Hom}_K(\sigma_{min}(\lambda),{\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(x)) \twoheadrightarrow \mathrm{Hom}_K(\sigma_{min}(\lambda),\pi)$. Moreover by Frobenius reciprocity we have that
$$\mathrm{Hom}_K(\sigma_{min}(\lambda),{\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(x))$$
$$=\mathrm{Hom}_G({\mathrm{c\text{--} Ind}}_K^{G}\sigma_{min}(\lambda),{\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(x))$$
\noindent and by \cite[Lemma 5.2]{Pyv1}:
\[\mathrm{Hom}_G({\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda),{\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(x)) \]
\[\simeq \mathrm{Hom}_G({\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda),{\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda)) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(x)\]
Since $\sigma_{min}(\lambda)$ occurs with multiplicity one in $\mathrm{Ind}_J^K \lambda$, then by \cite[Corollary 7.2]{Pyv1}, we have \[ \mathfrak{Z}_{\Omega} \simeq \mathrm{Hom}_G({\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda),{\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda)).\] It follows that \[ \mathrm{Hom}_K(\sigma_{min}(\lambda),{\mathrm{c\text{--} Ind}}_K^{G} \sigma_{min}(\lambda) \otimes_{\mathfrak{Z}_{\Omega}}\kappa(x)) \simeq \kappa(x).\]Hence we have a surjective map of $\kappa(x)$-vector spaces: \[\kappa(x)\twoheadrightarrow \mathrm{Hom}_K(\sigma_{min}(\lambda),\pi)\]
Then $1\geq \dim \mathrm{Hom}_K(\sigma_{min}(\lambda),\pi)$ and this space is non-zero, hence it must be one-dimensional.
\end{proof}
\subsection*{Acknowledgments} The results of this paper are a part of the author's PhD thesis. The author is tremendously grateful to his advisor Vytautas Pa\v{s}k\={u}nas for sharing his ideas with the author and for many helpful discussions. We would like to thank Peter Schneider for pointing out the reference \cite{MR1915088}, which allowed to simplify some of our arguments. The author would also like to thank the referee for useful comments and corrections, which improved considerably the exposition of this paper. This work was supported by SFB/TR 45 of the DFG.
\bibliographystyle{alpha}
\addcontentsline{toc}{section}{References}
|
{
"timestamp": "2019-06-04T02:22:40",
"yymm": "1803",
"arxiv_id": "1803.02693",
"language": "en",
"url": "https://arxiv.org/abs/1803.02693"
}
|
\section{Introduction}
Tremendous advances in natural language processing (NLP) have been enabled by novel deep neural network architectures and word embeddings. Historically, convolutional neural network (CNN)\cite{lecun1998gradient,Zhang:charcnn} and recurrent neural network (RNN)\cite{elman1990finding,Yang:han} topologies have competed to provide state-of-the-art results for NLP tasks, ranging from text classification to reading comprehension. CNNs identify and aggregate patterns with increasing feature sizes, reflecting our common practice of identifying patterns, literal or idiomatic, for understanding language; they are thus adept at tasks involving key phrase identification. RNNs instead construct a representation of sentences by successively updating their understanding of the sentence as they read new words, appealing to the formally sequential and rule-based construction of language. While both networks display great efficacy at certain tasks \cite{yin2017comparative}, RNNs tend to be the more versatile, have emerged as the clear victor in, e.g., language translation \cite{Britz:seq2seq, chiu2017monotonic, Vaswani:attn}, and are typically more capable of identifying important contextual points through attention mechanisms for, e.g., reading comprehension \cite{Cui:comprehension, Xiong:comprehension, Kumar:dmn, Xiong:dmn+}. With an interest in NLP, we thus turn to RNNs.
RNNs nominally aim to solve a general problem involving sequential inputs. For various more specified tasks, specialized and constrained implementations tend to perform better \cite{Ruben:nested, Arjovsky:unitary, Wisdom:unitary, Vaswani:attn, Hyland:unitary, costa2017cortical, Jing:gorthornn, Kumar:dmn, Xiong:dmn+, Cui:comprehension, Xiong:comprehension}. Often, the improvement simply mitigates the exploding/vanishing gradient problem \cite{ Chung:gru, Hochreiter:lstm}, but, for many tasks, the improvement is more capable of generalizing the network's training for that task. Understanding better how and why certain networks excel at certain NLP tasks can lead to more performant networks, and networks that solve new problems.
Advances in word embeddings have furnished the remainder of recent progress in NLP \cite{Mokolov:word2vec, Pennington:glove, Nickel:poincare, Speer:conceptnet, Mikolov:pretrained, Wu:starspace}. Although it is possible to train word embeddings end-to-end with the rest of a network, this is often either prohibitive due to exploding/vanishing gradients for long corpora, or results in poor embeddings for rare words \cite{Bahdanau:e2e}. Embeddings are thus typically constructed using powerful, but heuristically motivated, procedures to provide pre-trained vectors on top of which a network can be trained. As with the RNNs themselves, understanding better how and why optimal embeddings are constructed in, e.g., end-to-end training can provide the necessary insight to forge better embedding algorithms that can be deployed pre-network training.
Beyond improving technologies and ensuring deep learning advances at a breakneck pace, gaining a better understanding of how these systems function is crucial for allaying public concerns surrounding the often inscrutable nature of deep neural networks. This is particularly important for RNNs, since nothing comparable to DeepDream or Lucid exists for them \cite{olah2018the}.
To these ends, the goal of this work is two fold. First, we wish to understand any emergent algebraic structure RNNs and word embeddings, trained end-to-end, may exhibit. Many algebraic structures are well understood, so any hints of structure would provide us with new perspectives from which and tools with which deep learning can be approached. Second, we wish to propose novel networks and word embedding schemes by appealing to any emergent structure, should it appear.
The paper is structured as follows. Methods and experimental results comprise the bulk of the paper, so, for faster reference, \S \ref{sec:summary} provides a convenient summary and intrepretation of the results, and outlines a new class of neural network and new word embedding scheme leveraging the results. \S \ref{sec:motiv_and_setup} motivates the investigation into algebraic structures and explains the experimental setup. \S \ref{sec:results} Discusses the findings from each of the experiments. \S \ref{sec:discussion} interprets the results, and motivates the proposed network class and word embeddings. \S \ref{sec:closing} provides closing remarks and discusses followup work, and \S \ref{sec:acknowledgements} gives acknowledgments.
To make a matter of notation clear going forward, we begin by referring to the space of words as $W$, and transition to $G$ after analyzing the results in order to be consistent with notation in the literature on algebraic spaces.
\section{Summary of results} \label{sec:summary}
We embedded words as vectors and used a uni-directional GRU connected to a dense layer to classify the account from which tweets may have originated. The embeddings and simple network were trained end-to-end to avoid imposing any artificial or heuristic constraints on the system.
There are two primary takeaways from the work presented herein:
\begin{itemize}
\item Words naturally embed as elements in a Lie group, $G$, and end-to-end word vectors may be related to the generating Lie algebra.
\item RNNs learn to parallel transport nonlinear representations of $G$ either on the Lie group itself, or on a principal $G$-bundle.
\end{itemize}
The first point follows since 1) words are embedded in a continuous space; 2) an identity word exists that causes the RNN to act trivially on a hidden state; 3) word inverses exist that cause the RNN to undo its action on a hidden state; 4) the successive action of the RNN using two words is equivalent to the action of the RNN with a single third word, implying the multiplicative closure of words; and 5) words are not manifestly closed under any other binary action.
The second point follows given that words embed on a manifold, sentences traces out paths on the manifold, and the difference equation the RNN solves bears a striking resemble to the first order equation for parallel transport,
\begin{align}
(h_{t+1} - h_t) + \gamma_{w_t} h_t =& 0 \\
\gamma_{w_t} \equiv& 1 - R_{w_t},
\end{align}
where $h_t$ is the $t$-th hidden state encountered when reading over a sentence and $R_{w_t}$ is the RNN conditioned by the $t$-th word, $w_t$, acting on the hidden state. Since sentences trace out a path on the word manifold, and parallel transport operators for representations of the word manifold take values in the group, the RNN must parallel transport hidden states either on the group itself or on a base space, $M$, equipped with some word field, $w:M\to G$, that connects the path in the base space to the path on the word manifold.
Leveraging these results, we propose two new technologies.
First, we propose a class of recurrent-like neural networks for NLP tasks that satisfy the differential equation
\begin{equation}
D^n_t h(t) = 0,
\end{equation}
where
\begin{align}
D_t = \partial_t + v^\mu(t) \Gamma_\mu,
\end{align}
and where $\Gamma$ and $v$ are learned functions. $n=1$ corresponds to traditional RNNs, with $v^\mu \Gamma_\mu \propto \gamma$. For $n>1$, this takes the form of RNN cells with either nested internal memories or dependencies that extend temporally beyond the immediately previous hidden state. In particular, using $n=2$ for sentence generation is the topic of a manuscript presently in preparation.
Second, we propose embedding schemes that explicitly embed words as elements of a Lie group. In practice, these embedding schemes would involve representing words as constrained matrices, and optimizing the elements, subject to the constraints, according to a loss function constructed from invariants of the matrices, and then applying the matrix log to obtain Lie vectors. A prototypical implementation, in which the words are assumed to be in the fundamental representation of the special orthogonal group, $SO(N)$, and are conditioned on losses sensitive to the relative actions of words, is the subject of another manuscript presently in preparation.
The proposals are only briefly discussed herein, as they are the focus of followup work; the focus of the present work is on the experimental evidence for the emergent algebraic structure of RNNs and embeddings in NLP.
\section{Motivation and experimental setup} \label{sec:motiv_and_setup}
\subsection{Intuition and motivation} \label{sec:motivation}
We provide two points to motivate examining the potential algebraic properties of RNNs and their space of inputs in the context of NLP.
First, a RNN provides a function, $R$, that successively updates a hidden memory vector, $h \in H$, characterizing the information contained in a sequence of input vectors, $\left\{w_1, w_2, \dots \right\} \in W$, as it reads over elements of the sequence. Explicitly, $R : W \times H \to H$. At face value, $R$ takes the same form as a (nonlinear) representation of some general algebraic structure, $W$, with at least a binary action, $\cdot : W \times W \to W$, on the vector space $H$. While demanding much structure on $W$ generally places a strong constraint on the network's behavior, it would be fortuitous for such structure to emerge. Generally, constrained systems still capable of performing a required task will perform the task better, or, at least, generalize more reliably \cite{pascanu2013number, montufar2014number, livni2014computational, telgarsky2016benefits, poggio2017and, kawaguchi2017generalization}. To this end, the suggestive form RNNs assume invites further examination to determine if there exist any reasonable constraints that may be placed on the network. To highlight the suggestiveness of this form in what follows, we represent the $W$ argument of $R$ as a subscript and the $H$ argument by treating $R$ as a left action on $H$, adopting the notation $R(w, h) \to R_w h$. Since, in this paper, we consider RNNs vis-\`{a}-vis NLP, we take $W$ as the (continuous) set of words\footnote{Traditionally, words are treated as categorical objects, and embedding them in a continuous (vector) space for computational purposes is largely a convenience; however, we relax this categorical perspective, and treat unused word vectors as acceptable objects as far as the algebraic structure is concerned, even if they are not actively employed in language.}.
Second, in the massive exploration of hyperparameters presented in \cite{Britz:seq2seq}, it was noted that, for a given word embedding dimension, the network's performance on a seq2seq task was largely insensitive to the hidden dimension of the RNN above a threshold ($\sim$128). The dimension of admissible representations of a given algebraic structure is generally discrete and spaced out. Interpreting neurons as basis functions and the output of layers as elements of the span of the functions \cite{Raghu:svcca, csaji2001approximation, hornik1991approximation}, we would expect a network's performance to improve until an admissible dimension for the representation is found, after which the addition of hidden neurons would simply contribute to better learning the components of the proper representation by appearing in linear combinations with other neurons, and contribute minimally to improving the overall performance. In their hyperparameter search, a marginal improvement was found at a hidden dimension of 2024, suggesting a potentially better representation may have been found.
These motivating factors may hint at an underlying algebraic structure in language, at least when using RNNs, but they raise the question: what structures are worth investigating?
Groups present themselves as a candidate for consideration since they naturally appear in a variety of applications. Unitary weight matrices have already enjoyed much success in mitigating the exploding/vanishing gradients problem \cite{Arjovsky:unitary,Wisdom:unitary}, and RNNs even further constrained to act explicitly as nonlinear representations of unitary groups offer competitive results \cite{Hyland:unitary}. Moreover, intuitively, RNNs in NLP could plausibly behave as a group since: 1) the RNN must learn to ignore padding words used to square batches of training data, indicating an identity element of $W$ must exist; 2) the existence of contractions, portmanteaus, and the Germanic tradition of representing sentences as singular words suggest $W$ might be closed; and 3) the ability to backtrack and undo statements suggests language may admit natural inverses - that is, active, controlled ``forgetting" in language may be tied to inversion. Indeed, groups seem reasonably promising.
It is also possible portmanteaus only make sense for a finite subset of pairs of words, so $W$ may take on the structure of a groupoid instead; moreover, it is possible, at least in classification tasks, that information is lost through successive applications of $R$, suggesting an inverse may not actually exist, leaving $W$ as either a monoid or category. $W$ may also actually admit \textit{additional} structure, or an additional binary operation, rendering it a ring or algebra.
To determine what, if any, algebraic structure $W$ possesses, we tested if the following axiomatic properties of faithful representations of $W$ hold:
\begin{enumerate}
\item (Identity) $\exists I \in W$ such that $\forall h \in H$, $R_I h = h$
\item (Closure under multiplication) $\forall w_1, w_2 \in W$, $\exists w_3 \in W$ such that $\forall h \in H$, $R_{w_2} R_{w_1} h = R_{w_3} h$
\item (Inverse) $\forall w \in W$, $\exists w^{-1} \in W$ such that $\forall h \in H$, $R_{w^{-1}} R_w h = R_w R_{w^{-1}} h = h$
\item (Closure under Lie bracket) $\forall w_1, w_2 \in W$, $\exists w_3 \in W$ such that $\forall h \in H$, $R_{w_2} R_{w_1} h - R_{w_1} R_{w_2} h = R_{w_3} h$
\end{enumerate}
Closure under Lie bracket simultaneously checks for ring and Lie algebra structures.
Whatever structure, if any, $W$ possesses, it must additionally be continuous since words are typically embedded in continuous spaces. This implies Lie groups (manifolds), Lie semigroups with an identity (also manifolds), and Lie algebras (vector spaces with a Lie bracket) are all plausible algebraic candidates.
\subsection{Data and methods}
We trained word embeddings and a uni-directional GRU connected to a dense layer end-to-end for text classification on a set of scraped tweets using cross-entropy as the loss function. End-to-end training was selected to impose as few heuristic constraints on the system as possible. Each tweet was tokenized using NLTK TweetTokenizer and classified as one of 10 potential accounts from which it may have originated. The accounts were chosen based on the distinct topics each is known to typically tweet about. Tokens that occurred fewer than 5 times were disregarded in the model. The model was trained on 22106 tweets over 10 epochs, while 5526 were reserved for validation and testing sets (2763 each). The network demonstrated an insensitivity to the initialization of the hidden state, so, for algebraic considerations, $(\frac{1}{\sqrt{n}}, \frac{1}{\sqrt{n}}, \dots )$ was chosen for hidden dimension of $n$. A graph of the network is shown in Fig.(\ref{fig:network}).
Algebraic structures typically exhibit some relationship between the dimension of the structure and the dimension of admissible representations, so exploring the embedding and hidden dimensions for which certain algebraic properties hold is of interest. Additionally, beyond the present interest in algebraic properties, the network's insensitivity to the hidden dimension invites an investigation into its sensitivity to the word embedding dimension. To address both points of interest, we extend the hyperparameter search of \cite{Britz:seq2seq}, and perform a comparative search over embedding dimensions and hidden dimensions to determine the impact of each on the network's performance and algebraic properties. Each dimension in the hyperparameter pair, $(m,n) = (\text{embedding dim}, \text{hidden dim})$, runs from 20 to 280 by increments of 20.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.7]{./graphics/network.png}
\caption{\footnotesize The simple network trained as a classifier: GRU$\to$Dense$\to$Linear$\to$Softmax. There are 10 nonlinear neurons dedicated to each of the final 10 energies that are combined through a linear layer before softmaxing. This is to capitalize on the universal approximation theorem's implication that neurons serve as basis functions - i.e. each energy function is determined by 10 basis functions. The hidden dimension $n$ of the GRU, and the word embedding dimension, are hyperparameters that are scanned over.} \label{fig:network}
\end{center}
\end{figure}
After training the network for each hyperparameter pair, the GRU model parameters and embedding matrix were frozen to begin testing for emergent algebraic structure. To satisfy the common ``$\forall h\in H$" requirement stated in \S \ref{sec:motivation}, real hidden states encountered in the testing data were saved to be randomly sampled when testing the actions of the GRU on states. 7 tests were conducted for each hyperparameter pair with randomly selected states:
\begin{enumerate}
\item Identity (``arbitrary identity")
\item Inverse of all words in corpus (``arbitrary inverse")
\item Closure under multiplication of arbitrary pairs of words in total corpus (``arbitrary closure")
\item Closure under commutation of arbitrary pairs of words in total corpus (``arbitrary commutativity")
\item Closure under multiplication of random pairs of words from within each tweet (``intra-sentence closure")
\item Closure of composition of long sequences of words in each tweet (``composite closure")
\item Inverse of composition of long sequences of words in each tweet (``composite inverse")
\end{enumerate}
Tests 6 and 7 were performed since, if closure is upheld, the composition of multiple words must also be upheld. These tests were done to ensure mathematical consistency.
To test for the existence of ``words" that satisfy these conditions, vectors were searched for that, when inserted into the GRU, minimized the ratio of the Euclidean norms of the difference between the ``searched" hidden vector and the correct hidden vector. For concreteness, the loss function for each algebraic property from \S \ref{sec:motivation} were defined as follows:
\begin{enumerate}
\begin{subequations} \label{eqn:losses}
\item (Identity)
\begin{align}
\mathcal{L}_\text{id} =& \frac{|R_x h - h|}{|h|} \label{eqn:id_loss}
\end{align}
\item (Closure under multiplication)
\begin{align}
\mathcal{L}_\text{closure} =& \frac{|R_{x} h - R_{w_2} R_{w_1} h|}{|R_{w_2} R_{w_1} h|}. \label{eqn:clo_loss}
\end{align}
\item (Inverse)
\begin{align}
\mathcal{L}_\text{inverse} =& \frac{|R_{x} R_w h - h|}{|h|} \label{eqn:inv_loss}
\end{align}
\item (Closure under Lie bracket)
\begin{align}
\mathcal{L}_\text{com} =& \frac{|R_x h - (R_{w_2} R_{w_1} h - R_{w_1} R_{w_2} h)|}{\max\left(|(R_{w_2} R_{w_1} h - R_{w_1} R_{w_2} h)|,10^{-6}\right)} \label{eqn:com_loss}
\end{align}
\end{subequations}
\end{enumerate}
where $w_i$ are random, learned word vectors, $h$ is a hidden state, and $x$ is the model parameter trained to minimize the loss. We refer to Eqs.(\ref{eqn:losses}) as the ``axiomatic losses." It is worth noting that the non-zero hidden state initialization was chosen to prevent the denominators from vanishing when the initial state is selected as a candidate $h$ in Eqs.(\ref{eqn:id_loss})\&(\ref{eqn:inv_loss}). The reported losses below are the average across all $w$'s and $h$'s that were examined. Optimization over the losses in Eqs.(\ref{eqn:losses}) was performed over 5000 epochs. For the associated condition to be satisfied, there must exist a word vector $x$ that sufficiently minimizes the axiomatic losses.
If it is indeed the case that the GRU attempts to learn a representation of an algebraic structure and each neuron serves as a basis function, it is not necessary that each neuron individually satisfies the above constraints. For clarity, recall the second motivating point that the addition of neurons, once a representation is found, simply contributes to learning the representation better. Instead, only a linear combination of the neurons must. We consider this possibility for the most task-performant hyperparameter pair, and two other capricious pairs. The target dimension of the linear combination, $p$, which we refer to as the ``latent dimension," could generally be smaller than the hidden dimension, $n$. To compute the linear combination of the neurons, the outputs of the GRU were right-multiplied by a $n \times p$ matrix, $P$\footnote{There may be concern over the differing treatment of the output of the GRU and the input hidden state, since the former is being projected into a lower dimension while the latter is not. If the linear combination matrix were trained in parallel to the GRU itself, there would be a degeneracy in the product between it and the GRU weight matrices in successive updates, and it and the dense layer weight matrices after the final update, such that the effect of the linear combination would be absorbed by the weight matrices. To this end, since we are considering linear combinations after freezing the GRU weight matrices, it is unnecessary to consider the role the linear combination matrix would play on the input hidden states, and necessary for only the output of the GRU itself.}:
\begin{subequations}
\begin{align} \label{eqn:lin_comb}
h \to& h P \\
R_w h \to& (R_w h) P \\
R_{w_2} R_{w_1} h \to& (R_{w_2} R_{w_1} h) P
\end{align}
\end{subequations}
Since the linear combination is not \`{a} priori known, $P$ is treated as a model parameter.
The minimization task previously described was repeated with this combinatorial modification while scanning over latent dimensions, $p \in [20,n-20]$, in steps of 20. The test was performed 10 times and the reported results averaged for each value of $p$ to reduce fluctuations in the loss from differing local minima. $P$ was trained to optimize various combinations of the algebraic axioms, the results of which were largely found to be redundant. In \S \ref{sec:results}, we address the case in which $P$ was only trained to assist in optimizing a single condition, and frozen in other axiomatic tests; the commutative closure condition, however, was given a separate linear combination matrix for reasons that will be discussed later.
Finally, the geometric structure of the resulting word vectors was explored, naively using the Euclidean metric. Sentences trace out (discrete) paths in the word embedding space, so it was natural to consider relationships between both word vectors and vectors ``tangent" to the sentences' paths. Explicitly, the angles and distances between
\begin{enumerate}
\item random pairs of words
\item all words and the global average word vector
\item random pairs of co-occurring words
\item all words with a co-occurring word vector average
\item adjacent tangent vectors
\item tangent vectors with a co-occurring tangent vector average
\end{enumerate}
were computed to determine how word vectors are geometrically distributed. Intuitively, similar words are expected to affect hidden states similarly. To test this, and to gain insight into possible algebraic interpretations of word embeddings, the ratio of the Euclidean norm of the difference between hidden states produced by acting on a hidden state with two different words to the Euclidean norm of the original hidden state was computed as a function of the popular cosine similarity metric and distance between embeddings. This fractional difference, cosine similarity, and word distance were computed as,
\begin{align}
\mathfrak{E} =& \frac{|R_{w_1} h - R_{w_2} h|}{|h|} \label{eqn:frac_dist} \\
\cos(\theta_w) =& w_1^\alpha w_2^\alpha, \label{eqn:cos_sim}\\
|\Delta w| =& ||w_1 - w_2||_2 \label{eqn:dist_sim},
\end{align}
where Einstein summation is applied to the (contravariant) vector indices.
High-level descriptions of the methods will be briefly revisited in each subsection of \S \ref{sec:results} so that they are more self-contained and pedagogical.
\section{Results} \label{sec:results}
\subsection{Hyperparameters and model accuracy}
We performed hyperparameter tuning over the word embedding dimension and the GRU hidden dimension to optimize the classifier's accuracy. Each dimension ran from 20 to 280 in increments of 20. A contour plot of the hyperparameter search is shown in Fig.(\ref{fig:class_acc}).
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.5]{./graphics/classification_accuracy.png}
\caption{\footnotesize The range of the model accuracy is $[50.1\%, 89.7\%]$.} \label{fig:class_acc}
\end{center}
\end{figure}
For comparison, using pretrained, 50 dimensional GloVe vectors with this network architecture typically yielded accuracies on the order of $\mathcal{O}(50\%)$ on this data set, even for more performant hidden dimensions. Thus, training the embeddings end-to-end is clearly advantageous for short text classification. It is worth noting that training them end-to-end is viable primarily because of the short length of tweets; for longer documents, exploding/vanishing gradients typically prohibits such training.
The average Fisher information of each hyperparameter dimension over the searched region was computed to determine the relative sensitivities of the model to the hyperparameters. The Fisher information for the hidden dimension was $4.63 \times 10^{-6}$; the Fisher information for the embedding dimension was $2.62 \times 10^{-6}$. Evidently, by this metric, the model was, on average in this region of parameter space, 1.76 times more sensitive to the hidden dimension than the embedding dimension. Nevertheless, a larger word embedding dimension was critical for the network to realize its full potential.
The model performance generally behaved as expected across the hyperparameter search. Indeed, higher embedding and hidden dimensions tended to yield better results. Given time and resource constraints, the results are not averaged over many search attempts. Consequently, it is unclear if the pockets of entropy are indicative of anything deeper, or merely incidental fluctuations. It would be worthwhile to revisit this search in future work.
\subsection{Algebraic properties}
Seven tests were conducted for each hyperparameter pair to explore any emergent algebraic structure the GRU and word embeddings may exhibit. Specifically, the tests searched for 1) the existence of an identity element, 2) existence of an inverse word for each word, 3) multiplicative closure for arbitrary pairs of words, 4) commutative closure for arbitrary pairs of words, 5) multiplicative closure of pairs of words that co-occur within a tweet, 6) multiplicative closure of all sequences of words that appear in tweets, and 7) the existence of an inverse for all sequences of words that appear in tweets. The tests optimized the axiomatic losses defined in Eqs.(\ref{eqn:losses}).
In what follows, we have chosen $\mathcal{L}<0.01$ (or, $1\%$ error) as the criterion by which we declare a condition ``satisfied."
The tests can be broken roughly into two classes: 1) arbitrary solitary words and pairs of words, and 2) pairs and sequences of words co-occurring within a tweet. The results for class 1 are shown in Fig.(\ref{fig:arb_elems}); the results for class 2 are shown in Fig.(\ref{fig:sent_elems}).
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/arb_id.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/arb_inverse.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/arb_closure.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/arb_commutativity.png}}
\end{tabular}
\caption{\footnotesize The \% axiomatic error as a function of the word embedding and GRU hidden dimensions. (a) The existence of an identity element for multiple hidden states. Note the log scale. (b) The existence of an inverse word for every word acting on random hidden states. Linear scale. (c) The existence of a third, `effective' word performing the action of two randomly chosen words in succession, acting on random states. Linear scale. (d) The existence of a third word performing the action of the commutation of two randomly chosen words, acting on random states. Nonlinear scale.} \label{fig:arb_elems}
\end{center}
\end{figure}
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/comp_closure.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/comp_inverse.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/sent_closure.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/arb_inverse.png}}
\end{tabular}
\caption{\footnotesize The \% axiomatic error as a function of the word embedding and GRU hidden dimensions. (a) The existence of a third word performing the action of all, ordered words comprising a tweet, acting on the initial state. Linear scale. (b) The existence of a word that reverses the action of the ordered words comprising a tweet that acted on the initial state. Nonlinear scale. (c) The existence of a third word performing the action of two random words co-occurring within a tweet, acting on random states. Linear scale. (d) The existence of an inverse word for every word acting on random hidden states. This is the same as in Fig.(\ref{fig:arb_elems}), and is simply provided for side-by-side comparison.} \label{fig:sent_elems}
\end{center}
\end{figure}
The identity condition was clearly satisfied for virtually all embedding and hidden dimensions, with possible exceptions for small embedding dimensions and large hidden dimensions. Although we did not explicitly check, it is likely that even the possible exceptions would be viable in the linear combination search.
Arbitrary pairs of words were evidently not closed under multiplication without performing a linear combination search, with a minimum error of $7.51\%$ across all dimensions. Moreover, the large entropy across the search does not suggest any fundamentally interesting or notable behavior, or any connections between the embedding dimension, hidden dimension, and closure property.
Arbitrary pairs of words were very badly not closed under commutation, and it is unfathomable that even a linear combination search could rescue the property. One might consider the possibility that specific pairs of words might have still closed under commutation, and that the exceptional error was due to a handful of words that commute outright since this would push the loss up with a near-vanishing denominator. As previously stated, the hidden states were not initialized to be zero states, and separate experiments confirm that the zero state was not in the orbit of any non-zero state, so there would have been no hope to negate the vanishing denominator. Thus, this concern is in principle possible. However, explicitly removing examples with exploding denominators (norm$<10^{-3}$) from the loss when performing linear combination searches still resulted in unacceptable errors ($80\% +$), so this possibility is not actually realized. We did not explicitly check for this closure in class 2 tests since class 2 is a subset of class 1, and such a flagrant violation of the condition would not be possible if successful closure in class 2 were averaged into class 1 results. Even though commutative closure is not satisfied, it is curious to note that the error exhibited a mostly well-behaved stratification.
The most interesting class 1\footnote{The arbitrary inverse is neither, strictly, class 1 nor class 2 since it does not involve pairings with any other words. We simply group it with class 1 to keep it distinct from the composite inverse experiment, which is decidedly class 2.} result was the arbitrary inverse. For embedding dimensions sufficiently large compared to the hidden dimension, inverses clearly existed even without a linear combination search. Even more remarkable was the well-behaved stratification of the axiomatic error, implying a very clear relationship between the embedding dimension, hidden dimension, and emergent algebraic structure of the model. It is not unreasonable to expect the inverse condition to be trivially satisfied in a linear combination search for a broad range of hyperparameter pairs.
The same behavior of the inverse property is immediately apparent in all class 2 results. The stratification of the error was virtually identical, and all of the tested properties have acceptable errors for sufficiently large embedding dimensions for given hidden dimensions, even without a linear combination search.
\subsection{Linear combination search}
The optimal hyperparameter pair for this single pass of tuning was $(m,n)=(280,220)$, which resulted in a model accuracy of $89.7\%$. This was not a statistically significant result since multiple searches were not averaged, so random variations in validation sets and optimization running to differing local minima may have lead to fluctuations in the test accuracies. However, the selection provided a reasonable injection point to investigate the algebraic properties of linear combinations of the output of the GRU's neurons. For comparison, we also considered $(m,n)=(180,220)$ and $(m,n)=(100,180)$.
The tests were run with the linear combination matrix, $P$, trained to assist in optimizing the composite inverse. The learned $P$ was then applied to the output hidden states for the other properties except for commutative closure, which was given its own linear combination matrix to determine if any existed that would render it an emergent property.
The combination was trained to optimize a single condition because, if there exists an optimal linear combination for one condition, and there indeed exists an underlying algebraic structure incorporating other conditions, the linear combination would be optimal for all other conditions.
Initial results for the $(m,n) = (280,220)$ search is shown in Figs.(\ref{fig:lin_comb_grp_280_220})\&(\ref{fig:lin_comb_bad_280_220}). Well-optimized properties are shown in Fig.(\ref{fig:lin_comb_grp_280_220}), while the expected poorly-optimized properties are shown in Fig.(\ref{fig:lin_comb_bad_280_220}).
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/comp_clo_1.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/comp_inv_1.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/geo_clo_1.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/inv_1.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (280,220)$. Graphs of \% axiomatic error for the satisfied conditions after a linear combination search. The graphs are ordered as they were in Fig.(\ref{fig:sent_elems})} \label{fig:lin_comb_grp_280_220}
\end{center}
\end{figure}
The four conditions examined in Fig.(\ref{fig:lin_comb_grp_280_220}) are clearly satisfied for all latent dimensions. They all also reach a minimum error in the same region. Composite closure, intra-sentence closure, and arbitrary inverse are all optimized for $p\approx 100$; composite inverse is optimized for $p\approx 120$, though the variation in the range $[100,140]$ is small ($\sim 0.8\%$ variation around the mean, or an absolute variation of $\sim 0.006\%$ in the error).
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/arb_clo_1.png}}&
\sidesubfloat[]{\includegraphics[scale=0.45]{./graphics/lin_comb/280_220/arb_com_1.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (280,220)$. Graphs of \% axiomatic error for the unsatisfied conditions after a linear combination search.} \label{fig:lin_comb_bad_280_220}
\end{center}
\end{figure}
Arbitrary multiplicative closure and commutative closure are highly anti-correlated, and both conditions are badly violated. It is worth noting that the results in Fig.(\ref{fig:lin_comb_bad_280_220})(b) did not remove commutative pairs of words from the error, and yet the scale of the error in the linear combination search is virtually identical to what was separately observed with the commutative pairs removed. They both also exhibit a monotonic dependence on the latent dimension. Despite their violation, this dependence is well-behaved, and potentially indicative of some other structure.
Before discussing the linear combination searches for the other selected hyperparameter pairs, it is worthwhile noting that retraining the network and performing the linear combination search again can yield differing results. Figs.(\ref{fig:lin_comb_grp_280_220_2})\&(\ref{fig:lin_comb_bad_280_220_2}) show the linear combination results after retraining the model for the same hyperparameter pair, with a different network performance of $87.3\%$.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/comp_clo_2.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/comp_inv_2.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/geo_clo_2.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/inv_2.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (280,220)$, retrained. Graphs of \% axiomatic error for the satisfied conditions after a linear combination search. The graphs are ordered as they were in Fig.(\ref{fig:sent_elems})} \label{fig:lin_comb_grp_280_220_2}
\end{center}
\end{figure}
Qualitatively, the results are mostly the same: there is a common minimizing region of $p$, and conditions are satisfied, at least in the common minimal region. However, the minimizing region starkly shifted down, and became sharper for composite closure, intra-sentence closure, and arbitrary inverse.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/arb_clo_2.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/280_220/arb_com_2.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (280,220)$, retrained. Graphs of \% axiomatic error for the unsatisfied conditions after a linear combination search.} \label{fig:lin_comb_bad_280_220_2}
\end{center}
\end{figure}
Once more, the results are mostly the same. Arbitrary closure error drastically increased, but both are still highly anti-correlated, and mostly monotonic, despite the erratic fluctuations in the arbitrary closure error.
Figs.(\ref{fig:lin_comb_grp_180_220})\&(\ref{fig:lin_comb_bad_180_220}) show the linear combination search for $(m,n) = (180,220)$. The model was retrained, and achieved $90.1\%$ for the displayed results.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/180_220/comp_clo.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/180_220/comp_inv.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/180_220/geo_clo.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/180_220/inv.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (180,220)$. Graphs of \% axiomatic error for the satisfied conditions after a linear combination search. The graphs are ordered as they were in Fig.(\ref{fig:sent_elems})} \label{fig:lin_comb_grp_180_220}
\end{center}
\end{figure}
Interestingly, the optimal latent dimension occurs significantly higher than for the other reported hyperparameter pairs. This result, however, is not true for all retrainings at this $(m,n)$ pair.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/180_220/arb_clo.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/180_220/arb_com.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (180,220)$. Graphs of \% axiomatic error for the unsatisfied conditions after a linear combination search.} \label{fig:lin_comb_bad_180_220}
\end{center}
\end{figure}
The entropy in the arbitrary closure loss increased, and the commutative closure loss seemed to asymptote at higher latent dimension.
Figs.(\ref{fig:lin_comb_grp_100_180})\&(\ref{fig:lin_comb_bad_100_180}) show the linear combination search for $(m,n) = (100,180)$. The model was retrained, and achieved $87.1\%$ for the displayed results.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/100_180/comp_clo.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/100_180/comp_inv.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/100_180/geo_clo.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/100_180/arb_inv.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (100,180)$. Graphs of \% axiomatic error for the satisfied conditions after a linear combination search. The graphs are ordered as they were in Fig.(\ref{fig:sent_elems})} \label{fig:lin_comb_grp_100_180}
\end{center}
\end{figure}
At lower dimensions, the optimal latent dimension was no longer shared between the satisfied conditions.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/100_180/arb_clo.png}}&
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/lin_comb/100_180/arb_com.png}}
\end{tabular}
\caption{\footnotesize $(m,n) = (100,180)$. Graphs of \% axiomatic error for the unsatisfied conditions after a linear combination search.} \label{fig:lin_comb_bad_100_180}
\end{center}
\end{figure}
The unsatisfied conditions displayed mostly the same behavior at lower dimensions.
\subsection{Embedding structure}
To explore the geometric distribution of word vectors, the angles and distances between 1) random pairs of words, 2) all words and the global average word vector, 3) random pairs of co-occurring words, 4) all words with a co-occurring word vector average, 5) adjacent tangent vectors, 6) tangent vectors with a co-occurring tangent vector average were computed. The magnitudes of the average word vectors, average co-occurring word vectors, and average tangent vectors were also computed.
Additionally, the relative effect of words on states is computed verses their cosine similarities and relative distances, measured by Eqs.(\ref{eqn:frac_dist})-(\ref{eqn:dist_sim}).
In the figures that follow, there are, generally, three categories of word vectors explored: 1) random word vectors from the pool of all word vectors, 2) co-occurring word vectors, and 3) tangent vectors (the difference vector between adjacent words).
Fig.(\ref{fig:norm_dist}) shows the distribution in the Euclidean norms of the average vectors that were investigated.
\begin{figure}[H]
\begin{center}
\includegraphics[scale=0.5]{./graphics/subspace/norm_avg.png}
\caption{\footnotesize The frequency distribution of the norm of average vectors. There was one instance of a norm for the average of all word vectors, hence the singular spike for its distribution. The other vector distributions were over the average for different individual tweets.} \label{fig:norm_dist}
\end{center}
\end{figure}
The tangent vectors and average word vectors had comparable norms. The non-zero value of the average word vector indicates that words do not perfectly distribute throughout space. The non-zero value of the average tangent vectors indicates that tweets in general progress in a preferred direction relative to the origin in embedding space; albeit, since the magnitudes are the smallest of the categories investigated, the preference is only slight. The norm of the average of co-occurring word vectors is significantly larger than the norms of others categories of vectors, indicating that the words in tweets typically occupy a more strongly preferred region of embedding space (e.g. in a cone, thus preventing component-wise cancellations when computing the average).
Fig.(\ref{fig:cos_sim_dist}) shows the distribution of the Euclidean cosine similarities of both pairs of vectors and vectors relative to the categorical averages.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/subspace/cos_sim.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/subspace/cos_sim_avg.png}}
\end{tabular}
\caption{\footnotesize Distributions of cosine similarities of vectors with respect to (a) other vectors (b) category average vectors. Averages were taken as they were in Fig.(\ref{fig:norm_dist}). } \label{fig:cos_sim_dist}
\end{center}
\end{figure}
The cosine similarity of pairs of random words and co-occurring words shared a very common distribution, albeit with the notable spikes are specific angles and a prominent spike at $\cos(\theta_w) = 1$ for co-occurring pairs. The prominent spike could potentially be explained by the re-occurrence of punctuation within tweets, so it may not indicate anything of importance; the potential origin of the smaller spikes throughout the co-occurring distribution is unclear. Generally, the pairs strongly preferred to be orthogonal, which is unsurprising given recent investigations into the efficacy of orthogonal embeddings \cite{Choromanski:orthoembed}. Adjacent pairs of tangent vectors, however, exhibited a very strong preference for obtuse relative angles, with a spike at $\cos(\theta_w) = -1$.
Words tended to have at most a very slightly positive cosine similarity to the global average, which is again indicative of the fact words did not spread out uniformly. Co-occurring words tended to form acute angles with respect to the co-occurring average. Meanwhile, tangent vectors strongly preferred to be orthogonal to the average.
The strong negative cosine similarity of adjacent tangent vectors, and the strong positive cosine similarity of words with their co-occurring average, indicate co-occurring words \textit{tended} to form a grid structure in a cone. That is, adjacent words tended to be perpendicular to each other in the positive span of some set of word basis vectors. Of course, this was not strictly adhered to, but the preferred geometry is apparent.
Fig.(\ref{fig:dist_dist}) shows the distribution of the Euclidean distances of both pairs of vectors and vectors relative to the categorical averages.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/subspace/dist.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/subspace/dist_avg.png}}
\end{tabular}
\caption{\footnotesize Distributions of the Euclidean distances of vectors to (a) other vectors (b) category average vectors. Averages were taken as they were in Fig.(\ref{fig:norm_dist}).} \label{fig:dist_dist}
\end{center}
\end{figure}
Distributions of random pairs of words and co-occurring words were virtually identical in both plots, indicating that most of the variation is attributable to the relative orientations of the vectors rather than the distances between them.
Fig.(\ref{fig:word_sim}) shows the correlation of the similarity of the action of pairs of words to their cosine similarity and distances apart.
\begin{figure}[H]
\begin{center}
\begin{tabular}{cc}
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/subspace/word_sim.png}}\\
\sidesubfloat[]{\includegraphics[scale=0.5]{./graphics/subspace/word_sim_dist.png}}
\end{tabular}
\caption{\footnotesize Plots of $\mathfrak{E}$ with respect to (a) word cosine similarity, $\cos(\theta_w)$ (b) distance between words, $|\Delta w|$. Eqs.(\ref{eqn:frac_dist})-(\ref{eqn:dist_sim})} \label{fig:word_sim}
\end{center}
\end{figure}
Both plots confirm that the more similar words are, the more similar their actions on the hidden states are. The strongly linear, bi-modal dependence of the fractional difference on the distance between words indicates that word distance is a stronger predictor of the relative meaning of words than the popular cosine similarity.
\section{Discussion} \label{sec:discussion}
\subsection{Interpretation of results}
The important take-aways from the results are:
\begin{itemize}
\item The GRU trivially learned an identity `word'.
\item The action of the GRU for any individual word admits an inverse for sufficiently large embedding dimension relative to the hidden dimension.
\item The successive action of the GRU for any arbitrary pair of words is not, generally, equivalent to the action of the GRU for any equivalent third `word'.
\item The commutation of successive actions of the GRU for any arbitrary pair of words is not equivalent to the action of the GRU for any equivalent third `word'.
\item The successive action of the GRU for any co-occurring pair of words is equivalent to the action of the GRU for an equivalent third `word' for sufficiently large embedding dimension relative to the hidden dimension.
\item The successive action of the GRU for any series of co-occuring words is equivalent to the action of the GRU for an equivalent `word' for sufficiently large embedding dimension relative to the hidden dimension.
\item The action of the GRU for any series of co-occurring words admits an inverse for sufficiently large embedding dimension relative to the hidden dimension.
\item Any condition satisfied for a sufficiently large embedding dimension relative to the hidden dimension is true for any pair of dimensions given an appropriate linear combination of the outputs of the GRU projected into an appropriate lower dimension (latent dimension).
\item The axiomatic errors for all satisfied conditions for the most performant models are minimized for specific, shared latent dimensions, and increases away from these latent dimensions; the optimal latent dimension is not shared for sufficiently small embedding dimensions.
\item Models with lower test performance tend to optimally satisfy these conditions for lower latent dimensions.
\item Co-occurring word vectors tend to be perpendicular to each other and occupy a cone in embedding space.
\item The difference of the action of two word vectors on a hidden state increases linearly with the distance between the two words, and follows a generally bi-modal trend.
\end{itemize}
Although there are still several outstanding points to consider, we offer an attempt to interpret these results in this section.
Identity, inverse, and closure properties for co-occurring words are satisfied, and in such a way that they are all related under some algebraic structure. Since closure is not satisfied for arbitrary pairs of words, there are, essentially, two possible explanations for the observed structure:
\begin{enumerate}
\item The union of all sets of co-occurring words is the Cartesian product of multiple Lie groups:
\begin{equation}
W = G_1 \times G_2 \times \dots, \label{eqn:cartesian_grp}
\end{equation}
where $W$ is the space of words, and $G_i$ is a Lie group. Since multiplication between groups is not defined, the closure of arbitrary pairs of words is unsatisfied.
\item The GRU's inability to properly close pairs of words it has never encountered together is the result of the generalization problem, and all words consequently embed in a larger Lie group:
\begin{equation}
G_1 \times G_2 \times \dots \subset G = W. \label{eqn:total_grp}
\end{equation}
\end{enumerate}
In either case, words can be considered elements of a Lie group. Since Lie groups are also manifolds, the word vector components can be interpreted as coordinates on this Lie group. Traditionally, Lie groups are practically handled by considering the Lie algebra that generates them, $\mathfrak{g}, G = \exp(\mathfrak{g})$. The components of the Lie vectors in $\mathfrak{g}$ are then typically taken to be the coordinates on the Lie group. This hints at a connection between $\mathfrak{g}$ and the word vectors, but this connection was not made clear by any of the experiments. Furthermore, RNNs learn a nonlinear representation of the group on some latent space spanned by the hidden layer.
Since sentences form paths on the embedding group, it's reasonable to attempt to form a more precise interpretation of the action of RNNs. We begin by considering their explicit action on hidden states as the path is traversed:
\begin{subequations}
\begin{align}
h_{t+1} =& R_{w_t} h_t \implies \label{eqn:rnn_action} \\
(h_{t+1} - h_t) + \gamma_{w_t} h_t =& 0, \label{eqn:rnn_connection} \\
\gamma_{w_t} \equiv& 1 - R_{w_t}.
\end{align}
\end{subequations}
Eq.(\ref{eqn:rnn_connection}) takes the form of a difference equation. In particular, it looks very similar to the finite form of the differential equation governing the nonlinear parallel transport along a path, $m(t)$, on a principal fibre bundle with base space $M$ and group $G=\exp(\mathfrak{g})$. If the tangent vector at $m(t)$ is $v(t)$, and the vector being transported at $m(t)$ is $h(t)$ then we have
\begin{align}
\partial_t h(t) + v^\mu(t) \Gamma_\mu[m(t), h(t)] =&0, \label{eqn:parallel_transport}
\end{align}
where $\Gamma$ is the (nonlinear) connection at $m(t)$. If $v$ were explicitly a function of $m$, Eq.(\ref{eqn:parallel_transport}) would take a more familiar form:
\begin{subequations}
\begin{align}
\partial_t h(t) + \gamma_{m(t)}[h(t)] =& 0, \label{eqn:geodesic_transport} \\
\gamma_{m(t)} \equiv& v^\mu[m(t)] \Gamma_\mu[m(t), h(t)]. \label{eqn:rnn_connection_relation}
\end{align}
\end{subequations}
Given the striking resemblance between Eqs.(\ref{eqn:geodesic_transport})\&(\ref{eqn:rnn_connection}), is it natural to consider either
\begin{enumerate}
\item The word embedding group serving as the base space, $M=G$, so that the path $m(t)$ corresponds explicitly to the sentence path.
\item A word field on the base space, $w: M \to G$, so that there exists a mapping between $m(t)$ and the sentence path.
\end{enumerate}
The second option is more general, but requires both a candidate for $M$ and a compelling way to connect $m(t)$ and $v(t)$. This is also more challenging, since, generally, parallel transport operators, while taking values in the group, are not closed. If the path were on $G$ itself, closure would be guaranteed, since any parallel transport operator would be an element of the co-occurring subgroup, and closure arises from an equivalence class of paths.
To recapitulate the final interpretations of word embeddings and RNNs in NLP:
\begin{itemize}
\item Words naturally embed as elements in a Lie group, $G$, and end-to-end word vectors may be related to the generating Lie algebra.
\item RNNs learn to parallel transport nonlinear representations of $G$ either on the Lie group itself, or on a principal $G$-bundle.
\end{itemize}
\subsection{Proposal for class of recurrent-like networks}
The geometric derivative along a path parameterized by $t$ is defined as:
\begin{equation}
D_t = \partial_t + v^\mu(t) \Gamma_\mu,
\end{equation}
where $v(t)$ is the tangent vector at $t$, and $\Gamma$ is the connection. This implies RNNs learn the solution of the first-order geometric differential equation:
\begin{equation}
D_t h(t) = 0.
\end{equation}
It is natural, then, to consider neural network solutions to higher-order generalizations:
\begin{equation}
D^n_t h(t) = 0. \label{eqn:higher_order}
\end{equation}
Networks that solve Eq.(\ref{eqn:higher_order}) are recurrent-like. Updates to a hidden state will generally depend on states beyond the immediately preceding one; often, this dependence can be captured by evolving on the phase space of the hidden states, rather than on the sequences of the hidden states themselves. The latter results in a nested RNN structure for the recurrent-like cell, similar to the structure proposed in \cite{Ruben:nested}.
Applications of Eq.(\ref{eqn:higher_order}) are currently being explored. In particular, if no additional structure exists and RNNs parallel transport states along paths on the word embedding group itself (the first RNN interpretation), geodesics emerge as a natural candidate for sentence paths to lie on. Thus, sentence generation could potentially be modeled using the geodesic equation and a nonlinear adjoint representation: $n=2$, $h \in \mathfrak{g}$ in Eq.(\ref{eqn:higher_order}). This geodesic neural network (GeoNN) is the topic of a manuscript presently in preparation.
\subsection{Proposal for new word embeddings}
The embeddings trained end-to-end in this work provided highly performant results. Unfortunately, training embeddings on end-tasks with longer documents is challenging, and the resulting embeddings are often poor for rare words. However, it would seem constructing pre-trained word embeddings by leveraging the emergent Lie group structure observed herein could provide competitive results without the need for end-to-end training.
Intuitively, it is unsurprising groups appear as a candidate to construct word embeddings. Evidently, the proximity of words is governed by their actions on hidden states, and groups are often the natural language to describe actions on vectors. Since groups are generally non-commutative, embedding words in a Lie group can additionally capture their order- and context-dependence. Lie groups are also generated by Lie algebras, so one group can act on the algebra of another group, and recursively form a hierarchical tower. Such an arrangement can explicitly capture the hierarchical structure language is expected to exhibit. E.g., the group structure in the first interpretation given by Eq.(\ref{eqn:cartesian_grp}),
\begin{equation}
G = G_1 \times G_2 \times G_3 \times \dots,
\end{equation}
admits, for appropriately selected $G_N$, hierarchical representations of the form
\begin{subequations}
\begin{align}
R_N :& G_N \times \mathfrak{g}_{N-1} \to \mathfrak{g}_{N-1}, \\
R_1 :& G_1 \times H \to H,
\end{align}
\end{subequations}
where $G_N = \exp(\mathfrak{g}_N)$. Such embedding schemes have the potential to generalize current attempts at capturing hierarchy, such as Poincar\'{e} embeddings \cite{Nickel:poincare}. Indeed, hyperbolic geometries, such as the Poincar\'{e} ball, owe their structure to their isometry groups. Indeed, it is well known that the hyperbolic $N+1$ dimensional Minkowski space arises as a representation of $SO(1,N)$ + translation symmetries.
In practice, Lie group embedding schemes would involve representing words as constrained matrices and optimizing the elements, subject to the constraints, according to a loss function constructed from invariants of the matrices, and then applying the matrix log to obtain Lie vectors. A prototypical implementation, dubbed ``LieGr," in which the words are assumed to be in the fundamental representation of the special orthogonal group, $SO(N)$, and are conditioned on losses sensitive to the relative actions of words, is the subject of another manuscript presently in preparation.
\section{Closing remarks} \label{sec:closing}
The results presented herein offer insight into how RNNs and word embeddings naturally tend to structure themselves for text classification. Beyond elucidating the inner machinations of deep NLP, such results can be used to help construct novel network architectures and embeddings.
There is, however, much immediate followup work worth pursuing. In particular, the uniqueness of identities, inverses, and multiplicative closure was not addressed in this work, which is critical to better understand the observed emergent algebraic structure. The cause for the hyperparameter stratification of the error in, and a more complete exploration of, commutative closure remains outstanding. Additionally, the cause of the breakdown of the common optimal latent dimension for low embedding dimension is unclear, and the bi-model, linear relationship between the action of words on hidden states and the Euclidean distance between end-to-end word embeddings invites much investigation.
As a less critical, but still curious inquiry: is the additive relationship between words, e.g. ``king - man + woman = queen," preserved, or is it replaced by something new? In light of the Lie group structure words trained on end tasks seem to exhibit, it would not be surprising if a new relationship, such as the Baker-Campbell-Hausdorff formula\footnote{Since the BCH formula is simply an non-commutative correction to the additive formula usually applied for analogies, it may be possible that this relation would already better represent analogies for disparately-related words.}, applied.
\section{Acknowledgements} \label{sec:acknowledgements}
The author would like to thank Robin Tully, Dr. John H. Cantrell, and Mark Laczin for providing useful discussions, of both linguistic and mathematical natures, as the work unfolded. Robin in particular provided essential feedback throughout the work, and helped explore the potential use of free groups in computational linguistics at the outset. John furnished many essential conversations that ensured the scientific and mathematical consistency of the experiments, and provided useful insights into the results. Mark prompted the investigation into potential emergent monoid structures since they appear frequently in state machines.
|
{
"timestamp": "2018-03-09T02:00:25",
"yymm": "1803",
"arxiv_id": "1803.02839",
"language": "en",
"url": "https://arxiv.org/abs/1803.02839"
}
|
\section{Introduction}
Recently Andrews \cite{Andrews1} studied integer partitions in which all parts of a given parity
are smaller than those of the opposite parity. Furthermore, he considered
eight subcases based on the parity of the smaller parts
and parts of a given parity appearing at most once or an unlimited number of times.
Following Andrews, we use ``ed'' for evens distinct, ``eu'' for evens unlimited,
``od'' for odds distinct, and ``ou'' for odds unlimited. With
``zw'' and ``xy'' from the four choices above, we let
$F^{\rm zw}_{\rm xy}(q)$ denote the generating function of partitions where
zw specifies the parity and condition of the larger parts and
xy specifies the parity and condition of the smaller parts.
The eight relevant generating functions are
\begin{align*}
F^{\rm ou}_{\rm eu}(q)
&:=
\sum_{n=0}^\infty \frac{q^{2n}}{\left(q^2;q^2\right)_n\left(q^{2n+1};q^2\right)_\infty}
,\\
F^{\rm od}_{\rm eu}(q)
&:=
\sum_{n=0}^\infty \frac{q^{2n} \left(-q^{2n+1};q^2\right)_\infty }{\left(q^{2};q^2\right)_n}
,\\
F^{\rm ou}_{\rm ed}(q)
&:=
\sum_{n=0}^\infty \frac{\left(-q^2;q^2\right)_nq^{2n+2}}{\left(q^{2n+3};q^2\right)_\infty}
,\\
F^{\rm od}_{\rm ed}(q)
&:=
\sum_{n=0}^\infty q^{2n+2}\left(-q^2,q^2\right)_n\left(-q^{2n+3};q^2\right)_\infty
,\\
F^{\rm eu}_{\rm ou}(q)
&:=
\sum_{n=0}^\infty \frac{q^{2n+1}}{\left(q;q^2\right)_{n+1}\left(q^{2n+2};q^2\right)_\infty}
,\\
F^{\rm ed}_{\rm ou}(q)
&:=
\sum_{n=0}^\infty \frac{q^{2n+1}\left(-q^{2n+2};q^2\right)_\infty}{\left(q;q^2\right)_{n+1}}
,\\
F^{\rm eu}_{\rm od}(q)
&:=
\sum_{n=0}^\infty \frac{q^{2n+1}\left(-q;q^2\right)_n}{\left(q^{2n+2};q^2\right)_\infty}
,\\
F^{\rm ed}_{\rm od}(q)
&:=
\sum_{n=0}^\infty q^{2n+1}\left(-q;q^2\right)_n\left(-q^{2n+2};q^2\right)_{\infty}
.
\end{align*}
Here we are using the standard product notation
$(a;q)_n := \prod_{j=0}^{n-1}(1-aq^j)$ for $n\in\mathbb N_0\cup\{\infty\}$.
We note that with the exception of $F^{\rm ou}_{\rm eu}(q)$ and
$F^{\rm od}_{\rm eu}(q)$, we do not allow the subpartition consisting
of the smaller parts to be empty.
Andrews' identities (after minor corrections) can be stated as
\begin{align*}
F^{\rm ou}_{\rm eu}(q)
&=
\frac{1}{\left(1-q\right)\left(q^2;q^2\right)_\infty}
,\\
F^{\rm od}_{\rm eu}(q)
&=
\frac{1}{2}\left(\frac{1}{\left(q^2;q^2\right)_\infty}+\left(-q;q^2\right)_\infty^2 \right)
,\\
F^{\rm ou}_{\rm ed}(-q)
&=
\frac{1}{2\left(-q;q^2\right)_\infty}\left(
\left(-q;q\right)_\infty - 1 - \sum_{n=0}^\infty q^{\frac{n\left(3n-1\right)}{2}}\left(1-q^n\right)
\right)
,\\
F^{\rm eu}_{\rm ou}(q)
&=
\frac{1}{1-q}\left(
\frac{1}{\left(q;q^2\right)_\infty} - \frac{1}{\left(q^2;q^2\right)_\infty}
\right)
,\\
F^{\rm ed}_{\rm ou}(-q)
&=
-\frac{\left(-q^2;q^2\right)_\infty}{2}\left(
2 - \frac{1}{\left(-q;q\right)_\infty}
-
\sum_{n=0}^\infty \frac{q^{n^2+n}}{\left(-q;q\right)_n^2\left(1+q^{n+1}\right)}
\right)
,\\
F^{\rm eu}_{\rm od}(-q)
&=
-\frac{1}{\left(q^2;q^2\right)_\infty}\sum_{j=1}^\infty\sum_{n=j}^\infty
\left(-1\right)^{n+j}q^{\frac{n\left(3n+1\right)}{2}-j^2}\left(1-q^{2n+1}\right)
.
\end{align*}
Surprisingly, these identities are derived with little more than the $q$-binomial theorem,
Heine's transformation, and the Rogers-Fine identity. In the following
theorem, we give new identities for $F^{\rm od}_{\rm ed}(q)$,
$F^{\rm ed}_{\rm od}(q)$, and $F^{\rm ed}_{\rm ou}(-q)$.
\begin{theorem}\label{TheTheorem}
The following identities hold,
\begin{align}
\label{Eqoded}
F^{\rm od}_{\rm ed}(q)
&=
\frac{q\left(-q;q^2\right)_\infty}{1-q}\left(
1
-
\frac{(-q^2;q^2)_\infty}{(-q;q^2)_\infty}
\right)
,\\
\label{Eqedod}
F^{\rm ed}_{\rm od}(q)
&=
\frac{q(-q^2;q^2)_\infty}{1-q}\left( 2 - \frac{(-q;q^2)_\infty}{(-q^2;q^2)_\infty}\right)
,\\
\label{Eqedou}
F^{\rm ed}_{\rm ou}(-q)
&=
-\frac{\left(-q^2;q^2\right)_\infty}{2}\left(
2 - \frac{1}{\left(-q;q\right)_\infty}
-\frac{2}{\left(q;q\right)_\infty}\sum_{n\in\mathbb Z}
\frac{\left(-1\right)^nq^{\frac{3n\left(n+1\right)}{2}}}{1+q^n}
\right)
.
\end{align}
\end{theorem}
\begin{remark}
The functions $F^{\rm od}_{\rm ed}(q)$ and $F^{\rm ed}_{\rm od}(q)$ are basically
modular functions. Also we find that $F^{\rm ed}_{\rm ou}(-q)$ is related
to Ramanujan's third order mock theta function $f(q)$, as
\begin{align*}
f(q) &:=
\sum_{n=0}^\infty \frac{q^{n^2}}{(-q;q)_n^2}
=
\frac{2}{\left(q;q\right)_\infty}\sum_{n\in\mathbb Z}
\frac{\left(-1\right)^nq^{\frac{n\left(3n+1\right)}{2}}}{1+q^n}
\\&=
2-\frac{2}{\left(q;q\right)_\infty}\sum_{n\in\mathbb Z}
\frac{\left(-1\right)^nq^{\frac{3n\left(n+1\right)}{2}}}{1+q^n}
,
\end{align*}
where the final equality uses Euler's pentagonal numbers theorem.
\end{remark}
\section{Proof of Theorem \ref{TheTheorem}}
To prove equations \eqref{Eqoded} and \eqref{Eqedod}, we require the following
$q$-series identity,
\begin{align}\label{EqQseries1}
\sum_{n=0}^\infty \frac{(x;q)_nq^n}{(y;q)_n}
&=
\frac{q(x;q)_\infty}{y(y;q)_\infty \left(1-\frac{xq}{y}\right)}
+
\frac{\left(1-\frac{q}{y}\right)}{\left(1-\frac{xq}{y}\right)}.
\end{align}
We note that \eqref{EqQseries1} is (4.1) from \cite{AndrewsSubbaraoVidyasagar1}
and was proved with
Heine's transformation
\cite[page 241, (III.2)]{GasperRahman1}.
To prove equation \eqref{Eqedou} we require the
concept of a Bailey pair and Bailey's Lemma, which are described in
\cite[Chapter 3]{Andrews3}. A pair of sequences
$(\alpha,\beta)$ is called a \textit{Bailey pair relative} to $a=q$ if
\begin{align*}
\beta_n &= \sum_{j=0}^n \frac{\alpha_j}{(q;q)_{n-j}(q^2;q)_{n+j}}.
\end{align*}
A limiting form of Bailey's Lemma states that if $(\alpha_n,\beta_n)$
is a Bailey pair relative to $q$, then
\begin{align}\label{EqBaileysLemma}
\sum_{n=0}^\infty q^{n^2+n}\beta_n
&=
\frac{1}{(q^2;q)_\infty}\sum_{n=0}^\infty q^{n^2+n}\alpha_n.
\end{align}
The Bailey pair we use is given by
\begin{align}\label{EqBaileyPair}
\beta_n^\prime &:= \frac{1}{(-q;q)_n^2(1+q^{n+1})}
,&
\alpha_n^\prime &:= \frac{2(-1)^n q^{\frac{n(n+1)}{2}}(1-q^{2n+1})}
{(1-q)(1+q^n)(1+q^{n+1})}
,
\end{align}
which follows from taking the Bailey pair
from Theorem 8 of \cite{Lovejoy1} with
$a\rightarrow q$, $b=-1$, $c=-q$, and $d=-1$
and dividing both $\alpha_n$ and $\beta_n$
by $(1+q)$.
\begin{proof}[Proof of Theorem \ref{TheTheorem}]
We find that
\begin{align*}
F^{\rm od}_{\rm ed}(q)
&=
\left(-q;q^2\right)_\infty\sum_{n=1}^\infty \frac{\left(-q^2;q^2\right)_{n-1}q^{2n}}{\left(-q;q^2\right)_n}
\\
&=
\frac{\left(-q;q^2\right)_\infty}{2}\left(
-1
+\sum_{n=0}^\infty \frac{\left(-1;q^2\right)_{n}q^{2n}}{\left(-q;q^2\right)_n}
\right).
\end{align*}
With $q\mapsto q^2$, $x=-1$, and $y=-q$, equation \eqref{EqQseries1} implies that
\begin{align*}
\sum_{n=0}^\infty \frac{(-1;q^2)q^{2n}}{(-q;q^2)_n}
&=
-\frac{q\left(-1;q^2\right)_\infty}{\left(-q;q^2\right)_\infty(1-q)} + \frac{1+q}{1-q}.
\end{align*}
Equation \eqref{Eqoded} then follows after elementary simplifications.
Similarly, we have that
\begin{align*}
F^{\rm ed}_{\rm od}(q)
&=
\left(-q^2;q^2\right)_\infty\sum_{n=0}^\infty \frac{\left(-q;q^2\right)_{n}q^{2n+1}}{\left(-q^2;q^2\right)_n}.
\end{align*}
By applying \eqref{EqQseries1} with $q\mapsto q^2$, $x=-q$, and $y=-q^2$, we find that
\begin{align*}
\sum_{n=0}^\infty \frac{\left(-q;q^2\right)q^{2n}}{\left(-q^2;q^2\right)_n}
&=
-\frac{\left(-q;q^2\right)_\infty}{\left(-q^2;q^2\right)_\infty(1-q)} + \frac{2}{1-q},
\end{align*}
and \eqref{Eqedod} follows.
For $F^{\rm ed}_{\rm ou}(q)$, we begin with Andrews' identity \cite{Andrews1}
\begin{align*}
F^{\rm ed}_{\rm ou}(-q)
&=
-\frac{\left(-q^2;q^2\right)_\infty}{2}\left(
2- \frac{1}{(-q;q)_\infty}
-\sum_{n=0}^\infty \frac{q^{n^2+n}}{(-q;q)_n^2\left(1+q^{n+1}\right)}
\right).
\end{align*}
By applying \eqref{EqBaileysLemma} to the Bailey pair
$(\alpha^\prime,\beta^\prime)$ in \eqref{EqBaileyPair},
we have that
\begin{align*}
\sum_{n=0}^\infty \frac{q^{n^2+n}}{(-q;q)_n^2\left(1+q^{n+1}\right)}
&=
\frac{2}{(q;q)_\infty}
\sum_{n=0}^\infty \frac{(-1)^nq^{\frac{3n(n+1)}{2}} \left(1-q^{2n+1}\right)}{\left(1+q^n\right)\left(1+q^{n+1}\right)}.
\end{align*}
We use the partial fraction decomposition
\begin{align*}
\frac{1-q^{2n+1}}{\left(1+q^n\right)\left(1+q^{n+1}\right)}
&=
\frac{1}{1+q^n}-\frac{q^{n+1}}{1+q^{n+1}}
,
\end{align*}
to deduce that
\begin{align*}
\sum_{n=0}^\infty \frac{(-1)^nq^{\frac{3n(n+1)}{2}} \left(1-q^{2n+1}\right)}{\left(1+q^n\right)\left(1+q^{n+1}\right)}
&=
\sum_{n=0}^\infty (-1)^nq^{\frac{3n(n+1)}{2}}
\left( \frac{1}{1+q^n}-\frac{q^{n+1}}{1+q^{n+1}}\right)
\\
&=
\sum_{n\in\mathbb Z} \frac{(-1)^nq^{\frac{3n(n+1)}{2}}}{1+q^n}.
\end{align*}
Altogether this implies equation \eqref{Eqedou}.
\end{proof}
By applying Theorem 1.1 part $3$ of \cite{Lovejoy2} to the Bailey
pair $E(3)$ of \cite{Slater1},
we find that
\begin{multline*}
F^{\rm ed}_{\rm od}(-q) =
-\frac{q\left(q;q\right)_\infty\left(-q^2;q^2\right)_\infty}{\left(q^2;q^2\right)_\infty^2}
\\
\times \sum_{n=0}^\infty\sum_{m=0}^\infty \left(-1\right)^mq^{\frac{n\left(n+3\right)}{2}+2nm+2m^2+2m}\left(1+q^{2m+1}\right)
.
\end{multline*}
As such, we have that
\begin{multline*}
\left(\sum_{n,m\geq 0} - \sum_{n,m<0}\right) (-1)^m q^{\frac{n(n+3)}{2}+2nm +2m(m+1)}\\
=
\frac{2\left(q^2;q^2\right)_\infty}{(1+q)\left(q;q^2\right)_\infty}-\frac{\left(q^2;q^2\right)_\infty}{(1+q)\left(-q^2;q^2\right)_\infty}.
\end{multline*}
We note that the corresponding quadratic form is degenerate, and so a priori the
modularity properties of this theta function are unclear. More generally, one
can prove directly that, for $c\in\mathbb N$,
\begin{align*}
\sum_{n,m\geq 0}z^nw^mq^{n^2+2cnm +c^2m^2}
&=
\frac{1}{1-\frac{w}{z^c}}\sum_{k=0}^{c-1}\sum_{n=0}^\infty
z^{cn+k}q^{(cn+k)^2}
\left(1-\frac{w^{n+1}}{z^{cn+c}}\right).
\end{align*}
The above is a sum of partial theta functions, which sometimes
combine to give a modular form.
\section*{Acknowledgments}
The authors thank George Andrews, Karl Mahlburg, and the anonymous referee for their careful reading
and comments on an earlier version of this manuscript.
|
{
"timestamp": "2019-03-19T01:29:24",
"yymm": "1803",
"arxiv_id": "1803.02573",
"language": "en",
"url": "https://arxiv.org/abs/1803.02573"
}
|
\section{Introduction}
\label{sec:intro}
In recent years quantum mechanical systems of dipolar molecules have emerged as a fascinating platform for studying a number of interesting and novel phenomena in condensed matter physics, as reviewed in
\cite{koehlerRMP06,micheliNaturePhys06,baranovPhysRep08,kremsColdMoleculesBook,hazzard:many-body_2014,Carr2009}.
Possessing a permanent electric dipole moment (in the molecule body-fixed frame),
they interact at long range via the anisotropic dipole-dipole interaction potential
\begin{equation}
V_\mathrm{ij} = \frac{1}{4\pi\epsilon_0} \left( \frac{\mathbf{d}_i \cdot \mathbf{d}_j}{r_{ij}^3} - 3 \frac{(\mathbf{d}_i \cdot \mathbf{r}_{ij})(\mathbf{d}_j \cdot \mathbf{r}_{ij})}{r_{ij}^5} \right),
\end{equation}
where the $\mathbf{d}_i = d_i \mathbf{n}_i$ is the electric dipole moment of the $i$th molecule and $\mathbf{r}_{ij}$ is the displacement vector between the two molecules. Such interactions can be made highly tunable through the application of external fields~\cite{Buchler2007}.
Two-dimensional ensembles of
dipolar molecules have been shown to exhibit a variety of interesting behaviors,
with a rich phase diagram even
when the polarization is constrained to be perpendicular to the plane. Theoretical studies of fully polarized dipoles have been shown to possess a roton minimum~\cite{Mazzanti2009,Hufnagl2010,hufnaglPRL11}. They go through a series of distinct phases as the direction of the polarization is modulated~\cite{Macia2012,Macia2014}. For polarization normal to the plane of translational confinement, such systems have been shown to organize themselves into triangular lattices~\cite{Buchler2007,Astrakharchik2007}, to form a supersolid phase~\cite{Golomedov2011}, and even a unique crystalline phase that is stabilized by the zero-point motion of the dipoles~\cite{Boninsegni2013}.
Studies of fully polarized dipolar molecules confined to vertices of a lattice have been found to show
rich phase diagrams~\cite{Goral2002,Pollet2010,Trefzger2010,Capogrosso-Sansone2010,Yamamoto2012}. In this context, the dipolar arrays are generally described using an effective Hubbard model Hamiltonian~\cite{Carr2009}, often including long-ranged corrections~\cite{Goral2002}. In particular, on triangular lattices dipolar molecules have been shown to exhibit a normal fluid phase, a Mott insulating phase, a superfluid, and a supersolid phase~\cite{Pollet2010}.
Dipolar bilayers can exhibit a pairing phase transition to pair-superfluidity\cite{maciaPRA14,filinovPRA16,astraPRA16} as well as a
self-bound liquid state\cite{hebenstreitPRA16,raderPRA17}.
After mean field studies showed that quasi-two dimensional ensembles of harmonically confined dipoles can also exhibit roton excitations
\cite{santosPRL03,odellPRL03}, these systems have also been extensively studied. Most recently, the roton
excitations predicted by the mean field analysis have been experimentally confirmed~\cite{chomazarxiv17}.
Dipolar condensates of magnetic atoms have first been achieved with
$^{52}$Cr atoms~\cite{griesmaierPRL05,lahayeNature07}, and
more recently with Dy~\cite{luPRL11dysprosium}
and Er~\cite{aikawaPRL12} atoms. Of particular importance for experimental realizations is an
understanding under what circumstances such systems are stable~\cite{lahayePRL08,kochNaturePhys08}.
The influence
of trap geometry, dipole strength and short range repulsion on the stability has been
studied in the mean field approximation, indicating instability by a buckling of the condensate cloud~\cite{Ronen2007}. In this regard, an exciting new development is the experimental generation of
self-bound droplets of trapped dipolar bosons~\cite{kadauNature16,Ferrier-Barbut_2016,chomazPRX16} which is driven
by this instability and which has been generally confirmed by calculations
\cite{Bisset_2015,Blakie_2016,Wachtler_2016,Kui-Tian_2016,maciaPRL16,cintiPRA17} using a
variety of methods.
The long range and anisotropic nature of the interaction poses challenges for a full theoretical analysis of the expected phases and dynamics of large numbers of such dipolar molecules, whether magnetically or optically trapped as ensembles, or individually localized at vertices of a lattice.
Many of the above-mentioned
prior theoretical studies share a key common approximation, namely that molecules are treated as if they were perfectly oriented by application of an external field, with the effect of imperfect orientation in a finite external field being accommodated only through the use of an effective dipole moment.
In cases where dipolar molecules are used to design effective spin Hamiltonians, only a few molecular excitation levels are typically included~\cite{hazzardPRL14}.
To date the full molecular excitation structure of ensembles of polar molecules have been studied only
in the mean field approximation~\cite{Zillich2011}.
The assumption of "perfect" orientation with an external field is a reasonable assumption at molecular densities low enough that the energy scale set by the rotational degree of freedom, $hB = \hbar^2/2I$ where $I$ is the molecular moment of inertia, is much larger than the dipole-dipole interaction energy, $d^2/(4\pi\epsilon_0\left<r_{ij}\right>^3)$.
In this regime the dipole-dipole interaction can be considered
to be only a small perturbation and its effect on the orientation of the dipoles neglected, so that
dipole orientation can be considered to be only a function of applied external field strength.
Most prior and current experiments with atoms and molecules in optical lattices are in this regime due to their large typical lattice spacing, around 300~nm to 1000~nm. Conventionally, in a lattice formed by counterpropagating laser beams, the lattice spacing is given by $\lambda/2$, where $\lambda$ is the wavelength of the trapping laser. However, several recent proposals have been made for synthesis of lattices with significantly smaller lattice spacings \cite{Ritt2006,romeroisartPRL13,Nascimbene2015,Lkacki2016,Perczel2017}, and experimental demonstration of a $\lambda/4$ lattice has already been made~\cite{Ritt2006}. It is therefore timely to undertake a theoretical investigation of the effect of both the dipolar interactions and the external field on the molecular orientation for a general lattice of dipoles in which the lattice spacing is varied over a large range of values extending down to values where the dipole-dipole interaction becomes appreciable.
We explore in the present work how, as we increase the density, the dipole interaction starts to affects the rotational degree of freedoms of dipoles arranged on the sites of two dimensional lattices.
The present analysis
can potentially shed light also on the behavior of dipoles confined in
one-dimensional trapping potentials
at high densities and subjected to a transverse electric field, where self-assembled lattices can form in the other two directions due to repulsive dipole-dipole interactions~\cite{Buchler2007,Astrakharchik2007}.
We have previously shown that at sufficiently high densities in one dimensional
lattices, above a certain interaction strength, or conversely as the intermolecular spacing decreases and the molecular density increases, molecules will tend to spontaneously align with one another
~\cite{Abolins2011}. This gives rise to a fully polarized phase that reflects a breaking of the $O(3)$ symmetry of a distance-dependent quantum rotor Hamiltonian by the second, anisotropic term in the dipole-dipole interaction.
For a one-dimensional array, this introduces an Ising-like $\mathbb{Z}_2$ symmetry along the axis of the array and the ordered phase for large $g$ is then a 2-fold degenerate end-to-end ordering of dipoles along the lattice axis.
In this work we explore the possibility of
analogous transitions occurring in two dimensional systems of dipolar molecules at fixed lattice positions on square and triangular lattices.
In order the assess the full behavior of such systems and assess the limits of application for the assumption of perfect orientation, we study the behavior
here of dipolar rotors confined to a 2-dimensional lattice ($xy$-plane) for two lattice geometries. The first lattice considered in the current work is the 2-dimensional triangular lattice.
This choice is motivated by the theoretical predictions of dipolar molecules forming
planar triangular crystalline lattices~\cite{Buchler2007,Astrakharchik2007}
in presence of an explicitly defined external electric field perpendicular to the lattice directions. The second lattice considered is the 2-dimensional square lattice.
Both of these lattice structures, and many more, can be realized by confining dipolar molecules to the minima of 2-dimensional optical lattices of the corresponding geometry~\cite{windpassingerRPP13}.
In both cases the dipoles are described by a Hamiltonian of the form
\begin{equation}
H = \sum_{i=1}^N \frac{\mathbf{L}_i^2}{\hbar^2} - u \mathbf{n}_i \cdot \hat{\mathbf{e}}
+ g \sum_{j < i} \left[\frac{\mathbf{n}_i \cdot \mathbf{n}_j}{r_{ij}^3}
- 3 \frac{(\mathbf{n}_i \cdot \mathbf{r}_{ij})(\mathbf{n}_j \cdot \mathbf{r}_{ij})}{r_{ij}^5}\right], \label{eq:ham}
\end{equation}
where $\mathbf{L}_i$ is the usual quantum mechanical angular momentum of rotor $i$, $u = dE/hB$ with $\mathbf{E} = E\hat{\mathbf{e}}$ being the applied electric field, and $g = d^2/(4\pi\epsilon_0hBr_\mathrm{lat}^3)$ sets the strength of the dipole-dipole interacting relative to the rotational kinetic energy scale. Table~\ref{Table1:dipole parameters} shows the values of electric field and lattice spacing required to achieve the parameter values $u \geq 1$ and $g=1$, respectively, for a range of dipolar diatomic molecules. These values may be taken to correspond approximately to the onset of strong orientation and strong interparticle interactions, respectively. We see that for the alkali halides, the strong interaction regime may be accessed at lattice spacings of a few tens of nm, while a number of the other species become strongly interacting at lattice spacings on the order of 10 nm. For all species shown here, the electric field strengths required to reach the strong orientation regime are readily accessible with current experimental capabilities.
\begin{table*}[ht]
\caption{Permanent dipole moments, rotational constants, electric field strength required to realize $u=1$, and lattice spacing required to realize $g=1$ for a range of dipolar diatomic molecules.}
\label{Table1:dipole parameters}
\centering
\vspace{0.1in}
\begin{tabular}{ | c | c | c | c |c | c | }
\hline
{Molecule} & {d (Debye)} & {B (GHz)} & {E (kV/cm) at $u=1$} & {$r_{lat}$ (nm) for $g=1$} & {sources} \\
\hline
KRb & 0.57 & 1.10 & 3.80 & 3.56 & \cite{Gonzalez2017,Zuchowski2013} \\
LiCs & 5.46 & 6.53 & 2.37 & 8.83& \cite{Gonzalez2017} \\
NaCs & 4.70 & 1.74 & 0.73 & 12.42 & \cite{Gonzalez2017} \\
CsI & 11.69 & 0.71 & 0.12 & 30.70 & \cite{Honerjager1973,Story1976} \\
KBr & 10.60 & 2.43 & 0.46 & 19.10 & \cite{NISTdiatomics} \\
SrO & 8.87 & 10.13 & 2.27 & 10.53 & \cite{NISTdiatomics} \\
SrF & 3.47 & 7.52 & 4.30 & 6.22 & \cite{Gonzalez2017} \\
YO & 4.54 & 11.63 & 5.11 & 6.42& \cite{Gonzalez2017} \\
YbF & 9.93 & 9.19 & 1.44 & 12.93 & \cite{Bethlem2003} \\
\hline
\end{tabular}
\end{table*}
The Hamiltonian Eq.~(\ref{eq:ham}) is remarkably similar to the well-known quantum rotor model, of which there is no
known physical example~\cite{Sachdev}. In particular, when the anisotropic term in the dipolar interaction is omitted Eq.~(\ref{eq:ham}) becomes equivalent to the $O(3)$ quantum rotor model in an external field~\cite{Sachdev,Dutta2001}. The results
in this work show that when the short range spin-spin interaction characteristic of the conventional quantum rotor model is replaced by an anisotropic long range dipolar interaction, a new class of dipolar quantum rotor phases emerges.
\section{Methods}
\label{sec:methods}
To explicitly study the effects of the dipole interaction on all degrees of freedom of dipolar molecules,
we employ the path integral ground state quantum Monte Carlo (PIGS) method,
sometimes referred to as the variational path integral Monte Carlo method~\cite{Sarsa2000}. The PIGS method is a straightforward extension of the well known finite temperature path integral Monte Carlo (PIMC) method that has been used extensively in recent years to study ground states of quantum systems of importance in chemistry and in condensed matter physics. These studies include van der Waals complexes~\cite{Sarsa2000}, low temperature condensed phases of helium~\cite{Sarsa2000,Rossi2009}, and more recently the elementary excitation spectrum~\cite{Macia2012} as well as the
$T=0$~K phase diagram~\cite{Macia2014} of continuum systems of two dimensional fully polarized dipoles. The related reptation Monte Carlo method has been used to study the rotational and translational dynamics of small molecules embedded in $^4$He clusters~\cite{Cazzato2004}.
We have previously extended the PIGS method to a full simulation of both rotational and translational motion of ensembles of molecules~\cite{Abolins2011}. We used this technique to study the behavior of dipolar rotors confined to one dimensional lattices, e.g.\ without translational degrees of freedom, finding a crossover from an unpolarized phase at low dipole-dipole interaction strength to polarized behavior at higher dipole-dipole interaction strength as
mentioned above~\cite{Abolins2011}.
In the present study we extend this work to 2-dimensional lattices at unit filling, still fixing the translational
coordinates to the sites of a lattice. In the following we briefly summarize the computational approach for the most general
case that both rotational and translational degrees of freedom are allowed to fluctuate.
\subsection{Path Integral Ground State for Rotating and Translating Dipolar Molecules}
\label{subsec:PIGS}
The PIGS method is quite general, being broadly applicable to the study of the ground states of arbitrary bosonic systems. PIGS belongs to the broader family of projector Monte Carlo methods which begin with a trial state or wave function. This state can formally be written in terms of the eigenstates of the Hamiltonian of interest
\begin{equation}
\ket{\Psi_\mathrm{trial}} = \sum_k^\infty c_k \ket{\Phi_k}
\end{equation}
where $H\ket{\Phi_k} = E_k\ket{\Phi_k}$ are the eigenstates of $H$. By propagating this state in imaginary time for a duration $\beta/2$,
\begin{equation}
G(\beta/2)\ket{\Psi_\mathrm{trial}} = \sum_k c_k e^{-\beta E_k/2\hbar} \ket{\Phi_k},
\end{equation}
the trial state will asymptotically approach the ground state,
\begin{equation}
\lim_{\beta \rightarrow \infty} \frac{G(\beta/2) \ket{\Psi_\mathrm{trial}}}{\sqrt{\bra{\Psi_\mathrm{trial}}G(\beta)\ket{\Psi_\mathrm{trial}}}} = \ket{\Phi_0} \label{eq:gs_exact}
\end{equation}
assuming that $c_0 \neq 0$, where
$G(\beta/2) = e^{-\beta H /2\hbar}$ is the usual imaginary time evolution operator, simply referred to as propagator, and $\ket{\Phi_0}$ is the exact ground state of $H$. In this limit expectation values of
an operator $O$ can be expressed as
\begin{equation}
\expect{O} = \lim_{\beta \rightarrow \infty} \frac{\bra{\Psi_\mathrm{trial}}G(\beta / 2) O G(\beta / 2)\ket{\Psi_\mathrm{trial}}}{\bra{\Psi_\mathrm{trial}}G(\beta)\ket{\Psi_\mathrm{trial}}} = \frac{\bra{\Phi_0}O\ket{\Phi_0}}{\braket{\Phi_0}{\Phi_0}}.
\label{eq:expect_exact}
\end{equation}
Breaking up the propagation into many smaller steps,
\begin{equation}
G(\beta) = [G(\tau)]^M,
\end{equation}
where $\tau = \beta / M$, suggests the use of short time approximations to the propagator, such as the well known fourth-order Trotter-Suzuki propagator~\cite{Suzuki1995} and other related approximations~\cite{Chin1997}. In what follows we used a sixth-order ``any-order'' propagator~\cite{Zillich2010} of the form
\begin{equation}
G_{2n}(\tau) = \sum_{i=1}^n c_i \left(G_2(\tau/k_i)\right)^{k_i} = G(\tau) + O(\tau^{2n+1}),
\label{eq:any_order}
\end{equation}
where $k_i = \{1, 2, 4\}$. Here, $G_2(\tau)$ is the second order propagator approximation (so-called primitive approximation) given by
\begin{equation}
G_2(\tau) = e^{-\tau V/(2\hbar)} e^{-\tau T/\hbar} e^{-\tau V/(2 \hbar)},
\end{equation}
with $H = T + V$ and $T$ is the kinetic energy and $V$ is the potential energy.
Working in a representation with coordinates $\mathbf{X}$ (e.g. in the present case
$\mathbf{X}=(\mathbf{n}_1,\dots,\mathbf{n}_N)$) and assuming a sufficiently large number
$M$ of sufficiently small imaginary propagation time steps, one arrives at an approximate expression for the expectation value
\begin{equation}
\expect{O} \approx \frac{1}{N(\beta, M)} \int d^M\mathbf{X} \,\,
\left[\Psi_\mathrm{trial}^*(\mathbf{X}_1)
\left(\prod_{i=1}^{M-1} G_n(\mathbf{X}_i, \mathbf{X}_{i+1}, \tau)\right)
\Psi_\mathrm{trial}(\mathbf{X}_M)\right]
O(\mathbf{X}_{\lfloor M/2 \rfloor + 1}),l
\label{eq:expectO}
\end{equation}
where the integral is taken over all of the coordinates of the system, $\{ \mathbf{X}_1, \dots, \mathbf{X}_M \}$, $G_n(\mathbf{X}, \mathbf{X}', \tau) = \bra{\mathbf{X}} G_n(\tau) \ket{\mathbf{X}'}$, and $N(\beta, M)$ is a normalization constant. This form suggests the use of Monte Carlo integral evaluation of the high dimensional integral. This can be done using the well-known Metropolis algorithm~\cite{Metropolis1953} where the weight for a given path
through the muti-dimensional configuration space of integral Eq.~(\ref{eq:expectO}) is
\begin{equation}
W(\mathbf{X}_1, \dots, \mathbf{X}_M; \beta, M) =
\Psi_\mathrm{trial}^*(\mathbf{X}_1)
\left(\prod_{i=1}^{M-1} G_n(\mathbf{X}_i, \mathbf{X}_{i+1}, \tau)\right)
\Psi_\mathrm{trial}(\mathbf{X}_M)/N(\beta, M).
\label{eq:pdf}
\end{equation}
Since the Metropolis algorithm only depends on ratios of the weights for different configurations, the normalization $N(\beta, M)$ is of no consequence and need not be evaluated.
From this we see that PIGS has many desirable qualities, namely that it can be applied to any system where the weights $W(\mathbf{X}_1, \dots, \mathbf{X}_M; \beta, M) \geq 0$, and so is generally applicable to the ground state of bosonic
or distinguishable quantum systems. The only inputs are the system Hamiltonian, which enters through the expression for the effective propagator, and the trial wave function. This trial wave function can be as sophisticated or as simple as is desired
to balance the tradeoff between computational efficiency and complexity of evaluation of the integrand. In many situations even a constant trial wave function can be used without incurring too great a penalty in terms of efficiency~\cite{Rossi2009,rotaPRE10,Abolins2011}.
Unlike in variational Monte Carlo, the employed trial wave function does not bias the results, provided it has non-zero overlap with the ground state and the extrapolation to infinite path length is performed.
All of the approximations
made in implementing Eqs.~(\ref{eq:expectO}) - (\ref{eq:pdf}) are in principle controllable through extrapolation to the infinite path length limit, $\beta\to\infty$, and the zero time step limit, $\tau\to 0$. These properties make the PIGS method extremely useful for studying the ground state behavior of bosonic many-body systems, although we show in the Appendix that the
convergence of our results with increasing $\beta$ is problematic close to a quantum phase transition.
While the diffusion Monte Carlo (DMC) method usually give the ground state energy with a smaller
variance, obtaining estimators that are not biased by the trial functions is not straightforward
for expectations values of operators that do not commute with $H$ \cite{casulleras95}.
To sample
the imaginary time paths we utilized the multi-level bisection algorithm~\cite{Ceperley1995}. We sample the orientations of the rotors according to the procedure described in~\cite{Abolins2011},
utilizing the rotational kinetic energy propagator~\cite{Cui1997,Marx1999}
\begin{equation}
G_0(\mathbf{X}, \mathbf{X}', \tau) = \bra{\mathbf{X}}e^{-\tau T/\hbar}\ket{\mathbf{X}'} = \prod_{i=1}^N \sum_{l=0}^\infty \frac{2l+1}{4\pi} P_l(\mathbf{n}_i\cdot\mathbf{n}_i') e^{-2\pi\tau B l(l + 1)},
\label{eq:G_rot}
\end{equation}
where $\mathbf{n}_i$ is the orientation of the $i$th molecule and $P_l(x)$ is the Legendre polynomial of degree $l$.
For computational efficiency, Eq.~(\ref{eq:G_rot}) is tabulated on a grid of values of $\mathbf{n}_i \cdot \mathbf{n}_i'$ at the beginning of a simulation and then values are calculated using linear interpolation of the values on the pre-evaluated grid throughout the course of the simulations.
\subsection{Trial Functions}
\label{subsec:trialfunctions}
In the present work we employ a Hartree trial wave function of the form
\begin{equation}
\Psi_\text{trial}(\mathbf{X}) = \prod_{i=1}^N e^{\alpha \cos \theta_i}, \label{eq:approx_wf}
\end{equation}
where $\cos \theta_i = \mathbf{n}_i \cdot \hat{\mathbf{e}}$, with a variational parameter $\alpha$ optimized for $g = 0$ and the relevant value of $u$ for each simulation. The form of the wave function in Eq.~(\ref{eq:approx_wf}) is qualitatively similar to that of a single fixed dipole in an electric field
directed along the $z$-axis and as such is expected to capture much of the behavior
in regions of high $u$ and low $g$. This qualitative
argument is why this particular trial wave function was employed, however it should be noted that with sufficiently long imaginary time paths it is possible to retrieve the exact behavior of the system in question even
when using a constant trial wave function~\cite{Rossi2009,rotaPRE10,Abolins2011}. The convergence study
in the Appendix demonstrates, however, that a good trial wave function is preferable especially close
to a phase transition.
\subsection{Extended System Simulation Details}
\label{subsec:pbc}
To describe extended systems, we employ finite sized systems with periodic boundary conditions. In two dimensions the dipole-dipole interaction, which decays with distance as $1/r^3$, requires a large cutoff to ensure that the finite sized system is representative of an extended system~\cite{Weis2003}. For our calculations on 2-dimensional lattices
the cutoff required was found to be too large to make the conventional minimum image convention~\cite{FrenkelSmit} feasible for all system sizes and so a sum over extended periodic images within a suitable chosen cutoff was employed instead, {\em i.e.} we write
\begin{equation}
H = \sum_{i=1}^N \frac{\mathbf{L}_i^2}{\hbar^2} - u \mathbf{n}_i \cdot \hat{\mathbf{e}}
+ g \sum_{j < i} \sum_\mathbf{v} \left[\frac{\mathbf{n}_i \cdot \mathbf{n}_j}{|\mathbf{r}_{ij} + \mathbf{v}|^3}
- 3 \frac{(\mathbf{n}_i \cdot (\mathbf{r}_{ij} + \mathbf{v}))(\mathbf{n}_j \cdot (\mathbf{r}_{ij} + \mathbf{v}))}{|\mathbf{r}_{ij} + \mathbf{v}|^5}\right], \label{eq:ham_periodic}
\end{equation}
where the sum over $\mathbf{v}$ is the sum over vectors connecting the origin of the primary simulation box to that of its periodic images~\cite{FrenkelSmit}. The long range cutoff value of $v$, yielding
$r_{max} = \max_{ij} |r_{ij} + v_{max}|$, was chosen so that $d^2/(4\pi \epsilon_0 hB r_\mathrm{max}^3) < 10^{-8}$. To properly take the anisotropic nature of the dipole-dipole interaction into account, it is also essential that the primary simulation cell has the proper symmetry.
In the case of triangular lattice simulations a hexagonal periodic simulation box was employed, illustrated in Figure~\ref{fig:hexagonal_pbc}, while for square lattice simulations a 2 dimensional square periodic simulation box was employed.
The Hamiltonian (\ref{eq:ham_periodic}) can further be written as
\begin{equation}
H = \sum_{i=1}^{N} \mathbf{L}_i^2 - u \mathbf{n}_i \cdot \hat{\mathbf{e}} + g \sum_{j<i} \mathbf{n}_i \cdot \mathbf{S}_{ij} \cdot \mathbf{n}_j \label{eq:ham_simp}
\end{equation}
with $\mathbf{S}_{ij}$ given by
\begin{equation}
\mathbf{S}_{ij} = \sum_\mathbf{v} \left[\frac{1}{|\mathbf{r}_{ij} + \mathbf{v}|^3} \mathbb{I}
- \frac{3}{|\mathbf{r}_{ij} + \mathbf{v}|^5} (\mathbf{r}_{ij} + \mathbf{v})(\mathbf{r}_{ij} + \mathbf{v})^\intercal\right].
\end{equation}
Because dipoles are confined to fixed points on the triangular lattice, and as a result $r_{ij}$ for all $i$ and $j$ do not change during the simulation, $\mathbf{S}_{ij}$ may be precomputed at the beginning of each simulation, leading to considerable computational savings in calculating the periodic sums in Eq.(\ref{eq:ham_periodic}).
\begin{figure}[h]!
\centering
\includegraphics[width=3in]{{hexagonal_pbc}.pdf}
\caption{The geometry of the hexagonal periodic simulation box. Corresponding points on the boundary are labeled with the letters A -- E.}
\label{fig:hexagonal_pbc}
\end{figure}
\section{Results and Discussion}
\label{sec:results}
\subsection{Order Parameters}
\label{subsec:orderparams}
\subsubsection{Orientational phases and order parameters for triangular lattices}
In order to assess the ordering of the ground state in the strong interaction (classical) limit, several different types of orderings were considered. In the case of triangular lattice simulations the most plausible of these are a fully polarized ordering, where all dipoles are oriented in the same direction in the plane of the lattice, and a striped ordering where alternating rows or columns of dipoles are oriented in the same direction, and all dipoles are aligned with the same axes, leading to a vanishing net polarization. These orderings are illustrated in Figure~\ref{fig:tri_ord} for a hexagonal simulation box with periodic boundary conditions.
\begin{figure}[h]
\centering
\includegraphics{{tri_ordering}.pdf}
\caption{The considered orderings in the limit $g \gg 1$ and $u \ll 1$ for a system of 12 dipoles with the primary simulation box outlined for reference. On the left is a fully polarized ordering, which is degenerate with all other configurations that are fully polarized in the lattice plane, and on the right is one of six degenerate striped orderings, two along each of three triangular lattice axes. The dipoles not pictured on the right and top boundaries correspond to particles on the opposite boundary.}
\label{fig:tri_ord}
\end{figure}
The correct classical ground state ordering
of the different phases was determined by computing and comparing the potential energy per particle of each configuration as a function of the lattice size. The resulting classical energies on the triangular lattice are depicted in Figure~\ref{fig:tri_ord_en} as function of system size. We see that the polarized ordering is by far the lower energy of the two classical ordered configurations.
This classical analysis also reveals that the energy of the fully polarized state is invariant to arbitrary rotation of the orientation of the dipoles in the lattice ($xy$-)plane, exhibiting $O(2)$ symmetry with respect to dipole orientation in this plane. The left and right panel show the potential energy per particle calculated with the periodic sum convention explained in section~\ref{subsec:pbc} and with the conventional minimum image convention. The comparison illustrates that the latter is severely biased by system size in the polarized case, which would require a prohibitively large simulations size.
The evaluation of the potential energy with the periodic sum convention used in this work has essentially no finite size bias, therefore our simulations can be quite small, with typically 48 dipoles in the triangular case.
Notice that for the smallest system size, the nearest image convention erroneously predicts that the striped configurations have lower potential energy than the fully polarized configurations.
\begin{figure}[h]
\centering
\includegraphics{{triangular_orderings_energies}.pdf}
\caption{The classical potential energy per particle of striped and polarized configurations
in the triangular lattice as a function of system size, with $C_\text{dd}/(4\pi r_\text{lat}^3) = 1$~a.u. and $u = 0$ in a hexagonal unit cell under the periodic sum convention with a cutoff radius of $100$ lattice sites (left) and the minimum image convention (right).}
\label{fig:tri_ord_en}
\end{figure}
With the fully polarized in-plane ordering being established as a likely candidate for the ground state in the classical strong interaction limit, we can then construct an order parameter
for the triangular lattice that is maximal in this fully polarized state. The quantity
\begin{equation}
\phi_\mathrm{pol} = \sqrt{\expect{\frac{1}{N} \sum_{i=1}^N n_i^x}^2 + \expect{\frac{1}{N} \sum_{i=1}^N n_i^y}^2}, \label{eq:tri_orderp}
\end{equation}
where $n_i^\alpha$ is the $\alpha$ component of the orientation vector, $\mathbf{n}_i$, takes on a maximum in the fully ordered state and vanishes in a state with randomly distributed dipoles. $\phi_\mathrm{pol}$ can thus serve as an order parameter for formation of the fully polarized in-plane state during a quantum Monte Carlo simulation. It should be noted that since this quantity is unsigned, it will not fully vanish in a simulation of a disordered state since there will always be some non-vanishing net polarization in some direction, albeit small and randomly distributed. Consequently the average value over the course of a simulation will be small but finite. In addition, the following quantity indicative of the polarization along the applied field direction
\begin{equation}
\phi_z = \expect{\frac{1}{N} \sum_{i=1}^N n_i^z}
\end{equation}
provides an order parameter for polarization in the transverse direction.
\subsubsection{Orientational phases and order parameters for square lattices}
In the case of square lattice simulations similar
orientational orderings were considered, with the key difference from the orderings for the triangular lattice being that the resulting polarization and striping is now defined along the cartesian directions. Because of the symmetry of the square lattice there are only four equivalent fully polarized configurations: one in which all of the dipoles are completely polarized along the $x$-axis, and one in which the dipoles are completely polarized along the $y$-axis. All other polarizations are higher in energy, in contrast to what is found on triangular lattices.
On square lattices the symmetry of the lattice admits another possible attractive ordering in which dipoles are aligned with the $z$-axis and where nearest neighbor dipoles are oriented anti-parallel to one another.
We refer to this ordering as the checkerboard ordering. Striped, fully polarized, and checkerboard orderings for the square lattice are depicted in Figure~\ref{fig:sq_ord}.
\begin{figure}[h]
\centering
\includegraphics{{2d_square_orderings}.pdf}
\caption{The considered orderings in the limit $g \gg 1$ and $u \ll 1$ for a system of 16 dipoles with the primary simulation box outlined for reference. On top is a fully polarized ordering aligned along the $x$-axis, which is degenerate with the ordering in which all dipoles are aligned along the $y$-axis. On the bottom left is one of four degenerate striped orderings, two along each of two cartesian lattice axes. On the bottom right is one of two possible checkerboard orderings where an "x" denotes a dipole oriented in the negative $z$ direction and a dot denotes a dipole oriented in the positive $z$ direction.}
\label{fig:sq_ord}
\end{figure}
As in the case of triangular lattices, an analysis of the classical potential energy of various orderings was performed. In this case, however, it was found that for all lattice sizes the two striped orderings had the lowest potential energy, depicted in Figure~\ref{fig:sq_ord_en}. Again, the left and right panel show the potential energy per particle calculated with the periodic sum convention and with the conventional minimum image convention, which
again highlights the advantage of the periodic sum convention regarding finite size effects.
In order to detect the relevant striped orderings the following quantity
\begin{equation}
\phi_{xy} = \sqrt{\expect{n_{x\text{-stripe}}}^2 + \expect{n_{y\text{-stripe}}}^2}, \label{eq:sq_xystripe}
\end{equation}
where the operators
\begin{equation}
n_{x\text{-stripe}} = \frac{1}{N}\left| \sum_{i=1}^{\sqrt{N}} \sum_{j=1}^{\sqrt{N}} (-1)^i n^x_{i\sqrt{N} + j} \right| \label{eq:x_stripe}
\end{equation}
and
\begin{equation}
n_{y\text{-stripe}} = \frac{1}{N}\left| \sum_{i=1}^{\sqrt{N}} \sum_{j=1}^{\sqrt{N}} (-1)^j n^y_{i\sqrt{N} + j} \right| \label{eq:y_stripe}
\end{equation}
pick out striped configurations aligned along the $x$ and $y$ axes, respectively, takes on a maximum in the fully ordered state when $g$ is very large and vanishes when dipoles are oriented randomly. The quantity $\phi_z$ remains unchanged as a way to detect polarization in the $z$ direction. The quantity
\begin{equation}
\phi_\text{checkerboard} = \expect{\frac{1}{N}\left| \sum_{i=1}^{\sqrt{N}} \sum_{j=1}^{\sqrt{N}} (-1)^{i + j} n^z_{i\sqrt{N} + j} \right|}
\end{equation}
is maximal in the case of checkerboard ordering.
However this was not observed to any appreciable extent.
\begin{figure}[h]
\centering
\includegraphics[width=5in]{{square_orderings_energies}.pdf}
\caption{The classical potential energy per particle of striped, polarized, and checkerboard configurations as a function of system size, with $C_\text{dd}/(4\pi r_\text{lat}^3) = 1$~a.u. and $u = 0$ on a square lattice under the periodic sum convention with a cutoff radius of $100$ lattice sites (left) and the minimum image convention (right).}
\label{fig:sq_ord_en}
\end{figure}
\subsection{PIGS Results for Dipoles on Triangular Lattices}
\label{subsec:results}
Systems of 48 dipoles confined to triangular lattices were simulated using the PIGS method and varying the values of $u$ and $g$ from 0 to 3 each.
We chose a time step of $\tau = 0.0375$ $(2\pi B)^{-1}$ and a imaginary time path length of $\beta = 5.1$ $(2\pi B)^{-1}$.
A detailed study of the bias introduced by finite $\tau$ and $\beta$ can be found in the Appendix.
\begin{figure}[p]
\centering
\includegraphics{{triangular_np_48_phase_pol_z}.pdf}
\caption{The in-plane polarization $\phi_\text{pol}$ (left panel) and transverse polarization $\phi_z$ (right panel) vs. $g$ and $u$ for a system of 48 dipoles on a triangular lattice with $\beta = 5.1$ $(2\pi B)^{-1}$ and $\tau = 0.0375$ $(2\pi B)^{-1}$.}
\label{fig:tri_z}
\includegraphics[width=4in]{{tri_z-pol_slice}.pdf}
\caption{Transverse polarization, $\phi_z$, vs. $g$ for a system of 48 dipoles on a triangular lattice with $\beta = 5.1$ $(2\pi B)^{-1}$ and $\tau = 0.0375$ $(2\pi B)^{-1}$. This shows the same data as the right panel in Figure~\ref{fig:tri_z} but presented in a way to highlight the gradual decay of the $z$-polarization as the interaction strength $g$ is increased.}
\label{fig:tri_z_slice}
\end{figure}
The right and left panel of Figure~\ref{fig:tri_z} show the dependence of the transverse and in-plane polarizations, $\phi_z$ and $\phi_{pol}$, respectively, on the parameters $g$ and $u$. We see that there is a decrease in the $z$ polarization (right panel), both as $g$ is increased and as $u$ is decreased. The fact that the $z$ polarization decreases as $u$ is decreased is not surprising, since a decline in the polarization with decreasing applied field is generally expected. The dependence on interaction strength $g$ is less immediately intuitive.
However for classical dipoles, one can show that in the limit $g \gg 1$, depending on the lattice geometry, the minimal classical energy configuration can have all dipoles polarized in the plane of the lattice with zero polarization out of the plane. The decline in the out of plane polarization for the PIGS results as $u$ decreases appears to be gradual with respect to both $g$ and $u$, with no sharp drop-off. This gradual decrease can be seen more clearly in Figure~\ref{fig:tri_z_slice}.
The behavior of the in-plane $\phi_\mathrm{pol}$ evident in the left panel of Figure~\ref{fig:tri_z}, is quite different, showing a very sharp rise in $\phi_\mathrm{pol}$ at a critical value $g \approx 1.5$ that is very nearly independent of $u$. This transition occurs over a very narrow range of $g$ values, and does not coincide with the decrease in $\phi_z$. While there is a slight modulation of the position of this crossover behavior with respect to $u$, the precise location of the transition is nevertheless difficult to pin down in this range of $u$ values, with the variation with $u$ being less than the width of the crossover region for the lattice sizes considered here.
This sharp variation of the in plane polarization with respect to the interaction strength is highly suggestive of a quantum phase transition between an essentially unpolarized state and a state exhibiting macroscopic polarization. In light of the behavior of one dimensional lattice systems of rotors~\cite{Abolins2011} this is not altogether surprising. In fact, in one dimensional systems without electric fields this crossover from an unpolarized state to a polarized state occurs at a similar value of $g$. The large statistical fluctuations near the critical value of $g$ precludes a definite answer whether $\phi_\mathrm{pol}$ varies continuously
with $g$ (second order phase transition) or has a jump (first order phase transition). But a continuous
second order phase transition would be consistent with the large statistical fluctuations, and
in particular with the slow convergence
of $\phi_\mathrm{pol}$ close to the critical value of $g$ when the imaginary time path length
$\beta$ is increased. This is discussed in detail in the Appendix.
What is somewhat surprising is that this behavior in two dimensions on a triangular lattice appears to be only weakly dependent on the strength $u$ of the external field, at least for moderately strong external fields. This suggests that this behavior can have profound effects on systems over a wide range of experimentally accessible external field strengths.
\subsection{PIGS Results for dipoles on Square Lattices}
\label{subsec:sq_results}
Systems of 64 dipoles confined to square lattices were simulated using the PIGS method, varying both $u$ and $g$
over the range 0 to 3. The time step was $\tau = 0.0375$ $(2\pi B)^{-1}$ and the imaginary time path length
was $\beta = 4.2$ $(2\pi B)^{-1}$. For a discussion of the convergence with $\tau$ and $\beta$ we refer again to the Appendix.
The dependence of the transverse polarization $\phi_z$ in the case of a square lattice is shown in the right panel of Figure~\ref{fig:sq_z}, showing very similar behavior to that shown for the transverse polarization on a triangular lattice in the right panel of Figure~\ref{fig:tri_z}. The reasoning behind the trends in $\phi_z$ with $g$ and $u$
is identical to that described for the triangular lattice. As with the triangular lattice results, the decline in $\phi_z$, shown in the right panel of Figure~\ref{fig:sq_z}, is gradual and not sharp.
\begin{figure}[h]
\centering
\centering
\includegraphics{{square_np_64_phase_xystripe_z}.pdf}
\caption{The striped order parameter $\phi_{xy}$ (left panel) and transverse polarization $\phi_z$ (right panel) vs. $g$ and $u$ for a system of 64 dipoles on a square lattice with $\beta = 4.2$ $(2\pi B)^{-1}$ and $\tau = 0.0375$ $(2\pi B)^{-1}$.}
\label{fig:sq_z}
\end{figure}
The left panel of Figure~\ref{fig:sq_z} shows the $g$ and $u$ dependence of the in-plane striped order parameter, $\phi_{xy}$ for the square lattice. Though the ordering described in this plot is quite different to the in-plane ordering shown in the left panel of Figure~\ref{fig:tri_z} for the triangular lattice, {\em i.e.} showing striped ordering rather than a totally polarized ordering, the qualitative behavior exhibited is very similar,
with a sharp transition from a disordered phase to an ordered phase at a critical value of $g$ around $1.25$. Similar to the triangular lattice, there is a slight trend toward a higher critical value of $g$ as the strength $u$ of the transverse field is increased. It should be noted that the critical value of $g$ for the square lattice differs from the corresponding
value for the triangular lattice, although they are quite similar.
\subsection{Mean Field Analysis}
\label{subsec:meanfield}
To compare with the PIGS results, we also studied the phase diagrams using a self-consistent mean field theory. In this
approach the $N$-body wave function
is approximated by a product of single-body wave functions
\begin{equation}
\ket{\Psi_\text{mf}} = \prod_{i=1}^{N} \ket{\phi_i}. \label{eq:mf_wf}
\end{equation}
Under this approximation the expectation energy of Eq.(\ref{eq:ham_simp}) can be written as
\begin{align}
\expect{E} &= \bra{\Psi_\text{mf}} H \ket{\Psi_\text{mf}} \\
&= \sum_{i=1}^{N} \bra{\phi_i}\mathbf{L}_i^2 - u \mathbf{n}_i \cdot \hat{\mathbf{e}}\ket{\phi_i} + \frac{g}{2} \sum_{i = 1}^N \sum_{j = 1}^N \bra{\phi_i} \mathbf{n}_i \ket{\phi_i} \cdot \mathbf{S}_{ij} \cdot \bra{\phi_j} \mathbf{n}_j \ket{\phi_j} \\
&= \sum_{i=1}^N \bra{\phi_i} h_\text{eff}(i) \ket{\phi_i}.
\end{align}
Expanding the single particle wave functions in the basis of spherical harmonics centered on particle $i$,
\begin{equation}
\ket{\phi_i} = \sum_{l=0}^\infty\sum_{m=-l}^l c_{i,lm} \ket{lm},
\end{equation}
and minimizing the expectation value of the energy with respect to the expansion coefficients, $c_{i,lm}$,
while requiring self-consistency, provides an approximation to the ground state energy and associated wave function. A value of $l_{max}=4$ was found sufficient
to converge the energies and order parameters over the range of $g$ and $u$ studied here.
\begin{figure}[h]
\centering
\includegraphics{{triangular_np_12_mf_phase_pol_z}.pdf}
\caption{Mean field calculation of the in-plane order parameter $\phi_\text{pol}$ (left panel) and transverse polarization $\phi_z$ (right panel) as a function of $g$ and $u$ for 12 dipoles on a triangular lattice.}
\label{fig:tri_xy_mf}
\end{figure}
The left and right panels of Figure~\ref{fig:tri_xy_mf} show, respectively, plots of the order parameters
$\phi_\mathrm{pol}$ and $\phi_z$
as functions of the field strength $u$ and interaction strength $g$, for 12 dipoles on a triangular lattice, with a spatial cutoff for the range of the periodic sum of 100 nearest-neighbor distances and a maximum angular momentum $l_\text{max} = 4$, corresponding to 300 total basis functions, derived from the self-consistent field wave function, $\ket{\Psi_\text{mf}}$.
\begin{figure}[p!]
\centering
\includegraphics{{triangular_np_48_phase_pol_z_comp}.pdf}
\caption{Comparison of $\phi_\mathrm{pol}$ (left panel) and $\phi_z$ (right panel) on triangular lattices calculated from the mean field approximation (12 dipoles) and from PIGS (48 dipoles) as a function of $g$ at $u = 3$.}
\label{fig:tri_pol_comp}
\includegraphics{{triangular_np_48_phase_pol_z_comp_u}.pdf}
\caption{Comparison of $\phi_\mathrm{pol}$ (left panel) and $\phi_z$ (right panel) on triangular lattices calculated from the mean field approximation (12 dipoles) and from PIGS (48 dipoles) as a function of $u$ at $g = 1.5$.}
\label{fig:tri_z_comp}
\end{figure}
While the mean-field results show the same overall features as the Monte Carlo results, they show
quantitative differences arising from the neglect of correlation between particles in the mean field approach.
Only in the case of non-interacting dipoles ($g=0$), do the mean field results become exact.
The differences between the mean-field and PIGS results are quantified in Figures~\ref{fig:tri_pol_comp} and ~\ref{fig:tri_z_comp}, which show cuts of $\phi_\mathrm{pol}$ and $\phi_z$ at fixed transverse field strength $u$ and fixed interaction strength $g$, respectively.
Each of these cuts show significant differences between the mean field and PIGS values, with the mean field values lying systematically below the PIGS values over a range of values of $u$ and $g$ for the in-plane polarization $\phi_\mathrm{pol}$, and systematically above the PIGS values for the transverse polarization $\phi_z$.
We can qualitatively understand the different sign of the deviation of
the mean field results for $\phi_\mathrm{pol}$ and $\phi_z$ from those obtained by PIGS in terms of the dipole correlations.
A nonzero value of $\phi_\mathrm{pol}$ can result only from the interactions
between dipoles, which cause correlations between these. Since such correlations are
neglected in the mean field approximation, this underestimates $\phi_\mathrm{pol}$. In contrast,
$\phi_z$ results from the external field, which does not cause correlations. $\phi_z$ must
exhibit the opposite trend of of $\phi_\mathrm{pol}$: if there is less in-plane order as
quantified by $\phi_\mathrm{pol}$, this increases the out-of-plane components
of the orientation vectors of the dipolar rotor, thus allowing an increase of $\phi_z$.
The underestimation of $\phi_\mathrm{pol}$ in the mean field approximation
is therefore accompanied by an overestimation of $\phi_z$.
A mean-field analysis analogous to that undertaken for triangular lattices was also performed for square lattices. The differences between the mean field and the PIGS simulation results for the square lattice are qualitatively similar to that observed with triangular lattices, showing the same general trends as those in Figures~\ref{fig:tri_pol_comp} and \ref{fig:tri_z_comp} and can be rationalized by similar arguments as above.
\section{Discussion}
\label{sec:discussion}
Because of the
anisotropic nature of the dipole-dipole interaction, the predicted
classical orientational orderings of
ground states of dipolar ensembles on square and triangular lattices in two dimensions is quite different.
The quantum Monte Carlo results presented here show that dipoles on a triangular lattice prefer orderings featuring a net in-plane polarization of the system, with all such polarized phases being degenerate in the absence of any fields in the plane of the lattice. On a square lattice, dipoles are predicted to adopt a striped ordering, characterized by no
average net polarization in the lattice plane. These differences can be explained by differences in the symmetry of the lattices: the layout of the triangular lattice results in 6 nearest neighbor interactions for each dipole and 12 next-nearest neighbor interactions, while dipoles on the square lattice possess only 4 nearest and 4 next-nearest neighbors.
Because of differences in the lattice geometries, the number of neighbors in successive shells, as well as the distance between successive shells, differs between lattices, leading to different contributions to the
long-ranged dipole-dipole interaction. Its anisotropy causes to partial cancellation of
positive and negative contributions, which leads to the different phases favored by the dipole interaction for
the two lattices.
We stress that dipoles situated on a regular lattice described by the
Hamiltonian (\ref{eq:ham}) are a proper quantum many-body system, and the
orientational ordering is a quantum phase transition. The classical
limit at zero temperature (no rotational kinetic energy) is taken
by letting $B \to 0$, which in our energy units corresponds to
$g\to\infty$ and $u\to\infty$. In the classical limit the dipoles
orient themselves such that the total potential energy, consisting of the
interaction and the external field, is minimized. At $T=0$ K, classical dipoles
are therefore in an ordered phase for any nonzero value of $g$ and $u$.
The correct quantum description accounts for the quantum kinetic energy
which leads to a higher potential energy due to orientational delocalization.
As a measure of ``quantumness'',
we use the difference between the potential minimum $V_{\rm min}$ and the quantum
mechanical expectation value of the potential $\langle V\rangle$ calculated in our
PIGS simulations. Figure\ref{fig:quantum_PEcomparison} shows $V_{\rm min}$ per particle
(line) and the PIGS expectation value $\langle V\rangle$ (symbols) per particle
for the triangular lattice.
In the left panel, the strength of the external potential $u$ is varied,
with $g=0$, and in the right panel the strength of the interaction $g$
is varied with $u=0$. $V_{\rm min}$ is a straight line
because we simply scale the respective potential and thus the minimum
of the potential. The ratio between $\langle V\rangle$ and
$V_{\rm min}$ can be regarded as measure of quantumness. For small
potential strengths $g$ or $u$, the quantum kinetic energy (not shown)
is dominant leading to a large delocatization and thus an expectation
value $\langle V\rangle$ much higher and close to zero. As $g$ or $u$
are increased, the potential becomes more dominant. For $g\to\infty$
or $u\to\infty$ the ratio $\langle V\rangle / V_{\rm min}$ will converge
to unity, as expected, and in this limit the quantum system is in an ordered
phase like the classical system. However, we note in the whole range
of $g$ and $u$ studied in this work, the ratio is significantly less
than unity. Particularly, in the vicinity of the phase transition
with a critical value $g\approx 1.5$ (see the left panel of Figure~\ref{fig:tri_z}), the
left panel of Figure~\ref{fig:quantum_PEcomparison}
shows that $\langle V\rangle / V_{\rm min}\approx 0.5$. In the interesting
regime around the quantum phase transition to an ordered phase,
dipolar rotors on a lattice require a full quantum mechanical description.
\begin{figure}[h]
\centering
\includegraphics{{triangular_pot_e_comp_g_u}.pdf}
\caption{Comparison of the potential energy per particle from PIGS simulations of 48 dipoles on a triangular lattice, with the classical potential energy of a system of 675 dipoles on the same lattice, polarized in both cases along the $x$-axis. The potential energy is shown as a function of the dipole-dipole interaction strength parameter $g$, at $u = 0$ (left panel) and as a function of the strength of the electric field $u$, at $g=0$ (right panel).}
\label{fig:quantum_PEcomparison}
\end{figure}
Our PIGS results reveal that, including the full quantum effects, dipoles on a triangular lattice will still tend to form a polarized phase at sufficiently high interaction strength
$g$. Furthermore this tendency appears to be nearly independent of the applied field strength in the transverse direction.
Since some proposals of self-assembled two-dimensional crystals rely on
imposition of transverse fields to ensure dipole orientation~\cite{Buchler2007,Astrakharchik2007}, such a phase transition could limit the densities at which these two-dimensional crystals are stable. This is because at higher densities, which is to say higher values of $g$, dipoles will naturally tend to form attractive head-to-tail configurations, rather than the repulsive transverse polarized configurations
that lead to stable crystals, in which case the dipoles may cease to be trapped due to collisions~\cite{Ni2008}.
In contrast, when confined to square lattices, dipoles will tend to form striped phases at high values of $g$. Despite differences in the precise nature of the orientational ordering at the higher values of $g$, the location in $g$ at which the transition occurs is remarkably similar on the two lattices. The differences in the orientational orderings could potentially derive from a variety of reasons. One possible reason is differences in system size, since both systems exhibited slight finite size effects in the PIGS calculations. Differences in the potential energy per particle between the fully polarized state (on the triangular lattice) and the striped phase (on the square lattice) could also lead to differences in the precise value of the interaction strength, $g$, at which the transition occurs on each lattice. For a given $g$ value, the potential energy per particle of the fully polarized state on the triangular lattice is predicted to be marginally lower than that of the striped phase on the square lattice, a trend which carries over to the PIGS results and can be seen in Figure~\ref{fig:pot_en_comp}. This suggests that the value of $g$ required to fully polarize a system of dipoles on a triangular lattice should be smaller than that required to form a striped phase on a square lattice, which is precisely what was observed in the Monte Carlo calculations.
\begin{figure}[h!]
\centering
\includegraphics{{potential_energy_comparison}.pdf}
\caption{Comparison of the potential energy per particle from PIGS simulations of 64 dipoles on a square lattice and 48 dipoles on a triangular lattice at $u = 3$. At higher values of the interaction strength, $g$, the potential energy per particle on a triangular lattice is found to be marginally lower than in the equivalent case on a square lattice, possibly helping to explain differences in the value of $g$ at which the transition from a disordered state to an ordered state occurs.}
\label{fig:pot_en_comp}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
Since the interaction strength $g$ includes the effect of not only the dipole moment, $d$, but also of the average inter-particle distance, the results found in this work imply that such ordered phases should be expected to occur for sufficiently dense systems.
From the values in Table~\ref{Table1:dipole parameters}, it can be seen that for diatomics of potential interest to trapped cold molecule experiments, having permanent dipole moments between 5 and 10 Debye, the transition from weakly to strongly interacting behavior with onset of marked in-plane polarization is expected to occur for inter-particle distances on the order of 10 - 30 nm. The best candidates for observing these phases are molecules that also have small rotational constants, i.e., the heavier species. In particular, CsI achieves $g = 1$ at a lattice spacing as large as approximately 30 nm.
Experimental systems that have been realized to date have been located well within the realm of
the weakly interacting regime of unpolarized dipoles, having been formed at much lower densities~\cite{Ni2008,Yan2013}. Additionally, even predicted self-assembled crystalline phases of dipolar molecules are expected to occur in the low density, {\em i.e.} the low interaction strength, regime~\cite{Buchler2007}. This should allow these self-assembled crystals to be probed without fear of entering a phase with in-plane ordering which in the absence of an external lattice potential, would lead to a collapse of the observed crystal due to attractive interactions.
This effect may also be important in systems of dipolar coupled pseudo-spins or excitons within a strongly interacting regime~\cite{Kocherzhenko2014}. Such a strongly interacting regime of dipolar coupled pseudo-spins~\cite{Kocherzhenko2014} is precisely the same as the high density regime where the dipoles are likely to
interact strongly with one another.
More directly relevant to dipolar molecules, as was pointed out in Section~\ref{sec:intro}, new trapping schemes~\cite{Ritt2006,romeroisartPRL13,Nascimbene2015,Lkacki2016,Perczel2017} may lead to lattices with much smaller lattice constants than are currently implemented by optical lattices.
In the present study the limits of stability in the strong interaction regime were not truly probed, as the dipolar particles were treated as translationally frozen with no explicit spatial trapping potential.
Further study is warranted in this regard, since the dynamics of an unstable ensemble trapped by finite optical lattice potentials would require overcoming or tunneling through the additional optical lattice potential. However, the results of this study show that at the very least it would seem that molecules can not be reliably counted upon to remain transversely polarized in the high density limit, even in the presence of strong external potentials in the transverse direction.
\section{Acknowledgements}
B. P. A. and K. B. W. were supported by funding from the National Science Foundation Grant No. CHE-1213141. R. E. Z. acknowledges funding from the Austrian science fund FWF (grant number P23535-N20).
\begin{appendix}
\section{Parameter Convergence}
\label{subsec:parameter_convergence}
The PIGS method is an exact quantum Monte Carlo method for bosons, apart from the bias of finite time step, $\tau$, and
finite imaginary time path lengths, $\beta$. The bias can be made arbitrarily small by choosing small values for
$\tau$ and large values for $\beta$.
To benefit from the systematic nature of the approximations it is necessary to carry out simulations at varying $\tau$ and $\beta$. In general such studies must be carried out at different points in the space of parameters in the Hamiltonian of interest, in the present work this is the values of the parameters controlling the interaction strength with an external field, $u$ in Eq.~(\ref{eq:ham}), and the strength of the dipole-dipole interaction, $g$ in Eq.~(\ref{eq:ham}).
\begin{figure}[p!]
\centering
\includegraphics{{triangular_tau_prelim}.pdf}
\caption{The behavior of $\phi_\mathrm{pol}$ as a function of the small time step, $\tau$, for a system of 48 dipoles on a triangular lattice at a constant imaginary path length, $\beta = 4.2$ $(2\pi B)^{-1}$.}
\label{fig:triangular_tau}
\includegraphics{{square_tau_prelim}.pdf}
\caption{The behavior of $\phi_{xy}$ as a function of the small time step, $\tau$, for a system of 64 dipoles on a square lattice at a constant imaginary path length, $\beta = 4.2$ $(2\pi B)^{-1}$.}
\label{fig:square_tau}
\end{figure}
\begin{figure}[p!]
\centering
\includegraphics{{square_beta_prelim}.pdf}
\caption{The behavior of $\phi_{xy}$ as a function of the imaginary path length, $\beta$, for a system of 64 dipoles on a square lattice at a constant small time step, $\tau = 0.0375$ $(2\pi B)^{-1}$.}
\label{fig:square_beta}
\includegraphics{{triangular_beta_prelim}.pdf}
\caption{The behavior of $\phi_\mathrm{pol}$ as a function of the imaginary path length, $\beta$, for a system of 48 dipoles on a triangular lattice at a constant small time step, $\tau = 0.0375$ $(2\pi B)^{-1}$ and constant $u = 0.5$.}
\label{fig:triangular_beta}
\end{figure}
Figure~\ref{fig:triangular_tau} shows the behavior of the order parameter $\phi_\mathrm{pol}$ as a function of the time step, $\tau$, at constant path length, $\beta = 4.2$ $(2\pi B)^{-1}$ for six different points in the space of Hamiltonian parameters for dipoles on a triangular lattice with 48 dipoles in the periodic simulation cell. These points correspond to the limits of the present study with two additional points at intermediate dipolar interaction strength, $(g, u) = \{(0, 0)$, $(0, 3)$, $(3, 0)$, $(3, 3)$, $(1.45, 0)$, $(1.45, 3)\}$. The order parameter has converged within acceptable numerical precision by $\tau = 0.0375$ $(2\pi B)^{-1}$. Similar behavior is observed for the total energy, and for the out-of-plane polarization, $\phi_z$.
Analogously, Figure~\ref{fig:square_tau} shows the behavior of $\phi_{xy}$ as a function of $\tau$ at this same value of $\beta$ and with $(g, u) = \{(0, 0)$, $(0, 3)$, $(3, 0)$, $(3, 3)$, $(1.25, 0)$, $(1.25, 3)\}$ for dipoles on a square lattice with 64 dipoles in the periodic simulation cell. As with the triangular lattice simulations, the variation of $\phi_{xy}$ is converged within acceptable numerical precision by $\tau = 0.0375$ $(2\pi B)^{-1}$. Just as for the triangular lattice, similar behavior is also observed for the total energy and for $\phi_z$.
Having established acceptable values for the short time step, $\tau = 0.0375$ $(2\pi B)^{-1}$, balancing accuracy and efficiency, it is then necessary to establish the required path length in imaginary time to ensure sampling of the ground state, and to make sure that the trial wave function, Eq.~(\ref{eq:approx_wf}), is not biasing the results. Figure~\ref{fig:square_beta} shows the behavior of $\phi_{xy}$ as a function of $\beta$ for $(g, u) = \{(0, 0)$, $(0, 3)$, $(3, 0)$, $(3, 3)$, $(1.25, 0)$, $(1.25, 3)\}$. For $g = 0$ and $3$ it appears that the quantities of interest are converged within acceptable tolerances by $\beta = 4.2$ $(2\pi B)^{-1}$. What is interesting is the behavior of $\phi_{xy}$ with respect to $\beta$ at $g = 1.25$. While the decay of $\phi_{xy}$ with respect to $\beta$ is very rapid at $g = 0$ and $3$, at $g = 1.25$ the convergence is very slow. This slow convergence with respect to $\beta$ was not observed for the energy.
For a better idea of how widespread this slow convergence is, a series of simulations at $u = 0.5$ and various values of $g$ were undertaken with 48 dipoles on a triangular lattice. Figure~\ref{fig:triangular_beta} shows the behavior of $\phi_\mathrm{pol}$ as a function of $\beta$ and $g$. It appears that for values of $g \leq 1.25$ and $g \geq 1.75$, $\phi_\mathrm{pol}$ is effectively converged well before $\beta = 5.1$ $(2 \pi B)^{-1}$. However, for $g = 1.5$ the value of $\phi_\mathrm{pol}$ converges very slowly. One plausible explanation for this is the presence of a second order phase transition, one manifestation of which would be a diverging spatial correlation length at the transition. In this case the Hartree trial wave function, Eq.~(\ref{eq:approx_wf}), which takes the form of a product of single particle functions, would be qualitatively incorrect. As a result it would have very poor overlap with the true ground state wave function, and so the decay of relevant quantities toward their ground state values would expected to be very slow.
In addition, a second order phase transition is characterized by soft modes
(Goldstone modes) with a vanishing excitation energy as we approach the phase transition.
This is the reason for the slow dynamics near the second order phase transition.
In the imaginary time evolution used in PIGS, such soft modes decay with
a large time constant that, in the thermodynamic limit, diverges at the quantum phase transition,
leading to a slow convergence to the ground state as a function
of the imaginary time path length $\beta$. Note that due to the small excitation energy of a soft mode, the total energy is barely affected, which is consistent with its observed rapid convergence with
$\beta$. For this reasons, we propose that the quantum phase transitions studied in this work are of second order.
Although the goal of quantum Monte Carlo simulations is a quantiative description,
the convergence study shows that close to the quantum phase transition to the
orientationally ordered phase, our simulations are still biased
by the value of the imaginary time path length $\beta$.
From Figures~\ref{fig:square_beta} and \ref{fig:triangular_beta}
we see that at the critical $g$, $\beta$ would need to be much larger than the value $\beta = 4.2$ $(2 \pi B)^{-1}$ and $\beta = 5.1$ $(2 \pi B)^{-1}$ used in our simulations of square and triangular lattices, respectively. In fact, in the thermodynamic limit $\beta$ would need to be infinitely large for
the critical value of $g$. Determining the exact values of the order
parameters $\phi_\mathrm{pol}$ and $\phi_{xy}$ very close to the critical value of $g$ requires an extensive (and computationally
expensive) suite of simulations with increasing $\beta$
and careful extrapolations to $\beta\to\infty$, in addition to undertaking a finite size scaling analysis
typically used in studies of second order phase transition.
The most efficient solution to reduce the bias is by reducing the excited state contribution
to the trial wave function. Rather than relying on a Hartree trial wave function Eq.~(\ref{eq:approx_wf}),
correlated trial wave functions need to be designed and optimized. For example, an optimized Jastrow ansatz
would include pair correlations. This goes beyond the scope of this paper but we note that the
hypernetted-chain Euler-Lagrange method adapted to orientational degrees of freedoms\cite{hufnagldiss}
could provide a highly optimized Jastrow ansatz, with presumably very small
overlap with excited states, such that even small $\beta$ values give accurate results.
\end{appendix}
|
{
"timestamp": "2018-03-12T01:04:25",
"yymm": "1803",
"arxiv_id": "1803.02512",
"language": "en",
"url": "https://arxiv.org/abs/1803.02512"
}
|
\section{Introduction}
Wireless sensor networks (WSNs) have gained a lot of popularity during the past few decades and are being implemented in different applications such as military and medical sectors as a mean for monitoring, processing and disseminating data\cite{pantazis2009energy}. The ease of implementation and being cost and energy efficient are among the reasons that have made WSNs popular. Wireless sensor nodes are small size devices that can create dense networks that are randomly positioned and deployed which makes them a suitable choice for inaccessible locations or disaster relief operations as well. WSNs have different properties and applications compared to the traditional wireless ad hoc networks, hence, protocols and algorithms that are being used in those networks are not valid for WSNs anymore \cite{akyildiz2002survey}. This opens up a wide field of research regarding WSNs \cite{al2015internet}.
Having an energy efficient network is always challenging when dealing with wireless networks. WSNs are also no exception to this, specially since they usually have access to a limited power source both in terms of the available energy ($<0.5$ Ah, $1.2$ V) and size \cite{p2, vardhan2000wireless} and in many cases such as aforementioned hardly accessible locations, it is not possible to renew the power sources for sensor networks which are usually batteries, hence, the battery life in such networks play a crucial role in the sensors lifetime which makes the energy consumption of the network elements a very important factor that needs to be considered when dealing with WSNs \cite{akyildiz2002wireless,de2011energy}.
Although a sensor network consumes energy in all its three areas of responsibilities which were mentioned earlier, it is the data disseminating, which includes both transmitting and receiving data, that consumes the most energy in a WSN. Sensor networks are usually used in short range communications with low data rates and short packet size which makes the RF communications a suitable choice for them \cite{rabaey2000picoradio}. However, designing an energy efficient WSN is always one of the challenges engineers face since the radio technologies are not suitable for being used in all kinds of applications\cite{shih2001physical}. Thus, in this paper, we analyze the energy efficiency of a WSN as part of an unlicensed network which allows for retransmission in case of an outage event happening.
Energy efficiency (EE) studies have become very popular during the past years and researchers have been studying EE in different types of applications. In \cite{hasan2011green}, authors study the reduction of the energy consumption of the whole network while in \cite{wang2010energy,sadek2009energy}, the energy consumption of two non-cooperative and cooperative networks with considering different network densities have been studied. In \cite{wang2010energy}, optimizing the packet size is used as a mean for maximizing the energy efficiency of the two mentioned networks. In \cite{sadek2009energy}, energy efficiency of a cooperative network is studied constrained by an outage threshold. Moreover, EE is investigated in \cite{de2011energy} by setting an end-to-end throughput constraint on the network while in \cite{alves2014outage}, by studying the throughput and outage of a full-duplex and an incremental cooperative half-duplex networks respectively.
While the following studies are important and valuable, they do not consider EE of a sensor network, based on the optimal throughput of the system, constrained by an outage threshold. In this paper, we expand our previous works in \cite{nardelli2016maximizing,tome2016joint} to cover a more generalized model rather than focusing on only the smart grids application. We follow the same model described in \cite{nardelli2012optimal,nardelli2014throughput} where there is possibility for retransmissions of a message in case of an outage happening in the network and it is shown that having a limited number of retransmissions can enhance the spatial throughput and transmission capacity of ad hoc networks. We use the same network model for investigating the EE of the network by first optimizing the link throughput in the system subjected to a minimum outage requirement where an outage event happens if the transmitted message is not decoded correctly or is never received by the receiver. Note that the number of retransmissions is limited and if the message is not received after a certain number of retransmissions, it is dropped
The rest of the paper is divided as follows.
Section \ref{sec:model} introduces the system model, while Section \ref{sec:opt} details the proposed throughput optimization and energy efficiency analysis.
Section \ref{sec:res} presents the numerical results and Section \ref{sec:concl} concludes this paper.
\section{System model}
\label{sec:model}
Considering the network model introduced in \cite{nardelli2016maximizing,tome2016joint}, the same model which is also shown in Fig. \ref{fig-scheme}, is used here for the implementation of the communication network in which the sensors transmit their data to their corresponding aggregator/controller where the following assumptions hold.
\begin{itemize}
\item \textbf{Assumption 1}: A communication network consists of both licensed and unlicensed networks where the users of both of the networks share the frequency bands allocated to the uplink channel.
\item \textbf{Assumption 2}: Licensed link is the connection link between static cellular base-stations and mobile users while the sensor nodes with fixed positions consist of unlicensed users which communicate with their corresponding controller through the uplink channel.
\item \textbf{Assumption 3}: The amount of power used by the unlicensed users (sensors) for their transmission is limited. This limitation can be enforced by the licensed network or can also be related to the sensors' own capabilities.
\item \textbf{Assumption 4}: In this model, it is assumed that there are no packet collisions between sensors associated with the same aggregator/controller due to the fact that the size of the transmitted messages are assumed to be small and multiple access solutions are effective for the size of the unlicensed network.
\end{itemize}
By considering the above assumptions, we are able to simplify the model to some extend. This would make the analysis easier.
The use of directional antennas in unlicensed communication links is justified based on Assumption 2 which states that the unlicensed nodes are static. This would result in not having any orientation errors \cite{wildman2014joint}. Having a limited transmit power based on Assumption 3 means that there is also a limit to the maximum range that the signal sent by the sensors can reach. Hence, the radiation pattern created by this transmission can be modeled as a line segment with the starting point being the sensors and the end point being limited by the imposed power constraint.
The interference in this model based on Assumptions 1, 2 and 4 can have different sources. (i) from unlicensed users (sensors) to cellular base-stations, (ii) from sensors to controller that are not their corresponding controller and (iii) from licensed users (mobile users) to to aggregators/controllers. The first two interference sources can be avoided if when implementing the network, the position of either the licensed or unlicensed nodes are specified. Even if the positions are randomly implemented, it is highly unlikely that there would be a base station or controller in the same line as the sensors' transmitted signal.
This leaves out only one source of interference in this model which would be (iii) from licensed users to aggregators/controllers. In order to be able to analyze the effect of interference on the performance of the network, we need to model the uncertainty of the interferers positions. We use Poisson point process $\Phi$ to model the interfering nodes in this network which are distributed over an infinite two-dimensional plane with network density $\lambda$. Details of using stochastic geometry and Poisson point process in modeling the wireless networks can be found in \cite{haenggi2012stochastic}.
Different metrics such as distance-dependent path-loss and fast fading is considered when modeling the wireless channel. Consider $r_i$ to be the distance between the \textit{i}th interferer and the reference receiver which is located arbitrary at the origin. Based on the Slivynak theorem, a receiver can have an arbitrary fixed position at the center of the Euclidean distance. This would make the estimation of the other elements of the network surrounding the receiver easier \cite{haenggi2012stochastic}. In this model, $g_i$ is considered to be the channel gain. The reference receiver received power then would be $W g_ir_i^{-\alpha}$ where $W$ is the transmit power and $\alpha> 2$ is the path-loss exponent. This will result in the following signal to interference ratio (SIR$_0$).
\begin{equation}
\label{eq_SIR}
\textrm{SIR}_0 = \dfrac{W_\mathrm{s} g_{0} r_0^{-\alpha}}{W_\mathrm{p} \; \underset{i \in \Phi}{\displaystyle \sum} g_{i} r_i^{-\alpha}}.
\end{equation}
In this equation, $W_\mathrm{p}$ and $W_\mathrm{s}$ denote the licensed users and unlicensed users transmit power respectively.
It should be noted that although the noise is neglected here, even the presence of the noise would not make a qualitative difference as stated also in \cite{weber2010overview}.
\begin{figure}[!t]%
\centering
\includegraphics [width= \columnwidth]{newmodel.pdf}
%
\caption{An illustration of the proposed scenario, where licensed and unlicensed users share the up-link channel. The reference sensor (unlicensed transmitter) is depicted by the sensor, the controller (unlicensed receiver) by the CPU and its antenna, the handsets are the mobile licensed users (interferers to the controller) and the big antenna is the cellular base-station. As the sensors uses directional antennas with limited transmit power (bold arrow), its interference towards the base-station can be ignored. The thin black arrows represent the licensed users' desired signal, while the red ones represent their interference towards the controller.}
\label{fig-scheme}
%
\end{figure}
Considering that point-to-point Gaussian codes and interference-as-noise decoding rules \cite{nardelli2015throughput,baccelli2011interference} are used in the reference link, it means that obtaining the desired spectral efficiency of $\log_2(1+\beta)$ in bits/s/Hz depends on the fact that the SIR is greater than a given threshold or not $\beta$ (i.e $\textrm{SIR}>\beta$), Hence, the probability of an outage event happening, $P_\textrm{out}$, can be explained as the probability of $\textrm{SIR}\leq\beta$. If a transmitted message is decoded in outage, it is retransmitted with the maximum of $m$ attempts, meaning that the message is dropped if it is still not successfully decoded by the receiver after $1+m$ transmissions. Thus, the probability of a successful transmission is calculated as $P_\textrm{suc} = 1 - P_\textrm{out}^{1+m}$.
Since the licensed users (interferers) are not static, their position is constantly changing in each transmission. SIR$_0$ in this model can be statistically evaluated by considering different realizations of Poisson point process $\Phi$. In order to compute $P_\textrm{out} = \textup{Pr}\left[\textup{SIR}_0 \leq \beta\right]$ for each transmission attempt by considering quasi-static channel gains (squared envelopes) $g$ which are independent and identically distributed exponential random variables (Rayleigh fading) with mean $1$, the following equation is used \cite{nardelli2012optimal}.
\begin{equation}\label{eq:6}
P_\textrm{out}= 1 - e^{- k \lambda \beta^{2/\alpha}},
\end{equation}
where $k=\pi r_0^2 \Gamma{\left(1 - \frac{2}{\alpha} \right)} \Gamma{\left(1 + \frac{2}{\alpha} \right)}$.
The throughput of the reference link $T$ is then calculated as \cite{nardelli2012optimal}:
\begin{equation}\label{eq:5}
T=\frac{\log(1+\beta)}{1+\bar{m}}\left(1-P_\textrm{out}^{1+m}\right),
\end{equation}
where $m$ is the maximum number of retransmission attempts. It should be noted that in order to find $m$, we use an approximation of \cite[§17]{nardellimult2009i} which is also explained in \cite{Iran}, in order to calculate the average number of transmissions needed to successfully transmit a message ($1+\bar{m}$).
\section{Throughput optimization And Energy Efficiency}
\label{sec:opt}
Energy efficiency of a wireless network can be seen as a criteria that captures the trade-off between the total power consumption (PC) and the throughput of the network. Hence, we first start by defining the optimal throughput of the network which is obtained by the following optimization problem:
\begin{equation}
\begin{aligned}\label{eq:7}
& \underset{(\beta,m)}{\text{max}}
& & \frac{\log(1+\beta)}{1+\bar{m}}\times\left(1-P_\textrm{out}^{1+m}\right) \\
& \text{subject to}
& & P_\textrm{out}^{1+m} \leq \epsilon
\end{aligned}.
\end{equation}
In this problem, the throughput is constrained to a maximum acceptable error rate $\epsilon$, which shows how often a message is dropped after reaching the maximum number of allowed retransmissions. Here, the SIR threshold $\beta>0$ and the number of allowed retransmissions $m \in \mathbb{N}$ are the design variables.
\begin{proposition}
The throughput $T = f (\beta, m)$ in \eqref{eq:5} is a function of the variables $m>0$ and $\beta>0$. The function $f$ is then concave with respect to $\beta$ if $\frac{\partial^2 T}{\partial \beta^2} < 0$.
$\beta^\ast$ and $m^\ast$ represent the value of $\beta$ and maximum number of retransmissions that maximizes the link throughput respectively :
\end{proposition}
\begin{equation}\label{eq:8}
\beta^\ast=\left(- \frac{1}{k\lambda} \log{\left (1 - \epsilon^{\frac{1}{m + 1}} \right )}\right)^{\frac{\alpha}{2}}.
\end{equation}
\begin{align}\label{eq:90}
m^\ast= \max\limits_{m \in \mathbb{N}} & \;\; \log\left (- \frac{1}{k \lambda} \log\left (1 -\epsilon^{\frac{1}{m + 1}} \right ) \right ) + \nonumber \\
&\alpha \left(- \frac{1}{k\lambda}\right)^{\frac{\alpha}{2}} \; \frac{ \left(\log\left (1 - \epsilon^{\frac{1}{m + 1}} \right ) \right )^{\frac{\alpha}{2} - 1}}{2 - \frac{2}{k\lambda} \left (\log\left (1 - \epsilon^{\frac{1}{m + 1}} \right ) \right )^{\frac{\alpha}{2}}} .
\end{align}
\begin{IEEEproof}
As $m$ and $\beta$ are strictly positive variables and function $T$ is twice differentiable in terms of $\beta$, then $T$ is concave if and only if $\frac{\partial^2 T}{\partial \beta^2} < 0$. Eq. \eqref{eq:8} is then attained by solving the derivative equation $\partial T /\partial \beta = 0$, whose solution is $\beta^\ast$. From (\ref{eq:8}), we find $T$ as a function of $m$ considering $\beta^*$.
%
The optimal throughput $T^*$ in terms of both $m$ and $\beta$ is then given by the value of $m$ that maximizes the throughput, which is given in \eqref{eq:90}. Moreover, from (\ref{eq:8}), we find $T$ as a function of $m$ considering $\beta^*$.
%
The optimal throughput $T^*$ in terms of both $m$ and $\beta$ is then given by the value of $m$ that maximizes the throughput, which is given in \eqref{eq:90}.
\end{IEEEproof}
\begin{remark} The maximum number of retransmissions $m^*$ is a natural number that is usually small, which makes the evaluation of \eqref{eq:10} computationally simple.
\end{remark}
By having the optimal throughput, we can now analyze the energy efficiency of the network. As it was mentioned earlier, the EE depends on the total power consumption and throughput of the network where the total power consumption of the network in its turn, includes the distance dependent transmission power in addition to the total energy consumed by the RF components and bit rate \cite{de2011energy,alves2014outage}. Considering the above parameters, the total power consumption of our single hop model is derived as:
\begin{equation}
\textup{PC}=\sum_{1}^{m+1} \frac{PC_{PA}+PC_{T_x}+PC_{R_x}}{\textup{log} (1+\beta^*)},
\end{equation}
\noindent where $PC_{PA}$ denotes the power amplifier power consumption in a one-hop transmission which also depends on a parameter called the drain efficiency of the amplifier. We denote the drain efficiency by $\zeta$ which would result in $PC_{PA}=\frac{\beta^*}{\zeta}$ in this model. Moreover, $\textup{log} (1+\beta^*)$ represents the bit rate (bits/s) of the system while $PC_{T_x}$ and $PC_{R_x}$ are constants which depends on the current technologies and are equal to the energy consumed during the transmission and reception operations by the internal circuitry respectively. By having the above parameters, the EE is expressed as
\begin{equation}\label{eq:10}
\textup{EE}=\frac{T^*}{\textup{PC}},
\end{equation}
\noindent where $T^*$ denotes the optimal throughput previously calculated.
\section{Numerical results}
\label{sec:res}
In this part, the numerical results of our analysis is presented. It should be noted that the following parameters were considered for obtaining these results. The distance between the sensors and the receiver $r_0=1$ meter and path-loss exponent $\alpha=4$. Also, based on the parameter setting presented in \cite{de2011energy}, $PC_{T_x}=97.9$ mW, $PC_{R_x}=112.2$ mW and $\zeta=0.35$.
Fig. \ref{fig:pc} shows how the power consumption changes with the density of interferers (measured in node/$m^2$ ) and outage constraints of the network for both limited and unlimited number of retransmissions being allowed in the network. We can see that in both cases, the outage constraint has a big impact on the total power consumption of the network. As the outage requirement gets stricter, meaning that a lower level of outage is allowed in the network (higher reliability), the total power consumption of the network also increases, showing that more power is used by the network in order to have a successfully decoded transmission.
Moreover, we can see that increasing the density of interferers also affects the total power consumption. While $\epsilon=0.001$, for very low density of interferers ($\lambda\leq 0.07$), having limited number of retransmissions results in having lower total power consumption. However, as the density of interferers increases and expectedly the total power consumption also increases, limited and unlimited retransmissions consume almost the same amount of energy. As the outage requirement gets looser, the range of $\lambda$ for which the limited transmission consumes less energy also increases. For instance, while ($\lambda\leq 0.22$), having limited $m$ means having lower energy consumption when $\epsilon=0.01$. When the outage requirement of the system is very loose, $\epsilon=0.1$, for all of the considered $\lambda$ range in our analysis, having limited $m$ would consume less energy, since when the network density is lower, the interference level of the network is also lower. This means that even when $m$ is limited, the system can achieve its expected outage constraint without having to consume a lot of energy, hence, having limited $m$ consumes less energy, but as $\lambda$ increases, the level of interference also increases which would mean that when having limited $m$, the system needs to use more energy in order to reach the required $\epsilon$, thus, almost the same amount of energy as having unlimited $m$ would be used by the network.
Fig. \ref{fig:ee} illustrates the behavior of the energy efficiency of the network with respect to $\lambda$ and different outage requirements. As the optimal throughput reduces dramatically by $\lambda$, we can see that the same thing is happening in the case of EE. As the outage requirement gets more stringent, the system needs more retransmission in order to reach the optimal throughput. This means that if $m$ is limited, the system can not always reach the optimal throughput as $\lambda$ increases. Since as shown in \eqref{eq:10}, EE has a direct relationship with the optimal throughput, the throughput decrease will effect EE also, that is why we can see that for most of the $\lambda$ range, having a limited number of retransmissions has also reduced the energy efficiency of the network compared to when an unlimited number of retransmissions is allowed. Although energy efficiency also depends on the power consumption which was shown increases with $\lambda$, this change is not as high and as effective as the decrease is the throughput, hence, the EE eventually ends up decreasing.
It is also shown in Fig. \ref{fig:ee} that like PC, EE also has a different behavior when $\lambda$ is very low. For those cases, having limited $m$ results in having higher EE. As it was explained earlier, for low $\lambda$, the network would consumes less energy while $m$ is limited, resulting in higher energy efficiency in the network. However, as $\lambda$ increase, limited $m$ uses as much energy as unlimited $m$ in order to reach the required constraints, on the other hand, the throughput for the unlimited case decreases also since the network can not reach the optimal throughput anymore. All these would eventually result in the network having lower EE when the number of retransmissions is limited as $\lambda$ and interference level increase.
It should be noted that the sharp fall and rise in Figs. \ref{fig:pc} and \ref{fig:ee} respectively are the results of having very low $\lambda$ which means having a very low interference. Moreover, Fig.\ref{fig:eree} illustrates the EE behavior as a function of $\epsilon$ for different $\lambda$s which further proves our point showing that looser outage requirements and lower network densities results in a higher level of energy efficiency in the network.
\begin{figure}[!t]
\includegraphics[width=1.05\columnwidth]{compc.pdf}
\caption{Power Consumption PC versus the density of interferers $\lambda$ for $\alpha=4$, $r_0=1$ for both unlimited and limited ($m=5$) number of retransmissions.}
\label{fig:pc}
\end{figure}
\begin{figure}[!t]
\includegraphics[width=1.05\columnwidth]{comee.pdf}
\caption{Energy efficiency EE versus the density of interferers $\lambda$ for $\alpha=4$, $r_0=1$ for both unlimited and limited ($m=5$) number of retransmissions.}
\label{fig:ee}\vspace{-3mm}
\end{figure}
\begin{figure}[!t]
\includegraphics[width=1.05\columnwidth]{EEER.pdf}
\caption{Energy efficiency EE versus the error threshold $\epsilon$ for $\alpha=4$, $r_0=1$ for different densities of interferers $\lambda$ while the number of retransmissions is limited ($m=5$).}
\label{fig:eree}\vspace{-3mm}
\end{figure}
\section{Conclusion}
\label{sec:concl}
In this paper, we analyzed the energy efficiency of a wireless network where the licensed and unlicensed users share the uplink channel. However, the unlicensed users do not cause interference on the licensed users transmissions. In this model, retransmission is also allowed if a message is decoded in outage. Our results showed the effect of retransmission and outage constraint on the power consumption and energy efficiency of the network considering different network densities. It was shown that having stricter outage requirement in the network also means having higher power consumption during transmissions. Depending on $\epsilon$ and $\lambda$, having limited retransmissions means lower power consumption or at most as much power consumption compared to having unlimited $m$. We also showed that as $\lambda$ increases, the energy efficiency of the network decreases due to the decrease in the optimal throughput. Having higher outage requirement also results in needing more $m$ in order to reach the $T^*$, therefore, having limited $m$ means having lower $EE$ for most of the $\lambda$ range. We plan to continue and improve the work done in this paper by jointly optimizing the energy efficiency and throughput constrained by a minimum outage requirement.
\section*{Acknowledgments}
This work is partially supported by Aka Project SAFE (Grant n.303532) and Strategic Research Council/Aka BCDC Energy (Grant n.$292854$).
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-03-08T02:05:34",
"yymm": "1803",
"arxiv_id": "1803.02562",
"language": "en",
"url": "https://arxiv.org/abs/1803.02562"
}
|
\section{Introduction}
Recall the famous Rogers-Ramanujan identities \cite{And-book,Sills-book}:
\begin{thm*} For any positive integer $n$ we have:
\begin{enumerate}[noitemsep,topsep=0pt]
\item Number of partitions of $n$ in which adjacent parts differ by at least $2$
is the same as the number of partitions of $n$ in which each part is $\equiv \pm 1\pmod{5}$.
In generating function form, we have:
\begin{equation}
\sum\limits_{n\geq 0}\dfrac{q^{n^2}}{(1-q)(1-q^2)\cdots(1-q^n)} = \prod\limits_{
\substack{m\geq 1\\m\, \equiv\, \pm 1\,\,(\mathrm{mod}\, 5) }}\frac{1}{1-q^m}.
\end{equation}
\item Number of partitions of $n$ in which adjacent parts differ by at least $2$ and $1$ does not appear as a part
is the same as the number of partitions of $n$ in which each part is $\equiv \pm 2\pmod{5}$.
In generating function form, we have:
\begin{equation}
\sum\limits_{n\geq 0}\dfrac{q^{n^2+n}}{(1-q)(1-q^2)\cdots(1-q^n)} = \prod\limits_{
\substack{m\geq 1\\m\, \equiv\, \pm 2\,\,(\mathrm{mod}\, 5)}}\frac{1}{1-q^m}.
\end{equation}
\end{enumerate}
\end{thm*}
In the above, we interpret the $q$-series identities as identities of purely formal series.
We introduce the terms ``partition-theoretic sum-side'' to refer to the difference conditions,
``analytic sum-side'' to refer to the sum in the $q$-series expansions and ``product-side''
to the product in the $q$-series.
The present paper deals with experimentally finding many identities of Rogers-Ramanujan type as we now explain.
\subsection{Affine Lie algebras and integer partition identities}
Affine Lie algebras have been an important source of new and intriguing integer partition identities and $q$-series identities.
We refer the reader to \cite{Sills-book} for an excellent review.
Building on the work \cite{FL,LM}, in a series of papers \cite{LW1,LW2,LW3,LW4} Lepowsky and Wilson showed how to interpret and prove Rogers-Ramanujan type identities using principally specialized standard modules for affine Lie algebras.
In these works, the product-sides of the identities arise from Weyl-Kac character formula combined with Lepowsky's numerator formula and the
partition-theoretic sum-sides arise from Lepowsky-Wilson's $Z$-algebras.
In \cite{C1}, Capparelli used Meurman and Primc's variant \cite{MP1} of Lepowsky and Wilson's method of $Z$-algebras to conjecture new partition identities using level 3 standard modules for $A^{(2)}_2$.
In this setup, the sum-side partition conditions (generically, these are multi-color partitions) follow by reducing a given Poincar\'e-Birkhoff-Witt-type spanning set to a basis --- a reduction that is achieved by using vertex-algebraic ``relations.''
One can therefore be assured that representations of any affine Lie algebra at any positive integral level lead to some sort of identities involving partitions, with the caveat that such identities are generically extremely complicated. Nonetheless, the point is that the characters of standard modules of affine Lie algebras
are a treasure trove of many interesting and as yet unknown integer partition identities.
In principle, the process of conjecturing partition identities using $Z$-algebras (or other vertex-algebraic methods) for any given affine Lie algebra at any given (positive integral) level could be utilized to discover new identities, however this gets notoriously tedious when the rank
of the algebra and/or the level of the module become large. This necessitates the need for new techniques of investigation.
In \cite{KR}, we initiated a study to discover new partition identities using experimental methods, and we discovered six new conjectural identities. Three of these six turned out to be related to principally specialized characters of the level $3$ standard modules for the affine Lie algebra $D_4^{(3)}$. In a current work in progress \cite{KNR}, we have initiated a search for identities that mimic certain $Z$-algebraic considerations.
However, in these works, a priori we were not specifically looking for identities related to affine Lie algebras.
After the search, one had to check if the newly found identities would be the characters of standard modules for some affine Lie algebra.
In the present paper, we undertake the exploration in a fundamentally different philosophical direction.
We start with a specific algebra at a specific level (the affine Lie algebra $A_9^{(2)}$ at level 2) and we start with the principally specialized characters of the vacuum spaces (with respect to the principal Heiseberg algebra) of the corresponding standard modules. These characters naturally factor into an infinite product due to Weyl-Kac character formula and Lepowsky's numerator formula.
We then use experimental methods to conjecture corresponding partition-theoretic sum-sides.
For the algebras $A_{2n+1}^{(2)}$, the products arising from those level $2$ modules that are contained in the tensor product of two inequivalent level $1$ modules \cite{BM-crystal} seem promising from the partition-theoretic viewpoint:
among other things, their inverse Euler transform is periodic with only entries being $0$ or $\pm 1$. By a slight abuse of terminology, we say that the inverse Euler transform of the $q$-series $\prod_{m\geq 1}(1-q^{m})^{-a_m}$ is the sequence $\{a_m\}_{m\geq 1}$.
The following table summarizes the known information and explains why $A_9^{(2)}$ was a natural candidate to explore (see \cite{B} for the corresponding product-sides):
\begin{center}
\begin{small}
{
\def1.4{1.4}
\begin{tabular}{|c|c|c|}
\hline
\textit{Algebra} & \textit{Product Sides/Identities} & \textit{Sum-side status} \\
\hline \hline
$A_3^{(2)}$ & {Alladi's companion to Schur's identity \cite{And-310}} &
$Z$-algebraic interpretation given \cite{T} \\
\hline
$A_5^{(2)}$ & G\"ollnitz-Gordon identities &
$Z$-algebraic interpretation given \cite{K-GGZ} \\
\hline
$A_7^{(2)}$ & Rogers-Ramanujan identities &
$Z$-algebraic interpretation given in \cite{BM-RR} \\
\hline
$A_9^{(2)}$
& Present article
& $Z$-algebraic interpretation is a future work \\
\hline
$A_{11}^{(2)}$ & Nandi's products \cite{N} &
Sum-sides given in \cite{N} using level 4 modules for $A_2^{(2)}$ \\
\hline
\end{tabular}
}
\end{small}
\end{center}
To the best of our knowledge, for $A_{2n+1}^{(2)}$ with $n\geq 6$, the products do not correspond to known partition identities.
\subsection{Staircases to the sum-sides}
A striking feature of the identities found using $Z$-algebras is that the sum-sides of such identities are inherently partition-theoretic. Generically, it appears to be a hard task to find ``nice'' analytic sum-sides that count the sum-side partitions. In a few cases, certain candidates for analytic sum-sides are known using Bailey techniques (see for instance \cite{Sills-nandi}), but it is far from obvious how these analytic sum-sides are related to the partition theoretic sum-sides.
For the identities presented in this article, we are able to provide analytic companions to our partition-theoretic sum-sides, by using staircases and jagged partitions. Given a partition $\pi: \lambda_1+\cdots+\lambda_j$ written in a weakly increasing order and a positive integer $s$, the (ordered) sequence $\mu: \lambda_1, \lambda_2-s,\lambda_3-2s,\cdots, \lambda_{j}-(j-1)s$ is said to be obtained from $\pi$ by removing an $s$-staircase. As such, $\mu$ may not be in a weakly increasing order and
may also have non-positive entries. Such a sequence is called a jagged partition.
For all the identities in this paper, removing an appropriate staircase leads to interesting jagged partitions whose generating functions could be written down explicitly. One then replaces the staircases to arrive at analytical sum-sides for the original identities.
\begin{alignat*}{3}
&\begin{matrix}
\begin{tikzpicture}
\foreach \x in {0,...,11}
\draw (\x*.5, 0) node {$\bullet$};
\foreach \x in {0,...,10}
\draw (\x*.5, -0.5) node {$\bullet$};
\foreach \x in {0,...,3}
\draw (\x*.5, -1) node {$\bullet$};
\foreach \x in {0,...,3}
\draw (\x*.5, -1.5) node {$\bullet$};
\foreach \x in {0,...,2}
\draw (\x*.5, -2) node {$\bullet$};
\foreach \x in {0,...,0}
\draw (\x*.5, -2.5) node {$\bullet$};
\draw[dashed, thick, OliveGreen] (-0.25,0.25) -- (-0.25,-0.25) -- (4.75,-0.25) -- (4.75,0.25) -- (-0.25,0.25);
\draw[dashed, thick, OliveGreen] (-0.25,-0.25) -- (-0.25,-0.75) -- (3.75,-0.75) -- (3.75,-0.25);
\draw[dashed, thick, OliveGreen] (-0.25,-0.75) -- (-0.25,-1.25) -- (2.75,-1.25) -- (2.75,-0.75);
\draw[dashed, thick, OliveGreen] (-0.25,-1.25) -- (-0.25,-1.75) -- (1.75,-1.75) -- (1.75,-1.25);
\draw[dashed, thick, OliveGreen] (-0.25,-1.75) -- (-0.25,-2.25) -- (0.75,-2.25) -- (0.75,-1.75);
\end{tikzpicture}
\end{matrix}
&& \qquad \rightsquigarrow {\text{Remove a 2-staircase}} \rightsquigarrow\qquad
&&
\begin{matrix}
\begin{tikzpicture}
\foreach \x in {0,...,1}
\draw (\x*.5, 0) node {$\bullet$};
\foreach \x in {0,...,2}
\draw (\x*.5, -0.5) node {$\bullet$};
\foreach \x in {-2,...,-1}
\draw (\x*.5, -1) node {$\times$};
\foreach \x in {0,...,0}
\draw (\x*.5, -2) node {$\bullet$};
\foreach \x in {0,...,0}
\draw (\x*.5, -2.5) node {$\bullet$};
\end{tikzpicture}
\end{matrix}\\
&\pi: 1 + 3 + 4 + 4 + 11 + 12
&&\qquad\rightsquigarrow {\text{Remove a 2-staircase}} \rightsquigarrow\qquad
&&\mu: 1, 1, 0, -2, 3, 2
\end{alignat*}
The identities in $A_9^{(2)}$ lead to analytic sum-sides which differ only in the linear term in the exponent of $q$ in each summand. Varying this linear term further leads us to six more conjectural identities.
We then use this technique to provide analytic sum sides to certain previous conjectures, namely,
Identities $I_5$ and $I_6$ from \cite{KR} and Identities $I_{5a}$ and $I_{6a}$ from \cite{R}.
Again, variations on the analytic sum-sides for \cite[$I_6$]{KR} lead us to three more conjectural identities.
One of these three identities has an asymmetric product-side; we present
one further identity whose product-side has negatives of the residues from the aforementioned asymmetric product.
For every new (conjectural) identity presented in this paper, we are able to provide both the partition-theoretic and analytic sum-sides and prove that the analytic sum-sides are indeed generating functions of the partition-theoretic sum-sides.
Lastly, we also discuss analytic sum-sides to the two Capparelli identities.
As mentioned above, jagged partitions play a crucial role in the present paper. They first arose in the physics literature
\cite{FJM1} in the analysis of fermionic characters for certain superconformal minimal models.
Moreover, in \cite{FJM2,FJM3} some beautiful identities for jagged partitions were established.
In \cite{L}, Lovejoy established certain constant term identities related to generating functions for jagged partitions.
In \cite{ABM}, new analytic sum-sides for Schur's identity were found using staircases.
Very recently, in \cite{DL}, generalizations of the (first) Capparelli identity were found by utilizing jagged (over)partitions.
Our treatment of the first Capparelli identity is precisely a ``dilated'' version of the argument in \cite{DL}.
Notably, in \cite{Cap-higher}, Capparelli used staircases in his investigation of the identities related to the standard modules at levels $5$ and $7$ for the affine Lie algebra $A_2^{(2)}$.
\subsection{Verification}
All the new conjectural identities in this paper have been verified up to the coefficient of $q^{500}$.
Unlike \cite{KR} and \cite{R} where recursions based on partition-theoretic sum-sides were used for such a verification,
here we directly use the analytic sum-sides.
Maple code for verification can be found appended in the plain-text format at the end of the \texttt{.tex} file of this paper on arXiv.
\subsection{Future work and work in progress}
Proving the $q$-series identities in this paper is a work in progress.
We expect a vertex-operator-theoretic interpretation of the identities
arising from $A_9^{(2)}$ to be tedious. It is quite possible that such an investigation
may actually lead to completely different partition-theoretic sum-sides than the ones
conjectured here.
Investigation of identities related to analogous level 2 standard modules for all $A_{\text{odd}}^{(2)}$
and higher level standard modules for $A_2^{(2)}$ would be extremely interesting; see \cite{Cap-higher}, \cite{MS} and \cite[Section 6.4]{Sills-book}.
Nandi's identities (originating from level 4 standard modules for $A_2^{(2)}$) \cite{N} still remain quite difficult;
see \cite{Sills-nandi} for some recent results. These identities merit a closer study in the light of techniques presented
here.
We are working on experimentally finding more $q$-series identities of the kind presented here using jagged partitions
and staircases.
\subsection*{Acknowledgments} It is our pleasure to thank George E.\ Andrews, James Lepowsky, Mirko Primc, Andrew V.\ Sills and Doron Zeilberger
for their interest in our work and encouragement. We also thank Jeremy Lovejoy for a correspondence regarding \cite{DL}.
\section{Preliminaries}
We use the standard conventions regarding the $q$-Pochhammer symbols:
\begin{align}
(a;q)_n &= \prod_{1\leq t < n} (1-aq^t),\\
(a;q)_\infty &= \prod_{1\leq t } (1-aq^t),\\
(a_1,a_2,\dots,a_j;q)_m&=(a_1;q)_m(a_2;q)_m\cdots(a_j;q)_m.
\end{align}
For us, all identities presented in this paper are formal power series identities, and we shall expand expressions such as $\frac{1}{1-q}$ using geometric series.
We shall frequently use the following fundamental $q$-series identities due to Euler:
\begin{align}
\left(x;q\right)_\infty^{-1}&=
\sum_{n\geq 0}\dfrac{x^n}{(q;q)_n}\label{eqn:eulerp},\\
\left(-x;q\right)_\infty&=
\sum_{n\geq 0}\dfrac{x^nq^{n(n-1)/2}}{(q;q)_n}\label{eqn:eulerd}.
\end{align}
We write partitions of a positive integer in a \emph{weakly increasing} order. Sub-partitions of a partition $\pi$ will always refer
to \emph{contiguous} portions of $\pi$.
Suppose $\pi: p_1 + p_2 + \dots + p_n$ is a partition, where sometimes we may have to let $p_1=0$. Let $s$ be a positive integer.
Let $\mu$ be the sequence $p_1, p_2-s,p_3-2s,\dots,p_n-(n-1)s$.
We say that $\mu$ is obtained by deleting an $s$-staircase from $\pi$. Note that $\mu$ may have non-positive entries and may not be in a weakly increasing order (which is the reason why we separate entries of $\mu$ by a comma rather than a plus sign), however,
successive differences in $\mu$ are at least $-s$.
We call such $\mu$ jagged partitions.
Suppose that the generating function of a set $\mathcal{P}$ of partitions is $\sum\limits_{m,n\geq 0}a_{m,n}x^mq^n$, i.e., there are
$a_{m,n}$ partitions of $n$ of length $m$ in $\mathcal{P}$. Then,
the generating function of jagged partitions obtained by removing an $s$-staircase from each of the partitions of $\mathcal{P}$
is $\sum\limits_{m,n\geq 0}a_{m,n}x^mq^{-sm(m-1)/2}q^n$.
Jagged partitions emerging in this paper will have a very specific structure. We shall scan a jagged partition $\mu$ from left to right and identify maximum jagged portions of $\mu$ corresponding to each of the positive integers $1,2,\dots$ (sometimes $0$ will have to be considered as well). If $\mu$ is obtained by removing an $s$-staircase, the block
corresponding to $j$ is defined as the maximal contiguous block of $\mu$
starting with $j$ and containing only integers from $\{j,j-1,\dots,j-s\}$; either of these conditions may be sometimes slightly relaxed but it will be clear what we mean.
We shall designate such blocks using regular expressions.
If $\mathbf{P}$ is a pattern of integers (for example $\mathbf{P}=``2,1"$), then:
\begin{enumerate}
\item[$\mathbf{P}^*$] corresponds to a string of either $0$ or more contiguous blocks of $\mathbf{P}$.
\item[$\mathbf{P}^+$] will be a string of
$1$ or more contiguous blocks of $\mathbf{P}$.
\item[$\mathbf{P}^\bullet$] will be either the empty string or $\mathbf{P}$ itself.
\end{enumerate}
For instance, if the jagged partition
$2,1,2,1,3,3,4,3$
is obtained by removing a $1$-staircase from some partition $\pi$, its maximal block corresponding to $2$ matches $[2,1]^*$ (and also $[2,1]^+$ but not $[2,1]^\bullet$), the maximal block
corresponding to $3$ matches $3^*$ (and also $3^+$ but not
$3^\bullet$), the one for $4$ matches $[4,3]^\bullet$ etc.
We shall always denote a partition by $\pi$ and the jagged partition obtained by removing an $s$-staircase by $\mu$.
\section{Identities related to certain level $2$ modules for $A_9^{(2)}$}
\subsection{The symmetric conjectures}
We use the standard convention \cite{Kac} for designating the nodes in the Dynkin diagram of $A_9^{(2)}$,
see Figure \ref{fig:A92}.
In our conjectures, the product sides are precisely the principally specialized characters of the vacuum spaces
(with respect to the principal Heisenberg subalgebra) of certain level 2 standard modules as we specify below. These modules are contained in the tensor product of two inequivalent level $1$ modules \cite{BM-crystal}.
See \cite{LW1,LW2,LW3,LW4} and \cite{B} for the relevant terminology and the product-sides.
\begin{figure}\label{fig:A92}
\begin{tikzpicture
\draw [-] (1,1) -- (2.5,2);
\draw [-] (1,3) -- (2.5,2);
\draw [-] (2.5,2) -- (4,2);
\path (4,2) -- node {\(==\Leftarrow =\)} (5.5,2);
\draw [fill] (1,1) circle [radius=0.1];
\draw [fill] (1,3) circle [radius=0.1];
\draw [fill] (2.5,2) circle [radius=0.1];
\draw [fill] (4,2) circle [radius=0.1];
\draw [fill] (5.5,2) circle [radius=0.1];
\node [below] at (1,0.9) {$\alpha_1$};
\node [below] at (1,2.9) {$\alpha_0$};
\node [below] at (2.5,1.9) {$\alpha_2$};
\node [below] at (4,1.9) {$\alpha_3$};
\node [below] at (5.5,1.9) {$\alpha_4$};
\node [above] at (1,1.1) {$1$};
\node [above] at (1,3.1) {$1$};
\node [above] at (2.5,1.9+0.2) {$2$};
\node [above] at (4,1.9+0.2) {$2$};
\node [above] at (5.5,1.9+0.2) {$2$};
\end{tikzpicture}
\caption{Dynkin diagram of $A_9^{(2)}$. Labels $\alpha_\bullet$ enumerate the nodes. Numerical labels are coefficients for the canonical central element.}
\end{figure}
Ideas stemming from \cite{KNR, KR, N, R} suggest the following conjectures.
These conjectures have been checked up to the partitions of
$N=500$.
The following sum-side conditions are common to the three ensuing conjectures:
\begin{enumerate}
\item No consecutive parts allowed.
\item Odd parts do not repeat.
\item Even parts appear at most twice.
\item If a part $2j$ appears twice then $2j\pm 3,2j\pm 2$
(and an additional copy of $2j$, but this is subsumed in the third condition, also
$2j\pm 1$, but this is subsumed in the first condition) are forbidden to appear at all.
\end{enumerate}
Equivalently:
\begin{enumerate}
\item No consecutive parts allowed.
\item Odd parts do not repeat.
\item For a contiguous sub-partition $\lambda_i+\lambda_{i+1}+\lambda_{i+2}$, we have
$|\lambda_i - \lambda_{i+2}| \geq 4$ if
$\lambda_{i+1}$ is even and appears more than once.
\end{enumerate}
\subsubsection{Identity 1:}
Arises from the module $L(\Lambda_0 + \Lambda_1)$
of $A_9^{(2)}$.
Product:
$$\frac 1 {\left(q,q^4,q^6,q^8,q^{11};q^{12}\right)_\infty}.$$
Partition-theoretic sum-side: Above conditions, along with the initial condition that $2+2$ is not allowed as a sub-partition.
Example: There are ten partitions of each type for $n=12$.
\begin{minipage}[t]{3.7in}
Product side: \\
$1+11$ \\
$4+8$ \\
$1+1+1+1+8$ \\
$6+6$ \\
$1+1+4+6$ \\
$1+1+1+1+1+1+6$ \\
$4+4+4$ \\
$1+1+1+1+4+4$ \\
$1+1+1+1+1+1+1+1+4$ \\
$1+1+1+1+1+1+1+1+1+1+1+1$
\end{minipage}
\hfill
\begin{minipage}[t]{2.3in}
Partition-theoretic sum-side: \\
$12$ \\
$1+11$ \\
$2+10$ \\
$3+9$ \\
$4+8$ \\
$1+3+8$ \\
$5+7$ \\
$1+4+7$ \\
$6+6$ \\
$2+4+6$
\end{minipage}
\subsubsection{Identity 2:}
Arises from the module $L(\Lambda_3)$
of $A_9^{(2)}$.
Product:
\vspace{-.2in}
$$\frac {\left(q^6;q^{12}\right)_\infty}{\left(q^2,q^3,q^4,q^8,q^9,q^{10};q^{12}\right)_\infty}
= \frac {\left(-q^3;q^6\right)_\infty\left(q^6;q^6\right)_\infty}{\left(q^2;q^2\right)_\infty} $$
Partition-theoretic sum-side: Above conditions, along with the initial condition that $1$ is forbidden to appear.
\subsubsection{Identity 3:}
Arises from the module $L(\Lambda_5)$
of $A_9^{(2)}$.
Product:
$$\frac 1 {\left(q^4,q^5,q^6,q^7,q^8;q^{12}\right)_\infty}.$$
Partition-theoretic sum-side: Above conditions, along with the initial condition that $1$, $2$, and $3$ are all forbidden as parts.
\subsection{Analytic sum-sides for the symmetric conjectures}
\subsubsection{Identity 1}
Let $\pi$ be a partition counted in the sum-side and remove a $2$-staircase to obtain a jagged partition $\mu$.
Looking at the restrictions on $\pi$ it is clear the corresponding restrictions on $\mu$ amount to forbidding the appearance of the following blocks.
\begin{enumerate}
\item $j, j-1$. \label{it:id1noconsec}
\item $j, j-2$ if $j$ is odd\label{it:id1nooddrepeat}.
\item $j, j-2, j-4$ if $j$ is even.
\item $j, j-2, j-2$ if $j$ is even.
\item $j, j, j-2$ if $j$ is even.
\item $j, j+1, j-1$ if $j$ is odd.
\item $2,0$ is not allowed to appear in $\mu$.
\end{enumerate}
This implies that if $j$ is odd, the maximal block corresponding to $j$ in $\mu$ is of the form $j^*$.
Similarly, if $j\neq 2$ is even, the maximal block corresponding to $j$ in $\mu$ is of the form $j^*,[j-2, j]^*$.
Also, no block of the shape $j-1,j+1,j$ appears for $j$ odd.
It is now straightforward to obtain that the $(x,q)$-generating function for such $\mu$ is
given by:
\begin{align}
&\prod_{i\, \mathrm{odd}}\dfrac{1}{1-xq^i}
\prod_{j\,\mathrm{odd}}(1-xq^{j-1}\cdot xq^{j+1}\cdot xq^{j})
\prod_{k\,\mathrm{even}, k\geq 4}\dfrac{1}{1-xq^k\cdot xq^{k-2}}
\prod_{l\,\mathrm{even}}\dfrac{1}{1-xq^l}\nonumber\\
&
=\left(xq;q\right)^{-1}_\infty\left(x^2q^6;q^4\right)^{-1}_\infty\left(x^3q^9;q^6\right)_\infty\nonumber\\
&= \left(\sum_{i\ge 0} \frac{x^i q^i}{\left(q;q\right)_i}\right)
\left(\sum_{j\ge 0} \frac{x^{2j}q^{6j}}{\left(q^4;q^4\right)_j}\right)
\left(\sum_{k \ge 0} \frac{\left(-1\right)^kx^{3k}q^{3k^2+6k}} {\left(q^6;q^6\right)_k}\right) \nonumber\\
&= \sum_{i,j,k\ge 0} \left(-1\right)^k \frac{x^{i+2j+3k} q^{i+6j+3k^2+6k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k}.
\end{align}
Now we reinstate the $2$-staircase, i.e., we let $x^m\mapsto x^mq^{m(m-1)}$ to get that the required analytic sum side is:
\begin{align}
\sum_{i,j,k\ge 0} \left(-1\right)^k \frac{x^{i+2j+3k} q^{(i+2j+3k)(i+2j+3k-1) + i+6j+3k^2+6k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k}.
\end{align}
One now takes $x\mapsto 1$ to deduce the conjectures.
We shall omit this last step in the identities below.
\subsubsection{Identity 2} For this identity, only the initial conditions change. We get that $\mu$ must avoid blocks corresponding to $j=1$.
We have that the generating function for $\mu$ is:
\begin{align}
&\prod_{i\, \mathrm{odd}, i\geq 3}\dfrac{1}{1-xq^i}
\prod_{j\,\mathrm{odd}, j\geq 3}(1-xq^j\cdot xq^{j+1}\cdot xq^{j-1})
\prod_{k\,\mathrm{even} }\dfrac{1}{1-xq^k\cdot xq^{k-2}}
\prod_{l\,\mathrm{even}}\dfrac{1}{1-xq^l}\nonumber\\
&=\left(xq^2;q\right)^{-1}_\infty\left(x^3q^9;q^6\right)_\infty \left(x^2q^2;q^4\right)^{-1}_\infty\nonumber\\
&= \left(\sum_{i\ge 0} \frac{x^i q^{2i}}{\left(q;q\right)_i} \right)
\left(\sum_{k \ge 0} \frac{\left(-1\right)^kx^{3k}q^{3k^2+6k}}{\left(q^6;q^6\right)_k}\right) \left(\sum_{j\ge 0} \frac{x^{2j}q^{2j}}{\left(q^4;q^4\right)_j}\right) \nonumber\\
&= \sum_{i,j,k\ge 0} \frac{\left(-1\right)^k x^{i+2j+3k} q^{2i+2j+3k^2+6k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k}.
\end{align}
Reinstating the $2$-staircase, we obtain:
\begin{align}
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^k x^{i+2j+3k} q^{(i+2j+3k)(i+2j+3k-1)+2i+2j+3k^2+6k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k}.
\end{align}
\subsubsection{Identity 3}
For this identity, we omit the blocks corresponding to parts $1$, $2$ and $3$ from the generating function for $\mu$:
\begin{align}
&\prod_{i\, \mathrm{odd}, i\geq 5}\dfrac{1}{1-xq^i}
\prod_{j\,\mathrm{odd}, j\geq 5}(1-xq^j\cdot xq^{j+1}\cdot xq^{j-1})
\prod_{k\,\mathrm{even}, k\geq 4 }\dfrac{1}{1-xq^k\cdot xq^{k-2}}
\prod_{l\,\mathrm{even}, k\geq 4}\dfrac{1}{1-xq^l}\nonumber\\
&=\left(xq^4;q\right)^{-1}_\infty\left(x^3q^{15};q^6\right)_\infty\left(x^2q^6;q^4\right)^{-1}_\infty\nonumber\\
&=\left(\sum_{i\ge 0} \frac{x^i q^{4i}}{\left(q;q\right)_i}\right)\left(\sum_{k \ge 0} \frac{\left(-1\right)^kx^{3k}q^{3k^2+12k}}{\left(q^6;q^6\right)_k}\right)\left(\sum_{j\ge 0} \frac{x^{2j}q^{6j}}{\left(q^4;q^4\right)_j}\right) \nonumber\\
&=\sum_{i,j,k\ge 0} \frac{\left(-1\right)^k x^{i+2j+3k} q^{4i+6j+3k^2+12k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k}.
\end{align}
Reinstating the $2$-staircase, we obtain:
\begin{align}
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^k x^{i+2j+3k} q^{(i+2j+3k)(i+2j+3k-1)+4i+6j+3k^2+12k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k}.
\end{align}
\subsection{An intriguing relation}
We now deduce a relation that holds among the symmetric $A_9^{(2)}$ conjectures.
Let us denote the sum-side in the identity $i=1,2,3$ by $S_i(x,q)$ where $x$ counts number of parts
and $q$ corresponds to the number being partitioned.
\begin{thm}\label{thm:intrigueSum}
We have that:
\begin{align}
S_1(x,q)=S_2(x,q)+xqS_3(x,q).
\end{align}
\end{thm}
\begin{proof}
Recall the initial conditions:
\begin{itemize}
\item[$S_1$:] $2+2$ is forbidden,
\item[$S_2$:] $1$ is forbidden,
\item[$S_3$:] $1,2,3$ are forbidden.
\end{itemize}
Let $\lambda: \lambda_1+\lambda_2+\dots+\lambda_k$ be a partition
of $n$ counted in $S_1$ (recall that we have been using a \emph{weakly increasing} order)
and consider the maximal (possibly empty) contiguous string of odds starting with $1$ contained in $\lambda$.
Let us denote this string by $\sigma: \lambda_1=1 + \dots +\lambda_s$
(note that $\sigma$ may be an empty string, in which case we let $s=0$).
Also note that parts in $\sigma$ are strictly increasing, since odds are not allowed to repeat.
Now, depending on the parity of $s$, we transform the $\sigma$ substring of $\lambda$ (keeping the rest of $\lambda$ unchanged) to obtain new partitions.
\begin{enumerate}
\item[$s$ even:] Replace every pair $\lambda_{2i-1} + \lambda_{2i}$ of adjacent parts appearing in $\sigma$ by their average, i.e.,
$\frac{(\lambda_{2i-1} + \lambda_{2i})}{2} + \frac{(\lambda_{2i-1} + \lambda_{2i})}{2}$.
Example: $1+3+5+7 \rightsquigarrow 2+2 + 6+6$.
It is easy to check that the new partition thus obtained
has the same length and weight as $\lambda$ and
is counted in $S_2$.
\item[$s$ odd:] Replace every pair $\lambda_{2i} + \lambda_{2i+1}$ of adjacent parts appearing in $\sigma$ by their average, i.e.,
$\frac{(\lambda_{2i} + \lambda_{2i+1})}{2} + \frac{(\lambda_{2i} + \lambda_{2i+1})}{2}$
and then delete the initial $1$.
Example: $1+3+5+7+9 \rightsquigarrow 4+4 + 8+8$.
It is clear that the new partition obtained has weight and length one lower than that of $\lambda$ and is counted in $S_3$.
\end{enumerate}
\end{proof}
The product-side analogue of Theorem \ref{thm:intrigueSum} (with $x\mapsto 1$)
can be found in \cite{CH}:
\begin{thm}\label{thm:intrigueProd}
$$\frac 1 {\left(q,q^4,q^6,q^8,q^{11};q^{12}\right)_\infty}
=
\frac {\left(q^6;q^{12}\right)_\infty}{\left(q^2,q^3,q^4,q^8,q^9,q^{10};q^{12}\right)_\infty}
+ q\cdot
\frac 1 {\left(q^4,q^5,q^6,q^7,q^8;q^{12}\right)_\infty}.$$
\end{thm}
\begin{proof}
Follows from Equation (12) of \cite{CH}.
\end{proof}
\section{Asymmetric companions of the $A_9^{(2)}$ conjectures}
Observe that the analytic sum-sides presented above differ only in the linear terms in the exponents of $q$. A computer search for other possible linear terms reveals further conjectures.
\subsection{Analytic forms}
\begin{align}
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^kq^{i^2+4ij+6ik+4j^2+12jk+12k^2+j}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k} &= \frac 1 {\left(q,q^4,q^5,q^9,q^{11};q^{12}\right)_\infty} \label{conj:new1} \\
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^kq^{i^2+4ij+6ik+4j^2+12jk+12k^2+
i-3j}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k} &=
\frac 1 {\left(q,q^5,q^7,q^8,q^{9};q^{12}\right)_\infty} \label{conj:new1a} \\
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^k q^{i^2+4ij+6ik+4j^2+12jk+12k^2-2j-3k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k} & =
\frac {\left(q^3;q^{12}\right)_\infty}{\left(q,q^2,q^5,q^6,q^{9},q^{10};q^{12}\right)_\infty} \label{conj:new2} \\
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^k q^{i^2+4ij+6ik+4j^2+12jk+12k^2+i+2j+3k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k} & = \frac {\left(q^9;q^{12}\right)_\infty}{\left(q^2,q^3,q^6,q^7,q^{10},q^{11};q^{12}\right)_\infty} \label{conj:new2a} \\
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^k q^{i^2+4ij+6ik+4j^2+12jk+12k^2
-j}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k} & =
\frac 1 {\left(q,q^3,q^7,q^8,q^{11};q^{12}\right)_\infty}\label{conj:new3} \\
\sum_{i,j,k\ge 0} \frac{\left(-1\right)^k q^{i^2+4ij+6ik+4j^2+12jk+12k^2+2i+3j+6k}}{\left(q;q\right)_i\left(q^4;q^4\right)_j\left(q^6;q^6\right)_k} & = \frac 1 {\left(q^3,q^4,q^5,q^7,q^{11};q^{12}\right)_\infty}\label{conj:new3a}
\end{align}
Note the way in which the product sides come in pairs: \eqref{conj:new1} and \eqref{conj:new3},
\eqref{conj:new1a} and \eqref{conj:new3a}, and
\eqref{conj:new2} and \eqref{conj:new2a}. (The allowable congruence classes for \eqref{conj:new1} are $1,4,5,9,11 \pmod{12}$, while the allowable congruence classes for \eqref{conj:new3} are $-1,-4,-5,-9,-11 \pmod{12}$.)
\subsection{Partition-theoretic sum-sides}
\subsubsection{Identities 4 and 4a: \eqref{conj:new1} and \eqref{conj:new1a} }
The sum side of Conjecture \eqref{conj:new1} has the following difference conditions:
\begin{enumerate}
\item No part repeats.
\item Adjacent parts do not differ by 1 if the larger part is even.
\item $(2j) + (2j+1) + (2j+3)$ forbidden as a sub-partition.
\item $(2j) + (2j+2) + (2j+3)$ forbidden as a sub-partition.
\item $(2j) + (2j+2) + (2j+4)$ forbidden as a sub-partition.
\end{enumerate}
The difference conditions for Conjecture \eqref{conj:new1a} are the same, except for an additional initial condition (hence our 4/4a terminology):
\begin{enumerate}[resume]
\item None of $1+3$, $2+3$, $2+4$ can appear as sub-partitions. Alternately, assume that the partition starts with a fictitious $0$, and then these initial conditions are implied by the remaining difference conditions.
\end{enumerate}
We prove that these are the correct partition-theoretic sum-sides by using staircases.
Let $\pi$ be a partition counted by the sum-side for Identity 4.
Delete a $2$-staircase to obtain a jagged partition $\mu$.
It is clear that the conditions on $\pi$ amount to forbidding the following bocks in $\mu$.
\begin{enumerate}
\item $j,j-2$
\item $j,j-1$ if $j$ is odd.
\item $j,j-1,j-1$ if $j$ is even.
\item $j,j,j-1$ if $j$ is even.
\item $j,j,j$ if $j$ is even.
\end{enumerate}
Therefore, the maximal block in $\mu$ corresponding to an even value of $j$ is $[j, j-1,]^*j^\bullet, j^\bullet $ whereas the maximal block in $\mu$ corresponding to an odd $j$ is $j^*$.
We get that the generating function for $\mu$ is:
\begin{align}
&\prod_{j\,\mathrm{even}}\dfrac{1}{1-xq^j\cdot xq^{j-1}}
\prod_{k\,\mathrm{even}}(1+xq^k + xq^k\cdot xq^k)
\prod_{l\,\mathrm{odd}}\dfrac{1}{1-xq^l}\\
&=\left(xq;q\right)_{\infty}^{-1}\left(x^2q^3;q^4\right)_\infty^{-1}\left(x^3q^6;q^6\right)_\infty
\nonumber \\
&=\left(\sum_{i\geq 0}\dfrac{x^iq^{i}}{(q;q)_i}\right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{3j}}{(q^4;q^4)_j}\right)
\left(\sum_{k\geq 0}(-1)^k\dfrac{x^{3k}q^{3k^2+3k}}{(q^6;q^6)_k}\right)\nonumber\\
&=\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{i+3j+3k^2+3k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
Letting $x^m\mapsto x^mq^{m(m-1)}$ we get:
\begin{align}
\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{
(i+2j+3k)(i+2j+3k-1) + i + 3j+ 3k^2 + 3k}
}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
The analysis for Identity $4a$ is perhaps the most delicate of all identities considered in this article.
Let $\pi$ be a partition counted in the sum-side of Identity $4a$
and let $\tilde{\mu}$ be obtained by removing a $2$-staircase from $0+\pi$.
We make two cases and arrive at a jagged partition $\mu$ accordingly:
\begin{enumerate}[label={\alph*.}]
\item If every occurrence of $0$ in $\tilde{\mu}$ is immediately succeeded by $-1$ and no occurrence of $-1$ is immediately succeeded by $1$, we let $\mu=\tilde{\mu}$.
\item In all other cases, we let $\mu$ to be obtained by simply deleting a $2$-staircase from $\pi$.
\end{enumerate}
To explain the cases, we consider the following five examples:
\begin{align}
\pi= 1+6+7 \rightsquigarrow 0 + \pi = 0 + 1+6+7
\rightsquigarrow \tilde{\mu} = 0 , -1,2,1 \rightsquigarrow^{\text{case\,\,a}}
\mu&=0,-1,2,1.
\\
\pi= 1+5+8 \rightsquigarrow 0 + \pi = 0 + 1+5+8
\rightsquigarrow \tilde{\mu} = 0 , -1,1,2 \rightsquigarrow^{\text{case\,\,b}}
\mu&=1,3,4.
\\
\pi= 1+4+6 \rightsquigarrow 0 + \pi = 0 + 1+4+6
\rightsquigarrow \tilde{\mu} = 0 , -1,0,0 \rightsquigarrow^{\text{case\,\,b}}
\mu &= 1,2,2.\\
\pi= 2+6+7 \rightsquigarrow 0 + \pi = 0 + 2+6+7
\rightsquigarrow \tilde{\mu} = 0 , 0,2,1 \rightsquigarrow^{\text{case\,\,b}}
\mu&=2,4,3.
\\
\pi= 3+5+7 \rightsquigarrow 0 + \pi = 0 + 3+5+7
\rightsquigarrow \tilde{\mu} = 0 , 1,1,1 \rightsquigarrow^{\text{case\,\,b}}
\mu&=3,3,3.
\end{align}
Now, it turns out that the $\mu$ corresponding to case a have the following form. The maximal block corresponding to $0$ has the form $[0,-1]^+$,
the maximal block corresponding to an even number $j\geq 2$ is
$[j,j-1]^*,j^\bullet,j^\bullet$ and for an odd $j\geq 3$ it is $j^*$. The generating function for such $\mu$ is:
\begin{align*}
\dfrac{xq^0\cdot xq^{-1}}{1-xq^0\cdot xq^{-1}}
\left(\prod\limits_{j\,\,\text{even}, j\geq 2}
\dfrac{1}{1-xq^j\cdot xq^{j-1}}
(1+xq^j + xq^j\cdot xq^j)
\right)
\left(\prod\limits_{k\,\,\text{odd}, j\geq 3}
\dfrac{1}{1-xq^k}
\right)\\
=x^2q^{-1}\left(x^2q^{-1};q^4\right)_\infty^{-1}
\left(xq^2;q \right)_\infty^{-1}
\left(x^3q^6;q^6 \right)_\infty.
\end{align*}
The $\mu$ corresponding to case b have the following form.
The initial segment of $\mu$ is either
$[1,2]^*,2^\bullet$ or $[1,2]^*,[1,3]^\bullet$. After this initial segment, we have blocks corresponding to $j\geq 3$, which are the same as before, namely, for even $j\geq 2$ we have
$[j,j-1]^*,j^\bullet,j^\bullet$ and for an odd $j\geq 3$ it is $j^*$.
The generating function for such $\mu$ is therefore the following:
\begin{align*}
\left( \dfrac{1}{1-xq\cdot xq^2}(1 + xq^2 + xq\cdot xq^3) \right)
&\left(\prod\limits_{j\,\,\text{even}, j\geq 4}
\dfrac{1}{1-xq^j\cdot xq^{j-1}}
(1+xq^j + xq^j\cdot xq^j)
\right)
\left(\prod\limits_{k\,\,\text{odd}, j\geq 3}
\dfrac{1}{1-xq^k}
\right)\\
&=\left(x^2q^{3};q^4\right)_\infty^{-1}
\left(xq^2;q \right)_\infty^{-1}
\left(x^3q^6;q^6 \right)_\infty.
\end{align*}
Combining the cases, we arrive at:
\begin{align}
x^2q^{-1}&\left(x^2q^{-1};q^4\right)_\infty^{-1}
\left(xq^2;q \right)_\infty^{-1}
\left(x^3q^6;q^6 \right)_\infty
+\left(x^2q^{3};q^4\right)_\infty^{-1}
\left(xq^2;q \right)_\infty^{-1}
\left(x^3q^6;q^6 \right)_\infty\\
&=
\left(x^2q^{-1};q^4\right)_\infty^{-1}
\left(xq^2;q \right)_\infty^{-1}
\left(x^3q^6;q^6 \right)_\infty
\\
&=\left(\sum_{i\geq 0}\dfrac{x^iq^{2i}}{(q;q)_i}\right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{-j}}{(q^4;q^4)_j}\right)
\left(\sum_{k\geq 0}(-1)^k\dfrac{x^{3k}q^{3k^2+3k}}{(q^6;q^6)_k}\right)\nonumber\\
&=\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{2i-j+3k^2+3k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
Now letting $x^m\mapsto x^mq^{m(m-1)}$ we get:
\begin{align}
\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{(i+2j+3k)(i+2j+3k-1)+2i-j+3k^2+3k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
\begin{rmk}
This is the only identity in this paper where $x$ does not
exactly correspond to the number of parts in the sense that for certain partitions (namely those from case a) one has to include a fictitious zero and therefore the power of $x$ has to be shifted by $1$.
\end{rmk}
\subsubsection{Identities 5 and 5a: \eqref{conj:new2} and \eqref{conj:new2a} }
The sum-side of Conjecture \eqref{conj:new2} has the following difference conditions:
\begin{enumerate}
\item Adjacent parts do not differ by 1.
\item Even parts do not repeat.
\item A sub-partition of type $(2j+1) + (2j+1) + (2j+1 + t)$ is not allowed if $t\leq 3$.
\item A sub-partition of type $(2j+1-t) + (2j+1) + (2j+1)$ is not allowed if $t\leq 2$.
\item A sub-partition of type $(2j+1) + (2j+3) + (2j+5)$ is not allowed.
\end{enumerate}
The difference conditions for Conjecture \eqref{conj:new2a} are the same, except for an additional initial condition:
\begin{enumerate}[resume]
\item No 1s allowed. Alternately, assume that the partition starts with a fictitious $0$.
\end{enumerate}
Let $\pi$ be a counted in the sum-side for Identity $5$. Let $\mu$ be obtained by deleting a $2$-staircase from $\pi$.
Following patterns are forbidden in $\mu$:
\begin{enumerate}
\item $j, j-1$ for any $j$.
\item $j, j-2$ for even $j$.
\item $j, j-2,j-4$ for odd $j$.
\item $j, j-2, j-3$ for odd $j$.
\item $j, j-2, j-2$ for odd $j$.
\item $j, j-2, j-1$ for odd $j$.
\item $j, j, j-2$ for odd $j$.
\item $j, j, j$ for odd $j$.
\end{enumerate}
Therefore, the maximal block in $\mu$ corresponding to an even $j$ is of the form $j^*$, while
the one for an odd $j$ is of the form $[j,j-2,]^*,j^\bullet,j^\bullet$.
We have that the generating function for such $\mu$ is:
\begin{align}
&\prod_{i \mathrm{\,even}}\dfrac{1}{1-xq^i}
\prod_{j \mathrm{\,odd}}\dfrac{1}{1-xq^i\cdot xq^{i-2}}
\prod_{k \mathrm{\,odd}}(1+xq^k + xq^k\cdot xq^k)\\
&=\left(xq,q\right)_\infty^{-1}\left(x^2,q^4\right)_\infty^{-1}\left(x^3q^3;q^6\right)_\infty\nonumber\\
&=\left( \sum_{i\geq 0} \dfrac{x^iq^i}{(q;q)_i}\right)\left(\sum_{j\geq 0}\dfrac{x^{2j}}{(q^4;q^4)_j}\right)\left(\sum_{k\geq 0}(-1)^k\dfrac{x^{3k}q^{3k^2}}{(q^6;q^6)_k}\right)
\nonumber\\
&= \sum_{i,j,k\geq 0}(-1)^k \dfrac{x^{i+2j+3k}q^{i+3k^2}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
Reinstating the $2$-staircase, we obtain:
\begin{align}
\sum_{i,j,k\geq 0}(-1)^k \dfrac{x^{i+2j+3k}
q^{(i+2j+3k)(i+2j+3k-1)+i+3k^2}
}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
For Identity 5a, we simply change the initial conditions to get that the generating function for $\mu$ is:
\begin{align}
&\prod_{i \mathrm{\,even}}\dfrac{1}{1-xq^i}
\prod_{j \mathrm{\,odd}, j\geq 3}\dfrac{1}{1-xq^i\cdot xq^{i-2}}
\prod_{k \mathrm{\,odd}, k\geq 3}(1+xq^k + xq^k\cdot xq^k)\\
&=\left(xq^2,q\right)_\infty^{-1}\left(x^2q^4,q^4\right)_\infty^{-1}\left(x^3q^9;q^6\right)_\infty\nonumber\\
&=\left( \sum_{i\geq 0} \dfrac{x^iq^{2i}}{(q;q)_i}\right)\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{4j}}{(q^4;q^4)_j}\right)\left(\sum_{k\geq 0}(-1)^k\dfrac{x^{3k}q^{3k^2+6k}}{(q^6;q^6)_k}\right)
\nonumber\\
&= \sum_{i,j,k\geq 0}(-1)^k \dfrac{x^{i+2j+3k}q^{2i+4j+3k^2+6k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
Reinstating the $2$-staircase, we obtain:
\begin{align}
\sum_{i,j,k\geq 0}(-1)^k \dfrac{x^{i+2j+3k}q^{(i+2j+3k)(i+2j+3k-1)+2i+4j+3k^2+6k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
\subsubsection{Identities 6 and 6a: \eqref{conj:new3} and \eqref{conj:new3a}}
The sum-side of Conjecture \eqref{conj:new3} has the following difference conditions:
\begin{enumerate}
\item No parts repeat.
\item Adjacent parts do not differ by 1 if the smaller part is even.
\item A sub-partition of type $(2j) + (2j+2) + (2j+4)$ is not allowed.
\item A sub-partition of type $(2j+1) + (2j+2) + (2j+4)$ is not allowed.
\item A sub-partition of type $(2j+1) + (2j+3) + (2j+4)$ is not allowed.
\end{enumerate}
The difference conditions for Conjecture \eqref{conj:new3a} are the same, except for an additional initial condition:
\begin{enumerate}[resume]
\item Smallest part is at least 3.
\end{enumerate}
Let $\pi$ be a counted in the sum-side for Identity $6$. Let $\mu$ be obtained by deleting a $2$-staircase from $\pi$.
Following patterns are forbidden in $\mu$:
\begin{enumerate}
\item $j,j-2$ for any $j$.
\item $j,j-1$ if $j$ is even.
\item $j,j,j$ if $j$ is even.
\item $j,j-1,j-1$ if $j$ is odd.
\item $j,j,j-1$ if $j$ is odd.
\end{enumerate}
It is clear that the maximal block in $\mu$ corresponding to an even $j$ is $j^\bullet,j^\bullet$ while that corresponding to an odd $j$ is $[j,j-1]^*,j^*$.
Thus, the generating function for $\mu$ is:
\begin{align}
&\prod_{i\mathrm{\,even}}(1+xq^i+xq^i\cdot xq^i)
\prod_{j\mathrm{\,odd}}\dfrac{1}{(1-xq^j\cdot x q^{j-1})(1-xq^j) }\\
&=\left(xq;q\right)_\infty^{-1}\left(x^2q;q^4\right)_{\infty}^{-1}\left(x^3q^6;q^6\right)_\infty
\nonumber\\
&=\left(\sum_{i\geq 0}\dfrac{x^iq^i}{(q;q)_i}\right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^j}{(q^4;q^4)_j}\right)\left(\sum_{k\geq 0}(-1)^k\dfrac{x^{3k}q^{3k^2+3k}}{(q^6;q^6)_k}\right)\nonumber\\
&=\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{i+j+3k^2+3k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}
\end{align}
Letting $x^m\mapsto x^mq^{m(m-1)}$:
\begin{align}
\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{(i+2j+3k)(i+2j+3k-1)+i+j+3k^2+3k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}
\end{align}
For $6a$ we incorporate the initial conditions:
\begin{align}
&\prod_{i\mathrm{\,even}, i\geq 4}(1+xq^i+xq^i\cdot xq^i)
\prod_{j\mathrm{\,odd},j\geq 3}\dfrac{1}{(1-xq^j\cdot x q^{j-1})(1-xq^j) }\\
&=\left(xq^3;q\right)_\infty^{-1}\left(x^2q^5;q^4\right)_{\infty}^{-1}\left(x^3q^{12};q^6\right)_\infty
\nonumber\\
&=\left(\sum_{i\geq 0}\dfrac{x^iq^{3i}}{(q;q)_i}\right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{5j}}{(q^4;q^4)_j}\right)\left(\sum_{k\geq 0}(-1)^k\dfrac{x^{3k}q^{3k^2+9k}}{(q^6;q^6)_k}\right)\nonumber\\
&=\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{3i+5j+3k^2+9k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}
\end{align}
Letting $x^m\mapsto x^mq^{m(m-1)}$:
\begin{align}
\sum_{i,j,k\geq 0}(-1)^k\dfrac{x^{i+2j+3k}q^{(i+2j+3k)(i+2j+3k-1)+3i+5j+3k^2+9k}}{(q;q)_i(q^4;q^4)_j(q^6;q^6)_k}.
\end{align}
\section{Analytic sum-sides for some previous conjectures from \cite{KR} and \cite{R}}
In \cite{KR} we had conjectured six new partition identities and three further conjectures were given by one of the authors in \cite{R}.
We now provide analytic sum-sides for some of these identities.
For all the identities in this section, we shall use $1$-staircases to arrive at the analytic sum-sides.
To describe these identities, we first need a few definitions.
\begin{defi}
A partition $\pi=\lambda_1+\lambda_2+\cdots+\lambda_m$ written in a weakly increasing order
is said to have ``difference at least $k$ at distance $d$'' if for all $j$, $\lambda_{j+d}-\lambda_j\geq k$.
For example, the first Rogers-Ramanujan identity recalled above enumerates partitions satisfying
difference at least $2$ at distance $1$.
\end{defi}
\begin{defi} Fix an integer $i$.
A partition $\pi=\lambda_1+\lambda_2+\cdots+\lambda_m$ is said to satisfy ``Condition($i$)''
if $\pi$ has difference at least $3$ at distance $3$
such that if parts at distance two differ by at most $1$, then their sum (together with the intermediate part) is congruent to $i \pmod 3$.
\end{defi}
\subsection{Analytic forms for some identities from \cite{KR}}
We consider identities \cite[$I_5$]{KR} and \cite[$I_6$]{KR}.
\subsubsection{Identity $I_5$ from \cite{KR}}
This conjectural identity states that:
\begin{quote}
The number of partitions of a non-negative integer into parts congruent to $1,$ $3,$ $4,$ $6,$ $7,$ $10,$ or $11$ (mod $12$)
is the same as the number of partitions satisfying Condition($1$)
with at most one appearance of the part 1.
\end{quote}
The partition-theoretic sum-side
has the following equivalent formulation as counting the partitions
$\pi$ which forbid the following patterns:
\begin{enumerate}
\item $i+i+i$ for any $i$.
\item $i+(i+1)+(i+1)$ for any $i$.
\item $i+i+(i+1)+(i+2)$ for any $i$.
\item $i+i+(i+2)+(i+2)$ for any $i$.
\item $1$ is forbidden to appear more than once.
\end{enumerate}
Consider such a partition $\pi$ and delete a $1$-staircase to obtain a jagged partition $\mu$.
$\mu$ forbids the following patterns:
\begin{enumerate}
\item $j,j-1,j-2$ for any $j$.
\item $j,j,j-1$ for any $j$.
\item $j,j-1,j-1,j-1$ for any $j$.
\item $j,j-1,j,j-1$ for any $j$.
\item $\mu$ can not begin with $1,0$.
\end{enumerate}
We get that the maximal block corresponding to any $j\geq 2$ in $\mu$ is of the form $[j,j-1,j-1]^*,[j,j-1]^\bullet, j^*$.
The maximal block corresponding to $1$ is $1^*$.
We have the following generating function for such jagged partitions $\mu$.
\begin{align}
&\dfrac{1}{1-xq}\prod_{j\geq 2}\left(\dfrac{1}{1-xq^j\cdot xq^{j-1}\cdot xq^{j-1}}(1+xq^j\cdot xq^{j-1})\dfrac{1}{1-xq^j}\right)\\
&=\left(xq;q\right)_\infty^{-1}\left(-x^2q^{3};q^2\right)\left(x^3q^4;q^3\right)_\infty^{-1}\nonumber\\
&=\left(\sum_{i\geq 0}\dfrac{x^iq^i}{(q;q)_i}\right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{j^2+2j}}{(q^2;q^2)_j}\right)\left(\sum_{k\geq 0}\dfrac{x^{3k}q^{4k}}{(q^3;q^3)_k}\right)\nonumber\\
&=\sum_{i,j,k\geq 0}\dfrac{x^{i+2j+3k}q^{i+j^2+2j+4k}}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}.
\end{align}
Reinstating the $1$-staircase, i.e., $x^m\mapsto x^mq^{m(m-1)/2}$:
\begin{align}
\sum_{i,j,k\geq 0}\dfrac{x^{i+2j+3k}q^{
\frac{(i+2j+3k)(i+2j+3k-1)}{2} + i + j^2 + 2j + 3k
}}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}.
\end{align}
\subsubsection{Identity $I_6$ from \cite{KR}}
This conjectural identity states that:
\begin{quote}
The number of partitions of a non-negative integer into parts congruent to $2,$ $3,$ $5,$ $6,$ $7,$ $8,$ or $11$ (mod $12$)
is the same as the number of partitions
satisfying Condition($2$)
with smallest part at least 2 and at most one appearance of the part 2.
\end{quote}
The partition-theoretic sum-side has the following equivalent formulation as counting the partitions
$\pi$ which forbid the following patterns:
\begin{enumerate}
\item $i+i+i$ for any $i$.
\item $i+i+(i+1)$ for any $i$.
\item $i+(i+1)+(i+2)+(i+2)$ for any $i$.
\item $i+i+(i+2)+(i+2)$ for any $i$.
\item $1$ is forbidden to appear.
\item $2$ can appear at most once.
\end{enumerate}
Consider such a partition $\pi$ and delete a $1$-staircase to obtain a jagged partition $\mu$.
$\mu$ forbids the following patterns:
\begin{enumerate}
\item $j,j-1,j-2$ for any $j$.
\item $j,j-1,j-1$ for any $j$.
\item $j,j,j,j-1$ for any $j$.
\item $j,j-1,j,j-1$ for any $j$.
\item $\mu$ can not begin with $1$.
\item $\mu$ can not begin with $2,1$.
\end{enumerate}
We get that the maximal block corresponding to any $j\geq 3$ in $\mu$ is of the form $[j,j-1]^\bullet,[j,j,j-1]^*, j^*$.
The maximal block corresponding to $2$ is $[2,2,1]^*,2^*$.
We have the following generating function for such jagged partitions $\mu$.
\begin{align}
&\dfrac{1}{1-xq^2\cdot xq^2\cdot xq}\dfrac{1}{1-xq^2}
\prod_{j\geq 3}\left((1+xq^j\cdot xq^{j-1})\dfrac{1}{1-xq^j\cdot xq^{j}\cdot xq^{j-1}}\dfrac{1}{1-xq^j}\right)\\
&=\left(xq^2;q\right)_\infty^{-1}\left(-x^2q^{5};q^2\right)\left(x^3q^5;q^3\right)_\infty^{-1}\label{KRI6:removestair}\\
&=\left(\sum_{i\geq 0}\dfrac{x^iq^{2i}}{(q;q)_i}\right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{j^2+4j}}{(q^2;q^2)_j}\right)\left(\sum_{k\geq 0}\dfrac{x^{3k}q^{5k}}{(q^3;q^3)_k}\right)\nonumber\\
&=\sum_{i,j,k\geq 0}\dfrac{x^{i+2j+3k}q^{2i+j^2+4j+5k}}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}.
\end{align}
Reinstating the $1$-staircase, i.e., $x^m\mapsto x^mq^{m(m-1)/2}$:
\begin{align}
\sum_{i,j,k\geq 0}\dfrac{x^{i+2j+3k}q^{
\frac{(i+2j+3k)(i+2j+3k-1)}{2} + 2i + j^2+4j + 5k
}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}.
\end{align}
With a look ahead towards finding further companions to this identity, we now present a different analytic sum side
to \cite[$I_6$]{KR}.
Let us rewrite \eqref{KRI6:removestair} as follows:
\begin{align}
\left(xq^2;q\right)_\infty^{-1}&\left(-x^2q^{5};q^2\right)\left(x^3q^5;q^3\right)_\infty^{-1}
=\left(xq^2;q\right)_\infty^{-1}\dfrac{\left(x^4q^{10};q^4\right)_\infty}{\left(x^2q^5;q^2\right)_\infty}\left(x^3q^5;q^3\right)_\infty^{-1}\\
&=\left(\sum\limits_{i\geq 0}\dfrac{x^iq^{2i}}{(q;q)_i} \right)
\left(\sum\limits_{j\geq 0}\dfrac{x^{2j}q^{5j}}{\left(q^2;q^2\right)_j} \right)
\left(\sum\limits_{k\geq 0}\dfrac{x^{3k}q^{5k}}{\left(q^3;q^3\right)_k} \right)
\left(\sum\limits_{l\geq 0}(-1)^l\dfrac{x^{4l}q^{2l^2+8l}}{\left(q^4;q^4\right)_l} \right)
\nonumber\\
&=\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{2i+5j+5k+2l^2+8l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\end{align}
Putting back the $1$-staircase, i.e., $x^m\mapsto x^mq^{m(m-1)/2}$ we arrive at an alternate sum-side:
\begin{align}
\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+ 2i+5j+5k+2l^2+8l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\label{eqn:KRI6alternate}
\end{align}
Below, we shall vary the linear term in the exponent of $q$ to deduce further companions to this identity.
\subsection{Analytic forms for some identities from \cite{R}}
Motivated by the desire to find certain complementary identities to the conjectures in \cite{KR}, one of the authors
provided three further conjectures in \cite{R}. Now, we provide analytic sum-sides to \cite[$I_{5a}$]{R}
and \cite[$I_{6a}$]{R}.
\subsubsection{Identity $I_{5a}$ from \cite{R}}
This conjectural identity states that:
\begin{quote}
The number of partitions of a non-negative integer into parts congruent to $1, 2,
5, 6, 8, 9,$ or $11$ (mod $12$) is the same as the number of partitions
satisfying Condition($2$) such that $1+2+2$ is not allowed to appear in the partition.
\end{quote}
The conditions on the partition-theoretic sum-side of \cite[$I_{5a}$]{R}
are the same as the one for \cite[$I_6$]{KR}, except for the initial condition:
\cite[$I_{5a}$]{R} forbids the appearance of $1+2+2$ in $\pi$.
This translates to forbidding $1,1,0$ as an initial segment in $\mu$.
In effect, the block corresponding to $1$ in $\mu$ has to be $[1,0,[1,1,0]^*]^\bullet,1^*$
Therefore, modifying what we have above appropriately,
we have the following generating function for such jagged partitions $\mu$.
\begin{align}
&\left(1+\dfrac{x^2q}{1-x^3q^2}\right)\dfrac{1}{1-xq}
\prod_{j\geq 2}\left((1+xq^j\cdot xq^{j-1})\dfrac{1}{1-xq^j\cdot xq^{j}\cdot xq^{j-1}}\dfrac{1}{1-xq^j}\right)\\
&=(1+x^2q-x^3q^2)
\left(xq;q\right)_\infty^{-1}\left(-x^2q^3;q^2\right)\left(x^3q^2;q^3\right)_\infty^{-1}\nonumber\\
&=(1+x^2q-x^3q^2)\left(\sum_{i\geq 0}\dfrac{x^iq^{i}}{(q;q)_i}\right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{j^2+2j}}{(q^2;q^2)_j}\right)\left(\sum_{k\geq 0}\dfrac{x^{3k}q^{2k}}{(q^3;q^3)_k}\right)\nonumber\\
&=(1+x^2q-x^3q^2)\left(\sum_{i,j,k\geq 0}\dfrac{x^{i+2j+3k}q^{i+j^2+2j+5k}}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}\right)
\nonumber\\
&=
\sum_{i,j,k\geq 0}
\dfrac{x^{i+2j+3k}q^{i+j^2+2j+5k}+x^{i+2j+3k+2}q^{i+j^2+2j+5k+1}-x^{i+2j+3k+3}q^{i+j^2+2j+2k+2}}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}
\end{align}
Reinstating $1$-staircase, we have:
\begin{align}
\sum_{i,j,k\geq 0}
&\left\lbrace
\dfrac{
x^{i+2j+3k}q^{\frac{(i+2j+3k)(i+2j+3k-1)}{2}+i+j^2+2j+2k}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}
\right. \nonumber\\
&\qquad+\dfrac{
x^{i+2j+3k+2}q^{\frac{(i+2j+3k+2)(i+2j+3k+1)}{2}+i+j^2+2j+2k+1}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}
\nonumber\\
&\qquad-\left.
\dfrac{
x^{i+2j+3k+3}q^{\frac{(i+2j+3k+3)(i+2j+3k+2)}{2}+i+j^2+2j+2k+2}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}
\right\rbrace.
\end{align}
\subsubsection{Identity $I_{6a}$ from \cite{R}}
This conjectural identity states that:
\begin{quote}
The number of partitions of a non-negative integer into parts congruent to $1, 4,
5, 6, 7, 9$, or $10$ (mod $12$) is the same as the number of partitions
satisfying Condition($1$) such that
2 is not allowed to appear in the partition.
\end{quote}
The sum-side conditions on this identity are same as the one for \cite[$I_5$]{KR}, except for the initial conditions.
For \cite[$I_{6a}$]{R}, $2$ is forbidden to appear as a part.
Therefore, proceeding just like \cite[$I_5$]{KR} above, the maximal block in $\mu$ corresponding to $j\geq 3$, is of the form $[j,j-1,j-1]^*,[j,j-1]^\bullet,j^*$. However, if $1$ appears in $\mu$, the initial block in $\mu$ has to be either:
$1,0, 1^*,[2,1,1]^*,[2,1]^\bullet,2^*$ or
$1,[2,1,1]^*,[2,1]^\bullet, 2^*$.
The generating function for $\mu$ is:
\begin{align}
&
\left( \left(xq + xq\cdot x\cdot\dfrac{1}{1-xq}\right)\dfrac{1}{1-xq^2\cdot xq\cdot xq}
(1+xq^2\cdot xq) \dfrac{1}{1-xq^2}
+1\right)\cdot \nonumber\\
&\quad\prod_{j\geq 3}\left(\dfrac{1}{1-xq^j\cdot xq^{j-1}\cdot xq^{j-1}}(1+xq^j\cdot xq^{j-1})\dfrac{1}{1-xq^j}\right)\\
&=\left({\frac {xq \left(1+ {x}^{2}{q}^{3} \right) }{ \left(1- {x}^{3}{q}^{4}\right) \left(1- x{q}^{2} \right) }}
+{\frac{{x}^{2}q \left(1+ {x}^{2}{q}^{3} \right) }{ \left( 1-xq\right) \left(1- {x}^{3}{q}^{4} \right) \left(1- x{q}^{2} \right) }}
+1\right)
\left(xq^3;q\right)_\infty^{-1}\left(-x^2q^{5};q^2\right)\left(x^3q^7;q^3\right)_\infty^{-1}\nonumber\\
&=
xq\left(xq^2;q\right)_\infty^{-1}\left(-x^2q^{3};q^2\right)\left(x^3q^4;q^3\right)_\infty^{-1}
+x^2q\left(xq;q\right)_\infty^{-1}\left(-x^2q^{3};q^2\right)\left(x^3q^4;q^3\right)_\infty^{-1}\nonumber\\
&\quad +\left(xq^3;q\right)_\infty^{-1}\left(-x^2q^{5};q^2\right)\left(x^3q^7;q^3\right)_\infty^{-1}
\nonumber\\
&=\sum_{i,j,k\geq 0}
\dfrac{x^{i+2j+3k+1}q^{2i+j^2+2j+4k+1}
+ x^{i+2j+3k+2}q^{i+j^2+2j+4k+1}
+ x^{i+2j+3k}q^{3i+j^2+4j+7k}}
{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}.
\end{align}
Reinstating the $1$-staircase, i.e., $x^m\mapsto x^mq^{m(m-1)/2}$:
\begin{align}
\sum_{i,j,k\geq 0}
&\left\lbrace
\dfrac{x^{i+2j+3k+1}q^{\frac{(i+2j+3k+1)(i+2j+3k)}{2}
+2i+j^2+2j+4k+1}
}
{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}
\right.\nonumber\\
&\quad +
\dfrac{
x^{i+2j+3k+2}q^{\frac{(i+2j+3k+2)(i+2j+3k+1)}{2}+i+j^2+2j+4k+1}
}
{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}\nonumber\\
&\left.\quad +
\dfrac{
x^{i+2j+3k}q^{\frac{(i+2j+3k)(i+2j+3k-1)}{2}+3i+j^2+4j+7k}
}
{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k}\right\rbrace.
\end{align}
\section{Further companions of some previous conjectures from \cite{KR} and \cite{R}}
As promised above, we now vary the linear term in the exponent of $q$ in the expression \eqref{eqn:KRI6alternate}
to conjecture further companions to \cite[$I_6$]{KR}.
\subsection{Analytic forms}
We have the following conjectured identities.
\begin{align}
\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+2l^2+ i+3j+6k+6l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}
&=\dfrac{1}{\left(q,q^{3},q^{4},q^{6},q^{8},q^{9},q^{11} ;q^{12}\right)_\infty},
\label{eqn:KRI7}\\
\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+2l^2+ 3i+5j+6k+10l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}
&=\dfrac{1}{\left(q^3,q^{4},q^{5},q^{6},q^{7},q^{8},q^{9} ;q^{12}\right)_\infty},
\label{eqn:KRI7a}\\
\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+2l^2+ 2i+3j+5k+6l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}
&=\dfrac{1}{\left(q^2,q^{3},q^{4},q^{5},q^{8},q^{9},q^{11} ;q^{12}\right)_\infty}.
\label{eqn:KRI8}
\end{align}
Note that the product-sides of \eqref{eqn:KRI7} and \eqref{eqn:KRI7a} are symmetric.
Search for an identity in which the allowable congruences in the product-side are negatives of those
appearing in \eqref{eqn:KRI8} results in the following identity.
\begin{align}
\sum_{i,j,k,l\geq 0}
&\left\lbrace
(-1)^l\dfrac{
q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2} + 2l^2
+ 2i + 3j + 4k + 6l}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}
+
(-1)^l\dfrac{
q^{\frac{(i+2j+3k+4l+1)(i+2j+3k+4l)}{2} + 2l^2
+ 2i + 3j + 4k + 6l+1}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}\right.\nonumber\\
&\left.+
(-1)^l\dfrac{
q^{\frac{(i+2j+3k+4l+2)(i+2j+3k+4l+1)}{2} + 2l^2
+ 2i + 3j + 4k + 6l+2}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}
+
(-1)^l\dfrac{
q^{\frac{(i+2j+3k+4l+3)(i+2j+3k+4l+2)}{2} + 2l^2
+ 2i + 3j + 4k + 6l+4}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}
\right\rbrace\nonumber\\
&\qquad=\dfrac{1}{\left(q,q^3,q^4,q^7,q^8,q^9,q^{10} ;q^{12}\right)_\infty}.
\label{eqn:KRI8a}
\end{align}
\subsection{Partition-theoretic sum-sides} We now present a partition-theoretic interpretation for each of the sum-sides in the identities above.
\subsubsection{Identities $7$ and $7a$: \eqref{eqn:KRI7} and \eqref{eqn:KRI7a}}
The sum-side of these identities count partitions $\pi$ forbidding the following patterns:
\begin{enumerate}
\item $i + (i+1) + (i+1)$.
\item $i + i + (i+1)$.
\item $i + (i+2) + (i+2) + (i+2)$.
\item $i + i + i + (i+2)$.
\item $i + i + i + i$.
\end{enumerate}
For Identity $7$, i.e., \eqref{eqn:KRI7} the initial conditions are given by a fictitious zero:
\begin{enumerate}
\item[(6)] $1+1$ and $2+2+2$ are forbidden to appear.
\end{enumerate}
For Identity $7a$, i.e., \eqref{eqn:KRI7a} the initial conditions are given by:
\begin{enumerate}
\item[(6)] Smallest part is at least $3$.
\end{enumerate}
Let us first work with Identity $7$, i.e., \eqref{eqn:KRI7}.
Remove a $1$-staircase from $\pi$ to obtain a jagged partition $\mu$.
It is clear that $\mu$ forbids the following patterns:
\begin{enumerate}
\item $j,j,j-1$.
\item $j,j-1,j-1$.
\item $j,j+1,j,j-1$.
\item $j,j-1,j-2,j-1$.
\item $j,j-1,j-2,j-3$.
\item $\mu$ can not start with $1,0$ and $2,1,0$.
\end{enumerate}
The maximal block in $\mu$ corresponding to a part $j$
is of the shape $[j,j-1,j-2]^*,[j,j-1]^*,j^*$, with the following exception:
we may not have a string of $j,j+1,j,j-1$.
Moreover, due to the initial conditions on $\pi$, the blocks corresponding to $1$ and $2$ in $\mu$ must be of the form $1^*,[2,1]^*,2^*$.
This translates to the following generating function for $\mu$:
\begin{align}
&\dfrac{1}{1-xq}\cdot \dfrac{1}{1-xq^2\cdot xq}\cdot \dfrac{1}{1-xq^2}\cdot\nonumber \\
&\quad\quad\quad\cdot
\prod\limits_{j\geq 3}\left(\dfrac{1}{1-xq^j\cdot xq^{j-1} \cdot xq^{j-2}}\cdot
\dfrac{1}{1-xq^j\cdot xq^{j-1}}\cdot \dfrac{1}{1-xq^j} \right)
\cdot
\prod\limits_{j\geq 2}\left(1-xq^{j}\cdot xq^{j+1}\cdot xq^{j}\cdot xq^{j-1}
\right)\\
&=\left(xq;q\right)_\infty^{-1}\left(x^2q^3;q^2\right)_\infty^{-1}\left(x^3q^6;q^3\right)_\infty^{-1}
\left(x^4q^8;q^4\right)_\infty\nonumber\\
&=\left(\sum\limits_{i\geq 0}\dfrac{x^iq^{i}}{(q;q)_i} \right)
\left(\sum\limits_{j\geq 0}\dfrac{x^{2j}q^{3j}}{\left(q^2;q^2\right)_j} \right)
\left(\sum\limits_{k\geq 0}\dfrac{x^{3k}q^{6k}}{\left(q^3;q^3\right)_k} \right)
\left(\sum\limits_{l\geq 0}(-1)^l\dfrac{x^{4l}q^{2l^2+6l}}{\left(q^4;q^4\right)_l} \right)
\nonumber\\
&=\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{2l^2+i+3j+6k+6l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\end{align}
Putting back the $1$-staircase, i.e., $x^m\mapsto x^mq^{m(m-1)/2}$ we get:
\begin{align}
\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+2l^2+ i+3j+6k+6l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\end{align}
For Identity $7a$, i.e., \eqref{eqn:KRI7a} we merely change the initial conditions
to get the following generating function for $\mu$:
\begin{align}
&
\prod\limits_{j\geq 3}\left(\dfrac{1}{1-xq^j\cdot xq^{j-1} \cdot xq^{j-2}}\cdot
\dfrac{1}{1-xq^j\cdot xq^{j-1}}\cdot \dfrac{1}{1-xq^j} \right)
\cdot\prod\limits_{j\geq 3}\left(1-xq^{j}\cdot xq^{j+1}\cdot xq^{j}\cdot xq^{j-1}
\right)\\
&=\left(xq^3;q\right)_\infty^{-1}\left(x^2q^5;q^2\right)_\infty^{-1}\left(x^3q^6;q^3\right)_\infty^{-1}
\left(x^4q^{12};q^4\right)_\infty\nonumber\\
&=\left(\sum\limits_{i\geq 0}\dfrac{x^iq^{3i}}{(q;q)_i} \right)
\left(\sum\limits_{j\geq 0}\dfrac{x^{2j}q^{5j}}{\left(q^2;q^2\right)_j} \right)
\left(\sum\limits_{k\geq 0}\dfrac{x^{3k}q^{6k}}{\left(q^3;q^3\right)_k} \right)
\left(\sum\limits_{l\geq 0}(-1)^l\dfrac{x^{4l}q^{2l^2+10l}}{\left(q^4;q^4\right)_l} \right)
\nonumber\\
&=\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{2l^2+3i+5j+6k+10l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\end{align}
Putting back the $1$-staircase, i.e., $x^m\mapsto x^mq^{m(m-1)/2}$ we get:
\begin{align}
\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+2l^2 +3i+5j+6k+10l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\end{align}
\subsubsection{Identity $8$: \eqref{eqn:KRI8}}
The sum-side of this identity counts partitions $\pi$ forbidding the following patterns:
\begin{enumerate}
\item $i+i+i$.
\item $i+i+(i+1)$.
\item $i+(i+1)+(i+2)+(i+2)$.
\item $i+(i+1)+(i+2)+(i+3)$.
\item $i+(i+1)+(i+1)+(i+3)+(i+3)$.
\item Initial conditions are given by two fictitious zeros, i.e., $1$ is forbidden to appear as a part.
\end{enumerate}
Removing a staircase, we see that $\mu$ must forbid the following patters:
\begin{enumerate}
\item $j,j-1,j-2$.
\item $j,j-1,j-1$.
\item $j,j,j,j-1$.
\item $j,j,j,j$.
\item $j,j,j-1,j,j-1$.
\item $\mu$ does not start with a $1$.
\end{enumerate}
The maximal block corresponding to a part $j$ in $\mu$ is therefore of the shape
$[j,j-1]^*,[j,j,j-1]^*,j^\bullet, j^\bullet, j^\bullet$
and we get the following generating function for $\mu$:
\begin{align}
&
\prod\limits_{j\geq 2}\left(
\dfrac{1}{1-xq^j\cdot xq^{j-1}}\cdot
\dfrac{1}{1-xq^j\cdot xq^{j-1} \cdot xq^{j-1}}
\cdot\left(1 + xq^{j} + xq^j\cdot xq^j+ xq^j\cdot xq^j\cdot xq^j\right) \right)
\\
&=\left(xq^2;q\right)_\infty^{-1}\left(x^2q^3;q^2\right)_\infty^{-1}\left(x^3q^5;q^3\right)_\infty^{-1}
\left(x^4q^{8};q^4\right)_\infty\nonumber\\
&=\left(\sum\limits_{i\geq 0}\dfrac{x^iq^{2i}}{(q;q)_i} \right)
\left(\sum\limits_{j\geq 0}\dfrac{x^{2j}q^{3j}}{\left(q^2;q^2\right)_j} \right)
\left(\sum\limits_{k\geq 0}\dfrac{x^{3k}q^{5k}}{\left(q^3;q^3\right)_k} \right)
\left(\sum\limits_{l\geq 0}(-1)^l\dfrac{x^{4l}q^{2l^2+6l}}{\left(q^4;q^4\right)_l} \right)
\nonumber\\
&=\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{2l^2+2i+3j+5k+6l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\end{align}
Putting back the $1$-staircase, i.e., $x^m\mapsto x^mq^{m(m-1)/2}$ we have:
\begin{align}
\sum\limits_{i,j,k,l\geq 0}(-1)^l\dfrac{x^{i+2j+3k+4l}q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+2l^2 +2i+3j+5k+6l}}{(q;q)_i\left(q^2;q^2\right)_j\left(q^3;q^3\right)_k\left(q^4;q^4\right)_l}.
\end{align}
\subsubsection{Identity $8a$: \eqref{eqn:KRI8a}}
The sum-side of this identity counts partitions forbidding the following patterns.
\begin{enumerate}
\item $i+i+i$.
\item $i+(i+1) + (i+1)$.
\item $i+i + (i+1)+(i+2)$.
\item $i+(i+1)+(i+2)+(i+3)$.
\item $i+i+(i+2)+(i+2)+(i+3)$.
\item $1+1$.
\item $1+2+3$.
\item $2+2+3$.
\end{enumerate}
Removing a $1$-staircase, we see that $\mu$ forbids:
\begin{enumerate}
\item $j, j-1,j-2$.
\item $j,j,j-1$.
\item $j,j-1,j-1,j-1$.
\item $j,j,j,j$.
\item $j,j-1,j,j-1,j-1$.
\item $\mu$ does not start with $1,0$ or $1,1,1$ or $2,1,1$.
\end{enumerate}
It is now clear that for the maximal block in $\mu$ corresponding to a part $j\geq 3$ is $[j,j-1,j-1]^*,[j,j-1]^*,j^\bullet,j^\bullet,j^\bullet$.
If $\mu$ does not start with $1$, then the block corresponding to $2$ is $[2,1]^*,2^\bullet,2^\bullet,2^\bullet$.
However, if $\mu$ does start with $1$, then the blocks corresponding to $1$ and $2$ match:
$1,1^\bullet,[2,1,1]^*,[2,1]^*,2^\bullet,2^\bullet,2^\bullet$.
Combining, we get that the generating function of such $\mu$ is:
\begin{align}
&(xq+x^2q^2)\prod_{j\geq 2}
\left(\dfrac{1}{1-xq^j\cdot xq^{j-1}\cdot xq^{j-1}}\cdot
\dfrac{1}{1-xq^j\cdot xq^{j-1}}\cdot
(1+xq^j + xq^{j}\cdot xq^j +xq^{j}\cdot xq^{j}\cdot xq^j)
\right)\nonumber\\
&+
\dfrac{1+xq^2+x^2q^4+
x^3q^6}{1-xq^2\cdot xq}
\prod_{j\geq 3}\left(\dfrac{1}{1-xq^j\cdot xq^{j-1}\cdot xq^{j-1}}\cdot
\dfrac{1}{1-xq^j\cdot xq^{j-1}}\cdot
(1+xq^j + xq^{j}\cdot xq^j +xq^{j}\cdot xq^{j}\cdot xq^j)
\right)\\
& =
(xq+x^2q^2)\left(xq^2;q\right)_\infty^{-1}
\left(x^2q^3;q^2\right)_\infty^{-1}
\left(x^3q^4;q^3\right)_\infty^{-1}
\left(x^4q^8;q^4\right)_\infty
+
\left(xq^2;q\right)_\infty^{-1}
\left(x^2q^3;q^2\right)_\infty^{-1}
\left(x^3q^7;q^3\right)_\infty^{-1}
\left(x^4q^8;q^4\right)_\infty\nonumber\\
&=
\left(1+xq+x^2q^2-x^3q^4\right)
\left(xq^2;q\right)_\infty^{-1}
\left(x^2q^3;q^2\right)_\infty^{-1}
\left(x^3q^4;q^3\right)_\infty^{-1}
\left(x^4q^8;q^4\right)_\infty\nonumber\\
&=\left(1+xq+x^2q^2-x^3q^4\right)
\left(\sum_{i\geq 0}\dfrac{x^iq^{2i}}{\left(q;q\right)_i} \right)
\left(\sum_{j\geq 0}\dfrac{x^{2j}q^{3j}}{\left(q^2;q^2\right)_{j}} \right)
\left(\sum_{k\geq 0}\dfrac{x^{3k}q^{4k}}{\left(q^3;q^3\right)_{k}} \right)
\left(\sum_{l\geq 0}(-1)^l\dfrac{x^{4l}q^{2l^2+6l}}{\left(q^4;q^4\right)_{l}} \right)
\nonumber\\
&=\sum_{i,j,k,l\geq 0}
\left\lbrace
(-1)^l\dfrac{
x^{i+2j+3k+4l}q^{2l^2
+ 2i + 3j + 4k + 6l}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}
+
(-1)^l\dfrac{
x^{i+2j+3k+4l+1}q^{2l^2
+ 2i + 3j + 4k + 6l+1}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}\right.\nonumber\\
&\left. +
(-1)^l\dfrac{
x^{i+2j+3k+4l+2}q^{2l^2
+ 2i + 3j + 4k + 6l+2}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}
+
(-1)^l\dfrac{
x^{i+2j+3k+4l+3}q^{2l^2
+ 2i + 3j + 4k + 6l+4}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}
\right\rbrace.
\end{align}
Reinstating the $1$-staircase, we get:
\begin{align}
\sum_{i,j,k,l\geq 0}
&\left\lbrace
(-1)^l\dfrac{
x^{i+2j+3k+4l}q^{\frac{(i+2j+3k+4l)(i+2j+3k+4l-1)}{2}+2l^2
+ 2i + 3j + 4k + 6l}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}\right.
\nonumber\\
&+
(-1)^l\dfrac{
x^{i+2j+3k+4l+1}q^{\frac{(i+2j+3k+4l+1)(i+2j+3k+4l)}{2}+2l^2
+ 2i + 3j + 4k + 6l+1}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}\nonumber\\
& +
(-1)^l\dfrac{
x^{i+2j+3k+4l+2}q^{\frac{(i+2j+3k+4l+2)(i+2j+3k+4l+1)}{2}+2l^2
+ 2i + 3j + 4k + 6l+2}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}\nonumber\\
&\left. +
(-1)^l\dfrac{
x^{i+2j+3k+4l+3}q^{\frac{(i+2j+3k+4l+3)(i+2j+3k+4l+2)}{2}+2l^2
+ 2i + 3j + 4k + 6l+4}
}{(q;q)_i(q^2;q^2)_j(q^3;q^3)_k(q^4;q^4)_l}
\right\rbrace.
\end{align}
\section{Analytic sum-sides for Capparelli's identities}
Recall Capparelli's identities \cite{C1}
which arose from level $3$ standard modules for $A_2^{(2)}$:
\begin{thm*} For any positive integer $n$ we have that:
\begin{enumerate}
\item Number of partitions of $n$ into parts different from $1$
such that the difference of two consecutive parts is at least $2$, and is exactly $2$ or $3$ only if their sum is a multiple of $3$ is the same as the number of partitions in which every part is congruent to $\pm 2$ or $\pm 3 \,\,(\mathrm{mod}\,12)$.
\item Number of partitions of $n$ into parts different from $2$
such that the difference of two consecutive parts is at least $2$, and is exactly $2$ or $3$ only if their sum is a multiple of $3$ is the same as the number of partitions into distinct parts congruent to $1$, $3$, $5$, or $0$ $(\text{mod}\,\, 6)$.
\end{enumerate}
\end{thm*}
The difference conditions common to both the identities forbid consecutive differences equaling $0$ or $1$
and additionally following patterns are forbidden:
\begin{enumerate}
\item $3j + (3j+2)$.
\item $(3j+1) + (3j+3)$.
\item $(3j+1) + (3j+4)$.
\item $(3j+2) + (3j+5)$.
\end{enumerate}
For the first identity, the initial conditions stipulate that
\begin{enumerate}
\item[(7)] No $1$s are allowed.
\end{enumerate}
For the second identity, the initial conditions stipulate that
\begin{enumerate}
\item[(7)] No $2$s are allowed.
\end{enumerate}
Let us work with the first identity.
What follows is essentially a ``dilated'' version of the argument in \cite{DL}.
Deleting a $3$-staircase, $\mu$ forbids the following patterns:
\begin{enumerate}
\item $j$, $j-3$.
\item $j$, $j-2$.
\item $3j,3j-1$.
\item $3j+1, 3j$.
\item $3j+1, 3j+1$.
\item $3j+2, 3j+2$.
\end{enumerate}
Therefore, one can conclude that the maximal block in $\mu$ corresponding to $3j$ is $(3j)^*$;
the one for $3j+1$ ($j\geq 1$) is $(3j+1)^\bullet$; for $3j+2$ it is $[(3j+2),(3j+1)]^*,(3j+2)^\bullet$.
It is now easy to deduce that the generating function for $\mu$ is:
\begin{align}
&\left(\prod\limits_{j\geq 1} \dfrac{1}{1-xq^{3j}}\right)
\left(\prod\limits_{j\geq 1}1+xq^{3j+1}\right)
\left(\prod\limits_{j\geq 0} \dfrac{1}{1-xq^{3j+2}\cdot xq^{3j+1}}\cdot (1+xq^{3j+2})\right)
\nonumber\\
&= \left(xq^3;q^3\right)_\infty^{-1}
\left(-xq^2;q^3\right)_\infty\left(-xq^4;q^3\right)_\infty\left(x^2q^3;q^6\right)_\infty^{-1}\label{eqn:Cap1eq1}\\
&=\sum\limits_{i,j,k,l\geq 0}\dfrac{x^{i+j+k+2l}
q^{3i + 2j + \frac{3j(j-1)}{2} + 4k + \frac{3k(k-1)}{2} + 3l}
}{\left(q^3;q^3\right)_i\left(q^3;q^3\right)_j\left(q^3;q^3\right)_k\left(q^6;q^6\right)_l}.
\end{align}
Now we put back a $3$-staircase, i.e., $x^m\mapsto x^mq^{3m(m-1)/2}$:
\begin{align}
\sum\limits_{i,j,k,l\geq 0}\dfrac{x^{i+j+k+2l}
q^{\frac{3(i+j+k+2l)(i+j+k+2l-1)}{2}+3i + 2j + \frac{3j(j-1)}{2} + 4k + \frac{3k(k-1)}{2} + 3l}
}{\left(q^3;q^3\right)_i\left(q^3;q^3\right)_j\left(q^3;q^3\right)_k\left(q^6;q^6\right)_l},
\end{align}
which is exactly \cite[Equation (2.6)]{DL} with
$r\mapsto k, s\mapsto j, t\mapsto i, v \mapsto l, q\mapsto q^3,
a\mapsto q^{-2}, b\mapsto q^{-4}$.
We may instead arrive at a different analytic sum-side by first simplifying the expression \eqref{eqn:Cap1eq1}.
\begin{align}
&\dfrac{\left(-xq^2;q^3\right)_\infty\left(-xq^4;q^3\right)_\infty}{\left(xq^3;q^3\right)_\infty\left(x^2q^3;q^6\right)_\infty}
=\dfrac{\left(-xq^2;q^3\right)_\infty\left(-xq^4;q^3\right)_\infty}{\left(xq^3;q^3\right)_\infty\left(x^3q^3;q^6\right)_\infty}
\cdot\dfrac{\left(xq^2;q^3\right)_\infty\left(xq^4;q^3\right)_\infty}{\left(xq^2;q^3\right)_\infty\left(xq^4;q^3\right)_\infty}\nonumber\\
&=\dfrac{\left(x^2q^4;q^6\right)_\infty\left(x^2q^8;q^6\right)_\infty}
{\left(xq^2;q^3\right)_\infty\left(xq^3;q^3\right)_\infty\left(xq^4;q^3\right)_\infty\left(x^2q^3;q^6\right)_\infty}
=
\dfrac{\left(x^2q^4;q^6\right)_\infty\left(x^2q^8;q^6\right)_\infty}
{\left(xq^2;q\right)_\infty\left(x^2q^3;q^6\right)_\infty}
\cdot
\dfrac{\left(x^2q^6;q^6\right)_\infty}{\left(x^2q^6;q^6\right)_\infty}\nonumber\\
&=
\dfrac{\left(x^2q^4;q^2\right)_\infty}
{\left(xq^2;q\right)_\infty\left(x^2q^3;q^3\right)_\infty}
=\dfrac{\left(-xq^2;q\right)_\infty\left(xq^2;q\right)_\infty}
{\left(xq^2;q\right)_\infty\left(x^2q^3;q^3\right)_\infty}
=\dfrac{\left(-xq^2;q\right)_\infty}
{\left(x^2q^3;q^3\right)_\infty}
=\left(\sum\limits_{i\geq 0}\dfrac{x^iq^{2i+\frac{i(i-1)}{2}}}{(q;q)_i}\right)
\left(\sum\limits_{j\geq 0}\dfrac{x^{2j}q^{3j}}{\left(q^3;q^3\right)_j}\right)
\nonumber\\
&=\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{2i+\frac{i(i-1)}{2}+3j}}{(q;q)_i\left(q^3;q^3\right)_j}
\end{align}
Now letting $x^m\mapsto x^mq^{3m(m-1)/2}$:
\begin{align}
\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{\frac{3(i+2j)(i+2j-1)}{2}+2i+\frac{i(i-1)}{2}+3j}}{(q;q)_i\left(q^3;q^3\right)_j}
=\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{2i^2+6ij+6j^2}}{(q;q)_i\left(q^3;q^3\right)_j}.
\end{align}
It is straightforward to repeat the above steps to analyze the second identity of Capparelli.
Everything is the same as before except for the initial blocks in $\mu$.
If $1$ appears in $\mu$ then the maximal block corresponding to $1$ and $2$ in $\mu$ is of the shape
$1,[2,1]^*,2^\bullet$. However, if $1$ does not appear in $\mu$ then $\mu$ must start with a part at least $3$.
The generating function for $\mu$ is thus:
\begin{align}
&\left(xq\dfrac{1}{1-xq^2\cdot xq}(1+xq^2) +1\right)
\left(\prod\limits_{j\geq 1} \dfrac{1}{1-xq^{3j}}\right)
\left(\prod\limits_{j\geq 1}1+xq^{3j+1}\right)
\left(\prod\limits_{j\geq 1} \dfrac{1}{1-xq^{3j+2}\cdot xq^{3j+1}}\cdot (1+xq^{3j+2})\right)
\nonumber\\
&= \left(xq^3;q^3\right)_\infty^{-1}
\left(-xq^5;q^3\right)_\infty\left(-xq;q^3\right)_\infty\left(x^2q^3;q^6\right)_\infty^{-1}\label{eqn:Cap2eq1}\\
&=\sum\limits_{i,j,k,l\geq 0}\dfrac{x^{i+j+k+2l}
q^{3i + 5j + \frac{3j(j-1)}{2} + k + \frac{3k(k-1)}{2} + 3l}
}{\left(q^3;q^3\right)_i\left(q^3;q^3\right)_j\left(q^3;q^3\right)_k\left(q^6;q^6\right)_l}.
\end{align}
Putting back a $3$-staircase:
\begin{align}
\sum\limits_{i,j,k,l\geq 0}\dfrac{x^{i+j+k+2l}
q^{\frac{3(i+j+k+2l)(i+j+k+2l-1)}{2}+3i + 5j + \frac{3j(j-1)}{2} + k + \frac{3k(k-1)}{2} + 3l}
}{\left(q^3;q^3\right)_i\left(q^3;q^3\right)_j\left(q^3;q^3\right)_k\left(q^6;q^6\right)_l}.
\end{align}
One may again arrive at a different analytic sum-side by simplifying expression \eqref{eqn:Cap2eq1} first:
\begin{align}
&\dfrac{\left(-xq^5;q^3\right)_\infty\left(-xq;q^3\right)_\infty}{\left(xq^3;q^3\right)_\infty\left(x^2q^3;q^6\right)_\infty}
=\dfrac{\left(-xq^5;q^3\right)_\infty\left(-xq;q^3\right)_\infty}{\left(xq^3;q^3\right)_\infty\left(x^3q^3;q^6\right)_\infty}
\cdot\dfrac{\left(xq^5;q^3\right)_\infty\left(xq;q^3\right)_\infty}{\left(xq^5;q^3\right)_\infty\left(xq;q^3\right)_\infty}\nonumber\\
&=\dfrac{\left(x^2q^2;q^6\right)_\infty\left(x^2q^{10};q^6\right)_\infty}
{\left(xq;q^3\right)_\infty\left(xq^3;q^3\right)_\infty\left(xq^5;q^3\right)_\infty\left(x^2q^3;q^6\right)_\infty}
=\dfrac{(1-xq^2)}{(1-x^2q^4)}\dfrac{\left(x^2q^2;q^6\right)_\infty\left(x^2q^{4};q^6\right)_\infty}
{\left(xq;q^3\right)_\infty\left(xq^2;q^3\right)_\infty\left(xq^3;q^3\right)_\infty\left(x^2q^3;q^6\right)_\infty}
\nonumber\\
&=\dfrac{(1-xq^2)}{(1-x^2q^4)}\dfrac{\left(x^2q^2;q^6\right)_\infty\left(x^2q^{4};q^6\right)_\infty}
{\left(xq;q\right)_\infty\left(x^2q^3;q^6\right)_\infty}\cdot\dfrac{\left(x^2q^6;q^6\right)_\infty}{\left(x^2q^6;q^6\right)_\infty}
=\dfrac{(1-xq^2)}{(1-x^2q^4)}\dfrac{\left(x^2q^2;q^2\right)_\infty}
{\left(xq;q\right)_\infty\left(x^2q^3;q^3\right)_\infty}
=\dfrac{(-xq;q)_\infty}{(1+xq^2)\left(x^2q^3;q^3\right)_\infty}
\nonumber\\
&=(1+xq)\dfrac{(-xq^3;q)_\infty}{\left(x^2q^3;q^3\right)_\infty}
=(1+xq)\left(\sum\limits_{i\geq 0}\dfrac{x^iq^{3i + \frac{i(i-1)}{2}}}{(q;q)_i}\right)
\left(\sum\limits_{j\geq 0}\dfrac{x^{2j}q^{3j}}{\left(q^3;q^3\right)_j}\right)\nonumber\\
&=\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{3i + \frac{i(i-1)}{2}+3j } + x^{i+2j+1}q^{3i + \frac{i(i-1)}{2}+3j+1 }}{(q;q)_i\left(q^3;q^3\right)_j}.
\end{align}
Putting back a $3$-staircase:
\begin{align}
&\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{\frac{3(i+2j)(i+2j-1)}{2}+3i + \frac{i(i-1)}{2}+3j } + x^{i+2j+1}q^{
\frac{3(i+2j+1)(i+2j)}{2}+3i + \frac{i(i-1)}{2}+3j+1 }}{(q;q)_i\left(q^3;q^3\right)_j}\nonumber\\
&=
\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j}q^{2i^2+6ij+6j^2+i}}{(q;q)_i\left(q^3;q^3\right)_j}
+\sum\limits_{i,j\geq 0}\dfrac{x^{i+2j+1}q^{2i^2+6ij+6j^2+4i+6j+1}}{(q;q)_i\left(q^3;q^3\right)_j}.
\end{align}
Since Capparelli's identities are true \cite{And-cap,AAG,C2,DL,MP2, TX}, we have:
\begin{thm} The following formal power series identities hold:
\begin{align}
\sum\limits_{i,j\geq 0}\dfrac{q^{2i^2+6ij+6j^2}}{(q;q)_i\left(q^3;q^3\right)_j}
&=\dfrac{1}{\left(q^2,q^3,q^9,q^{10};q^{12}\right)_\infty},\\
\sum\limits_{i,j,k,l\geq 0}\dfrac{
q^{\frac{3(i+j+k+2l)(i+j+k+2l-1)}{2}+3i + 5j + \frac{3j(j-1)}{2} + k + \frac{3k(k-1)}{2} + 3l}
}{\left(q^3;q^3\right)_i\left(q^3;q^3\right)_j\left(q^3;q^3\right)_k\left(q^6;q^6\right)_l}
&=
\sum\limits_{i,j\geq 0}\dfrac{q^{2i^2+6ij+6j^2+i}}{(q;q)_i\left(q^3;q^3\right)_j}
+\sum\limits_{i,j\geq 0}\dfrac{q^{2i^2+6ij+6j^2+4i+6j+1}}{(q;q)_i\left(q^3;q^3\right)_j}\nonumber\\
&=\left(-q,-q^3,-q^4,-q^6;q^6\right)_\infty.
\end{align}
\end{thm}
|
{
"timestamp": "2018-03-08T02:03:59",
"yymm": "1803",
"arxiv_id": "1803.02515",
"language": "en",
"url": "https://arxiv.org/abs/1803.02515"
}
|
\section{Introduction}\label{intro}
E. M. Stein proposed in the seventies the problem of restriction of the Fourier
transform to hypersurfaces. Given a smooth hypersurface $S$ in $\R^n$ with surface
measure $d\sigma_S,$ he asked for the range of exponents
$\tilde p$ and $\tilde q$ for which the estimate
\begin{align}
\bigg(\int_S|\widehat{f}|^{\tilde q}\,d\sigma_S\bigg)^{1/\tilde q}\le C\|f\|_{L^{\tilde
p}(\R^n)}
\end{align}
holds true for every Schwartz function $f\in\mathcal S(\R^n),$ with a constant $C$
independent of $f.$
The sharp range in dimension $n=2$ for curves with non-vanishing curvature was determined
through work by C. Fefferman, E. M. Stein and A. Zygmund \cite{F1}, \cite{Z}. In higher
dimension, the sharp $L^{\tilde p}-L^2$ result for hypersurfaces with
non-vanishing Gaussian curvature was obtained by E. M. Stein and P. A. Tomas \cite{To},
\cite{St1} (see also Strichartz \cite{Str}). Some more general classes of surfaces were
treated by A. Greenleaf \cite{Gr}. Many years later, general finite type surfaces in
$\R^3$
(without assumptions on the curvature) have been considered in work by I. Ikromov, M.
Kempe and D. M\"uller \cite{ikm} and Ikromov and M\"uller \cite{IM-uniform}, \cite{IM},
and the sharp range of Stein-Tomas type $L^{\tilde p}-L^2$ restriction
estimates has been determined for a large class of smooth, finite-type hypersurfaces,
including all analytic hypersurfaces.
The question about general $L^{\tilde p}-L^{\tilde q}$ restriction
estimates is nevertheless still wide open. Fundamental progress has been made since the
nineties, with major new ideas introduced by J. Bourgain (see for instance
\cite{Bo1}, \cite{Bo2}) and T. Wolff (\cite{W1}), which led to a better understanding of
the case of non-vanishing Gaussian curvature. These ideas and methods were further
developed by A. Moyua, A. Vargas, L. Vega and T. Tao (\cite{MVV1}, \cite{MVV2}
\cite{TVV}),
who established the so-called bilinear approach (which had been anticipated in the work
of C. Fefferman \cite{F1} and had implicitly been present in the work of J. Bourgain
\cite{Bo3}) for hypersurfaces with non-vanishing Gaussian curvature for which all
principal
curvatures have the same sign. The same method was applied to the light cone by
Tao-Vargas (see \cite{TV1}, \cite{TV2}). A culmination of the application of the
bilinear method to such types of surfaces was reached in work by T. Tao \cite{T2} (for
positive principal curvatures), and T. Wolff \cite{W2} and T. Tao \cite{T4} (for the
light cone). In particular, in these last two papers the sharp linear restriction
estimates for the light cone in $\R^4$ were obtained.
In the last years, J. Bourgain and L. Guth \cite{BoG} made further important progress on
the case of non-vanishing curvature by making use also of multilinear restriction
estimates due to J. Bennett, A. Carbery and T. Tao \cite{BCT}. Later L. Guth \cite{Gu16},
\cite{Gu17} improved these results by using the polynomial partitioning method.
For the case of non-vanishing curvature but principal curvatures of different signs, the
bilinear method was applied independently by S. Lee \cite{lee05}, and A. Vargas
\cite{v05}, to a specific surface, the hyperbolic paraboloid (or "saddle"). They obtained
a
result which is analogous to Tao's theorem \cite{T2} except for the end-point. B. Stovall
\cite{Sto} recently proved the end-point case. Also, C. H. Cho and J. Lee \cite{chl17} and
J. Kim \cite{k17} improved the range. Perhaps surprisingly at a first thought, their
methods did not give the desired result for any other surface with negative Gaussian
curvature. A key element in their proofs is the fact that the hyperbolic paraboloid is
invariant under certain anisotropic dilations, and no other surface satisfies this same
invariance.
\medskip
Our aim in this article is to provide some first steps towards gaining an understanding
of Fourier restriction to more general hyperbolic surfaces, by generalizing Lee's and
Vargas' result on the saddle to certain model surface $S,$ namely the graph of the
function
$\phi(x,y):=xy+\frac{1}{3}y^3$ over a given small neighborhood of the origin. Observing
that our specific $\phi$ is homogeneous under the parabolic scalings $(x,y)\mapsto (r^2
x,
ry), \, r>0,$ we may here assume as well that
\begin{align}\label{surface}
S:=\{(x,y,xy+y^3/3):(x,y)\in Q:=I\times I\},
\end{align}
where $I:=[-1,1].$
\smallskip
\noindent{\bf Remark.} We do concentrate here on the seemingly particular case of a perturbation of the form $y^3/3$ in
order
to not to have to deal with additional technical issues which come up when studying more general perturbations,
say, of the form
$f(y),$ but should like to mention that the understanding of this particular perturbation is indeed the key to
understanding more
general ones depending on $x,$ or $y,$ only. Indeed, in the follow-up preprint \cite{bmv19}, we show how finite
type perturbations $f(y)$ can be essentially reduced to the special case studied here, and in further article
we shall also study flat perturbations $f(y).$
\smallskip
As usual, it will be more convenient to use duality and work in the adjoint setting. If
${\cal R}$ denotes the Fourier restriction operator $g\mapsto {\cal R} g:=\hat g|_S$ to the surface
$S,$ its adjoint operator ${\cal R}^*$ is given by ${\cal R}^*f(\xi)=\ext f(-\xi),$ where
$\ext$ denotes the ``Fourier extension'' operator given by
\begin{align}\label{defop}
\ext f(\xi):=\widehat{f\,d\sigma_S}(\xi)= \int_S f(x)e^{-i\xi\cdot x}\,d\sigma_S(x),
\end{align}
with $f\in L^q(S,\sigma_S).$ The restriction problem is therefore
equivalent to the question of finding the appropriate range of exponents for which the
estimate
$$
\|\mathcal E f\|_{L^r(\R^3)}\le C\|f\|_{L^q(S,d\sigma_S)}
$$
holds true with a constant $C$ independent of the function $f\in L^q(S,d\sigma_s).$
By identifying a point $(x,y)\in Q$ with the corresponding point $(x,y,\phi(x,y))$ on $S,$
we may regard our Fourier extension operator $\ext$ as well as an operator mapping
functions on $Q$ to functions on $\R^3,$ which in terms of our phase function
$\phi(x,y)=xy+y^3/3$ can be expressed more explicitly in the form
$$
\ext f(\xi)=\int_Q f(x,y) e^{-i(\xi_1 x+\xi_2 y+\xi_3\phi(x,y))} \eta(x,y) \, dx dy,
$$
if $\xi= (\xi_1,\xi_2,\xi_3)\in \R^3,$ with a smooth density $\eta.$
Our main result is the following
\begin{thmnr}\label{mainresult}
Assume that $r>10/3$ and $1/q'>2/r,$ and let $\ext$ denote the Fourier extension
operator associated to the graph $S$ of the above phase function $\phi$. Then
\begin{align*}
\|\ext f\|_{L^r(\R^3)} \leq C_{r,q} \|f\|_{L^q(Q)}
\end{align*}
for all $f\in L^q(Q)$.
\end{thmnr}
In the remaining part of this section, we shall describe our strategy of proof, and some
of the obstacles that have to be dealt with.
We are going to follow the bilinear approach, which is based on bilinear estimates of the
form
\begin{align}\label{bil1}
\|\ext_{U_1}(f_1)\,\ext_{U_2}(f_2)\|_p \leq C(U_1,U_2) \|f_1\|_2\|f_2\|_2.
\end{align}
Here, $\ext_{U_1}$ and $\ext_{U_2}$ are the Fourier extension operators associated to
patches of sub-surfaces $S_i:=\text{\rm graph\,} \phi|_{U_i}\subset S,\ i=1,2,$ with $U_i\subset Q.$
What is crucial for obtaining useful bilinear estimates is that the two patches of
surface $S_1$ and $S_2$ satisfy certain {\it transversality conditions,} which are
stronger than just assuming that $S_1$ and $S_2$ are transversal as hypersurfaces (i.e.,
that all normals to $S_1$ are transversal to all normals to $S_2$). Indeed, what is
needed in addition is the following:
\smallskip
Translate the patches $S_1$ and $S_2$ so that they intersect in a smooth curve. Then the
normals to, say, $S_1$ for base points varying along this intersection curve form a
cone
$\Gamma_1.$ What is needed in the bilinear argument is that all the normals to the
surfaces $S_2$ pass transversally through this cone $\Gamma_1,$ and that the analogous
condition holds true with the roles of $S_1$ and $S_2$ interchanged.
For more details on this condition, we refer to the corresponding literature dealing
with
bilinear estimates, for instance \cite{lee05}, \cite{v05}, \cite{lv10}, or \cite{be16}.
In particular, according to Theorem 1.1 in \cite{lee05}, transversality is achieved if the
modulus of the
following quantity
\begin{align}\label{transs}
\Gamma^\phi_{z}(z_1,z_2,z_1',z_2'):= \left\langle
(H\phi)^{-1}(z)(\nabla\phi(z_2)-\nabla\phi(z_1)),\nabla\phi(z_2')-\nabla\phi(z_1')\right\rangle
\end{align}
is bounded from below for any $z_i=(x_i,y_i),\, z_i'\in U_i$, $i=1,2$, $z=(x,y)\in U_1\cup
U_2,$ $H\phi$
denoting the Hessian of $\phi$.
If this inequality holds, then we have \eqref{bil1} for $p>5/3,$ with a constant $C$ that
depends only
on an lower bound of (the modulus of) \eqref{transs}, and on upper bounds for the
derivatives of $\phi.$ If $U_1$ and
$U_2$ are sufficiently small (with sizes depending on upper bounds of the first and second
order derivatives of $\phi$ and a lower bound for the determinant of $H\phi$) this
condition reduces to the estimate
\begin{align}
|\Gamma^\phi_{z}(z_1,z_2)|\geq c,
\end{align}
for $z_i=(x_i,y_i)\in U_i$, $i=1,2$, $z=(x,y)\in U_1\cup U_2$, where
\begin{align}\label{trans}
\Gamma^\phi_{z}(z_1,z_2):= \left\langle
(H\phi)^{-1}(z)(\nabla\phi(z_2)-\nabla\phi(z_1)),\nabla\phi(z_2)-\nabla\phi(z_1)\right\rangle.
\end{align}
It is easy to check that for $\phi(x,y)=xy+y^3/3$, we have
\begin{eqnarray} \label{gammaz}
\Gamma^\phi_{z}(z_1,z_2)
&=& 2(y_2-y_1)[x_2-x_1+(y_1+y_2-y)(y_2-y_1)]\\
&=:& 2(y_2-y_1)\tau_{z}(z_1,z_2). \label{TV1}
\end{eqnarray}
This should be compared to the case of the non-perturbed hyperbolic paraboloid (the
``saddle'') $\phi_0(x,y)=xy,$ where we would simply get $2(y_2-y_1)(x_2-x_1)$ in place
of
\eqref{TV1}.
Since $z=(x,y)\in U_1\cup U_2,$ it will be particularly important to look at the
expression
\eqref{TV1} when $z=z_1\in U_1,$ and $z=z_2\in U_2.$ As above, if $U_1$ and
$U_2$ are sufficiently small, we can actually reduce to this case. We then see that for
our perturbed saddle, still the difference $y_2-y_1$ in the $y$-coordinates plays an
important role as for the unperturbed saddle, but in place of the difference $x_2-x_1$
in the $x$-coordinates now the quantities
\begin{align}\label{TV2}
\tau_{z_1}(z_1,z_2):=x_2-x_1+y_2(y_2-y_1)\\
\tau_{z_2}(z_1,z_2):=x_2-x_1+y_1(y_2-y_1) \label{TV3}
\end{align}
become relevant. Observe also that
\begin{align}\label{symtrans}
\tau_{z}(z_1,z_2)=-\tau_{z}(z_2,z_1).
\end{align}
It is important to notice that the constants $C(U_1,U_2)$ in the bilinear estimates
\eqref{bil1} will strongly depend on the sizes of the quantities appearing in \eqref{TV1}
-
\eqref{TV3}, as well as on the size of the derivatives of $\phi.$
\smallskip
In the case of the saddle, since the ``transversality'' is here given by
$2(y_2-y_1)(x_2-x_1),$ it is natural to perform a kind of Whitney decomposition with
respect to the diagonal of $Q\times Q$ into direct products $U_1\times U_2$ of pairs
of bi-dyadic rectangles $U_1,U_2$ of the same dimension $\lambda_1\times \lambda_2,$
which are separated in each coordinate by a distance proportional to the sizes
$\lambda_1$ respectively $\lambda_2,$ and then apply a scaling transformation of the
form $(x,y)=(\lambda_1 x',\lambda_2 y')$ -- this is exactly what had been done in
\cite{lee05} and \cite{v05}. Since the phase $xy$ is homogeneous under such kind of
scalings, in the new coordinates $(x',y'),$ one has then reduced the bilinear estimates
to the case of normalized patches $U_1,U_2$ of size $1\times 1$ for which the
transversalities are also of size $1,$ and from there on one could essentially apply the
``uniform'' bilinear estimates (similar to those known from the elliptic
case) and re-scale them to go back to the original coordinates.
\smallskip
Coming back to our perturbed saddle, we shall again try to scale in order to make both
transversalities become of size $\sim\pm1.$ However, as it will turn out, the ``right''
patches $U_1,U_2$ to be used will in general no longer be bi-dyadic rectangles, but in
some case a dyadic {\it square} $U_1$ and a bi-dyadic {\it curved box} $U_2.$
The ``right'' scalings, which will reduce matters to situations where all
transversalities are of size $\sim\pm1,$ will be anisotropic, and this will create a
new problem: while the saddle is invariant under such type of scaling (i.e., for the
function $\phi_0(x,y):=xy,$ we have $\frac1{\lambda_1\lambda_2}\phi_0(\frac
x{\lambda_1},\frac y{\lambda_2})=\phi_0(x,y)),$ for our function $\phi,$ the scaled
function $\phi^s(x,y):=\frac1{\lambda_1\lambda_2}\phi(\frac x{\lambda_1},\frac
y{\lambda_2})$ is given by
$$ \phi^s(x,y)=xy+\frac {\lambda_2^2 y^3}{3\lambda_1}.
$$
Thus, if $\lambda_1\gg \lambda_2^2,$ then the second term can indeed be view as a small
perturbation of the leading term $xy,$ and we can proceed in a very similar way as for the
saddle. However, if $\lambda_1\lesssim\lambda_2^2,$ then the second, cubic term, can
assume
a dominant role, and the treatment of this case will require further arguments.
In conclusion, the na\"\i f approach which would try to treat our surface $S$ as a
perturbation of the saddle, and which does
indeed work for Stein-Tomas type $L^q-L^2$ restriction estimates, breaks down if we
want to derive restriction estimates of more general type by means of the bilinear method.
\medskip
In order to get some better idea on how to suitably devise the ``right'' pairs of
patches $U_1, U_2$ for our Whitney type decomposition, note that the quantities
$\tau_{z_2}(z_1,z_2)$ and $\tau_{z_1}(z_1,z_2)$ can be of quite different size. For
instance, for $z_1^0=(0,0)$ and $z_2^0=(-1+\delta,1),$ we have
$\tau_{z_1^0}(z_1^0,z_2^0)=\delta$ while $\tau_{z_2^0}(z_1^0,z_2^0)=-1.$ Therefore there
can be a strong imbalance in the two transversalities for the perturbed saddle.
This is quite different from the situation of the saddle, where the two quantities
\eqref{TV2}, \eqref{TV3} are the same, namely $x_2-x_1.$
Nevertheless, observe that, due to the following important relation between the two
transversalities
\begin{align}\label{TV2+TV3}
\tau_{z_1}(z_1,z_2)-\tau_{z_2}(z_1,z_2)=(y_2-y_1)^2
\end{align}
(which is immediate from \eqref{TV2}, \eqref{TV3}),
we see that at least one of the two transversalities $\tau_{z_1}(z_1,z_2)$ or
$\tau_{z_2}(z_1,z_2)$ cannot be smaller than $|y_2-y_1|^2/4.$
This naturally leads to two cases, which we will discuss in detail in the next chapter: Either
$\tau_{z_1}(z_1,z_2)$ and $\tau_{z_2}(z_1,z_2)$ have similar size, and thus both are not much smaller than
$|y_2-y_1|^2$, or they are of quite different size, in which case one of the two transversalities has to be
comparable to $|y_2-y_1|^2$, while the other one can be much smaller.
\smallskip
In Section \ref{sect:admissible} we shall make a finer analysis of the transversality
conditions and introduce
the pairs of sets fit to them, \it admissible pairs\rm. In Section \ref{sect:bilin} we shall
state and prove the
sharp bilinear estimates for those pairs. In Section \ref{whitn} we will use the admissible
pairs to build a
Whitney type decomposition of $Q\times Q.$
\smallskip
Further difficulties arise in the passage from bilinear to linear Fourier extension, which
will be discussed in Section \ref{bilinlin}. In order to exploit all of the underlying
almost orthogonality, we shall have to further decompose the curved boxes $U_2$ (whenever
they appear) into smaller squares and also make use of some disjointness properties of the
corresponding pieces of the surface $S.$ In contrast to what is done in the case of
elliptic surfaces, as well as for the saddle, it turns out that for the perturbed saddle
it
is not sufficient to exploit disjointness properties with respect to the first two
coordinates, but also with respect to the third one. A further novelty is that we have to
improve on a by now standard almost orthogonality relation in $L^p$ between the
pieces
arising in our Whitney type decomposition, which needs to be employed before applying our
bilinear estimates to each of these pieces. As it turns out, this ``classical'' estimate
is insufficient for our curved boxes, and we improve on it by applying a classical
square
function estimate associated to partitions into rectangular boxes, due to Rubio de
Francia, which has had its roots in a square function estimate obtained independently by
L. Carleson \cite{c67}, and A. Cordoba \cite{co81}.
Finally, we can reassemble the smaller squares and pass back to curved boxes by means of
Khintchine's inequality.
\medskip
\noindent{\bf Guide to the reader:} The real thrust of the precise definition of admissible pairs given in
Subsection \ref{preciseadmissible} will become relevant in an essential way only later in Section
\ref{whitn}, when we shall show that our
admissible pairs $(U_1,U_2)$ will allow to perform a Whitney-type decomposition of $Q\times Q.$ To a smaller
degree, they are also relevant in Subsection \ref{scaling transform}, which prepares for the reduction of our
general bilinear estimates for admissible pairs in Theorem \ref{bilinear2} (for the curved box case) to the
crucial prototypical case introduced in Subsection \ref{proto} and studied in Theorem \ref{bilinear}.
For a first reading, we therefore suggest to skip Subsections \ref{preciseadmissible} and \ref{scaling
transform}, as well as the reduction of Theorem \ref{bilinear2} to Theorem \ref{bilinear} in Section
\ref{sect:bilin}, and first read Subsections \ref{TVS} and \ref{proto}, and then in Subsection \ref{bilinarg}
the proof of the bilinear estimates of Theorem \ref{bilinear}, which deals with the prototypical case,
before
coming back to Subsection \ref{preciseadmissible}.
\medskip
\noindent\textsc{Convention:}
Unless stated otherwise, $C > 0$ will stand for an absolute constant whose value may vary
from occurrence to occurrence. We will use the notation $A\sim_c B$ to express that
$\frac{1}{c}A\leq B \leq c A$. In some contexts where the size of $c$ is irrelevant we
shall drop the index $c$ and simply write $A\sim B.$ Similarly, $A\lesssim B$ will express
the fact that there is a constant $c$ (which does not depend on the relevant quantities
in
the estimate) such that $A\le c B,$ and we write $A\ll B,$ if the constant $c$ is
sufficiently small.
\medskip
\begin{center}
{\textsc{Acknowledgments}}
\end{center}
Part of this work was developed during the stay of the second and third author at the
Mathematical Sciences Research Institute at Berkeley during the Harmonic Analysis Program
of 2017. They wish to express their gratitude to the organizers and to the Institute and
its staff for their hospitality and for providing a wonderful working atmosphere.
\smallskip
We would also like to express our sincere gratitude to the referee for many valuable suggestions which have
greatly helped
to improve the presentation of the material in this article.
\setcounter{equation}{0}
\section{Transversality conditions and admissible pairs of sets}\label{sect:admissible}
\subsection{Admissible pairs of sets $U_1,$ $U_2$ on which transversalities are of a fixed
size: an informal discussion}\label{pairs of sets}
Recall again from the introduction that the crucial
``transversality quantities'' arising in
Lee's estimate \eqref{trans} are given by $y_2-y_1$ and \eqref{TV2}, \eqref{TV3}, i.e.,
\begin{align*}
\tau_{z_1}(z_1,z_2):=x_2-x_1+y_2(y_2-y_1), \\
\tau_{z_2}(z_1,z_2):=x_2-x_1+y_1(y_2-y_1).
\end{align*}
We shall therefore try to devise neighborhoods $U_1$ and $U_2$ of two given points
$z_1^0=(x_1^0,y_1^0)$ and $z_2^0=(x_2^0,y_2^0)$ on which these quantities are roughly
constant for $z_i=(x_i,y_i)\in U_i,$ $i=1,2$,
and which are also essentially chosen as large as possible. The corresponding pair $(U_1,U_2)$ of
neighborhoods of $z^0_1$ respectively $z^0_2$ will be called an {\it admissible pair}.
The goal of this subsection is to present some of the basic ideas, without being precise about details, such
as
constants that will be hidden in the arguments, in order to motivate the precise definition of admissible pairs
that will be given in the next subsection (which might otherwise appear a bit strange).
\medskip
In a first step, we choose a large constant $C_0\gg 1$, which will be made precise only later,
and assume that
$|y^0_2-y^0_1|\sim C_0\rho$ for some $\rho>0.$ It is then natural to allow $y_1$ to vary on $U_1$ and $y_2$
on $U_2$ by at most $\rho$ from $y^0_1$ and
$y^0_2,$ respectively, i.e., we shall assume that
\begin{align*
|y_i-y^0_i|\lesssim \rho, \qquad \text{for}\quad z_i\in U_i,\, i=1,2,
\end{align*}
so that indeed
\begin{equation}\label{rhosize}
|y_2-y_1|\sim C_0\rho \qquad\text{for}\quad z_i\in U_i,\, i=1,2.
\end{equation}
Recall next the identity \eqref{TV2+TV3}, which in particular implies that
\begin{align}\label{TV2+TV3'}
|\tau_{z^0_1}(z^0_1,z^0_2)-\tau_{z^0_2}(z^0_1,z^0_2)|{\sim C_0^2}\rho^2.
\end{align}
We begin with
\medskip
\noindent {\bf Case 1: Assume that $ |\tau_{z_1^0}(z_1^0,z_2^0)|\le |\tau_{z_2^0}(z_1^0,z_2^0)|.$ }
Let us then write
\begin{equation}\label{defdelta}
|\tau_{z_1^0}(z_1^0,z_2^0)|= \rho^2{\delta},
\end{equation}
where ${\delta}\ge 0.$ Note, however, that obviously $ \rho^2{\delta}\lesssim1.$ From \eqref{TV2+TV3'} one then easily
deduces that there are two subcases:
\medskip
\noindent {\bf Subcase 1(a): (the ``straight box'' case),} where $|\tau_{z_1^0}(z_1^0,z_2^0)| \sim
|\tau_{z_2^0}(z_1^0,z_2^0)|,$ or, equivalently, ${\delta}\gtrsim1.$ In this case, also
$|\tau_{z_2^0}(z_1^0,z_2^0)|\sim
\rho^2{\delta}.$
\medskip
\noindent {\bf Subcase 1(b): (the ``curved box'' case),} where $|\tau_{z_1^0}(z_1^0,z_2^0)| \ll
|\tau_{z_2^0}(z_1^0,z_2^0)|,$ or, equivalently, ${\delta}\ll 1.$ In this case, $|\tau_{z_2^0}(z_1^0,z_2^0)|\sim
\rho^2.$
\medskip
Given $\rho$ and $\delta,$ we shall then want to devise $U_1$ and $U_2$ so that the same kind of conditions hold
for all $z_1\in U_1$ and $z_2\in U_2,$
i.e.,
$$
|\tau_{z_1}(z_1,z_2)|\sim \rho^2{\delta},\text{ and} \quad |\tau_{z_2}(z_1,z_2)|\sim \rho^2(1\vee {\delta}).
$$
Note that in view of \eqref{TV2+TV3} and \eqref{rhosize} the second condition is redundant,
and so the only additional condition that needs to be satisfied is that, for all
$z_1=(x_1,y_1)\in
U_1$ and $z_2=(x_2,y_2)\in U_2,$ we have
\begin{align*
|\tau_{z_1}(z_1,z_2)|=|x_2-x_1+y_2(y_2-y_1)|\sim \rho^2{\delta}.
\end{align*}
{As said before, }we want to choose $U_2$ as large as possible w.r. to $y_2,$ i.e., we only assume that
$|y_2-y_2^0|\lesssim\rho.$ Let
\begin{equation}\label{a0}
a^0:=\tau_{z_1^0}(z_1^0,z_2^0)=x^0_2-x^0_1+y^0_2(y^0_2-y^0_1),
\end{equation} so that $|a^0|\sim
\rho^2{\delta}.$ Then we shall assume that {for $z_2\in U_2$} we have, say,
$|\tau_{z_1^0}(z_1^0,z_2)-\tau_{z_1^0}(z_1^0,z_2^0)|=|\tau_{z_1^0}(z_1^0,z_2)-a^0|\ll \rho^2{\delta}.$ This means
that we shall choose $U_2$ to be
of the form
\begin{equation}\label{U2def}
U_2=\{(x_2,y_2): |y_2-y_2^0|\lesssim \rho,\ |x_2-x^0_1+y_2(y_2-y_1^0)- a^0|\ll \rho^2{\delta}\}.
\end{equation}
As for $U_1,$ given our choice of $U_2,$ what we still need is that
$|\tau_{z_1}(z_1,z_2)-\tau_{z_1^0}(z_1^0,z_2)|\ll \rho^2{\delta}$ for all $z_1\in U_1$ and $z_2\in U_2,$ for then
also
$|\tau_{z_1}(z_1,z_2)-\tau_{z_1^0}(z_1^0,z_2^0)|\ll \rho^2{\delta}$ for all such $z_1, z_2.$ We therefore must
require that $|x_1-x_1^0+y_2(y_1-y_1^0)|\ll \rho^2{\delta}$ on $U_1\times U_2.$
But, since $y_2$ is allowed to vary within an interval of size $\rho,$ we
see that this requires a condition of the form $\rho|y_1-y_1^0|\ll\rho^2{\delta}.$
Assuming this,
we next see that we are allowed to replace $y_2$ in the condition $|x_1-x_1^0+y_2(y_1-y_1^0)|\ll \rho^2{\delta}$
by
$y_1^0.$ This leads to the condition $|x_1-x_1^0+y_1^0(y_1-y_1^0)|\ll \rho^2{\delta},$ which depends on $z_1$
only
and thus gives our second conditions on $U_1.$
Our discussion suggests that we should finally choose $U_1$ of the form
\begin{equation}\label{U1def}
U_1=\{(x_1,y_1): |y_1-y_1^0|\lesssim \rho(1\wedge {\delta}), |x_1-x_1^0+y_1^0(y_1-y_1^0)|\ll
\rho^2{\delta}\}.
\end{equation}
\noindent{\bf Note:} $U_1$ is essentially the affine image of a rectangular box of dimension $\rho^2{\delta} \times
\rho(1\wedge {\delta}).$ However, when ${\delta}\ll 1,$ then $U_2$ is a thin curved box, namely the segment of a
$\rho^2{\delta}$-neighborhood of a parabola lying within the horizontal strip where $|y_2-y_2^0|\lesssim \rho.$ On
the other hand, when ${\delta}\gtrsim 1,$ then it is easily seen that $U_2$ is essentially a rectangular box of
dimension $\rho^2{\delta}\times \rho.$ This explains why we called Subcase 1(b) where ${\delta}\ll1$ the ``curved box
case'', and Subcase 1(a) where ${\delta}\gtrsim 1$ the ``straight box case.''
\medskip
\noindent {\bf Case 2: Assume that $|\tau_{z_1^0}(z_1^0,z_2^0)|\ge |\tau_{z_2^0}(z_1^0,z_2^0)|.$ }
This case can easily be reduced to the previous one by symmetry. Indeed, in view of \eqref{symtrans}, we just
need to interchange the roles of $z_1$ and $z_2$ in the previous discussion, so that it is natural here to
define
an {\it admissible pair} $(\tilde U_1,\tilde U_2)$ {\it of type 2} of neighborhoods $\tilde U_1$ of $z^0_1$
respectively $\tilde U_2$ of $z^0_2$ by setting
\begin{eqnarray}\label{tildeU}
\begin{split}
\tilde U_1=\{(x_2,y_2): |y_1-y_1^0|\lesssim \rho,\ |x_1-x^0_2+y_1(y_1-y_2^0)- \tilde a^0|\ll \rho^2{\delta}\},\\
\tilde U_2=\{(x_1,y_1): |y_2-y_2^0|\lesssim \rho(1\wedge {\delta}), |x_2-x_2^0+y_2^0(y_2-y_2^0)|\ll
\rho^2{\delta}\},
\end{split}
\end{eqnarray}
where $\tilde a^0:=\tau_{z_2^0}(z_2^0,z_1^0).$
\bigskip
\medskip
\subsection{Precise definition of admissible pairs within $Q\times Q$}\label{preciseadmissible}
In view of our discussion in the previous subsection, we shall here devise more precisely
certain \lq\lq dyadic'' subsets of $Q\times Q$ which will
assume the roles of the sets $U_1,$ respectively $U_2, $ in such a way that on every pair of such sets each of
our transversality functions is essentially of some fixed dyadic size, and which will moreover lead to a kind
of Whitney decomposition of $Q\times Q$ (as will be shown in Section \ref{whitn})
To begin with, as before we fix a large dyadic constant $C_0\gg1.$
\smallskip
In a first step, we perform a classical {\bf dyadic decomposition in the
$y$-variable}
which is a variation of the one in \cite{TVV}:
For a given dyadic number $0<\rho\lesssim 1,$ we denote for
$j\in\mathbb Z$ such that $|j|\rho\le 1$ by $I_{j,\rho}$ the dyadic interval
$I_{j,\rho}:=[j\rho,j\rho+\rho)$ of length $\rho,$ and by $V_{j,\rho}$ the corresponding
horizontal ``strip'' $V_{j,\rho}:=[-1,1]\times I_{j,\rho}$ within $Q.$
Given two dyadic intervals $J,\,J'$ of the same size, we say that they are
{\it related} if their parents are adjacent but they are not adjacent. We divide each
dyadic interval $J$ in a disjoint union of dyadic subintervals $\{I_J^k\}_{1\le k\le
C_0/8},$ of length $8|J|/C_0.$ Then, we define $(I,I')$ to be an \it
admissible pair of dyadic intervals \rm if and only if there are $J$ and $J'$ related
dyadic intervals and $1\le k,\,j\le C_0/8$ such that $I=I_J^k$ and $I'=I_{J'}^j.$
We say that a pair of strips $(V_{j_1,\rho},V_{j_2,\rho})$ is {\it admissible } and write
$V_{j_1,\rho}\backsim V_{j_2,\rho},$ if $(I_{j_1,\rho},I_{j_2,\rho})$ is a pair of
admissible dyadic intervals. Notice that in this case,
\begin{align}\label{admissibleV}
C_0/8< |j_2-j_1|< C_0/2.
\end{align}
One can easily see that this leads to the following disjoint decomposition of $Q\times
Q:$
\begin{align}\label{whitney1}
Q\times Q= \overset{\cdot}{\bigcup\limits_{\rho}}
\,\Big(\overset{\cdot}{\bigcup\limits_{V_{j_1,\rho}\backsim
V_{j_2,\rho}}}V_{j_1,\rho}\times V_{j_2,\rho}\Big),
\end{align}
where the first union is meant to be over all such dyadic $\rho$'s.
\medskip
In a second step, we perform a non-standard {\bf Whitney type decomposition of any given
admissible pair of strips}, to obtain subregions in which the transversalities are roughly
constant.
To simplify notation, we fix $\rho$ and an admissible pair $(V_{j_1,\rho},V_{j_2,\rho}),$
and simply write $I_i:=I_{j_i,\rho},\, V_i:=V_{j_i,\rho}, \, i=1,2,$
so that $I_i$ is an interval of length $\rho$ with left endpoint $j_i\rho,$ and
\begin{align}\label{Vi}
V_1=[-1,1]\times I_1, \qquad V_2=[-1,1]\times I_2,
\end{align}
are rectangles of dimension $2\times \rho,$ which are vertically separated at scale
$C_0\rho.$
More precisely, for $z_1=(x_1,y_1)\in V_1$ and $z_2=(x_2,y_2)\in V_2$ we have
$|y_2-y_1|\in
|j_2\rho-j_1\rho|+[-\rho,\rho],$ i.e.,
\begin{align}\label{yseparation}
C_0\rho/2\le |y_2-y_1|\le C_0\rho.
\end{align}
Let $0<\delta\lesssim \rho^{-2}$ be a dyadic number (note that $\delta$ could be big,
depending on $\rho$), and let ${\cal I}$ be the set of points
which partition the interval $I$ into (dyadic) intervals of the same length $\rho^2{\delta}.$
\smallskip
Similarly, for $i=1,2,$ we choose a finite equidistant partition ${\cal I}_i$ of width
$\rho(1\wedge\delta)$ of the interval $I_i$ by points $y_i^0\in {\cal I}_i.$
Note: if ${\delta}>1,$ then $\rho(1\wedge\delta)=\rho,$ and we can choose for ${\cal I}_i$ just
the
singleton ${\cal I}_i=\{y_i^0\},$ {where $y_i^0$ is the left endpoint of $I_i.$}
\medskip
\begin{definr}
For any parameters $x^0_1,t^0_2\in{\cal I},$ $y^0_1\in{\cal I}_1$ defined in the previous lines and $y^0_2$ the left
endpoint of $I_2,$ we define the
sets
\begin{eqnarray}\label{whitneybox1}
\begin{split}
&U_1^{x^0_1,y_1^0,\delta} :=\{(x_1,y_1): 0\le y_1-y_1^0< \rho(1\wedge\delta),\,
0\le x_1-x^0_1+y_1^0(y_1-y_1^0)< \rho^2\delta \}, \\\\
&U_2^{t^0_2,y_1^0,y^0_2,\delta} :=\{(x_2,y_2): 0\le y_2-y^0_2<\rho, 0\le x_2-t^0_2+y_2(y_2-y_1^0)<
\rho^2\delta\}
\end{split}
\end{eqnarray}
and the points
\begin{equation}\label{pointsinU}
z^0_1:=(x^0_1,y^0_1), \qquad z^0_2=(x^0_2,y^0_2):=(t_2^0-y_2^0(y_2^0-y_1^0), y_2^0).
\end{equation}
\end{definr}
Observe that then
$$
z^0_1\in U_1^{x^0_1,y_1^0,\delta}\subset V_1\quad \text{ and } \quad z^0_2\in
U_2^{t^0_2,y_1^0,y^0_2,\delta}\subset V_2.
$$
Indeed, $z^0_i$ is in some sense the ``lower left'' vertex of $U_i,$ and the horizontal projection of
$U_2^{t^0_2,y_1^0,y^0_2,\delta}$ equals $I_2.$
Moreover, if we define $a_0$ by \eqref{a0}, we have that $x_1^0+a^0=t^0_2,$ so that our definitions of the sets
$U_1^{x^0_1,y_1^0,\delta}$ and $U_2^{t^0_2,y_1^0,y^0_2,\delta}$ are very close to the ones for the sets $U_1$
and
$U_2$ (cf. \eqref{U1def}, \eqref{U2def}) in the previous subsection.
In particular, $U_1^{x^0_1,y_1^0,\delta}$ is again essentially a paralellepiped of sidelengths $\sim
\rho^2{\delta}\times \rho(1\wedge {\delta}),$ containing the point $(x^0_1,y_1^0),$ whose
longer side has slope $y_1^0$ with respect to the $y$-axis.
Similarly, if ${\delta}\ll 1,$
then $U_2^{t^0_2,y_1^0,y^0_2\delta}$ is a thin curved box of width $\sim\rho^2 {\delta}$ and
length $\sim\rho,$ contained in a rectangle of dimension $\sim \rho^2\times \rho$ whose axes are
parallel to the coordinate axes (namely the part of a $\rho^2\delta$-neighborhood of a parabola containing the
point $(x^0_1,y^0_1)$ which lies within the horizontal strip $V_2$). If ${\delta}\gtrsim 1,$ then
$U_2^{t^0_2,y_1^0,y^0_2\delta}$ is essentially a rectangular box of dimension $\sim \rho^2{\delta}\times \rho$
lying
in the same horizontal strip.
Note also that we have chosen to use the parameter $t^0_2$ in place of using $x^0_2$ here, since with this
choice the identity
\begin{equation}\label{t2mean}
\tau_{z_1^0}(z_1^0,z_2^0)=t^0_2-x_1^0
\end{equation}
holds true, which will become quite useful in the sequel. We next have to relate the parameters $x_1^0,t^0_2,
y_1^0,y_2^0$ in order to give a precise definition of an admissible pair.
\smallskip
Here, and in the sequel, we shall always assume that the points $z_1^0,z_2^0$ associated to these parameters
are given by \eqref{pointsinU}.
\smallskip
\begin{definr}
Let us call a pair
$(U_1^{x_1^0,y_1^0,\delta},U_2^{t_2^0,y_1^0,y_2^0,\delta})$ an {\it admissible
pair of type 1
(at scales ${\delta},\;\rho$ and contained in $V_1\times V_2$),} if the following two conditions hold
true:
\begin{align}\label{admissible1}
\frac {C_0^2}4\rho^2{\delta}\le
|\tau_{z_1^0}(z_1^0,z_2^0)|&=|t_2^0-x_1^0|<4 \,C_0^2\rho^2{\delta},\\
\frac{C_0^2}{512}\rho^2(1\vee
{\delta})\le|\tau_{z_2^0}(z_1^0,z_2^0)|&< 5\,
C_0^2\rho^2(1\vee {\delta}).\label{admissible2}
\end{align}
By ${\cal P}^{{\delta}}$ we shall denote the set of all admissible pairs of
type 1 at scale ${\delta}$ (and $\rho$, contained in $V_1\times V_2,$), and by ${\cal P}$
the corresponding union over all dyadic
scales ${\delta}.$
\end{definr}
Observe that, by \eqref{TV2+TV3}, we have
$\tau_{z_2^0}(z_1^0,z_2^0)=\tau_{z_1^0}(z_1^0,z_2^0)-(y_2^0-y_1^0)^2.$
In view of \eqref{admissible1} and \eqref{yseparation} this shows that condition
\eqref{admissible2} is automatically satisfied, unless ${\delta}\sim 1.$
We remark that it would indeed be more appropriate to denote the sets ${\cal P}^{{\delta}}$ by ${\cal P}^{{\delta}}_{V_1\times V_2},$ but we
want to simplify the notation. In all instances in the rest of the paper ${\cal P}^{{\delta}}$ will be associated to a
fixed admissible pair of strips $(V_1,V_2),$ so that our imprecision will not cause any ambiguity.
\begin{lemnr}\label{sizeofdeltas}
If $(U_1^{x_1^0,y_1^0,\delta},U_2^{t_2^0,y_1^0,y_2^0,\delta})$ is an admissible
pair of type 1, then for
all $(z_1,z_2)\in (U_1^{x_1^0,y_1^0,\delta},U_2^{t_2^0,y_1^0,y_2^0,\delta})$ ,
$$
|\tau_{z_1}(z_1,z_2)|\sim_{8} C_0^2 \rho^2{\delta}\mbox{ and }
|\tau_{z_2}(z_1,z_2)|\sim_{1000} C_0^2 \rho^2(1\vee{\delta}).
$$
\end{lemnr}
\begin{proof}
Note that
\begin{eqnarray}\label{tauid}
&&\tau_{z_1}(z_1,z_2)=x_2-x_1+y_2(y_2-y_1)=t_2^0-x_1^0\\
&&\hskip0.5cm+ \big[x_2-t_2^0+y_2(y_2-y_1^0)\big]-
\big[x_1-x_1^0+y_1^0(y_1-y_1^0)\big]+(y_1^0-y_2)(y_1-y_1^0),
\nonumber
\end{eqnarray}
where, by \eqref{yseparation} and our definition of $U_1^{x_1^0,y_1^0,\delta}$ and
$U_2^{t_2^0,y_1^0,y_2^0,\delta},$ we have
$|x_2-t_2^0+y_2(y_2-y_1^0)|<\rho^2{\delta},\ \big|(x_1-x_1^0)+y_1^0(y_1-y_1^0)\big|<\rho^2{\delta}$
and
$|(y_1^0-y_2)(y_1-y_1^0)|<C_0\rho \cdot\rho(1\wedge\delta)\le C_0\rho^2{\delta}.$
This shows that
\begin{align}\label{tausize1}
|\tau_{z_1}(z_1,z_2)-\tau_{z_1^0}(z_1^0,z_2^0)|\le 2C_0
\rho^2{\delta},
\end{align}
and in particular in combination with \eqref{admissible1} that
$|\tau_{z_1}(z_1,z_2)|\sim_{8} C_0^2 \rho^2{\delta},$
if we choose $C_0$ sufficiently large.
Similarly, because of \eqref{TV2+TV3'}, we have
$$
\tau_{z_2}(z_1,z_2)-\tau_{z_2^0}(z_1^0,z_2^0)=\tau_{z_1}(z_1,z_2)-\tau_{z_1^0}(z_1^0,z_2^0)
-(y_2-y_1)^2+(y_2^0-y_1^0)^2,
$$
where
\begin{align*}
|-(y_2-y_1)^2+(y_2^0-y_1^0)^2|&=|(y_2^0-y_2)+(y_1-y_1^0)|\,|(y_2^0-y_1^0)+(y_2-y_1)|\\
&\le 2\rho \cdot 2 C_0\rho=4 C_0\rho^2.
\end{align*}
In combination with \eqref{tausize1} this implies
\begin{align}\label{tausize2}
|\tau_{z_2}(z_1,z_2)-\tau_{z_2^0}(z_1^0,z_2^0)|\le
6C_0 \rho^2(1\vee{\delta}).
\end{align}
Invoking also \eqref{admissible2} this implies $|\tau_{z_2}(z_1,z_2)|\sim_{1000} C_0^2
\rho^2(1\vee{\delta}).$
\end{proof}
\medskip
Finally, as in Case 2 of the previous subsection, we also need to consider the symmetric case and define
admissible pairs where
$|\tau_{z_1}(z_1,z_2)|\gtrsim|\tau_{z_2}(z_1,z_2)|.$ By
interchanging the roles of $z_1$ and $z_2$ (compare also with \eqref{symtrans}) we define
accordingly for any $t^0_1, x^0_2\in{\cal I},$ $y^0_1$ the left endpoint of $I_1$ and $y^0_2\in{\cal I}_2$ the sets
\begin{align*}
{\tilde U}_1^{t^0_1,y_1^0,y_2^0,\delta} &:=\{(x_1,y_1): 0\le y_1-y^0_1<\rho, 0\le x_1-t^0_1+y_1(y_1-y_2^0)<
\rho^2\delta\},\\
{\tilde U_2}^{x^0_2,y_2^0,\delta} &:=\{(x_2,y_2): 0\le y_2-y_2^0< \rho(1\wedge\delta),\,
0\le x_2-x^0_2+y_2^0(y_2-y_2^0)< \rho^2\delta \}.
\end{align*}
The corresponding points $z^0_1\in {\tilde U}_1^{t^0_1,y_1^0,y_2^0,\delta}$ and $z^0_2\in {\tilde
U_2}^{x^0_2,y_2^0,\delta}$ are here defined by
$$
z^0_1=(x^0_1,y^0_1):=(t_1^0-y_1^0(y_1^0-y_2^0), y_1^0), \qquad z^0_2:=(x^0_2,y^0_2).
$$
In analogy to our previous definition, if the conditions
$$
\frac {C_0^2}4\rho^2{\delta}\le
|\tau_{z_2^0}(z_1^0,z_2^0)|=|x_2^0-t_1^0|<4 \,C_0^2\rho^2{\delta}$$
(in place of \eqref{admissible1}) and
$$
C_0^2\rho^2(1\vee {\delta})/512\le|\tau_{z_1^0}(z_1^0,z_2^0)|< 5
C_0^2\rho^2(1\vee {\delta})
$$ (in place of \eqref{admissible2}) are satisfied, we shall call the
pair $({\tilde U}_1^{t^0_1,y_1^0,y_2^0,\delta} , {\tilde U_2}^{x^0_2,y_2^0,\delta})$ an
{\it
admissible pair of type 2 (at scales ${\delta},\;\rho$ and contained in $V_1\times V_2$)\rm}. Note that this means
that $({\tilde U_2}^{x^0_2,y_2^0,\delta}, {\tilde U}_1^{t^0_1,y_1^0,y_2^0,\delta})$ is an
admissible pair of type 1 in $V_2\times V_1$ at the same scales.
\medskip
By $\tilde {\cal P}^{{\delta}},$ we shall denote the set of all admissible pairs of
type 2 at scale ${\delta}$ (and $\rho$, contained in $V_1\times V_2,$), and by $\tilde {\cal P}$
the corresponding unions over all dyadic
scales ${\delta}.$
\medskip
In analogy with Lemma \ref{sizeofdeltas}, we have
\begin{lemnr}\label{sizeofdeltas2}
If $(\tilde U_1,\tilde U_2)=({\tilde U}_1^{t^0_1,y_1^0,y_2^0,\delta},{\tilde U_2}^{x^0_2,y_2^0,\delta})\in \tilde {\cal P}^{\delta}$ is an admissible pair of type 2, then for all $(z_1,z_2)\in(\tilde U_1,\tilde U_2)$ we have
$$
|\tau_{z_1}(z_1,z_2)|\sim_{1000} C_0^2 \rho^2(1\vee{\delta})\mbox{ and }
|\tau_{z_2}(z_1,z_2)|\sim_8 C_0^2 \rho^2{\delta}.
$$
\end{lemnr}
\subsection{The exact transversality conditions}\label{TVS}
In the curved box case, i.e., when ${\delta} \ll 1,$ it turns out that one cannot directly reduce the bilinear
Fourier extension estimates over admissible pairs to Lee's Theorem 1.1 in \cite{lee05}, since that would not
give us the optimal dependence on $\delta$. We shall therefore
have to be more precise about the required transversality conditions. So, let us recall in more detail the
exact transversality conditions mentioned in the Introduction that we need for the bilinear argument. As
references to this (by now
standard) argument we refer for instance to \cite{lee05}, and \cite{v05}.
\medskip
In this bilinear argument, we assume that we are given two patches of subsurfaces
$S_i=\{(z_i,\phi(z_i)): z_i\in U_i\}, i=1,2,$ of $S.$ For fixed points $z_1'\in U_1$ and
$z_2'\in U_2,$ we consider the translated surfaces
$$
\tilde S_1:=S_1+(z_2',\phi(z_2'))\mbox{ and } \tilde S_2:=S_2+(z_1',\phi(z_1')).
$$
We will assume in this subsection that there is a constant $C>0$ such that
$|\nabla\phi(z)|\le C$ for all $z\in U_1\cup U_2.$ The implicit constants in the argument
will depend on this $C.$ We will not make any assumption about other derivatives of $\phi$
and will keep track of the dependence on them of the transversalities. That will be
important in the rest of the section.
\medskip
If the normals to these two surfaces $S_i$ at the points $(z_i',\phi(z_i')),$ $i=1,2,$
are not parallel, we can locally define the {\it intersection curve}
$$
\Pi_{z_1',z_2'}:=\tilde S_1\cap\tilde
S_2=[S_1+(z_2',\phi(z_2'))]\cap[S_2+(z_1',\phi(z_1'))].
$$
Note that
$$
\tilde S_1=\{(z,\phi(z-z_2')+\phi(z_2')): z\in U_1+z_2'\} \mbox{ and }\tilde
S_2=\{(z,\phi(z-z_1')+\phi(z_1')): z\in U_2+z_1'\}.
$$
Set $\psi(z):= \phi(z-z_1')+\phi(z_1')-\phi(z-z_2')-\phi(z_2').$ Then, the orthogonal
projection of the curve $\Pi_{z_1',z_2'}$ on the $z$ - plane is the curve given by
$\{z:\;\psi(z)=0\}.$
We introduce a parametrization by arc length $\gamma(t),\, t\in J,$ of this curve, where
$t$ is from an open interval $ J.$
Notice that $\gamma(t)$ depends on the choices of $z_1'$ and $z_2'.$
By $N(x,y)$ we denote the following normal to our surface $S$ at $(x,y,\phi(x,y))\in S:$
$N(x,y):=
\left(
\begin{array}{c}
{\,}^t\nabla\phi(x,y) \\
-1
\end{array}
\right).
$
Note that these normal vectors are of size $|N(x,y)|\sim 1.$
Then the vector $N_2(z):=N(z-z_1')$ is normal to the translated surface $\tilde S_2$ at
the point $(z,\phi(z_1-z_1')+\phi(z_1')),$ and we consider the ``cone of normals of type
2 along the intersection curve'' $\Gamma_2:=\{sN_2(\gamma(t)): s\in \mathbb R,\,t\in
J\}.$ In an analogous way, we define the ``cone $\Gamma_1$ of normals of type 1 along the
intersection curve''.
In the bilinear argument (see, for instance, \cite{v05}, final remark on page 110), the
condition which is needed is that the normal vectors to $S$ at all points of $S_1$ are
transversal to the cone $\Gamma_2,$ more precisely that
\begin{align}\label{transv}
\left|\det\left(\frac{N(z_1)}{|N(z_1)|}\, \frac{N_2(\gamma(t))}{|N_2(\gamma(t))|}
\,\frac{\frac{d}{dt}N_2(\gamma(t))}{|\frac{d}{dt}N_2(\gamma(t))|}
\right)\right|\ge C>0,
\end{align}
for all $z_1\in U_1$ and all $t\in J,$ and that the symmetric condition holds true when
the roles of $S_1$ and $S_2$ are interchanged, i.e., for $S_2$ and $\Gamma_1.$ The above
determinant is equal to
\begin{align}\label{transv1}
TV_2(z_2,z_1):= \frac{\det \left(\begin{array}{ccc}
{\,}^t\nabla\phi( z_1) & {\,}^t\nabla\phi( z_2)& H\phi( z_2)\cdot{\,}^t\omega \\
-1 & -1 & 0
\end{array}\right)}
{\sqrt{1+|\nabla\phi( z_1)|^2}\sqrt{1+|\nabla\phi(z_2)|^2}\,
|H\phi(z_2)\cdot{\,}^t\omega|},
\end{align}
where $\omega=\gamma'(t),$ for $t$ such that $\gamma(t)=z_2+z_1'.$ A similar expression is
obtained for $S_2$ and $\Gamma_1,$ namely
\begin{align*
TV_1(z_1,z_2):= \frac{\det \left(\begin{array}{ccc}
{\,}^t\nabla\phi( z_2) & {\,}^t\nabla\phi( z_1)& H\phi( z_1)\cdot{\,}^t\omega \\
-1 & -1 & 0
\end{array}\right)}
{\sqrt{1+|\nabla\phi( z_2)|^2}\sqrt{1+|\nabla\phi(z_1)|^2}\,
|H\phi(z_1)\cdot{\,}^t\omega|},
\end{align*}
where here we assume that $z_2\in U_2$ and $z_1\in U_1$ is such that
$z_1+z_2'=\gamma(s)$
for some $s\in J,$ and then
$\omega:=\gamma'(s).$
\medskip
Condition \eqref{transv} can be written as
\begin{align}\label{transv5}
|TV_1(z_1,z_2)|,\; |TV_2(z_2,z_1)|\ge C>0.
\end{align}
\medskip
Notice that, formally, $TV_2(z_2,z_1)=TV_1(z_2,z_1)$ (though, $z_1'$ and
$z_2'$ should also be interchanged). Moreover, one easily shows that
\begin{align}\label{transv3}
TV_2(z_2,z_1)= \frac{(\nabla\phi( z_1)-\nabla\phi( z_2))\cdot J \cdot H\phi(
z_2)\cdot{\,}^t\omega}
{\sqrt{1+|\nabla\phi( z_1)|^2}\sqrt{1+|\nabla\phi(z_2)|^2}\,
|H\phi(z_2)\cdot{\,}^t\omega|},
\end{align}
where $J$ denotes the symplectic matrix
$J:=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0 \\
\end{array}
\right).$
\medskip
At this point, it is interesting to explain how the quantities $TV_1$ and $TV_2$ relate to
$\Gamma^\phi_{z}$ defined by \eqref{transs}. Going back to \eqref{transv1}, note that
\begin{align*}
\det\left(\begin{array}{ccc}
{\,}^t\nabla\phi( z_1) & {\,}^t\nabla\phi( z_2)& H\phi( z_2)\cdot{\,}^t\omega \\
-1 & -1 & 0
\end{array}\right)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\\
=\det\left(\begin{array}{cc}
H\phi( z_2) & 0 \\
0 & 1
\end{array}\right)\det\left(\begin{array}{ccc}
(H\phi( z_2))^{-1}{\,}^t\nabla\phi( z_1) & (H\phi( z_2))^{-1}{\,}^t\nabla\phi( z_2)&
{\,}^t\omega \\
-1 & -1 & 0
\end{array}\right) \\
=\det H\phi( z_2) \det\left(\begin{array}{ccc}
(H\phi( z_2))^{-1}({\,}^t\nabla\phi( z_1)-{\,}^t\nabla\phi( z_2)) & (H\phi(
z_2))^{-1}{\,}^t\nabla\phi( z_2)& {\,}^t\omega \\
0 & -1 & 0
\end{array}\right).
\end{align*}
Set $z_1'' \in U_1,$ such that $\gamma(t)=z_2+z_1'=z_1''+z_2'.$ Then, $\omega=\gamma'(t)$
is unitary and orthogonal to $\nabla\phi( z_1'')-\nabla\phi( z_2).$
Hence,
\begin{align}\label{transv4}
|TV_2(z_2,z_1)|= \frac{\big|\det H\phi( z_2)\left\langle
(H\phi)^{-1}(z)(\nabla\phi(z_2)-\nabla\phi(z_1)),
\frac{\nabla\phi(z_2)-\nabla\phi(z_1'')}{|\nabla\phi(z_2)-\nabla\phi(z_1'')|}
\right\rangle\big|}{\sqrt{1+|\nabla\phi(
z_1)|^2}\sqrt{1+|\nabla\phi(z_2)|^2}\,|H\phi(z_2)\cdot{\,}^t\omega|}\\
=\frac{\big|\det H\phi( z_2)\,\Gamma^\phi_{z_2}(z_1,z_2,z_1'',z_2)\big|}
{\sqrt{1+|\nabla\phi( z_1)|^2}\sqrt{1+|\nabla\phi(z_2)|^2}\,
|H\phi(z_2)\cdot{\,}^t\omega||\nabla\phi(z_2)-\nabla\phi(z_1'')|}.
\end{align}
\medskip
\subsection{A prototypical admissible pair in the curved box case and the crucial scaling
transformation}\label{proto}
In this section we shall present a \lq\lq prototypical"
case where $U_1$ and $U_2$ will form an admissible pair of type 1 centered at $z_1^0=0\in U_1$ and $z_2^0\in
U_2,$ with $\rho\sim1,$ i.e., $|y_1^0-y_2^0|\sim 1,$ and $|\tau_{z_2^0}(z_1^0,z_2^0)|\sim 1$ but
$|\tau_{z_1^0}(z_1^0,z_2^0)|\sim \delta\ll 1.$ This means that we shall be in the curved box case. Recall from
the Introduction the identity
given by \eqref{gammaz} and \eqref{TV1}. As we will show in Subsection \ref{scaling transform} in
detail, we can
always reduce to this particular situation
when the two transversalities $\tau_{z_2^0}(z_1^0,z_2^0)$ and $\tau_{z_1^0}(z_1^0,z_2^0)$
are of quite different sizes.
Fix a small number $0<c_0\ll1$ ($c_0=10^{-10}$ will, for instance, work). Assume that
$0<\delta\le1/10,$ and put
\begin{align}
U_1:=&[0,c_0^2\delta)\times [0,c_0\delta) \label{U1}\\
U_2:=&\{(x_2,y_2):0\le y_2-b< c_0,0\le x_2+y_2^2-a<c_0^2\delta\},\label{U2}
\end{align}
where $|b|\sim_21$ and $|a|\sim_4\delta$.
\
\noindent\bf Remark. \rm Note that, if we set $C_0=1/{c_0},$ $\rho=c_0,$ then any admissible pair
$(U_1,U_2)=(U_1^{0,0,\delta},U_2^{a,0,b,\delta})$ would satisfy \eqref{U1} and \eqref{U2} with the above conditions on $a$ and $b.$
\medskip
Observe that for $z_1=(x_1,y_1)\in U_1$ and $z_2=(x_2,y_2)\in U_2,$ we have
$|y_1-y_2|\sim1.$
Moreover, $(0,0)\in U_1,$ and $\tau_{(0,0)}((0,0),(x_2,y_2))\sim\delta$ for all
$(x_2,y_2)\in U_2,$ which easily implies that, more generally,
$\tau_{(x_1,y_1)}((x_1,y_1),(x_2,y_2))\sim\delta$ for all $(x_1,y_1)\in U_1$ and
$(x_2,y_2)\in U_2.$ Moreover, $\tau_{(x_2,y_2)}((x_1,y_1),(x_2,y_2))\sim1$ for all
$(x_1,y_1)\in U_1$ and $(x_2,y_2)\in U_2.$
One easily computes also that
$TV_1(z_1,z_2)\sim \delta$ and $TV_2(z_2,z_1)\sim 1,$ respectively. Thus there is an
unlucky discrepancy between those two transversalities, since $\delta\ll 1,$ and a
straight-forward application of the bilinear method would lead to a worse dependency on
$\delta$ of the constant in the bilinear estimate for these sets than the estimate \eqref{bilinest} in
Theorem \ref{bilinear}, which we shall need.\smallskip
In the following lines, we shall therefore apply a suitable scaling which will turn both
transversalities to become of size $\sim 1.$ The price, however, that we shall have to
pay
is that, after scaling, the curvature of one of the two patches of surface will become
large (compare \eqref{hessphis}), so that we still cannot apply standard bilinear
estimates, but shall have to go into more detail into the proof of those estimates. Those
details will be given in Subsection \ref{bilinarg}.
\medskip
To overcome the afore-mentioned problem, we introduce the scaling
$$\phi^s(\bar z):=\frac{1}{\mathfrak a}\phi(A\bar z),$$
where $A$ is a regular matrix, $\mathfrak a$ real.
Concretely, we choose $(x,y)=z=A\bar z:=(\delta\bar x,\bar y),$ i.e.,
\begin{align*}
\bar x&=\delta^{-1}x,\\
\bar y&=y,
\end{align*}
and $\mathfrak a:=\det A=\delta$. Denote by $U_i^s:=A^{-1}U_i$ the re-scaled domains
$U_i,$
and by $S_i^s$ the scaled surface patches
$$
S_i^s:=\{(\bar x,\bar y,\phi^s(\bar x,\bar y)): (\bar x,\bar y)\in U_i^s\}.
$$
Explicitly, we then have
$$
\phi^s(\bar x,\bar y)=\bar x\bar y+\frac{1}{3\delta}\bar y^3,
$$
hence
\begin{align}\label{nablaphis}
\nabla\phi^s(\bar x,\bar y)=(\bar y,\bar x+\frac{1}{\delta}\bar y^2)
=(y,\frac{1}{\delta}(x+y^2))
\end{align}
and
\begin{align}\label{hessphis}
H\phi^s(\bar x,\bar y)=\left(\begin{array}{cc}
0 & 1 \\
1 & 2\delta^{-1}\bar y
\end{array}\right)
=\left(\begin{array}{cc}
0 & 1 \\
1 & 2\delta^{-1} y
\end{array}\right).
\end{align}
In particular, we see that
\begin{align}\label{bddgrad}
|\nabla\phi^s(\bar z)|\lesssim 1
\end{align}
for all $z=A\bar z\in U_1\cup U_2$.
Let us also put $\bar a:=a/\delta,\, \bar b:=b, $ so that $|\bar b|\sim_{2}1$ and $|\bar
a|\sim_4 1$ and choose
$z_i=A\bar z_i\in U_i,$ $i=1,2.$
Then, by \eqref{U1}, \eqref{U2}, we have that $y_2-y_1=\bar b+{\cal O}(c_0)$ and
$x_1+y_1^2\leq 2 c_0\delta$, while $(x_2+y_2^2)/\delta=\bar a+{\cal O}(c_0)$. We conclude
by \eqref{nablaphis} that
\begin{align*}
\nabla\phi^s(\bar z_2)-\nabla\phi^s(\bar z_1) =(\bar b, \bar a)+{\cal O}(c_0).
\end{align*}
This shows that if the vector $\omega=(\omega_1,\omega_2)$ is tangential to the intersection curve of
$ S_1^s$ and $S_2^s,$ then \begin{align}\label{tangent}
\omega=(-\bar a,\bar b)+{\cal O}(c_0).
\end{align}
In combination with \eqref{hessphis} this implies that
\begin{align*}
H\phi^s(\bar z_i)\cdot{\,}^t\omega = {\,}^t (\bar b, -\bar a+2\bar
b\delta^{-1}y_i)+{\cal O}\big((1+|\delta^{-1}y_i|)c_0\big)
\end{align*}
and
\begin{align}\label{sep}
|H\phi^s(\bar z_i)\cdot{\,}^t\omega| \sim 1+|\delta^{-1}y_i|.
\end{align}
By \eqref{transv3}, the transversalities for the scaled patches of surface $S_i^s,$ and
the scaled function, $\phi^s
\,i=1,2,$ are thus given by
\begin{align*}
\Big|TV^s_i(\bar z_1,\bar z_2)\Big|=&\Big|(-1)^{i +1}\frac{(\nabla\phi^s(\bar
z_1)-\nabla\phi^s( \bar z_2))\cdot J \cdot H\phi^s( \bar z_i)\cdot{\,}^t\omega}
{\sqrt{1+|\nabla\phi^s( \bar z_1)|^2}\sqrt{1+|\nabla\phi^s(\bar z_2)|^2}\, |H\phi^s(\bar
z_i)\cdot{\,}^t\omega|}\Big| \\
\sim& \Big|\frac{2\bar b(-\bar a+\bar b \delta^{-1} y_i) +
{\cal O}\big((1+|\delta^{-1}y_i|)c_0)\big)}{ 1+|\delta^{-1}y_i|}\Big|.
\end{align*}
Now, if $i=1,$ then $|y_1|\leq c_0\delta$, i.e., $|\delta^{-1}y_i|\le c_0\ll1,$ so that
clearly $\Big|TV^s_1(\bar z_1,\bar z_2)\Big|\sim 1.$
And, if $i=2,$ then $|y_i-\bar b|\le c_0\ll1,$ which easily implies that $|2\bar b(-\bar
a+\bar b \delta^{-1} y_i) + {\cal O}\big((1+|\delta^{-1}y_i|)c_0\big)|\sim \delta^{-1},$
and
also $1+|\delta^{-1}y_i|\sim \delta^{-1},$ so that $\Big|TV^s_2(\bar z_1,\bar
z_2)\Big|\sim 1.$
We have thus shown
\begin{lemnr}\label{transvscaled}
The transversalities for the scaled patches of surface $S_i^s, \,i=1,2,$ satisfy
$$
\Big|TV^s_i(\bar z_1,\bar z_2)\Big|\sim 1, \quad i=1,2.
$$
\end{lemnr}
\medskip
\subsection{Reduction to the prototypical case}\label{scaling transform}
Let $(U_1,U_2)\in{\cal P}^{\delta}$ be an admissible pair of type 1, where
$U_1=U_1^{x^0_1,y_1^0,\delta}$ and $U_2= U_2^{t^0_2,y_1^0,y^0_2\delta}.$
In this section, we shall see that the bilinear estimates associated to the sets
$U_1,U_2$ can easily be
reduced by means of a suitable
affine-linear transformation to either the classical bilinear estimate in \cite{lee05},
when $\delta\ge 1/10,$
or
to the estimate for the special ``prototype'' situation given in Subsection
\ref{proto}, when $\delta\le 1/10.$
\medskip
Recall from \eqref{pointsinU} that we then had put
$$
z^0_1:=(x^0_1,y^0_1)\in U_1, \qquad z^0_2=(x^0_2,y^0_2):=(t_2^0-y_2^0(y_2^0-y_1^0), y_2^0)\in U_2.
$$
We translate the point $z_1^0$ to the origin. Under the corresponding
translation, the phase function $\phi$ changes as follows:
\begin{align*}
\phi(z_1^0+\tilde z)=&(x_1^0+\tilde x)(y_1^0+\tilde y)+\frac{1}{3}(y_1^0+\tilde y)^3 \\
=& \tilde x\tilde y+\frac{1}{3}{\tilde y}^3+y_1^0 {\tilde y}^2+ \text{affine linear terms} \\
=& (\tilde x+y_1^0 \tilde y)\tilde y+\frac{1}{3}{\tilde y}^3+ \text{affine linear terms}.
\end{align*}
It is therefore convenient to introduce new coordinates $z''=(x'',y''),$ by putting
\begin{align}\label{changeofvariables1}
x'':=& \tilde x+y_1^0 \tilde y = x-x_1^0 + y_1^0 (y-y_1^0), \nonumber\\
y'':=&\tilde y = y-y_1^0.
\end{align}
Then, in these new coordinates, the phase function is given by
\begin{align}\label{changeofvariables2}
\tilde \phi(z'')=\phi(z'')+ \text{affine linear terms}.
\end{align}
Noticing that affine linear terms in the phase function play no role in our Fourier
extension estimates, we may thus assume that $\tilde \phi=\phi.$
In view of the size of the sets $U_i,$ we perform a further
scaling transformation by writing $x''= \rho^2 (1\vee\delta)x',\, y''=\rho y'.$
Then,
clearly
$\phi(x'',y'')=\rho^3 (1\vee\delta)\phi_{\delta}(x',y'),$ if we put
$$
\phi_\delta(z):=xy+\frac{y^3}{3(1\vee\delta)}.
$$
Thus, altogether we define the change of coordinates $z'=T(z)$ by
\begin{align*}
x':=&(1\vee\delta)^{-1}\rho^{-2}(x-x_1^0+y_1^0(y-y_1^0)),\\
y':=&\rho^{-1}(y-y_1^0).
\end{align*}
Notice that from the following lemma, in the case $\delta\le 1/10,$ it is easy to pass to the prototypical case by another harmless scaling $(x',y')=(C_0^{2}x''',C_0y''').$ We shall skip the details.
\begin{lemnr}\label{U'}
We have
\begin{align}\label{phi-phide}
\phi(z)=\rho^3(1\vee{\delta})\phi_{\delta}(Tz)+L(z),
\end{align}
where $L$ is an affine-linear map. Moreover, in these new coordinates, $U_1,U_2$
correspond
to the sets
\begin{align}\label{scaledsets1}
U_1':=T(U_1)=&\{(x',y'):0\le y'< 1\wedge\delta,\, 0\le x'< 1\wedge\delta\}
=[0,1\wedge\delta)^2,\\
U_2':=T(U_2)=&\{(x',y'):0\le y'-b< 1,x'+\frac{y'^2}{1\vee\delta}-a<
1\wedge\delta\}, \label{scaledsets2}
\end{align}
where $|b|:=|\rho^{-1}(y_2^0-y_1^0)|\sim_2 C_0$ and
$|a|:=|\rho^{-2}(1\vee\delta)^{-1}(t_2^0-x_1^0)|\sim_{4} C_0^2\frac{\delta}{1\vee\delta} =
C_0^2(1\wedge\delta).$
Moreover, for Lee's transversality expression $\Gamma^{\phi_{\delta}}$ in \eqref{transs} for
$\phi_{\delta}$ , we have that
\begin{align}\label{transscaled}
|\Gamma^{\phi_{\delta}}_{\tilde z'_1}(z'_1,z'_2)|\sim C_0^3(1\wedge {\delta})\quad \text{for all }
\tilde
z'_1\in U'_1, \quad |\Gamma^{\phi_{\delta}}_{\tilde z'_2}(z'_1,z'_2)|\sim C_0^3 \quad
\text{for
all }
\tilde z'_2\in U'_2,
\end{align}
for every $z'_1\in U'_1$ and every $z'_2\in U'_2.$ Also, for $\delta\ge 1/10,$ the derivatives
of
$\phi_\delta$ can be uniformly
(independently of $\delta$) bounded from above.
\end{lemnr}
\begin{proof}
The first identity \eqref{phi-phide} is clear from our previous discussion.
The identities \eqref{scaledsets1}, \eqref{scaledsets2} and the formulas for $a$ and $b$
follow by straight-forward computation, and the statements about the sizes of $a$ and $b$
follow from \eqref{yseparation} and \eqref{admissible1}.
Recall that $\Gamma^\phi_{z_i}(z_1,z_2):=\left\langle
(H\phi)^{-1}(z_i)(\nabla\phi(z_2)-\nabla\phi(z_1)),\nabla\phi(z_2)-\nabla\phi(z_1)\right\rangle,$
and denote by $\Gamma^{\phi_{\delta}}_{z'_1}(z_1,z'_2)$ the corresponding quantity associated
to $\phi_{\delta}.$ These are obviously related by
$$
\Gamma^{\phi_{\delta}}_{z'_i}(z'_1,z'_2)=\frac
1{\rho^3(1\vee{\delta})}\Gamma^{\phi}_{T^{-1}z'_i}(T^{-1}z'_1,T^{-1}z'_2).
$$
Recall also from \eqref{TV1} that
$\Gamma^{\phi}_{z_1}(z_1,z_2)=2(y_2-y_1)\tau_{z_1}(z_1,z_2)$ and
$\Gamma^{\phi}_{z_2}(z_1,z_2)=2(y_2-y_1)\tau_{z_2}(z_1,z_2).$ Thus, by
Lemma \ref{covering} and \eqref{yseparation}, we have that for $z'_1\in U'_1, z'_2\in
U'_2,$
$$
|\Gamma^{\phi}_{T^{-1}z'_1}(T^{-1}z'_1,T^{-1}z'_2)|\sim \rho \, C_0^3\rho^2{\delta},\quad
|\Gamma^{\phi}_{T^{-1}z'_2}(T^{-1}z'_1,T^{-1}z'_2)|\sim \rho \, C_0^3\rho^2(1\vee {\delta}),
$$
hence
$$
|\Gamma^{\phi_{\delta}}_{z'_1}(z'_1,z'_2)|\sim
C_0^3(1\wedge {\delta}), \quad
|\Gamma^{\phi_{\delta}}_{z'_2}(z'_1,z'_2)|\sim
C_0^3.
$$
Moreover, by \eqref{gammaz}, if $z_1,\bar z_1\in U_1$ and $z_2,\bar z_2\in U_2,$ then
$$
|\Gamma^{\phi}_{\bar z_1}(z_1,z_2)-\Gamma^{\phi}_{z_1}(z_1,z_2)|=(y_2-y_1)^2|\bar
y_1-y_1|\lesssim C_0^2 \rho^2\rho (1\wedge \delta)\le C_0^2\rho^3\delta,
$$
and
$$
|\Gamma^{\phi}_{\bar z_2}(z_1,z_2)-\Gamma^{\phi}_{z_2}(z_1,z_2)|=(y_2-y_1)^2|\bar
y_2-y_2|\lesssim C_0^2 \rho^2\rho\le C_0^2 \rho^3(1\vee \delta).
$$
Hence, for $C_0$ sufficiently large,
$$
|\Gamma^{\phi}_{T^{-1}\bar z'_1}(T^{-1}z'_1,T^{-1}z'_2)|\sim \, C_0^3\rho^3{\delta},\quad
|\Gamma^{\phi}_{T^{-1}\bar z'_2}(T^{-1}z'_1,T^{-1}z'_2)|\sim \, C_0^3\rho^3(1\vee {\delta}),
$$
hence
$$
|\Gamma^{\phi_{\delta}}_{\bar z'_1}(z'_1,z'_2)|\sim
C_0^3(1\wedge {\delta}), \quad
|\Gamma^{\phi_{\delta}}_{\bar z'_2}(z'_1,z'_2)|\sim
C_0^3.
$$
This proves \eqref{transscaled}. For $\delta>1,$ $\phi_\delta$ does not depend on
$\delta,$
and the claim in the last statement of the lemma is trivially verified.
\end{proof}
\setcounter{equation}{0}
\section{Statements of the bilinear estimates. The proofs.}\label{sect:bilin}
We are now in a position to establish the following sharp bilinear Fourier extension
estimates for admissible pairs:
\begin{thmnr}\label{bilinear2}
Let $p>5/3,$ $q\ge2.$ Then, for every admissible pair $(U_1,U_2)\in {\cal P}^{\delta} $
at scale ${\delta},$ the following bilinear estimates hold true:
\begin{align*}
\|\ext_{U_1}(f)\ext_{U_2}(g)\|_p
\leq C_{p,q} \delta^{2(1-1/p-1/q)}\rho^{6(1-1/p-1/q)}
\|f\|_q\|g\|_q,\quad\text{if}\quad \delta> 1,
\end{align*}
and
\begin{align*}
\|\ext_{U_1}(f)\ext_{U_2}(g)\|_p
\leq C_{p,q}\, \delta^{5-3/q-6/p}\rho^{6(1-1/p-1/q)}
\|f\|_q\|g\|_q,\quad\text{if}\quad
\delta\le 1,
\end{align*}
with constants that are independent of the given pair, of $\rho,$ and of ${\delta}.$
\end{thmnr}
\begin{remnr}\label{remarkbilin}
Recall that for ${\delta}>1$ the sets $U_1$ and $U_2$ are essentially rectangular boxes of
dimension $\rho^2{\delta}\times \rho,$ and notice that our estimates for this case do agree
with the ones given in Proposition 2.1 in \cite{v05} for the case of the saddle.
\end{remnr}
By the considerations in the previous section, Theorem \ref{bilinear2} reduces to the
following statement for the prototypical case.
\begin{thmnr}[prototypical case]\label{bilinear}
Let $p>5/3,$ $\delta\le1/10,$ and let $(U_1,U_2)$ be an admissible pair given by \eqref{U1},\eqref{U2}.
Then
\begin{align}\label{bilinest}
\|\ext_{U_1}(f_1),\ext_{U_2}(f_2)\|_p \leq C_p \, \delta^{\frac{7}{2}-\frac{6}{p}}
\|f_1\|_2\|f_2\|_2
\end{align}
for every $f_1\in L^2(U_1)$ and $f_2\in L^2(U_2).$
\end{thmnr}
We begin by explaining in more detail here the reduction of Theorem \ref{bilinear2} to
Theorem \ref{bilinear}. Subsection \ref{bilinarg} will then be devoted to the proof of
Theorem \ref{bilinear}.
\bigskip
\noindent {\it Reduction of Theorem \ref{bilinear2} to Theorem \ref{bilinear}.} Fix $p>5/3$ and $q\ge 2,$ and
assume without loss of generality that
$U_1=U_1^{x_1^0,y_1^0,\delta}$ and $U_2= U_2^{t_2^0,y_1^0,y^0_2,\delta}$ form an admissible pair
of
type 1.
Assume first that $\delta\ge1/10.$ In this case, Lemma \ref{U'} shows that
the conditions of Lee's Theorem 1.1 in \cite{lee05} are satisfied for the patches of
surface $S'_1$ and $S'_2$ which are the graphs of $\phi_{\delta}$ over the sets $U'_1$ and
$U'_2$ given by \eqref{scaledsets1} and \eqref{scaledsets2}, and we can conclude that
there is a constant $C'>0$ which does not depend on $U'_1,U'_2$ such that $
\|\ext^{\delta}_{U'_1}(\tilde f)\ext^{\delta}_{U'_2}(\tilde g)\|_p\le C_p\|\tilde f\|_2\|\tilde
g\|_2.$ Here, the operators $\ext^{\delta}_{U'_i}$ are essentially given by
$$
\ext_{U'_i}^{\delta} h(\xi)=\int_{U'_i} h(x',y') e^{-i(\xi_1 x'+\xi_2 y'+\xi_3\phi_{\delta}(x',y'))}
\, dx' dy',\quad i=1,2.
$$
Since $|U'_1|=1$ and $|U'_2|\sim 1,$ by H\"older's inequality this implies that
\begin{align*
\|\ext^{\delta}_{U'_1}(\tilde f)\ext^{\delta}_{U'_2}(\tilde g)\|_p\le C_{p,q}\|\tilde
f\|_q\|\tilde g\|_q.
\end{align*}
By undoing the change of coordinates and using \eqref{phi-phide}, this leads to the
estimates
$$
\|\ext_{U_1}(f)\ext_{U_2}(g)\|_p
\leq C_{p,q} \big[\rho^3 (1\vee {\delta})\big]^{2(1-1/p-1/q)} \|f\|_q\|g\|_q.
$$
Assume next that ${\delta}\le 1/10 .$ Then Lemma \ref{U'} shows that $ U'_1=[0,\delta)^2,$ and
$U'_2=\{(x',y'):0\le y'-b< 1,\,0\le x'+\frac{y'^2}{1\vee\delta}-a< \delta\},$ where
$\phi_{\delta} (x',y')=\phi(x',y').$ Thus, in this situation we may reduce to the prototype
situation studied in Theorem \ref{bilinear}, by means of a change of coordinates of the
form $(x',y')=(C_0^2 x'', C_0 y''),$ and thus obtain the estimate
$$
\|\ext^{\delta}_{U'_1}(\tilde f)\ext^{\delta}_{U'_2}(\tilde g)\|_p\le C_{p}{\delta}^{7/2-6/p}\|\tilde
f\|_2\|\tilde g\|_2.
$$
Since here $|U'_1|={\delta}^2$ and $|U'_2|\sim {\delta},$ H\"older's inequality then implies that
$$
\|\ext^{\delta}_{U'_1}(\tilde f)\ext^{\delta}_{U'_2}(\tilde g)\|_p\le
C_{p,q}{\delta}^{3(1/2-1/q)}{\delta}^{7/2-6/p}\|\tilde f\|_q\|\tilde g\|_q=
C_{p,q}{\delta}^{5-3/q-6/p}\|\tilde f\|_q\|\tilde g\|_q.
$$
By undoing the change of coordinates, we again pick up an extra factor $\big[\rho^3
(1\vee {\delta})\big]^{2(1-1/p-1/q)}=\rho^{6(1-1/p-1/q)}$ and arrive at the claimed estimate
for the case ${\delta}\le {1/10}.$
\qed
\medskip
\subsection{The bilinear argument: proof of Theorem \ref{bilinear}}\label{bilinarg}
As the bilinear method is by now standard, we will only give a brief sketch of the proof
of Theorem \ref{bilinear}, pointing out the necessary modifications compared to the
classical case of elliptic surfaces.
First, recall from Subsection \ref{proto} that we had passed from our original coordinates $(x,y)$ to the coordinates $(\bar x, \bar y)$ by means of the scaling transformation $(x,y)=A(\bar x, \bar y)=(\delta\bar x,\bar y),$ and had put
$\phi^s(\bar z):=\phi(A\bar z)/{\mathfrak a},$ with $\mathfrak a:=\det A=\delta$.
We are going to establish the following bilinear Fourier extension estimate for the
scaled patches of surface $S_i^s$ which where defined as the graphs of $\phi^s$ over the sets $U^s_i, i=1,2:$
\begin{align}\label{bilinestscaled}
\|\ext_{U^s_1}(f_1)\,\ext_{U^s_2}(f_2)\|_p \leq C_p \,
\delta^{\frac{5}{2}-\frac{4}{p}} \|f_1\|_2\|f_2\|_2,
\end{align}
for every $f_1\in L^2(U^s_1)$ and every $f_2\in L^2(U^s_2).$ Here, $\ext_{U^s_i}(f_i)$
denotes the Fourier transform of $f_i d \sigma_i,$ where $\sigma_i$ is the pull-back of
the Lebesgue measure on $U^s_i$ to the surface $S^s_i$ by means of the projection onto
the $(\bar x,\bar y)$- plane, i.e.,
$$
\ext_{U^s_i}(f_i)(\xi,\tau)=\widehat {f_i d \sigma_i}(\xi,\tau)=\int_{U^s_i}f_i(\bar z)
e^{-i[\xi \bar z+\tau \phi^s(\bar z)]}\, d\bar z.
$$
Scaling back to our original coordinates, we obtain from this estimate that
\begin{align*}
\|\ext_{U_1}(f_1)\, \ext_{U_2}(f_2)\|_p \le&(\det A)^{1-1/p}{\mathfrak a}^{-1/p}
\delta^{5/2-4/p}
\|f_1\|_2\|f_2\|_2\\
=& \delta^{7/2-6/p} \|f_1\|_2\|f_2\|_2,
\end{align*}
hence \eqref{bilinest}.
\medskip
We start by recalling that a first important step in the bilinear argument consists in a
wave packet decomposition of the functions $\widehat {f_i d \sigma_i}$ (compare, e.g.,
\cite{lee05} for details on wave packet decompositions).
For the construction of the wave packets at scale $R$ in the bilinear argument, one first
decomposes the functions $f_i$ into well-localized (modulated) bump functions at scale
$1/R,$ localized near points $v,$ say $\varphi_v(\bar z)=\varphi(R(\bar z-v)),$
and then considers
their Fourier extensions $\widehat{\varphi_vd\sigma}$ (here $\sigma$ denotes any of the
measures $\sigma_i$).
Then, by means of a Taylor expansion, one finds that (essentially)
\begin{align*}
\widehat{\varphi_vd\sigma}(\xi,\tau)=&\int \varphi(R\bar z)e^{-i[\xi(v+\bar z) +
\tau\phi^s(v+\bar z)]} \,d\bar z \\
=\ & R^{-2}e^{-i\xi v}
\int \varphi(\bar z)e^{-i[R^{-1}\xi\bar z+\tau\phi^s(v+R^{-1} \bar z)]}\, d\bar z\\
= R^{-2}& e^{-i[\xi v + \tau\phi^s(v)]}
\int \varphi(\bar z)e^{-i[R^{-1}(\xi+\tau\nabla\phi^s(v))\bar z +
\frac{1}{2}R^{-2}\tau
\bar z^tH\phi^s(v')\bar z]}\, d\bar z,
\end{align*}
where $v'$ is on the line segment between $v$ and
$v+R^{-1}\bar z.$
Following \cite{lee05}, Lemma 2.3, integration by parts shows that the wave packet is
then
associated to the region where the complete phase satisfies
$|(R^{-1}\xi\bar z+\tau\phi^s(v+R^{-1}\bar z))|\le c$ for every $\bar z$ with $|\bar
z|\leq
1,$ say with $c$ small, on which $\widehat{\psi_vd\sigma}$ is essentially constant.
This condition requires in particular that the usual condition $|\xi+\tau
\nabla\phi^s(v)|\leq R$ holds true. Recall here also from \eqref{bddgrad} that for $v\in
U_1^s\cup U_2^s,$ we have $|\nabla\phi^s(v)|\lesssim1.$
Moreover, if we assume that $v=(\bar x_v,\bar y_v)\in U_1^s$, then $|\bar y_v|\leq
c_0\delta$, so that, if $R^{-1}\le\delta,$ by \eqref{hessphis},
$\|H\phi^s(v')\|\lesssim 1$. Hence we obtain the usual condition
$|\tau|\leq R^2$.
Note also that the higher order derivatives of the phase, $R^{-1}\xi\bar
z+\tau\phi^s(v+R^{-1} \bar z),$ are bounded by constants.
This means that the wave packets associated to $f_1$ and the patch of hypersurface
$S_1^s$ are essentially supported in tubes $T_1$ of the form
\[ T_{1}=\{(\xi,\tau):|\xi+\tau\nabla\phi^s(v)|\leq R, |\tau|\leq R^2\},
\]
respectively ``horizontal'' translates in $\xi$ of them (due to modulations). Notice the
standard fact that $T_1$ is a tube of dimension $R\times R\times R^2$ whose long axis is
pointing in the direction of the normal vector $N(v):=(\nabla\phi^s(v),-1)$ to $S_1^s$ at
the point $(v,\phi^s(v)).$
\smallskip
However, if $v=(\bar x_v,\bar y_v)\in U_2^s$, then $|\bar y_v|\sim 1$, so that, by
\eqref{hessphis}, if $R^{-1}\le\delta,$ $\|H\phi^s(v')\|\sim 1/\delta$. To
bound $R^{-2}|\tau \bar z^tH\phi(v')\bar z|\lesssim 1$, we thus here need to assume that
$|\tau|\leq \delta R^2,$ and the wave packets associated to $f_2$ and patch of
hypersurface $S_2^s$ are thus essentially supported in shorter tubes $T_2$ of the form
\[ T_{2}=\{(\xi,\tau):|\xi+\tau\nabla\phi^s(v)|\leq R, |\tau|\leq\delta R^2\},
\]
respectively ``horizontal'' translates in $\xi$ of them.
\smallskip
The wave packets associated to such tubes will be denoted by $\phi_{T_i},\, i=1,2.$
\smallskip
There is a technical obstacle here to be noticed, which is of a similar nature as a
related
problem that had arisen in \cite{bmv16}: to ensure that the wave packets are tubes, we
need that $\delta R^2>R$, i.e., $R>\delta^{-1}.$ For the usual induction on scales
argument this creates the difficulty that we cannot simply induct in the standard way
on
the scales $R>1.$
Instead, we change variables $R'=R\delta,$ and induct on the scales $R'>1$. The wave
packets $T_2$ are of then of dimension
$\frac{R'}{\delta}\times\frac{R'}{\delta}\times\frac{R'^2}{\delta},$ where now indeed
$\frac{R'^2}{\delta}>\frac{R'}{\delta}$.
\medskip
Following further on the bilinear method, we have to consider localized estimates at
scale $R'$ of the form
\begin{align}\label{inducthypo}
\|\ext_{U_1}(f_1)\, \ext_{U_2}(f_2)\|_{L^p(Q(R'))} \leq C(\delta) C_\alpha
(R')^\alpha
\|f_1\|_2\|f_2\|_2,
\end{align}
where $Q(R')$ is a cuboid determined by the wave packets, and need to push down the
exponent $\alpha$ by means of induction on scales. Wave packet decompositions then allow
to
reduce these estimates to bilinear estimates for the associated wave packets.
The corresponding $L^1$-estimate is trivial (compare \cite{lee05} , or \cite{bmv16}, for
details): since the wave packets of a given type arising in these decompositions are
almost
orthogonal, one easily finds by Cauchy-Schwarz' inequality that
\begin{align}\nonumber
\|\sum_{T_1,T_2}\phi_{T_1}\phi_{T_2}\|_{1} \leq
&\|\sum_{T_1}\phi_{T_1}\|_2\|\sum_{T_2}\phi_{T_2}\|_{2}\leq \Big(\frac{R'^2}{\delta^2}\#
T_1\Big)^{1/2} \Big(\frac{R'^2}{\delta} \# T_2\Big)^{1/2}\\
=& \delta^{-3/2} R'^2(\# T_1\, \# T_2)^{1/2}. \label{bilin1}
\end{align}
As for further $L^p$-estimates, grossly oversimplifying, the bilinear method allows to
devise some ``bad'' subset of $Q(R')$ whose contributions are simply controlled by means
of the induction on scales hypothesis, and a ``good'' subset, on which we can obtain a
strong $L^2$- estimate by means of sophisticated geometric-combinatorial considerations,
essentially of the form (without going into details)
\begin{align}\label{bilin2}
\|\sum_{T_1,T_2}\phi_{T_1}\phi_{T_2}\|_{2} \leq R^{-1/2} (\# T_1\, \# T_2)^{1/2}
= \delta^{1/2}R'^{-1/2} (\# T_1\, \# T_2)^{1/2}.
\end{align}
For $1<p<2,$ interpolation between these estimates in \eqref{bilin1} and \eqref{bilin2}
gives
\begin{align*}
\|\sum_{T_1,T_2}\phi_{T_1}\phi_{T_2}\|_{p}
\leq& (\delta^{-3/2}R'^2)^{2/p-1}(\delta ^{1/2}R'^{-1/2})^{2-2/p} (\# T_1\#
T_2)^{1/2}
\\
=& \delta^{5/2-4/p} (R')^{5/p-3} (\# T_1\# T_2)^{1/2}.
\end{align*}
Since $R'\ge 1,$ again grossly simplifying, this (very formal) argument will in the end
show that for $p>5/3$ indeed the bilinear estimate \eqref{bilinestscaled} holds true.
\medskip
In this very rough description of the bilinear approach in our setting we have suppressed
a number of subtle and important issues which we shall explain next in some more detail.
\medskip
Indeed, the proof of the crucial $L^2$- estimate \eqref{bilin2} requires more careful
considerations.
For the combinatorial argument to work, we not only need the lower bounds on the
transversalities given by Lemma \ref{transvscaled}, but also have to make sure that
the
tubes $T_i$ of a given type (on which the wave packets are essentially supported) are
separated as the base point varies along the intersection curves at distances of order
$1/R.$
Let us explain this in more detail. We take a collection of $1/R$ - separated points
along the projection onto the $\bar z=(\bar x,\bar y)$ - plane of an intersection curve
$\Pi^s_{\bar z_1',\bar z_2'}$ associated to the patches of surface $S_1^s$ and $S_2^s.$
For each point $\bar z$ of this collection we consider the point $\bar z_1:=\bar z-\bar
z_2'\in U_1^s.$ We fix another point $p_0\in\mathbb R^3$ and consider all the tubes
$T_1$
which are associated to such base points $\bar z_1$ and which pass through the given
point $p_0.$
What the geometric-combinatorial argument then requires is that the directions of these
tubes be separated so that the tubes
$T_1\cap B(p_0, R^2/2)^c$ have bounded overlap (see \cite{v05}, \cite{lee05},
\cite{bmv16}).
Given the dimensions of the tubes $T_1$ of type $1,$ it is clear that this kind of
separation is achieved if the directions of the normals to the re-scaled surface $S^s$ at
those points $(\bar z_1,\phi^s(\bar z_1))$ are $R/R^2=1/R$ - separated.
Similarly, given the dimensions of the tubes $T_2$ of type $2,$ we shall also need that
the directions of the normals to the surface at the points $(\bar z_2,\phi(\bar z_2)),$
for $\bar z_2:=\bar z-\bar z_1',$ are $R/(\delta R^2)=(1/\delta)(1/ R)$ - separated. The
sizes of the entry $\partial^2 \phi^s/\partial \bar y^2$ of the Hessian of $\phi^s$ in
\eqref{hessphis} at the points of $U_1^s$ respectively $U_2^s$ and the fact that the
tangents $\omega$ to the curve $\gamma$ are essentially diagonal (compare \eqref{tangent})
guaranty that the desired separation condition is indeed satisfied.
\medskip
Another obstacle, which again already arose in \cite{bmv16}, consists in setting up the
base case for the induction on scales argument, i.e., setting up a suitable estimate of
the
form \eqref{inducthypo} for some initial, possibly very large value of
the exponent $\alpha.$
In the classical setting of elliptic surfaces, the na\"\i f and easily established
estimate of the form
\begin{align*}
\|\ext_{U_1}(f_1)\, \ext_{U_2}(f_2)\|_{L^p(Q(R'))} \leq |Q(R')|^{1/p}
\|f_1\|_1\|f_2\|_1
\end{align*}
would work, but in our setting, this would not give the right power of $\delta$ needed
to
establish \eqref{inducthypo}.
We therefore follow our approach in \cite{bmv16}(compare Lemma 2.10, 2.11), which
provides us with the following a-priori $L^2$-estimate (not relying on the
afore-mentioned geometric argument):
\begin{align*}
\|\sum_{T_1,T_2}\phi_{T_1}\phi_{T_2}\|_{2} \leq R^{-1/2} \sup_{i=1,2}\sup_{\Pi}(\#
{\cal T}_i^\Pi)^{1/2}(\# T_1\, \# T_2)^{1/2},
\end{align*}
where $ {\cal T}_i^\Pi$ denotes the set of all tubes of type $i$ associated to base points
along a given intersection curve $\Pi$ of translates of $S_1^s$ and $S_2^s$. We are done
if we can show that $\# {\cal T}_i^\Pi$ is bounded by some power of $R'$ but independently of
$\delta$.\\
We already saw in \eqref{tangent} that the tangent $\omega$ of the intersection curve is
essentially diagonal. After scaling, $U_2^s$ is a set of dimensions $1\times 1$, but
$U_1^s$ is a rectangle of dimensions $1\times\delta$, so an essentially diagonal
intersection curve can have length at most ${\cal O}(\delta)$. Since the separation of the
base points of our wave packets along this curve is of size $1/R=\delta/R'$, we see
that
indeed we must have
$\# {\cal T}_i^\Pi\leq C R'$.
This completes our sketch of proof of Theorem \ref{bilinear}.
\hfill $\Box$\\
After distributing our preprint though the ArXiv, we learned from Timothy Candy that the
bilinear estimate in Theorem \ref{bilinear} could also be deduced from his more
general bilinear estimates in Theorem 1.4 of \cite{can17}, after applying the crucial
scaling in $x$ that we use in Subsection \ref{proto}. The convexity assumptions on the sets $\Lambda_j$
in his theorem is not really necessary, as he pointed out to us. We wish
to thank him for informing us about this.
\setcounter{equation}{0}
\section{The Whitney-type decomposition and its overlap}\label{whitn}
\subsection{The Whitney-type decomposition of $V_1\times V_2$}\label{whitneydecomp}
Let $(V_1,V_2)$ be an admissible pair of strips as defined in Subsection \ref{preciseadmissible}.
Recall the definition of admissible pairs of sets from the same subsection, and that we had also introduced
there the sets ${\cal P}^{\delta}$ respectively $\tilde {\cal P}^{\delta}$ of admissible pairs of type
1 respectively type 2 at scale
${\delta},$ and by ${\cal P}$ respectively $\tilde {\cal P}$ we had denoted the corresponding unions over all dyadic
scales ${\delta}.$
\begin{lemnr}\label{covering}
The following covering and overlapping properties hold true:
\begin{itemize}
\item[(i)] For fixed dyadic scale ${\delta},$ the subsets $U_1\times U_2, \, (U_1,U_2)\in
{\cal P}^{\delta},$ of $V_1\times V_2\subset Q\times Q$ are pairwise disjoint, as likewise
are the subsets $\tilde U_1\times \tilde U_2, \, (\tilde U_1,\tilde U_2)\in
\tilde{\cal P}^{\delta}.$
\item[(ii)] If ${\delta}$ and ${\delta}'$ are dyadic scales, and if $(U_1,U_2)\in {\cal P}^{\delta}$ and
$(U'_1,U'_2)\in {\cal P}^{{\delta}'},$ then the sets $U_1\times U_2$ and $U'_1\times U'_2$
can only intersect if ${\delta}/{\delta}' \sim_{2^7} 1.$ In the latter case, there is only
bounded overlap. I.e., there is a constant $M\le2^6$ such that for
every $(U_1,U_2)\in {\cal P}^{\delta} $ there are at most $M$ pairs $(U'_1,U'_2)\in
{\cal P}^{{\delta}'}$ such that
$(U_1\times U_2)\cap ( U'_1\times U'_2)\ne \emptyset,$ and vice versa.
The analogous statements apply to admissible pairs in $\tilde{\cal P}.$
\item[(iii)] If $(U_1,U_2)\in {\cal P}^{\delta}$ and $(\tilde U_1,\tilde U_2)\in
\tilde{\cal P}^{{\delta}'},$ then $U_1\times U_2$ and $\tilde U_1\times \tilde U_2$ are
disjoint too, except possibly when both ${\delta},{\delta}'\ge 1/800$ and
${\delta}\sim_{2^{10}}{\delta}'.$ In the latter case, there is only bounded overlap. I.e.,
there is a constant $N={\cal O} (C_0)$ such that for every $(U_1,U_2)\in {\cal P}^{\delta}
$ there are at most $N$ pairs $(\tilde U_1,\tilde U_2)\in \tilde{\cal P}^{{\delta}'}$
such that
$(U_1\times U_2)\cap (\tilde U_1\times \tilde U_2)\ne \emptyset,$ and vice versa.
\item[(iv)] The product sets associated to all admissible pairs cover $V_1\times V_2$
up to a set of measure $0,$ i.e.,
$$
V_1\times V_2=\Big(\bigcup\limits_{(U_1,U_2)\in {\cal P}}U_1\times U_2
\Big)\cup\Big(\bigcup\limits_{(\tilde U_1,\tilde U_2)\in {\cal P}}\tilde U_1\times \tilde
U_2\Big)
$$
in measure.
\end{itemize}
\end{lemnr}
\begin{proof}
(i) Let $(U_1,U_2),(U'_1,U'_2)\in {\cal P}^{\delta}$ be different admissible pairs of type 1 at
scale ${\delta}.$ Then we shall see that already the sets $U_1$ and $U'_1$ will be disjoint.
Indeed, this will obviously be true if the corresponding points $y_1^0$ are different
for $U_1$ and $U'_1,$ and if they do agree, then this will be true because of
different
values of $x_1^0.$
\medskip
(ii) The first statement follows immediately from Lemma \ref{sizeofdeltas}. To prove the second one, assume
that ${\delta}/{\delta}' \sim_{2^7} 1,$ and that $U_1=U_1^{x_1^0,y_1^0,\delta},
U'_1=U_1^{(x')_1^0,(y')_1^0,\delta'}.$ Recall that $U_1=U_1^{x_1^0,y_1^0,\delta}$ is
essentially a rectangular box of dimension $\sim \rho^2{\delta}\times \rho(1\wedge {\delta}),$
containing the point $(x_1^0,y_1^0),$ whose longer side has slope $y_1^0$ with respect
to
the $y$-axis, and an analogous statement is true of $U'_1.$ If $U_1\cap U'_1\ne
\emptyset,$ then we must have $|y_1^0-(y')_1^0|\le\rho(1\wedge {\delta}),$ and so
the slopes for these two boxes can only differ of size at $\rho(1\wedge {\delta}),$ which
implies that both boxes must be essentially of the same direction and dimension. This
easily implies the claimed overlapping properties, because of the separation properties
of the points in ${\cal I}_1$ and ${\cal I}.$
\medskip
(iii) If either ${\delta}<1/800,$ or ${\delta}'<1/800,$ or if ${\delta}\nsim_{2^{10}}{\delta}',$ then Lemma \ref{sizeofdeltas} and Lemma \ref{sizeofdeltas2} show that $U_1\times U_2$ and $\tilde U_1\times \tilde U_2$ must be disjoint. In
the
remaining case where both ${\delta},{\delta}'\ge 1/800$ and ${\delta}\sim_{2^{10}}{\delta}',$ assume that
$U_1=U_1^{x_1^0,y_1^0,\delta}$ and
$\widetilde{U}_1=\widetilde{U}_1^{x^0,y_2^0,\delta'}$ are so that $U_1\cap \tilde
U_1\ne
\emptyset.$ Then observe that both $U_1$ and $\tilde U_1$ are essentially rectangular
boxes of dimension $\sim \rho^2{\delta}\times \rho,$ where $U_1$ has slope $y_1^0$ with
respect
to the $y$-axis, and $\tilde U_1$ has slope $y_2^0$ with respect to the $y$-axis. Since
$|y_1^0-y_2^0|\sim C_0 \rho\lesssim C_0 \rho{\delta},$ this shows that $\tilde U_1$ must be
contained within a rectangular box of dimension $\sim (C_0\rho^2{\delta})\times \rho$ around
$U_1,$ so that there are at most ${\cal O}(C_0)$ sets $\tilde U_1$ of this type which
can intersect $U_1.$
\medskip
(iv) Let $(z_1,z_2)\in V_1\times V_2$. Without loss of generality, we may assume by
symmetry that $|\tau_{z_1}(z_1,z_2)|\le |\tau_{z_2}(z_1,z_2)|,$ i.e., that
$$
|x_2-x_1+y_2(y_2-y_1)|\leq|x_2-x_1+y_1(y_2-y_1)|.
$$
We shall then show that there is an admissible pair $(U_1,U_2)\in {\cal P}$ such that
$(z_1,z_2)\in U_1\times U_2$ (in the other case, we would accordingly find an admissible
pair of type 2 with this property). We shall also assume that $|\tau_{z_1}(z_1,z_2)|>0,$
since the set of pairs $(z_1,z_2)$ with $\tau_{z_1}(z_1,z_2)=0$ forms a set of measure
$0.$
Then there is a unique dyadic $0<\delta\lesssim\rho^{-2}$ such that
$$C_0^2\rho^2{\delta}\le |\tau_{z_1}(z_1,z_2)|<2C_0^2\rho^2{\delta}.
$$
Chose $y_1^0\in {\cal I}_1$ such that $0\le y_1-y_1^0< \rho(1\wedge{\delta}),$ and then
$x_1^0,t_2^0\in {\cal I}$ such that
\begin{align*}
0\le x_1-x_1^0+y_1^0(y_1-y_1^0)< \rho^2{\delta} \quad
\text{ and } \quad 0\le x_2-t_2^0+y_2(y_2-y_1^0)< \rho^2{\delta}.
\end{align*}
Define as in \eqref{pointsinU} $z_1^0:=(x_1^0,y_1^0)$ and $z_2^0=(t_2^0-y_2^0(y_2^0-y_1^0),
y_2^0).$
We observe that, as in the proof of Lemma \ref{sizeofdeltas}, these estimates imply that
the estimates \eqref{tausize1} and \eqref{tausize2}
remain valid. In particular, we immediately see that
$|\tau_{z_1^0}(z_1^0,z_2^0)|\sim_4 C_0^2\rho^2{\delta},$ so that
\eqref{admissible1} is satisfied.
\smallskip
As for condition \eqref{admissible2}, if ${\delta}>8,$ by means of \eqref{TV2+TV3'} and
\eqref{yseparation} we can estimate
$$
|\tau_{z_2^0}(z_1^0,z_2^0)|\ge
|\tau_{z_1^0}(z_1^0,z_2^0)|-C_0^2\rho^2\ge
C_0^2\rho^2({\delta}/4-1)\ge
C_0^2\rho^2(1\vee {\delta})/8.
$$
On the other hand, if ${\delta}<1/32,$ then we may estimate
$$
|\tau_{z_2^0}(z_1^0,z_2^0)|\ge
C_0^2\rho^2/4 -|\tau_{z_1^0}(z_1^0,z_2^0)|\ge
C_0^2\rho^2(1/4-4{\delta})\ge C_0^2\rho^2(1\vee {\delta})/8.
$$
What remains is the case where $1/32\le {\delta}\le 8.$ Here we use the estimate
$$
|\tau_{z_2}(z_1,z_2)|\ge |\tau_{z_1}(z_1,z_2)|\ge C_0^2\rho^2{\delta}\ge C_0^2\rho^2/32,
$$
which by \eqref{tausize2} implies
$$
|\tau_{z_2^0}(z_1^0,z_2^0)|\ge
C_0^2\rho^2/32-6C_0 \rho^2(1\vee{\delta})\ge C_0^2\rho^2/32-48C_0 \rho^2\ge
C_0^2\rho^2/64\ge
C_0^2\rho^2(1\vee {\delta})/512,
$$
if we choose $C_0$ sufficiently large.
Moreover, note that we always have
$$
|\tau_{z_2^0}(z_1^0,z_2^0)|\le
|\tau_{z_1^0}(z_1^0,z_2^0)|+C_0^2\rho^2\le 5C_0^2\rho^2 (1\vee
{\delta}).
$$
Thus we have also verified \eqref{admissible2}. Hence,
$(U_1^{x_1^0,y_1^0,\delta},U_2^{t^0_2,y_1^0,y^0_2,\delta})$ is an admissible pair of type 1 at
scale $\delta.$
\end{proof}
\medskip
\subsection{Handling the overlap in the Whitney-type decomposition of $V_1\times V_2$}
For $r=0,\dots, 9,$ we define the subset ${\cal P}_r:=\bigcup_j {\cal P}^{2^{10j+r}}$ of ${\cal P}.$ To
these subsets of admissible pairs, we associate the subsets
\begin{align}\label{Ar}
A_r:=\bigcup\limits_{(U_1,U_2)\in {\cal P}_r}U_1\times U_2, \qquad \tilde
A_r:=\bigcup\limits_{(\tilde U_1,\tilde U_2)\in \tilde {\cal P}_r}\tilde U_1\times \tilde
U_2,\qquad (r=0,\dots, 9),
\end{align}
of $V_1\times V_2.$ Then Lemma \ref{covering} shows the following:
\begin{itemize}
\item[(i)] The unions in \eqref{Ar} are disjoint unions of the sets $U_1\times U_2,$
respectively $\tilde U_1\times \tilde U_2.$
\item[(ii)] There is a fixed number $N\gg 1$ such that the following hold true:
For given $r\ne r'$ and $(U_1,U_2)\in {\cal P}_r,$ there are at most $N$ admissible pairs
$(U'_1,U'_2)\in {\cal P}_{r'} $ such that
$(U_1\times U_2)\cap ( U_1'\times U_2')\ne \emptyset,$ and vice versa. Similarly,
for given $r,r'$ and $(U_1,U_2)\in {\cal P}_r,$ there are at most $N$ admissible pairs
$(\tilde U_1,\tilde U_2)\in\tilde {\cal P}_{r'} $ such that
$(U_1\times U_2)\cap (\tilde U_1\times \tilde U_2)\ne \emptyset,$ and vice versa.
\item[(iii)] \hskip 2cm $V_1\times V_2=\bigcup\limits_{r=0}^9(A_r\cup \tilde A_r).$
\end{itemize}
We shall make use of the following identity, which follows easily by induction on $m:$ If
$B_1,\dots,B_m$ are subsets of a given set $X,$ then
\begin{align*
\chi_{B_1\cup\cdots\cup B_m}=\sum_{\emptyset\ne J\subset \{1,\dots, m\}} (-1)^{\# J+1}
\chi_{\bigcap_{j\in J}B_j}.
\end{align*}
Applying this to (iii), we find that
\begin{align}\label{chiunion}
\chi_{V_1\times V_2}=\sum_{\emptyset\ne J,J'\subset \{0,\dots, 9\}}(-1)^{\#J+\#J'+1}
\chi_{(\bigcap_{r\in J}A_r)\cap( \bigcap_{r'\in J'}\tilde A_{r'}) }.
\end{align}
This will allow to reduce our considerations to finite intersections of the set $A_r$ and
$\tilde A_{r'}.$ Indeed, let us define
$$
E(F)(\xi):=\int_{Q\times Q} F(z,z') e^{-i[\xi\cdot(z,\phi(z))+\xi\cdot(z',\phi(z'))]}
\eta(z)\eta(z') \, dz dz',
$$
for any integrable function $F$ on $Q\times Q,$ so that in particular $E(f\otimes
g)=\ext(f) \ext(g).$ Then, by \eqref{chiunion}, if $f$ is supported in $V_1$ and $g$ in
$V_2,$
\begin{align}\label{chiunionE}
\ext(f) \ext(g)=\sum_{\emptyset\ne J,J'\subset \{0,\dots, 9\}}(-1)^{\#J+\#J'+1}E\Big(
(f\otimes g)\, \chi_{(\bigcap_{r\in J}A_r)\cap( \bigcap_{r'\in J'}\tilde A_{r'}))}\Big).
\end{align}
We may thus reduce considerations to restrictions of $f\otimes g$ to any of the
intersection
sets in \eqref{chiunionE}. So, let us fix non empty subsets $J,J'\subset \{0,\dots, 9\}$
and put
$$
B:= (\bigcap_{r\in J}A_r)\cap( \bigcap_{r'\in J'}\tilde A_{r'}).
$$
We then choose $r_0\in J$ and note that $B\subset A_{r_0},$ where $A_{r_0}$ is the
disjoint union of the product sets $U_1\times U_2$ over all admissible pairs $(U_1,U_2)\in
{\cal P}_{r_0},$ so that
\begin{align*
B=\overset{\cdot}{\bigcup\limits_{(U_1,U_2)\in {\cal P}_{r_0}}}B\cap (U_1\times U_2).
\end{align*}
And, by (i),(ii), each intersection
$B\cap (U_1\times U_2)$ is the finite disjoint union
\begin{align*
B\cap (U_1\times U_2)=\overset{\cdot}{\bigcup\limits_{\iota}}\, \Omega_1^\iota\times
\Omega_2^\iota, \qquad (U_1,U_2)\in {\cal P}_{r_0},
\end{align*}
of at most $N^{\#J+\#J'}$ measurable product subsets $\Omega_1^\iota\times
\Omega_2^\iota\subset U_1\times U_2.$
Since we shall have to estimate the $L^p$- norm of $E((f\otimes g)\, \chi_B)$ later,
note that
$$
E((f\otimes g)\chi_{B\cap (U_1\times U_2)})=\sum\limits_\iota \ext(f\chi_{\Omega_1^\iota})
\ext(g\chi_{\Omega_2^\iota}).
$$
Thus, if we write our estimates in Corollary \ref{bilinear2} in the form
$ \|\ext_{U_1}(f)\ext_{U_2}(g)\|_p \leq C_{p,q}(U_1,U_2) $ \hfill $\times \|f\|_q\|g\|_q,$
since $\Omega_i^\iota\subset U_i,$ we see that also
$$
\|\ext(f\chi_{\Omega_1^\iota}) \ext(g\chi_{\Omega_2^\iota})\|_p \leq C_{p,q}(U_1,U_2)
\|f\chi_{U_1}\|_q\|g\chi_{U_2}\|_q,
$$
hence, for every $(U_1,U_2)\in {\cal P}_{r_0},$
\begin{align*
\|E((f\otimes g)\chi_{B\cap (U_1\times U_2)}\|_p\leq N^{\#J+\#J'} C_{p,q}(U_1,U_2)
\|f\chi_{U_1}\|_q\|g\chi_{U_2}\|_q.
\end{align*}
This shows that on the set $B,$ we essentially get the same estimates for the
contributions by subsets $U_1\times U_2$ that we get on $A_{r_0}.$ This will allow us
to
reduce our considerations in the next section to the sets $A_r$ (respectively $\tilde
A_r$), which are already disjoint unions of product set $U_1\times U_2$ associated to
admissible pairs.
Observe finally that, for given $r,$
\begin{align}\label{ChiAr}
(f\otimes g)\chi_{A_r}=\sum\limits_{(U_1,U_2)\in {\cal P}_{r}} (f\chi_{U_1}) \otimes
(f\chi_{U_2}).
\end{align}
\setcounter{equation}{0}
\section{Passage to linear restriction estimates and proof of Theorem
\ref{mainresult}}\label{bilinlin}
To prove Theorem \ref{mainresult}, assume that $r>10/3$ and $1/q'>2/r,$ and put $p:=r/2,$
so that $p>5/3$, $1/q'>1/p.$ By interpolation with the trivial estimate for
$r=\infty,q=1,$
it is enough to prove the result for $r$ close to $10/3$ and $q$ close to 5/2, i.e.,
$p$
close to $5/3$ and $q$ close to 5/2. Hence, we may assume that $p<2,$ $p<q<2p.$ Also, we
can assume that $\text{\rm supp\,} f\subset\{(x,y)\in Q: y>0\}.$
We prove the linear estimates in two steps, following our steps in the construction in
Subsection \ref{pairs of sets}.
\smallskip
In a first step, we fix a scale $\rho$ and shall prove uniform bilinear Fourier
extension estimates for admissible pairs $V_1\backsim V_2$ of strips at scale $\rho$
(as defined in \eqref{Vi} of Subsection \ref{pairs of sets}). Our goal will be to prove
the following
\begin{lemnr}\label{V1V2bilin}
If $V_1\backsim V_2$ form an admissible pair of ``strips''
$V_i=V_{j_i,\rho}=I_{j_i,\rho}\times [-1,1], \, i=1,2,$ at scale $\rho$ within
$Q,$ and if $f\in L^q(V_1)$ and $g\in L^q(V_2),$ then for the range of $p$'s and $q$'s
described above we have
\begin{align}\label{VVbilin}
\|\ext_{V_1} (f) \ext_{V_2} (g)\|_p\lesssim C_{p,q} \,\rho^{2(1-1/p-1/q)} \|f\|_q\,
\|g\|_q \ \text{for all} \ f\in L^q(V_1), g\in L^q(V_2).
\end{align}
\end{lemnr}
\
We remark that, eventually, we shall choose $f=g$, but for the arguments to follow it is
helpful to distinguish between $f$ and $g$.
\medskip
\begin{proof}
To begin with, observe that by means of an affine linear transformation of the form
\eqref{changeofvariables1}, we may ``move the strips $V_1, V_2$ vertically'' so that
$j_1=0,$ which means that $V_1$ contains the origin. {Also, by \eqref{admissibleV}, $C_0/8<|j_2|<C_0/2.$} This
we shall assume throughout
the
proof. \smallskip
We recall from the previous section that it is essentially sufficient to consider
$E((f\otimes g)\chi_{A_r})$ in place of $\ext_{V_1} (f) \ext_{V_2} (g)$ (and similarly
for
$\tilde A_r$ in place of $A_r$). But then \eqref{ChiAr} shows that we may decompose
$(f\otimes g)\chi_{A_r}=\sum_\delta \sum_{i,i',j} f_{i,j}^\delta\otimes g_{i',j}^\delta$,
where $f_{i,j}^\delta:=f\chi_{U_{1}^{i\rho^2{\delta},j\rho(1\wedge {\delta}),\delta}}$,
$g_{i',j}^\delta:=g\chi_{
U_{2}^{i'\rho^2{\delta},j\rho(1\wedge {\delta}),{j_2\rho},\delta}},$ and where each
$\big(U_{1}^{i\rho^2{\delta},j\rho(1\wedge {\delta}),\delta},
U_{2}^{i'\rho^2{\delta},j\rho(1\wedge {\delta}),{j_2\rho},\delta}\big)$ forms an
admissible pair, i.e., \eqref{admissible1}, \eqref{admissible2} are satisfied. This
means
in particular, {by \eqref{admissible1}} that $ |i-i'|\sim C_0^2.$ The summation in ${\delta}$ is here meant as
summation
over all dyadic ${\delta}$ such that ${\delta}\lesssim \rho^{-2}.$ We may and shall also assume that
$f$ and $g$ are supported on the set $\{y>0\}.$ Then
\begin{align}\label{EfgA}
E((f\otimes g)\chi_{A_r})
= \sum_{\delta{\ge1/10}}\sum_{i,i'} \widehat{f_{i}^\delta
d\sigma}\widehat{g_{i'}^\delta d\sigma}+ \sum_{{\delta< 1/10}}\sum_{i,i',j} \widehat{f_{i,j}^\delta
d\sigma}\widehat{g_{i',j}^\delta d\sigma}.
\end{align}
The first sum (which collects all straight box terms) can be treated by more classical arguments (compare,
e.g., \cite{lee05} or \cite{v05}), which in view of the first estimate in Theorem \ref{bilinear2} then leads
to
a bound for the contribution of that sum to $\|\ext_{V_1} (f) \ext_{V_2} (g)\|_p$ in \eqref{VVbilin} of the
order
$$
\sum_{1\lesssim {\delta}\lesssim (\rho^2)^{-1}}C_{p,q}\, \,({\delta}\rho^3)^{2(1-\frac 1p-\frac 1q)} \|f\|_q\|g\|_q
\lesssim\rho^{2(1-\frac 1p-\frac 1q)} \|f\|_q\|g\|_q,
$$
as required. We leave the details to the interested reader.
We shall therefore concentrate on the second sum in \eqref{EfgA} (which collects all {curved} box terms) in
which the admissibility conditions reduce to $|i-i'|\sim C_0^2.$
\smallskip
Let us then fix $\delta\le1/10,$ and simplify notation by writing $f_{i,j}:=f_{i,j}^\delta,$
$g_{i,j}:=g_{i,j}^\delta,$ and $U_{1,i,j}:=U_{1}^{i\rho^2{\delta},j\rho(1\wedge {\delta}),\delta}$,
$
U_{2,i',j}:=
U_{2}^{i'\rho^2{\delta},j\rho(1\wedge {\delta}),j_2\rho,\delta}$.
As a first step in proving estimate \eqref{VVbilin}, we exploit some almost orthogonality
with respect to the $x$-coordinate:
\begin{lemnr}\label{lpsquare} For $1\le p\le 2,$ we have
\begin{align}\label{psquarefunc}
\big\|\sum_{j,\,i,|i-i'|\sim C_0^2}\, \widehat{f_{i,j}d\sigma}\,
\widehat{g_{i',j}d\sigma}\big\|_{p}^p
\lesssim \sum_{N=0}^{\rho^{-2}}\Big\|\sum_{i\in[N{\delta}^{-1},(N+1){\delta}^{-1}]\, ,\atop
|i-i'|\sim C_0^2,\,j} \widehat{f_{i,j}d\sigma}\,\widehat{g_{i',j}d\sigma}\Big\|_{p}^p.
\end{align}
\end{lemnr}
\noindent {\it Proof of Lemma \ref{lpsquare}:} Assume that $
i\in[N{\delta}^{-1},(N+1){\delta}^{-1}],$ and that $z_1=(x_1,y_1)\in U_{1,i,j}$ and
$z_2=(x_2,y_2)\in U_{2,i',j},$ where $|i-i'|\sim C_0^2,$ so that $(U_{1,i,j},
U_{2,i',j})\in {\cal P}^{\delta}$ is an admissible pair. Then, by Lemma \ref{covering} (a), we have
$$
|x_2-x_1+y_2(y_2-y_1)|\sim_8 C_0^2\rho^2{\delta},
$$
where $|y_2||y_2-y_1|\le C_0\rho\, C_0 \rho$ by \eqref{yseparation}, so that we may
assume
that
$
|x_2-x_1|\le 9\, C_0^2\rho^2.
$
This implies that $x_1+x_2=2x_1+{\cal O}(\rho^2),$ and thus
$x_1+x_2=2N\rho^2+{\cal O}(\rho^2),$ where the constant in the error term is of order
$C_0^2,$ hence
$$
U_{1,i,j}+U_{2,i',j}\subset [2N\rho^2-10\, C_0^2\rho^2,2N\rho^2+10\,C_0^2\rho^2]\times
[0,2C_0\rho].
$$
Notice that the family of intervals $\big\{[2N\rho^2-10\,
C_0^2\rho^2,2N\rho^2+10\,C_0^2\rho^2]\big\}_{N=0}^{\rho^{-2}}$ is almost pairwise
disjoint.
Therefore we may argue as in the proof of Lemma 6.1 in \cite{TVV} in order to derive the
desired estimate.\hfill $\Box$
\smallskip
For our next step, recall that $U_{1,i,j}$ is essentially a rectangular box of dimension
$\sim
\rho^2{\delta}\times \rho{\delta},$ and $U_{2,i',j}$ is a thin curved box of width $\sim\rho^2 {\delta}$
and length $\sim\rho,$ contained in a rectangle of dimension $\sim \rho^2\times \rho$
whose axes are parallel to the coordinate axes. Notice also that if $\rho=1,$ then
$U_{1,i,j}$ is essentially a square of size
${\delta}\times{\delta},$ whereas $U_{2,i',j}$ is a thin curved box of width $ {\delta}$ and length
$\sim 1.$ In this special case, it will be useful to further decompose $U_{2,i',j}$ into
squares
of size ${\delta}\times{\delta}.$
Analogously, since we may pass from the case $\rho=1$ to the case of general $\rho$ in
our
definition of admissible pairs
by means of the dilations $D_\rho(x,y):=(\rho^2 x,\rho y),$ for arbitrary $\rho$ we
decompose $U_{2,i',j}$
in the $y$-coordinate into intervals of length $\rho{\delta},$ by putting
$U^k_{2,i',j}:=\{(x,y)\in U_{2,i',j}: 0\le y-k\rho{\delta}< \delta\rho\}.$
Then
\begin{align*
U_{2,i',j}=\overset{\cdot}{\bigcup\limits_k} \,U^k_{2,i',j},
\end{align*}
where the union is over a set of ${\cal O} (1/{\delta})$ indices $k.$ Accordingly, we
decompose
$g_{i',j}=\sum_k g_{i',j}^k,$ where
$g_{i',j}^k:=g\chi_{U^k_{2,i',j}}.$
Then we have the following uniform square function estimate:
\begin{lemnr}\label{squaref} For $1<p\le 2$ there exists a constant $C_p>0$ such that for
every $N=0,\dots,\rho^{-2}$ we have
\begin{align}\label{squarefunc}
\Big\|\sum_{i\in[N{\delta}^{-1},(N+1){\delta}^{-1}], \atop |i-i'|\sim C_0^2,\,j}
\widehat{f_{i,j}d\sigma}\,
\widehat{g_{i',j}d\sigma}\Big\|_{p}
\le C_p \Big\|\Big(\sum_{i\in[N{\delta}^{-1},(N+1){\delta}^{-1}],\atop |i-i'|\sim C_0^2,\,j\,
,k}
|\widehat{f_{i,j}d\sigma}\,
\widehat{g_{i',j}^kd\sigma}|^2\Big)^{1/2}\Big\|_{p}.
\end{align}
\end{lemnr}
\noindent {\it Proof of Lemma \ref{squaref}:} Notice first that a translation in $x$ by
$N\rho^2$ allows to reduce to the case $N=0,$ which we shall thus assume. Then the
relevant sets $U_{1,i,j}$ and $U_{2,i',j}$ will all have their $x$-coordinates in the
interval $[0,\rho^2].$
For $i,\,i',\,j,\,k$ as above, set
$S_{1,i,j}:=\{(\xi,\phi(\xi)): \xi\in U_{1,i,j}\},$
$S_{2,i',j}^k:=\{(\xi,\phi(\xi)):\xi\in
U_{2,i',j}^k\}.$
The key to the square function estimate \eqref{squarefunc} is the following almost
orthogonality lemma:
\begin{lemnr}\label{ao}
Assume $N=0,$ and denote by $\tilde D_\rho, \rho >0,$ the dilations on $\mathbb R^3$ given
by
$\tilde D_\rho(x,y,z):=(\rho^2 x, \rho y, \rho^3 z).$ Then there is a family of cubes
$\{Q_{i,i',j}^k\}_{i\in[0,{\delta}^{-1}], |i-i'|\sim C_0\,,j\, ,k}$ in $\mathbb R^3$ with
bounded overlap, whose sides are parallel to the coordinate axes and of length
$\sim\delta,$ such that
$S_{1,i,j}+S_{2,i',j}^k\subset
\tilde D_\rho( Q_{i,i',j}^k).$
\end{lemnr}
\noindent {\it Proof of Lemma \ref{ao}:} Recalling the parabolic scalings
$D_\rho(x,y):=(\rho^2 x,\rho y)$ (under which
the phase $\phi$ is homogeneous of degree 3), we may apply the scaling by $\tilde
D_{\rho^{-1}}$ in $\mathbb R^3$ order to reduce our considerations to the case $\rho=1.$
Then, as we have already seen, $S_{1,i,j}$ and $S_{2,i',j}^k$ are contained in boxes of
side
length, say, $2\delta$ and sides parallel to the axes, whose projections to the $x$-axis
lie within the unit interval $[0,1].$ Therefore we can choose for
$Q_{i,i',j}^k$ a square of of side length $4\delta,$ with sides parallel to the axes,
with the property that $S_{1,i,j}+S_{2,i',j}^k\subset Q_{i,i',j}^k.$
We need to prove that the overlap is bounded.
Note that, if $(x_1,y_1)\in U_{1,i,j}$ and $(x_2,y_2)\in U_{2,i',j}^k$ with $|i-i'|\sim
C^2_0,$ then, by Lemma \ref{sizeofdeltas}, since $\rho=1$,
we have
\begin{align*
|x_2-x_1+y_2(y_2-y_1)|=|\tau_{z_1}(z_1,z_2)|\sim C_0^2 \delta.
\end{align*}
It suffices to prove the following: if $(x_1,y_1),(x_2,y_2)$ and $(x_1',y_1'),(x_2',y_2')$
are so that each coordinate of these points is bounded by a large multiple of $C_0,$ the
$y$-coordinates are positive and satisfy $y_2-y_1\gtrsim C_0 $ (by the $y$-separation
\eqref{yseparation}), and
\begin{eqnarray*}
x_2-x_1+y_2(y_2-y_1)&\sim&C_0^2{\delta},\\
x_2'-x_1'+y_2'(y_2'-y_1')&\sim&C_0^2{\delta},\\
x_1+x_2&=&x_1'+x_2'+{\cal O}(\delta),\\
y_1+y_2&=&y_1'+y_2'+{\cal O}(\delta),\\
x_1y_1+\frac{y_1^3}3+x_2y_2+\frac{y_2^3}3&=&x_1'y_1'+\frac{(y_1')^3}3+x_2'y_2'+\frac{(y_2')^3}3
+{\cal O}(\delta),
\end{eqnarray*}
then
\begin{align}\label{overlap2}
x_1'=x_1+{\cal O}(\delta), \ y_1'=y_1+{\cal O}(\delta),\ x_2'=x_2+{\cal O}(\delta),\
y_2'=y_2+{\cal O}(\delta).
\end{align}
Set
$$
a:=x_1+x_2, \quad b:=y_1+y_2,\qquad a':=x'_1+x'_2, \quad b':=y'_1+y'_2,
$$
and
$$
t_1:=x_1y_1+\frac{y_1^3}3,\qquad t_2:=x_2y_2+\frac{y_2^3}3.
$$
The analogous quantities defined by $(x_1',y_1'),(x_2',y_2')$ are denoted by $t'_1$ and
$t'_2.$
Notice that by our assumptions, $a$ and $b$ only vary of order ${\cal O} ({\delta})$ if we
replace $(x_1,y_1),(x_2,y_2)$ by $(x_1',y_1'),(x_2',y_2').$
Then,
$$
t_1+t_2=2x_1y_1-bx_1-ay_1+ab+\frac{b^3}3-b^2y_1+by_1^2.
$$
We choose $c$ with $|c|\sim C^2_0,$ such that $x_2-x_1+y_2(y_2-y_1)=c\delta.$ Then we may
re-write $x_1=\big (a-c\delta+(b-y_1)(b-2y_1)\big)/2.$
Therefore,
$$
t_1+t_2=C(a,b)+\frac{bc\delta}2+y_1[-c\delta+\frac32
b^2]-3by_1^2+2y_1^3:=C(a,b)+{\cal O}(\delta)+\psi(y_1),
$$
where $C(a,b)$ is a polynomial in $a,b$ and where we have put $\psi(y):=\frac32
b^2y-3by^2+2y^3.$ Similarly, $t'_1+t'_2=C(a',b')+{\cal O}(\delta)+\psi(y'_1),$
and thus, since $a=a'+{\cal O}(\delta), b=b'+{\cal O}(\delta),$ hence
$C(a,b)=C(a',b')+{\cal O}(\delta),$ we have
$$
\psi(y_1)=\psi(y'_1)+ {\cal O}(\delta).
$$
We may assume without loss of generality that $y_2-y_1>0$ (the other case can be treated
in
a similar way). Then, because of the $y$-separation \eqref{yseparation}, we have
$y_2-y_1\gtrsim C_0 $ and $y_1\ge 0$ and $b=y_2+y_1,$ so that $y_1\le \frac{b-1}2$ and
$b\ge1.$ It is a calculus exercise to prove that in this situation, for $y\le\frac{b-1}2,$
we have $\psi'(y)\ge3/2.$
This shows that we must have $y_1'=y_1+{\cal O}(\delta),$ hence also
$y_2'=y_2+{\cal O}(\delta),$ and then our first two assumptions imply also the remaining
assertions in \eqref{overlap2}.
This finishes the proof of the almost orthogonality Lemma \ref{ao}.
\hfill $\Box$
\smallskip
To complete the proof of Lemma \ref{squaref}, define the linear operators
$$\widehat{T_{i,i',j}^kh}(\xi):=\chi_{\tilde D_\rho(Q_{i,i',j}^k)}(\xi)\hat h(\xi),
$$
and
$$
S(h):=\{T_{i,i',j}^k h\}_{i\in [0,{\delta}^{-1}]\,,|i-i'|\sim C^2_0,\,j,k},
$$ for $h\in L^{p'}(\mathbb R^3).$ By Lemma \ref{ao} and Rubio de Francia's estimate
\cite{rdf}, we have
$$
\Big\|\Big(\sum_{i\in [0,{\delta}^{-1}],\,|i-i'|\sim
C_0^2,\,j,k}|T^k_{{i,i',j}}h|^2\Big)^{1/2}\Big\|_{p'}\le C
\|h\|_{L^{p'}(\mathbb R^3)},
$$
for $2\le p'<\infty$ (this is clearly true when $\rho=1,$ and the linear change of
coordinates given by $\tilde D_\rho$ does not change this estimate). Then, by duality, we
have
the adjoint operator estimate
$$
\Big\|\sum_{i\in [0,{\delta}^{-1}],\,|i-i'|\sim
C_0^2,\,j,k}T^k_{i,i',j}F_{i,i',j}^k\Big\|_{L^p(\mathbb R^3)}\le
C\Big\|\Big(\sum_{i\in [0,{\delta}^{-1}],\,|i-i'|\sim
C_0^2,\,j,k}|F_{i,i',j}^k|^2\Big)^{1/2}\Big\|_{L^p(\mathbb
R^3)}.
$$
The estimate \eqref{squarefunc} follows by applying the estimate above to the family of
functions
$F_{i,i',j}^k:= \widehat{f_{i,j}d\sigma}\,\widehat{g_{i',j}^k d\sigma},$ since
$F_{i,i',j}^k$ is the Fourier transform of a function supported in $\tilde D_\rho(
Q_{i,i',j}^k).$
\hfill $\Box$
\smallskip
Since $p/2<1$ in Lemma \ref{V1V2bilin}, from \eqref{psquarefunc} and \eqref{squarefunc}
we deduce that
\begin{align*}
\big\|\sum_{j,\,i,|i-i'|\sim C_0^2}\, \widehat{f_{i,j}d\sigma}\,
\widehat{g_{i',j}d\sigma}\big\|_{p}^p
&\lesssim\sum_{N=0}^{\rho^{-2}} \Big\|\Big(\sum_{j, \, i\in[N{\delta}^{-1},(N+1){\delta}^{-1}],\atop
|i-i'|\sim C_0^2}\sum_k
|\widehat{f_{i,j}d\sigma}\,\widehat{g_{i',j}^kd\sigma}|^2\Big)^{1/2}\Big\|_{p}^p\\
&\lesssim \sum_{N=0}^{\rho^{-2}}\sum_{j, \,i\in[N{\delta}^{-1},(N+1){\delta}^{-1}],\atop |i-i'|\sim
C_0^2}\Big\|\Big(\sum_{k}
|\widehat{f_{i,j}d\sigma}\,\widehat{g_{i',j}^kd\sigma}|^2\Big)^{1/2}\Big\|_{p}^p\\
&=\sum_{j,\, i, |i-i'|\sim C_0^2}\Big\|\Big(\sum_{k}
|\widehat{f_{i,j}d\sigma}\,\widehat{g_{i',j}^kd\sigma}|^2\Big)^{1/2}\Big\|_{p}^p.
\end{align*}
Using Khintchine's inequality, it suffices to bound
\begin{align*}
\sum_{j, \,i,|i-i'|\sim C_0^2}\|\widehat{f_{i,j}d\sigma}\,\widehat{\tilde
g_{i',j}d\sigma}\|_{p}^p
\end{align*}
for all $\tilde g_{i',j} = \sum_k \epsilon_k g_{i',j}^k$, $\epsilon_k=\pm 1$. Note that
$\|\tilde g_{i',j}\|_q=\| g_{i',j}\|_q.$ Now, by Theorem \ref{bilinear2},
\begin{align*}
\big(\sum_{j, \,i,|i-i'|\sim C_0^2} \|\widehat{f_{i,j}d\sigma}\widehat{\tilde
g_{i',j}d\sigma}\big\|_{p}^p\big)^{1/p}
\lesssim& \delta^{5-3/q-6/p} \rho^{6(1-1/p-1/q)}\left(\sum_{j,\, i,|i-i'|\sim C_0^2}
\big\|f_{i,j}\big\|_q^p\big\|\tilde g_{i',j}\big\|_q^p\right)^{1/p}\\
=& \delta^{5-3/q-6/p} \rho^{6(1-1/p-1/q)}\left(\sum_{j,\, i,|i-i'|\sim C_0^2}
\big\|f_{i,j}\big\|_q^p\big\| g_{i',j}\big\|_q^p\right)^{1/p}.
\end{align*}
Assume for a moment that $j$ is fixed. Then notice the following obvious, but
crucial
facts:
\begin{itemize}
\item[(i)] For every $i,$ $U_{1,i,j}$ is contained in the horizontal ``strip''
$V_{j,\rho} $ given by $0\le y-j{\delta}\rho<\rho{\delta}.$
\item[(ii)] The sets $U_{1,i,j}$ are mutually disjoint, and that the same holds true
for the sets $U_{2,i',j}.$
\end{itemize}
Let us therefore put $f_j:=f\chi_{V_{j,\rho}}.$ Then, by Cauchy-Schwarz' inequality,
\begin{align*}
\sum_{i,|i-i'|\sim C^2_0} \big\|f_{i,j}\big\|_q^p\big\|g_{i',j}\big\|_q^p
\lesssim& \left(\sum_i \big\|f_{i,j}\big\|_q^{2p}\right)^{1/2}
\left(\sum_{i'}\big\|g_{i',j}\big\|_q^{2p}\right)^{1/2}\\
\lesssim& \left(\sum_i \big\|f_{i,j}\big\|_q^{q}\right)^{p/q}
\left(\sum_{i'}\big\|g_{i',j}\big\|_q^{q}\right)^{p/q}\\
\lesssim& \big\|f_{j}\big\|^p_q\big\|g\big\|^p_q,
\end{align*}
where we have also used that $2p\geq q.$ Consequently, since the total number of $j$'s
over which we are summing is of order ${\cal O}(\rho/\rho{\delta})={\cal O}(1/{\delta}),$ we get
\begin{align*}
\big\|\sum_{i,|i-i'|\sim C^2_0,\, j} \widehat{f_{i,j}d\sigma}\,
\widehat{g_{i',j}d\sigma}\big\|_{p}
\lesssim& \delta^{5-3/q-6/p} \rho^{6(1-1/p-1/q)}\big(\sum_{i, |i-i'|\sim C^2_0,\,j}
\big\|f_{i,j}\big\|_q^p\big\| g_{i',j}\big\|_q^p\big)^{1/p}\\
\lesssim& \delta^{5-3/q-6/p} \rho^{6(1-1/p-1/q)}\big(\sum_j\|f_j\|_q^p\big)^{1/p}\|g\|_q\\
\lesssim& \delta^{5-3/q-6/p}
\rho^{6(1-1/p-1/q)}(1/{\delta})^{1/p-1/q}\big(\sum_j\|f_j\|_q^q\big)^{1/q}\|g\|_q\\
\lesssim& \delta^{5-2/q-7/p} \rho^{6(1-1/p-1/q)}\|f\|_q\|g\|_q.
\end{align*}
Now observe that the exponent in our dyadic parameter $\delta$ is
\begin{align*}
5-2/q-7/p = 3+2/q'-7/p \geq 3-5/p > 0,
\end{align*}
so that we can sum in \eqref{ChiAr} over all dyadic scales $\delta\ll 1$ and arrive at
the
estimate \eqref{VVbilin}.
\end{proof}
Note that, up to this point, we have been working with a fixed admissible pair if strips $V_1\sim V_2$ in which
all functions where supported.
\smallskip
In a final step, we shall prove the linear Fourier extension estimate of Theorem
\ref{mainresult}
by summing over the contributions by all admissible pairs of strips $V_1\backsim V_2,$
over all scales $\rho.$ To this end, let us write $f_{j,\rho}:=f\chi_{V_{j,\rho}}.$ In
view of our Whitney-decomposition \eqref{whitney1}, we may therefore write
$$
\widehat{f\,d\sigma}\, \widehat{f\,d\sigma}=\sum_{\rho\lesssim 1}\sum_{j, j\backsim
j'}\widehat{f_{j,\rho}\,d\sigma}\widehat{f_{j',\rho}\,d\sigma},
$$
where summation in $\rho$ is meant to be over all dyadic $0<\rho\lesssim 1$ and we wrote
$j\backsim j' $ to denote
$V_{j,\rho}\backsim V_{j',\rho}.$
Assume that $r=2p$ and $q$ satisfy the hypotheses of Theorem \ref{mainresult}, so that
$p>5/3$ and $1/q'>1/p.$
Then, by Lemma \ref{V1V2bilin}, for fixed $j,\,j',$
$$
\|\widehat{f_{j,\rho}\,d\sigma}\,\widehat{f_{j',\rho}\,d\sigma}\|_{L^p}\le C_{p,q}\,
\rho^{2(1-1/p-1/q)}\|f_{j,\rho}\|_q\|f_{j',\rho}\|_q.
$$
Therefore, since $p<2,$ using Minkowski's inequality and Lemma 6.1 in \cite{TVV}
\begin{align*}
\|\widehat{f\,d\sigma}\|_{2p}^{2}=\|\widehat{f\,d\sigma}\,\widehat{f\,d\sigma}\|_p
&=\big\|\sum_{\rho\lesssim 1}\sum_{j, j\backsim
j'}\widehat{f_{j,\rho}\,d\sigma}\widehat{f_{j',\rho}\,d\sigma}\big\|_p\\
&\le\sum_{\rho\lesssim 1} \big \| \sum_{j,j\backsim
j'}\widehat{f_{j,\rho}\,d\sigma}\widehat{f_{j',\rho}\,d\sigma}\big\|_p\\
&\le\sum_{\rho\lesssim 1} \bigg(\sum_{j,j\backsim j'}\big \|
\widehat{f_{j,\rho}\,d\sigma}\widehat{f_{j',\rho}\,d\sigma}\big\|_p^p\bigg)^{1/p}.
\end{align*}
For fixed $\rho,$ by Lemma \ref{V1V2bilin}
and Cauchy-Schwarz' inequality,
\begin{align*}
\sum_{j,j\backsim j'}\big \|
\widehat{f_{j,\rho}\,d\sigma}\widehat{f_{j',\rho}\,d\sigma}\big\|_p^p
&\lesssim \rho^{2(1-1/p-1/q)p}\sum_{j,j\backsim j'}\|f_{j,\rho}\|_q^p\,
\|f_{j',\rho}\|_q^p\\
&\lesssim \rho^{2(1-1/p-1/q)p}\bigg(\sum_j\|f_{j,\rho}\|_q^{2p}\bigg)^{1/{2}}
\bigg(\sum_{j'}\|f_{j',\rho}\|_q^{2p}\bigg)^{1/{2}}\\
&\lesssim \rho^{2(1-1/p-1/q)p}\bigg(\sum_j\|f_{j,\rho}\|_q^{q}\bigg)^{p/{q}}
\bigg(\sum_{j'}\|f_{j',\rho}\|_q^{q}\bigg)^{p/{q}}\\
&\lesssim \rho^{2(1-1/p-1/q)p}\|f\|_q^p\, \|f\|_q^p.
\end{align*}
where we have again used that $2p\geq q.$
Therefore, since $1-1/p-1/q>0,$
$$
\|\widehat{f\,d\sigma}\|_{2p}^{2}
\lesssim\|f\|_q^2,
$$
This completes the proof of our Fourier extension (respectively restriction) Theorem
\ref{mainresult}.
\hfill$\Box$
\thispagestyle{empty}
\renewcommand{\refname}{References}
|
{
"timestamp": "2019-07-04T02:09:12",
"yymm": "1803",
"arxiv_id": "1803.02711",
"language": "en",
"url": "https://arxiv.org/abs/1803.02711"
}
|
\subsection{Experimental Setup}
Through all of our experiments, weighted cross entropy loss from \cite{paszke2016enet} is used, to overcome the class imbalance. Adam optimizer \cite{kingma2014adam} learning rate is set to 1$e^{-4}$. Batch normalization \cite{ioffe2015batch} is incorporated. L2 regularization with weight decay rate of 5$e^{-4}$ is utilized to avoid over-fitting. The feature extractor part of the network is initialized with the pre-trained corresponding encoder trained on Imagenet. A width multiplier of 1 for MobileNet to include all the feature channels is performed through all the experiments. The number of groups used in ShuffleNet is 3. Based on previous \cite{zhang2017shufflenet} results on classification and detection three groups provided adequate accuracy.
Results are reported on Cityscapes dataset \cite{cordts2016cityscapes} which contains 5000 images with fine annotation, with 20 classes including the ignored class. Another section of the dataset contains coarse annotations with 20,000 labeled images. These are used in the case of Coarse pre-training that improves the results of the segmentation. Experiments are conducted on images with resolution of 512x1024.
\begin{table}[ht!]
\centering
\caption{Comparison of the most promising models in our benchmarking framework in terms of GFLOPs and frames per second, this is computed on image resolution 512x1024.}
\label{table:performance}
\begin{tabular}{|l|l|}
\hline
Model & GFLOPs \\ \hline
SkipNet-MobileNet & 13.8 \\ \hline
UNet-MobileNet & 55.9 \\ \hline
\end{tabular}
\end{table}
\begin{table*}[ht!]
\centering
\caption{Comparison of some of the models from our benchmarking framework with the state of the art segmentation networks on cityscapes test set. GFLOPs is computed on image resolution 360x640.}
\label{table:quant_city}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
Model & GFLOPs & Class IoU & Class iIoU & Category IoU & Category iIoU \\ \hline
SegNet\cite{badrinarayanan2015segnet} & 286.03 & 56.1 & 34.2 & 79.8 & 66.4 \\ \hline
ENet\cite{paszke2016enet} & 3.83 & 58.3 & 24.4 & 80.4 & 64.0 \\ \hline
DeepLab\cite{chen2016deeplab} & - & \textbf{70.4} & \textbf{42.6} & \textbf{86.4} & 67.7 \\ \hline
SkipNet-VGG16\cite{long2015fully} & - & 65.3 & 41.7 & 85.7 & \textbf{70.1} \\ \hline
SkipNet-ShuffleNet & \textbf{2.0} & 58.3 & 32.4 & 80.2 & 62.2 \\ \hline
SkipNet-MobileNet & 6.2 & 61.5 & 35.2 & 82.0 & 63.0 \\ \hline
\end{tabular}
\end{table*}
\begin{figure*}[ht!]
\centering
\begin{subfigure}{0.38\textwidth}
\includegraphics[scale= 0.18]{Images/original}
\caption{}
\end{subfigure}%
\begin{subfigure}{0.38\textwidth}
\includegraphics[scale= 0.18]{Images/mobilenet}
\caption{}
\end{subfigure}
\begin{subfigure}{0.38\textwidth}
\includegraphics[scale= 0.18]{Images/resnet}
\caption{}
\end{subfigure}%
\begin{subfigure}{0.38\textwidth}
\includegraphics[scale= 0.18]{Images/shufflenet}
\caption{}
\end{subfigure}
\caption{Qualitative Results on CityScapes. (a) Original Image. (b) SkipNet-MobileNet pretrained with Coarse Annotations. (c) UNet-Resnet18. (d) SkipNet-ShuffleNet pretrained with Coarse Annotations.}
\label{fig:qual_city}
\end{figure*}
\subsection{Semantic Segmentation Results}
Semantic segmentation is evaluated using mean intersection over union (mIoU), per-class IoU, and per-category IoU. Table\ref{table:ablation} shows the results for the ablation study on different encoders-decoders with mIoU and GFLOPs to demonstrate the accuracy and computations trade-off. The main insight gained from our experiments is that, UNet decoding method provides more accurate segmentation results than Dilation Frontend. This is mainly due to the transposed convolution by 8x in the end of the Dilation Frontend, unlike the UNet stage-wise upsampling method. The SkipNet architecture provides on par results with UNet decoding method. In some architectures such as SkipNet-ShuffleNet it is less accurate than UNet counter part by 1.5\%.
The UNet method of incrementally upsampling with-in the network provides the best in terms of accuracy. However, Table \ref{table:performance} clearly shows that SkipNet architecture is more computationally efficient with 4x reduction in GFLOPs. This is explained by the fact that transposed convolutions in UNet are applied in the feature space unlike in SkipNet that are applied in label space. Table \ref{table:ablation} shows that Coarse pre-training improves the overall mIoU with 1-4\%. The underrepresented classes are the ones that often benefit from pre-training.
Experimental results on the cityscapes test set are shown in Table \ref{table:quant_city}. Although, DeepLab provides best results in terms of accuracy, it is not computationally efficient. ENet \cite{paszke2016enet} is compared to SkipNet-ShuffleNet and SkipNet-MobileNet in terms of accuracy and GFLOPs. SkipNet-ShuffleNet outperforms ENet in terms of GFLOPs, yet it maintains on par mIoU. However, we have not been able to outperform ENet in terms of inference speed. Both SkipNet-ShuffleNet and SkipNet-MobileNet outperform SegNet \cite{badrinarayanan2015segnet} in terms of computational cost and accuracy with reduction up to 143x in GFLOPs. Figure \ref{fig:qual_city} shows qualitative results for different encoders including MobileNet, ShuffleNet and ResNet18. It shows that MobileNet provides more accurate segmentation results than the later two. SkipNet-MobileNet is able to correctly segment the pedestrian and the signs on the right unlike the others.
\subsection{Meta-Architectures}
Three meta-architectures are integrated in our benchmarking software: (1) SkipNet meta-architecture\cite{long2015fully}. (2) U-Net meta-architecture\cite{ronneberger2015u}. (3) Dilation Frontend meta-architecture\cite{yu2015multi}.
The meta-architectures for semantic segmentation identify the decoding method for in the network upsampling. All of the network architectures share the same down-sampling factor of 32. The downsampling is achieved either by utilizing pooling layers, or strides in the convolutional layers. This ensures that different meta architectures have a unified down-sampling factor to assess the effect of the decoding method only.
\begin{table*}[ht!]
\centering
\caption{Comparison of different encoders and decoding methods in accuracy on cityscapes validation set. The modular decoupled design in RTSeg enabled such comparison. Coarse indicates whether the network was pre-trained on the coarse annotation or not.}
\label{table:ablation}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Decoder & Encoder & Coarse & mIoU & Road & Sidewalk & Building & Sign & Sky & Person & Car & Bicycle & Truck\\ \hline
SkipNet & MobileNet & No & 61.3 & \textbf{95.9} & 73.6 & \textbf{86.9} & 57.6 & 91.2 & 66.4 & \textbf{89.0} & 63.6 & \textbf{45.9}\\ \hline
SkipNet & ShuffleNet & No & 55.5 & 94.8 & 68.6 & 83.9 & 50.5 & 88.6 & 60.8 & 86.5 & 58.8 & 29.6\\ \hline
UNet & ResNet18 & No & 57.9 & 95.8 & 73.2 & 85.8 & 57.5 & 91.0 & 66.0 & 88.6 & 63.2 & 31.4 \\ \hline
UNet & MobileNet & No & 61.0 & 95.2 & 71.3 & 86.8 & \textbf{60.9} & \textbf{92.8} & \textbf{68.1} & 88.8 & \textbf{65.0} & 41.3 \\ \hline
UNet & ShuffleNet & No & 57.0 & 95.1 & 69.5 & 83.7 & 54.3 & 89.0 & 61.7 & 87.8 & 59.9 & 35.5\\ \hline
Dilation & MobileNet & No & 57.8 & 95.6 & 72.3 & 85.9 & 57.0 & 91.4 & 64.9 & 87.8 & 62.8 & 26.3 \\ \hline
Dilation & ShuffleNet & No & 53.9 & 95.2 & 68.5 & 84.1 & 57.3 & 90.3 & 62.9 & 86.6 & 60.2 & 23.3\\ \hline
SkipNet & MobileNet & Yes & \textbf{62.4} & 95.4 & \textbf{73.9} & 86.6 & 57.4 & 91.1 & 65.7 & 88.4 & 63.3 & 45.3 \\ \hline
SkipNet & ShuffleNet & Yes & 59.3 & 94.6 & 70.5 & 85.5 & 54.9 & 90.8 & 60.2 & 87.5 & 58.8 & 45.4\\ \hline
\end{tabular}
\end{table*}
\textbf{SkipNet} architecture denotes a similar architecture to FCN8s \cite{long2015fully}. The main idea of the skip architecture is to benefit from feature maps from higher resolution to improve the output segmentation. SkipNet applies transposed convolution on heatmaps in the label space instead of performing it on feature space. This entails a more computationally efficient decoding method than others. Feature extraction networks have the same downsampling factor of 32, so they follow the 8 stride version of skip architecture. Higher resolution feature maps are followed by 1x1 convolution to map from feature space to label space that produces heatmaps corresponding to each class. The final heatmap with downsampling factor of 32 is followed by transposed convolution with stride 2. Elementwise addition between this upsampled heatmaps and the higher resolution heatmaps is performed. Finally, the output heat maps are followed by a transposed convolution for up-sampling with stride 8. Figure \ref{fig:architecture}(a) shows the SkipNet architecture utilizing a MobileMet encoder.
\textbf{U-Net} architecture denotes the method of decoding that up-samples features using transposed convolution corresponding to each downsampling stage. The up-sampled features are fused with the corresponding features maps from the encoder with the same resolution. The stage-wise upsampling provides higher accuracy than one shot 8x upsampling. The current fusion method used in the framework is element-wise addition. Concatenation as a fusion method can provide better accuracy, as it enables the network to learn the weighted fusion of features. Nonetheless, it increases the computational cost, as it is directly affected by the number of channels. The upsampled features are then followed by 1x1 convolution to output the final pixel-wise classification. Figure \ref{fig:architecture}(b) shows the UNet architecture using MobileNet as a feature extraction network.
\textbf{Dilation Frontend} architecture utilizes dilated convolution instead of downsampling the feature maps. Dilated convolution enables the network to maintain an adequate receptive field, but without degrading the resolution from pooling or strided convolution. However, a side-effect of this method is that computational cost increases, since the operations are performed on larger resolution feature maps. The encoder network is modified to incorporate a downsampling factor of 8 instead of 32. The decrease of the downsampling is performed by either removing pooling layers or converting stride 2 convolution to stride 1. The pooling or strided convolutions are then replaced with two dilated convolutions\cite{yu2015multi} with dilation factor 2 and 4 respectively.
\subsection{Feature Extraction Architectures}
In order to achieve real-time performance multiple network architectures are integrated in the benchmarking framework. The framework includes four state of the art real-time network architectures for feature extraction. These are: (1) VGG16\cite{simonyan2014very}. (2) ResNet18\cite{he2016deep}. (3) MobileNet\cite{howard2017mobilenets}. (4) ShuffleNet \cite{zhang2017shufflenet}. The reason for using \textbf{VGG16} is to act as a baseline method to compare against as it was used in \cite{long2015fully}. The other architectures have been used in real-time systems for detection and classification. \textbf{ResNet18} incorporates the usage of residual blocks that directs the network toward learning the residual representation on identity mapping.
\textbf{MobileNet} network architecture is based on depthwise separable convolution. It is considered the extreme case of the inception module, where separate spatial convolution for each channel is applied denoted as depthwise convolutions. Then 1x1 convolution with all the channels to merge the output denoted as pointwise convolutions is used. The separation in depthwise and pointwise convolution improve the computational efficiency on one hand. On the other hand it improves the accuracy as the cross channel and spatial correlations mapping are learned separately.
\textbf{ShuffleNet} encoder is based on grouped convolution that is a generalization of depthwise separable convolution. It uses channel shuffling to ensure the connectivity between input and output channels. This eliminates connectivity restrictions posed by the grouped convolutions.
\section{Introduction}
\input{core/intro}
\section{Benchmarking Framework}
\input{core/method}
\label{sec:method}
\section{Experiments}
\input{core/exps}
\label{sec:exps}
\section{Conclusion}
\input{core/conc}
\label{sec:conc}
\bibliographystyle{IEEEbib}
|
{
"timestamp": "2019-10-21T02:01:17",
"yymm": "1803",
"arxiv_id": "1803.02758",
"language": "en",
"url": "https://arxiv.org/abs/1803.02758"
}
|
\section{Introduction}
\IEEEPARstart{W}{ith} the progress of image processing and pattern recognition techniques, computer-assisted diagnosis(CAD) has been widely utilized to assist medical professionals to interpret medical images. Digital pathology,as an important aspect of CAD application, is earning more and more attention from both image analysis researchers and pathologists due to the advent of whole-slide imaging. Its aim is to acquire, manage and interpret pathology information generated from digitized glass slides, among which the development of computational algorithms to automatically analyze digital tissue images is the key. The potential applications of digital pathology span a wide range such as segmentation of desired regions or objects, counting normal or cancel cells, recognizing tissue structures, classifying cancer grades, prognosis of cancers, etc \cite{pmid24759275,pmid28066683}. It is able to dramatically decrease human's workload and has the potential to work better than pathologists due to its objectiveness in the interpretation.
As an essential part of digital pathology, histopathology image analysis is playing increasingly important role in cancer diagnosis, which can provide direct and reliable evidence to diagnose the grade and type of cancer. This paper deals with nuclei segmentation, an important step in histopathological image analysis. The purpose of nuclei semgentation is not only counting the number of nuclei but also obtaining the detailed information of each nucleus. So unlike nuclei detection, here the outputs are the contour of each nucleus instead of only the position of their central points. Hence we can exactly extract each nucleus from the image and make it available for further analysis. For example, the features of the individual nucleus and the distribution of nuclei clusters can be used to grade and classify status of breast cancers \cite{nawaz2015computational,chen2015new}. Because of appearance variation such as color, shape, and texture, nuclei segmentation from histopathological images could be very challenging, as illustrated in Fig.\ref{fig:image_sample}, in which it is very challenging even for human to recognize and segment all nuclei within the images. Fig.\ref{colon} and Fig.\ref{prostate} illustrate two histopathological images from different organs.
Fig.\ref{breast_cancer1} and Fig.\ref{breast_cancer2} are two histopathological images from same organ but have different cancer grade.
\begin{figure}[hbp]
\begin{center}
\begin{subfigure}[t]{0.20\textwidth}
\includegraphics[width=3.5cm]{./figures/calon_sample.png}
\caption{}
\label{colon}
\end{subfigure}
\begin{subfigure}[t]{0.20\textwidth}
\includegraphics[width=3.5cm]{./figures/prostate_sample.png}
\caption{}
\label{prostate}
\end{subfigure}
\begin{subfigure}[t]{0.20\textwidth}
\includegraphics[width=3.5cm]{./figures/breast_cancer_1.png}
\caption{}
\label{breast_cancer1}
\end{subfigure}
\begin{subfigure}[t]{0.20\textwidth}
\includegraphics[width=3.5cm]{./figures/breast_cancer_3.png}
\caption{}
\label{breast_cancer2}
\end{subfigure}
\end{center}
\caption{(a)Colon cancer (b)Prostate cancer (c)Breast cancer (grade I) (d)Breast cancer(grade III)}
\label{fig:image_sample}
\end{figure}
The study of nuclei segmentation in histopathological images can be traced back to 10 years ago. A large number of methods have been proposed to pursue accurate segmentation on images of a variety of categories. The procedure of most traditional nuclei segmentation methods can be divided into two separate steps: first,detecting the nuclei and then obtaining each nucleus' contour. The detection step is expected to generate the area of nuclei or the seed of each nucleus. One popular and convenient method to detect the nucleus is intensity thresholding as used in Otsu's method\cite{otsu1975threshold}. However, it has an obvious limitation: it only works under the scenario that all the nuclei in the images have consistent intensity differences against the background. Another popular approach for nuclei detection is clustering including K-mean clusting\cite{filipczuk2011automatic}, Grab Cut\cite{rother2004grabcut} and etc. Furthermore, a few filtering based on methods have been proposed by utilizing the features of the nuclei\cite{al2010improved, veta2013automatic}. All of above methods have one common weakness: they are only effective for one or a few specific types of nuclei or images and are usually highly sensitive to manually set parameters. Since the appearances of nuclei are so diverse that we can hardly develop a single model or method suitable for all these different images. In recent years, supervised learning based approaches are becoming more and more attractive. They classify each pixel into one of two categories: nuclei or background\cite{mouelhi2013automatic,xu2016stacked,sirinukunwattana2016locality}. After the nuclei detection stage yields the nuclei area, the next step would be splitting the touching and overlapped nuclei areas. This could be achieved by methods such as bottleneck detection\cite{liao2016automatic} and ellipse fitting\cite{su2014automatic,kharma2007automatic}. If the seed of a nucleus is generated, its contour could be obtained by using marker controlled watershed\cite{veta2013automatic, qu2014two} or region growing\cite{xing2016automatic}.
Recently, deep learning based methods are becoming increasingly popular in image segmentation due to their dominating performance in many tasks of computer vision. They have significantly impacted all the research areas in computer vision such as object classification, object detection and segmentation. Since 2014, numerous convolutional neural network based image segmentation methods have been proposed. Long. et al firstly introduced
fully convolutional neural network (FCN)\cite{long2015fully} to semantic segmentation. Compared to prior models, it is demonstrated that the FCN algorithm is much more efficient and accurate. Converting fully connected layers into convolutional neural networks makes it possible to predict the heatmap of the objects in the image that needs to be segmented. U-net \cite{ronneberger2015u}, an FCN based network architecture won the Grand Challenge for Computer-Automated Detection of Caries in Bitewing Radiography at ISBI 2015. Later, a skip-architecture first introduced in residual networks \cite{he2016deep} is also applied to fuse different levels of semantic information.
Inspired by the U-net algorithm\cite{ronneberger2015u}, we propose to apply the FCN network to the nuclei segmentation problem. Current deep learning methods for nuclei segmentation usually need complex post-processing procedure to obtain the final nuclei boundries. Naylor \cite{naylor2017nuclei} employs FCN to discriminate the nuclei from background and then applies the watershed method to split the nuclei. However the resulting boundaries are not accurate. Xing \cite{xing2016automatic} proposed a sophisticated shape deformation method to generate each nucleus's boundary. Kumar\cite{kumar2017dataset} designed a CNN3 model to predict the nuclei and its boundary from the image. But a time consuming post-processing step is needed. Here we designed an end-to-end fully convolutional neural network architecture for nuclei segmentation. Unlike prior binary classifiers \cite{mouelhi2013automatic,xu2016stacked,sirinukunwattana2016locality}, which only discriminate nuclei against the background, our nuclei-boundary segmentation model predicts the nuclei and their contours at the same time. Due to the accurate prediction of nucleus and boundary in our approach, the final segmentation can be generated by a simple and fast post-processing procedure. To segment the whole-slide image, a pixel-wise segmentation strategy is necessary. However the border area of each patch cannot be predicted accurately because of lacking contextual information. A seamless patch extraction and assembling method is proposed to handle this problem.
The main contributions of this paper are as follows:
\begin{itemize}
\item We propose a nuclei-boundary model to explicitly detect nuclei and their boundaries simultaneously from histopathology images. Detecting boundary is able to improve the accuracy of nuclei detection and help split the touched and overlapped nuclei. Given the raw segmentation results by our nuclei-boundary model, only a simple dilation operation and noise removing steps are needed to produce the final segmentation results.
\item We develop an effective approach to segment extra-large high-resolution images that U-net cannot handle due to limited GPU memory using a seamless patch-wise segmentation. A weighted loss map is utilized to train the model and a vote mechanism is used to assemble the patches.
\item Extensive studies on the effects of a variety of data augmentation methods for nuclei segmentation are provided.
\item We introduce four evaluation criteria for more accurate nuclei segmentation performance evaluation: missing detection rate, false detection rate, under-segmentation rate, and over-segmentation rate. They are designed to help the pathologist obtain more in-depth understanding of the performance of automatic segmentation methods and choose the right one for their specific application.
\end{itemize}
\section{Method}
\subsection{Overview}
Our nuclei segmentation method adopts an end-to-end deep learning framework. The only preprocessing procedure is image color normalization. In the training phase, without extracting any features, even the H-channel, we directly apply the histopathology images in normalized RGB colors to the deep neural network to train the nucleus-boundary model. During the testing phase, the prediction result of raw normalized images yielded by the nucleus-boundary detector shows clear inside nuclei area and the boundaries. At last, we will obtain the area of each nucleus via a simple, fast and parameterless post-processing procedure. Fig.\ref{fig:overview} shows the procedure to segment nuclei from color normalized images in our algorithm.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.9\linewidth]{./figures/framework.png}
\end{center}
\caption{The overview of segmenting nuclei on histopathological images.}
\label{fig:overview}
\end{figure}
\subsection{Data Preprocessing}
H\&E stain is the most widely used stain protocol in medical diagnosis. Typically, the nuclei of cells are stained to blue by Haematoxylin while cytoplasm is colored to pink by Eosin. But in practice, the color of H\&E stained images could vary a lot due to variation in the H\&E reagents, staining process, scanner and the specialist who performs the staining, as shown in Fig.\ref{fig:image_sample}. A few H\&E stain normalization methods\cite{vahadane2015structure, khan2014nonlinear, macenko2009method} have been proposed to eliminate the negative interference caused by color variation. We tried two of them\cite{vahadane2015structure, macenko2009method} to normalize the raw H\&E stained images. For our segmentation algorithm, we did not find any considerable difference between these two normalization methods. Particularly, the result shown in experiment section \ref{Experiment} is generated based on the images normalized by the method in \cite{vahadane2015structure}. Given a target image, this method is able to convert one image's color into the target image's color space based on sparse non-negative matrix factorization(NMF). We choose one best stained H\&E image as the target and convert other images into its color space. According to the recommendation in \cite{vahadane2015structure}, the hyper-parameter $\lambda$ should be set between 0.01 and 0.1. In our experiment, $\lambda$ is set to 0.1.
Intuitively, the pure Haematoxylin-channel grayscale image would be much easier than RGB images to distinguish the foreground (nuclei) from the background (cytoplasm). A large number of nuclei segmentation methods\cite{cui2016self, qu2014two, wang2016automatic} employ some deconvolution algorithms to extract the H-channel from H\&E stained images. However, based on our experiments, we noticed that our deep fully convolutional neural network extracts the nuclei from raw RGB images better than from H-channel grayscale images. The reason would be that the H-channel might miss some information that might be helpful for distinguishing nuclei and the cytoplasm. Given well-labelled training images, the deep neural network can then learn the optimal way to extract the features that discriminate between each category of samples. So we skip extracting H-channel and directly apply the RGB color image as the input to our deep neural network.
\subsection{Nucleus-boundary model}
Traditional supervised nuclei segmentation methods usually apply a binary classifier to segment the nuclei area and the background area by classifying each pixel. These methods usually predict the category of the central pixel given a small patch. To segment the whole image, it needs to extract all the sliding windows(patches) with stride of 1 pixel and predict each of these patches. The most limitation of this strategy is high computational complexity. For example, if there is an image of size 1000X1000 pixel, this method needs to process one million sliding windows in order to segment this single image. Nevertheless, the typical whole slide of histopathology image may have billions of pixels, making it impossible to process it in an acceptable time using this strategy. Instead, our method is based on fully convolutional network (FCN) framework, which allows predicting the category of all the pixels of an image with only one pass. The input of the network is one image, the output is the estimated class map.
The task of nuclei segmentation can be roughly divided into two stages: the first stage is extracting the foreground(nuclei), the second stage is segmenting the connected foreground area into separated nuclei and finding out the boundary of each nucleus. Our method intends to merge these two steps by extracting the nuclei and their edges at the same time. That is the reason why it is named "nuclei-boundary(NB) model". As shown in Fig.\ref{fig:unet}, the output of the NB model has three channels, each has the same height and width with the input image. Its values represent the probabilities of each pixel being $background$, $boundary$ or $inside$ class, respectively. The manual annotation for our segmentation problem is the boundary of each nucleus. A pixel belonging to the $boundary$ class means that it is on or inside an annotated boundary and within 2 pixel from the boundary. Pixels of the $inside$ class are those that are inside annotated boundary but are not $boundary$ pixels. Correspondingly, the output can be regarded as an RGB image and the estimated maps of the $background$, $boundaries$ and $nuclei$ are represented by red, green and blue, respectively, as shown in Fig.\ref{fig:unet}. To generate the ternary mask for training, we apply a morphology operator to each nucleus to obtain the $inside$ pixels, and then subtract $inside$ pixels from the nucleus to get $boundary$ pixels.
\label{dense-unet}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=.8\linewidth]{./figures/unet.png}
\end{center}
\caption{The structure of our network. The size of each layer is shown in $height * width * channels$. The height and width of each layer are not fixed, which are determined by the size of input images. Here we assume the input image is of size $128 * 128$.}
\label{fig:unet}
\end{figure*}
\subsubsection{The architecture of our NB network}
Fig.\ref{fig:unet} shows the network architecture of our algorithm, which consists of a couple of encoding and decoding layers. The encoding layers are used to extract different levels of contextual feature maps. The decoding layers are designed to combine these feature maps produced by the encoding layers to generate the desired segmentation maps.
Due to the memory limitation of our GPU, the size of the input layer is set to 128X128 in our experiments. But we noticed that larger input layer may lead to better performance.
The weight of each convolutional layer is initialized by glorot uniform\cite{glorot2010understanding} and bias is initialized to 0. The glorot uniform is defined as:
\begin{equation}
W \sim U\left[ \frac { -\sqrt { 6 } }{ \sqrt { { n }_{ j } + { n }_{ j+1 } } } ,\frac { \sqrt { 6 } }{ \sqrt { { n }_{ j }+{ n }_{ j+1 } } } \right]
\label{eq:init}
\end{equation}
where $W$ means the initialized weight, $n_{j}$ means the size of the convolutional layer $j$.
The scaled exponential linear units(SELUs) \cite{pmid29259059} activation function is used in all convolutional layers. SELUs is designed to make the forward neural network(FNN) to have self-normalizing capability\cite{klambauer2017self}. The FNN using SELUs are shown to be able to outputperform the ones using explicit normalization methods, such as batch normalization, layer normalization, and weight normalization. This is why our network does not have any normalization layers. The selu activation function is defined as:
\begin{equation*}
selu(x)=\lambda \begin{cases} x\quad \quad \quad \quad \quad \quad if\quad x\quad <\quad 0 \\ \alpha { e }^{ x }\quad -\quad \alpha \quad if\quad x\quad \le \quad 0\quad \end{cases}
\label{eq:selu}
\end{equation*}
where
$\lambda = 1.0507$ and $\alpha = 1.6733$. The padding property of each convolutional layer is the 'same' in order to ensure it keeps the same size with its previous layer. The size of all convolutional filters is $3 X 3$. Each convolutional layer is followed by a dropout layer with 0.2 drop rate. The network is trained by Adam optimizer\cite{kingma2014adam}. This stochastic optimization method is able to compute adaptive learning rate for each parameter. It automatically controls the learning rate along the training, so it is not necessary to manually set the momentum and decay.
\subsubsection{Data Augmentation}
\label{data_augment}
Deep learning models often have millions of parameters so that it needs large-scale sample dataset to avoid the overfitting problem. In fact, the datasets of our nuclei segmentation task often contain only tens of images. Moreover, labeling an 1000*1000 image which contains hundreds of nuclei usually cost a specialist at least 5 hours. Hence it is impossible to manually label sufficient and nuclei boundaries accurately for training deep learning models. Data augmentation is an essential approach to overcome the over-fitting problem caused by lacking samples. The training samples, \textit{i}.\textit{e}. the patches, are randomly extracted from the H\&E stained images in the training datasets. Five augmentation techniques are used together in our experiments including random elastic transformation, rescale, affine transformation, shift, flip and rotate. Each training sample(one patch extracted from a whole image) as well as the corresponding target are processed by the data augmentation procedure.
Given a training sample, which is a RGB image $I$ with its corresponding ground truth $I_{gt}$, we transform $I$ to $I^{'}$ and $I_{gt}$ to $I_{gt}^{'}$.
$I^{'}$ and $I_{gt}^{'}$ are the real input and target of the neural network. The rescaling factors are set as a random number between 0.5-1.5.
We employ Simard's method\cite{simard2003best} to do elastic transforming. Two hyper-parameters $\alpha$ and $\sigma$ need to be manually set to control how dramatic the original image is transformed. In our experiment, $\alpha$ is set to a random number between 100-200, $\sigma$ is set to 12.
Besides transforming the input sample, it is necessary to do the same transformation on targets to maintain consistency. The one-hot encoding target consists of only binary values. However, the transformed target has some float-point numbers caused by bilinear interpolation we used for elastic transformation. They need to be binarized by the following rules:
Let the value of one pixel is $(t_{i}, t_{b}, t_{o})$, where $t_{i}$, $t_{b}$ and $t_{o}$ represents the label for $inside$, $boundary$ and $background$ respectively.
1. if $t_{b} > 0.5$, $t_{b} = 1$, else $t_{b} = 0$
2. if $t_{i} > 0 $and $t_{b} == 0$, $t_{i} = 1$, else $t_{i} = 0$
3. if $t_{i} == 1 $or $t_{b} == 1$, $t_{o} = 1$, else $t_{o} = 0$
An example of data augmentation is illustrated in Fig.\ref{fig:augment}.
\begin{figure}[!]
\begin{center}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4.5cm]{./figures/img_orig.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4.5cm]{./figures/gt_orig.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4.5cm]{./figures/img_deform.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4.5cm]{./figures/gt_transform.png}
\caption{}
\end{subfigure}
\end{center}
\caption{Example of data augmentation, (a) one patch extracted from a normalized image (b) corresponding ground truth of (a). (c) A training sample generated by data augmentation procedure based on patch(a). (d) the corresponding ground truth of (c).}
\label{fig:augment}
\end{figure}
\subsubsection{Weighted loss}
The U-net\cite{ronneberger2015u} model tends to predict the pixels with full context in the input image, which leads to generation of a smaller segmentation map than the input image. The border area of the input image is not predicted because of lacking enough context information. This strategy can solve the problem that the prediction of the border area is not accurate to some extent. One issue of this is that this U-net defines a fixed-size border area whose size is not changable without modifying the network structure. However, in practice, the border area size could vary in different histopathological images and it mainly depends on the size of the nuclei. Another limitation is that we have to do some cropping operation in neural network training to make the size of layers match each other, which might lose useful surrounding information.
As a trade-off of these issues, we designed a weighted loss and a scheme for patch extraction and assembling to allow the neural network to predict an segmentation map of equal size without concerning the lack of context issue in the border area.
The model is trained by minimizing the categorical softmax cross-entropy loss between predictions and targets, which is described in eq.\ref{eq:loss}:
\begin{equation}
L=\sum _{ i }^{ }{ \sum _{ j }^{ }{ { W }_{ i,j }log({ p }_{ t(i,j) }(i,j)) } }
\label{eq:loss}
\end{equation}
where
$t(i,j)$ denotes the true label of the pixel at (i,j) position, $p_{t(i,j)}(i,j)$ is the output of soft-max activation layer which indicates the probability of the pixel at (i,j) being $t(i,j)$. $W$ is the proposed weight map, which is defined as:
\begin{equation}
\begin{aligned}
&{ W }_{ i,j }={ \alpha \frac { { D }_{ i,j }^{ e } }{ { (D }_{ i,j }^{ c }+{ D }_{ i,j }^{ e }) } }\\
&{ \alpha =\frac { h\cdot w }{ \sum _{ i=1 }^{ h }{ \sum _{ j=1 }^{ w }{ \frac { { D }_{ i,j }^{ e } }{ { D }_{ i,j }^{ c }+{ D }_{ i,j }^{ e } } } } } }
\end{aligned}
\label{eq:loss_weight}
\end{equation}
where $W_{i,j}$ is the weight of position i, j,
, $D_{i,j}^{e}$ is the distance from border, $D_{i,j}^{c}$ denotes the distance from center.$h$ and $w$ are the height and width of the map, respectively.
\begin{figure}[hbp]
\begin{center}
\includegraphics[width=.4\linewidth]{./figures/loss_map.png}
\end{center}
\caption{The weighted loss map generated by Eq.\ref{eq:loss_weight}}
\label{fig:weight}
\end{figure}
\subsubsection{Extra-large Image Segmentation Using Overlapped Patch Extraction and Assembling}
Current medical image segmentation algorithms based on U-net and its derivatives has an unsolved problem for segmenting extra-large high-resolution histopathological images: due to the limited memory of the GPU, it is possible to feed the whole slide image into the deep neural network. It has to be cut into patches and perform patch-wise training and prediction. However, there is no reported solution for deal with this issue.
With close examination, we found the the main issue of U-net algorithm on patch-based segmentation is that the prediction at the border area is not accurate as demonstrated in \ref{fig:loss_weight_compare}. Here we propose an overlapped patch extraction and assembling method. The patches are extracted by sliding window with a stride. For assembling, a vote mechanism is applied to predict each pixel using
\begin{equation*}
P(i,j)\quad =\quad \frac { \sum _{ k }^{ }{ { W }_{ k(i,j) }p(k(i,j)) } }{ \sum _{ k }^{ }{ { W }_{ k(i,j) } } }
\label{eq:assembling}
\end{equation*}
where $P(i,j)$ is the final prediction of the pixel at position (i,j) in an image. $k(i,j)$ means the position of
it in the $kth$ patch.
\subsubsection{Post-processing}
From Fig.\ref{fig:result_step}, we can see that the raw prediction results already show clear $inside$ nucleus areas and boundaries. Due to this reliable prediction results, we no longer need the complex region growing algorithms \cite{kumar2017dataset, xing2016automatic} and splitting algorithms \cite{wang2016automatic} to extract the final segmented areas. These methods usually strongly rely on manual parameter tuning to get good performance and is computationally demanding. Instead, we use a parameter-free postprocessing procedure that runs in a negligibly short time. Since our NB model detects both $inside$ and $boundary$ classes, all we need is the $inside$ class map. Then the $inside$ class map is transformed to a binary map using a constant threshold $0.5$.
In this way, each connected component in the binary image indicates the $inside$ area of one nucleus. At the end, in order to recover the shape, we can simply apply the dilation operation to each connected component.
\section{Experiment}
\label{Experiment}
\subsection{Evaluation criteria}
Two level of criteria are usually used to measure the performance of nuclei segmentation methods: one is object-level criteria, another is pixel-level criteria.
The most common object-level criteria for object detection tasks include $precision$, $recall$, $F1 score$. $precision$ is defined as:
\begin{equation*}
precision\quad =\quad \frac { TP }{ TP\quad +\quad FP }
\end{equation*}
$recall$ is defined as:
\begin{equation*}
recall\quad =\quad \frac { TP }{ FN\quad +\quad TP }
\end{equation*}
$F1 score$ considers both of the $precision$ and $recall$, as shown in following equation.
\begin{equation*}
F1=2\cdot \frac { precision\cdot recall }{ precision+recall }
\end{equation*}
where the $TP$ is true positives, $FP$ means false positives and $FN$ means false negatives. Given a manually labelled ground truth nucleus $T_{i}$, if there is one nucleus $P_{j}$ in automatic segmentation result that matches $T_{i}$, $P_{j}$ can be counted as one $TP$.
$F1 \quad score$ is the harmonic average of $precision$ and $recall$ and its value is in the range of [0,1].
We noticed that $FN$ can be caused by two different types of errors: one is miss-detection(nuclei is predicted as cytoplasm), another is under-segmentation(Multiple ground truth nuclei are detected as one nucleus, hence only one of these nuclei ground truth nucleus has corresponding detected nucleus.).Similarly, $FP$ consists two types of errors: one is false detection (Cytoplasm is detected as nuclei), another is over-segmentation(One ground-truth nucleus is segmented into several nuclei. Each of them is a part of the ground truth nucleus and at most only one among them can be considered as the corresponding detected nucleus).
Let us think about this situation: one segmentation method is weak on discriminating the nuclei and cytoplasm while another one is weak on splitting the nuclei area.
But they may have similar $precision$ and $recall$, even $F1 score$. Apparently, $precision$, $recall$, $F1 score$ and their combination fail to differentiate the performance of these two segmentation methods. To handle this issue, we introduce four new criteria to evaluate automatic nuclei segmentation methods: missing detection rate(MDR), false detection rate(FDR), under-segmentation rate(USR), over-segmentation rate(OSR), as shown in Eq. \ref{eq:1}.
\begin{align}
&MDR=\frac { MD }{ FN\quad +\quad TP } \nonumber\\
&FDR=\frac { FD }{ TP\quad + \quad FP } \nonumber\\
&USR=\frac { US }{ P }\nonumber\\
&OSR=\frac { OS }{ S } \label{eq:1}
\end{align}
where $MD$ is the number of missing detections, $FD$ indicates the number of false detections, $US$ means the number of nuclei which are not detected caused by undersegmentation. $P$ is the number of ground truth nuclei in the region of $TP$, which can be defined as $Fn + TP - MD$. $OS$ means the number of false positives caused by oversegmentation
and $S$ means the number of segmented nuclei in the region of $TP$'s corresponding ground truth nuclei, which can be defined as $FP + TP + FD$.
The combination of $MDR$ and $FDR$ measures the capacity of discriminating the nuclei and cytoplasm while the combination of $USR$ and $OSR$ measures the performance of handling overlapped nuclei area. On the other hand, $recall$ value is negatively correlated with $MD$ and $USR$ while $precision$ is negatively correlated with $FDR$ and $OSR$. These four criteria are able to help pathologists to select proper automatic segmentation methods for specific tasks.
The pixel-level criteria are used to measure the accuracy of a segmentation algorithms in predicting the shape and size of the detected nuclei. The most essential one is Dice's coefficient, which is defined as:
\begin{equation}
D(X,Y)=2\frac { \left| X\cap Y \right| }{ \left| X \right| +\left| Y \right| }
\end{equation}
where $X$ indicates a manual segmentation and $Y$ means its corresponding automatic segmentation. That is, a manual segmentation is considered as a FP if there is no corresponding automatic segmentation with a Dice coefficient of at least 0.2.
\subsubsection{Datasets}
We evaluate the performance our method on three public available nuclei segmentation datasets. One is a multiple organ H\&E stained image dataset\cite{kumar2017dataset}(MOD). It consists of 30 images which were captured from 7 organs: breast, liver, kidney, prostate, bladder, colon and stomach. The resolution of each image is 1000X1000. Totally, about 21,000 nuclear boundaries are manually annotated. These 30 images are split into two subsets: the training set with 16 images composed of 4 from breast, 4 from liver, 4 from kidney and 4 from prostate and the test set with 14 images composed of 2 images from each organ.
The second dataset is the breast cancer histopathology image dataset(BCD). It contains two subsets: subsetA and subsetB. SubsetA includes 21 images and subsetB has 18 images. In \cite{veta2013automatic}, SubsetA is used to tune the parameters. In a similar way, we utilize subsetA as the training set and subsetB as the test set. Since one image may contains thousands of nuclei, it is impractical to manually label all the training images. We randomly select five images from subsetA and crop a 1000*1000 subimage from each of them to build the training set. It is manually annotated under the supervision of a specialist.
The third one is also a breast cancer image dataset(BNS)\cite{naylor2017nuclei}. It is composed of 33 H\&E stained images of size 512X512 from 7 triple negative breast cancer patients. There are totally 2754 manually annotated nuclei.
\subsection{Experiment result}
Figure \ref{fig:result_step} shows how our method segments the nuclei step by step. The color variety is well controlled by the color normalization procedure. The prediction result shows clear nuclear areas and nucleus boundaries. In the final segmentation result and ground truth image, each nucleus is represented by a different color.
\begin{figure*}[htbp]
\begin{center}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/orig_1.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/norm_1.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/prediction_1.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/result_1.png}
\end{subfigure}
\hspace{3mm}
~
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/orig_2.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/norm_2.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/prediction_2.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/result_2.png}
\end{subfigure}
\hspace{3mm}
~
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/orig_3.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/norm_3.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/prediction_3.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/result_3.png}
\caption{}
\end{subfigure}
\caption{(a) examples of original histopathology images; (b) corresponding images after color normalization. (c) raw segmentation results by our algorithm. (d) final segmentation result.}
\label{fig:result_step}
\end{center}
\end{figure*}
First, We test our method on the MOD dataset. Unfortunately, the dataset publicly provided online doesn't explicitly divide the whole dataset into the training set and test set. We do not know which image belongs to the training set exactly as introduced in their paper \cite{kumar2017dataset}. To make a fair comparison, we randomly select 16 images from breast, liver, kidney and prostate. Then we combine the remaining 8 images of these four types and the 6 images from bladder, colon and stomach to build the test images. 12000 patches are randomly extracted from 12 training images to train our model. To eliminate the bias caused by random selection, 5 different training sets and the corresponding test sets are randomly generated. Then the model is trained and tested on the 5 pairs of training set and test set separately. All of the models are trained for 300 epoch in 7.5 hours. For testing, the stride of overlapped patch extraction is set to 64. The quantitative comparison is listed in Table \ref{table:1}, which demonstrates that our method outperforms the state-of-the-art method CNN3 as reported in \cite{kumar2017dataset} in terms of both F1 score and Dice's Coefficient. Moreover, it shows that the under-segmentation error is much more significant than over-segmentation error and it achieves a balance between the false detection error and missing detection error.
Figure \ref{fig:compare_TMI} shows a visual comparison between our method and \cite{kumar2017dataset}. As shown in the sample images, our segmentation result has fewer false negatives and higher accuracy in terms of nuclei boundaries than \cite{kumar2017dataset}. Our method is not only more accurate but also much faster. It takes about 5 seconds to predict a 1000 * 1000 image by one Nvidia Titan X GPU and the time used for post-processing is less than 0.1 seconds. Given the same hardware environment and test images, \cite{kumar2017dataset} takes about 4 minutes to predict one image and 80 seconds to do the post-processing.
Additionally, a 10-folder cross-validation is performed to validate our method. The result is shown in Table \ref{table:1} NB model *.
\begin{figure*}[htbp]
\begin{center}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/orig_4.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/gt_4.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/TMI_4.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/result_4.png}
\end{subfigure}
\hspace{3mm}
~
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/orig_5.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/gt_5.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/TMI_5.png}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/result_5.png}
\end{subfigure}
\hspace{3mm}
~
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/orig_13.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/gt_13.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/TMI_13.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/result_13.png}
\caption{}
\end{subfigure}
\caption{The comparison between our method and CNN3\cite{kumar2017dataset}. (a): raw images; (b):ground truth; (c): CNN3 results; (d): our results }
\label{fig:compare_TMI}
\end{center}
\end{figure*}
\begin{table*}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
methods & precision & recall & F1 & Dice's Coefficient &
MDR & FDR & USR & OSR \\
\hline
\hline
CNN3 \cite{kumar2017dataset} & - & - & 0.827 & 0.762 & - & - & - & - \\
\hline
NB model 1(our method) & 0.813 & 0.914 & 0.854 & 0.812 & 0.09 & 0.09 & 0.09 & 0.01\\
\hline
NB model 2 & 0.861 & 0.856 & 0.846 & 0.808 & 0.05 & 0.13 & 0.08 & 0.03\\
\hline
NB model 3 & 0.880 & 0.864 & 0.854 & 0.818 & 0.07 & 0.11 & 0.05 & 0.03\\
\hline
NB model 4 & 0.812 & 0.925 & 0.861 & 0.805 & 0.09 & 0.07 & 0.09 & 0.01\\
\hline
NB model 5 & 0.814 & 0.910 & 0.846 & 0.803 & 0.10 & 0.08 & 0.10 & 0.01\\
\hline
NB model * & 0.845 & 0.892 & 0.850 & 0.81 & 0.06 & 0.11 & 0.02 & 0.08\\
\hline
\end{tabular}
\end{center}
\caption{Quantitative comparison results of segmentation performance on MOD dataset. }
\label{table:1}
\end{table*}
To show the benefit of our proposed evaluation metrics for nuclei segmentation, we compared the performance of our algorithm and the baseline CNN3 over two images with similar precision and recall, but different segmentation quality. As shown in Fig. \ref{fig:OSR}, the CNN3 algorithm got similar precision and recall scores on these two images.
From our proposed criteria, we can find that the segmentation error on the first image is mainly caused by under-segmentation and false detections while that it is mainly caused by oversegmentation, missing detection and false detection in the second image. This observation can be verified by the sample segmentation result.
\begin{figure}[htbp]
\begin{center}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/orig_15.png}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/gt_15.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/result_15.png}
\end{subfigure}
~
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/orig_16.png}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/gt_16.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/result_16.png}
\end{subfigure}
\caption{Cropped portions of two images. (a) precision = 0.76 recall = 0.83, OSR = 0.05 USR = 0.15 MDR = 0.02 FDR = 0.20 (b) precision = 0.78 recall = 0.83 OSR = 0.13 USR = 0.05 MDR = 0.12 FDR = 0.10}
\label{fig:OSR}
\end{center}
\end{figure}
Second, we test our method on the BCD dataset.The manually labeled training set consists of five 1000*1000 images. Instead of training the models from random initialization, we use the training data to fine-tune the network model trained on the MOD dataset. Thus the model would adjust to a new dataset with much shorter time by training on a limited training set for a small number of epochs. In this experiment, only 2000 patches are extracted to fine-tune the pre-trained model. It takes about 10 seconds to train one epoch and the training is terminated after 70 epochs.
Figure \ref{fig:result1} shows the visual comparison between our algorithm and algorithm in \cite{veta2013automatic} in terms of segmentation results.
At last, we follow the same strategy in \cite{naylor2017nuclei} to validate our method. The strategy is called leave-one-patient-out cross-validation. That is every time we train the model on 6 patient and use the rest one for validation.
Table \ref{table:2} shows that our method outperforms the state-of-the-art breast cancer nuclei segmentation method by a large margin in terms of precision, recall and F1 score.
\begin{figure*}[htbp]
\begin{center}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/orig_6.png}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/plos_6.png}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/result_6.png}
\end{subfigure}
\hspace{3mm}
~
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/orig_7.png}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/plos_7.png}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/result_7.png}
\end{subfigure}
\hspace{3mm}
~
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/orig_11.png}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/plos_11.png}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/result_11.png}
\end{subfigure}
\hspace{3mm}
~
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/orig_12.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/plos_12.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[width=5.5cm]{./figures/result_show/result_12.png}
\caption{}
\end{subfigure}
\caption{Nuclei segmentation result over the BCD dataset. (a) two breast cancer image samples. (b) automatic segmentation result of \cite{veta2013automatic}. (c) result of our method.}
\label{fig:result1}
\end{center}
\end{figure*}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
dataset & method & precision & recall & F1 & DC\\
\hline
\hline
BCD & Veta's method\cite{veta2013automatic} & 0.863 & 0.886 & 0.874 & ~0.88 \\
\hline
& TV-MRF-BP \cite{paramanandam2016automated} & 0.801 & 0.823 & 0.811 & 0.84 \\
\hline
& NB model & 0.942 & 0.915 & 0.923 & 0.862 \\
\hline
BNC & FCN\cite{long2015fully} & 0.823 & 0.752 & 0.763 & - \\
\hline
& DeconvNet\cite{noh2015learning} & 0.864 & 0.773 & 0.805 &-\\
\hline
& Ensemble\cite{naylor2017nuclei} & 0.741 & 0.9 & 0.802& -\\
\hline
& NB model & 0.920 & 0.7835 & 0.84 & 0.83\\
\hline
\end{tabular}
\end{center}
\caption{Quantitative comparison of segmentation performance on the BCD dataset}
\label{table:2}
\end{table}
\subsection{Discussion}
\subsubsection{Data augmentation for fully convolutional networks }
Data augmentation is a widely used technique to handle the overfitting issue caused by limited training samples. In image segmentation tasks, one can generate more images from one image using image transformation methods. The most common methods include rotation, flipping, shifting and rescaling. Elastic deformation transform, a higher level transformation method, is also employed in some image segmentation works. Ronneberger \textit{et al.} \cite{ronneberger2015u} claim that elastic deformation is the key method to do data augmentation for a segmentation network with very limited annotated images.
However, to the best of our knowledge, there is no systematic study of the effectiveness of these image transformation methods for nuclei segmentation using a fully convolutional network. We compare different training processes using rotation, flipping, shifting, rescaling and elastic deformation transform to augment the training data. To make fair comparisons, we let the training set and validation set have similar appearances by splitting each whole image into two sub-images and placing one in the training set and another one in the validation set. We randomly extract 6000 patches from the training set to train our neural networks and 6000 patches from the validation set for validation. The setting of these transformation methods is same with those reported in section \ref{data_augment}. The comparison is shown in Fig.\ref{fig:overfitting}. 'no' means don't apply data augmentation. 'combination' means data augmentation is performed by combining elastic deformation, flip, rotate, shift and rescale. It is very clear that without data augmentation, the network has severe overfitting issue, validation loss starts to increase rapidly from epoch 5. Unexpectedly, rotating rather than elastic deformation has achieved the best performance in performance improvement. But only rotating operation still cannot prevent the overfitting. One has to combine all of these transform methods together to do data augumentation to get good performance as done in this paper.
\begin{figure}[!ht]
\begin{center}
\begin{subfigure}[t]{0.40\textwidth}
\includegraphics[width=6cm]{./figures/data_augmentation.jpg}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.40\textwidth}
\includegraphics[width=6cm]{./figures/data_augmentation_1.jpg}
\caption{}
\end{subfigure}
\end{center}
\caption{(a) shows how the training loss changes during training. (b) indicates the validation loss.}.
\label{fig:overfitting}
\end{figure}
\subsubsection{Nuclei Segmentation on Extra-large Images}
To evaluate the effectiveness of the proposed weight map and overlapped patch extraction and assembling method for extra-large image segmentation, we compared the segmentation results with and without the proposed method in Fig. \ref{fig:loss_weight_compare}. We can see that the raw segmentation results without using those two techniques contain obvious seams between the patches. It also demonstrates that the predictions in the border area is not accurate. As shown in Fig. \ref{fig:loss}, if we employ the overlapped patch extraction and assembling but without the weight map (which means all the pixels in a patch have the same weight) the segmentation result still shows noticeable seams. Fig.\ref{fig:pred} and Fig. \ref{fig:loss} has the same stride, which is 64.
\begin{figure}[!h]
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/orig_10.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/predict_10.png}
\caption{}
\label{fig:pred}
\end{subfigure}
~
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/loss_10.png}
\caption{}
\end{subfigure}
\begin{subfigure}[t]{0.24\textwidth}
\includegraphics[width=4cm]{./figures/result_show/loss_10_1.png}
\caption{}
\label{fig:loss}
\end{subfigure}
\caption{(a) shows an H\&E stained image. (b) shows the raw segmentation results of our method. (c) shows the prediction result without applying weight map and overlapped patch extraction and assembling. (d) shows the prediction results using overlapped patch extraction and assembling but without weight map.}
\label{fig:loss_weight_compare}
\end{figure}
\subsubsection{NB model versus the mixed nucleus model + boundary model}
\label{double}
An alternative way to detect nuclei and their boundaries is training two binary classifiers to detect $inside$ and the boundary separately and then merge the detections together. We apply the same method with our NB model to train the nucleus model and boundary model except that the three-class classification is replaced by binary classification.
Fig. \ref{fig:three_two} depicts why the NB model outperforms the mixed nucleus model + boundary model. The NB model is able to learn the latent relationships between $inside$, $boundary$ and $background$. That is, there should be no gaps between $inside$ and $boundary$ class and $inside$ should not cross the $boundary$ class. From the samples shown in Fig. \ref{fig:three_two}, we can easily find out that NB model predicts the $inside$ class and $boundary$ class more precisely.
\begin{figure}[!ht]
\begin{center}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/orig_8.png}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/predict_8.png}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/two_8.png}
\end{subfigure}
\hspace{3mm}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/orig_9.png}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/predict_9.png}
\end{subfigure}
\begin{subfigure}[t]{0.15\textwidth}
\includegraphics[width=2.5cm]{./figures/result_show/two_9.png}
\end{subfigure}
\caption{The comparison of NB model against the mixed nucleus model and boundary model. First column shows the histophysiological images. Second column shows estimated nuclei and boundaries using our NB model. Third column shows the estimated result generated by the mixed nucleus model and boundary model. }
\label{fig:three_two}
\end{center}
\end{figure}
\section{Conclusion}
In this paper, we have presented a state-of-the-art supervised fully convolutional neural network method for nuclei segmentation in histopathological images. First, the histopathological images are normalized into the same color space. To handle the extra-large image issue, one whole image is split into overlapping patches for succeeding processing. Next, we propose a novel nucleus-boundary model to detect nuclei and boundaries on each patch. Then the predictions of all the patches are seamlessly reassembled to build the raw prediction result of the whole image. At the end, we apply a fast and non-parameter post-processing to generate the final nuclei segmentation results. The nucleus-boundary model is trained on very limited number of images and has been tested on the images that may have different appearances. Comparison with the state-of-the-art algorithm shows that our proposed method is accurate, robust, and fast. It is also found that our idea of simultaneous nucleus-boundary identification model can be applied to other image segmentation tasks such as cell segmentation, bacteria segmentation and so on.
\section{Acknowledgement}
We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X Pascal GPU used for this research.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-03-08T02:11:52",
"yymm": "1803",
"arxiv_id": "1803.02786",
"language": "en",
"url": "https://arxiv.org/abs/1803.02786"
}
|
\section{Introduction}
\paragraph{Introduction--}
Many new light bosons, such as axion \cite{axion-wilczek.1978, axion-weinberg.1978, axion-PhysRevLett.38.1440}, dark photon \cite{darkPhoton.2010, Dark-Photon-AN2015331}, paraphoton \cite{paraphoton-PhysRevLett.94.151802}, familon and majoron \citep{PDG16}, have been introduced by theories beyond the Standard Model.
If they exist, these kinds of new bosons may mediate new types of long-range fundamental forces, or the so-called 5th forces.
These possible new forces may break the C, P, or T (or their combinations) symmetry \cite{axion-PhysRevLett.38.1440}, and they have been suspected to be answers to questions like the strong CP violation problem \cite{axion-PhysRevLett.38.1440}.
The possibility of the existence of 5th forces has been extensively investigated experimentally \cite{Axion-Review-2010ARNPS,HILL1988253}.
Many forms of technology have been used to search for these
long-range spin- and/or velocity-dependent forces, including
the torsion balance \cite{ritter1990experimental, Torsion-Balance2006PRL,terrano2015short,hammond2007new},
the resonance spring \cite{Loong-Nature2003, Loong-PRD2015},
the spin exchange relaxation free (SERF) co-magnetometer \cite{Romalis-PRL2009,heckel2013limits,terrano2015short,wineland1991search},
nuclear magnetic resonance (NMR) based methods \cite{petukhov2010polarized,yan2015searching,chu2013laboratory}, and other high sensitivity technologies \cite{tullney2013constraints,serebrov2010search,ficek2017constraints,CPT2010a}.
In all of these experiments, in order to increase detecting sensitivities,
one of the key issues was how to improve the test matter's polarized spin density.
This is due to the fact that a Yukawa-like force is proportional to $e^{-r/\lambda}$\cite{16Forms2006Dobrescu}, where r is the source to probe distance and $\lambda$ is the force range.
For small $\lambda$, like in the range of about $1<\lambda <100$ cm, because of the limited volume $O(\lambda ^3)$ , increasing the polarized spin density is very critical to improve the detecting sensitivity.
It has been pointed out that interactions between two spin-1/2 fermions, which are mediated by spin 0 or spin 1 bosons, could be classified to 16 terms,
and 9 of them are spin-spin dependent \cite{16Forms2006Dobrescu}.
Among the spin-spin dependent terms, 3 of them are static, and 6 of them depend on the relative velocity between two polarized objects. Compared with the static terms \cite{Romalis-PRL2009,heckel2013limits,terrano2015short,wineland1991search,hunter2013using}, the experimental constraints on
the spin-spin-velocity-dependent forces (SSVDFs) are still rare today \cite{hunter2014using,ficek2017constraints}.
For the latter terms, not only a relative velocity between the source and the probe is required, but also both of them must be spin-polarized. Therefore, they are more difficult to study experimentally.
In Ref. \cite{ji2017searching},
an experimental scheme with high electron spin-density sources, iron-shielded SmCo$_5$ (ISSCs), was proposed to detect the SSVDFs.
By taking advantage of the high electron spin density of ISSC and the high sensitivity of SERF co-magnetometer \cite{Kornack:2005},
the proposed system had a potential to detect several SSVDFs with record sensitivities.
In this letter, we report new experimental studies on the SSVDFs
by using ISSCs and a SERF co-magnetometer \cite{2016yao, Yaodynamics2016,magnetic2016Yao,Smiciklas:2011}.
\paragraph{The SERF's Response to the SSVDFs--}
SSVDFs to be studied here are,
following the notation in Ref. \cite{16Forms2006Dobrescu,leslie2014prospects},
$V_{6+7}$, $V_8$, $V_{15}$ and $V_{16}$.
For example, $V_{16}$ can be written as,
\begin{align}
\begin{split}
V_{16}=&- \frac{ f_{16}\hbar^2}{8\pi m_\mu c^2}
\big\{
(\hat{\boldsymbol\sigma_2}\cdot\mathbf{v})\left[ \hat{\boldsymbol\sigma_1}\cdot(\mathbf{v}\times\hat{\boldsymbol r}) \right]\\&+(\hat{\boldsymbol\sigma_1}\cdot\mathbf{v})\left[ \hat{\boldsymbol\sigma_2}\cdot(\mathbf{v}\times\hat{\boldsymbol r})\right]
\big\}
\left(\frac{1}{\lambda r}+\frac{1}{r^2}\right)e^{-r/\lambda},
\end{split}
\end{align}
where $f_{16}$ is a dimensionless coupling constant,
$\hat{\sigma}_1$, $\hat{\sigma}_2$ are the spins of the two particles respectively, and
$\mathbf{v}$ is the relative velocity between the two interacting fermions.
For this new interaction, the corresponding effective magnetic field $\mathbf{B}_{eff}$ experienced by the polarized spin due to the spin source can be deduced from
$V_{16}=-\mathbf{\mu} \cdot \mathbf{B}_{eff}$, where $\mu$ is the magnetic momentum of the probing particle. In a typical polarized noble gas experiment,
the probe particles could be nuclei, e.g. $^{21}$Ne, or valence electrons of alkali.
If this $\mathbf{B}_{eff}$ exists,
the SERF's response can be estimated by the Bloch equations\cite{Kominis:2003},
\begin{equation}\label{eqn.pe}
\frac{\partial \mathbf{P}^e}{\partial t}=
\frac{\gamma_e}{Q(P^e)}\left[ \mathbf{B}_{eff}^e+\mathbf{B}+\lambda M^n \mathbf{P}^n+\mathbf{L} \right ]\times \mathbf{P}^e
+ \frac{P_0^e\hat{\mathbf{z}}-\mathbf{P}^e}{T_e Q(P^e)},
\end{equation}
\begin{equation}\label{eqn.pn}
\frac{\partial \mathbf{P}^n}{\partial t}=
\gamma_n \left[ \mathbf{B}_{eff}^n+\mathbf{B}+\lambda M^e \mathbf{P}^e \right ]\times \mathbf{P}^n
+ \frac{P_0^n\hat{\mathbf{z}}-\mathbf{P}^n}{\{T_{2n}, T_{2n}, T_{1n}\}},
\end{equation}
where $\mathbf{B}_{eff}^{e,n}$ are the effective magnetic fields due to the possible new SSVDFs coupling to the electron (or nucleon) spin;
$\mathbf{P}^{e,n}$ are the polarization of electron or nucleon respectively;
$\mathbf{B}$ is the external magnetic field;
$T_e$, $T_{1n}$, and $T_{2n}$ are the electron spin's relaxation time,
nucleon spin's longitudinal and transverse relaxation times respectively;
$M^{e,n}$ are the magnetization associated with the electron or the nucleon spin;
$P_0^e$ ($P_0^n$) is the equilibrium polarization of the electron ( nucleon);
$\mathbf{L}$ is the pumping light induced effective magnetic field experienced by the electron spin;
$Q(P^e)$ is the electron slow-down factor associated with the hyperfine interaction and spin-exchange collisions \cite{Kornack:2002};
and $\gamma_e$ ($\gamma_n$) is the gyromagnetic ratio of the electron (nucleon).
It is worth noticing that the Eq.(\ref{eqn.pe}) and (\ref{eqn.pn}) are coupled together.
For example, if $B^{n}_{eff}= 0$, but $B^{e}_{eff}\neq 0$,
the SERF still has nonzero output $S^{sim}(t)$.
By solving the equation set (\ref{eqn.pe}) and (\ref{eqn.pn}) numerically,
one can convert the SERF's response $\mathbf{P}^e(t)$ to a variable field
$\mathbf{B}^{e,n}_{eff}$. The numerical results, together with the experimental measurements, are shown in Fig. \ref{Fig.SERF.response}.
As shown in Fig. \ref{Fig.SERF.response}, the sensitivity of the co-magnetometer's response was frequency dependent.
\begin{figure}
\begin{center}
\includegraphics[width=9.cm]{SERF-response.pdf}
\caption{
(Color online)
SERF's response to magnetic fields. The red crosses and red dash line are experimental and simulation results for $B_x$ respectively, while the black triangles and black solid line are for $B_y$. The simulation results were obtained by solving Eq. \ref{eqn.pe}\&\ref{eqn.pn} numerically. The pink dot line and blue dash-dot line are simulations with assumptions that the exotic force only affected electrons.
}
\label{Fig.SERF.response}
\end{center}
\end{figure}
\paragraph{Experimental Setup--}
The experiment was carried out at Beihang University, Beijing, China.
The setup is shown in Fig. \ref{Fig.Exp.Setup} schematically.
The left side is a SERF co-magnetometer. A detailed description of the device can be found in Ref. \cite{2016yao}.
A spherical aluminosilicate glass vapor cell with a diameter of 14 mm was located at the center of the SERF.
It was filled with 3 bar of $^{21}$Ne gas (isotope enriched to 70$\%$ ), 53 mbar of N${_2}$ gas, and a small amount of K-Rb mixture. The mixture mole ratio was about 0.05 for the hybrid pumping purpose \cite{Smiciklas:2011}.
The cell was shielded by four layers of $\mu$-metal and a layer of 10-mm-thick ferrite \cite{Kornack:2007} magnetic field shielding to reduce the ambient magnetic field.
As shown in Fig. \ref{Fig.Exp.Setup},
a linearly polarized probe laser beam, which was modulated by a 50 kHz signal, passed through the cell, and
its Faraday rotation angle was then measured by using photo-elastic modulation (PEM).
The signals from photo-diodes were amplified by a lock-in amplifier,
which had a reference frequency of 50 kHz, the same as the probe's modulation.
The lock-in output was then recorded by a data-acquisition system.
As shown at the right side in Fig. \ref{Fig.Exp.Setup} , there were two ISSCs, the electron spin sources.
They were identical iron-shielded SmCo$_5$ (ISSC) magnets \cite{ji2017searching}.
Each ISSC had a cylindrical SmCo$_5$ magnet inside, which was covered by 3 layers of pure iron.
The magnets were cylindrical with diameter $40.00$ mm and height $40.00 $ mm. Thicknesses of the iron shielding layers were $15.00$ mm, $5.00 $ mm, and $5.00 $ mm respectively.
The internal magnetic field of the SmCo$_5$ magnet was about 1 T.
\begin{figure}
\begin{center}
\includegraphics[width=9.cm]{structure-nov11-ink.pdf}
\caption{
(Color online)The experimental setup.
The left side was the SERF co-magnetometer.
The two ISSC spin sources noted as 2A and 2B
were driven by a servo motor, and they could rotate CW and CCW along y axis with a given frequency.
}
\label{Fig.Exp.Setup}
\end{center}
\end{figure}
Driven by a servo motor, the ISSCs rotated with a frequency of $f_{0}=5.25$ Hz clockwise (CW) or counter-clockwise (CCW).
Because the two ISSCs were mounted centrosymmetrically,
the frequency of the possible SSVDF signals detected by the SERF was doubled, i.e. 10.5 Hz. This frequency was chosen due to the facts that the SERF co-magnetometer had relatively large responses to both $B^e_{eff, x}$ and $B^e_{eff, y}$ (Fig. \ref{Fig.SERF.response}), as well as relatively low noise level here (Fig. \ref{fig.power.spectrum}).
When rotating to a given angle, the ISSCs could trigger an optoelectronic pulse,
and this signal was recorded by the data-acquisition system.
This signal was used as the starting point of a new cycle for data analysis.
Similar to the SERF, the ISSCs as well as the servo motor were both shielded by 4 layers of $\mu$-metal to further reduce possible magnetic field leakage from the ISSCs and servo motor.
\begin{figure}
\begin{center}
\includegraphics[width=9cm]{power_spectrum_baex.pdf}
\caption{
A typical power spectrum measured in the experiment.
The ISSCs rotated in the frequency of 5.25 Hz, and the motor power system may cause the peak at 5.25 Hz in the spectrum.
The 50 Hz peak in the spectrum came from the power supply of the equipment, which was 50 Hz, 220V.
}
\label{fig.power.spectrum}
\end{center}
\end{figure}
\paragraph{Data Analysis--}
The experimental raw data was recorded as $\mathbf{S}^{exp}_{i,raw}(t_{j})$, where $i$ and $j$ mean the $j$-th point in the $i$-th cycle, $t_j=j*\Delta t$, and $ \Delta t$ is the data sampling period.
Then, $\mathbf{S}^{exp}_{i,raw}(t_{j})$ was first transformed to frequency domain by using Fast Fourier Transformations (FFT).
A typical SERF power spectrum is shown in Fig.\ref{fig.power.spectrum}.
Then Gaussian filters were applied to remove the peaks corresponding to 5.25 and 50 Hz.
After that, the signals were transformed back to the time domain with inverse FFT.
Furthermore, DC components in $\mathbf{S}^{exp}_{i,raw}(t_{j})$ were also removed. After the steps above, the raw signals $\mathbf{S}^{exp}_{i,raw}(t_{j})$ were then transferred to $\mathbf{S}^{exp}_{i}(t_{j})$ for further analysis.
Expected signals $S^{sim}(t)$ sensed by the SERF could be simulated by solving the equation sets (\ref{eqn.pe}) and (\ref{eqn.pn}) with the experimental parameters and a tentative coupling constant $f_{16}^{(tn)}$ in $V_{16}$ as inputs.
In the parameter space that we were interested in, i.e., $B_{eff}< 1 $ nT,
$P_x^e(t)$ approximately linearly dependent on $B_{eff}$, and thereafter the coupling constant $f_{16}$.
The $S^{sim}(t)$ then linearly depended on $B_{eff}$, i.e. $S^{sim}(t) \simeq \kappa f_{16}^{(tn)}\,B_{eff}$,
where $\kappa$ is the calibration constant, which was measured to be $110\pm 5$ V/nT.
The input $B_{eff}$ for solving Eq. \ref{eqn.pe}\&\ref{eqn.pn}
were simulated by the finite element analysis method\cite{ji2017searching}.
Two examples of simulated signals for $V_{16}$, the $\mathbf{S}^{sim}_{16}$, with motor rotating CW and CCW are shown in Fig. \ref{fig.signal}(a).
The experimental signals $\mathbf{S}^{exp}_{i}(t_{j})$ were then compared with the simulated ones $S^{sim}(t)$.
A cosine similarity score $k_i$ was used to weigh the similarity between $\mathbf{S}^{exp}_i$ and a given reference signal $\mathbf{S}^{ref}(t)$,
which can be written as \cite{PhysRevE.92.042927},
\begin{equation}\label{eq.ki}
k_i
\equiv\frac{\sum_j \mathbf{S}^{ref}(t_j)\cdot {\mathbf{S}}_i^{exp}(t_j)}
{\sqrt{\sum_j \left[\mathbf{S}^{ref}(t_j)\right]^2}
\sqrt{\sum_j \left[{\mathbf{S}_i}^{exp}(t_j)\right]^2}
}.
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=9.5cm,trim=0 0 0 0]{dist_mV.pdf}
\caption{(Color online)
(a) The simulated $\mathbf{S}^{sim}_{16}$ signals with $f^{(tn)}_{16}=1 \times 10^{-4}$, $\lambda=1000$ when motor rotates CW (blue solid line) and CCW (red dashed line).
(b) The distribution of the $f_i^{exp}$ with $S^{ref}=S^{sim}_{16}$.
The black area represents the ISSCs rotating CW, and green area CCW.
The red dashed line and blue solid line are their Gaussian fit respectively.
}
\label{fig.signal}
\end{center}
\end{figure}
The coupling constant in $V_{16}$ measured experimentally in $i$-th cycle, $f^{exp}_{i,16}$, can be written as,
\begin{equation}
f^{exp}_{i,16}=k_i f_{16}^{(tn)}\, \sqrt{\frac{\sum_j \left[\mathbf{S}^{exp}_i(t_j)\right]^2}
{\sum_j \left[\mathbf{S}^{ref}(t_j)\right]^2}}.
\end{equation}
Distributions of $f^{exp}_{i,16}$ are shown in Fig. \ref{fig.signal}(b). They agree with Gaussian shapes well.
The final experimentally measured coupling constant $f^{exp}_{16}$ was obtained by averaging all rotating cycles including CW and CCW, i.e.
\begin{equation}\label{eq.average}
f^{exp}_{16} =
\frac{\langle f^{exp}_{i,16}\rangle_+
+
\langle f^{exp}_{i,16}\rangle_-}
{2}
,
\end{equation}
where $\langle f^{exp}_{i,16}\rangle_+=\frac{1}{n}\sum_{i=1}^{n} f^{exp}_{i,16}$
is the average over the CW cycles,
and $\langle f^{exp}_{i,16}\rangle_-$, the CCW cycles.
\begin{figure*}[ht]
\begin{center}
\includegraphics[width=18.cm,height=10 cm,trim=0 0 0 0]{limits.pdf}
\caption{
Limits on the SSVDFs' coupling constants between two electrons measured in this work, and comparison with those in literature. The "Hunter2014" comes from Ref. \cite{hunter2014using}, in which polarized geo-electrons were used.
}
\label{fig.results}
\end{center}
\end{figure*}
\paragraph{Results and Discussion--}
The other terms of SSVDFs \cite{16Forms2006Dobrescu,leslie2014prospects} were analyzed by the same method. These interactions were:
\begin{align}
V_{\rm 6+7}=&\frac{ -f_{6+7} \hbar^2}{4\pi m_\mu c}
(\hat{\boldsymbol\sigma_1} \cdot \mathbf{v})(\hat{\boldsymbol\sigma}_2\cdot\hat{\boldsymbol r})
\left(\ \frac{1}{\lambda r}+\frac{1}{r^2} \right) e^{-r/\lambda},\\
\begin{split}
V_{8}&=\frac{ f_8 \hbar}{4 \pi c}
(\hat{\boldsymbol\sigma_1} \cdot\mathbf{v})
(\hat{\boldsymbol\sigma_2} \cdot\mathbf{v})
\frac{e^{-r/\lambda}}{r}
\end{split},\\
\begin{split}
V_{15}=& \frac{-f_{15}\hbar^3 }{8\pi m_1m_2 c^2}
\big\{
(\hat{\boldsymbol\sigma_2}\cdot\hat{\boldsymbol r}) \left[ \hat{\boldsymbol\sigma_1}\cdot(\mathbf{v}\times\hat{\boldsymbol r}) \right]+(\hat{\boldsymbol\sigma_1}\cdot\hat{\boldsymbol r})\\&\left[ \hat{\boldsymbol\sigma_2}\cdot(\mathbf{v}\times\hat{\boldsymbol r})\right]
\big\}
\left(\frac{1}{\lambda^2r }+\frac{3}{\lambda r^2} + \frac{3}{r^3}\right)e^{-r/\lambda}.
\end{split}
\end{align}
The parameters of the setup and their errors are shown in Tab. \ref{parameters.tab}.
Considering these errors, together with the statistical error,
the constraints on the SSVDFs between two electrons could be set. The results are shown in Fig. \ref{fig.results}.
The gray areas are excluded with 95\% confidence level.
For $V_{6+7}$, $V_{8}$, $V_{15}$, and $V_{16}$, our experiment can set up new record limits at the range of 5 cm -- 1 km.
Especially for $V_{15}$, our result is over 3 orders of magnitude better than \citep{hunter2014using} in force range between 5 cm and 1 km.
The error budget for $f^{exp}_{i,16}$
at $\lambda=1.1$ m
is shown in Tab. \ref{parameters.tab}.
The major systematic error came from
the cross-talking between the servo motor power system and the SERF system.
The 5.25 Hz peak shown in Fig. \ref{fig.power.spectrum} might come from cross-talking effect.
However, the major frequency considered here was 10.5 Hz, whose amplitude was about 40 times smaller than 5.25 Hz.
The secondary harmonics of 5.25 Hz could also contribute to systematic error.
In fact, the correlation between 5.25 Hz and 10.5 Hz could be calculated
by applying $\mathbf{S}^{ref}_A(t_j)=\sin[5.25t_j]$ or $\mathbf{S}^{ref}_B(t_j)=\sin[10.5t_j]$ to Eq.(\ref{eq.ki}).
A correlation between 5.25 and 10.5 Hz was indeed found this way, which confirmed the cross-talking effect.
The cross-talking was the dominant effect in our experiment.
Another major consideration was the magnetic leakage from the ISSCs.
With the iron shielding, at a distance of 20 cm away from the ISSC's mass center,
its residual magnetic field was measured to be $< 10$ mG.
The magnetic shielding factors for the mu-metals outside the ISSCs were measured to be $>10^{6}$, and shielding for SERF magnetometer, $>2\times 10^{6}$.
Considering all factors together, we conservatively expect the magnetic leakage from the ISSCs to SERF's center to be smaller than $10^{-2}$ aT, which was insignificant in regards to the error budget.
It is worth pointing out that only the errors of the parameters when doing the calculation of $f^{sim}$ as well as the statistic uncertainty could affect the limit curves drawn in Fig. \ref{fig.results}.
The magnetic field leakage and cross-talking were not subtracted in these plots.
\begin{table}[!h]
\begin{ruledtabular}
\caption {Input parameters for the FEA simulation and their error contribution to the final uncertainties. The origin of coordinates was at the center of the pumping cell.}
\label{parameters.tab}
\begin{tabular}{c c c c}
Parameter & Value & $\Delta f^{exp}_{16} (\times 10^{-8})$
\footnote{The contribution to the error budget of $V_{16}$
at $\lambda=1.1$ m
} \\
\hline
ISSC net spin ($\times10^{24}$) &$1.75\pm 0.21 $ & $^{+ 0.34}_{-0.32}$\\
Position of ISSCs y(m)& $-0.624\pm 0.005$ & $^{+ 0.13}_{-0.12} $\\
Position of ISSCs z(m)& $0.278\pm 0.005$ & $\pm 0.04$\\
D between 2 ISSCs(m) & $0.251\pm 0.001$ & $\pm 0.03$\\
Rotating frequency(Hz) &$5.250\pm 0.001$ & $ < 0.001$\\
Calib. const. $\kappa$ (V/nT) & $330\pm20$ & $\pm 0.23$ \\
phase uncertainty ($\deg$) &$\pm 5 $& $\pm 0.27$\\
\hline
Final $f^{exp}_{16} ( \times 10^{-8})$ & $4.0$
& $\pm 1.9\ (statistic)$\\
$(\lambda=1.1 m)$ & & $\pm 0.5
\footnote{
Error contribution from the uncertainties of the parameters listed above.
}
$
\end{tabular}
\end{ruledtabular}
\end{table}
\paragraph{Summary--}
In summary,
by using specially designed iron-shielded SmCo$_5$ permanent magnets, a high electron spin density source of about $ 1.7\times 10^{21}$ cm$^{-3}$ has been achieved, while still keeping its magnetic leakage down to about mG level.
The similarity analysis have been proved to be successful,
which gives a boost to the detecting sensitivities.
With help from the high spin density, the high sensitive SERF co-magnetometer, and the similarity analysis,
new constraints on possible new exotic potentials of $V_{6+7}$, $V_8$, $V_{15}$, and $V_{16}$
were derived for force range of 5 cm -- 1 km. To the best of our knowledge, it is the first time these results have beem attained.
By dedication to improving the SERF sensitivities, and reducing the crossing-talking effect,
a higher sensitivity by a factor of over 1000 is expected in future studies with a similar experimental setup.
\begin{acknowledgments}
This work is supported by Tsinghua University Initiative Scientific Research Program,
and the National Natural Science Foundation of China (NSFC) under Grant No. 11375114, 91636103, and 11675152. This work is also supported by the Key Programs of the NSFC under Grant No. 61227902.
\end{acknowledgments}
|
{
"timestamp": "2018-04-04T02:10:31",
"yymm": "1803",
"arxiv_id": "1803.02813",
"language": "en",
"url": "https://arxiv.org/abs/1803.02813"
}
|
\section{Introduction}
Recent advances in storage and networking technologies have resulted in many applications with interconnected relationships between objects.
This has led to the forming of gigantic inter-related and multi-typed heterogeneous information networks (HINs) across a variety of domains, such as e-government, e-commerce, biology, social media, etc. HINs provide an effective graph model to characterize the diverse relationships among different types of nodes. Understanding the vast amount of semantic information modeled in HINs has received a lot of attention. In particular, the concept of metapaths~\cite{Sun:2011:pathsim}, which connect two nodes through a sequence of relations between node types, is widely used to exploit rich semantics in HINs. In the last few years, many metapath-based algorithms are proposed to carry out data mining tasks over HINs, including similarity search~\cite{Sun:2011:pathsim}, personalized recommendation~\cite{Jamali:2013:HeteroMF,Shi:2015:semantic}, and object clustering~\cite{Sun:2012:integrating}.
Despite their great potential, data mining tasks in HINs often suffer from high complexity, because real-world HINs are very large and have very complex network structure. For example, when measuring metapath similarity between two distant nodes, all metapath instances need to be enumerated. This makes it very time-consuming to perform mining tasks, such as link prediction or similarity search, across the entire network. This inspires a lot of research interests in network embedding that aims to embed the network into a low-dimensional vector space, such that the proximity (or similarity) between nodes in the original network can be preserved. Analysis and search over large-scale HINs can then be applied in the embedding space, with the help of efficient indexing or parallelized algorithms designed for vector spaces.
Conventional network embedding techniques~\cite{cao2015grarep,grover2016node2vec,perozzi2014deepwalk,tang2015line,wang2016structural,zhang2016homophily,zhang2017user,Zhang:2018:Survey}, however, focus on homogeneous networks, where all nodes and relations are considered to have a single type. Thus, they cannot handle the heterogeneity of node and relation types in HINs. Only very recently, metapath-based approaches~\cite{Chen:2017:task,Dong:2017:metapath2vec}, such as MetaPath2Vec~\cite{Dong:2017:metapath2vec}, are proposed to exploit specific metapaths as guidance to generate random walks and then to learn heterogeneous network embedding. For example, consider a DBLP bibliographic network, Fig.~\ref{fig1:schema} shows the HIN schema, which consists of three node types: Author (A), Paper (P) and Venue (V), and three edge types: an author writes a paper, a paper cites another paper, and a paper is published in a venue. The metapath $\mathcal{P}_{1}$: $A \rightarrow P \rightarrow V \rightarrow P \rightarrow A$ describes the relationship where both authors have papers published in the same venue, while $\mathcal{P}_{2}$: $A \rightarrow P \rightarrow A \rightarrow P \rightarrow A$ describes that two authors share the same co-author. If $\mathcal{P}_{1}$ is used by MetaPath2Vec to generate random walks, a possible random walk could be: $a_1 \rightarrow p_1 \rightarrow v_1 \rightarrow p_2 \rightarrow a_2$. Consider a window size of 2, authors $a_1$ and $a_2$ would share the same context node $v_1$, so they should be close to each other in the embedding space. This way, semantic similarity between nodes conveyed by metapaths is preserved.
\begin{figure}[!htbp]
\begin{scriptsize}
\centering
\subfigure[Schema]{
\label{fig1:schema}
\begin{tikzpicture}
\node (V) at (0,0) [circle, draw, thick] {V};
\node[circle, draw, thick, above = 0.9 cm of V] (P) {P};
\node[circle, draw, thick, above = 0.9 cm of P] (A) {A};
\node[circle, below = 0.3 cm of V] (V0) {};
\draw [->,thick] (A) -- node[midway,left] {$write$} (P);
\draw [->,thick] (V) -- node[midway,left] {$publish$} (P);
\draw [->,thick] (P) edge [loop, out=45, in=315, looseness=7] node[midway,right] {$cite$} (P);
\end{tikzpicture}}
\subfigure[Metapah and Metagraph]{
\label{fig1:meta}
\begin{tikzpicture}
\node (A1) at (0,0) [circle, draw, thick] {A};
\node[circle, draw, thick, right = 1 cm of A1] (P1) {P};
\node[circle, right = 1 cm of P1] (N) {};
\node[circle, draw, thick, right = 1 cm of N] (P2) {P};
\node[circle, draw, thick, right = 1 cm of P2] (A2) {A};
\node[circle, draw, thick, above =0.3 cm of N] (V1) {V};
\node[circle, draw, thick, below = 0.3 cm of N] (A3) {A};
\node[circle, below = 0.1 cm of A3] (V0) {};
\node[circle, draw, thick, above = 0.5 cm of V1] (A4) {A};
\node[circle, draw, thick, left = 1 cm of A4] (P3) {P};
\node[circle, draw, thick, left = 1 cm of P3] (A5) {A};
\node[circle, draw, thick, right = 1 cm of A4] (P4) {P};
\node[circle, draw, thick, right = 1 cm of P4] (A6) {A};
\node[circle, draw, thick, above = 0.5 cm of A4] (V2) {V};
\node[circle, draw, thick, left = 1 cm of V2] (P5) {P};
\node[circle, draw, thick, left = 1 cm of P5] (A7) {A};
\node[circle, draw, thick, right = 1 cm of V2] (P6) {P};
\node[circle, draw, thick, right = 1 cm of P6] (A8) {A};
\node[left = 0.3cm of A1] (metagraph) {$\mathcal{G}:$};
\node[left = 0.3cm of A5] (metapath1) {$\mathcal{P}_{2}:$};
\node[left = 0.3cm of A7] (metapath2) {$\mathcal{P}_{1}:$};
\draw [->,thick] (A1) -- node[midway,above] {\tiny$write$} (P1);
\draw [->,thick] (P2) -- node[midway,above] {\tiny$write^{-1}$} (A2);
\draw [->,thick] (P1) -- node[near end,left] {\tiny$publish^{-1}$} (V1);
\draw [->,thick] (P1) -- node[near end,left] {\tiny$write^{-1}$} (A3);
\draw [->,thick] (V1) -- node[near start,right] {\tiny$publish$} (P2);
\draw [->,thick] (A3) -- node[near start,right] {\tiny$write$} (P2);
\draw [->,thick] (A5) -- node[midway,above] {\tiny$write$} (P3);
\draw [->,thick] (P3) -- node[midway,above] {\tiny$write^{-1}$} (A4);
\draw [->,thick] (A4) -- node[midway,above] {\tiny$write$} (P4);
\draw [->,thick] (P4) -- node[midway,above] {\tiny$write^{-1}$} (A6);
\draw [->,thick] (A7) -- node[midway,above] {\tiny$write$} (P5);
\draw [->,thick] (P5) -- node[midway,above] {\tiny$publish^{-1}$} (V2);
\draw [->,thick] (V2) -- node[midway,above] {\tiny$publish$} (P6);
\draw [->,thick] (P6) -- node[midway,above] {\tiny$write^{-1}$} (A8);
\end{tikzpicture}}
\caption{Schema, Metapath and Metagraph}
\label{fig1}
\end{scriptsize}
\end{figure}
Due to difficulties in information access, however, real-world HINs often have sparse connections or many missing links. As a result, metapath-based algorithms may fail to capture latent semantics between distant nodes. As an example, consider the bibliographic network, where many papers may not have venue information, as they may be preprints submitted to upcoming venues or their venues are simply missing. The lack of paper-venue connection would result in many short random walks, failing to capture hidden semantic similarity between distant nodes. On the other hand, besides publishing papers on same venues, distant authors can also be connected by other types of relations, like sharing common co-authors or publishing papers with similar topics. Such information should be taken into account to augment metapath-based embedding techniques.
\begin{figure}[!htbp]
\begin{scriptsize}
\centering
\begin{tikzpicture}
\node (a1) at (0,0) [thick, scale = 0.8, circle, draw] {$a_{1}$};
\node[scale = 0.8, circle, draw, right = 0.5cm of a1] (p1) {$p_{1}$};
\node[scale = 0.8, circle, draw, right = 0.5cm of p1] (v1) {$v_{1}$};
\node[scale = 1.2, right = of a1] (cross) {\color{red}$\mathbf{\times}$};
\node[scale = 0.8, circle, draw, above = 0.5cm of v1] (a2) {$a_{2}$};
\node[scale = 0.8, circle, draw, right = 0.5cm of v1] (p2) {$p_{2}$};
\node[thick, scale = 0.8, circle, draw, right = 0.5cm of p2] (a3) {$a_{3}$};
\node[scale = 0.8, circle, draw, right = 0.5cm of a3] (p4) {$p_{4}$};
\node[scale = 0.8, circle, draw, right = 0.5cm of p4] (v2) {$v_{2}$};
\node[scale = 0.8, circle, draw, right = 0.5cm of v2] (p5) {$p_{5}$};
\node[thick, scale = 0.8, circle, draw, right = 0.5cm of p5] (a4) {$a_{4}$};
\draw [->] (a1) -- (p1);
\draw [->] (p1) -- (v1);
\draw [->] (v1) -- (p2);
\draw [->] (p2) -- (a3);
\draw [->] (a3) -- (p4);
\draw [->] (p4) -- (v2);
\draw [->] (v2) -- (p5);
\draw [->] (p5) -- (a4);
\draw [->] (p1) edge [out=90, in=180, looseness=0.8] (a2);
\draw [->] (a2) edge [out=0, in=90, looseness=0.8] (p2);
\end{tikzpicture}
\caption{An example of random walk from $a_1$ to $a_4$ based on metagraph $\mathcal{G}$, which cannot be generated using metapaths $\mathcal{P}_1$ and $\mathcal{P}_2$. This justifies the ability of MetaGraph2Vec to provide richer structural contexts to measure semantic similarity between distant nodes.}
\label{fig2:example}
\end{scriptsize}
\end{figure}
Inspired by this observation, we propose a new method for heterogeneous network embedding, called MetaGraph2Vec, that learns more informative embeddings by capturing richer semantic relations between distant nodes. The main idea is to use metagraph~\cite{Huang:2016:metastructure} to guide random walk generation in an HIN, which fully encodes latent semantic relations between distant nodes at the network level. Metagraph has its strength to describe complex relationships between nodes and to provide more flexible matching when generating random walks in an HIN. Fig.~\ref{fig1:meta} illustrates a metagraph $\mathcal{G}$, which describes that two authors are relevant if they have papers published in the same venue or they share the same co-authors. Metagraph $\mathcal{G}$ can be considered as a union of metapaths $\mathcal{P}_1$ and $\mathcal{P}_2$, but when generating random walks, it can provide a superset of random walks generated by both $\mathcal{P}_1$ and $\mathcal{P}_2$. Fig.~\ref{fig2:example} gives an example to illustrate the intuition behind. When one uses metapath $\mathcal{P}_1$ to guide random walks, if paper $p_1$ has no venue information, the random walk would stop at $p_1$ because the link from $p_1$ to $v_1$ is missing. This results in generating too many short random walks that cannot reveal semantic relation between authors $a_1$ and $a_3$. In contrast, when metagraph $\mathcal{G}$ is used as guidance, the random walk $a_1 \rightarrow p_1 \rightarrow a_2 \rightarrow p_2 \rightarrow a_3$, and $a_3 \rightarrow p_4 \rightarrow v_2 \rightarrow p_5 \rightarrow a_4$ is generated by taking the path en route $A$ and $V$ in $\mathcal{G}$, respectively. This testifies the ability of MetaGraph2Vec to provide richer structural contexts to measure semantic similarity between distant nodes, thereby enabling more informative network embedding.
Based on this idea, in MetaGraph2Vec, we first propose metagraph guided random walks in HINs to generate heterogeneous neighborhoods that fully encode rich semantic relations between distant nodes. Second, we generalize the Skip-Gram model~\cite{mikolov2013distributed} to learn latent embeddings for multiple types of nodes. Finally, we develop a heterogeneous negative sampling based method that facilitates the efficient and accurate prediction of a node's heterogeneous neighborhood. MetaGraph2Vec has the advantage of offering more flexible ways to generate random walks in HINs so that richer structural contexts and semantics between nodes can be preserved in the embedding space.
The contributions of our paper are summarized as follows:
\begin{enumerate}
\item We advocate a new \textit{metagraph} descriptor which augments metapaths for flexible and reliable relationship description in HINs. Our study investigates the ineffectiveness of existing metapath based node proximity in dealing with sparse HINs, and explains the advantage of metagraph based solutions.
\item We propose a new network embedding method, called MetaGraph2Vec, that uses metagraph to capture richer structural contexts and semantics between distant nodes and to learn latent embeddings for multiple types of nodes in HINs.
\item We demonstrate the effectiveness of our proposed method through various heterogeneous network mining tasks such as node classification, node clustering, and similarity search, outperforming the state-of-the-art.
\end{enumerate}
\section{Preliminaries and Problem Definition}
In this section, we formalize the problem of heterogeneous information network embedding and give some preliminary definitions.
\begin{definition}A \textbf{heterogeneous information network (HIN)} is defined as a directed graph $G=(V,E)$ with a node type mapping function $\phi:V\rightarrow\mathcal{L}$ and an edge type mapping function $\psi:E\rightarrow\mathcal{R}$. $T_{G}=(\mathcal{L},\mathcal{R})$ is the network schema that defines the node type set $\mathcal{L}$ with $\phi(v)\in\mathcal{L}$ for each node $v\in V$, and the allowable link types $\mathcal{R}$ with $\psi(e)\in\mathcal{R}$ for each edge $e\in E$.
\end{definition}
\begin{example}For a bibliographic HIN composed of authors, papers, and venues, Fig.~\ref{fig1:schema} defines its network schema. The network schema contains three node types, author (A), paper (P) and venue (V), and defines three allowable relations, $A\xrightarrow{write}P$, $P\xrightarrow{cite}P$ and $V\xrightarrow{publish}P$. Implicitly, the network schema also defines the reverse relations, i.e., $P\xrightarrow{write^{-1}}A$, $P\xrightarrow{cite^{-1}}P$ and $P\xrightarrow{publish^{-1}}V$.
\end{example}
\begin{definition}Given an HIN $G$, \textbf{heterogeneous network embedding} aims to learn a mapping function $\mathrm{\Phi}:V\rightarrow\mathbb{R}^{d}$ that embeds the network nodes $v\in V$ into a low-dimensional Euclidean space with $d\ll|V|$ and guarantees that nodes sharing similar semantics in $G$ have close low-dimensional representations $\mathrm{\Phi}(v)$.
\end{definition}
\begin{definition} A \textbf{metagraph} is a directed acyclic graph (DAG) $\mathcal{G}=(N,M,n_{s},n_{t})$ defined on the given HIN schema $T_{G}=(\mathcal{L},\mathcal{R})$, which has only a single source node $n_{s}$ (\textit{i.e.}, with 0 in-degree) and a single target node $n_{t}$ (\textit{i.e.}, with 0 out-degree). $N$ is the set of the occurrences of node types with $n\in\mathcal{L}$ for each $n\in N$. $M$ is the set of the occurrences of edge types with $m\in\mathcal{R}$ for each $m\in M$. \end{definition}
As metagraph $\mathcal{G}$ depicts complex composite relations between nodes of type $n_{s}$ and $n_{t}$, $N$ and $M$ may contain duplicate node and edge types. To clarify, we define the \textit{layer} of each node in $N$ as its topological order in $\mathcal{G}$ and denote the number of layers by $d_{\mathcal{G}}$. According to nodes' layer, we can partition $N$ into disjoint subsets $N[i]\ (1\leq i\leq d_{\mathcal{G}})$, which represents the set of nodes in layer $i$. Each $N[i]$ does not contain duplicate nodes. Now each element in $N$ and $M$ can be uniquely described as follows. For each $n$ in $N$, there exists a unique $i$ with $1\leq i\leq d_{\mathcal{G}}$ satisfying $n\in N[i]$ and we define the layer of node $n$ as $l(n)=i$. For each $m\in M$, there exist unique $i$ and $j$ with $1\leq i<j\leq d_{\mathcal{G}}$ satisfying $m\in N[i]\times N[j]$.
\begin{example}Given a bibliographic HIN $G$ and a network schema $T_{G}$ shown in Fig.~\ref{fig1:schema}, Fig.~\ref{fig1:meta} shows an example of metagraph $\mathcal{G}=(N,M,n_{s},n_{t})$ with $n_{s}=n_{t}=A$. There are $5$ layers in $\mathcal{G}$ and node set $N$ can be partitioned into 5 disjoint subsets, one for each layer, where $N[1]=\{A\}, N[2]=\{P\}, N[3]=\{A, V\}, N[4]=\{P\}, N[5]=\{A\}$.
\end{example}
\begin{definition} For a metagraph $\mathcal{G}=(N,M,n_{s},n_{t})$ with $n_{s}=n_{t}$, its \textbf{recursive metagraph} $\mathcal{G}^{\infty}=(N^{\infty},M^{\infty},n^{\infty}_{s},n^{\infty}_{t})$ is a metagraph formed by tail-head concatenation of an arbitrary number of $\mathcal{G}$. $\mathcal{G}^{\infty}$ satisfies the following conditions:
\begin{enumerate}
\item $N^{\infty}[i]=N[i]$ for $1\leq i< d_{\mathcal{G}}$, and $N^{\infty}[i]={N[i\ \mathrm{mod}\ d_{\mathcal{G}}+1]}$ for $i\geq d_{\mathcal{G}}$.
\item For each $m\in N^{\infty}[i]\times N^{\infty}[j]$ with any $i$ and $j$, $m\in M^{\infty}$ if and only if one of the following two conditions is satisfied:
\begin{enumerate}
\item $1\leq i<j\leq d_{\mathcal{G}}$ and $m\in M\bigcap(N[i]\times N[j])$;
\item $i\geq d_{\mathcal{G}}$, $1\leq j-i\leq d_{\mathcal{G}}$ and $m\in M\bigcap(N[i\mod d_{\mathcal{G}}+1]\times {N[j\mod d_{\mathcal{G}}+1]})$.
\end{enumerate}
\end{enumerate}In the recursive metagraph $\mathcal{G}^{\infty}$, for each node $n\in N^{\infty}$, we define its layer as $l^{\infty}(n)$.
\end{definition}
\begin{definition}\label{randwalkseq}
Given an HIN $G$ and a metagraph $\mathcal{G}=(N,M,n_{s},n_{t})$ with $n_{s}=n_{t}$ defined on its network schema $T_{G}$, together with the corresponding recursive metagraph $\mathcal{G}^{\infty}=(N^{\infty},M^{\infty},n^{\infty}_{s},n^{\infty}_{t})$, we define the random walk node sequence constrained by metagraph $\mathcal{G}$ as $\mathcal{S}_{\mathcal{G}}=\{v_{1},v_{2},\cdots,v_{L}\}$ with length $L$ satisfying the following conditions:
\begin{enumerate}
\item For each $v_{i}\ (1\leq i\leq L)$ in $\mathcal{S}_{\mathcal{G}}$, $v_{i}\in V$ and for each $v_{i}\ (1< i\leq L)$ in $\mathcal{S}_{\mathcal{G}}$, $(v_{i-1},v_{i})\in E$. Namely, the sequence $\mathcal{S}_{\mathcal{G}}$ respects the network structure in $G$.
\item $\phi(v_{1})=n_{s}$ and $l^{\infty}(\phi(v_{1}))=1$. Namely, the random walk starts from a node with type $n_{s}$.
\item For each $v_{i}\ (1< i\leq L)$ in $\mathcal{S}_{\mathcal{G}}$, there exists a unique $j$ satisfying $(\phi(v_{i-1}),\phi(v_{i}))\in M^{\infty}\bigcap (N^{\infty}[l^{\infty}(\phi(v_{i-1}))]\times N^{\infty}[j])$ with $j>l^{\infty}(\phi(v_{i-1}))$, $\phi(v_{i})\in N^{\infty}[j]$ and $l^{\infty}(\phi(v_{i}))=j$. Namely, the random walk is constrained by the recursive metagraph $\mathcal{G}^{\infty}$.
\end{enumerate}
\end{definition}
\begin{example}
Given metagraph $\mathcal{G}$ in Fig.~\ref{fig1:meta}, a possible random walk is $a_1 \rightarrow p_1 \rightarrow v_1 \rightarrow p_2 \rightarrow a_2 \rightarrow p_3 \rightarrow a_3 \rightarrow p_4 \rightarrow a_5$. It describes that author $a_1$ and $a_2$ publish papers in the same venue $v_1$ and author $a_2$ and $a_5$ share the common co-author $a_3$. Compared with metapath $\mathcal{P}_1$ given in Fig.~\ref{fig1:meta}, metagraph $\mathcal{G}$ captures richer semantic relations between distant nodes.
\end{example}
\section{Methodology}
In this section, we first present metagraph-guided random walk to generate heterogeneous neighborhood in an HIN, and then present the MetaGraph2Vec learning strategy to learn latent embeddings of multiple types of nodes.
\subsection{MetaGraph Guided Random Walk}
In an HIN $G=(V,E)$, assuming a metagraph $\mathcal{G}=(N,M,n_{s},n_{t})$ with $n_{s}=n_{t}$ is given according to domain knowledge, we can get the corresponding recursive metagraph $\mathcal{G}^{\infty}=(N^{\infty},M^{\infty},n^{\infty}_{s},n^{\infty}_{t})$. After choosing a node of type $n_{s}$, we can start the metagraph guided random walk. We denote the transition probability guided by metagraph $\mathcal{G}$ at $i$th step as $\mathrm{Pr}(v_{i}|v_{i-1};\mathcal{G}^{\infty})$. According to Definition $\ref{randwalkseq}$, if $(v_{i-1},v_{i})\notin E$, or $(v_{i-1},v_{i})\in E$ but there is no link from node type $\phi(v_{i-1})$ at layer $l^{\infty}(\phi(v_{i-1}))$ to node type $\phi(v_{i})$ in the recursive metagraph $\mathcal{G}^{\infty}$, the transition probability $\mathrm{Pr}(v_{i}|v_{i-1};\mathcal{G}^{\infty})$ is $0$. The probability $\mathrm{Pr}(v_{i}|v_{i-1};\mathcal{G}^{\infty})$ for $v_{i}$ that satisfies the conditions of Definition $\ref{randwalkseq}$ is defined as
{\small\begin{equation}
\mathrm{Pr}(v_{i}|v_{i-1};\mathcal{G}^{\infty})=\frac{1}{T_{\mathcal{G}^{\infty}}(v_{i-1})}\times\frac{1}{|\{u|(v_{i-1},u)\in E, \phi(v_{i})=\phi(u)\}|}.
\end{equation}}Above, $T_{\mathcal{G}^{\infty}}(v_{i-1})$ is the number of edge types among the edges starting from $v_{i-1}$ that satisfy the constraints of the recursive metagraph $\mathcal{G}^{\infty}$, which is formalized as
{\footnotesize\begin{equation}
T_{\mathcal{G}^{\infty}}(v_{i-1})={|\{j| (\phi(v_{i-1}),\phi(u))\in M^{\infty}\bigcap (N^{\infty}[l^{\infty}(\phi(v_{i-1}))]\times N^{\infty}[j]), (v_{i-1},u)\in E \}|},
\end{equation}}and $|\{u|(v_{i-1},u)\in E, \phi(v_{i})=\phi(u)\}|$ is the number of $v_{i-1}$'s 1-hop forward neighbors sharing common node type with node $v_{i}$.
At step $i$, the metagraph guided random walk works as follows. Among the edges starting from $v_{i-1}$, it firstly counts the number of edge types satisfying the constraints and randomly selects one qualified edge type. Then it randomly walks across one edge of the selected edge type to the next node. If there are no qualified edge types, the random walk would terminate.
\subsection{MetaGraph2Vec Embedding Learning}
Given a metagraph guided random walk $\mathcal{S}_{\mathcal{G}}=\{v_{1},v_{2},\cdots,v_{L}\}$ with length $L$, the node embedding function $\mathrm{\Phi}(\cdot)$ is learned by maximizing the probability of the occurrence of $v_{i}$'s context nodes within $w$ window size conditioned on $\mathrm{\Phi}(v_{i})$:
{\small\begin{equation}
\min_{\mathrm{\Phi}}-\log\mathrm{Pr}(\{v_{i-w},\cdots,v_{i+w}\}\setminus v_{i}|\mathrm{\Phi}(v_{i})),
\end{equation}}where,
{\small\begin{equation}
\mathrm{Pr}(\{v_{i-w},\cdots,v_{i+w}\}\setminus v_{i}|\mathrm{\Phi}(v_{i}))=\prod_{j=i-w,j\neq i}^{i+w}\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i})).
\end{equation}}Following MetaPath2Vec~\cite{Dong:2017:metapath2vec}, the probability $\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}))$ is modeled in two different ways:
\begin{enumerate}
\item \textbf{Homogeneous Skip-Gram} that assumes the probability $\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}))$ does not depend on the type of $v_{j}$, and thus models the probability $\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}))$ directly by softmax:
{\small\begin{equation}
\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}))=\frac{\exp(\mathrm{\Psi}(v_{j})\cdot\mathrm{\Phi}(v_{i}))}{\sum_{u\in V}\exp(\mathrm{\Psi}(u)\cdot\mathrm{\Phi}(v_{i}))}.
\end{equation}}
\item \textbf{Heterogeneous Skip-Gram} that assumes the probability $\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}))$ is related to the type of node $v_{j}$:
{\small\begin{equation}
\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}))=\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}),\phi(v_{j}))\mathrm{Pr}(\phi(v_{j})|\mathrm{\Phi}(v_{i})),
\end{equation}}where the probability $\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}),\phi(v_{j}))$ is modeled via softmax:
{\small\begin{equation}
\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i}),\phi(v_{j}))=\frac{\exp(\mathrm{\Psi}(v_{j})\cdot\mathrm{\Phi}(v_{i}))}{\sum_{u\in V, \phi(u)=\phi(v_{j})}\exp(\mathrm{\Psi}(u)\cdot\mathrm{\Phi}(v_{i}))}.
\end{equation}}
\end{enumerate}
To learn node embeddings, the MetaGraph2Vec algorithm first generates a set of metagraph guided random walks, and then counts the occurrence frequency $\mathbb{F}(v_{i},v_{j})$ of each node context pair $(v_{i},v_{j})$ within $w$ window size. After that, stochastic gradient descent is used to learn the parameters. At each iteration, a node context pair $(v_{i},v_{j})$ is sampled according to the distribution of $\mathbb{F}(v_{i},v_{j})$, and the gradients are updated to minimize the following objective,
{\small\begin{equation}\label{SGD_obj}
\mathcal{O}_{ij}=-\log\mathrm{Pr}(v_{j}|\mathrm{\Phi}(v_{i})).
\end{equation}}To speed up training, negative sampling is used to approximate the objective function:
{\small\begin{equation}\label{SGD_obj_1}
\mathcal{O}_{ij}=\log\sigma(\mathrm{\Psi}(v_{j})\cdot\mathrm{\Phi}(v_{i}))+\sum_{k=1}^{K}\log\sigma(-\mathrm{\Psi}(v_{N_{j,k}})\cdot\mathrm{\Phi}(v_{i})),
\end{equation}}where $\sigma(\cdot)$ is the sigmoid function, $v_{N_{j,k}}$ is the $k$th negative node sampled for node $v_{j}$ and $K$ is the number of negative samples.
For Homogeneous Skip-Gram, $v_{N_{j,k}}$ is sampled from all nodes in $V$; for Heterogeneous Skip-Gram, $v_{N_{j,k}}$ is sampled from nodes with type $\phi(v_{j})$. Formally, parameters $\mathrm{\Phi}$ and $\mathrm{\Psi}$ are updated as follows:
{\small\begin{equation}\label{update_para}
\begin{aligned}
\mathrm{\Phi}=\mathrm{\Phi}-\alpha\frac{\partial\mathcal{O}_{ij}}{\partial \mathrm{\Phi}};\ \ \ \ \mathrm{\Psi}=\mathrm{\Phi}-\alpha\frac{\partial\mathcal{O}_{ij}}{\partial \mathrm{\Psi}},
\end{aligned}
\end{equation}}where $\alpha$ is the learning rate.
The pseudo code of the MetaGraph2Vec algorithm is given in Algorithm~\ref{alg:metgraph2vec}.
\begin{algorithm}[htb]
\caption{The MetaGraph2Vec Algorithm}
\label{alg:metgraph2vec}
\begin{algorithmic}[1]
\REQUIRE ~~\\
(1) A heterogeneous information network (HIN): $G=(V,E)$;\\
(2) A metagraph: $\mathcal{G}=(N,M,n_{s},n_{t})$ with $n_{s}=n_{t}$;\\
(3) Maximum number of iterations: $MaxIterations$;
\ENSURE ~~\\
Node embedding $\mathrm{\Phi}(\cdot)$ for each $v\in V$;
\STATE $\mathbb{S}$ $\leftarrow$ generate a set of random walks according to $\mathcal{G}$;
\STATE $\mathbb{F}(v_i,v_j)$ $\leftarrow$ count frequency of node context pairs ($v_{i},v_{j})$ in $\mathbb{S}$;
\STATE $Iterations \leftarrow 0;$
\REPEAT
\STATE $(v_i,v_j) \leftarrow$ sample a node context pair according to the distribution of $\mathbb{F}(v_i,v_j)$;
\STATE $(\mathrm{\Phi}, \mathrm{\Psi}) \leftarrow$ update parameters using $(v_i,v_j)$ and Eq.~(\ref{update_para});
\STATE $Iterations \leftarrow Iterations+1$;
\UNTIL {$convergence$ or $Iterations\ge MaxIterations$ }\label{code:iteration}
\STATE \textbf{return} $\mathrm{\Phi}$;
\end{algorithmic}
\end{algorithm}
\section{Experiments}
In this section, we demonstrate the effectiveness of the proposed algorithms for heterogeneous network embedding via various network mining tasks, including node classification, node clustering, and similarity search.
\subsection{Experimental Settings}
For evaluation, we carry out experiments on the DBLP\footnote{https://aminer.org/citation (Version 3 is used)} bibliographic HIN, which is composed of papers, authors, venues, and their relationships. Based on paper's venues, we extract papers falling into four research areas: \textit{Database}, \textit{Data Mining}, \textit{Artificial Intelligence}, \textit{Computer Vision}, and preserve the associated authors and venues, together with their relations. To simulate the paper-venue sparsity, we randomly select 1/5 papers and remove their paper-venue relations. This results in a dataset that contains 70,910 papers, 67,950 authors, 97 venues, as well as 189,875 paper-author relations, 91,048 paper-paper relations and 56,728 venue-paper relations.
To evaluate the quality of the learned embeddings, we carry out multi-class classification, clustering and similarity search on author embeddings. Metapaths and metagraph shown in Fig.~\ref{fig1:meta} are used to measure the proximity between authors. The author's ground true label is determined by research area of his/her major publications.
We evaluate MetaGraph2Vec with Homogeneous Skip-Gram and its variant MetaGraph2Vec++ with Heterogeneous Skip-Gram. We compare their performance with the following state-of-the-art baseline methods:
\begin{itemize}
\item DeepWalk~\cite{perozzi2014deepwalk}: It uses the uniform random walk that treats nodes of different types equally to generate random walks.
\item LINE~\cite{tang2015line}: We use two versions of LINE, namely LINE\_1 and LINE\_2, which models the first order and second order proximity, respectively. Both neglect different node types and edge types.
\item MetaPath2Vec and MetaPath2Vec++~\cite{Dong:2017:metapath2vec}: They are the state-of-the-art network embedding algorithms for HINs, with MetaPath2Vec++ being a variant of MetaPath2Vec that uses heterogeneous negative sampling. To demonstrate the strength of metagraph over metapath, we compare with different versions of the two algorithms: $\mathcal{P}_1$ MetaPath2Vec, $\mathcal{P}_2$ MetaPath2Vec and Mixed MetaPath2Vec, which uses $\mathcal{P}_1$ only, $\mathcal{P}_2$ only, or both, to guide random walks, as well as their counterparts, $\mathcal{P}_1$ MetaPath2Vec++, $\mathcal{P}_2$ MetaPath2Vec++, and Mixed MetaPath2Vec++.
\end{itemize}
For all random walk based algorithms, we start random walks with length $L=100$ at each author for $\gamma=80$ times, for efficiency reasons. For the mixed MetaPath2Vec methods, $\gamma/2=40$ random walks are generated by following metapaths $\mathcal{P}_1$ and $\mathcal{P}_2$, respectively. To improve the efficiency, we use our optimization strategy for all random walk based methods: After random walks are generated, we first count the co-occurrence frequencies of node context pairs using a window size $w=5$, and according to the frequency distribution, we then sample one node context pair to do stochastic gradient descent sequentially. For fair comparisons, the total number of samples (iterations) is set to 100 million, for both random walk based methods and LINE. For all methods, the dimension of learned node embeddings $d$ is set to $128$.
\subsection{Node Classification Results}
We first carry out multi-class classification on the learned author embeddings to compare the performance of all algorithms. We vary the ratio of training data from 1\% to 9\%. For each training ratio, we randomly split training set and test set for 10 times and report the averaged accuracy.
\begin{table}[!htbp]
\centering
\caption{Multi-class author classification on DBLP}
\label{author_classification}
\begin{scriptsize}
\renewcommand{\arraystretch}{1}
\setlength\tabcolsep{3pt}
\begin{tabular}{lccccccccc}
\toprule
Method & 1\% & 2\% & 3\% & 4\% & 5\% & 6\% & 7\% & 8\% & 9\%\\
\midrule
DeepWalk & 82.39 &86.04 &87.16 &88.15 &89.10 &89.49 &90.02 &90.25 &90.56\\
LINE\_1 &71.25 &79.25 &83.11 &85.60 &87.17 &88.29 &89.05 &89.45 &89.63\\
LINE\_2 &75.70 &80.80 &82.49 &83.88 &84.83 &85.71 &86.58 &86.90 &86.93\\
$\mathcal{P}_1$ MetaPath2Vec & 83.24 &87.70 &88.42 &89.05 &89.26 &89.46 &89.51 &89.76 &89.69\\
$\mathcal{P}_1$ MetaPath2Vec++& 82.14 &86.02 &87.04 &87.96 &88.47 &88.66 &88.90 &88.91 &89.02\\
$\mathcal{P}_2$ MetaPath2Vec & 49.59 &52.12 &53.76 &54.67 &55.68 &55.49 &55.83 &55.68 &56.07\\
$\mathcal{P}_2$ MetaPath2Vec++& 50.31 &52.50 &53.72 &54.47 &55.53 &55.78 &56.30 &56.36 &57.02\\
Mixed MetaPath2Vec & 83.86 &87.34 &88.37 &89.22 &89.70 &90.01 &90.37 &90.42 &90.71 \\
Mixed MetaPath2Vec++& 83.08 &86.91 &88.13 &89.07 &89.69 &90.09 &90.58 &90.68 &90.87\\
MetaGraph2Vec &\textbf{85.76} &\textbf{89.00} &89.79 &90.55 &91.02 &91.30 &91.72 &92.13 &92.25\\
MetaGraph2Vec++ & 85.20 &88.97 &\textbf{89.99} &\textbf{90.78} &\textbf{91.42} &\textbf{91.65} &\textbf{92.13} &\textbf{92.42} &\textbf{92.46}\\
\bottomrule
\end{tabular}
\end{scriptsize}
\end{table}
Table \ref{author_classification} shows the multi-class author classification results in terms of accuracy (\%) for all algorithms, with the highest score highlighted by $\textbf{bold}$. Our MetaGraph2Vec and MetaGraph2vec++ algorithms achieve the best performance in all cases. The performance gain over metapath based algorithms proves the capacity of MetaGraph2Vec in capturing complex semantic relations between distant authors in sparse networks, and the effectiveness of the semantic similarity in learning informative node embeddings. By considering methpaths between different types of nodes, MetaPath2Vec can capture better proximity properties and learn better author embeddings than DeepWalk and LINE, which neglect different node types and edge types.
\subsection{Node Clustering Results}
We also carry out node clustering experiments to compare different embedding algorithms. We take the learned author embeddings produced by different methods as input and adopt $K$-means to do clustering. With authors' labels as ground truth, we evaluate the quality of clustering using three metrics, including Accuracy, F score and NMI. From Table \ref{clustering}, we can see that MetaGraph2Vec and MetaGraph2Vec++ yield the best clustering results on all three metrics.
\begin{table}[!htbp]
\centering
\caption{Author clustering on DBLP}
\label{clustering}
\begin{scriptsize}
\begin{tabular}{lccc}
\toprule
Method & Accuracy(\%) & F(\%) & NMI(\%)\\
\midrule
DeepWalk & 73.87 & 67.39 & 42.02\\
LINE\_1 & 50.26 & 46.33 & 17.94\\
LINE\_2 & 52.14 & 45.89 & 19.55\\
$\mathcal{P}_1$ MetaPath2Vec & 69.39 & 63.05 & 41.72\\
$\mathcal{P}_1$ MetaPath2Vec++ & 66.11 & 58.68 & 36.45\\
$\mathcal{P}_2$ MetaPath2Vec & 47.51 &43.30 & 6.17\\
$\mathcal{P}_2$ MetaPath2Vec++& 47.65 & 41.48 & 6.56\\
Mixed MetaPath2Vec & 77.20 & 69.50 & 49.43\\
Mixed MetaPath2Vec++& 72.36 & 65.09 & 42.40\\
MetaGraph2Vec & \textbf{78.00} & \textbf{70.96}& \textbf{51.40}\\
MetaGraph2Vec++ &77.48&70.69&50.60\\
\bottomrule
\end{tabular}
\end{scriptsize}
\end{table}
\subsection{Node Similarity Search}
Experiments are also performed on similarity search to verify the ability of MetaGraph2Vec to capture author proximities in the embedding space. We randomly select 1,000 authors and rank their similar authors according to cosine similarity score. Table \ref{search} gives the averaged precision@100 and precision@500 for different embedding algorithms. As can be seen, our MetaGraph2Vec and MetaGraph2Vec++ achieve the best search precisions.
\begin{table}[!htbp]
\centering
\caption{Author similarity search on DBLP}
\label{search}
\begin{scriptsize}
\begin{tabular}{lccc}
\toprule
Methods & Precision$@100$ (\%) & Precision$@500$ (\%) \\
\midrule
DeepWalk & 91.65 & 91.44 \\
LINE\_1 & 91.18 & 89.88 \\
LINE\_2 & 91.92 & 91.38 \\
$\mathcal{P}_1$ MetaPath2Vec & 88.21 & 88.64 \\
$\mathcal{P}_1$ MetaPath2Vec++& 88.68 & 88.58 \\
$\mathcal{P}_2$ MetaPath2Vec & 53.98 & 44.11 \\
$\mathcal{P}_2$ MetaPath2Vec++& 53.39 & 44.11 \\
Mixed MetaPath2Vec & 90.94 & 90.27 \\
Mixed MetaPath2Vec++& 91.49 & 90.69 \\
MetaGraph2Vec & 92.50& \textbf{92.17} \\
MetaGraph2Vec++ & \textbf{92.59} &91.92\\
\bottomrule
\end{tabular}
\end{scriptsize}
\end{table}
\subsection{Parameter Sensitivity}
We further analyze the sensitivity of MetaGraph2vec and MetaGraph2Vec++ to three parameters: (1) $\gamma$: the number of metagraph guided random walks starting from each author; (2) $w$: the window size used for collecting node context pairs; (3) $d$: the dimension of learned embeddings. Fig.~\ref{fig:parameter} shows node classification performance with 5\% training ratio by varying the values of these parameters. We can see that, as the dimension of learned embeddings $d$ increases, MetaGraph2Vec and MetaGraph2Vec++ gradually perform better and then stay at a stable level. Yet, both algorithms are not very sensitive to the the number of random walks and window size.
\begin{figure}[!htbp]
\centering
\subfigure[$\gamma$]{
\label{fig:parameter:subfig:gamma}
\includegraphics[width=1.5in]{para_gamma.eps}}
\subfigure[$w$]{
\label{fig:parameter:subfig:w}
\includegraphics[width=1.5in]{para_w.eps}}
\subfigure[$d$]{
\label{fig:parameter:subfig:d}
\includegraphics[width=1.5in]{para_d.eps}}
\caption{The effect of parameters $\gamma$, $w$, and $d$ on node classification performance}
\label{fig:parameter}
\end{figure}
\section{Conclusions and Future Work}
This paper studied network embedding learning for heterogeneous information networks. We analyzed the ineffectiveness of existing \textit{metapath} based approaches in handling sparse HINs, mainly because metapath is too strict for capturing relationships in HINs. Accordingly, we proposed a new \textit{metagraph} relationship descriptor which augments metapaths for flexible and reliable relationship description in HINs. By using metagraph to guide the generation of random walks, our new proposed algorithm, MetaGraph2Vec, can capture rich context and semantic information between different types of nodes in the network. The main contribution of this work, compared to the existing research in the field, is twofold: (1) a new metagraph guided random walk approach to capturing rich contexts and semantics between nodes in HINs, and (2) a new network embedding algorithm for very sparse HINs, outperforming the state-of-the-art.
In the future, we will study automatic methods for efficiently learning metagraph structures from HINs and assess the contributions of different metagraphs to network embedding. We will also evaluate the performance of MetaGraph2Vec on other types of HINs, such as heterogeneous biological networks and social networks, for producing informative node embeddings.\\
\noindent\textbf{Acknowledgments}. This work is partially supported by the Australian Research Council (ARC) under discovery grant DP140100545, and by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning. Daokun Zhang is supported by China Scholarship Council (CSC) with No. 201506300082 and a supplementary postgraduate scholarship from CSIRO.
\bibliographystyle{plain}
|
{
"timestamp": "2018-03-08T02:04:39",
"yymm": "1803",
"arxiv_id": "1803.02533",
"language": "en",
"url": "https://arxiv.org/abs/1803.02533"
}
|
\section*{Appendix}
\subsection{Proof of Theorem~\ref{thm:surjectivity}}
Conditions $2, 3, 4$ are easily seen to be equivalent.
To see that 2 and 3 are equivalent, note that the matrix
\[
\begin{bmatrix}
Q_i^{1/2} & 0 \\
0 & R_i^{1/2}
\end{bmatrix}
\]
is symmetric, so its nullspace is perpendicular to its range.
Therefore surjectivity of $D_i$ is equivalent to the condition that
the range of $\begin{bmatrix} I\\ H_i\end{bmatrix}$ covers this nullspace. \\
To see the equivalence of 2 and 4, recall that $B$ is surjective if and only if $BB^T$
is invertible, so $D_i$ is surjective exactly when the matrix
\[
\begin{bmatrix}
Q_i + I & H_i^T \\
H_i & R_i + H_i H_i^T
\end{bmatrix}
\]
is invertible. $Q_i +I$ is always invertible, so invertibility of the block $2\times2$ matrix is equivalent to the
invertibility of the Schur complement $R_i + H_i\left(I-(Q_i+I)^{-1}\right)H_i^T$.\\
It remains to show that conditions 1 and 2 are equivalent. We proceed by induction on $N$.
The base case is trivial, since for $N =1$, $A = D_1$. For the inductive case, consider that
for $N=k$ the result holds, and write the $N=k+1$ case as
\[
\begin{bmatrix}
A_k & 0 \\
[ 0 \quad B_k] & D_{k+1}
\end{bmatrix}
\begin{bmatrix} z_1 \\ z_2 \end{bmatrix}
= \begin{bmatrix} w_1 \\ w_2 \end{bmatrix},
\]
and assume that $A_k$ is surjective. We then know that there exists $z_1$ that satisfies $A_k z_1 = w_1$.
The second row can now be written explicitly as
\[
D_{k+1}z_2 = w_2 +G_{k+1}x_k,
\]
where $x_k$ is the last component of $z_1$. Thus $A_{k+1}$ is surjective exactly when $D_{k+1}$ is, as desired.
\subsection{Proof of Lemma~\ref{lemma:property}.}
\textit{Proof:}
\textit{As $D$ is monotone we have}
\[
\langle \eta^* - T\eta, D\eta^* - DT\eta\rangle \geq 0
\]
\textit{as $0 \in (D+M)\eta^*$ this implies}
\[
\langle \eta^* - T\eta,-M\eta^* - DT\eta\rangle \geq 0
\]
\textit{Now} $DT\eta = DT\eta + HT\eta - HT\eta = (H-M)\eta- HT\eta$. \textit{Thus}
\[
0 \leq \langle \eta^* - T\eta,-M\eta^* + HT\eta - (H-M)\eta\rangle
\]
\[
= \langle \eta^* - T\eta,-M(\eta^* - \eta) + H(T\eta-\eta)\rangle
\]
\[
= \langle \eta^* - \eta, -M(\eta^*-\eta) + H(T\eta-\eta)\rangle
\]
\[\
\indent+ \langle \eta-T\eta, -M(\eta^*-\eta) + H(T\eta-\eta)\rangle
\]
\textit{By definition of $M$ we have}
\[
\langle M\eta,\eta\rangle = 0
\]
\textit{for any $\eta$. Therefore}
\[
0 \leq \langle \eta^*-\eta,H(T\eta-\eta)\rangle + \langle \eta-T\eta,-M(\eta^*-\eta)\rangle
\]
\[
+ \langle \eta-T\eta,H(T\eta-\eta)\rangle - \langle \eta-T\eta,M(T\eta-\eta)\rangle
\]
\[
= \langle \eta^* - \eta, H(T\eta-\eta) \rangle + \langle \eta-T\eta, -M(\eta^*-\eta)\rangle - ||T\eta-\eta||_{H-M}^2
\]
\[
= \langle \eta^* - \eta, H(T\eta-\eta) \rangle + \langle M(\eta-T\eta), \eta^* - \eta\rangle - ||T\eta-\eta||_{H-M}^2
\]
\[
= \langle \eta^* - \eta, (H-M)(T\eta-\eta)\rangle - ||T\eta-\eta||_{H-M}^2
\]
\subsection{Statement of \cite{AG}, Theorem 3.3.}
For a proper closed convex function $f$, the subdifferential $\partial f$ is metrically subregular at $\bar{x}$ for $\bar{y}$ with $(\bar{x}, \bar{y}) \in $ gra $\partial f$ if and only if there exists a positive constant $c$ and a neighborhood $\mathcal{U}$ of $\bar{x}$ such that
\[
f(x) \geq f(\bar{x}) + \langle \bar{y}, x-\bar{x}\rangle + cd^2(x, (\partial f)^{-1}(\bar{y})), \quad \forall x \in \mathcal{U}.
\]
\subsection{Computing with Prox Operators}
In this section, we collect the proximal operators used in the paper.
From simple calculus, we have
\begin{itemize}
\item
\(
\mbox{prox}_{\frac{\gamma}{2}\|\cdot\|^2}(z) = \frac{1}{1+\gamma}z.
\)
\end{itemize}
This generalizes to easily invertible least squares terms:
\begin{itemize}
\item
\(
\mbox{prox}_{\alpha \frac{1}{2}\|Ax-b\|^2}(z) = (I + \alpha A^TA)^{-1}(\alpha A^Tb+z).
\)
\end{itemize}
For $\rho(z) = \delta_{C}(z)$, we have
\[
\mbox{prox}_{\gamma \rho}(z) = \mbox{proj}_{C}(z).
\]
This gives simple formulas for the following operators:
\begin{itemize}
\item $\mbox{proj}_{\gamma\mathbb{B}_2}(z) = \min(\|z\|,\gamma)\frac{z}{\|z\|}$.
\item $\mbox{proj}_{\gamma\mathbb{B}_\infty}(z) = \min(\max(z,-\gamma),\gamma)$.
\item $\mbox{proj}_{\mathbb{R}_+}(z) = \max(z,0)$.
\end{itemize}
We also have fast implementations for the following operators:
\begin{itemize}
\item $\mbox{proj}_{\gamma\mathbb{B}_1}(z)$, the 1-norm projection
\item $\mbox{proj}_{\gamma \Delta}(z)$, the scaled simplex projection
\item $\mbox{proj}_{\gamma \Delta_1}(z)$, the capped simplex projection.
\end{itemize}
Next, the Moreau identity relates the prox operators for $f$ and $f^*$:
\[
\mbox{prox}_{\alpha f^*}(z) = z - \alpha\mbox{prox}_{\alpha^{-1}}(\alpha^{-1}z)
\]
This identity together with previous results yields the following operators:
\begin{itemize}
\item $\mbox{prox}_{\gamma\|\cdot\|_2}(z)$
\item $\mbox{prox}_{\gamma\|\cdot\|_1}(z)$
\item $\mbox{prox}_{\rho_h}(z)$, prox of hinge loss.
\item $\mbox{prox}_{\gamma \|\cdot\|_\infty}$
\end{itemize}
Often we add a simple quadratic to a penalty; the prox of the sum
can be expressed in terms of the prox of the original penalty.
\begin{itemize}
\item
\(
\mbox{prox}_{\alpha (f + \gamma/2 \|\cdot\|^2)}(x) = \mbox{prox}_{\frac{\alpha}{1+2\alpha\gamma} f}\left(\frac{1}{1+2\alpha\gamma}x\right).
\)
\end{itemize}
This immediately gives the prox of the elastic net, which is the sum of the 1-norm and a simple quadratic.
Likewise, we can compute the prox of a Moreau envelope of a given penalty.
\begin{itemize}
\item $\mbox{prox}_{\gamma e_\alpha \rho}(z) = \frac{\alpha}{\gamma + \alpha} z + \frac{\gamma}{\gamma + \alpha} \mbox{prox}_{(\gamma + \alpha)\rho }(z)$
\end{itemize}
This immediately gives us formulas for prox of the Huber, as well as smoothed variants of any other
penalty in the collection.
\section{Analysis of Mooring Data}
\label{sec:numerics}
We are interested in the ability to maintain an accurate position
estimate on-board an autonomous underwater vehicle using acceleration
measurements from a low cost inertial measurement unit (IMU), given periodic
position fixes. To test this, we use the singular general Kalman framework to analyze data collected from a surface mooring
equipped with an IMU that was deployed off the coast of Florida during spring 2017.
We use the mooring, which is drifting with the current, as a proxy for a slowly
moving underwater vehicle subject to unknown disturbances.
In particular we are looking at the position uncertainty and error accrued over time between the
periodic, world-referenced position fixes that are provided by the ultra short
baseline (USBL) system.
The new capabilities are useful because
\begin{enumerate}
\item Navigation models are singular
\item Data are noisy
\item IMU has biases, captured using singular models
\item data can be quantized, motivating a special loss.
\end{enumerate}
In this analysis, we use the singular linear kinematics model
in Section \ref{sec:vehicleModel}, the Huber loss from Section~\ref{sec:intro},
and the DRS algorithm from Section~\ref{sec:Opt} to solve the final smoothing
problem.
\subsection{Experimental Setup}
\label{sec:exp_setup}
\vspace{-.1in}
As shown in Figure \ref{fig:depl_setup}, the
mooring comprises an articulated spar buoy on the surface, supporting a cable
with various instruments attached. The mooring can be shortened using
yale grips shown in figure. We are using a portion of data from when the mooring
is at its max length of approximately 715m.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\columnwidth]{Figures/RxArrayHardwareDetailv4.pdf}
\caption{Sketch of the mooring and tender ship showing major buoyancy and
counterweight components and their relative depths. The USBL system
provides position fixes of the USBL transponder. The
navigation module provides roll, pitch, heading, and linear accelerations.}
\label{fig:depl_setup}
\end{center}
\end{figure}
At 121 meters above the bottom of the mooring is an ultra short baseline (USBL)
receiver which, in concert with a nearby tender ship, provided three dimensional
position updates for the mooring (latitude, longitude, and depth).
Below the main clump weight is a 4.25 meter section of
Spectra\textsuperscript{\textregistered} line with its own smaller clump weight
of approximately 45 kg. This supports a 25 cm diameter spherical glass housing
containing the navigation module.\\
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.7\columnwidth]{Figures/navsphere.jpg}
\caption{The navigation module, housed in a 25 cm diameter glass sphere.
}
\label{fig:navmodule}
\end{center}
\end{figure}
The self-contained navigation module, shown in Figure \ref{fig:navmodule},
consists of a RaspberryPi-based logger supporting a precision clock (Adafruit
ChronoDot RTC v2.1, based on the DS3231 temperature compensated crystal
oscillator), gyro (L3GD20H), and accelerometer and compass (LSM303D). The
navigation module carries its own batteries and recorded continuously throughout the
deployment, providing the time-stamped attitude and acceleration data used in this analysis.
Quantization in the attitude (roll, pitch, and yaw) and
linear acceleration measurements resulted in a degradation of the native
accuracy of the sensors. Table \ref{tbl:specs} provides a summary of the
measurements and associated resolutions as recorded during this experiment.
In this capacity the navigation module data serves as a proxy for a low cost autonomous
underwater vehicle using a low grade commercial IMU.\\
\begin{table}[ht]
\renewcommand{\arraystretch}{0.97} \scriptsize
\begin{tabular}{p{0.25\columnwidth}p{0.22\columnwidth}p{0.25\columnwidth}}
\hline
\textbf{Measurement} & \textbf{Resolution} & \textbf{Sample Freq.} \\
\hline
time & 3.5 ppm & n/a\\
roll, pitch, yaw & 0.1$^{\circ}$ & 25 Hz\\
lin. acceleration & 0.00766 m/s$^2$ & 25 Hz\\
\hline
\end{tabular}
\caption{~Navigation module sensor specifications.}
\label{tbl:specs}
\end{table}
The articulated spar buoy was tethered to the ship through an umbilical that
supplied power, two way communications and data transfer. During operations,
the intent was to decouple the motion of the ship from that of the surface
mooring, keeping slack in this umbilical. This is accomplished by using the ship
to tow the mooring into position and then allowing both the ship and mooring to
drift with the current.
Ground truth for the position of the mooring was provided by the Sonardyne
Ranger 2, a USBL system that provided 3-D position fixes every 2 seconds. The
USBL system self reports its measurement uncertainties at each
measurement. These ranged from 3.7 to 7.5 m uncertainty in x and y, and 0.8 to 4.0 m uncertainty in depth.
\subsection{Model and Experimental Results}
\vspace{-.1in}
Two challenges in the experimental setup required the flexibility of the modeling framework.
The depth acceleration data, some of which is plotted in Figure~\ref{fig:acceldata},
is extremely discretized and appears to have mean shifted away from zero.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.5]{Figures/Depthacceldat.pdf}
\caption{A snippet of the depth acceleration data, rotated into the world frame,
shows the quantization of the acceleration data.}
\label{fig:acceldata}
\end{center}
\end{figure}
To counteract this, a constant bias for acceleration measurements was fit and removed.
In the singular framework, we easily include a constant term, by imposing equality constraints
across all time points using the process model. The measurement maps are then modified
to directly subtract the estimated bias. \\
Because of the level of discretization we want to
use the Vapnik loss function (Figure~\ref{fig:vap}) that does not penalize in a small interval around the data.
The `deadzone' region is set according to the quantization of the data, which is $.05$.
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}
\begin{axis}[
thick,
height=2cm,
xmin=-2,xmax=2,ymin=0,ymax=1,
no markers,
samples=50,
axis lines*=left,
axis lines*=middle,
scale only axis,
xtick={-0.5,0.5},
xticklabels={},
ytick={0},
]
\addplot[red,domain=-2:-0.5,densely dashed] {-x-0.5};
\addplot[domain=-0.5:+0.5] {0};
\addplot[red,domain=+0.5:+2,densely dashed] {x-0.5};
\addplot[blue,mark=*,only marks] coordinates {(-0.5,0) (0.5,0)};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[
thick,
height=2cm,
xmin=-2,xmax=2,ymin=0,ymax=1,
no markers,
samples=50,
axis lines*=left,
axis lines*=middle,
scale only axis,
xtick={-0.5,0.5},
xticklabels={},
ytick={0},
]
\addplot[red,domain=-2:-1.5,densely dashed] {-1.5*x-1.75};
\addplot[red,domain=-1.5:-.5,densely dashed] {0.5*(x+0.5)^2};
\addplot[domain=-0.5:+0.5] {0};
\addplot[red,domain=+0.5:+1.5,densely dashed] {0.5*(x-0.5)^2};
\addplot[red,domain=1.5:2,densely dashed] {1.5*x-1.75};
\addplot[blue,mark=*,only marks] coordinates {(-0.5,0) (0.5,0)};
\end{axis}
\end{tikzpicture}
\end{center}
\caption{Vapnik loss function and a smoothed variant.}
\label{fig:vap}
\end{figure}
The `corners' of the Vapnik encourage the errors to be exactly equal to the quantization value, an unnecessary artifact.
We therefore use a Huberized version of the Vapnik, smoothing the corners but leaving the deadzone.
In addition to the deadzone, this loss is robust, as it has linear tail growth. \\
{\bf Results for $10$ Minute Track:}
We begin by considering $10$ minutes of IMU data with occasional USBL position data. The position data are available approximately every $2$ seconds, but we test performance with intervals of $30, 60, 120$ seconds.
The $x_0$ given to the algorithm is as follows: position is set to the first
position fix and acceleration is set to zero, while velocity is taken
to be the slope from the last available position data to the starting time.
The algorithm is initialized by propagating this $x_0$ through the entire model and then run for $500$ iterations. \\
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.5]{Figures/experiment_plots/accel_short_depth.pdf}
\caption{Depth acceleration data and fit after debiasing.}
\label{fig:biasremoved}
\end{center}
\end{figure}
Figure~\ref{fig:biasremoved} shows the depth acceleration data after the bias is removed, now centered around $0$.
Biases computed for 30, 60, and 120 second intervals were all near $0.073$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.5]{Figures/experiment_plots/pos_short_easting.pdf}
\includegraphics[scale=.5]{Figures/experiment_plots/pos_short_northing.pdf}
\includegraphics[scale=.5]{Figures/experiment_plots/pos_short_depth.pdf}
\caption{Fitted position for three frequencies of position data. With supplemental position data the estimates perform much better than when only acceleration data is used.}
\label{fig:posshort}
\end{center}
\end{figure}
Figure~\ref{fig:posshort} has the fitted position plots for all three frequencies. The depth plot shows why using only acceleration data is can lead to large errors; small errors in acceleration data build up to have a large effect over time. However when the acceleration data is combined with a small amount of position data all three perform very well. In fact there is not a large difference in the estimates produced; this gives a promising view
toward an online implementation. Figure~\ref{fig:velshort} shows the fitted
velocity for all three models. Here the small differences in the fit become apparent with lower frequency position data leading to much larger changes in velocity over time. \\
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.5]{Figures/experiment_plots/vel_short_easting.pdf}
\includegraphics[scale=.5]{Figures/experiment_plots/vel_short_northing.pdf}
\includegraphics[scale=.5]{Figures/experiment_plots/vel_short_depth.pdf}
\caption{Velocity fit for different frequencies of position data. The build up of acceleration errors can be seen clearly here, especially when no position data is used.}
\label{fig:velshort}
\end{center}
\end{figure}
\\
{\bf Results for $50$ Minute Track}
At this scale, we consider position data at intervals of $3$ and $5$ minutes. There is also a gap in the position data near minute $27$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.5]{Figures/experiment_plots/pos_long_easting.pdf}
\includegraphics[scale=.5]{Figures/experiment_plots/pos_long_northing.pdf}
\includegraphics[scale=.5]{Figures/experiment_plots/pos_long_depth.pdf}
\caption{Position fit over $50$ minutes with $3$ and $5$ minute gaps in position data. With very low frequency of position data poor acceleration in the depth data leads to much less stable position estimates.}
\label{fig:poslong}
\end{center}
\end{figure}
Figure~\ref{fig:poslong} shows the position estimates for the longer time period. When the position data is only seen every $5$ minutes the estimate becomes unstable, especially for depth, where the acceleration data quality is poorest.
However even with three minute gaps in between position data the model performs
fairly well. Modern underwater vehicles are well-instrumented in depth, and
typically have some model for velocity (e.g. hydrodynamic velocity model in
gliders, prop counts in propeller-drive vehicles); an extension of the methods proposed here could enable an online navigation system that requires
ever fewer high-fidelity external position fixes (such as those provided here from the USBL data).\\
\section{Discussion}
\vspace{-.1in}
We propose a singular Kalman smoothing framework that can use
singular covariance models for process and measurements, convex robust
losses, and state-space constraints. The modeler can use any convex
loss that has an implementable prox; in particular any piecewise linear-quadratic
loss and simple polyhedral constraint can be used. The framework offers a range of tools
that we illustrated using a sea survey analysis. Future work will consider real-time implementation,
as well as extension to nonlinear models. \\
Numerical experiments illustrate that the local linear rate we have in theory requires
a good initialization in practice. All experiments in the paper were initialized by propagating the state
estimate forward; this worked far better than an arbitrary initialization (e.g. at the $0$ vector).
Smarter initialization can be developed for streaming/online contexts, where recent estimates play
a key role in initializing smoothing subproblems.
\section{Acknowledgements}
This material is based upon work supported by the Defense Advanced Research
Agency (DARPA) and Space and Naval Warfare Systems Center Pacific (SSC Pacific)
under Contract No. N66001-16-C-4001.
The work of Dr. Aravkin was supported by the Washington Research Foundation Data
Science Professorship.
\section{General Singular Kalman Smoothing}
\label{sec:reformulation}
\vspace{-.1in}
Following the ideas proposed by~\cite{OGLB}, we introduce variables $u_k$ for the normalized process innovations, and $t_k$ for the normalized residuals. We also introduce a penalty $\rho_3$ for the states. In the examples we consider,
$\rho_3$ is an indicator function for the known feasible regions $X_k$:
\[
\rho_3(x_k) = \begin{cases} 0 & x_k \in X_k \\ \infty & x_k \not \in X_k\end{cases}.
\]
The reformulated singular Kalman smoothing problem is given by
\begin{equation}
\label{eq:genKalman}
\begin{aligned}
\min_{u,t,x} & \sum_{k=1}^N \rho_1(u_k) + \rho_2(t_k) + \rho_3(x_k)\\
&\text{s.t.} \quad
\begin{aligned}
Q_k^{1/2}u_k &= G_kx_{k-1} - x_k\\
R_k^{1/2}t_k &= y_k - H_kx_k
\end{aligned}
\end{aligned}.
\end{equation}
This problem is equivalent to~\eqref{eq:genSmoother} when $Q_k$ and $R_k$ are nonsingular.
For singular models,~\eqref{eq:genKalman} requires only that roots $Q^{1/2}$ and $R^{1/2}$ are available.
{\bf Constrained Robust DC motor}. Recall the DC motor example in the introduction~\eqref{eq:DCmotor}.
The data used to make Figure~\ref{fig:DCmotor} is contaminated with outliers, so we want to use the robust Huber loss
for the measurement errors. Suppose we also know upper and lower bounds on the states, $B := \{x: l \leq x \leq u\}$.
Then the formulation of the robust constrained singular DC motor is given by
\[
\begin{aligned}
\min_{u,t,x} & \sum_{k=1}^N \|u_k\|^2 + \rho_h(t_k) + \delta_{B}(x_k), \quad \sigma t_k = a_k - x_{2,k},\\
&
\begin{aligned}
\begin{bmatrix}11.8 & 0 \\ 0.62 & 0 \end{bmatrix} u_k &= x_{k+1} - \left(\begin{array}{cc} 0.7 & 0 \\0.084 & 1\end{array}\right) x_k
-\left(\begin{array}{c}11.81 \\0.62 \end{array}\right) c_k\\
\end{aligned}
\end{aligned}.
\]
{\bf Structure-preserving Reformulation.}
We now rewrite~\eqref{eq:genKalman} into a more compact form.
Define
\begin{equation}
\label{eq:DjBj}
\begin{aligned}
D_i &= \begin{pmatrix}Q_i^{1/2} & 0 & I\\ 0 & R_i^{1/2} & H_i\end{pmatrix} \text{ for } i = 1, \dots N,\\
B_j &= \begin{pmatrix}0 & \qquad 0 & -G_{j+1}\\ 0 & \qquad 0 & 0\end{pmatrix}, \text{ for } j = 1, \dots, N-1,
\end{aligned}
\end{equation}
and let
\begin{equation}
\label{eq:A}
A =
\begin{pmatrix} D_1 & 0 & \dots & 0 \\
B_1 & D_2 & 0 & \vdots \\
0& \ddots & \ddots & 0\\
0 & 0 & B_{N-1} & D_N\end{pmatrix}.
\end{equation}
Define also
\begin{equation}
\label{eq:zw}
\begin{aligned}
z^T &= \begin{pmatrix} u_1^T & t_1^T & x_1^T & \dots u_N^T & t_N^T & x_N^T\end{pmatrix} \\
\hat{w}^T &= \begin{pmatrix} x_0^T & y_1^T & 0 & y_2^T & \dots & 0 & y_N^T\end{pmatrix}.
\end{aligned}
\end{equation}
Now we can write~\eqref{eq:genKalman} compactly as
\begin{equation}
\label{eq:full}
\begin{aligned}
\min_{z} &\quad \rho(z) \quad \text{s.t. } Az = \hat{w}, \\
\rho(z) &= \sum_{k=1}^N \rho_1(u_k) + \rho_2(t_k) + \rho_3(x_k).
\end{aligned}
\end{equation}
The order of blocks in $z$ is chosen to the constraint matrix $A$ in~\eqref{eq:A} lower block bi-diagonal. \\
The constraint $Az = \hat w$ raises a natural question: when is a singular Kalman smoothing model solvable?
Clearly we want $\hat w \in\mathrm{Ran}(A)$, but we want this condition to hold for any realization of the data $\hat w$,
so we want to know when $A$ is surjective. We can characterize this condition precisely
in terms of a simple conditions on the individual blocks $R_i, Q_i, H_i$.
\begin{theorem}[Surjectivity of $A$]
\label{thm:surjectivity}
The following are equivalent.
\begin{enumerate}
\item $A$ is surjective.
\item Each block $D_i$ is surjective.
\item $\mathrm{null}
\left(\begin{bmatrix} Q_i^{1/2} & 0 \\ 0& R_i^{1/2} \end{bmatrix}\right)
\subset \mathrm{Ran} \left(\begin{bmatrix} I \\ H_i\end{bmatrix}\right)$ for all $i$.
\item $R_i + H_i\left(I-(Q_i+I)^{-1}\right)H_i^T$ is invertible for all $i$.
\end{enumerate}
\end{theorem}
The proof is given in the Appendix.
\section{Douglas-Rachford Splitting for General Singular Kalman Smoothing}
\label{sec:Opt}
\vspace{-.1in}
Consider problem~\eqref{eq:full} as a sum of two functions, $\rho +g$, with $\rho$ as in~\eqref{eq:full}
and $g$ the indicator function of the affine constraint $Az = \hat w$:
\begin{equation}
\label{eq:indicator}
g(z) = \begin{cases} 0 & Az = \hat w\\ \infty & Az \not = \hat w\end{cases}.
\end{equation}
Douglas-Rachford splitting (DRS) is a classic algorithm for this problem.
For a convex function $f$, define the proximity operator (see e.g.~\cite{combettes2011proximal}) as
\[
\mbox{prox}_{\alpha f}(\zeta) = \arg\min_x \frac{1}{2\alpha} \|\zeta - x\|^2 + f(x).
\]
The DRS algorithm for~\eqref{eq:full} detailed in Algorithm~\ref{DRS}.
For more on splitting methods and their convergence rates see the survey~\cite{davis2016convergence}.
\begin{algorithm}[H]
\caption[Caption]{Douglas-Rachford Splitting (DRS
\label{DRS}}
\begin{algorithmic}[1]
\Require{Initialize at any $z^0$, $\zeta^0$.}
\Loop
\State {$z^k = \mbox{prox}_{\tau g}(z^{k-1}-\tau \zeta^{k-1})$}
\State {$\zeta^k = \mbox{prox}_{\sigma \rho^*}(\zeta^{k-1}+\sigma(2z^{k} - z^{k-1}))$}
\EndLoop
\Return{$z^k$}
\end{algorithmic}
\end{algorithm}
Implementing DRS in our case requires computing two proximity operators at each iteration.
One proximity operator is $\mbox{prox}_{\rho^*}$, where $\rho^*$ denotes the \textit{convex conjugate}:
\[
\rho^*(y) = \sup_x \langle y,x \rangle - \rho(x)
\]
The prox of of a function is related to the prox of its conjugate by Moreau's decomposition:
\[
\mbox{prox}_\rho(x) + \mbox{prox}_{\rho^*}(x) =x.
\]
Thus it suffices to compute $\mbox{prox}_\rho$.
The function $\rho$ captures all user-supplied models, including losses used process and measurement
transitions, as well as penalties or constraints on the state, $\rho_1, \rho_2$ and $\rho_3$.
The proximity operators of these individual elements must be provided; then $\mbox{prox}_\rho$
is a stack of these input functions.
Proximity operators for many common functions are easily available~\cite{combettes2011proximal}, and we include a small library with our implementation\footnote{https://github.com/UW-AMO/KalmanJulia.}. \\
The second proximity operator is $\mbox{prox}_g$, which is \underline{independent} of user choice for process, measurement, and prior models:
\[
\mbox{prox}_g(\eta) = \arg\min_{Az = \hat w} \frac{1}{2}\|\eta-z\|^2.
\]
This is a simple quadratic with affine constraints, with optimality conditions given by
\[
\begin{bmatrix}I & A^T\\A & 0\end{bmatrix}\begin{bmatrix}z\\ \nu\end{bmatrix} = \begin{bmatrix}\eta\\ \hat{w}\end{bmatrix}.
\]
There are many ways to solve this system. We opt to reduce the problem to solving a block tridiagonal system:
\[
\begin{bmatrix}I & A^T\\0 & AA^T\end{bmatrix} \begin{bmatrix}z\\ \nu\end{bmatrix} = \begin{bmatrix}\eta\\ A\eta - \hat{w}\end{bmatrix}
\]
We solve $AA^T \nu = A\eta - \hat{w}$, then back-substitute to get the optimal $z$. The system $AA^T$ does not change
over iterations; only the right hand side changes. We can therefore compute a single factorization, then use it in each iteration.
Since $A$ is block bidiagonal~\eqref{eq:A}, $AA^T$ is block tridiagonal;
when $A$ is surjective, $AA^T$ is nonsingular, and
we can find a lower block diagonal Cholesky factorization $L$ with $LL^T = AA^T$:
\begin{equation}
\label{eq:Structure}
AA^T =
\begin{bmatrix}
a_1& b_1^T & & \\
b_1 & a_2 & b_2^T & \\
& b_2 & a_3 & b_3^T \\
&&b_3 & a_4
\end{bmatrix}, \quad L =
\begin{bmatrix}
c_1& & & \\
d_1 & c_2 & & \\
& d_2 & c_3 & \\
&&d_3 & c_4
\end{bmatrix}
\end{equation}
The factorization is detailed in Algorithm~\ref{chol}.
\begin{algorithm}[H]
\caption[Caption]{Block bi-diagonal Cholesky factorization for a block tri-diagonal positive definite matri
\label{chol}}
\begin{algorithmic}[1]
\Require{Input block diagonals $\{a_i\}$ and lower off-diagonals $\{b_i\}$ of block tridiagonal matrix $AA^T$~\eqref{eq:Structure}.}
\State{$s_0 = 0, b_0 = 0$}
\Loop{ $k=1, \dots, N$}
\State{$s_k = a_k - b_{k-1}s_{k-1}^{-1}b_{k-1}^T$}
\State {$c_k = \mbox{chol}(s_k)$}
\State {$d_k = b_1 c_k^{-^T}$}
\EndLoop
\Return{Diagonal blocks $\{c_i\}$ and lower-diagonal blocks $d_i$ of block $L$~\eqref{eq:Structure}}
\end{algorithmic}
\end{algorithm}
Algorithm~\ref{chol} is derived as follows. Multiplying out $LL^T$ we have
\[
a_1 = c_1c_1^T, \quad d_1 = b_1 c_1^{-T}
\]
To compute $c_1$ we need the standard the Cholesky factorization of $a_1$. Then
\[
c_2c_2^T = a_2 - b_1a_1^{-1}b_1^T, \quad d_2 = b_2c_2^{-1}.
\]
For convenience, we introduce the recursively defined auxiliary terms $s_k$, with $s_1 = a_1$, and
\[
s_k = a_k - b_{k-1}s_{k-1}^{-1}b_{k-1}^T.
\]
Then each $c_k$ is the standard Cholesky factorization of $s_k$, and $d_k$ is immediately computed as in Algorithm~\ref{chol}.
The overall complexity required for the single factorization is $O(n^3N)$.
Once $L$ has been pre-computed, we need only $O(n^2N)$ arithmetic operations to solve $LL^T \nu = Ac - \hat{w}$
for any right hand side. This is the same complexity as that of a matrix-vector multiply with $A$.
\begin{figure}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.5]{Figures/dc_loss_rel-eps-converted-to}
\end{tabular}
\caption{\label{fig:local_rate} Convergence rate for DRS splitting is locally linear when the objectives are piecewise linear-quadratic (PLQ). The convergence plot show here corresponds to the robust DC motor example in Figure~\ref{fig:DCmotor}.}
\end{center}
\end{figure}
\noindent
{\bf Local Linear Rate.}
When $\rho$ is piecewise linear-quadratic~\cite{RTRW,JMLR:v14:aravkin13a}, the DRS algorithm converges locally linearly to a solution, see Figure~\ref{fig:local_rate}. More precisely, there is a real number $R >0$ such that if $||\eta^K - \eta^*|| < R$ then there is a constant $\kappa$ with $0 < \kappa < 1$ such that for all $k > K$,
\[
\|\eta^{k+1} - \eta^*\| < \kappa\|\eta^k - \eta^*\|,
\]
where $\eta=\begin{bmatrix}z & \zeta \end{bmatrix}^T$, is the primal and dual pair.
\begin{theorem}
Algorithm 1 converges with a locally linear rate.
\end{theorem}
\vspace{-.1in}
\textit{ Proof:}
Following the proof technique of ~\cite[Theorem 5]{LFP}, Algorithm 1 has a local linear convergence rate if the following two conditions are satisfied:
\begin{enumerate}
\item Algorithm~\ref{DRS} can be written as the action of a nonlinear operator satisfying a regularity property (see Lemma~\ref{lemma:property}).
\item The functions $g, \rho$ are \textit{subdiffererentially metrically subregular}\footnote{ A mapping $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is called \textit{metrically subregular} at $\bar{x}$ for $\bar{y}$ if $(\bar{x}, \bar{y}) \in $ graph $F$ and there exists $\eta \in [0 , \infty)$, neighborhoods $\mathcal{U}$ of $\bar{x}$, and $\mathcal{Y}$ of $\bar{y}$ such that
\[
d(x, F^{-1}\bar{y}) \leq \eta d(\bar{y}, Fx \cap \mathcal{Y}), \quad \forall x \in \mathcal{U}
\]}.
\end{enumerate}
We show that these conditions hold for Algorithm~\ref{DRS}. Define
\[
Dx \mapsto \begin{bmatrix}\partial g(z) \\ \partial \rho^*(\zeta)\end{bmatrix}, \quad
M = \begin{bmatrix} 0 & I\\ -I & 0\end{bmatrix}, \quad
H = \begin{bmatrix} \frac{1}{\tau}I & 0 \\ -2I & \frac{1}{\sigma}I\end{bmatrix}.
\]
Define the nonlinear operator $T$ by
\begin{equation}
\label{eq:T}
T = (H+D)^{-1}(H-M).
\end{equation}
$T$ captures the iteration in Algorithm~\ref{DRS}, which can be written as $\eta^{k} = T\eta^{k-1}$, for
$\eta = \begin{bmatrix} z^T, \zeta ^T\end{bmatrix}^T$. Then we have the following lemma.
\begin{lemma}
\label{lemma:property}
Suppose that $\tau, \sigma < 1$. Then
\[||T\eta-\eta||_{H-M}^2 \leq \langle \eta^*-\eta,(H-M)(T\eta-\eta)\rangle
\]
where $\eta^*$ is such that $0 \in (D+M)\eta^*$.
\end{lemma}
The proof is given in the Appendix. \\
\begin{figure}
\begin{center}
\includegraphics[scale=0.6]{Figures/compareFinal}
\caption{\label{fig:comparison} Objective vs. iteration counts of Algorithm~\ref{DRS} for~\eqref{eq:full} (black),
vs. accelerated gradient descent (AGD) (blue) and L-BFGS (red) for~\eqref{eq:genSmoother}.
Both $\rho_1$ and $\rho_2$ are Huber losses, with $\rho_3 \equiv 0$, $N=200, n=2$ and $Q, R$ nonsingular, so~\eqref{eq:full} and~\eqref{eq:genSmoother} are equivalent. All iterations require $O(n^2N)$ operations.
DRS splitting is much faster than methods with linear convergence rates and similar iteration complexity. }
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{Figures/GradCompare-eps-converted-to}
\caption{\label{fig:timecompare} Timed run of Algorithm~\ref{DRS} vs. IPsolve for the same setup as presented in Figure~\ref{fig:comparison}.
At this scale, we see the locally linear convergence rate of the DRS. Even though IPsolve has superlinear rate, DRS wins because the slope
of the rate is very steep, and each iteration is fast. By the time DRS is done, IPsolve has had time for only taken a few iterations.}
\end{center}
\end{figure}
This establishes condition (1).
Condition (2) requires the concept of metric subregularity. This property holds for PLQ functions~\cite{LFP},
and holds for indicators of convex sets by~\cite[Theorem 3.3]{AG}, reproduced in the Appendix. This completes the proof of the theorem.
\noindent
{\bf Comparison on Smooth Nonsingular Problems.}
If the covariances, $Q, R$ are non-singular and the penalties $\rho_{1,2}$ are $\mathcal{C}^1$-smooth,
then the Kalman smoothing problem can be written as a smooth convex problem. In this case the same reformulation will work and Algorithm~\ref{DRS} will still give a local linear rate. However more common algorithms such as gradient descent and L-BFGS can also be applied. We compare the performance of these three algorithms to track a particle moving along a smooth path with $N=200$ and $n=2$.
We use non-singular $Q_k$, and Huber penalty functions.\\
As seen in Figure~\ref{fig:comparison}, Algorithm~\ref{DRS} for~\eqref{eq:full} converges far faster than either accelerated gradient descent
or LBFGS method on the equivalent nonsingular smoothing formulation~\eqref{eq:genSmoother} . This is because its convergence rate does not
depend on the condition number of the matrix $A$, so each iteration makes a lot of progress, and
we can keep the complexity of each iteration at $O(n^2N)$, same as for a matrix-vector multiply needed for a gradient evaluation,
if we factor the sparse block tridiagonal matrix $AA^T$ once at the start of the algorithm. \\
We also compare with the second-order interior point method, implemented in the IPsolve package\footnote{\url{https://github.com/UW-AMO/IPsolve}.}. Use-cases and performance of IPsolve for nonsingular Kalman smoothing is discussed in~\cite{aravkin2017generalized}. The results are shown in Figure~\ref{fig:timecompare}, where IPsolve and
DRS for the equivalent reformulation are compared for the nonsingular Huber model.
Even though DRS has at best a linear rate, the constants are very good, as they do not depend on the conditioning
of the Kalman smoothing problem.
The other advantage is that DRS can use a pre-factorized matrix, while IPsolve has to solve a modified linear system
every time; there is no simple strategy to pre-factor as with DRS.\\
The numerical experiments suggest that Algorithm~\ref{DRS} should be used regardless of whether $Q$ and $R$ are singular or not. In the next section, we focus on a rich class of singular noise models found in navigation.
\section{Introduction}
\label{sec:intro}
\vspace{-.1in}
The linear state space model is widely used in tracking and navigation~\cite{ybarshalom-2001a}, control~\cite{anderson2007optimal}, signal processing \cite{AndersonMoore}, and other time series~\cite{hyndman2002state,tsay2005analysis}.
The model assumes linear relationships between latent states with noisy observations:
\begin{equation} \label{eq:statespace}
\begin{aligned}
x_1 & = x_0 + w_1
\\
x_k & = G_kx_{k-1} + w_k, \quad k = 2, \dots, N
\\
y_k & = H_kx_k + v_k, \quad k = 1, \dots, N,
\end{aligned}
\end{equation}
where $x_0$ is a given initial state estimate, $x_1, \dots, x_N$ are unknown latent states with known linear process models $G_k$, and $y_1, \dots, y_N$ are observations obtained using known linear models $H_k$.
Data must be in the range of $H_k$; so we assume $H_k$ are surjective. \\
The errors $w_k$ and $v_k$ are assumed to be mutually independent random variables with known covariances $Q_k$ and $R_k$.
In tracking and navigation, the end goal is the estimation of the latent states $\{x_k\}$. In autocorrelated time series models (e.g. Holt-Winters c.f.~\cite{hyndman2002state}, ARMA c.f.~\cite{tsay2005analysis}), estimating the state is a necessary step to
estimating additional parameters on which $G_k$, $H_k$, $Q_k$ and $R_k$ may depend. In both settings, estimating the state sequence $\{x_k\}$ efficiently is essential. \\
{\bf Singular Covariances}. We are particularly interested in models where $Q_k$ and $R_k$ may be singular. These models arise in all settings where state-space formulations are used.
In navigation, the simplest example is the DC motor~\cite[pp. 95-97]{Ljung:99}:
\begin{equation}
\label{eq:DCmotor}
\begin{aligned}
x_{k+1} &= \left(\begin{array}{cc} 0.7 & 0 \\0.084 & 1\end{array}\right) x_k
+\left(\begin{array}{c}11.81 \\0.62 \end{array}\right) (c_k + d_k)\\
y_k &= \left(\begin{array}{cc}0 & 1\end{array}\right) x_k + v_k.
\end{aligned}
\end{equation}
Here, $y_k$ are noisy samples of the angle of the motor shaft, $c_k$ are known inputs, and $d_k$ denote random process disturbances. The covariance matrix $Q_k$ associated to $w_k$ has dimension 2 and rank 1. This example is general in the sense that singular models appear any time a single source of error is integrated
into multiple states; a pervasive phenomenon in navigation models~\cite{AndersonMoore}. \\
\begin{figure}
\begin{tabular}{l}
\hspace{-.3in}
\includegraphics[scale=0.6]{Figures/dc_plot-eps-converted-to}
\end{tabular}
\caption{\label{fig:DCmotor} DC motor~\eqref{eq:DCmotor} with outliers, generated from a Gaussian with high variance. The process covariance $Q$ is singular, but the standard RTS smoother still finds the linear minimum variance estimate (red). Our reformulation allows using robust penalties (in this case, Huber) with a singular covariances to obtain a better solution (blue).}
\end{figure}
The classic Kalman filter~\cite{kalman} and RTS smoother~\cite{RTS} assume that $w_k, v_k$ are Gaussian, and
find the {minimum variance} estimates of the state, conditioned on the observations~\cite{AndersonMoore}.
More generally, the RTS smoother finds the linear minimum variance estimator. This procedure is well defined for singular covariances $Q_k$ and $R_k$, and the smoother can be derived as a sequence of least squares projections~\cite{Ansley}.
However, when the noise is not Gaussian (e.g. in the presence of outliers), these estimates are not satisfactory;
and far better estimates can be obtained through a maximum a posteriori (MAP) estimator~\cite{aravkin2017generalized}.
The results in Figure~\ref{fig:DCmotor} are obtained using the Huber loss, which is a convex penalty function that is quadratic near the origin, but with linear tails:
\[
\rho(x) = \begin{cases}
\frac{1}{2}x^2 & \quad \text{ if } |x| \leq \kappa \\
\kappa(|x| - \frac{1}{2}\kappa) & \quad \text{ if } |x| \geq \kappa
\end{cases}
\]
Implementing a general MAP estimator for singular covariances requires a new approach.
{\bf General Kalman Smoothing}. Classic Gaussian formulations fail when outliers are present in the data, are unable to track abrupt state changes, and cannot incorporate side information through constraints. To develop effective approaches in these cases, generalized Kalman smoothing formulations have been proposed in the last few years, see~\cite{aravkin2017generalized} and the references within. The conditional mean is no longer tractable to compute these estimates, and {\it maximum likelihood} (ML) formulations are much more natural. The general form of Kalman smoothing considered in~\cite{aravkin2017generalized} is given by
\begin{equation}
\label{eq:genSmoother}
\min_{x\in X} \sum_{i=0}^n \rho_1(Q_k^{-1/2}(x_k - G_kx_{k-1}))+ \rho_2(R_k^{-1/2}(y_k - H_k x_k)),
\end{equation}
where $\rho_1, \rho_2$ are convex penalties, and $x \in X$ is a set of state-space constraints.
The two approaches agree in the nonsingular Gaussian case, where~\eqref{eq:genSmoother} becomes
a least squares (LS) problem that can be solved with classic RTS or Mayne-Fraser smoothing algorithms~\cite{aravkin2017generalized}.
{\bf Contribution}. We develop a new reformulation to extend~\eqref{eq:genSmoother} to {singular covariance models} $Q_k$ and $R_k$, and implement a Douglas-Rachford splitting (DRS) algorithm to solve this reformulation. The result in Figure~\ref{fig:DCmotor} uses Huber penalties for process and measurement, with the singular process covariance model from~\eqref{eq:DCmotor}.\\
We analyze the DRS for the singular reformulation, and show that it converges locally linearly for any piecewise linear quadratic (PLQ) loss, and that the rate does not depend on the conditioning of the system. {Even when the model is nonsingular, the new approach is potentially much faster} than first-order and second-order methods for~\eqref{eq:genSmoother}. The advantage increases as the models become more ill-conditioned; however the {\it local}
linear rate means that initialization becomes very important. \\
The paper proceeds as follows. In Section~\ref{sec:survey} we discuss prior approaches to singular models.
In Section~\ref{sec:reformulation}, we develop a constrained reformulation
of~\eqref{eq:genSmoother}, building on early work of~\cite{Paige79} for singular least squares.
In Section~\ref{sec:Opt}, we show how to efficiently optimize a wide range of
singular smoothing problems using DRS. The algorithm we use has a {\it local linear rate of convergence} for any piecewise
linear-quadratic penalties $\rho_1, \rho_2$ in~\eqref{eq:genSmoother}, and each iteration is efficiently and stably computed by exploiting dynamic problem structure. We compare the new algorithm to first-order methods, L-BFGS, and IPsolve, a toolbox specifically developed for PLQ Kalman smoothing (for nonsingular formulations).
In Section~\ref{sec:vehicleModel}, we present a navigation model that uses singular errors. In Section~\ref{sec:numerics} we apply
the methodology to analyze data from a drifting mooring as a proxy for an
autonomous underwater vehicle.
\section{Related Work}
\label{sec:survey}
\vspace{-.1in}
Several approaches in the literature deal with singular models. We give a brief description and
references for each. To ground the discussion,
consider tracking a particle moving along a smooth path in space, where state comprises velocity and position.
Singular models arise naturally in this situation.
We can model velocity as subject to error, and position as a deterministic integral:
\begin{equation}
\label{eq:integral}
\begin{aligned}
x_{k+1} &= x_{k} + \Delta t \dot x_{k} \\
\dot x_{k+1} &= \dot x_{k} + \epsilon_{k}.
\end{aligned}
\end{equation}
Here, the process covariance matrix $Q_k$ has rank one.\\
{\bf Using the original Kalman filter.} In the linear Gaussian setting, the original Kalman
filter does not require $Q$ and $R$ to be invertible. Applying the Kalman filter (and RTS smoother)
will return the minimum variance estimate for singular innovation/measurement errors~\cite{AndersonMoore}.
The limitation is that we cannot consider the general optimization context~\eqref{eq:genSmoother}, which we need
to incorporate robustness to outliers and constraints for prior information (see example in Figure~\ref{fig:DCmotor}). \\
{\bf Changing the model.} A common approach is to modify the model to make $Q_k, R_k$ nonsingular. Treating~\eqref{eq:integral}
as a discretization of a stochastic differential equation (SDE), many authors opt for a nonsingular error model~\cite{Jaz,Oks,Bell2008,YAA}
\[
Q_k = \begin{bmatrix} \Delta t_k & \Delta t_k^2/2 \\ \Delta t_k^2/2 & \Delta t_k^3/3\end{bmatrix},
\]
derived by computing the variance of a discretized process noise term, similar to what is done in Section~\ref{sec:vehicleModel}, see~\eqref{eq:Qderiv}.
The approach has limitations for navigation models with high-dimensional states driven by low-dimensional errors.
The low-dimensional error structure should simplify estimation, but instead this approach introduces full-dimensional
and ill-conditioned $Q_k$. In addition, making $Q_k$ nonsingular is antithetical to state-space formulations for models such as ARMA, which use singularity to enforce auto-regressive constraints. \\
{\bf Change of coordinates.}
When only $R_k$ are singular, \cite{AndersonMoore} suggests making a change of coordinates in the measurement variables and then projecting to remove the extra dimensions. The projections can vary between time points, and the approach does not extend to the singular state equation~\eqref{eq:integral}.\\
{\bf Pseudo-inverse with orthogonality constraints.}
The formulation that is closest to ours is that of~\cite{OGLB}, who replace the inverse of $Q_k$ by a pseudo-inverse, and add orthogonality constraints (namely that projection onto the null space of $Q_k$ is zero). With potentially singular $Q_k$ and $R_k$, the maximum likelihood estimate for the Gaussian/LS model can be formulated as
\begin{equation}
\label{eq:singularOne}
\begin{aligned}
\min_x &\sum_k ||Q_k^{\dagger/2}(x_k - G_kx_{k-1})||^2 + ||R_k^{\dagger/2}(y_k - H_kx_k)||^2\\
\quad &\text{s.t. } Q_k^\perp(x_k - G_kx_{k-1}) = 0, \quad R_k^\perp(y_k - H_kx_k) = 0 \\
\quad & \quad \text{ for all } k = 1, \dots, N,
\end{aligned}
\end{equation}
see~\cite[Appendix A]{aravkin2017generalized}. This requires computing both the pseudo-inverse and orthogonality constraints. \\
{\bf Constrained reformulation.}
The reformulation we choose was first used by Paige \cite{Paige79}. Given the singular least squares problem
\[
\min_{x} \|Q^{\dagger/2}(Ax-b)\|^2 \quad \mbox{s.t.} \quad Q^\perp (Ax-b) = 0,
\]
we can instead write it as
\begin{equation}
\label{eq:singularTwo}
\min_{x, u} \|u\|^2 \quad \mbox{s.t.} \quad \mbox Q^{1/2} u = Ax-b.
\end{equation}
It is easy to see~\eqref{eq:singularOne} and~\eqref{eq:singularTwo} are equivalent; the latter is more elegant, and only requires computing a root of $Q$,
rather than using both $Q$ and $Q^\dagger$.
When $Q$ is invertible, we can eliminate $u$ from both formulations
and reduce to a least squares problem in $x$.
Splitting the affine constraint from the original penalty has theoretical and practical advantages for general Kalman smoothing,
as shown in the next sections.
\section{Efficient Splitting Methods for Singular Kalman Smoothing}
\label{sec:Opt}
We consider problems of the form
\begin{equation}
\min_{z} \rho(z) \quad \mbox{s.t.} \quad Az = \hat w.
\label{eq:main}
\end{equation}
Where
\[
\rho(z) = \rho_1(u) + \rho_2(t) + \rho_3(x),
\]
With $\rho_1, \rho_2, \rho_3$ al PLQ (piecewise linear quadratic) functions.
\subsection{Overall optimization: DRS}
To solve (\ref{eq:main}) we employ a Douglas-Rachford splitting algorithm. This is an iterative primal-dual algorithm with each iteration consisting of two sub-problems:
\begin{equation}
\lambda^{n+1} = \mbox{prox}_{\sigma \rho^*}(\lambda^n + \sigma \bar{z}^n) = \argmin_x \frac{1}{2\sigma}\|x-\lambda^n - \sigma \bar{z}^n\|^2 + \rho*(\lambda^n + \sigma \bar{z}^n )
\label{eq:proxsub}
\end{equation}
\begin{equation}
z^{n+1} = \arg\min_t \|t - z^n + \tau \lambda^{n+1}\|^2 \quad \text{ s.t. } At = \hat{w}
\label{eq:lssub}
\end{equation}
And a momentum term:
\[
\bar{z}^{n+1} = 2z^{n+1} - z^n
\]
For problems of type (\ref{eq:main}) the DRS algorithm converges locally linearly to a solution. This can be shown by replicating an argument in \cite{LFP} which uses metric subregularity of operators to show linear convergence. We recall the definition of metric subregulairty:
\begin{definition} A mapping $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ is called \textit{metrically subregular} at $\bar{x}$ for $\bar{y}$ if $(\bar{x}, \bar{y}) \in $ gra $F$ and there exists $\eta \in [0 , infty)$, neighborhoods $\mathcal{U}$ of $\bar{x}$, and $\mathcal{Y}$ of $\bar{y}$ such that
\[
d(x, F^{-1}\bar{y}) \leq \eta d(\bar{y}, Fx \cap \mathcal{Y}), \quad \forall x \in \mathcal{U}
\]
\end{definition}
In \cite{LFP} they argue that all PLQ functions have subdifferentials that are metrically subregular. To apply their argument to the case of (\ref{eq:main}) we additionally need to verify that the subdifferential of the indicator function for a affine subspace is also metrically subregular. However this follows from the quadratic growth condition for subdifferentials of convex functions , namely
\begin{theorem}[\cite{AG}, Theorem 3.3]
For a proper closed convex function $f$, the subdifferential $\partial f$ is metrically subregular at $\bar{x}$ for $\bar{y}$ with $(\bar{x}, \bar{y}) \in $ gra $\partial f$ if and only if there exists a positive constant $c$ and a neighborhood $\mathcal{U}$ of $\bar{x}$ such that
\[
f(x) \geq f(\bar{x}) + \langle \bar{y}, x-\bar{x}\rangle + cd^2(x, (\partial f)^{-1}(\bar{y})), \quad \forall x \in \mathcal{U}
\]
\end{theorem}
In the case that $f$ is an indicator function there is nothing to check, it is true by definition of $\partial f$. Now the same proof as in \cite{LFP} will show local linear convergence.
With a fast convergence rate established we now turn to per-iteration complexity as the best way to increase performance.
\subsection{Proximity operators for $\rho$}
We define
\[
\mbox{prox}_{\gamma \rho}(z) = \arg\min_x \frac{1}{2\gamma}\|x-z\|^2 + \rho(x).
\]
The proximity operator appearing in the sub-problem of DRS involves $\rho^*$ instead of $\rho$ so we apply the Moreau decomposition to relate the proximity operator of $\rho$ and $\rho^*$ by
\[
\mbox{prox}_{\sigma \rho}(z) + \sigma \mbox{prox}_{\rho^*/\sigma}(z/\sigma) = z
\]
Thus only $\mbox{prox}_{\rho}(z)$ needs to be computed. By assumption $\rho(z) = \rho_1(u) + \rho_2(t) + \rho_3(x)$ thus computing $\mbox{prox}_{\rho}(z)$ reduces to separately computing $\mbox{prox}_{\rho_1}(u)$, $\mbox{prox}_{\rho_2}(t)$, $\mbox{prox}_{\rho_3}(x)$. For complexity analysis of DRS we assume that all of these proximal functions can be computed via an algorithm that converges linearly.
This assumption is automatically satisfied if $\rho_i, i=1,2,3$ are PLQ (piecewise linear quadratic) by using an interior point method. This assures that our assumption is not very restrictive as the class of PLQ functions is quite large and includes many common choices of penalty functions such as $\|\cdot\|_1, \|\cdot\|_2^2$, the huber loss function, and many others. A number of PLQ functions and their computed proximal operators are discussed in the appendix.
\subsection{Proximity operator for constraint}
In each iteration (\ref{eq:lssub}) takes the form:
\[
\min_{At = w} \frac{1}{2}\|t - c\|^2.
\]
Where $c$ is a constant that changes from iteration to iteration. Examining optimality conditions we see that finding the optimum is equivalent to solving the linear system
\[
\begin{bmatrix}I & A^T\\A & 0\end{bmatrix}\begin{bmatrix}t\\ \nu\end{bmatrix} = \begin{bmatrix}c\\ \hat{w}\end{bmatrix}
\]
Which can be written equivalently
\[
\begin{bmatrix}I & A^T\\0 & AA^T\end{bmatrix} \begin{bmatrix}t\\ \nu\end{bmatrix} = \begin{bmatrix}c\\ Ac - \hat{w}\end{bmatrix}
\]
Thus solving $AA^T \nu = Ac - \hat{w}$ then back substitution will yield the optimal $t$. As $AA^T$ appears in every iteration we compute a factorization of this matrix once before implementing the algorithm and use this to efficiently solve (\ref{eq:lssub}) in every iteration.
\subsection{Factorization of $AA^T$}
Recall from the definition that $A$ is lower block bi-diagonal and thus $AA^T$ is block tri-diagonal. We will compute the Cholesky factorization of $AA^T = LL^T$ Where $L$ is a square block bi-diagonal matrix. This can be done efficiently by iterating through the blocks of $AA^T$ as follows. Let
\[
AA^T =
\begin{bmatrix}
a_1& b_1^T & & \\
b_1 & a_2 & b_2^T & \\
& b_2 & a_3 & b_3^T \\
&&b_3 & a_4
\end{bmatrix}
\quad
L =
\begin{bmatrix}
c_1& & & \\
d_1 & c_2 & & \\
& d_2 & c_3 & \\
&&d_3 & c_4
\end{bmatrix}
\]
Multiplying out $LL^T$ it follows that
\[
a_1 = c_1c_1^T, \quad d_1 = b_1 c_1^{-T}
\]
Thus to compute $c_1$ we simple compute the Cholekly factorization of $a_1$. Then
\[
c_2c_2^T = a_2 - b_1a_1^{-1}b_1^T, \quad d_2 = b_2c_2^{-1}.
\]
Take $s_1 = a_1$, and proceeding recursively, we have
\[
s_k = a_k - b_{k-1}s_{k-1}^{-1}b_{k-1}^T
\]
and
\[
c_kc_k^T = s_k, \quad d_k = b_kc_k^{-1}.
\]
$c_k$ is then found by finding the Cholesky factorization of $s_k$, then $c_k$ is inverted to find $d_k$.
The overall speed of computing this factorization is not of great importance as it only will be computed a single time.
Once $L$ has been computed we can solve $LL^T \nu = Ac - \hat{w}$ in $O(n^2N)$ operations as the blocks of $L$ are triangular.
\section{Navigation Models}
\label{sec:vehicleModel}
\vspace{-.1in}
Autonomous navigation requires high-fidelity tracking using occasional GPS and frequent depth/height, gyrocompass, and linear acceleration data. Gyro, compass, and linear acceleration are readily available from inertial measurement units (IMUs). \\
In this section, we develop a simple kinematic model that is trivially applicable to any
vehicle, and is particularly appropriate for many underwater vehicle applications, where
accelerations are heavily damped and autonomous vehicles often travel in long straight lines (e.g. for survey work). When the attitude is known or changing slowly, the model can be linearized effectively and the situation simplifies considerably;
our synthetic examples and underwater survey application use linearized models. \\
{\bf Linear Singular Navigation Model.}
For a vehicle that is well-instrumented in attitude, the uncertainty in position
(and the x-y states in particular) is typically orders of magnitude larger than
the uncertainty in attitude. In practice, we simplify the full
nonlinear vehicle process model to track only position states
$( x, y, z)$, while assuming that
the attitude states $(r, p,h)$ are directly available from the most recent sensor
measurements. To make the model linear, the position and its derivatives are
referenced to the local-level frame.\\
To incorporate linear acceleration measurements from
an inertial measurement unit (IMU), we must track both linear velocities
and linear acceleration in the state vector. This leads to the augmented state
\begin{equation}
\vect{x}_s = [x,y,z,\dot{x},\dot{y},\dot{z},\ddot{x},\ddot{y},\ddot{z}]^\top.
\end{equation} \label{lin_pm}
The linear kinematic process model is given by
\begin{align}
\dot{\vect{x}}_s &=
\underbrace{\left[\begin{array}{ccc}
\mtrx{0} & \mtrx{I} & \mtrx{0} \\
\mtrx{0} & \mtrx{0} & \mtrx{I} \\
\mtrx{0} & \mtrx{0} & \mtrx{0} \end{array}\right]}_{\mbox{$\mtrx{F}_s$}} \vect{x}_s +
\underbrace{\left[\begin{array}{c}
\mtrx{0} \\
\mtrx{I} \\
\mtrx{0} \end{array}\right]}_{\mbox{$\mtrx{G}_s$}} \vect{w}_s \label{x_s-dot},
\end{align}
where $\vect{w}_s \sim \mathcal{N}(0,\mtrx{Q}_s)$ is zero-mean
Gaussian noise. \\
The linear process model \eqref{x_s-dot} is usually
discretized using a Taylor series:
\begin{align}
\vect{x}_{s_{k+1}} &= \mtrx{F}_{s_k} \vect{x}_{s_k} + \vect{w}_{s_k} \label{disc_s}\\
\mtrx{F}_{s_k} &= e^{\mtrx{F}_s T}\\
&= \mtrx{I} + \mtrx{F}_sT +{\frac{1}{2!}\mtrx{F}_s^2T^2} + \cancelto{0}{\frac{1}{3!}\mtrx{F}_s^3T^3} + \cdots \nonumber\\
&= \left[\begin{array}{ccc}
\mtrx{I}&\mtrx{I}T&\frac{1}{2}\mtrx{I}T^2\\
\mtrx{0}&\mtrx{I}&\mtrx{I}T\\
\mtrx{0}&\mtrx{0}&\mtrx{I} \end{array} \right] \nonumber
\end{align}
where the higher order terms are {\it identically zero} because of the
structure of $\mtrx{F}_s$, resulting in a simple closed-form solution
for $\mtrx{F}_{s_k}$.
The discretized process noise
\begin{equation}
\vect{w}_{s_k} = \int^T_0 e^{\mtrx{F}_s(T - \tau)}\mtrx{G}_s \vect{w}_s(\tau) d\tau,
\end{equation}
is a zero-mean Gaussian, with covariance given by
\begin{equation}
\label{eq:Qderiv_0}
\mtrx{Q}_{s_k} = \int^T_0 e^{\mtrx{F} (T-\tau)} \mtrx{G} \mtrx{Q} \mtrx{G}^\top
e^{\mtrx{F}^\top (T-\tau)} d\tau,
\end{equation}
which simplifies to
\begin{equation}
\label{eq:Qderiv}
\mtrx{Q}_{s_k} =
\left[\begin{array}{ccc}
\frac{1}{3}T^3&\frac{1}{2}T^2&0 \\
\frac{1}{2}T^2&T &0 \\
0 &0 &0 \end{array} \right] \mtrx{Q}_s,
\end{equation}
for
\begin{equation*}
e^{\mtrx{F} (T-\tau)} = \left[\begin{array}{ccc}
\mtrx{I}&\mtrx{I}(T-\tau)&\frac{1}{2}\mtrx{I}(T-\tau)^2\\
\mtrx{0}&\mtrx{I}&\mtrx{I}(T-\tau)\\
\mtrx{0}&\mtrx{0}&\mtrx{I} \end{array} \right],\ \ \mtrx{G} = \left[\begin{array}{c}
\mtrx{0}\\
\mtrx{I}\\
\mtrx{0} \end{array} \right].
\end{equation*}
In practice this can lead to wildly incorrect results. In Figure~\ref{fig:badexample},
we show the estimate of position obtained from a subset of the navigation data.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=1]{Figures/bad_example_small.pdf}
\caption{Full position data is plotted using green crosses; the smoother only uses
a subset highlighted with gray disks. The model is used to provide the
initialization for the solver, shown in blue, by forward propagating
from the first data point. With $Q$ as in~\eqref{eq:Qderiv}, the model forces a zero
acceleration constraint. Combined with a non-zero initial velocity, this results
in an erroneous initialization, which even high
confidence in the observed datapoints is unable to overcome, yielding a counterintuitive
result.}
\label{fig:badexample}
\end{center}
\end{figure}
The model is defined with constraints
\[
Q_{s_k}^{1/2}u_{s_k} = F_{s_k}x_{s_{k+1}} - x_{s_k}.
\]
The $Q$ in~\eqref{eq:Qderiv} forces the acceleration to be $0$ across the entire model because the lower right corner
is set to $0$. As a result, the initialized track can be biased away from the
data by a fixed
velocity, obtained by finding the slope from the most recent position data. The available data do not agree, but the constraint is stronger; the
information is integrated in a counter-intuitive way. \\
Instead, we model the covariance as if the error were the next term in the Taylor series approximation,
a technique suggested by~\cite{YAA}. More precisely we set covariance to be the outer product, $\Gamma^T \Gamma$ where
\[
\Gamma = \begin{bmatrix}\frac{1}{3!}\mtrx{I}T^3 & \frac{1}{2!}\mtrx{I}T^2 & \mtrx{I}T\end{bmatrix}
\]
This leads to a rank 3 covariance for a $9\times 9$ matrix for a model that comprises
position, velocity, and acceleration in 3D space. This model avoids the issue in Figure~\ref{fig:badexample}.\\
{\bf Measurement Models for the IMU.}
The inertial measurement unit (IMU) does not measure position or velocity, just linear and angular accelerations.
To use these measurements, we track linear acceleration as part of the state. However, the
acceleration measured by the IMU is relative to the physical frame of the
vehicle on which it is mounted, while the acceleration of the state
is relative to the navigation frame. A coordinate transformation between these frames is required for a comparison;
we use heading, pitch, and roll of the vehicle for the linear
model.
The transformation from body-frame to local-level is given by \textbf{$R(\vect{\varphi})$},
where $\vect{\varphi}$ comprises heading $h$, pitch $p$, and roll $r$:
\begin{equation}
\mtrx{R}(\vect{\varphi}) = \mtrx{R}^\top_h \mtrx{R}^\top_p \mtrx{R}^\top_r,
\end{equation}
where $R_h$, $R_p$, and $R_r$ are given by
\begin{equation}
\left[\begin{array}{ccc}
c{h} & s{h} & 0 \\
-s{h} & c{h} & 0 \\
0 & 0 & 1 \end{array}\right],\quad
\left[\begin{array}{ccc}
c{p} & 0 & -s{p} \\
0 & 1 & 0 \\
sp & 0 & cp \end{array}\right], \quad
\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 &c{r} & s{r} \\
0 & -s{r} & c{r} \end{array}\right]
\end{equation}
with $c\cdot$ and $s\cdot$ shorthand for $\cos(\cdot)$
and $\sin(\cdot)$. \\
Any navigation system that relies on an IMU needs occasional measurements that inform the position
(e.g. GPS), otherwise the error in position estimates grows without bound. We are given
these data from a separate source, sampled at a lower update rate than that of the IMU.
For any $s$ where such data is available, we have the measurement model
\[
H_s = \begin{bmatrix} I_{3 \times 3} & 0_{3 \times 6}\\ 0_{3 \times 6} & R(\vect{\varphi})\end{bmatrix}, \quad
\vect{z}_s = \begin{bmatrix} \vect{b}^\top & \ddot{x}_{\textrm{meas}} & \ddot{y}_{\textrm{meas}} & \ddot{z}_{\textrm{meas}}\end{bmatrix}^\top.
\]
If there is no position data measured at time $s$ then we use the model
\[
H_s = \begin{bmatrix} 0_{3 \times 3} & 0_{3 \times 6}\\ 0_{3 \times 6} & R(\vect{\varphi})\end{bmatrix}, \quad
\vect{z}_s = \begin{bmatrix} 0& \ddot{x}_{\textrm{meas}} & \ddot{y}_{\textrm{meas}} & \ddot{z}_{\textrm{meas}}\end{bmatrix}^\top.
\]
The covariance used for measurement data depends on whether there was position data available:
\[
R_s = \begin{bmatrix} 0_{3 \times 3} & 0_{3 \times 3}\\ 0_{3 \times 3} & r_s I_{3 \times 3}\end{bmatrix}, \quad R_s = \begin{bmatrix} U_s & 0_{3 \times 3}\\ 0_{3 \times 3} & r_s I_{3 \times 3}\end{bmatrix}
\]
wher ethe top $3\times 3$ block is either $0$ (position not available) or $U$, a diagonal matrix reflecting position uncertainty (position is available).
The scalar $r_s$ models uncertainty in IMU measurements.
\section{Vehicle Model}
\label{sec:vehicleModel}
The filter fuses depth, gyrocompass, and linear acceleration data from
the vehicle. A description of the full nonlinear process model is
presented in Section \ref{sec:nonlin_pm}, the linear simplification in
Section \ref{sec:lin_pm}, the process prodiction in in Section
\ref{sec:proc_pred}, and the measurement models in Section
\ref{sec:meas_model}.
\subsection{Nonlinear Process Model}
\label{sec:nonlin_pm}
The complete state vector, denoted $\vect{x}$, contains pose and attitude, as
well as body-frame linear and angular velocities
\begin{equation}
\vect{x} = [\vect{s}^\top,\vect{\varphi}^\top,\vect{\upsilon}^\top,\vect{\omega}^\top]^\top
\end{equation}
\begin{equation}
\vect{s} = \left[\begin{array}{c}
x \\
y \\
z \end{array}\right],~
\vect{\varphi} = \left[\begin{array}{c}
r \\
p \\
h \end{array}\right],~
\vect{\upsilon} = \left[\begin{array}{c}
u \\
v \\
w \end{array}\right],~
\vect{\omega} = \left[\begin{array}{c}
a \\
b \\
c \end{array}\right]
\end{equation}
where $\vect{s}$ is the vehicle pose in the local frame,
$\vect{\varphi}$ is the vehicle attitude (Euler roll,
pitch, heading), $\vect{\upsilon}$ is the body-frame linear velocity,
and $\vect{\omega}$ is the body-frame angular velocity.
The filter uses a vehicle-agnostic constant-velocity process model for the
vehicle, which is defined as
\begin{align}
\dot{\vect{x}} &=
\underbrace{\left[\begin{array}{cccc}
\mtrx{0}&\mtrx{0}&\mtrx{R}(\vect{\varphi})&\mtrx{0} \\
\mtrx{0}&\mtrx{0}&\mtrx{0}&\mtrx{\mathcal{J}(\vect{\varphi})} \\
\mtrx{0}&\mtrx{0}&\mtrx{0}&\mtrx{0} \\
\mtrx{0}&\mtrx{0}&\mtrx{0}&\mtrx{0} \end{array}\right] \vect{x}}_{\mbox{$\vect{f}(\vect{x}(t))$}} +
\underbrace{\left[\begin{array}{cc}
\mtrx{0}&\mtrx{0} \\
\mtrx{0}&\mtrx{0} \\
\mtrx{I}&\mtrx{0} \\
\mtrx{0}&\mtrx{I} \end{array}\right]}_{\mbox{$\mtrx{G}$}} \vect{w} \label{x_v-dot}
\end{align}
where $\mtrx{R}(\vect{\varphi})$ is the transformation from body-frame
to local-level linear velocities, $\mtrx{\mathcal{J}}(\vect{\varphi})$
is the transformation from body-frame angular velocities to Euler
rates, and $\vect{w} \sim \mathcal{N}(0,\mtrx{Q})$ is zero-mean
Gaussian process noise in the velocity
term. $\mtrx{R}(\vect{\varphi})$ and
$\mtrx{\mathcal{J}}(\vect{\varphi})$ are found by solving
\begin{equation}
\mtrx{R}(\vect{\varphi}) = \mtrx{R}^\top_h \mtrx{R}^\top_p \mtrx{R}^\top_r
\end{equation}
where
\begin{equation}
\mtrx{R}_{h} = \left[\begin{array}{ccc}
\cos{h} & \sin{h} & 0 \\
-\sin{h} & \cos{h} & 0 \\
0 & 0 & 1 \end{array}\right],
\mtrx{R}_{p} = \left[\begin{array}{ccc}
\cos{p} & 0 & -\sin{p} \\
0 & 1 & 0 \\
\sin{p} & 0 & \cos{p} \end{array}\right], ~
\mtrx{R}_{r} = \left[\begin{array}{ccc}
1 & 0 & 0 \\
0 &\cos{r} & \sin{r} \\
0 & -\sin{r} & \cos{r} \end{array}\right]
\end{equation}
and
\begin{align}
\vect{\omega} &= \left[\begin{array}{c} \dot{r} \\ 0 \\
0 \end{array}\right] + R_r \left[\begin{array}{c} 0 \\
\dot{p} \\ 0 \end{array}\right] + R_r R_p
\left[\begin{array}{c}
0 \\ 0 \\ \dot{h} \end{array}\right] \nonumber\\
&= \underbrace{\left[\begin{array}{ccc}
1 & 0 & -\sin{p} \\
0 & \cos{r} & \sin{r} \cos{p} \\
0 & -\sin{r} & \cos{r} \cos{p} \end{array}\right]
}_{\mbox{$\mtrx{\mathcal{J}}^{-1}$}} \dot{\varphi}
\end{align}
\begin{align}
\mtrx{\mathcal{J}}
&= \left[\begin{array}{ccc}
1 & \sin{r}\tan{p} & \cos{r}\tan{p} \\
0 & \cos{r} & -\sin{r} \\
0 & \sin{r}\sec{p} & \cos{r} \sec{p} \end{array}\right] .
\end{align}
Note that the vehicle process model does not include a control input term
$\vect{u}(t)$ because we do not assume a dynamic model for the vehicle. The use
of a simple kinematic model makes the algorithm trivially applicable to any
vehicle, and is appropriate for many underwater vehicle applications because
accelerations are heavily damped by the water and autonomous vehicle's often are
used for survey work, which involves travelling in long straight lines.
We linearize the vehicle process model \eqref{x_v-dot} about
$\vect{\mu}$, our estimate of the vehicle state at time $t$, using
the Taylor series expansion
\begin{equation}\begin{split}
\dot{\vect{x}}(t) &= \vect{f}(\vect{\mu}) +
\mtrx{F}(\vect{x}(t) - \vect{\mu})\\
&\quad+ HOT + \mtrx{G} \vect{w}(t)
\end{split}\end{equation}
where
\begin{equation}
\mtrx{F} = \left. \dfrac{\partial \vect{f}(\vect{x})}{\partial
\vect{x}}\right|_{\vect{x}(t) = \vect{x}(t_k)}
\end{equation}
and $HOT $ denotes higher order terms. Dropping the $HOT$ and rearranging we get
\begin{align}
\dot{\vect{x}}(t) &\approx \mtrx{F} \vect{x}(t) + \underbrace{\vect{f}(\vect{\mu}) - \mtrx{F} \vect{\mu}}_{\mbox{$\vect{u}(t)$}} + \mtrx{G} \vect{w}(t) \\
&= \mtrx{F} \vect{x}(t) + \vect{u}(t) + \mtrx{G} \vect{w}(t) \label{linz}
\end{align}
where $\vect{f}(\vect{\mu}) - \mtrx{F} \vect{\mu}$
is treated as a constant pseudo-input term $\vect{u}(t) $.
In order to discretize the linearized vehicle process model
we rewrite \eqref{linz} as
\begin{equation}
\dot{\vect{x}}(t) = \mtrx{F} \vect{x}(t) + \mtrx{B} \vect{u}(t) + \mtrx{G} \vect{w}(t)
\end{equation}
where $\mtrx{B} = \mtrx{I}$.
Assuming zero-order hold and using the standard method
\cite{ybarshalom-2001a} to discretize over a
time step $T$ we solve for $\mtrx{F}_{v_k}$ and $\mtrx{B}_{v_k}$ in
the discrete form of the process model:
\begin{align}
\vect{x}_{v_{k+1}} &= \mtrx{F}_{v_k} \vect{x}_{v_k} + \mtrx{B}_{v_k} \vect{u}_k + \vect{w}_{v_k} \label{disc_v}\\
\mtrx{F}_{v_k} &= e^{\mtrx{F} T} \\
\mtrx{B}_{v_k} &= \int^T_0 e^{\mtrx{F} (T-\tau)}\mtrx{B} d\tau \nonumber\\
&= e^{\mtrx{F} T}\int^T_0 e^{-\mtrx{F} \tau}d\tau.
\end{align}
The discretized process noise $\vect{w}_{v_k}$ has the form
\begin{equation}
\vect{w}_{v_k} = \int^T_0 e^{\mtrx{F} (T-\tau)}\mtrx{G} \vect{w}(\tau) d\tau
\end{equation}
for which we can calculate the mean:
\begin{align}
E\left[\vect{w}_{v_k}\right] &= E\left[ \int^T_0 e^{\mtrx{F} (T-\tau)}\mtrx{G} \vect{w}(\tau) d\tau \right] \label{mean}\\
&= \int^T_0 e^{\mtrx{F} (T-\tau)}\mtrx{G} \cancelto{0}{E[\vect{w}(\tau)]} d\tau \nonumber\\
&= 0. \nonumber
\end{align}
To calculate the variance of the discretized vehicle process noise we
make use of the facts that the expected value can be brought inside
the integral because it is a linear operator and that the noise vector
$\vect{w}$ is independent and identically distributed in time so that
the covariance $E\left[\vect{w}(\tau) \vect{w}^\top_v(\gamma)\right]$
is zero except when $\gamma = \tau$:
\begin{align}
\mtrx{Q}_{v_k} &= E\left[\vect{w}_{v_k}\vect{w}_{v_k}^\top\right] \label{var}\\
&= E\left[\int^T_0 e^{\mtrx{F} (T-\tau)}\mtrx{G} \vect{w}(\tau) d\tau \int^T_0 \left(e^{\mtrx{F} (T-\gamma)}\mtrx{G} \vect{w}(\gamma) \right)^\top d\gamma \right] \nonumber\\
&= E\left[\int^T_0 \int^T_0 e^{\mtrx{F} (T-\tau)} \mtrx{G} \vect{w}(\tau) \vect{w}^\top_v(\gamma) \mtrx{G}^\top e^{\mtrx{F}^\top (T-\gamma)} d\tau d\gamma\right] \nonumber\\
&= \int^T_0 \int^T_0 e^{\mtrx{F} (T-\tau)} \mtrx{G} \underbrace{E\left[\vect{w}(\tau) \vect{w}^\top_v(\gamma)\right]}_{\mbox{$\mtrx{Q} \delta(\tau-\gamma)$}} \mtrx{G}^\top e^{\mtrx{F}^\top (T-\gamma)} d\tau d\gamma \nonumber\\
&= \int^T_0 e^{\mtrx{F} (T-\tau)} \mtrx{G} \mtrx{Q} \mtrx{G}^\top e^{\mtrx{F}^\top (T-\tau)} d\tau. \nonumber
\end{align}
\subsection{Linear Process Model}
\label{sec:lin_pm}
For a vehicle that is well-instrumented in attitude, the uncertainty in position
(and the x-y states in particular) is typically orders of magnitude larger than
the uncertainty attitude. As such, in practice, we can often simplify the full
nonlinear vehicle process model to track only position, $\vect{s} =
\left[\begin{array}{ccc} x & y & z\end{array}\right]^\top$, while assuming that
the attitude, $\vect{\varphi} = \left[\begin{array}{ccc} r & p &
h\end{array}\right]^\top$, can be known from the most recent sensor
measurements. To make the model linear, the position and its derivatives will all be
referenced to the local-level frame.
In addition, in order to incorporate linear accleration measurements from
an inertial measurement unit (IMU), we will include linear acceleration in
the state vector in addition to the linear velocities. This leads to the state
vector
\begin{equation}
\vect{x}_s = [x,y,z,\dot{x},\dot{y},\dot{z},\ddot{x},\ddot{y},\ddot{z}]^\top.
\end{equation} \label{lin_pm}
We still employ a linear constant-velocity process model, which simplifies to
\begin{align}
\dot{\vect{x}}_s &=
\underbrace{\left[\begin{array}{ccc}
\mtrx{0} & \mtrx{I} & \mtrx{0} \\
\mtrx{0} & \mtrx{0} & \mtrx{I} \\
\mtrx{0} & \mtrx{0} & \mtrx{0} \end{array}\right]}_{\mbox{$\mtrx{F}_s$}} \vect{x}_s +
\underbrace{\left[\begin{array}{c}
\mtrx{0} \\
\mtrx{I} \\
\mtrx{0} \end{array}\right]}_{\mbox{$\mtrx{G}_s$}} \vect{w}_s \label{x_s-dot},
\end{align}
where $\vect{w}_s \sim \mathcal{N}(0,\mtrx{Q}_s)$ is zero-mean
Gaussian process noise, which we still inlcude in the veloity term.
The linear process model \eqref{x_s-dot} is
discretized in the same fashion as the nonlinear vehicle process model:
\begin{align}
\vect{x}_{s_{k+1}} &= \mtrx{F}_{s_k} \vect{x}_{s_k} + \vect{w}_{s_k} \label{disc_s}\\
\mtrx{F}_{s_k} &= e^{\mtrx{F}_s T}\\
&= \mtrx{I} + \mtrx{F}_sT + \cancelto{0}{\frac{1}{2!}\mtrx{F}_s^2T^2} + \cancelto{0}{\frac{1}{3!}\mtrx{F}_s^3T^3} + \cdots \nonumber\\
&= \left[\begin{array}{ccc}
\mtrx{I}&\mtrx{I}T&\mtrx{0}\\
\mtrx{0}&\mtrx{I}&\mtrx{I}T\\
\mtrx{0}&\mtrx{0}&\mtrx{I} \end{array} \right] \nonumber
\end{align}
where the higher order terms are {\it identically zero} because of the
structure of $\mtrx{F}_s$, resulting in a simple closed-form solution
for $\mtrx{F}_{s_k}$. Note that $\mtrx{B}_{s_k} = \mtrx{0}$ because
$\mtrx{B}_s = \mtrx{0}$.
The discretized process noise
\begin{equation}
\vect{w}_{s_k} = \int^T_0 e^{\mtrx{F}_s(T - \tau)}\mtrx{G}_s \vect{w}_s(\tau) d\tau,
\end{equation}
can also be shown to be zero-mean Gaussian using formulas \eqref{mean}
and \eqref{var}, such that $\vect{w}_{s_k} \sim
\mathcal{N}(0,\mtrx{Q}_{s_k})$. Due to the structure of
$\mtrx{F}_{s_k}$, we can also significantly simplify the covariance matrix:
\begin{align}
\mtrx{Q}_{s_k} &= \int^T_0 e^{\mtrx{F} (T-\tau)} \mtrx{G} \mtrx{Q} \mtrx{G}^\top e^{\mtrx{F}^\top (T-\tau)} d\tau \nonumber \\
&= \int^T_0
\left[\begin{array}{ccc}
\mtrx{I}&\mtrx{I}(T-\tau)&\mtrx{0}\\
\mtrx{0}&\mtrx{I}&\mtrx{I}(T-\tau)\\
\mtrx{0}&\mtrx{0}&\mtrx{I} \end{array} \right]
\left[\begin{array}{c}
\mtrx{0}\\
\mtrx{I}\\
\mtrx{0} \end{array} \right]
\mtrx{Q}_s
\left[\begin{array}{c}
\mtrx{0}\\
\mtrx{I}\\
\mtrx{0} \end{array} \right]^\top
\left[\begin{array}{ccc}
\mtrx{I}&\mtrx{I}(T-\tau)&\mtrx{0}\\
\mtrx{0}&\mtrx{I}&\mtrx{I}(T-\tau)\\
\mtrx{0}&\mtrx{0}&\mtrx{I} \end{array} \right]
d\tau \nonumber \\
&= \left[\begin{array}{ccc}
\frac{1}{3}T^3&\frac{1}{2}T^2&0 \\
\frac{1}{2}T^2&T &0 \\
0 &0 &0 \end{array} \right] \mtrx{Q}_s,
\end{align}
which, as expected, is singular, beacuse the velocity term is an exact
derivative of the acceleration term with no additive noise.
\subsection{Process Prediction}
\label{sec:proc_pred}
The complete state process prediction is found by substituting the
discrete-time linearized or linear vehicle process model (\eqref{disc_v} or \eqref{disc_s} respectively) into the
discrete-time linearized Kalman process prediction
equation \eqref{ekfpred1lin},
\begin{align}
\vect{\mu}_{k+1|k} &= \mtrx{F}_k \vect{\mu}_{k|k} + \mtrx{B}_{v_k} \vect{u}_k
\label{nonlin_procpred}\\
\mtrx{\Sigma}_{k+1|k} &= \mtrx{F}_k \mtrx{\Sigma}_{k|k} \mtrx{F}_k^\top + \mtrx{Q}_k
\label{nonlin_procpredP}
\end{align}
where $\vect{\mu}$ and $\mtrx{\Sigma}$ are the mean and covariance,
respectively, of the estimate of the state $\vect{x}$ and
$\mtrx{Q}_k$ is defined above.
\subsection{Measurement Models}
\label{sec:meas_model}
To be updated.
\begin{itemize}
\item accelXYZ -- this is the only measurement used by the filter, but its in body
frame, must be rotated into local-level frame to inform
$\ddot{x}\ddot{y}\ddot{z}$ states
\item gyroXYZ -- ignore this and filter only RPH
\item magXYZ -- ignore this as well
\item head, pitch, roll -- processpressing to smooth RPH and then use ``as is''
in order to transform linear accelerations from body frame to local-level
frame
\item USBL -- provides ground truth for position $xyz$ at much lower update rate
than accelXYZ data
\end{itemize}
|
{
"timestamp": "2018-07-02T02:01:56",
"yymm": "1803",
"arxiv_id": "1803.02525",
"language": "en",
"url": "https://arxiv.org/abs/1803.02525"
}
|
\section*{abstract}
We verify the existence of Generalized Sudden Future
Singularities (GSFS) in quintessence models with scalar field
potential of the form $V(\phi)\sim \vert \phi\vert^n$ where $0<n<1$
and in the presence of a perfect fluid,
both numerically and analytically, using a proper generalized expansion ansatz for the
scale factor and the scalar field close to the singularity. This
generalized ansatz includes linear and quadratic terms, which
dominate close to the singularity and cannot be ignored when
estimating the Hubble parameter and the scalar field energy
density; as a result, they are important for analysing the
observational signatures of such singularities. We derive analytical
expressions for the power (strength) of the singularity in terms of
the power $n$ of the scalar field potential. We then extend the
analysis to the case of scalar tensor quintessence models with the
same scalar field potential in the presence of a perfect fluid, and
show that a Sudden Future Singularity (SFS) occurs in this case. We derive both analytically and numerically the strength of the singularity in terms of the power $n$ of the scalar field potential.
\end{abstract}
\maketitle
\section{Introduction}
Latest evidence of an accelerating Universe \cite{1, 2, 3, 4, 5, 6},
has opened new windows in the context of the study of physics in
cosmological scales, and has lead to the consideration of models alternative to \lcdm. Such models include modifications of GR (modified Gravity) \cite{7, 8}, scalar field dark energy (quintessence) \cite{9, 10}, physically motivated forms of fluids \eg Chaplygin gas \cite{11, 12} etc.
Some of these dark energy models predict the existence of exotic cosmological singularities, involving divergences of the scalar spacetime curvature and/or its derivatives. These singularities can be either geodesically complete \cite{13, 14, 15, 16} (geodesics continue beyond the singularity and the Universe may remain in existence) or geodesically incomplete \cite{17, 18} (geodesics do not continue beyond the singularity and the Universe ends at the classical level). They appear in various physical theories such as superstrings \cite{19}, scalar field quintessence with negative potentials \cite{20}, modified gravities and others \cite{21, 22}.
The divergence of the scale factor and/or its derivatives leads to divergence of scalar quantities like the Ricci scalar, thus to different types of singularities or `cosmological milestones' \cite{23, 25, 26}. However geodesics do not necessarily end at these singularities and if the scale factor remains finite, they are extended beyond these events \cite{22} even though a diverging impulse may lead to dissociation of all bound systems in the Universe at the time $t_s$ of these events\cite{24}.
Thus, singularities can be classified \cite{27} according to the behaviour of the scale factor $a(t)$, and/or its derivatives at the time $t_s$ of the event or equivalently, and the energy density and pressure of the content of the universe at the time $t_s$. A classification of such singularities and their properties is shown in Table \ref{TabI}.
\begin{table}[h]
\caption{Classification and properties of cosmological singularities.}\label{TabI}
\resizebox{1\textwidth}{!}
{
\begin{tabular}{c c c c c c c c c c} \\
\hline
Name & $t_{sing}$ & $a(t_{s})$ & $\rho(t_{s})$ & $p(t_{s})$ & $\dot p(t_{s})$ & $w(t_{s})$ & T & K & Geodesically \\
\hline\hline
Big-Bang (BB) & 0 & 0 & $\infty$ & $\infty$ & $\infty$ & finite & strong & strong & incomplete \\
\hline
Big-Rip (BR) & $t_{s}$ & $\infty$ & $\infty$ & $\infty$ & $\infty$ & finite & strong & strong & incomplete \\
\hline
Big-Crunch (BC) & $t_{s}$ & 0 & $\infty$ & $\infty$ & $\infty$ & finite & strong & strong & incomplete \\
\hline
Little-Rip (LR) & $\infty$ & $\infty$ & $\infty$ & $\infty$ & $\infty$ & finite & strong & strong & incomplete \\
\hline
Pseudo-Rip (PR) & $\infty$ & $\infty$ & finite & finite & finite & finite & weak & weak & incomplete \\
\hline
Sudden Future (SFS) & $t_{s}$ & $a_{s}$ & $\rho_{s}$ & $\infty$ & $\infty$ & finite & weak & weak & complete \\
\hline
Big-Brake (BBS) & $t_{s}$ & $a_{s}$ & 0 & $\infty$ & $\infty$ & finite & weak & weak & complete \\
\hline
Finite Sudden Future (FSF) & $t_{s}$ & $a_{s}$ & $\infty$ & $\infty$ & $\infty$ & finite & weak & strong & complete \\
\hline
Generalized Sudden Future (GSFS) & $t_{s}$ & $a_{s}$ & $\rho_{s}$ & $p_{s}$ & $\infty$ &finite & weak & strong & complete \\
\hline
Big-Separation (BS) & $t_{s}$ & $a_{s}$ & 0 & 0 & $\infty$ & $\infty$ & weak & weak & complete \\
\hline
w-singularity (w) & $t_{s}$ & $a_{s}$ & 0 & 0 & 0 & $\infty$ & weak & weak & complete \\
\hline
\end{tabular}
}
\end{table}
A particularly interesting type of singularities are the Sudden Future Singularities \cite{21}, which involve violation of the dominant energy condition $\rho \geq |p|$, and divergence of the cosmic pressure of the Ricci Scalar and of the second time derivative of the cosmic scale factor Table \ref{TabI}. The scale factor can be parametrized as
\be \label{sfab}
a(t)=\left (\frac{t}{t_{s}} \right )^{m} (a_{s}-1)+1-\left (1-\frac{t}{t_{s}} \right )^{q},
\ee
\noindent where $a_{s}$ is the scale factor at the time $t_{s}$ and $1<q<2$. For this range of the parameter $q$, the scale factor and its first derivative, \ie $a, \dot a$ respectively, and $\rho$ remain finite at $t_{s}$. However, the quantities $p, \dot \rho$ and $\ddot a$ become infinite. Thus, when the first derivative of the scale factor is finite at the singularity, but the second derivative diverges (SFS singularity \cite{21, 28}), the energy density is finite but the pressure diverges.
In the following, we focus on the quintessence models with a perfect fluid, and investigate the strength of the GSFS both analytically and numerically. We extend the analysis to the case of scalar-tensor quintessence and investigate the modification of the strength of the singularity both analytically (using a proper expansion ansatz) and numerically, by explicitly solving the dynamical cosmological equations.
\section{The setup}
In FRW spacetime with metric
\be
ds^2=-dt^2+a^{2}(t)\bigg[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^{2}\theta d\phi^2)\bigg]
\ee
\noindent the most general action involving gravity, nonminimally coupled with a scalar field $\phi$, and a perfect fluid is
\be \label{staction}
S=\int \left [\frac{1}{2}F(\phi)R+\frac{1}{2}g^{\mu \nu} \phi_{;\mu} \phi_{;\nu}-V(\phi)+\mathcal{L}_{(fluid)} \right ]\sqrt{-g}d^{4}x.
\ee
\noindent where $F(\phi)$ is the nonminimal coupling of gravity to the
scalar field and $\mathcal{L}_{(fluid)}$ the fluid term. We have set
$8\pi G=c=1$ and assume spatial flatness ($k = 0$). In the case of the
scalar-tensor models, corresponding to the action (\ref{staction}), we
assume a non-minimal coupling linear in the scalar field
$F(\phi)=1-\lambda \phi$, even though the results on the type of the
singularity in this class of models
are unaffected by the particular choice of the non-minimal coupling.
\noindent In the special case where the non-minimal coupling $F(\phi)=1$, the action (\ref{staction}) reduces to the simple case of quintessece models with a perfect fluid
\be \label{qaction}
S=\int \left [\frac{1}{2}R+\frac{1}{2}g^{\mu \nu} \phi_{;\mu} \phi_{;\nu}-V(\phi)+\mathcal{L}_{(fluid)} \right ]\sqrt{-g}d^{4}x.
\ee
The potential $V(\phi)$ is of the form
\be \label{potential}
V(\phi)=A|\phi|^{n},\ \ \ \ \ A>0,
\ee
\noindent with $0<n<1$ and $A$ a constant parameter. The dynamical evolution of the scalar field due to the potential is shown in Fig. \ref{fig:fig1}
\begin{figure}[!h]
\centering
\vspace{0.3cm}\rotatebox{0}{\vspace{0cm}\hspace{0cm}{\includegraphics{fig1}}}
\caption{Dynamical evolution of the scalar field potential $V(\phi)=A|\phi|^{n}$}
\label{fig:fig1}
\end{figure}
It was shown, through a qualitative analysis \cite{30}, that the power law scalar potential (\ref{potential}) leads to singularities at any scale factor derivative order larger than three, depending on the value of the power $n$. In particular, for $k<n<k+1$, with $k>0$, the $(k+2)^{th}$ derivative of the scale factor diverges at the singularity. This is in fact the simplest extension of \lcdm with geodesically complete cosmic singularities and occurs at the time $t_s$, when the scalar field becomes zero ($\phi=0$).
\section{The Quintessence case}
The action in this class of models, is of the form (\ref{qaction}). The energy density and pressure of the scalar field $\phi$, may be written as
\be \label{rphipphi}
\rho_{\phi}=\frac{1}{2}\dot \phi ^{2}+V(\phi) \ \ \ \ \ \ \ \ and \ \ \ \ \ \ \ \ p_{\phi}=\frac{1}{2}\dot \phi ^{2}-V(\phi).
\ee \
\noindent and we assume that the perfect fluid is pressureless ($p_{m}=0$).
Variation of the action (\ref{qaction}) leads to the dynamical equations
\be \label{qde1}
3H^2=\frac{3\Omega_{0,m}}{a^{3}}+ \frac{1}{2}\dot{\phi}^2+V(\phi)
\ee
\be \label{qde2}
\ddot{\phi}=-3H\dot{\phi}-An|\phi|^{n-1} \Theta(\phi)
\ee
\be \label{qde3}
2\dot H=-\frac{3\Omega_{0,m}}{a^{3}}-\dot \phi^2
\ee
\noindent where $a$ is the scale factor, $H=\frac{\dot a}{a}$ is the Hubble parameter, $\rho_{m}=\frac{\rho_{0,m}}{a^{3}}=\frac{3\Omega_{0,m}}{a^{3}}$, $\Omega_{0,m}=0.3$ and
\be
\Theta(\phi)=
\begin{cases}
1, & \phi>0 \\
-1, & \phi<0
\end{cases}
\ee.
From eqs (\ref{qde1}), (\ref{qde3}), it follows that when $t\to t_{s}$ \ie $\phi \to 0$, the Hubble parameter $H$ and its first derivative $\dot H$ remain finite and so does $\dot \phi$. But in eq. (\ref{qde2}) there is a divergence of the term $\phi^{n-1}$ for $0<n<1$ and thus $\ddot \phi \to \infty$ as $\phi \to 0$. $\ddot H$ also diverges at this point due to the divergence of $\ddot \phi$, as follows by differentiating eq. (\ref{qde3}). This implies that the third derivative of the scale factor diverges, and a GSFS occurs at this point (\ie $a_{s}, \rho_{s}, p_{s}$ remain finite but $\dot p \to \infty)$. Thus, the constraints on the power exponents $q,r$ of the diverging terms in the expansion of the scale factor ($\sim (t_{s}-t)^q$ ) and of the scalar field ($\sim (t_{s}-t)^r$ ) are $2<q<3$ and $1<r<2$ respectively (see eqs (\ref{sfa}), (\ref{sfi}) below). It has been shown in \cite{31} that by choosing $q$ to lie in the intervals $(N, N+1)$ for $N\geq 2$, where $N\in \mathbb{Z}^+$, a finite-time singularity occurs in which
\be
\frac{d^{N+1}a}{dt^{N+1}}\to \infty
\ee
\noindent but
\be
\frac{d^{s}a}{dt^{s}}\to 0, \ \ for \ \ s\leq N\in \mathbb{Z}^+
\ee
\noindent This allows for pressure singularities which are accompanied by divergence of higher time derivatives of the scale factor (divergence of the fourth-order derivative of the scale factor \cite{31} when $p\to \infty$), in Friedmann solutions of higher-order gravity $(f(R))$ theories \cite{32}.
The above qualitative analysis can be extended to a quantitative level by introducing a new ansatz for the scale factor and the scalar field, containing linear and quadratic terms of $(t_{s}-t)$. These terms play an important role, since they dominate in the first and second derivative of the scale factor as the singularity is approached.
The new ansatz for the scale factor which generalizes (\ref{sfab}), by introducing linear and quadratic terms in $(t_{s}-t)$, is of the form \cite{29}
\be \label{sfa}
a(t)=1+(a_s-1)\left(\frac{t}{t_s}\right)^m+b(t_{s}-t) + c(t_{s}-t)^2 +d(t_{s}-t)^q,
\ee
\noindent where $m=\frac{2}{3(1+w)}$, $w$ the state parameter, $b, c, d$ are real constants to be determined, and $2<q<3$ so that $\dddot a$ diverges at the GSFS.
\noindent The corresponding expansion of the scalar field $\phi(t)$ in the vicinity of the singularity is of the form
\be \label{sfi}
\phi(t)=f(t_{s}-t)+h(t_{s}-t)^{r}
\ee
\noindent where $1<r<2$ so that $\ddot \phi$ diverges at the singularity and $f,h$ are real constants to be determined.
\begin{figure}[!h]
\centering
\vspace{0.3cm}\rotatebox{0}{\vspace{0cm}\hspace{0cm}{\includegraphics{fig5}}}
\caption{Numerical solutions of the second time derivative of the scalar field for $n=0.5, 0.7, 0.9$. Notice the divergence at the time of the singularity when the scalar field vanishes.}
\label{fig:fig2}
\end{figure}
From eq. (\ref{qde2}) and differentiated eq. (\ref{qde3}), using the forms of the scale factor (\ref{sfa}) and the scalar field (\ref{sfi}), we get two equations that contain only dominant terms in $(t_{s}-t)$, in which both the left and right-hand sides diverge at the singularity for $0<n<1$, $2<q<3$ and $1<r<2$. Equating the power laws $q$ and $r$ of the divergent terms we obtain
\be \label{qr}
r=n+1
\ee
\be \label{qq}
q=r+1.
\ee
and it follows that
\be \label{qqn}
q=n+2.
\ee
Figure \ref{fig:fig2} shows the divergence of the second derivative of
the scalar field at the time of the singularity. In figures
\ref{fig:fig3}, \ref{fig:fig4} we plot the numerically verified derived power law dependence (eqs (\ref{qr}), (\ref{qqn})) of the scalar field and the scale factor respectively, as the singularity is approached. It is clear that eqs (\ref{qr}), (\ref{qqn}) are consistent with the qualitatively expected range of $r,q$, for $0<n<1$.
\begin{figure}[!h]
\centering
\begin{subfigure}{.5\textwidth}
\includegraphics[width=.95\linewidth]{fig2}
\caption{3a}
\label{fig:fig3}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.95\linewidth]{fig3}
\caption{3b}
\label{fig:fig4}
\end{subfigure}
\caption{Plots of numerical verification of the $q$-exponent (3a) and $r$-exponent (3b) for 3 values of $n$ ($n=0.5, n=0.7$ and $n=0.9$). The orange dashed line, denotes the analytical, while the blue line denotes the numerical solution. As expected the slopes for each n for both $q$ and $r$ are identical.}
\end{figure}
The additional linear and quadratic terms in $(t_{s}-t)$, in the expression of the scale factor (\ref{sfa}), play an important role in the estimation of the Hubble parameter and its derivative as the singularity is aproached. An interesting result arises from the derivation of the relation between the coefficients $b, c$. The relations between these coefficients can lead to relations between the Hubble parameter and its derivative close to the singularity, which in turn correspond to observational predictions, that may be used to identify the presence of these singularities in angular diameter of luminosity distance data. The relation between $b, c$ is of the form
\be \label{qc}
c=\frac{\rho_{0,m}}{4a^{2}_{s}}-\frac{1}{2}(a_{s}-1)m(m-1)-\frac{[(a_{s}-1)m-b]^{2}}{a_{s}},
\ee
\noindent and thus
\be \label{qdotH}
\dot H=\frac{3\Omega_{0,m}}{2a^{3}_{s}}-3H^{2}
\ee
\noindent and as a function of redshift parameter $z$ at present time
\be \label{qHz}
H^{2}(z)= \Omega_{0,m}(1+z)^{3}[1-(1+z)^{3}(1+z_{0})^{-3}]+(1+z)^{6}(1+z_{0})^{-6}H^{2}_{0},
\ee
\noindent where $H_{0}, z_{0}$ are the Hubble and redshift parameter respectively at present time. This result may be used as observational signature of such singularities in this class of models.
In the absence of the perfect fluid, the strength of the singularity
remains unaffected. This means that the evaluated relations of $r$ and
$q$ (eqs (\ref{qr}), (\ref{qqn})) respectively, are exactly the same. The Hubble parameter and its derivative in this case is
\be \label{qdotHwm}
\dot H=-3H^2
\ee
\noindent and as a function of redshift parameter $z$ at present time
\be \label{qHzwm}
H(z)=\frac{H_{0}(1+z)^{3}}{(1+z_{0})^3}.
\ee
\noindent These are the reduced relations of eqs (\ref{qdotH}) and (\ref{qHz}) respectively, for $\rho_{0,m}=0$.
\section{Modified Gravity: The Scalar-Tensor Quintessence case}
The action of the theory, in this class of models, is of the form (\ref{staction}). The corresponding dynamical equations are
\be \label{stde1}
3FH^{2}=\frac{3\Omega_{0,m}}{a^{3}}+\frac{\dot \phi^2}{2}+V-3H\dot{F}
\ee
\be \label{stde2}
\ddot \phi+3H\dot \phi-3F_{\phi}\bigg(\frac{\ddot a}{a}+H^2\bigg)+An|\phi|^{(n-1)} \Theta(\phi)=0
\ee
\be \label{stde3}
-2F\bigg(\frac{\ddot a}{a}-H^{2}\bigg)=\frac{3\Omega_{0,m}}{a^{3}}+\dot \phi^{2}+\ddot F-H\dot F,
\ee
\noindent where $F_{\phi}=\frac{dF}{d\phi}$. From eq. (\ref{stde1}), it is clear that $H, \dot \phi, F, \dot F$ all remain finite when $\phi \to 0$ ($t\to t_{s}$). However, in eq. (\ref{stde2}) there is a divergence of the term $V_{\phi}$ for $0<n<1$ and $\ddot \phi \to \infty$ as $\phi \to 0$. This means that $\ddot F\to \infty$ because of the generation of the second derivative of $\phi$ that leads to a divergence of $\ddot a$ in eq. (\ref{stde3}). Clearly, an SFS singularity (Table \ref{TabI}) is expected to occur in scalar-tensor quintessence models, as opposed to the GSFS singularity in the corresponding quintessence models. Thus, the constraints on the power exponents $q,r$ in this case are $1<q<2$ and $1<r<2$ respectively.
From the above dynamical equations, using the same parametrizations
(\ref{sfa}), (\ref{sfi}) for the scale factor and the scalar field respectively and keeping only the dominant terms, the values for $r$ and $q$ are
\be \label{stq}
q=r
\ee
\be \label{str}
r=n+1,
\ee
\noindent which leads to
\be \label{stqn}
q=n+1.
\ee
In figures \ref{fig:fig5}, \ref{fig:fig6} we illustrate the numerically verified derived power law dependence eqs (\ref{str}), (\ref{stqn}) of the scalar field and the scale factor respectively, as the singularity is approached. Figures \ref{fig:fig7}, \ref{fig:fig8} depict the divergence of the second derivative, of both the scale factor and the scalar field, at the time of the singularity.
\begin{figure}[!h]
\centering
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=0.95\linewidth]{fig6}
\caption{4a}
\label{fig:fig5}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{fig7}
\caption{4b}
\label{fig:fig6}
\end{subfigure}
\caption{Numerical verification of the $q$-exponent (4a) and $r$-exponent (4b), in the scalar-tensor case, for 3 values of $n$ ($n=0.2, n=0.4$ and $n=0.6$). The orange dashed line, denotes the analytical, while the blue line denotes the numerical solution. As expected the slopes for each n for both $q$ and $r$ are identical.}
\end{figure}
\begin{figure}[!h]
\centering
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=0.95\linewidth]{fig8}
\caption{5a}
\label{fig:fig7}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.95\linewidth]{fig9}
\caption{5b}
\label{fig:fig8}
\end{subfigure}
\caption{Numerical solutions of the second time derivative of the scale factor (5a) and the scalar field (5b) for $n=0.2, 0.4, 0.6$. Notice the divergence of both the scale factor and scalar field at the time of the singularity.}
\end{figure}
The results (\ref{str}) and (\ref{stqn}) are consistent with the above
qualitative discussion for the expected strength of the
singularity. Thus, in the case of the scalar-tensor theory, we have a
stronger singularity at $t_{s}$, as compared to the singularity that occurs in quintessence models. This is a general result, valid not only for the coupling constant of the form $F=1-\lambda \phi$ but also for other forms of $F(\phi)$ (\eg $F\sim \phi^r$), because the second derivative of $F$ with respect to time, in the dynamical equations, will always generate a second derivative of $\phi$ with divergence, leading to a divergence of $\ddot a$.
The quadratic term of $(t_{s}-t)$, in the expression of the scale factor (\ref{sfa}), is now subdominant as the second derivarive of the scale factor diverges. The only additional term of $(t_{s}-t)$ that can play an important role in the estimation of the Hubble parameter, is the linear term. Clearly, for the first derivative of (\ref{sfa}), as $t\to t_{s}$ from below, the linear term dominates over all other terms, while the quadratic term is subdominant in the second derivative, in the divergence of the $q$-term. Thus, in the case of the scalar-tensor quintessence models $H$ remain finite and dominated by the term $b(t_{s}-t)$, while $\dot H \to \infty$ as $t\to t_{s}$.
As in quintessence case of the previous section, in the absence of the perfect fluid, the strength of the singularity remains unaffected. This means that the evaluated relations of $r$ and $q$, eqs (\ref{str}), (\ref{stqn}) respectively, are exactly the same.
\section{Conclusions and Discussion}
We have derived analytically and numerically the cosmological solution close to a future-time singularity for both quintessence and scalar-tensor quintessence models. For quintessence, we have shown that there is a divergence of $\dddot a$ and a GSFS singularity occurs ($a_{s}, \rho_{s}, p_{s}$ remain finite but $\dot p \to \infty)$ , while in the case of scalar-tensor quintessence models there is a divergence of $\ddot a$ and an SFS singularity occurs ($a_{s}, \rho_{s}$ remain finite but $p_{s}\to \infty$, $\dot p \to \infty)$. In the absence of the perfect fluid in the dynamical equations, in both cases, we have shown that this result is still valid in our cosmological solution.
These are the simplest non-exotic physical models where GSFS and SFS singularities naturally arise. In the case of scalar-tensor quintessence models, there is a divergence of the scalar curvature $R=6\left (\frac{\ddot {a}}{a}+\frac{\dot a^{2}}{a^{2}} \right ) \to \infty$ because of the divergence of the second derivative of the scale factor. Thus, a stronger singularity occurs in this class of models. Such divergence of the scalar curvature is not present in the simple quintessence case.
We have also shown the important role of the additional linear and quadratic terms of $t_{s}-t$ in the form of the scale factor as $t\to t_{s}$. However, in the scalar-tensor case the quadratic term becomes subdominant close to the singularity.
For quintessence models, we derived relations of the Hubble parameter, $H^{2}(z)= \Omega_{0,m}(1+z)^{3}[1-(1+z)^{3}(1+z_{0})^{-3}]+(1+z)^{6}(1+z_{0})^{-6}H^{2}_{0}$ (for the fluid case) and $H(z)=\frac{H_{0}(1+z)^{3}}{(1+z_{0})^3}$ (for the no fluid case), close to the singularity. These relations may be used as observational signatures of such singularities in this class of models.
Interesting extensions of the present analysis include the study of the strength of these singularities in other modified gravity models \eg string-inspired gravity, Gauss-Bonnet gravity etc. and the search for signatures of such singularities in cosmological luminosity distance and angular diameter distance data.
\section{Acknowledgments}
I would like to thank my collaborators S. Lola and L. Perivolaropoulos
for their stimulating and fruitful contribution and collaboration that
led to this work. I also thank the organizers for the opportunity to
present these results and for the stimulating atmosphere they have created during the conference. Financial support from the COST Action CA15108, is gratefully acknowledged.
|
{
"timestamp": "2018-03-08T02:05:08",
"yymm": "1803",
"arxiv_id": "1803.02554",
"language": "en",
"url": "https://arxiv.org/abs/1803.02554"
}
|
\section{Introduction}
AI agents
will increasingly find assistive roles in
homes, labs, factories and public places. The widespread adoption of
conversational agents such as Alexa, Siri and Google Home demonstrate
the natural demand for such assistive agents.
To go beyond supporting the simplistic ``what is the weather?''
queries however, these agents need domain-specific
knowledge such as the recipes and standard operating procedures. While
it is possible to hand-code such knowledge (as is done by most of the
``skills'' used by Alexa-like agents), ultimately that is too labor
intensive an option. One idea is to have these agents automatically ``read''
instructional texts, typically written for human workers, and
convert them into action sequences and plans for later use (such as learning domain models \cite{DBLP:journals/ai/ZhuoM014,DBLP:journals/ai/Zhuo014} or model-lite planning \cite{DBLP:journals/ai/ZhuoK17}).
Extracting action sequences from natural language texts meant for
human consumption is however challenging, as it requires agents to understand
complex contexts of actions.
For example, in Figure \ref{examples}, given a document of action
descriptions (the left part of Figure \ref{examples}) such as
``\emph{Cook the rice the day before, or use leftover rice in the
refrigerator. The important thing to remember is not to heat up the
rice, but keep it cold.}'', which addresses the procedure of making
egg fired rice, an action sequence of ``\emph{cook(rice), keep(rice,
cold)}'' or ``\emph{use(leftover rice), keep(rice, cold)}'' is
expected to be extracted\ignore{ from the action descriptions}. This
task is challenging. For \ignore{example, for }the first sentence, the
agent needs to learn to figure out that ``cook'' and ``use'' are
\emph{exclusive} (denoted by ``EX'' in the middle of Figure
\ref{examples}), meaning that we could extract only one of them; for
the second sentence, we need to learn to understand that among the
three verbs ``remember'', ``heat'' and ``keep'', the last one is the
best because the goal of this step is to ``keep the rice cold''
(denoted by ``ES'' indicating this action is \emph{essential}). There
is also another action ``Recycle'' denoted by ``OP'' indicating this
action can be extracted \emph{optionally}. We also need to consider
action arguments which can be either ``EX'' or ``ES'' as well (as
shown in the middle of Figure \ref{examples}). The possible action
sequences extracted are shown in the right part of Figure
\ref{examples}. This action sequence extraction problem is different
from sequence labeling and dependency parsing, since we aim
to extract ``meaningful'' or ``correct'' action sequences (which
suggest some actions should be ignored because they are exclusive),
such as ``\emph{cook(rice), keep(rice, cold)}'', instead of
``\emph{cook(rice),use(leftover rice), remember(thing), heat(rice),
keep(rice, cold)}'' as would be extracted by LSTM-CRF models\cite{DBLP:conf/acl/MaH16} or external NLP tools.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{examples.pdf}
\caption{Illustration of our action sequence extraction problem}
\label{examples}
\end{center}
\end{figure}
There has been work on extracting action sequences from action
descriptions. For example, \cite{DBLP:conf/acl/BranavanCZB09} propose
to map instructions to sequences of executable actions using
reinforcement learning. \cite{Mei2016Listen,Daniele2017Navigational}
interpret natural instructions as action sequences or generate
navigational action description using an encoder-aligner-decoder
structure. Despite the success of those approaches, they all require a
limited set of action names given as input, which are mapped to action
descriptions. Another approach, proposed by
\cite{DBLP:conf/aips/LindsayRFHPG17}, builds action sequences from
texts based on dependency parsers and then builds planning models,
assuming texts are in restricted templates when describing actions.
In this paper, we aim to extract meaningful action sequences from
texts in \emph{free} natural language, i.e., without any restricted
templates, even when the candidate set of actions is unknown. We
propose an approach called {{\tt EASDRL}}, which stands for
\textbf{E}xtracting \textbf{A}ction \textbf{S}equences from texts
based on \textbf{D}eep \textbf{R}einforcement \textbf{L}earning. In
{{\tt EASDRL}}, we view texts associated with actions
as ``states'', and associating words in texts with labels as
``actions'', and then build deep Q-networks to extract action
sequences from texts. We capture complex relations among actions
by considering previously extracted actions as parts of states for
deciding the choice of next operations. In other words, once we know
action ``cook(rice)'' has been extracted and included as parts of
states, we will choose to extract next action ``keep(rice, cold)''
instead of ``use(leftover rice)'' in the above-mentioned example.
In the remainder of paper, we first review previous work related to our approach. After that we give a formal definition of our plan extraction problem and present our {{\tt EASDRL}} approach in detail. We then evaluate our {{\tt EASDRL}} approach with comparison to state-of-the-art approaches and conclude the paper with future work.
\section{Related Work}
There have been approaches related to our work besides the ones we mentioned in the introduction section. Mapping SAIL route instructions \cite{Macmahon2006Walk} to action sequences has aroused great interest of in natural language processing community. Early approaches, like \cite{DBLP:conf/aaai/ChenM11,DBLP:conf/acl/Chen12,Kim2013Unsupervised,Kim2013Adapting}, largely depend on specialized resources, i.e. semantic parsers, learned lexicons and re-rankers. Recently, LSTM encoder-decoder structure \cite{Mei2016Listen} has been applied to this problem and gets decent performance in processing single-sentence instructions, however, it could not handle multi-sentence texts well.
There is also a lot of work on learning STRIPS representation actions \cite{DBLP:conf/ijcai/FikesN71,DBLP:conf/ki/PomarlanKB17} from texts. \cite{DBLP:conf/aaaifs/SilHY10,DBLP:conf/ranlp/SilY11} learn sentence patterns and lexicons or use off-the-shelf toolkits, i.e., OpenNLP\footnote{https://opennlp.apache.org/} and Stanford CoreNLP\footnote{http://stanfordnlp.github.io/CoreNLP/}. \cite{DBLP:conf/aips/LindsayRFHPG17} also build action models with the help of LOCM \cite{cresswell2009acquisition} after extracting action sequences by using NLP tools. These tools are trained for universal natural language processing tasks, they cannot solve the complex action sequence extraction problem well, and their performance will be greatly affected by POS-tagging and dependency parsing results. In this paper we aim to build a model that learns to directly extract action sequences without external tools.
\ignore{
Recently \cite{Mnih2015Human} applied deep neural network (DQN) to solve
reinforcement learning problems and obtain state-of-the-art
results. \cite{Silver2016Mastering,kulkarni2016hierarchical} propose
DQN models to play more challenging games with enormous search space
or sparse
feedback. \cite{Lillicrap2015Continuous,Schaul2015Prioritized} develop
DQN structure to continuous action space and improve the experience
replay trick. Unlike those works which mainly focus on video games, \cite{DBLP:conf/emnlp/NarasimhanKB15,DBLP:conf/acl/HeCHGLDO16} take as input texts descriptions of games, which also shed light on our task.}
\section{Problem Definition}\label{def}
Our training data can be defined by $\Phi=\{\langle X,Y\rangle\}$, where $X=\langle w_1,w_2,\ldots,w_N\rangle$ is a sequence of words and $Y=\langle y_1,y_2,\ldots,y_N\rangle$ is a sequence of annotations. If $w_i$ is not an action name, $y_i$ is $\emptyset$. Otherwise, $y_i$ is a tuple $(ActType, \{ExActId\}, \{\langle ArgId, ExArgId\rangle\})$ to describe \emph{type} of the action name and its corresponding arguments. $ActType$ indicates the type of action $a_i$ corresponding to $w_i$, which can be one of \emph{essential}, \emph{optional} and \emph{exclusive}. The type $essential$ suggests the corresponding action $a_i$ to be extracted, $optional$ suggests $a_i$ that can be ``optionally'' extracted, $exclusive$ suggests $a_i$ that is ``exclusive'' with other actions indicated by the set $\{ExActId\}$ (in other words, either $a_i$ or exactly one action in $\{ExActId\}$ can be extracted). $ExActId$ is the index of the action exclusive with $a_i$. We denote the size of $\{ExActId\}$ by $M$, i.e., $|\{ExActId\}|=M$. Note that ``$M=0$'' indicates the type $ActType$ of action $a_i$ is either \emph{essential} or \emph{optional}, and ``$M\neq 0$'' indicates $ActType$ is \emph{exclusive}.
$ArgId$ is the index of the word composing arguments of $a_i$, and $ExArgId$ is the index of words exclusive with $ArgId$.
\emph{
For example, as shown in Figure \ref{exampleXY}, given a text denoted by $X$, its corresponding annotation is shown in the figure denoted by $Y$. In $y_1$, ``\{11\}'' indicates the action exclusive with $w_1$ (i.e., ``Hang'') is ``opt'' with index 11. ``$\{\langle 3,5\rangle,\langle 9,\rangle\}$'' indicates the corresponding arguments ``engraving'' and ``lithograph'' are exclusive, and the other argument ``frame'' with index 9 is essential since it is exclusive with an empty index, likewise for $y_{11}$. For $y_2,\ldots,y_{10}$ and $y_{12},\ldots,y_{15}$, they are empty since their corresponding words are not action names. From $Y$, we can generate three possible actions as shown at the bottom of Figure \ref{exampleXY}.
}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.45\textwidth]{exampleXY.pdf}
\caption{Illustration of text X and its corresponding annotation Y}
\label{exampleXY}
\end{center}
\end{figure}
As we can see from the training data, it is uneasy to build a supervised learning model to directly predict annotations for new texts $X$, since annotations $y_i$ is complex and the size $|y_i|$ varies with respect to different $w_i$ (different action names have different arguments with different lengths). We seek to build a \emph{unified} framework to predict \emph{simple} ``labels'' (corresponding to ``actions'' in reinforcement learning) for extracting action names and their arguments. We exploit the framework to learn two models to predict action names and arguments, respectively. Specifically, given a new text $X$, we would like to predict a sequence of operations $O=\langle o_1,o_2,\ldots,o_N\rangle$ (instead of annotations in $\Phi$) on $X$, where $o_i$ is an $operation$ that $selects$ or $eliminates$ word $w_i$ in $X$. In other words, when predicting action names (or arguments), $o_i=Select$ indicates $w_i$ is extracted as an action name (or argument), while $o_i=Eliminate$ indicates $w_i$ is not extracted as an action name (or argument).
In summary, our action sequence extraction problem can be defined by: given a set of training data $\Phi$, we aim to learn two models (with the same framework) to predict action names and arguments for new texts $X$, respectively. The two models are
\begin{equation}\label{m1}
\mathcal{F}^1_{\Phi}(O|X; \theta_1)
\end{equation}
and
\begin{equation}\label{m2}
\mathcal{F}^2_{\Phi}(O|X,a; \theta_2),
\end{equation}
where $\theta_1$ and $\theta_2$ are parameters to be learnt for predicting action names and arguments, respectively. $a$ is an action name extracted based on $\mathcal{F}^1_{\Phi}$. We train $\mathcal{F}^2_{\Phi}$ for extracting arguments based on ground-truth action names. When testing, we extract arguments based on the action names extracted by $\mathcal{F}^1_{\Phi}$. We will present the details of building these two models in the following sections.
\section{Our {{\tt EASDRL}} Approach}
In this section we present the details of our {{\tt EASDRL}} approach. As mentioned in the introduction section, our action sequence extraction problem can be viewed as a reinforcement learning problem. We thus first describe how to build \emph{states} and \emph{operations} given text $X$, and then present deep Q-networks to build the Q-functions. Finally we present the training procedure and give an overview of our {{\tt EASDRL}} approach. Note that we will use the term \emph{operation} to represent the meaning of ``action'' in reinforcement learning since the term ``action'' has been used to represent an action name with arguments in this work.
\subsection{Generating State Representations}
In this subsection we address how to generate state representations from texts.
As defined in the problem definition section, the space of operations is $\{Select, Eliminate\}$. We view texts associated with operations as ``states''. Specifically, we represent a text $X$ by a sequence of vectors $\langle \mathbf{w}_1,\mathbf{w}_2,\ldots,\mathbf{w}_N\rangle$, where $\mathbf{w}_i \in \mathcal{R}^{K_1}$ is a $K_1$-dimension real-valued vector \cite{word2vec}, representing the $i$th word in $X$. Words of texts stay the same when we perform operations, so we embed operations in state representations to generate state transitions. We extend the set of operations to $\{NULL,Select,Eliminate\}$ where ``NULL'' indicates a word has not been processed. We represent the operation sequence $O$ corresponding to $X$ by a sequence of vectors $\langle \mathbf{o}_1,\mathbf{o}_2, \ldots, \mathbf{o}_N\rangle$, where $\mathbf{o}_i \in \mathcal{R}^{K_2}$ is a $K_2$-dimension real-valued vector. In order to balance the dimension of $\mathbf{o}_i$ and $\mathbf{w}_i$, we generate each $\mathbf{o}_i$ by a \emph{repeat-representation} $[\cdot]_{K_2}$, i.e., if $K_2=1$, $\mathbf{o}_i\in\{[0],[1],[2]\}$, and if $K_2=3$, $\mathbf{o}_i\in\{[0,0,0],[1,1,1],[2,2,2]\}$, where $\{0,1,2\}$ corresponds to $\{NULL,Select,Eliminate\}$, respectively.
We define a \emph{state} $s$ as a tuple $\langle \mathbf{X},\mathbf{O}\rangle$, where $\mathbf{X}$ is a matrix in $\mathcal{R}^{K_1\times N}$, $\mathbf{O}$ is a matrix in $\mathcal{R}^{K_2\times N}$. The $i$th row of $s$ is denoted by $[\mathbf{w}_i,\mathbf{o}_i]$. The space of states is denoted by $\mathcal{S}$. A state $s$ is changed into a new state $s'$ after performing an operation $\mathbf{o}'_i$ on $s$, such that $s'=\langle \mathbf{X},\mathbf{O}'\rangle$, where $\mathbf{O}'=\langle \mathbf{o}_1,\ldots,\mathbf{o}_{i-1},\mathbf{o}'_i,\mathbf{o}_{i+1},\ldots,\mathbf{o}_N\rangle$.
\emph{For example, consider a text ``Cook the rice the day before..." and a state $s$ corresponding to it is shown in the left part of Figure \ref{text_matrix}. After performing an operation $\mathbf{o}_1=Select$ on $s$, a new state $s'$ (the right part) will be generated.} In this way, we can learn $\theta_1$ in $\mathcal{F}^1_{\Phi}$ (Equation (\ref{m1})) based on $s$ with deep Q-networks as introduced in the next subsection.
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.38\textwidth]{text_matrix.pdf}
\caption{Illustration of states and operations}
\label{text_matrix}
\end{center}
\end{figure}
After $\mathcal{F}^1_{\Phi}$ is learnt, we can use it to predict action names, and then exploit the predicted action names to extract action arguments by training $\mathcal{F}^2_{\Phi}$ (Equation (\ref{m2})). To do this, we would like to encode the predicted action names in states to generate a new state representation $\hat{s}$ for learning $\theta_2$ in $\mathcal{F}^2_{\Phi}$. We denote by $w_a$ the word corresponding to the action name. We build $\hat{s}$ by appending the distance between $w_a$ and $w_j$ based on their indices, such that $\hat{s}=\langle \mathbf{X},\mathbf{D},\mathbf{O}\rangle$, where $\mathbf{D}=\langle \mathbf{d}_1,\mathbf{d}_2,\ldots, \mathbf{d}_N\rangle$, where $\mathbf{d}_j=[d_j]_{K_3}$ and $d_j=|a-j|$. Note that $\mathbf{d}_j$ is a $K_3$-dimension real-valued vector using \emph{repeat-representation} $[\cdot]_{K_3}$. In this way we can learn $\mathcal{F}^2_{\Phi}$ based on $\hat{s}$ with the same deep Q-networks.
Note that in our experiments, we found that the results were the best when we set $K_1=K_2=K_3$, suggesting the impact of word vectors, distance vectors and operation vectors was generally identical.
\subsection{Deep Q-networks for Operation Execution}
Given the formulation of states and operations, we aim to extract action sequences from texts. We construct sequences by repeatedly choosing operations given current states, and applying operations on current states to achieve new states.
In Q-Learning, this process can be described by a Q-function and updating the Q-function iteratively according to Bellman equation. In our action sequence extraction problem, actions are composed of action names and action arguments. We need to first extract action names from texts and use the extracted action names to further extract action arguments. Specifically, we define two Q-functions $Q(s,o)$ and $Q(\hat{s},o)$, where $\hat{s}$ contains the information of extracted action names, as defined in the last subsection. The update procedure based on Bellman equation and deep Q-networks can be defined by:
\begin{equation}
Q_{i+1}(s,o;\theta_1) = E\left\lbrace r+\gamma\max_{o'}Q_i(s',o';\theta_1)|s,o\right\rbrace
\label{Bellman}
\end{equation}
\begin{equation}
Q_{i+1}(\hat{s},o;\theta_2) = E\left\lbrace r+\gamma\max_{o'}Q_i(\hat{s}',o';\theta_2)|\hat{s},o\right\rbrace
\label{Bellman}
\end{equation}
where $Q_{i+1}(s,o;\theta_1)$ and $Q_{i+1}(\hat{s},o;\theta_2)$ correspond to the deep Q-networks \cite{Mnih2015Human} for extracting action names and arguments, respectively. As $i \rightarrow \infty$, $Q_i \rightarrow Q^*$. In this way, we can define $\mathcal{F}^1_{\Phi}=Q^*(s,o;\theta_1)$ and $\mathcal{F}^2_{\Phi}=Q^*(\hat{s},o;\theta_2)$ in Equations (\ref{m1}) and (\ref{m2}), and then use $\mathcal{F}^1_{\Phi}$ and $\mathcal{F}^2_{\Phi}$ to extract action names and arguments, respectively.
Since Convolutional Neural Networks (CNNs) are widely applied in natural language processing \cite{DBLP:conf/emnlp/Kim14,DBLP:journals/corr/ZhangW15b,ijcai2017-406}, we build CNN models to learn Q-functions $Q(s,o,\theta_1)$ and $Q(\hat{s},o,\theta_2)$. We adopt the CNN Architecture of \cite{DBLP:journals/corr/ZhangW15b}. To build the kernels of our CNN models, we test from uni-gram context to ten-gram context and observe that five-word context works well in our task. We thus design four types of kernels, which correspond to bigram, trigram, four-gram and five-gram, respectively.
\subsection{Computing Rewards}
In this subsection we compute the reward $r$ based on state $s$ and operation $o$. Specifically, $r$ is composed of two parts, i.e., \textbf{basic reward} and \textbf{additional reward}. For the basic reward at time step $\tau$, denoted by $r_{b,\tau}$, if a word is not an \emph{item} (we use \emph{item} to represent action name or action argument when it is not confused), $r_{b,\tau}$ is $+50$ when the operation is correct and $-50$ otherwise. If a word is an \emph{essential item}, $r_{b,\tau}=+100$ when the operation is correct and $r_{b,\tau}=-100$ when it is incorrect. If the word is an \emph{optional item}, $r_{b,\tau}=+100$ when the operation is correct and $r_{b,\tau}=0$ when it is incorrect. If a word is an \emph{exclusive item}, $r_{b,\tau}=+150$ when the operation is correct and $r_{b,\tau}=-150$ when it is incorrect. We denote that an operation is correct when it selects essential items, selects optional items, selects only one item of exclusive items or eliminates words that are not items.
Note that action names are key verbs of a text and action arguments
are some nominal words, so the percentage of these words in a text is
closely related to action sequence extraction process. We thus calculate
the percentage, namely an \emph{item rate}, denoted by
$\delta=\frac{\#Item}{\#Word}$, where $\#Item$ indicates the amount of
action names or action arguments in all the annotated texts and
$\#Word$ indicates the total number of words of these texts. We define a \emph{real-time item rate} as $\delta_\tau$ to denote the percentage of words that have been selected as action names or action arguments in a text \emph{after $\tau$ training steps}, and $\delta_0=0$. On one hand, when $\delta_{\tau-1} \leq \delta$, a positive additional reward is added to $r_{b,\tau}$ if $r_{b,\tau} \geq 0$ (i.e., the operation is correct), otherwise a negative additional reward is added to $r_{b,\tau}$. On the other hand, when $\delta_\tau > \delta$, which means that words selected as action names or action arguments are out of the expected number and it is more likely to be incorrect if subsequent words are selected, then a negative additional reward should be added to the basic reward. In this way, the reward $r_\tau$ at time step $\tau$ can be obtained by Equation (\ref{complete reward}),
\begin{equation}
r_\tau=\begin{cases}
r_{b,\tau}+\mathop{\mathrm{sgn}}{r_{b,\tau}} \cdot c\delta_{\tau-1} & \text{$\delta_{\tau-1} \leq \delta$},\\
r_{b,\tau}-c\delta_{\tau-1} & \text{$\delta_{\tau-1} > \delta$}.
\end{cases}
\label{complete reward}
\end{equation}
where $c$ is a positive constant and $0 \leq \delta_{\tau-1} < 1$. \ignore{We empirically use $c\delta_\tau$ to represent the additional reward, which can be replaced by others. It is, however, effective based on our experiments. }
\subsection{Training Our Model}
To learn the parameters $\theta_1$ and $\theta_2$ of our two DQNs, we
store transitions $\langle s,o,r,s'\rangle$ and $\langle
\hat{s},o,r,\hat{s}'\rangle$ in replay memories $\Omega$ and
$\hat{\Omega}$, respectively, and exploit a mini-batch sampling
strategy. As indicated in \cite{DBLP:conf/emnlp/NarasimhanKB15},
transitions that provide positive rewards can be used more often to
learn optimal Q-values faster. We thus develop a \emph{positive-rate
based experience replay} instead of randomly sampling transitions
from $\Omega$ (or $\hat{\Omega}$), where \emph{positive-rate}
indicates the percentage of transitions with positive rewards. To do
this, we set a positive rate $\rho (0 < \rho < 1 )$ and require the
proportion of positive samples in each mini-batch be $\rho$.
We present the learning procedure of our {{\tt EASDRL}} approach in Algorithm \ref{our_algorithm}, for building $\mathcal{F}^1_{\Phi}$. We can simply replace $s_1$, $\Omega$ and $\theta_1$ with $\hat{s}_1$, $\hat{\Omega}$ and $\theta_2$ for building $\mathcal{F}^2_{\Phi}$. In Step 4 of Algorithm \ref{our_algorithm}, we generate the initial state $s_1$ ($\hat{s}_1$ for learning $\mathcal{F}^2_{\Phi}$) for each training data $\Phi=\{\langle X,Y\rangle\}$ by setting all operations $o_i$ in $s_1$ to be $NULL$. We perform $N$ steps to execute one of the operations $\{Select, Eliminate\}$ in Steps 6, 7 and 8. From Steps 10 and 11, we do a \emph{positive-rate based experience replay} according to positive rate $\rho$. From Steps 12 and 13, we update parameters $\theta_1$ using gradient descent on the loss function $\mathcal{L}(\theta_1)=(y_j - Q(s_j,o_j; \theta_1))^2$ as shown in Step 13.
With Algorithm \ref{our_algorithm}, we are able to build the Q-function $Q(s,o;\theta_1)$ and execute operations $\{Select, Eliminate\}$ to a new text by iteratively maximizing the Q-function. Once we obtain operation sequences, we can generate action names and use the action names to build $Q(\hat{s},o;\theta_2)$ with $\hat{\Omega}$ and the same framework of Algorithm \ref{our_algorithm}. We then exploit the built $Q(\hat{s},o;\theta_2)$ to extract action arguments. As a result, we can extract action sequences from texts using both of the built $Q(s,o;\theta_1)$ and $Q(\hat{s},o;\theta_2)$.
\begin{algorithm}[!ht]
\caption{Our {{\tt EASDRL}} algorithm}
\label{our_algorithm}
\textbf{Input:} a training set $\Phi$, positive rate $\rho$, item rate $\delta$ \\
\textbf{Output:} the parameters $\theta_1$
\begin{algorithmic}[1]
\STATE Initialize $\Omega=\emptyset$, CNN with random values for $\theta_1$
\FOR{epoch = 1: $H$}
\FOR {each training data $\langle X,Y\rangle\in\Phi$}
\STATE Generate the initial state $s_1$ based on $X$
\FOR{$\tau$ = 1: $N$}
\STATE Perform an operation $o_\tau$ with probability $\epsilon$
\STATE Otherwise select $o_\tau = \max\limits_{o} Q(s_\tau, o; \theta_1)$
\STATE Perform $o_\tau$ on $s_\tau$ to generate $s_{\tau+1}$
\STATE Calculate $r_\tau$ based on $s_{\tau+1}$, $o_\tau$, $Y$ and $\delta$
\STATE Store transition $(s_\tau, o_\tau, r_\tau, s_{\tau+1})$ in $\Omega$
\STATE Sample $(s_j,o_j,r_j,s_{j+1})$ from $\Omega$ based on $\rho$
\STATE Set \\$y_j=\begin{cases} r_j \qquad\qquad\qquad \text{for terminal $s_{j+1}$} \\ r_j + \gamma\max\limits_{o'}Q(s_{j+1},o';\theta_1) \ \text{otherwise} \end{cases}$
\STATE Update $\theta_1$ based on loss function $\mathcal{L}(\theta_1)$
\ENDFOR
\ENDFOR
\ENDFOR
\RETURN The parameters $\theta_1$
\end{algorithmic}
\end{algorithm}
\section{Experiments}
\subsection{Datasets and Evaluation Metric}
We conducted experiments on three datasets, i.e., ``Microsoft Windows
Help and Support'' (WHS) documents \cite{DBLP:conf/acl/BranavanCZB09},
and two datasets collected from ``WikiHow Home and
Garden''\footnote{https://www.wikihow.com/Category:Home-and-Garden}
(WHG) and ``CookingTutorial''\footnote{http://cookingtutorials.com/}
(CT). Details are presented in Table \ref{Statistics of
Datasets}. Supervised learning models require that training data are
one-to-one pairs (i.e. each word has a unique label), so we generate
input-texts-to-output-labels based on annotation $Y$ (as defined in
Section \ref{def}). In our task, a single text with $n$ optional items
or $n$ exclusive pairs can generate more than $2^n$ potential label sequences (i.e. each item of them can be extracted or not be extracted). Especially, we observe that $n$ is larger than 30 in some texts of our datasets, which means more than 1 billion sequences will be generated. We thus restrict $n\leq 8$ (no more than $2^{8}$ label sequences) to generate reasonable number of sequences.
\begin{table}[!htbp]
\caption{Datasets used in our experiments }
\label{Statistics of Datasets}
\centering
\begin{tabular}{lccc}
\hline
& \textbf{WHS} & \textbf{CT} & \textbf{WHG} \\
\hline
Labeled texts & 154 & 116 & 150 \\
Input-output pairs &1.5K & 134K & 34M \\
Action name rate (\%) & 19.47 & 10.37 & 7.61 \\
Action argument rate (\%) & 15.45 & 7.44 & 6.30 \\
Unlabeled texts & 0 & 0 & 80 \\
\hline
\end{tabular}
\end{table}
For evaluation, we first feed test texts to each model to output sequences of labels or operations. We then extract action sequences based on these labels or operations. After that, we compare these action sequences to their corresponding annotations and calculate $\#TotalTruth$ (total ground truth items), $\#TotalTagged$ (total extracted items), $\#TotalRight$ (total correctly extracted items). Finally we compute metrics: $precision=\frac{\#TotalRight}{\#TotalTagged}$, $recall=\frac{\#TotalRight}{\#TotalTruth}$, and $F1=\frac{2\times precision \times recall}{precision + recall}$. We randomly split each dataset into 10 folds, calculated an average of performance over 10 runs via 10-fold cross validation, and used the F1 metric to validate the performance in our experiments.
\subsection{Experimental Results}
We compare {{\tt EASDRL}} to four baselines, as shown below:
\begin{itemize}
\item{STFC:} Stanford CoreNLP, an off-the-shelf tool, denoted by STFC, extracts action sequences by viewing root verbs as action names and objects as action arguments \cite{DBLP:conf/aips/LindsayRFHPG17}.
\item{BLCC:} Bi-directional LSTM-CNNs-CRF model \cite{DBLP:conf/acl/MaH16,DBLP:conf/emnlp/ReimersG17} is a state-of-the-art sequence labeling approach. We fine-tuned parameters of the approach, including character embedding, embedding size, dropout rate, etc., and denoted the resulting approach by BLCC.
\item{EAD:} The Encoder-Aligner-Decoder approach maps instructions to action sequences proposed by \cite{Mei2016Listen}, denoted by EAD.
\item{CMLP:} We consider a Combined Multi-layer Perceptron (CMLP), which consists of $N$ MLP classifiers. $N=500$ for action names extraction and $N=100$ for action arguments extraction. Each MLP classifier focuses on not only a single word but also the k-gram context.
\end{itemize}
When comparing with baselines, we adopt the settings used by \cite{DBLP:journals/corr/ZhangW15b} to build our CNN networks. We set the input dimension to be $(500 \times 100)$ for action names and $(100 \times 150)$ for action arguments, the number of feature-maps to be $32$. We used $0.25$ dropout on the concatenated max pooling outputs and exploited a $256$ dimensional fully-connected layer before the final two dimensional outputs. We set the replay memory $\Omega=100000$, discount factor $\gamma=0.9$. We varied $\rho$ from $0.05$ to $0.95$ with the interval of $0.05$ and found the best value is $0.80$ (that is why we set $\rho=0.80$ in the experiment). We set $\delta=0.10$ for action names, $\delta=0.07$ for arguments according to Table \ref{Statistics of Datasets}, the constant $c =50$, learning rate of adam to be 0.001, probability $\epsilon$ for $\epsilon$-greedy decreasing from 1 to 0.1 over 1000 training steps.
\subsubsection{Comparison with Baselines}
\begin{table}[!ht]
\caption{F1 scores of different methods in extracting all types of action names and all types of action arguments}
\label{best reaults}
\begin{small}
\begin{tabular}{c|ccc|ccc}
\hline
& \multicolumn{3}{c|}{\textbf{Action Names}} & \multicolumn{3}{c}{\textbf{Action Arguments}} \\
Method & \textbf{WHS} & \textbf{CT} & \textbf{WHG} & \textbf{WHS} & \textbf{CT} & \textbf{WHG} \\
\hline
EAD-2 & 86.25 & 64.74 & 53.49 & 57.71 & 51.77 & 37.70 \\
EAD-8 & 85.32 & 61.66 & 48.67 & 57.71 & 51.77 & 37.70 \\
CMLP-2 & 83.15 & 83.00 & 67.36 & 47.29 & 34.14 & 32.54 \\
CMLP-8 & 80.14 & 73.10 & 53.50 & 47.29 & 34.14 & 32.54 \\
BLCC-2 & 90.16 & 80.50 & 69.46 & 93.30 & \textbf{76.33} & 70.32 \\
BLCC-8 & 89.95 & 72.87 & 59.63 & 93.30 & \textbf{76.33} & 70.32 \\
STFC & 62.66 & 67.39 & 62.75 & 38.79 & 43.31 & 42.75 \\
{{\tt EASDRL}} & \textbf{93.46} & \textbf{84.18} & \textbf{75.40} &
\textbf{95.07} & 74.80 & \textbf{75.02} \\
\hline
\end{tabular}
\end{small}
\end{table}
We set the restriction $n=2$ and $n=8$ for EAD, CMLP and BLCC which need one-to-one sequence pairs, and no restriction for STFC and {{\tt EASDRL}}. In all of our datasets, the arguments of an action are either all essential arguments or one exclusive argument pair together with all other essential arguments, which means at most $2^1$ sequences can be generated. Therefore, the results of action arguments extraction are identical when $n=2$ and $n=8$. The experimental results are shown in Table \ref{best reaults}. From Table \ref{best reaults}, we can see that our {{\tt EASDRL}} approach performs the best on extracting both action names and action arguments in most datasets, except for CT dataset. We observe that the number of arguments in most texts of the CT dataset is very small, such that BLCC performs well on extracting arguments in the CT dataset.
On the other hand, we can also observe that BLCC, EAD and CMLP get
worse performance when relaxing the restriction on $n$ ($n=2$ and $n=8$). The reason is that when given a single text with many possible output sequences, these models learn common parts (essential items) of outputs, neglecting the different parts (optional or exclusive items). We can also see that both sequence labeling method and encoder-decoder structure do not work well, which exhibits that, in this task, our reinforcement learning framework can indeed perform better than traditional methods.
\begin{table}[!ht]
\caption{F1 scores of different methods in extracting exclusive action names and exclusive action arguments}
\label{exclusive study}
\begin{small}
\begin{tabular}{c|ccc|ccc}
\hline
& \multicolumn{3}{c|}{\textbf{Action Names}} & \multicolumn{3}{c}{\textbf{Action Arguments}} \\
Method & \textbf{WHS} & \textbf{CT} & \textbf{WHG} & \textbf{WHS} & \textbf{CT} & \textbf{WHG} \\
\hline
EAD-2 & 26.60 & 21.76 & 22.75 & 40.78 & 47.91 & 39.81 \\
EAD-8 & 22.12 & 17.01 & 23.12 & 40.78 & 47.91 & 39.81 \\
CMLP-2 & 31.54 & 54.75 & 51.29 & 35.52 & 25.07 & 29.78 \\
CMLP-8 & 26.90 & 51.80 & 41.03 & 35.52 & 25.07 & 29.78 \\
BLCC-2 & 16.35 & 38.27 & 54.34 & 12.50 & 13.45 & 18.57 \\
BLCC-8 & 19.55 & 35.01 & 41.27 & 12.50 & 13.45 & 18.57 \\
STFC & 46.40 & 50.28 & 44.32 & 50.00 & 46.40 & 50.32 \\
{{\tt EASDRL}} & \textbf{56.19} & \textbf{66.37} & \textbf{68.29} &
\textbf{66.67} & \textbf{54.24} & \textbf{55.67} \\
\hline
\end{tabular}
\end{small}
\end{table}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.2\textwidth]{names_ablation.pdf}
\includegraphics[width=0.2\textwidth]{arguments_ablation.pdf}
\caption{Results of {{\tt EASDRL}} ablation studies}
\label{ablation study}
\end{figure}
In order to test and verify whether or not our {{\tt EASDRL}} method can deal with complex action types well, we compare with baselines in extracting \emph{exclusive action names} and \emph{exclusive action arguments}. Results are shown in Table \ref{exclusive study}. In this part, our {{\tt EASDRL}} model outperforms all baselines and leads more than $5\%$ absolutely, which demonstrates the effectiveness of our {{\tt EASDRL}} model in this task.
We would like to evaluate the impact of additional reward and positive-rate based experience replay. We test our {{\tt EASDRL}} model by removing \emph{positive-rate based experience replay} (denoted by ``-PR'') or \emph{additional reward} (denoted by ``-AR''). Results are shown in Figure \ref{ablation study}. We observe that removing either \emph{positive-rate based experience replay} or \emph{additional reward} degrades the performance of our model.
\subsubsection{Online Training Results}
To further test the robustness and self-learning ability of our approach, we design a human-agent interaction environment to collect the feedback from humans. The environment takes a text as input (as shown in the upper left part of Figure \ref{gui}) and present the results of our {{\tt EASDRL}} approach in the upper right part of Figure \ref{gui}. Humans adjust the output results by inputting values in the ``function panel'' (as shown in the middle row) and pressing the buttons (in the bottom). After that, the environment updates the deep Q-networks of our {{\tt EASDRL}} approach based on humans' adjustment (or feedback) and output new results in the upper right part. Note that the parts indicated by $\langle 1\rangle,\langle 2\rangle,\ldots, \langle 6\rangle$ in the upper right part comprise the extracted action sequence. For example, the action ``Remove(tape)'', which is indicated in the upper right part with orange color, should be ``Remove(tape, deck)''. The user can delete, revise or insert words (corresponding to the buttons with labels ``Delete'', ``Revise'' and ``Insert'', respectively) by input ``values'' in the middle row, where ``Act/Arg'' is used to decide whether the inputed words belong to action names or action arguments, ``ActType/ArgType'' is used to decide whether the inputed words are essential, optional or exclusive, ``SentId'' and ``ActId/ArgId'' are used to input the sentence indices and word indices of inputed words, ``ExSentId'' and ``ExActId/ExArgId'' are used to input the indices of exclusive action names or arguments. After that, the modified text with its annotations will be used to update our model.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.48\textwidth]{gui.png}
\caption{A snapshot of our human-agent interacting environment}
\label{gui}
\end{figure}
Before online training, we pre-train an initial model of {{\tt EASDRL}} by combining all labeled texts of WHS, CT and WHG, with $30$ labeled texts of WHG for testing. The accuracy of this initial model is low since it is domain-independent. We then use the unlabeled texts in WHG (i.e., 80 texts as indicated in the last row in Table \ref{Statistics of Datasets}) for online training. We ``invited'' humans to provide feedbacks for these 80 texts (with an average of 5 texts for each human). When a human finishes the job assigned to him, we update our model (as well as the baseline model). We compare {{\tt EASDRL}} to the best offline-trained baseline BLCC-2. Figure \ref{online test} shows the results of online training, where ``online collected texts'' indicates the number of texts on which humans provide feedbacks. We can see that {{\tt EASDRL}} outperforms BLCC-2 significantly, which demonstrates the effectiveness of our reinforcement learning framework.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.2\textwidth]{names_online.pdf}
\includegraphics[width=0.2\textwidth]{arguments_online.pdf}
\caption{Online test results of WHG dataset}
\label{online test}
\end{figure}
\section{Conclusion}
In this paper, we proposed a novel approach {{\tt EASDRL}} to automatically
extract action sequences from texts based on deep reinforcement
learning. To the best of our knowledge, our {{\tt EASDRL}} approach is the
first approach that explores deep reinforcement learning to extract
action sequences from texts. We empirically demonstrated that our {{\tt EASDRL}} model outperforms state-of-the-art baselines on three datasets. We showed that our {{\tt EASDRL}} approach could better handle complex action types and arguments. We also exhibited the effectiveness of our {{\tt EASDRL}} approach in an online learning environment. In the future, it would be interesting to explore the feasibility of learning more structured knowledge from texts such as state sequences or action models for supporting planning.
\section*{Acknowledgements}
Zhuo thanks the support of the National Key Research and Development
Program of China (2016YFB0201900), National Natural Science Foundation
of China (U1611262), Guangdong Natural Science Funds for Distinguished
Young Scholar (2017A030306028), Pearl River Science and Technology New
Star of Guangzhou, and Guangdong Province Key Laboratory of Big Data
Analysis and Processing for the support of this research.
Kambhampati's research is supported in part by the AFOSR grant FA9550-18-1-0067, ONR grants N00014161-2892, N00014-13-1-0176, N00014- 13-1-0519, N00014-15-1-2027, and the NASA grant NNX17AD06G.
\bibliographystyle{named}
|
{
"timestamp": "2018-05-14T02:10:26",
"yymm": "1803",
"arxiv_id": "1803.02632",
"language": "en",
"url": "https://arxiv.org/abs/1803.02632"
}
|
\section{Introduction}
Planets form in protoplanetary disks. When observed at high resolution with ALMA, a number of these disks show interesting patterns in their millimeter emission profiles, in some cases including series of bright and dark rings \citep{Brogan2015,Dong2015,Andrews2016,Isella2016,Loomis2017,Fedele2017,Fedele2017b,Dipierro2018}. Some of these disks may even be quite young ($\lesssim1$ Myr) \citep{Brogan2015,Dong2015,Fedele2017b,Dipierro2018}. There are a number of explanations for such features, including chemical processes that alter dust opacities and sticking/fracturing processes near snow lines \citep[e.g.][]{Ros2013,Zhang2015,Banzatti2015} as well as vortices created at the edges of magnetic dead zones \citep[e.g.][]{Simon2014,Flock2015}, but the most exciting possibility is that these features are tracing gaps opened in disks by forming planets \citep[e.g.][]{Dong2015}.
GY 91 is a M4 protostar \citep{Doppmann2005} in the L1688 region of the $\rho$ Ophiuchus molecular cloud, located at a distance of 137 pc \citep{OrtizLeon2017}. GY 91's broadband spectral energy distribution (SED) rises sharply in the infrared and appears to peak at far-infrared wavelengths, although it has not been detected between 35 $\mu$m and 870 $\mu$m. The infrared spectral index ($\alpha_{IR} = 0.45$) and bolometric temperature ($T_{bol} = 370$ K), as well as its association with a 1.1 mm core, classify GY 91 as a Class I protostar \citep[e.g.][]{Enoch2008,McClure2010,Dunham2015}. This indicates that the protostar is surrounded by a protoplanetary disk still embedded in its natal envelope of collapsing cloud material, and is young \citep[$\lesssim0.5$ Myr;][]{Evans2009}. The Spitzer IRS spectrum of the source shows both silicate and ice absorption features, which are also commonly associated with embedded protostars \citep[e.g.][]{Watson2004}.
A few studies that consider alternate classification schemes have suggested that GY 91 may not be embedded. \citet{McClure2010} find that the 5--12 $\mu$m spectral index is within the range found for disks with foreground extinction ($n_{5-12} = -0.25$). However, their measured value is also on the border between disks with foreground extinction and disks with envelopes (of $n_{5-12} = -0.2$), and the extinction corrected spectral index ($\alpha_{IR}' = 0.31$) and bolometric temperature ($T_{bol}' = 470$ K) still qualify the source as a Class I protostar \citep{Dunham2015}. \citet{vanKempen2009} also found HCO$^+$ emission towards GY 91 that was bright enough to be above the cutoff for an embedded source, but that emission seems to be associated with a patch of cloud that peaks 30" away from the source.
\floattable
\input{table1.tex}
Here we present new ALMA data, which when combined with the observed SED, show that GY 91 is indeed a Class I protostar with a circumstellar disk embedded in an envelope. Our 3 mm and 870 $\mu$m images also reveal the presence of three narrow dark rings in its disk that resemble those seen in HL Tau and a handful of other disks \citep{Brogan2015,Dong2015,Andrews2016,Isella2016,Loomis2017,Fedele2017,Fedele2017b,Dipierro2018}. We compare the circumstellar structure of GY 91 to HL Tau, and argue that GY 91 is the youngest source in which disk gaps have been detected. If caused by planets, these features provide evidence for giant planet formation within ~0.5 Myr.
\section{Observations \& Data Reduction}
\subsection{ALMA}
GY 91 was observed with ALMA Band 3 (100 GHz/3 mm) in three tracks from 31 October 2015 to 17 April 2016, with baselines ranging from 14 m -- 15.3 km. All four basebands were tuned for continuum observations centered at 90.5, 92.5, 102.5, 104.5 GHz, each with 128 15.625 MHz channels for 2 GHz of continuum bandwidth per baseband. In all the observations had 8 GHz of total continuum bandwidth. We also observed GY 91 with ALMA Band 7 (345 GHz/870 $\mu$m) on 19 May 2016 and 11 September 2016, with baselines ranging from 15 -- 3140 m. Two of four basebands were configured for continuum observations centered at 343 GHz and 356.25 GHz, with a total of 4 GHz of continuum bandwidth. The remaining basebands were devoted to spectral line observations, although nothing was detected. We list details of the observations in Table \ref{table:alma_obs}.
The data were reduced in the standard way with the \texttt{CASA} pipeline and the calibrators listed in Table \ref{table:alma_obs}. After calibrating, we imaged the data by Fourier transforming the visibilities with the \texttt{CLEAN} routine. After our initial imaging, we found that we could improve the sensitivity of the 345 GHz image by self-calibrating. We ran four iterations of phase-only self-calibration on the compact configuration track and a single iteration of phase-only self-calibration on the extended configuration track. This improved the rms in an image produced with natural weighting (i.e. a robust parameter of 2) from 0.36 mJy /beam to 0.27 mJy/beam. We were unable to improve the 100 GHz image by self-calibration.
Our final images were produced using Briggs weighting with a robust parameter of 0.5, which provides a good balance between sensitivity and resolution, to weight the visibilities for both datasets. The 3 mm image has a beam of size 0.06" by 0.05" with a P.A. of 81.9$^{\circ}$ and an rms of 36 $\mu$Jy/beam. The 870 $\mu$m image nas a beam size of 0.134" by 0.129" with a P.A. of -9.4$^{\circ}$ and an rms of 0.31 mJy/beam. We show the images in Figure \ref{fig:alma_data}.
\begin{figure*}
\centering
\figurenum{1}
\includegraphics[width=7in]{figure1.pdf}
\caption{Our ALMA 345 GHz ({\it left}) and 100 GHz ({\it right}) maps of the GY 91 protoplanetary disk. Two dark lanes are readily apparent in the 345 GHz map, while a third dark lane is also apparent in inner regions of the disk at 100 GHz because of the higher resolution of our 100 GHz maps.}
\label{fig:alma_data}
\end{figure*}
\subsection{SED from the Literature}
We compile a broadband spectral energy distribution (SED) for GY 91 from a thorough literature search. The data includes {\it Spitzer} IRAC and MIPS photometry as well as fluxes from the literature at a range of wavelengths \citep{Wilking1983,Lada1984,Greene1992,Andre1994,Strom1995,Barsony1997,Johnstone2000,Allen2002,Natta2006,Stanke2006,AlvesdeOliveira2008,Jorgensen2008,Padgett2008,Wilking2008,Evans2009,Gutermuth2009,Barsony2012}. When modeling the SED, as we discuss below, we assume a constant 10\% uncertainty on any photometry from the literature to account for any flux calibration uncertainties between the measurements.
In addition to the broadband photometry, we also download the Spitzer IRS spectrum of GY 91 from the CASSIS database \citep{Lebouteiller2011,Lebouteiller2015}. Rather than consider the entire SED, which can be computationally prohibitive for the radiative transfer calculations described below, we sample the IRS spectrum at 25 points ranging from 5 to 35 $\mu$m. We also assume a 10\% uncertainty on these fluxes, like we do for the broadband photometry.
\begin{figure}[b]
\centering
\figurenum{2}
\includegraphics[width=3.3in]{figure2.pdf}
\caption{The one dimensional, azimuthally averaged, de-projected radial brightness profile of the GY 91 disk at 345 GHz and 100 GHz, with the locations of the dark lanes marked by vertical dashed lines. These gaps are readily seen in the brightness profile. We also show the azimuthally averaged brightness profile of our gapped disk+envelope model (see Figure \ref{fig:disk_ulrich}, Table 2) at each wavelength.}
\label{fig:radial_profile}
\end{figure}
\section{Results}
We show our 3mm and 870 $\mu$m images of GY 91 in Figure \ref{fig:alma_data}. The 870 $\mu$m image has a much higher signal-to-noise ratio, and it is fairly easy to identify, by-eye, two concentric dark lanes that appear in the disk. The 3 mm image also reveals a third dark lane in the inner regions of the disk that is not visible at 870 $\mu$m because of the factor of two poorer resolution. As the disk is massive (see below), high optical depth in the inner disk could also help to hide the inner gap at 870 $\mu$m. To better illustrate the presence of these features, in Figure \ref{fig:radial_profile} we show a one dimensional brightness profile for both the 870 $\mu$m and 3 mm images, averaged in ellipses defined by the position angle and inclination of the disk to be constant radius bins. The outer two dark lanes show up clearly in the 870 $\mu$m radial profile, while the inner lane shows up clearly in the 3 mm profile. Moreover, there appears to be a break in the 870 $\mu$m profile at the location of the inner dark lane, and there appears to be a dip in the 3 mm brightness profile that is consistent with the location of the middle dark lane, despite the noisiness of the 3 mm image that prevents it from being detected by eye.
\begin{figure*}
\centering
\figurenum{3}
\includegraphics[width=7in]{figure3.pdf}
\caption{The best fit simple geometrical model for GY 91 compared with the data. The model assumes the disk is flat, with a surface density described by Equation 1 and $M_{disk} = 0.36$ M$_{\odot}$, $r_c = 71$ AU, $\gamma = 0.3$, $i = 39^{\circ}$, and $p.a. = -19^{\circ}$. The model uses a power-law temperature distribution with $T = 46 (R / 1 \, \text{AU})^{-0.4}$. We use the power-law millimeter opacity function described in \citet{Beckwith1990}, $\kappa({\nu}) = 0.1 \, (\nu / 1000 \, \text{GHz})^\beta$ cm$^2$ g$^{-1}$ with $\beta = 1.8$. Our model includes three gaps with the following parameters: $R_{gap,1} = 10.4$ AU, $w_{gap,1} = 5.9$ AU, $\delta_{gap,1} \approx 0$, $R_{gap,2} = 40.3$ AU, $w_{gap,2} = 27.5$ AU, $\delta_{gap,2} = 0.15$, $R_{gap,3} = 68.9$ AU, $w_{gap,3} = 10.7$ AU, and $\delta_{gap,3} \approx 0$. Our modeling indicates that the first and third gaps are deep, however as the data is noisy and not high enough resolution to well resolve the gaps, the actual depths are quite uncertain. As the disk is quite massive, it is somewhat optically thick at $870$ $\mu$m. We show the one dimensional, azimuthally averaged, visibility amplitudes on the left, the raw model images on the center-left, the model images convolved with the beam in the center-right column, and the residuals on the right. The colorbar shows the color scale for the center-right and right-most panels. The peak residuals are $1.7\sigma$ at 345 GHz and $3.5\sigma$ at 100 GHz.}
\label{fig:geometrical_model}
\end{figure*}
In order to study these features in greater detail, we fit a model to the data to determine disk properties such as radius, position angle, and inclination, as well as the locations, widths, and depths of the gaps. We use Monte Carlo radiative transfer codes to produce synthetic observations of model protostars that can be fit to our combined millimeter visibility and broadband SED dataset of GY 91. This modeling procedure is described in further detail in \citet{Sheehan2014} and Sheehan \& Eisner (submitted), but we give a brief overview here.
Our model includes a flared protoplanetary disk with a physically motivated surface density profile \citep[e.g.][]{LyndenBell1974} surrounded by a rotating collapsing envelope \citep[e.g.][]{Ulrich1976},
\begin{equation}
\Sigma = \Sigma_0 \, \left(\frac{R}{r_c}\right)^{-\gamma} \, \exp\left[-\left(\frac{R}{r_c}\right)^{2-\gamma}\right],
\end{equation}
\begin{equation}
\rho_{disk} =\frac{\Sigma}{\sqrt{2\pi}\,h} \, \exp\left(-\frac{1}{2}\left[\frac{z}{h}\right]^2\right),
\end{equation}
\begin{equation}
h = h_0 \left(\frac{R}{1 \text{ AU}}\right)^{\beta},
\end{equation}
\begin{equation}
\footnotesize
\rho_{env} = \frac{\dot{M}}{4\pi}\left(G M_* r^3\right)^{-\frac{1}{2}} \left(1+\frac{\mu}{\mu_0} \right)^{-\frac{1}{2}} \left(\frac{\mu}{\mu_0}+2\mu_0^2\frac{R_c}{r}\right)^{-1}.
\end{equation}
In Equations 1, 2 \& 3, $R$ and $z$ are in cylindrical coordinates, while in Equation 4, $\mu = \cos\,\theta$ and $r$ and $\theta$ are in spherical coordinates.
In this model the disk mass, $M_{disk}$, inner and outer radii, $R_{in}$ \& $R_{disk}$, surface density power-law exponent, $\gamma$, scale height power-law exponent, $\beta$, and scale height at 1 AU, $h_0$, are left as free parameters. We also leave the envelope mass, $M_{env}$, and radius, $R_{env}$, as free parameters, and give the envelope an outflow cavity described by $f_{cav}$, the fraction by which the density is reduced in the cavity, and $\xi$, which relates to the cavity opening angle. We supply the density structure with opacities described in \citet{Sheehan2014}, leaving the maximum dust grain size, $a_{max}$, and grain size distribution power-law exponent, $p$, as free parameters.
We model the dark lanes as gaps in the surface density profile, which are described by the radius at the center of the gap ($R_{gap,i}$), width ($w_{gap,i}$), and depth ($\delta_{gap,i}$). The depth of the gap is a multiplicative factor that represents the amount by which the surface density is reduced in the gap. $\delta = 0$ corresponds to a complete absence of material in the gap.
Because full radiative transfer modeling is very computationally intensive, we make initial estimates of disk properties and the gap widths and depths by fitting a simple geometric model to the $870$ $\mu$m and 3 mm visibilities (see Figure 3) using the MCMC code \texttt{emcee} \citep{ForemanMackey2013}. The parameters found from this simple geometrical fit are then used as initial guesses to generate a detailed radiative transfer model that simultaneously matches both visibility datasets and the broadband SED.
The purpose of the radiative transfer modeling is, in particular, to determine whether an envelope component is needed to explain the observations. We use the Monte Carlo radiative transfer codes RADMC-3D \citep{Dullemond2012} and Hyperion \citep{Robitaille2011} to calculate the temperature throughout the density structure described above, and then to produce synthetic millimeter visibilities and broadband SEDs. We fit these synthetic observations simultaneously to all three (870 $\mu$m visibilities, 3 mm visibilities, and broadband SED) of our datasets, again using the \texttt{emcee} code.
The MCMC walkers are allowed to move through parameter space for an extended burn-in period. We consider the fit to have converged when the walkers have reached a steady state, with the best fit value for each parameter changing minimally over a large number of steps. We then calculate the best-fit value for each parameter as the median position of the walkers after discarding the burn-in steps. We estimate the uncertainty on these parameter values as the range around the median containing 68\% of the walkers.
\begin{figure*}
\centering
\figurenum{4}
\includegraphics[width=7in]{figure4.pdf}
\caption{The gapped disk+envelope model for GY 91 compared with the data. We show the one dimensional, azimuthally averaged, 870 $\mu$m visibility amplitudes on the left, the 3 mm visibilities in the center, and the SED on the right.}
\label{fig:disk_ulrich}
\end{figure*}
We compare the gapped disk+envelope model to the observed visibilities and SED in Figure \ref{fig:disk_ulrich} and list the model parameters in Table \ref{table:best_fits}. The images for our gapped disk+envelope model look almost identical to those shown in Figure \ref{fig:geometrical_model}, although the residuals between data and model are higher, not surprising since we are fitting the visibilities and SED simultaneously here.
This model can simultaneously reproduce the 870 $\mu$m visibilities, 3 mm visibilities, and broadband SED for GY 91. The GY 91 disk appears to be embedded in an envelope with $M_{env} = 0.12 \, M_{disk}$.
Gaps are found at radii of $\sim$7 AU, $\sim$40 AU, and $\sim$69 AU, with widths of $\sim$7 AU, $\sim$30 AU, and $\sim$10 AU. The gap depths for the inner and outer gaps are not well constrained because they are not resolved well by our observations. The middle gap appears to be wide and somewhat shallow, although with higher resolution it is possible that it will break up into multiple gaps.
\input{table2.tex}
\section{Discussion \& Conclusion}
GY 91 appears to be part of a growing population of protoplanetary disks that have ring-like features in their millimeter emission profiles. The 870 $\mu$m image resembles the disks of HL Tau, AA Tau, TW Hya, HD 163296, HD 169142, AS 209, and Elias 24, all of which have several gaps visible in their millimeter emission profiles \citep{Brogan2015,Andrews2016,Isella2016,Loomis2017,Fedele2017,Fedele2017b,Dipierro2018}. Closer inspection of these systems, however, reveals differences in the appearance of the features in each disk. The bright and dark rings seen in TW Hya are narrow (sizes $<2$ AU) and shallow \citep{Andrews2016}. Only the innermost gap, at 2 AU, has a significant depth. The gaps found in AA Tau, HD 163296, and HD 169142, on the other hand, are all very wide and deep, with widths of 22 -- 55 AU \citep{Isella2016,Loomis2017,Fedele2017}. The gaps found in HL Tau and AS 209 appear to be deep, with moderate widths of $\sim5-30$ AU \citep[e.g.][]{Brogan2015,Zhang2015}.
The innermost and outermost gaps we find in GY 91's disk appear to be quantitatively the most similar to the HL Tau gaps as they are somewhat narrow, with widths of $\sim$7 AU and $\sim$10 AU, while the middle gap appears to be large like the gaps found in AA Tau, HD 163296, and HD 169142.
\subsection{Planets Carving Gaps?}
Although there are a number of potential origins of these features, the most exciting possibility is, perhaps, that these gaps are carved by proto-planets embedded in the disk. \citet{Dong2015} found that the gaps in the HL Tau disk could be sculpted by planets with masses as small as a Saturn-mass. \citet{Isella2016} found similar results for HD 163296, although the gaps are much larger in that disk.
We can estimate the masses of planets that are needed to produce the gaps we see in GY 91's disk. Simulations suggest that planets should open gaps whose widths are a few times larger than the Hill radius of the planet,
\begin{equation}
W \approx 8 \times R_p \, \left(\frac{M_p}{M_*}\right)^{1/3}
\end{equation}
\citep{Rosotti2016}. Although the protostellar mass of GY 91 is not constrained well, it is thought to be a M4 protostar with a temperature of 3300 K \citep{Doppmann2005}, which evolutionary models predict should have a mass of $\sim0.25$ M$_{\odot}$ at $\sim0.5$ Myr \citep{Baraffe2015}. Using these assumptions, we estimate that planets of masses $\sim0.2$ M$_J$, $\sim$0.2 M$_J$, and $\sim$0.002 M$_J$ are needed to produce the observed gaps.
The mass estimated for the outermost planet highlights the limitations of these simple estimates, as it seems unlikely that an Earth-mass planet is opening such a gap. Recent studies have suggested that for low-mass planets, the gap width may be a constant multiple of the scale height and therefore independent of planet mass \citep[e.g.][]{Duffell2013,Dong2017}. Further studies suggest that the mass of a gap-opening planet is best constrained by measurements of the gap width and depth in the gas distribution, combined with a measurement of disk viscosity \citep{Fung2014,Kanagawa2015,Dong2017}. Without knowledge of the gas distribution, however, we cannot place stronger constraints on the potential planet masses. It also should be noted that a single planet can open multiple gaps in a disk \citep{Bae2017}.
\subsection{Other Causes of Dark Lanes}
Planets aren't the only possible cause of these features. One alternative that should be common in protoplanetary disk is the variation in dust opacities and collisional fragmentation/coagulation properties that is expected to occur at snow lines. As dust grains radially drift inwards due to the loss of angular momentum from a headwind of sub-Keplerian gas \citep{Weidenschilling1977b}, they will cross a series of snow lines for various volatiles. When they cross a snow line, that volatile sublimates back into the gas phase. As the sublimated gas radially diffuses, it can re-condense onto particles outside of the snow line. The icy particles outside of the snow line can efficiently grow to decimeter or larger sizes, while solids inside the snow line tend to fragment \citep{Cuzzi2004,Ros2013,Banzatti2015}. The change in the optical properties of dust grains across the snow line could cause features like those seen in HL Tau or GY 91 \citep[e.g.][]{Zhang2015}.
Alternatively, the ``sintering" of dust grains just below the sublimation temperature produces brittle grains that fragment more readily and therefore grow to smaller sizes just outside the snow line. Because the sintered grains have smaller sizes, they undergo slower radial drift, causing pile ups near snow lines. This process could also produce features similar to those seen in HL Tau or GY 91 \citep{Okuzumi2016}.
\begin{figure*}[t]
\centering
\figurenum{5}
\includegraphics[width=6in]{figure5.pdf}
\caption{The disk+envelope model for HL Tau compared with the data. In the first row we show the one dimensional, azimuthally averaged, 870 $\mu$m visibility amplitudes on the left and the SED on the right, with the model as a green curve in both. The second row shows the 345 GHz model and residual images. We did not include gaps in this model, which is why they can be seen in the residual map. The model has a disk with a mass of 0.2 M$_{\odot}$ a radius of 120 AU, a surface density power law exponent of $\gamma = 1.7$, and an inclination of 44$^{\circ}$. The envelope has a mass of $0.04$ M$_{\odot}$ and a radius of 1800 AU.}
\label{fig:hltau}
\end{figure*}
We compare the midplane disk temperature inferred for GY 91 with the temperatures of snow lines of common volatiles \citep[e.g.][]{Zhang2015}. The outermost gap does roughly match the freeze out region of $N_2$. However no obvious counterparts are seen for the inner two gaps.
Zonal flows produced by magneto-rotational instability driven turbulence \citep{Johansen2009} have also been shown to produce axisymmetric pressure bumps that can trap large dust grains and may produce gap-like features in millimeter images \citep{Pinilla2012,Dittrich2013,Simon2014}. In this case, the pressure bumps are created by large scale variations in the magnetically driven turbulence that produce variations in the mass accretion rate that in turn causes material to pile up. This effect can also be seen at the outer edge of magnetic dead zones, where there is strong radial variation in the mass accretion rate. These flows can produce gap-like features in disks \citep{Pinilla2012,Flock2015}.
\subsection{Comparison with HL Tau}
If planets are indeed carving gaps in GY 91's disk, the masses of those planets would place strong constraints on the timescales for planet formation in disks. As a Class I protostar, GY 91 likely has an age of $\sim0.5$ Myr \citep{Evans2009}, so planets must grow to masses of $\sim0.2$ M$_J$ on these short timescales.
Similar constraints have been placed on the timescale for planet formation by the gaps in other young disks. AS 209 and Elias 24 are also in $\rho$ Ophiuchus, which has been estimated to be quite young \citep[$0.5-1$ Myr; e.g.][]{Natta2006}. Both are, however, found to be Class II sources \citep[e.g.][]{Barsony2005,Andrews2009}, and therefore are likely to be older than GY 91. HL Tau's disk may be even younger than AS 209 and Elias 24, as it is also possibly still embedded, but comparable to or perhaps older than GY 91 \citep[e.g.][]{Robitaille2007}.
To the best of our knowledge, however, a detailed radiative transfer modeling fit to the combined HL Tau millimeter visibilities and SED has not been done since the ALMA Science Verification data was acquired. We use the disk+envelope modeling procedure described above to fit a disk+envelope model to the HL Tau ALMA millimeter visibilities and SED. For simplicity, though, we ignore the gaps and consider only a smooth density distribution. Furthermore, as it is possible that we are resolving out large scale envelope structure with our visibility datasets, we have limited the HL Tau visibilities to a minimum baseline of 16 k$\lambda$ to match our GY 91 dataset and ensure a fair comparison of the two sources. The best fit model is shown in Figure \ref{fig:hltau}.
\begin{figure}[t]
\centering
\figurenum{6}
\includegraphics[width=3.25in]{figure6.pdf}
\caption{Histograms of the positions of MCMC walkers in $M_{env}/M_{disk}$ over the last 10 steps of the MCMC fit, indicating the allowed range $M_{env}/M_{disk}$ for each source.}
\label{fig:menv_mdisk}
\end{figure}
Our model for HL Tau has $M_{env} = 0.106 \, M_{disk}$, similar to what we find for GY 91 ($M_{env} = 0.121 \, M_{disk}$). This is in agreement with the classification of HL Tau as a ``flat spectrum" object, indicating it is likely in transition from the Class I to the Class II stage. The range of allowed values for $M_{env}/M_{disk}$ are shown in \ref{fig:menv_mdisk}. Although the distributions are similar, suggesting that both sources are at a similar stage of their evolution, our modeling finds a slightly larger value of $M_{env}/M_{disk} = 0.121$ for GY 91. This may indicate that a larger fraction of the HL Tau envelope has been depleted onto the disk or central protostar, although not by much.
If we assume that $M_{env}/M_{disk}$ is an evolutionary indicator \citep[e.g.][]{Crapsi2008}, this suggests that GY 91 is of similar age to, or possibly younger than HL Tau.
The identification of multiple young disks with gaps is a clear indication that planet formation, or at the very least processes that contribute to planet formation, are happening in the very early stages of stellar formation. If GY 91 is even younger than HL Tau, then measurements of the masses of planets embedded in it's disk could place even stronger constraints on the timescales of planet formation than planets in the HL Tau disk.
\vspace{10pt}
Regardless of whether these dark lanes are formed by planets, zonal flows, or chemical variations produced by radial drift, the presence of these features is likely an indication that planet formation is well underway at early times. Both zonal flows and chemical effects have been suggested to enhance the growth of particles in disks \citep[e.g.][]{Simon2014,Ros2013}, and may be key elements in how planets are formed. Further high resolution studies of these young disks are crucial for understanding the early stages of planet formation.
\acknowledgements
The authors would like to thank the anonymous referee, whose comments helped to improve this work. This work was supported by NSF AAG grant 1311910. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2015.1.00761.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
\software{CASA \citep{McMullin2007}, RADMC-3D \citep{Dullemond2012}, Hyperion \citep{Robitaille2011}, emcee \citep{ForemanMackey2013}}
|
{
"timestamp": "2018-03-09T02:00:42",
"yymm": "1803",
"arxiv_id": "1803.02847",
"language": "en",
"url": "https://arxiv.org/abs/1803.02847"
}
|
\section{Introduction}
It has been known since the late 60’s that any non-linear interaction of radiation with electrons depends on the quantum statistical properties of the radiation \cite{ref1,ref2,ref3,ref4,ref5,ref6,ref7,ref8,ref9,ref10,
ref11,ref12,ref13,ref14,ref15}. In particular, a transition from a bound state to a continuum, such as ionization, offers the simplest and most directly observable process, in which intensity correlation functions of the radiation are involved. A standard derivation of the transition probability per unit time (rate) for N-photon ionization, leads to a rate proportional to some effective N-photon matrix element multiplied by the Nth order intensity correlation function. However, any N-photon transition from the ground state to the continuum inevitably involves transition amplitudes through virtual or real bound atomic intermediate states. Strictly speaking, the above statement on the dependence of the process on the ${N^{th}}$-order intensity correlation function is valid, as long as the intermediate states can be assumed to be sufficiently far from resonance, so that they can be eliminated adiabatically, which leads to the effective N-photon matrix element.
To make further discussion more concrete, we consider for the moment 2-photon ionization, whose rate would be proportional to the 2nd order intensity correlation function. The transition amplitude for the first photon involves all non-vanishing matrix elements between the ground and excited states. A closely related problem, namely the strong driving between two bound states, by stochastic radiation, represents a very fascinating problem, which cannot be treated in terms of a single transition probability per unit time. That problem has received considerable attention in the past \cite{ref16,ref17} and can be considered, for all practical purposes, solved. In what follows, we will be concerned with non-resonant 2-photon ionization. We shall assume that the chosen photon frequency is far from resonance with the nearest allowable intermediate state; an assumption to be qualified later on. More precisely, we assume that the laser bandwidth is much smaller than the detuning from the nearest state. Taking this formally to a limiting case, we shall cast this discussion in terms of a monochromatic source, which implies zero bandwidth.
For the sake of simplicity, which does not entail a significant limitation of generality, we stay with the assumption that initially the electron is in the ground state. The 2-photon transition amplitude involves a summation over the complete manifold of intermediate states connected to the ground state with non-vanishing matrix elements. In the limit of small intensity, the transition probability per unit time is indeed proportional to the 2nd order intensity correlation function multiplied by an effective 2-photon matrix element in which all intermediate states are the bare atomic states \cite{ref8}. To the extent that the above condition is satisfied, the rate of ionization is simply proportional to the 2nd order intensity correlation function, which for a chaotic state is larger by a factor of 2 from that for a coherent state. For an N-photon process, the ratio is N!, which hereafter shall be referred to as the chaotic state enhancement. It bears emphasizing at this point that the above analysis is valid only within perturbation theory, in the form of Fermi's golden rule, describing the 2-photon transition in terms of a single rate from the ground state to the continuum. Modifications to that simple case are discussed in the sections that follow.
However, for 2-photon ionization (or any non-linear process for that matter) to be observable, the laser intensity cannot be too low. As a consequence, even if the photon frequency is “sufficiently” far from resonance with the nearest intermediate state, as the intensity rises, the Rabi frequency connecting that state to the ground state may reach a value for which the non-resonant condition is no longer valid; which will occur when the Rabi frequency becomes comparable to the detuning from that state. Obviously, this implies that the validity of the “non-resonant” condition, which is the basis for the adiabatic elimination of the intermediate states, is not independent of the intensity. When that condition is violated, the simple dependence of the process on the 2nd order correlation function becomes at best questionable. It becomes therefore necessary to examine the possible modification of the role of photon statistics as a function of the intensity.
One can explore the issue starting from the other end, by considering 2-photon ionization in the presence of one or even two nearby intermediate resonances, which has in fact been addressed quite some time ago \cite{ref18,ref19,ref20,ref21,ref22} . As one might have expected, the simple proportionality of the rate to the 2nd order intensity correlation function was found to be modified significantly. Further richness was found in the vicinity of two neighbouring intermediate states. Aiming at the generalization of that exploration by including a squeezed state, we consider 2-photon and 3-photon ionization as a function of intensity, at various detunings from the intermediate resonances. We focus in particular on the role of photon correlations as the Rabi frequency becomes comparable to the detuning.
The initial motivation for this work seemed academic, aiming at the calibration of the possibility of using non-linear photoabsorption to obtain information on the photon statistics of squeezed light sources and its role on non-linear photoabsorption \cite{ref23,ref24,ref25,ref26,ref27, ref28,ref29,ref30}. This led us to the reexamination of that issue in the context of standard sources, such as coherent and chaotic, in the process of which we realized that certain assumptions, taken until now for granted, are highly questionable. As discussed in detail later on, it turns out that, in practical terms, the notion of non-resonant few-photon ionization is an abstraction difficult, if possible at all, to implement in an experiment. This may explain why, over the last forty years or so, there are hardly any definitive experimental data exhibiting the chaotic field enhancement, even in the simple case of 2-photon ionization. In the few existing experimental data, the observed enhancement factor for chaotic light has in most cases been less than the expected factor of 2 \cite{ref9,ref10,ref11,ref12}. The relevance of this work to present day possibilities, as far as squeezed radiation is concerned, has been underscored by very recent experimental results on the effect of squeezed light on harmonic generation \cite{ref31}; albeit at quite low intensities, a limitation which may be lifted in the future.
The theoretical problem can be cast in terms of the time-dependent wavefunction or the density matrix. If the quantity we need is the rate (transition probability per unit time), depending on the values of the parameters, the time-dependent wavefunction may exhibit rapid oscillations. Although they can be eliminated, through suitable approximations, their meaning tends to be somewhat opaque, even with extensive discussion \cite{ref18}. The density matrix on the other hand lends itself to the derivation of rate equations which are free of such oscillations, although their validity may become questionable, for certain values of parameters that will be discussed in detail. For the sake of completeness, both of these two mutually complementary approaches are explored in this work.
\section{General Theoretical Background}
\subsection{Multi-photon ionization rate}
Consider an atom in its ground state $\left| g \right\rangle$, coupled to a monochromatic radiation field. Assume that upon the absorption of N photons of frequency $\omega$ the atom is ionized ejecting one electron. Denoting the final continuum state by $\left| f \right\rangle$ and all the intermediate states by $\left| {{a_i}} \right\rangle $, the transition probability per unit time describing the N-photon ionization of the atom has been shown to be of the form \cite{ref8}
\begin{equation}
W_{fg}^{\left( N \right)} = {\hat \sigma _N}{G_N}
\end{equation}
where ${{\hat \sigma }_{\rm N}}$ is a generalized cross section given by
\begin{equation}
{{\hat \sigma }_{\rm N}} = \frac{{{{(2\pi \alpha )}^{\rm N}}}}{{4{\pi ^2}}}\frac{{mK}}{\hbar }{\omega ^{\rm N}}{\int {\left| {{\rm M}_{fg}^{(N)}} \right|} ^2}d{\Omega _\textbf{K}}
\end{equation}
and ${G_N}$ the ${N^{th}}$-order intensity correlation function, which contains information about the coherence properties of the radiation field \cite{ref23}. By $\alpha $ we denote the hyperfine constant, m is the mass of the outgoing electron and \textbf{K} its wavevector. The generalized cross section is given by integration of the differential generalized cross section $\frac{{d{{\hat \sigma }_{\rm N}}}}{{d{\Omega _{\rm \textbf{K}}}}}$ over all possible directions $\Omega _\textbf{K}$ of the outgoing electron. As discussed in the sections that follow, this expression for the transition probability per unit time is valid only in the off-resonance weak field limit, where the effect of the intermediate states on the transition is negligible, in the sense that they acquire no population during the process. If this condition is not satisfied, then the process is not describable in terms of a single rate as in eqn.(1) and a time-dependent approach is necessary. The matrix elements ${\rm M}_{fg}^{(N)}$ contain all of the information about the atomic structure and are defined via
\begin{equation}
{\rm M}_{fg}^{(N)} = \sum\limits_{{a_{N - 1}}...{a_1}} {\frac{{\left\langle f \right|{r^{(\lambda )}}\left| {{a_{N - 1}}} \right\rangle \cdot \cdot \cdot \left\langle {{a_1}} \right|{r^{(\lambda )}}\left| g \right\rangle }}{{({\omega _{{a_{N - 1}}}} - {\omega _g} - (N - 1)\omega ) \cdot \cdot \cdot ({\omega _{{a_1}}} - {\omega _g} - \omega )}}}
\end{equation}
where ${{r^{(\lambda )}}}$ is the projection of \textbf{r} on the polarization vector of the field and \textbf{r} the position operator of the electron.
\subsection{Density Matrix Formalism}
As will become apparent in the course of the treatment, we do at some point need the density matrix formulation. Its general advantages are that it can account for the repopulation of lower states due to spontaneous decay, as well as allowing for the derivation of rate equations which often are quite convenient. If we call $\tilde \rho (t)$ the density operator of the compound system (Atom + Radiation Field) in the interaction picture, its equation of motion is given by:
\begin{equation}
\frac{\partial }{{\partial t}}\tilde \rho (t) = - \frac{i}{\hbar }[\tilde V(t),\tilde \rho (t)]
\end{equation}
where
\begin{equation}
\tilde V(t) = {e^{\frac{i}{\hbar }{H^0}t}}V{e^{ - \frac{i}{\hbar }{H^0}t}}
\end{equation}
and V the atom-radiation interaction which can be time-dependent in general. ${H^0}$ is the unperturbed Hamiltonian of the system and is expressed as the sum of the atomic and the radiation Hamiltonian, i.e. ${H^A}$ and ${H^R}$, respectively.
The density operator of our system ${{\tilde \rho }^S} \equiv T{r_R}(\tilde \rho )$ is obtained via "tracing out" the degrees of freedom of the reservoir (radiation field) to which is coupled, leading to spontaneous decay. The equation of motion of the ${{\tilde \rho }^S}$ operator is known as the Master Equation.
In the case of a two-level atom coupled in addition to an externally imposed radiation field, it can be shown that the Master Equation (to compress notation we have dropped the superscript S) is
\begin{equation}
\frac{\partial}{\partial t}\tilde \rho = - \frac{i}{\hbar}\left[ {\tilde V}^{AF}, \tilde \rho \right] + \mathscr{L}\tilde \rho
\end{equation}
where
\begin{equation}
\mathscr{L} \equiv (2{\sigma _ - }\tilde \rho {\sigma _ + } - {\sigma _ + }{\sigma _ - }\tilde \rho - \tilde \rho {\sigma _ + }{\sigma _ - })
\end{equation}
If we denote the ground state by $\left| g \right\rangle $ and the excited state by $\left| e \right\rangle $, the raising and lowering operators ${\sigma _ + }$ and ${\sigma _ - }$ are defined via ${\sigma _ + } \equiv \left| e \right\rangle \left\langle g \right|$ and ${\sigma _ - } \equiv \left| g \right\rangle \left\langle e \right|$, respectively.
By introducing an electric field of the form $E(t) = E {e^{ - i\omega t}} + c.c.$, the interaction Hamiltonian in the rotating-wave approximation can be written as:
\begin{equation}
{{\tilde V}^{AF}} = - \hbar [{\sigma _ + }\Omega {e^{ - i\Delta t}} + {\sigma _ - }{\Omega ^*}{e^{i\Delta t}}]
\end{equation}
where $\Delta \equiv \omega - {\omega _{eg}}$, the detuning from resonance and $\Omega = {\wp _{eg}}E /\hbar $ the Rabi frequency of the field, expressed as the product of the dipole operator matrix element and the field amplitude, divided by $\hbar$.
If we denote by $\gamma$ the spontaneous decay of the upper state, then by taking matrix elements of eqn. (6) in the base that diagonalizes ${H^A}$, we get the following set of equations \cite{ref18}:
\begin{equation}
\frac{\partial }{{\partial t}}{{\tilde \rho }_{gg}} = \gamma {{\tilde \rho }_{ee}} + i({{\tilde \rho }_{eg}}{\Omega ^*}{e^{i\Delta t}} - {{\tilde \rho }_{ge}}\Omega {e^{ - i\Delta t}})
\end{equation}
\begin{equation}
\frac{\partial }{{\partial t}}{{\tilde \rho }_{ee}} = - \gamma {{\tilde \rho }_{ee}} + i({{\tilde \rho }_{ge}}\Omega {e^{ - i\Delta t}} - {{\tilde \rho }_{eg}}{\Omega ^*}{e^{i\Delta t}})
\end{equation}
\begin{equation}
\frac{\partial }{{\partial t}}{{\tilde \rho }_{eg}} = - {\gamma _{eg}}{{\tilde \rho }_{eg}} + i({{\tilde \rho }_{gg}}\Omega {e^{ - i\Delta t}} - {{\tilde \rho }_{ee}}\Omega {e^{ - i\Delta t}})
\end{equation}
where $\gamma _{eg}= \frac{\gamma }{2} + 2{\gamma _{ph}}$ is the coherence relaxation rate which in general includes the spontaneous decay and all the decays due to other relaxation mechanisms such as collisions, field phase fluctuations, etc.
It is often convenient to solve the equations in a frame rotating with the frequency of the external field. This implies the transformation ${{\tilde \rho }_{gg}} = {\rho _{gg}}$, ${{\tilde \rho }_{ee}} = {\rho _{ee}}$ and ${{\tilde \rho }_{eg}} = {\rho _{eg}}{e^{ - i\Delta t}}$, yielding:
\begin{equation}
\frac{\partial }{{\partial t}}{\rho _{gg}} = \gamma {\rho _{ee}} + i({\Omega ^*}{\rho _{eg}} - {\rho _{ge}}\Omega )
\end{equation}
\begin{equation}
\frac{\partial }{{\partial t}}{\rho _{ee}} = - \gamma {\rho _{ee}} + i(\Omega {\rho _{ge}} - {\rho _{eg}}{\Omega ^*})
\end{equation}
\begin{equation}
\frac{\partial }{{\partial t}}{\rho _{eg}} = (i\Delta - {\gamma _{eg}}){\rho _{eg}} - i\Omega ({\rho _{ee}} - {\rho _{gg}})
\end{equation}
Formal integration of equation (14) leads to
\begin{equation}
{\rho _{eg}}(t) = - i\Omega \int_0^t {{e^{(i\Delta - {\gamma _{eg}})(t - t')}}} D(t')dt'
\end{equation}
in which $D(t')$ is the population inversion, defined via $D(t') \equiv {\rho _{ee}}(t') - {\rho _{gg}}(t')$. If $D(t')$ does not change significantly for times $t > \gamma _{eg}^{ - 1}$, then we can approximately replace $D(t')$ by $D(t)$. Now the integral can be calculated, yielding:
\begin{equation}
{\rho _{eg}}(t) = \frac{{\Omega D(t)}}{{\Delta + i{\gamma _{eg}}}}
\end{equation}
This approximation is valid in the weak field limit and is widely used for transforming equations (12) and (13) to a set of a differential rate equations:
\begin{equation}
\frac{\partial }{{\partial t}}{\rho _{gg}} = \gamma {\rho _{ee}} + RD
\end{equation}
\begin{equation}
\frac{\partial }{{\partial t}}{\rho _{ee}} = - \frac{\partial }{{\partial t}}{\rho _{gg}} = - \gamma {\rho _{ee}} - RD
\end{equation}
where
\begin{equation}
R = \frac{{2{\gamma _{eg}}{{\left| \Omega \right|}^2}}}{{{\Delta ^2} + \gamma _{eg}^2}}
\end{equation}
represents the rate of the transition. As discussed in section III, the above equations can be modified so as to account in addition for the ionization of the upper state.
\subsection{Photon Probability Distributions}
The statistical properties of the radiation, embodied in its correlation functions, depend on the physical processes that occur within its source \cite{ref23,ref24}. The statistical nature of such processes are reflected in the photon probability distributions that express the probability of finding n photons at a given intensity (mean photon number). In this subsection we outline the basic features of three types of electromagnetic field states that can be generated experimentally by means of modern light sources.
\subsubsection{Coherent State}
A coherent state is the eigenstate of the annihilation operator $\hat a$, therefore by definition it satisfies the equation
\begin{equation}
\hat a \left| a \right\rangle = a \left| a \right\rangle
\end{equation}
Expanding a coherent state in the orthonormal set of eigenstates of the number operator (Fock states), we get:
\begin{equation}
\left| a \right\rangle = {e^{ - \frac{1}{2}{{\left| a \right|}^2}}}\sum\limits_{n = 0}^\infty {\frac{{{a^n}}}{{\sqrt {n!} }}} \left|n\right\rangle
\end{equation}
The probability of finding n photons in the field is
\begin{equation}
{P_{coh}}(n) = {\left| {\left\langle {n}
\mathrel{\left | {\vphantom {n a}}
\right. \kern-\nulldelimiterspace}
{a} \right\rangle } \right|^2} = {e^{ - {{\left| a \right|}^2}}}\frac{{{{\left| a \right|}^{2n}}}}{{n!}}
\end{equation}
The average photon number is given by $\bar n = \sum\limits_{n = 0}^\infty {n{P_{coh}}(n)} = {\left| a \right|^2}$, in terms of which the photon probability distribution assumes the form
\begin{equation}
{P_{coh}}(n) = {e^{ - \bar n}}\frac{{{{\bar n}^n}}}{{n!}}
\end{equation}
\subsubsection{Chaotic State}
A chaotic state, as a statistical mixture of different states, is only describable in terms of the density operator, given by
\begin{equation}
\rho = \frac{{{e^{ - \hat H/{k_B}T}}}}{{Tr[{e^{ - \hat H/{k_B}T}}]}}
\end{equation}
where T is the temperature of the black body source emitting the radiation. In principle, a black body emits radiation over the whole spectrum, according to Planck's Law. If we are concerned about the statistics of a single mode in that state, the density operator is given by
\begin{equation}
\rho = \frac{{{e^{ - \hbar \omega {a^\dag }a/{k_B}T}}}}{{Tr({e^{ - \hbar \omega {a^\dag }a/{k_B}T}})}}
\end{equation}
Therefore the probability of finding n photons is
\begin{equation}
{P_{Chao}}(n) = Tr(\rho \left| n \right\rangle \left\langle n \right|) = Tr({\rho _{nn}}) = $$
$$ \frac{{{e^{ - \hbar \omega n/{k_B}T}}}}{{\sum\limits_{n = 0}^\infty {{e^{ - \hbar \omega n/{k_B}T}}} }} = {e^{ - \hbar \omega n/{k_B}T}}(1 - {e^{ - \hbar \omega /{k_B}T}})
\end{equation}
Since the mean number of photons is $\bar n = \sum\limits_{n = 0}^\infty n {P_{Chao}}(n) = \frac{1}{{{e^{\hbar \omega /{k_B}T}} - 1}}$ we can express the photon probability distribution of a chaotic state in terms of ${\bar n}$ as
\begin{equation}
{P_{chao}}(n) = \frac{1}{{1 + \bar n}}{(\frac{{\bar n}}{{1 + \bar n}})^n} = \frac{{{{\bar n}^n}}}{{{{(1 + \bar n)}^{n + 1}}}}
\end{equation}
\subsubsection{Squeezed Vacuum State}
Since the properties of squeezed radiation are not as commonly found in the literature, in this subsection we provide a somewhat extended summary of its properties.
A squeezed state is a state with phase-sensitive quantum fluctuations, which at certain phase angles are less than those of a coherent or the vacuum field. Squeezed states of radiation field are generated in nonlinear processes in which an electromagnetic field drives a nonlinear medium. In such a medium, pairs of correlated photons of the same frequency are generated. In the interaction picture, this process can be described by the effective Hamiltonian \cite{ref32,ref33,ref34}
\begin{equation}
{\hat H_I} = \varepsilon {({\hat \alpha ^\dag })^2} + {\varepsilon ^*}{\hat \alpha ^2}
\end{equation}
This Hamiltonian describes how a pump field is down-converted to its sub-harmonics at half the driving frequency, with the parameter $\varepsilon $ containing the amplitude of the driving field and the second-order susceptibility for the down-conversion. Since the total Hamiltonian is time-independent, the time evolution operator (also called squeeze operator) is
\begin{equation}
\hat U(t) = \exp ( - \frac{{i\hat Ht}}{\hbar }) = \exp (\xi \frac{{{{({{\hat \alpha }^\dag })}^2}}}{2} - {\xi ^*}\frac{{{{\hat \alpha }^2}}}{2}) \equiv S(\xi )
\end{equation}
where $\xi = - \frac{{i\varepsilon t}}{\hbar }$ is the so-called squeezing parameter, which can also be written as $\xi = r\exp (i\varphi )$. The squeezing parameter characterizes the degree of squeezing and depends on the amplitude of the driving field and the interaction time, i.e. the time that takes for light to travel via the non-linear medium.
The action of the squeeze operator on the vacuum state $\left| 0 \right\rangle$, results the so-called squeezed vacuum state denoted by
\begin{equation}
\left| \xi \right\rangle \equiv S(\xi )\left| 0 \right\rangle
\end{equation}
In order to obtain the photon number probability distribution of the squeezed vacuum state \cite{ref32,ref33,ref34}, we decompose $\left| \xi \right\rangle$ in the Fock basis,
\begin{equation}
\left| \xi \right\rangle = \sum\limits_{n = 0}^\infty {{C_n}\left| n \right\rangle }
\end{equation}
and seek an expression for the relevant coefficients. Starting with the vacuum state, which satisfies the relation
\begin{equation}
\hat a\left| 0 \right\rangle = 0
\end{equation}
we multiply by $\hat S(\xi )$ from the left and use the fact that $\hat S(\xi )$ is unitary, to obtain
\begin{equation}
\hat S(\xi )\hat a{{\hat S}^\dag }(\xi )\hat S(\xi )\left| 0 \right\rangle = 0 \Leftrightarrow \hat S(\xi )\hat a{{\hat S}^\dag }(\xi )\left| \xi \right\rangle = 0
\end{equation}
By the definition of $\xi$ we find that
\begin{equation}
\hat S(\xi )\hat a{{\hat S}^\dag }(\xi ) = \hat a\cosh r + {e^{i\theta }}{{\hat a}^\dag }\sinh r
\end{equation}
In view of eqn.(34), eqn.(33) becomes:
\begin{equation}
(\hat a\cosh r + {{\hat a}^\dag }{e^{i\theta }}\sinh r)\left| \xi \right\rangle = 0
\end{equation}
By substituting eqn.(31) in eqn.(35), we obtain a recursion relation for the coefficients ${C_n}$:
\begin{equation}
{C_{n + 1}} = - {e^{i\theta }}\tanh r{(\frac{n}{{n + 1}})^{1/2}}{C_{n - 1}}
\end{equation}
whose solution is
\begin{equation}
{C_{2n}} = {( - 1)^n}{({e^{i\theta }}\tanh r)^n}{\left[ {\frac{{(2n - 1)!!}}{{(2n)!!}}} \right]^{1/2}}{C_0}
\end{equation}
If we demand from $C_{2n}$ to satisfy the normalization condition ${\sum\limits_{n = 0}^\infty {\left| {{C_{2n}}} \right|} ^2} = 1$, we obtain
\begin{equation}
{\left| {{C_0}} \right|^2}\left( {1 + \sum\limits_{n = 0}^\infty {\frac{{{{[tanhr]}^{2n}}(2n - 1)!!}}{{(2n)!!}}} } \right) = 1
\end{equation}
Using the identity
\begin{equation}
1 + \sum\limits_{n = 0}^\infty {{z^n}\left( {\frac{{(2n - 1)!!}}{{(2n)!!}}} \right)} = {(1 - z)^{ - 1/2}}
\end{equation}
eqn.(38) reduces to ${C_0} = \sqrt {\cosh r}$. Finally, in view of the following two identities
\begin{equation}
(2n)!! = {2^n}n!
\end{equation}
\begin{equation}
(2n - 1)!! = \frac{1}{{{2^n}}}\frac{{(2n)!}}{{n!}}
\end{equation}
one obtains the final expression for the coefficients
\begin{equation}
{C_{2n}} = {( - 1)^n}\frac{{\sqrt {(2n)!} }}{{{2^n}n!}}\frac{{{{({e^{i\theta }}\tanh r)}^n}}}{{\sqrt {\cosh r} }}
\end{equation}
Substitution of eqn.(42) back to eqn.(31), gives the decomposition of the squeezed vacuum state in the Fock basis:
\begin{equation}
\left| \xi \right\rangle = \frac{1}{{\sqrt {\cosh r} }}\sum\limits_{n = 0}^\infty {{{( - 1)}^n}\frac{{\sqrt {(2n)!} }}{{{2^n}n!}}{e^{in\theta }}{{(\tanh r)}^n}} \left| {2n} \right\rangle
\end{equation}
The probability of detecting 2n photons in the field is
\begin{equation}
{P_{2n}} = {\left| {\left\langle {{2n}}
\mathrel{\left | {\vphantom {{2n} \xi }}
\right. \kern-\nulldelimiterspace}
{\xi } \right\rangle } \right|^2} = \frac{{(2n)!}}{{{2^{2n}}{{(n!)}^2}}}\frac{{{{(\tanh r)}^{2n}}}}{{\cosh r}}
\end{equation}
and the probability of the detection of 2n+1 photons, is
\begin{equation}
{P_{2n + 1}} = {\left| {\left\langle {{2n + 1}}
\mathrel{\left | {\vphantom {{2n + 1} \xi }}
\right. \kern-\nulldelimiterspace}
{\xi } \right\rangle } \right|^2} = 0
\end{equation}
Equations (44) and (45) indicate that the photon probability distribution of a squeezed vacuum state is oscillatory, with the probability for all odd photon numbers to be zero. The probability can also be expressed in terms of the mean photon number $\bar n = \sum\limits_{n = 0}^\infty {{P_{2n}}(2n) = } {\sinh ^2}r$, as:
\begin{equation}
{P_{2n}} = \frac{1}{{\sqrt {1 + \bar n} }}\frac{{(2n)!}}{{{{(n!)}^2}{2^{2n}}}}{\left( {\frac{{\bar n}}{{1 + \bar n}}} \right)^n}
\end{equation}
\section{Photon Correlation Effects in Near-Resonant Two-Photon Ionization}
In this section, we present a self-contained formulation and discussion of the effect of photon statistics on 2-photon ionization, with emphasis on the
near-resonant process. It is useful to solve the problem assuming that the field is initially prepared in a number state and then use the photon probability distributions derived in the previous section, to obtain results for the cases of coherent, chaotic or squeezed vacuum radiation.
For reasons discussed in the introduction, we have approached the problem through two different formulations; specifically, the resolvent operator, as well as the density matrix.
\subsection{Resolvent Operator Formalism}
Consider the atom initially in its ground state $\left| g \right\rangle $, in the presence of an external field prepared in a Fock state $\left|n\right\rangle$. The initial state of the compound system (atom + field) is $\left| I \right\rangle = \left| g \right\rangle \left| n \right\rangle$. The initial atomic state is connected to an intermediate atomic state $\left| a \right\rangle$ via a single-photon electric dipole transition of frequency $\omega$, which brings the compound system to the intermediate state $\left| A \right\rangle = \left| a \right\rangle \left| {n - 1} \right\rangle$. The absorption of a second photon takes the atom to the final state $\left| f \right\rangle$ which belongs to continuum. Therefore the final state of the compound system is $\left| F \right\rangle = \left| f \right\rangle \left| {n - 2} \right\rangle$. The energies of the above three system states are ${\omega _I} = {\omega _g} + n\omega $, ${\omega _{\rm A}} = {\omega _a} + (n - 1)\omega$ and ${\omega _F} = {\omega _f} + (n - 2)\omega $. Note that all energies are measured in units of frequency, since all Hamiltonians are assumed divided by $\hbar$. The detuning from the intermediate resonance is $\Delta = \omega - {\omega _{ag}} = \omega - ({\omega _a} - {\omega _g})$.
The Hamiltonian $H$ of the system is the sum of the unperturbed Hamiltonian ${H^0}$ and the field-atom interaction Hamiltonian $V$. The wavefunction of the system at times $t>0$, is given by $\left| {\Psi (t)} \right\rangle = U(t)\left| I \right\rangle $, where $U(t)$ is the time evolution operator.
\begin{figure}[!hbt]
\centering
\includegraphics[width=8cm]{two-photons}
\caption[Two-Photon Ionization]{Ionization via two single-photon electric dipole transitions}
\end{figure}
In order to obtain the probability of ionization as a function of the time t, we need the equations of motion of the matrix elements ${U_{II}}$ and ${U_{AI}}$ or ${U_{FI}}$ of the time evolution operator, in terms of which the ionization probability is expressed as
\begin{equation}
{P_{ion}}(t) = \int {d{\omega _F}{{\left| {{U_{FI}}(t)} \right|}^2}} = 1 - {\left| {{U_{AI}}(t)} \right|^2} - {\left| {{U_{II}}(t)} \right|^2}
\end{equation}
This equation, based on the conservation of probability, simply states that what is missing from the two bound states is in the continuum.
The time evolution of these matrix elements can be obtained analytically with the help of the resolvent operator $G(z)\equiv (z-H)^{-1}$. The procedure involves finding of the equations that govern the time evolution of the matrix elements ${G_{II}}$, ${G_{AI}}$ and ${G_{FI}}$ of the resolvent operator and relating them with the respective time evolution operator matrix elements via
\begin{equation}
{U_{ij}}(t) = - \frac{1}{{2\pi i}}\int_{ - \infty }^{ + \infty } {{e^{ - ixt}}{G_{ij}}({x^ + })dx}
\end{equation}
where ${x^ + } = x + i\eta $, with $\eta \to {0^ + }$.
The mathematical details of this procedure are presented in the Appendix.
The matrix element $2{V_{AI}}$ reflects the Rabi frequency of the $\left| g \right\rangle \leftrightarrow \left| a \right\rangle$ transition:
\begin{equation}
\Omega = 2{V_{AI}}
\end{equation}
while ${V_{FA}}$ is related to the rate of ionization of the intermediate state ${\Gamma _A}$ as \cite{ref24}:
\begin{equation}
{\Gamma _A} = 2\pi {\left| {{V_{FA}}} \right|^2}
\end{equation}
Note that for equation (47) to represent the probability of ionization, the spontaneous decay of the intermediate state has to be negligible compared to the ionization width.
Since we are working in the number state representation, we can express the Rabi frequency and the ionization rate in terms of the number of photons of the states $\left| I \right\rangle $ and $\left| A \right\rangle $ as
\begin{equation}
{\Omega} = {\mu}\sqrt n
\end{equation}
\begin{equation}
{\Gamma _a} = \sigma (n - 1)
\end{equation}
where ${\mu}$ the single-photon dipole matrix element of the $\left| g \right\rangle \leftrightarrow \left| a \right\rangle$ transition and $\sigma $ the cross section associated with the ionization of $\left| a \right\rangle $.
To account for the effects of photon statistics, we average the ionization probability over the photon probability distributions of a coherent, a chaotic and a squeezed vacuum state:
\begin{equation}
Pcoh(t) = \sum\limits_{n = 2}^\infty {{P_{coh}}} \left( {n,\left\langle n \right\rangle } \right)Pion(t,n)
\end{equation}
\begin{equation}
Pchao(t) = \sum\limits_{n = 2}^\infty {{P_{chao}}} \left( {n,\left\langle n \right\rangle } \right)Pion(t,n)
\end{equation}
\begin{equation}
PSqVac(t) = \sum\limits_{n = 1}^\infty {{P_{SqVac}}} \left( {2n,\left\langle n \right\rangle } \right)Pion(t,2n)
\end{equation}
Notice that in eqns. (49) and (50) the summation begins from n=2, since it is the lowest number of photons necessary for the process to be completed. In eqn. (51) the summation begins from n=1, since the argument of the squeezed vacuum distribution is 2n.
We are interested in the behaviour of the ratios $Pchao(T)/Pcoh(T)$ and $PSqVac(T)/Pcoh(T)$ as a function of the mean photon number for various detunings, where T is a time sufficiently larger than the time it takes for the atom to get ionized. The results are presented and discussed in section V.
\subsection{Density Matrix Formalism}
We follow the procedure developed in chapter II, using the same density matrix equations but modified properly to account for the ionization of state $\left| a \right\rangle $. This is accomplished through the introduction of an ionization rate ${\Gamma _{ion}}$ that describe the transfer of population from state $\left| a \right\rangle $ to the continuum. If we neglect the spontaneous decay of the excited state, as it usually is negligible compared to the ionization rate, the density matrix rate equations after applying the approximation discussed in section II, are
\begin{equation}
\frac{\partial }{{\partial t}}{\rho _{gg}}(t) = R[{\rho _{aa}}(t) - {\rho _{gg}}(t)]
\end{equation}
\begin{equation}
\frac{\partial }{{\partial t}}{\rho _{aa}}(t) = - {\Gamma _{ion}}{\rho _{aa}}(t) - R[{\rho _{aa}}(t) - {\rho _{gg}}(t)]
\end{equation}
where
\begin{equation}
R = \frac{{{\Gamma _{ion}} {{\left| \Omega \right|}^2}}}{{{\Delta ^2} + \frac{{{\Gamma _{ion}}^2}}{4}}}
\end{equation}
is the rate of the process. Note that since we neglected the spontaneous decay of the excited state and have no additional relaxation mechanism, the off-diagonal relaxation rate is $\gamma _{\alpha g}= \frac{\Gamma }{2}$.
In view of equations (51) and (52), that relate the Rabi frequency and the ionization width to the number of photons, the rate is expressed in terms of the photon number n, as
\begin{equation}
R = \frac{{{\sigma }{\mu}^2{n(n-1)}}}{{{\Delta ^2} + \frac{{\sigma ^2}}{4}{(n-1)}^2}} \equiv W(n)
\end{equation}
Again, we obtain the effects of photon correlations, after averaging equation (59) over the photon probability distributions of a coherent, a chaotic and a squeezed vacuum state, as described in the previous subsection.
\section{Photon Correlation Effects in Near-Resonant Three-Photon Ionization}
In this section, we explore the photon statistics effects on 3-photon near-resonant ionization. The problem is cast in both the resolvent operator and density matrix formalisms.
\subsection{Resolvent Operator Formalism}
Using the same notation introduced in the previous section, we denote the states of the compound system (atom + radiation field) as $\left| I \right\rangle = \left| g \right\rangle \left| n \right\rangle$, $\left| A \right\rangle = \left| a \right\rangle \left| n-1 \right\rangle$, $\left| B \right\rangle = \left| b \right\rangle \left| n-2 \right\rangle$, $\left| F \right\rangle = \left| f \right\rangle \left| n-3 \right\rangle$, where $\left| g \right\rangle $ the initial atomic state, $\left| a \right\rangle $ and $\left| b \right\rangle $ the two intermediate states, and $\left| f \right\rangle $ the final atomic state that belongs to the continuum. Every state is coupled to its lower one via a single-photon electric dipole transition in presence of a driving field with frequency $\omega$. The respective energies of the compound states are ${\omega _I} = {\omega _g} + n\omega $, ${\omega _{\rm A}} = {\omega _a} + (n - 1)\omega$, ${\omega _{\rm B}} = {\omega _b} + (n - 2)\omega$ and ${\omega _F} = {\omega _f} + (n - 3)\omega $. We introduce the two detunings from the intermediate resonances as ${\Delta _1} = \omega - {\omega _{ag}} = \omega - {\omega _\alpha } + {\omega _g}$ and ${\Delta _2} = 2\omega - {\omega _{bg}} = 2\omega - {\omega _b} + {\omega _g}$.
\begin{figure}[!hbt]
\centering
\includegraphics[width=8cm]{three-photons}
\caption[Three-Photon Ionization]{Ionization via three single-photon electric dipole transitions}
\end{figure}
We choose the energies of the atomic states and the driving frequency such that they result to a detuning ${\Delta _1}$ sufficiently larger than the energy difference ${\omega _{ag}}$, i.e. ${\Delta _1} \gg {\omega _{ag}}$, implying that the first transition is off-resonant, focusing on the problem for various detunings from the second resonance. The problem is formulated in terms of the resolvent operator $G(z)\equiv (z-H)^{-1}$, where $H$ the Hamiltonian of the system which is equal to the sum of the unperturbed Hamiltonian ${H^0}$ and the atom-field interaction Hamiltonian $V$. The equations of motion of the resolvent operator's matrix elements are four but they can be reduced to three after eliminating the continuum as described in the Appendix. The coupling of $\left| b \right\rangle $ to $\left| f \right\rangle $ leads to an ionization rate ${\Gamma _b}$ and a shift whose effect is neglected for sake of simplicity.
If the spontaneous decays of the intermediate states are negligible compared to this rate, the ionization probability at times $t>0$ is given by
\begin{equation}
{P_{ion}}(t) = 1 - {\left| {{U_{II}}(t)} \right|^2} - {\left| {{U_{AI}}(t)} \right|^2} - {\left| {{U_{BI}}(t)} \right|^2}
\end{equation}
where ${U_{ii}}$, $i = I,A,B$, the matrix elements of the time evolution operator between the states of the compound system considered in our problem.
The matrix elements $2{V_{GA}} \equiv {\Omega _1}$ and $2{V_{AB}} \equiv {\Omega _2}$ reflect the Rabi frequencies of the two transitions, while the ionization rate is equal to ${\Gamma _b} = 2\pi {\left| {{V_{FB}}} \right|^2}$.
Since we are working in the number state representation we express the two Rabi frequencies and the ionization rate in terms of the number of photons of the states $\left| I \right\rangle $, $\left| A \right\rangle $ and $\left| B \right\rangle $, i.e.
\begin{equation}
{\Omega _1} = {\mu _1}\sqrt n
\end{equation}
\begin{equation}
{\Omega _2} = {\mu _2}\sqrt {n - 1}
\end{equation}
\begin{equation}
{\Gamma _b} = \sigma (n - 2)
\end{equation}
where ${\mu _1}$, ${\mu _2}$ are the single-photon dipole matrix elements of the transitions $\left| g \right\rangle \leftrightarrow \left| a \right\rangle $ and $\left| a \right\rangle \leftrightarrow \left| b \right\rangle $, respectively, while $\sigma $ is the cross section characterizing the ionization of state $\left| b \right\rangle $.
Now we average the ionization probability over the coherent, chaotic and squeezed vacuum photon distributions starting the sums from the least number of photons needed for the process to occur. Note that in the squeezed vacuum average the argument of the ionization probability is 2n and the sum begins from n=2, since the squeezed vacuum photon probability distribution is zero for odd number of photons.
\begin{equation}
Pcoh(t) = \sum\limits_{n = 3}^\infty {{P_{coh}}} \left( {n,\left\langle n \right\rangle } \right){P_{ion}}(t,n)
\end{equation}
\begin{equation}
Pchao(t) = \sum\limits_{n = 3}^\infty {{P_{chao}}} \left( {n,\left\langle n \right\rangle } \right){P_{ion}}(t,n)
\end{equation}
\begin{equation}
PSqVac(t) = \sum\limits_{n = 2}^\infty {{P_{SqVac}}} \left( {2n,\left\langle n \right\rangle } \right){P_{ion}}(t,2n)
\end{equation}
In section V we plot the ratios $Pchao/Pcoh$ and $PSqVac/Pcoh$ as a function of the mean number of photons for various detunings from the second resonance, at times larger than the time it takes for the atom to get ionized.
\subsection{Density Matrix Formalism}
In this subsection, we use the density matrix formalism to derive an effective 3-photon rate describing the process of ionization. Again we choose the frequencies of the atomic states and the external frequency such that they result ${\Delta _1} \gg {\omega _{ag}}$. In this case, since the first transition is off-resonance, the 3 photon process can be realized as a "2+1" process where the two photon process is driven by an effective two-photon Rabi frequency ${\Omega _{eff}}$.
This allows us to make use of the rate we derived for the two-photon ionization with the effective Rabi frequency and the ionization rate given by:
\begin{equation}
{\Omega _{eff}} = {\mu ^{(2)}}\sqrt {n(n - 1)}
\end{equation}
\begin{equation}
{\Gamma _{ion}} = \sigma (n - 2)
\end{equation}
where $\sigma$ the ionization cross section and $\mu ^{(2)}$ the effective two-photon dipole matrix element.
The rate of the "2+1" process becomes:
\begin{equation}
R = \frac{{{\Gamma _{ion}}{{\left| {{\Omega _{eff}}} \right|}^2}}}{{{\Delta ^2} + \frac{{\Gamma _{ion}^2}}{4}}} = \frac{{\sigma {{\left( {{\mu ^{(2)}}} \right)}^2}n(n - 1)(n - 2)}}{{{\Delta ^2} + \frac{{{\sigma ^2}{{\left( {n - 2} \right)}^2}}}{4}}} \equiv W(n)
\end{equation}
Since we are working in the number state representation, the ionization probability in presence of a chaotic or squeezed vacuum radiation can be derived by summing over the corresponding distributions in the same manner as in the previous sections.
\section{Results and Discussion}
In this section, we present the main results of our calculations, with an interpretation of the underlying physics. Although most of the physics is common to both cases, we discuss 2- and 3-photon separately. In the plots that follow we study the ionization yield, as a function of intensity, for different quantum states of the driving field. For the comparative study that we are interested in, we plot the ratio of the yield for either chaotic or squeezed vacuum state to that for a coherent state, in order to assess the modification of the enhancement due to bunching, as the intensity rises. The respective ionization probabilities are denoted by the self-evident notation Pcoh, Pchao and PSqVac and the respective transition rates by Wcoh, Wchao and WSqVac.
\subsection{Two-photon Results and Discussion}
In figure 3, we have chosen the dipole matrix element coupling to the intermediate state and the ionization cross section so that they result to Rabi frequency equal to the ionization rate $\Gamma $, for small mean photon numbers. This picture changes with increasing intensity, because the Rabi frequency is proportional to the square root of intensity, while the ionization rate scales linearly with intensity. Note that in the single-mode approximation, the mean photon number is approximately related to the average intensity I via $\left\langle n \right\rangle = \left( {8{\pi ^3}{c^2}/{\omega ^2}} \right)\left( {{\rm I}/\Delta\omega } \right)$ \cite{ref8}, where $\Delta\omega$ is the bandwidth of the source. At low intensities the ratio Pchao/Pcoh is equal to 2, in agreement with the expected enhancement factor due to the linear dependence of the ionization on the second order correlation function, which is 2 for chaotic field.
As the intensity increases, however, we notice a rapid decrease of the ratio below the value of 2. This decrease from 2 can be attributed to the fact that, with saturation approaching, the ionization yield begins
depending on all higher order correlation functions and not only on the second-order one. As a result, we observe a drastic change of the ratio.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{2photon1}
\caption[Two-Photon Ionization]{Ratio of chaotic over coherent 2-photon ionization probability as a function of the mean photon number, for various detunings from the intermediate resonance. The values of the dipole matrix element and the cross section used are $\mu = \sigma = 0.0003$ a.u. Blue Line: $\Delta /{\omega _\alpha } = 0.0001$, Red Line: $\Delta /{\omega _\alpha } = 0.01$, Olive Line: $\Delta /{\omega _\alpha } = 0.05$, Green Line: $\Delta /{\omega _\alpha } = 0.1$. Blue and Red lines coincide.}
\end{figure}
The different curves in figure 3 correspond to different values of the detuning from the intermediate resonance. In the limit of large intensities all curves end up to unity as expected due to saturation, but the decrease of the ratio below the N! factor is faster when the external frequency is tuned on resonance with the $\left| a \right\rangle \leftrightarrow \left| g \right\rangle $ transition; because that is when the validity of the non-resonant scheme breaks down faster with increasing intensity. It is interesting to note that there is a rather broad regime of mean photon numbers for which the ratio drops below unity. Evidently, for that range of intensities, chaotic radiation is less effective in two-photon ionization than coherent radiation; a rather surprising result. Actually, these results are in agreement with those of earlier work by one of the authors, on saturation in atomic transitions \cite{ref16,ref17}, where it was shown that the initially observed monotonical decrease of the ratio to unity was due to the assumption of a chaotic field within the decorrelation approximation (DA). When the DA is adopted to the case of an N-photon transition, it results to equations of motion that contain information only about ${N^{th}}$-order correlation function. Therefore for low intensities where the process depends only on the ${N^{th}}$-order correlation function, the DA is valid. However, for stronger fields the simple proportionality of the process to the ${N^{th}}$-order correlation function breaks down since higher order correlations functions come into play, at which point the DA can lead to false predictions.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{2photon2}
\caption[Two-Photon Ionization]{Ratio of squeezed vacuum over coherent 2-photon ionization probability as a function of the mean photon number, for various detunings from the intermediate resonance. The values of the dipole matrix element and the cross section used are $\mu = \sigma = 0.0003$ a.u. Blue Line: $\Delta /{\omega _\alpha } = 0.0001$, Red Line: $\Delta /{\omega _\alpha } = 0.01$, Olive Line: $\Delta /{\omega _\alpha } = 0.05$, Green Line: $\Delta /{\omega _\alpha } = 0.1$. Blue and Red lines coincide.}
\end{figure}
In figure 4, we plot the ratio of the two-photon ionization probability for squeezed vacuum over coherent, as a function of the mean photon number, again for different values of the detuning from the intermediate resonance. Although the overall behaviour of the ratios, as depicted in those curves, appears similar to those of figure 3, an important difference arises at small mean photon numbers,
for which the ratio diverges. This is compatible with the fact that at weak fields the process should depend linearly on the second-order field correlation function, which in the case of a squeezed vacuum field is equal to ${\left\langle n \right\rangle ^2}\left( {3 + \frac{1}{{\left\langle n \right\rangle }}} \right)$. This result can be obtained by averaging the second order correlation function of a field in a Fock state, i.e. $G_2^{Fock} = n(n - 1)$, over the squeezed vacuum photon probability distribution, given by equation (46). The ratio of the squeezed vacuum over coherent second-order correlation functions would then be equal to ${3 + \frac{1}{{\left\langle n \right\rangle }}}$, which apparently diverges when $\left\langle n \right\rangle \to 0$. Be that as it may, a non-linear process, such as a two-photon transition, becomes meaningless in the limit of zero intensity.
In the strong field limit as expected, owing to saturation, all curves approach unity, reaching that value at approximately $\left\langle n \right\rangle \simeq 200$. As in the case of chaotic field, there is a broad region of intensities between the weak field and the saturation limits, where the ratio drops below unity. Therefore, in two-photon near-resonant ionization, squeezed vacuum radiation is more effective than coherent radiation only in the vicinity of small mean photon numbers. For the parameters used in the problem at hand, in view of the relation between the mean photon number and the intensity shown above, the notion of "small mean photon numbers" correspond to field intensities in the vicinity of $I = {10^5}W/c{m^2}$. Although the loss of the enhancement due to chaotic radiation, in a certain range of intensities, had been noted in earlier work \cite{ref16,ref17}, finding the same behavior for superbunched squeezed radiation could not have been anticipated.
Before continuing with the rate equations' results, we need to clarify the issue regarding the possible efficiency of squeezed vacuum radiation, in near-resonant few-photon ionization. It is known that the ${N^{th}}$-order correlation function of a field in a strongly squeezed vacuum state is equal to $\left({2N - 1} \right)!!{\left\langle n \right\rangle ^N}$ \cite{ref35}. For $N=2$ this is equal to $3{\left\langle {n} \right\rangle ^2}$. One might be tempted to infer that the strongly squeezed vacuum light is 3 times more efficient than coherent light, in two-photon ionization. We should, however, keep in mind that the notion of strongly squeezed vacuum refers to a squeezed vacuum state with a high squeezing parameter r and therefore a high mean photon number, according to $\left\langle n \right\rangle = {\left( {\sinh r} \right)^2}$ \cite{ref36}. In fact, the enhancement factor 3 can also be seen by considering $\left\langle n \right\rangle \gg 1$ in the two-photon squeezed vacuum correlation function $G_2^{SqVac} = {\left\langle n \right\rangle ^2}\left( {3 + \frac{1}{{\left\langle n \right\rangle }}} \right)$. Due to the exponential character of the ${\sinh r}$ function, small changes of the squeezing parameter are equivalent to large changes in the mean photon number. For example an increase of r from 1 to 2 is equivalent to an increase of the mean photon number from about 1.4 to 13.2. Therefore, in view of the results of figure 4, we should not expect to observe the $(2N - 1)!!$ enhancement in near-resonant ionization, due to the rapid approach to the saturation limit. In others words, the $(2N - 1)!!$ enhancement requires intensities in the range where the simple dependence of the process on the ${N^{th}}$-order intensity correlation function has already become invalid near resonance. However, the near-resonant process may be enhanced significantly, if the atom is exposed to weak squeezed vacuum radiation; assuming observability is feasible.
As discussed in the previous sections, apart from the ionization probability using the time-dependent wavefunction, a transition probability (rate) can also be derived with the help of the density matrix equations. A sample of results is shown in figure 5, with parameters identical to those of figure 3. The behaviour of the various curves of figure 5 is in overall agreement with the respective behaviour of the curves of figure 3, reaching eventually the value of unity. A difference can be noticed, however, in that all curves, with the exception of the blue one corresponding to detuning 0.0001 times the frequency of the intermediate state, now will reach unity at much higher mean photon numbers $\left( {\left\langle n \right\rangle \simeq 800} \right)$, which has been verified numerically, although not shown in the figure; a result that should be viewed with precaution. Because, the derivation of the two-photon rate is based on the assumption of a Rabi frequency not much larger than the ionization rate $\Gamma $ and/or the detuning from the intermediate resonance. However, owing to the linear dependence of the ionization rate on the photon number, it increases much faster than the Rabi frequency, which is proportional to the square root of the photon number. Therefore, even if they are comparable for small mean photon numbers, the necessary condition $\Omega < \Gamma $ can be reached fairly quickly, as the intensity rises. However, since the detuning is fixed, for large mean photon numbers the Rabi frequency will eventually become larger than the detuning, with the validity of the rate approximation breaking down.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{2photonrate1}
\caption[Two-Photon Ionization]{Ratio of chaotic over coherent 2-photon ionization rate as a function of the mean photon number, for various detunings from the intermediate resonance. The values of the dipole matrix element and the cross section used are $\mu = \sigma = 0.0003$ a.u. Blue Line: $\Delta /{\omega _\alpha } = 0.0001$, Red Line: $\Delta /{\omega _\alpha } = 0.01$, Olive Line: $\Delta /{\omega _\alpha } = 0.05$, Green Line: $\Delta /{\omega _\alpha } = 0.1$.}
\end{figure}
In figure 6, we present results on the ratio WSqVac/Wcoh as a function of the mean photon number, as obtained through the rate equations. Comparing the results of this figure to those of Figure 4, we note that now the ratio of the rates is more sensitive to detuning than the ratio of the ionization probabilities. This is reflected in the startling difference between the blue and red lines (which in figure 4 are indistinguishable), as well as in the different behaviour of the green and olive lines, with increasing intensity. Still, the overall trend of the curves in the two figures is similar. We should point out that owing to the specific form of equation (59), in averaging over a photon probability distribution, the dipole matrix element and the ionization cross section appearing in the numerator of (59) are factored out and cancel when the ratios are taken. This leads to ratios that depend only on the detuning and the cross section appearing in the denominator of (59). As a result, changes in the dipole matrix element will not affect the ratio of the rates. But since the derivation of the rate equations is based on the approximation discussed above, the results would be meaningful only when the intensities are such that they conform to a Rabi frequency within the limits of the approximation. It is very interesting to notice that equation (59) within the limit $\Delta \gg \frac{\sigma }{2}(n - 1)$ reduces to the second-order correlation function of a field in a number state, multiplied by the factor $\frac{{\sigma {\mu ^2}}}{{{\Delta ^2}}}$. This suggests that the $(2N-1)!!$ enhancement of the two-photon ionization under squeezed vacuum radiation, over that under coherent, would appear when the above condition is satisfied, according to the ratio of the respective correlation functions. But we must keep in mind that the summations over a photon probability distribution includes photon numbers up to infinity. Even if the high photon number terms enter with less weight, when high mean photon numbers are considered, the condition $\Delta \gg \frac{\sigma }{2}(n - 1)$ would no longer be satisfied. This actually is another way to see why the simple proportionality of an N-photon process to the field ${N^{th}}$-order correlation field will eventually break down with increasing intensity.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{2photonrate2}
\caption[Two-Photon Ionization]{Ratio of squeezed vacuum over coherent 2-photon ionization rate as a function of the mean photon number, for various detunings from the intermediate resonance. The values of the dipole matrix element and the cross section used are $\mu = \sigma = 0.0003$ a.u. Blue Line: $\Delta /{\omega _\alpha } = 0.0001$, Red Line: $\Delta /{\omega _\alpha } = 0.01$, Olive Line: $\Delta /{\omega _\alpha } = 0.05$, Green Line: $\Delta /{\omega _\alpha } = 0.1$.}
\end{figure}
\subsection{Three-photon Results and Discussion}
In three-photon near-resonant ionization, there are two intermediate states, which means a double near-resonance is possible. In this work, we have chosen the photon frequency so that the detuning ${\Delta _1} \gg {\omega _{ag}}$, of the first transition is sufficiently large, for the two-photon transition to the second intermediate state to satisfy the non-resonant condition, for all intensities employed in the calculations. The reason is that we wanted to explore the role of the non-linearity in the bound-bound transition, in contrast to the two-photon case, where the bound-bound transition depends linearly on the radiation. Thus we have only one near-resonance state to study.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{3photon1}
\caption[Three-Photon Ionization]{Ratio of chaotic over coherent 3-photon ionization probability as a function of the mean photon number, for various detunings from the second intermediate resonance. The values of the dipole matrix elements and the cross section used are ${\mu _1} = {\mu _2} = 0.0004$ a.u. and $\sigma = 0.0008$ a.u. Blue Line: ${\Delta _2}/{\omega _b} = 0.0001$, Red Line: ${\Delta _2}/{\omega _b} = 0.01$, Olive Line: ${\Delta _2}/{\omega _b} = 0.05$, Green Line: ${\Delta _2}/{\omega _b} = 0.1$.}
\end{figure}
In direct analogy with the two-photon case, we a have a Rabi frequency coupling the bound states and an ionization cross section. For the results of figure 7, we have chosen a cross section $\sigma = 0.0008$ a.u. two times higher (expressed in atomic units) than the dipole matrix elements ${\mu _1},{\mu _2} = 0.0004$ a.u. of the transitions $\left| g \right\rangle \leftrightarrow \left| a \right\rangle $ and $\left| a \right\rangle \leftrightarrow \left| b \right\rangle $, respectively. At low intensities, the ratio of chaotic over coherent ionization transition probabilities is equal to 6, which is compatible with the expected weak field $N!$ enhancement for $N=3$, arising from the ratios of the respective correlation functions. With increasing intensity, the ratio drops below N! rather rapidly, approaching unity, as expected. The approach to unity turns out to be faster, as the photon frequency is tuned closer to resonance with the second transition. In contrast to the two-photon case, the ratio does not drop below unity, at any intensity. It appears that the non-linearity in the two-photon Rabi frequency, in this case is responsible for this behaviour. Recall that now, both Rabi frequency and ionization rate have the same dependence on intensity.
As in the two-photon case, the squeezed vacuum over coherent ratio of ionization transition probabilities exhibits a behaviour similar to that chaotic over coherent ratio, with the exception of the divergence for weak fields, noted also for two-photon ionization. Again, this is connected to the form of the squeezed vacuum third-order correlation function $G_3^{SqVac} = {\left\langle n \right\rangle ^3}\left( {15 + \frac{9}{{\left\langle n \right\rangle }}} \right)$, which diverges in the vicinity of $\left\langle n \right\rangle = 0$, when divided by the coherent third order correlation function $G_3^{coh} = {\left\langle n \right\rangle ^3}$. For high mean photon numbers, the correlation function is equal to ${15{{\left\langle n \right\rangle }^3}}$, capturing the $(2N-1)!!$ strongly squeezed vacuum enhancement factor for $N=3$. However, if tuned near-resonance, saturation is approached much faster, with the observation of the expected enhancement being problematic.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{3photon2}
\caption[Three-Photon Ionization]{Ratio of squeezed vacuum over coherent 3-photon ionization probability as a function of the mean photon number, for various detunings from the second intermediate resonance. The values of the dipole matrix elements and the cross section used are ${\mu _1} = {\mu _2} = 0.0004$ a.u. and $\sigma = 0.0008$ a.u. Blue Line: ${\Delta _2}/{\omega _b} = 0.0001$, Red Line: ${\Delta _2}/{\omega _b} = 0.01$, Olive Line: ${\Delta _2}/{\omega _b} = 0.05$, Green Line: ${\Delta _2}/{\omega _b} = 0.1$.}
\end{figure}
In order to illustrate cases contrasting to the above results, in figures 8 and 9 we show the bahaviour for the relatively large detuning of ${\Delta _2}/{\omega _b} = 0.5$ from two-photon resonance. These results have been obtained through the rate equations by taking averages of equation (69) over the respective photon probability distributions. Although a detuning of this magnitude may be a bit too large, within the constrains of our model, let us nevertheless examine the dependence of ionization as a function of intensity.
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{3photonrate1}
\caption[Three-Photon Ionization]{Ratio of chaotic over coherent 3-photon ionization rate as a function of the mean photon number, for various detunings from the second intermediate resonance. The value of the cross section used is $\sigma = 0.0003$ a.u. Blue Line: ${\Delta _2}/{\omega _b} = 0.0001$, Red Line: ${\Delta _2}/{\omega _b} = 0.01$, Olive Line: ${\Delta _2}/{\omega _b} = 0.05$, Green Line: ${\Delta _2}/{\omega _b} = 0.5$.}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=8cm]{3photonrate2}
\caption[Three-Photon Ionization]{Ratio of squeezed vacuum over coherent 3-photon ionization rate as a function of the mean photon number, for various detunings from the second intermediate resonance. The value of the cross section used is $\sigma = 0.0003$ a.u. Blue Line: ${\Delta _2}/{\omega _b} = 0.0001$, Red Line: ${\Delta _2}/{\omega _b} = 0.01$, Olive Line: ${\Delta _2}/{\omega _b} = 0.05$, Green Line: ${\Delta _2}/{\omega _b} = 0.5$.}
\end{figure}
For $\sigma = 0.0003$ a.u. corresponding approximately to a cross section about ${10^{ - 20}}c{m^2}$, the ratios tend to stabilize to values with the enhancements factors 6 and 15, reflecting the chaotic and squeezed vacuum correlation functions, respectively. Eventually, even the off-resonant curves will fall to unity but much more slowly than the near-resonant ones. As in the two-photon case, the specific form of the three-photon ionization rate (69), implies that the ratios will only depend on $\sigma $ and ${\Delta _2}$. The effective two-photon dipole matrix element ${\mu ^{(2)}}$ doesn't appear in the ratios, but has to be such that it does not invalidate the rate approximation. If the cross section
$\sigma$ is chosen, for example, one order of magnitude larger than $0.0003$ a.u., even the green curves approach unity very rapidly. But for cross sections smaller than $0.0003$ a.u. the ratios do indeed stabilize to the theoretical enhancement factors for a broad range of intensities, in the non-resonant limit. Since the values of parameters we have chosen in the above illustrations are not unphysical, the message that emerges is that the conditions satisfying the non-resonant assumption are quite sensitive to the interplay between intensity and parameters.
\section{Conclusion and Closing Remarks}
N-photon transition from a bound state to a continuum, such as ionization, involves summation over intermediate states. As long as it may be justified to assume that all of them are "sufficiently" far from resonance with the absorption of one or more photons, a transition probability proportional to the Nth order intensity correlation function is meaningful. Theoretically, the matter stops there, as has been the case with much of the related literature \cite{ref1,ref2,ref3,ref4,ref5,ref6,ref19,ref26}. Given, however, that any non-linear process is observable, only if the intensity is sufficiently strong, the non-resonant condition cannot be taken for granted; beyond a theoretical academic exercise or at best poof of principle.
Two or three-photon process should be optimal for non-resonant ionization, as it may be possible to select the photon frequency so as to satisfy the non-resonant requirement, up to some intensity enabling observability. For four or higher order processes, it is practically impossible to avoid near resonance with intermediate states, because with increasing energy their spacing becomes progressively denser. As we have shown in this paper, however, even for two and three photon ionization, it is only at quite low intensity that the condition of non-resonance can be taken for granted. As a consequence, the enhancement expected for chaotic or squeezed radiation will more often than not be smaller than predicted on the basis of the relevant intensity correlation function. This may well be the reason that, over the 50 years or so that have elapsed since the first predictions of the chaotic enhancement \cite{ref1,ref2,ref3,ref4,ref5,ref6}, even for two-photon ionization a definitive enhancement by the expected factor of 2, has been very difficult to observe; let alone for order three or higher. There is of course always the nagging issue of whether the radiation is truly chaotic or truly coherent \cite{ref10,ref11,ref12}, which in the light of our results poses a dilemma. On the one hand, an N-photon process would be an ideal tool for the measurement of an Nth order intensity correlation function. On the other hand, the possible influence of intermediate near-resonances are apt to be misleading as to the underlying reason for departure from the expected enhancement factor. It seems to us that, given the specific atomic system employed in an experiment, only the quantitative evaluation of the role of intermediate states can offer a way out of the dilemma.
The very recent achievement in measuring the enhancement in harmonic generation, due to superbunched squeezed radiation reported by Kirill et al. \cite{ref31}, appears to be in contrast to the above dilemma. Actually, for two reasons, the contrast may be only apparent. First, owing to the long wavelength of the radiation in that experiment, the few-photon absorption was within the bound spectrum, satisfying the non-resonant condition. Second, the intensity was quite low; too low to induce a Rabi frequency comparable to the detuning. And the observation of up to the 4th harmonic, at such low intensity, attests to the elegance of that experiment. It could therefore be argued that there is no contradiction with our results, as they address a different range of intensities. But even in harmonic generation, at shorter wavelengths, reaching into the continuum \cite{ref37}, the issue of intermediate states is of extreme importance. Hoping that it will eventually be possible to explore the effect of superbunching on non-linear processes at shorter wavelengths and higher intensities, our results can serve
as a guide to the planning of relevant experiments.
Departing for the moment from transitions to a continuum, the effect of superbunching on a strongly driven 2-photon bound-bound transition, an extension of its single-photon counterpart solved quite some time ago by Ritsch and Zoller \cite{ref28,ref29}, poses a daunting theoretical challenge. In early work \cite{ref16,ref17}, it has been found that, in contrast to bound-continuum transitions, chaotic radiation is less effective than coherent in saturating a two-photon bound-bound transition. Would superbunched squeezed radiation be even less effective in that situation? It may well be that squeezed light at wavelengths and intensities appropriate for the strong driving of a bound-bound two-photon transition may be available not too long from now.
Finally, aside from using multiphoton ionization as a "detector" of an intensity correlation function, from the standpoint of enhancing the process induced by bunched radiation, the exact factor of enhancement may not be as important, especially for higher order processes. On that aspect, the results by Lecompte et al, quite some time ago \cite{ref15}, may well be the most dramatic example on record, in which and in line with our analysis the enhancement was not exactly 11!. Still, it was less than two orders of magnitude lower. The enhancement of multiphoton ionization under chaotic radiation has re-emerged during the last ten years or so, for systems driven by FEL sources which are known to exhibit strong intensity fluctuations, similar to those of chaotic radiation \cite[and references therein]{ref38}. The theoretical problem, using realistic simulation of the FEL radiation, has been addressed to some extent \cite{ref39}. Given that in several experiments, fairly high order ionization processes have been observed \cite[and references therein]{ref39}, the intensity fluctuations must surely have played a very significant role. Up to this point, however, there has not been any systematic investigation aiming at the quantification of the effect on experimental data.
\section*{Acknowledgements}
The work on this paper was motivated by questions posed to one of us (PL) by Dr. Gerd Leuchs. For this as well as occasional useful discussions we are very grateful. As always, discussions with Dr. George Nikolopoulos have been quite helpful during our work.
|
{
"timestamp": "2018-03-08T02:11:08",
"yymm": "1803",
"arxiv_id": "1803.02761",
"language": "en",
"url": "https://arxiv.org/abs/1803.02761"
}
|
\section{Introduction}
For $t$ and $r$ non-negative integers, we define
\[
p_{r,t}(x) = \sum_{j=0}^{r} \binom{t+j}{t} x^j = \sum_{j=0}^{r} \binom{t+j}{j} x^j.
\]
This polynomial arises from a normalization of the $t^{\rm th}$ derivative of $1+x+ \cdots +x^{t+r}$.
The polynomial is connected to a factor of the Shabat polynomials of a family of
\emph{dessins d'enfant} which are trees and have passport size one (cf.~\cite[Example 3.3]{NMA}).
The polynomial $p_{r,t}(x)$ was conjectured to be irreducible, and the irreducibility was studied in \cite{BFLT}.
In particular, we find there that if $r$ is fixed, then $p_{r,t}(x)$ is irreducible for $t$ sufficiently large.
A generalization of this irreducibility result can be found in \cite{FKP}, where this polynomial was
considered in a different form. There, the irreducibility of the polynomial
\[
q_{r,n}(x) = \sum_{j=0}^{r} \binom{n}{j} x^{j},
\]
which is a truncated binomial expansion of $(x+1)^{n}$, was investigated.
As noted there, this truncated binomial expansion came up in
investigations of the Schubert calculus in Grassmannians \cite{ISch}.
Other results concerning these polynomials can be found in \cite{DK, DS, KKL}.
There are some identities involving $p_{r,t}(x)$ and $q_{r,n}(x)$ which helped establish the
results found in \cite{BFLT} and \cite{FKP}. If we define
\[
\tilde{p}_{r,t}(x) = x^{r}p_{r,t}(1/x) = \sum_{j=0}^{r} \binom{t+j}{j} x^{r-j},
\]
then according to \cite{BFLT} we have
\[
\tilde{p}_{r,t}(x+1) = \sum_{j=0}^{r} \binom{t+r+1}{j} x^{r-j}.
\]
Thus, $\tilde{p}_{r,t}(x+1) = x^{r} q_{r,t+r+1}(1/x)$. We have from \cite{FKP} that
\[
q_{r,n}(x-1) = \sum_{j=0}^{r} c_{j} x^{j},
\quad \text{ where }
c_{j} = \binom{n}{j} \binom{n-j-1}{r-j} (-1)^{r-j}.
\]
As noted in \cite{FKP}, we can write
\begin{equation*}
c_{j} = \dfrac{(-1)^{r-j} n (n-1) \cdots (n-j+1) (n-j-1) \cdots (n-r+1) (n-r)}{j! (r-j)!}.
\end{equation*}
These identities are of interest as the irreducibility over $\mathbb Q$ of one of
$p_{r,t}(x)$, $\tilde{p}_{r,t}(x)$, $\tilde{p}_{r,t}(x+1)$ and $q_{r,t+r+1}(x-1)$ implies the irreducibility of the other three.
Furthermore, it is not difficult to see that these polynomials all have the same discriminant
(as reversing the coefficients of a polynomial and translating do not affect the discriminant).
Also, as the roots for each all generate the same number field, we have that for
a fixed $r$ and $t$, the Galois groups
over $\mathbb Q$ associated with these polynomials are all the same.
The main goal of this paper is to show that these polynomials give rise to examples
of polynomials having Galois group over $\mathbb Q$ the symmetric group.
\begin{theorem}\label{mainthm}
Let $r$ be an integer $\ge 2$ with $r \ne 6$. If $t$ is a sufficiently large positive integer, then the Galois group
associated with any one of $p_{r,t}(x)$, $\tilde{p}_{r,t}(x)$, $\tilde{p}_{r,t}(x+1)$ and $q_{r,t+r+1}(x-1)$
over $\mathbb Q$ is the symmetric group $S_{r}$. In the case that $r = 6$, there are at most $O(\log T)$
values of $t \le T$ for which the Galois group of any one of $p_{r,t}(x)$, $\tilde{p}_{r,t}(x)$, $\tilde{p}_{r,t}(x+1)$ and $q_{r,t+r+1}(x-1)$
over $\mathbb Q$ is not the symmetric group $S_{6}$. In these cases, for sufficiently large $t$,
the Galois group is $PGL_{2}(5)$, a transitive subgroup of $S_{6}$ isomorphic to $S_{5}$.
\end{theorem}
\noindent
Observe that Theorem~\ref{mainthm} has as an immediate consequence that
for a fixed integer $r \ge 2$ and $r \ne 6$, the Galois group of $q_{r,n}(x)$ over $\mathbb Q$
is $S_{r}$ provided $n$ is sufficiently large with a similar result for almost all $n$ in the case that $r = 6$.
We note that $q_{6,10}(x)$ has Galois group $PGL_{2}(5)$.
The proof of Theorem~\ref{mainthm} will give, up to a finite number of exceptions,
an explicit description of the set $\mathcal N$ of the $O(\log T)$ values of $t \le T$
where the Galois group $PGL_{2}(5)$ might occur.
We explain computations that verify directly that for $10 < n \le 10^{10}$ and $n \in \mathcal N$,
the Galois group of $q_{6,n}(x)$ is $S_{6}$. The bound of $10^{10}$ can easily be extended much
further. However, we note that there may still be $n \in (10,10^{10}]$ for which $q_{6,n}(x)$ is reducible
so that $q_{6,n}(x)$ does not have Galois group $S_{6}$ since
the explicitly given $\mathcal N$ does not take into account that our proof
that $q_{6,n}(x)$ has Galois group $S_{6}$ requires $n$ to be sufficiently large so that, in particular, the
results from \cite{BFLT} and \cite{FKP} imply $q_{6,n}(x)$ is irreducible. Nevertheless, based on further
computations, we conjecture that $q_{6,n}(x)$ has Galois group $S_{6}$ for all $n \ge 11$.
\section{Preliminary Material}
We will make use of Newton polygons, which we describe briefly here.
Let $f(x) = \sum_{j=0}^{r} a_{j}x^{j} \in \mathbb Z[x]$ with $a_{0} a_{r} \ne 0$, and
let $p$ be a prime.
For an integer $m \ne 0$, we use $\nu_{p}(m)$ to denote the exponent in the largest power of $p$ dividing $m$.
Let $S$ be the set of lattice points $\big(j, \nu_{p}(a_{r-j})\big)$, for $0 \le j \le r$ with $a_{r-j} \ne 0$.
The polygonal path along the lower edges of the convex hull of these points from
$\big(0, \nu_{p}(a_{r})\big)$ to $\big(r, \nu_{p}(a_{0})\big)$
is called the Newton polygon of $f(x)$ with respect to the prime $p$.
The left-most edge has an endpoint $\big(0, \nu_{p}(a_{r})\big)$ and the right-most edge has an endpoint
$\big(r, \nu_{p}(a_{0})\big)$. The endpoints of every edge belong to the set $S$,
and each edge has a distinct slope that increases as we move along the Newton polygon from left to right.
Newton polygons provide information about the factorization of $f(x)$ over the $p$-adic field $\mathbb Q_{p}$
and, hence, information about the Galois group of $f(x)$ over $\mathbb Q_{p}$. As this Galois group is a subgroup
of the Galois group of $f(x)$ over $\mathbb Q$, we can use Newton polygons to obtain information about
the Galois group of $f(x)$ over $\mathbb Q$. Recalling that we are viewing edges of Newton polygons
as having distinct slopes, each edge of the Newton polygon of $f(x)$ with respect to
a prime $p$ corresponds to a factor of $f(x)$ in $\mathbb Q_{p}[x]$ that is not necessarily irreducible.
More precisely, if an edge has endpoints $(x_{1},y_{1})$ and $(x_{2},y_{2})$, then its slope
$a/b = (y_{2}-y_{1})/(x_{2}-x_{1})$ with $\gcd(a,b) = 1$ is such that $f(x)$ has a factor $g(x)$ in $\mathbb Q_{p}[x]$
of degree $x_{2}-x_{1}$ and with each irreducible factor of $g(x)$ in $\mathbb Q_{p}[x]$ of degree a multiple of $b$.
We comment here that a theorem of Dedekind (cf.~\cite{DAC}) allows one to obtain information about the
Galois group associated with a polynomial $f(x)$ over $\mathbb Q$ by looking at the polynomials factorization
modulo a prime $p$. More precisely, suppose $f(x)$ is an irreducible polynomial in $\mathbb Z[x]$ and $p$ is a prime
which does not divide its discriminant. Suppose further that $f(x)$ factors modulo $p$
as a product of $r$ irreducible polynomials of degrees $d_{1}, \ldots, d_{r}$. Then Dedekind's Theorem asserts
that the Galois group of $f(x)$ over $\mathbb Q$ contains an element that is the product of $r$ disjoint cycles
with cycle lengths $d_{1}, \ldots, d_{r}$.
Our main tool for establishing Theorem~\ref{mainthm} is based on combining some of the above ideas with
a theorem of C.~Jordan \cite{CJ} and noted in work of R.~Coleman \cite{RFC}. It has been cast in a
convenient form by F.~Hajir \cite{FH}, which we summarize as follows.
\begin{lemma}\label{hajirlemma}
Let $f(x)$ be an irreducible polynomial of degree $r$, and suppose $q$ is a prime in the interval $(r/2,r-2)$
such that the Newton polygon with respect to some prime $p$ has an edge with slope $a/b$ where
$a$ and $b$ are relatively prime integers and $q|b$. Let $\Delta$ be the discriminant of $f(x)$.
Then the Galois group of $f(x)$ over $\mathbb Q$ is the alternating group $A_{r}$ if $\Delta$ is a square
and is the symmetric group $S_{r}$ if $\Delta$ is not a square.
\end{lemma}
We also make use of the following result from \cite{KC} and \cite{DM} (Theorem 3.3A).
\begin{lemma}\label{kconradlemma}
Let $f(x)$ be an irreducible polynomial of degree $r \ge 2$.
If the Galois group of $f(x)$ over $\mathbb Q$ contains a $2$-cycle and a $q$-cycle for some
prime $q > r/2$, then the Galois group is $S_{r}$. Alternatively, if the Galois group of $f(x)$ over
$\mathbb Q$ contains a $3$-cycle and a $q$-cycle for some
prime $q > r/2$, then the Galois group is either the alternating group $A_{r}$ or
the symmetric group $S_{r}$.
\end{lemma}
Note that in general
if the Galois group of an $f(x) \in \mathbb Z[x]$ over $\mathbb Q$ is contained in an alternating group, then its
discriminant is a square.
Thus, in the statement of Lemma~\ref{kconradlemma}, one can conclude that the Galois group is the
symmetric group by showing that the discriminant $\Delta$ of $f(x)$ is not a square.
In the next section, we give an explicit formula for the discriminant $\Delta$ of our polynomials in
Theorem~\ref{mainthm} and show that $\Delta$ is not a square for fixed $r$
and for $t$ sufficiently large. In the last section, for $r \ge 8$,
we show the existence of primes $q$ and $p$ as in Lemma~\ref{hajirlemma}. For $r \le 7$, we appeal to
Lemma~\ref{kconradlemma} to finish off the proof of Theorem~\ref{mainthm}.
\section{The Discriminant}
\begin{lemma}\label{discrimvaluelemma}
Let $t$ and $r$ be integers with $t \ge 0$ and $r \ge 2$.
Let $\Delta$ be the common discriminant of $p_{r,t}(x)$, $\tilde{p}_{r,t}(x)$, $\tilde{p}_{r,t}(x+1)$ and $q_{r,t+r+1}(x-1)$. Then
\[
\Delta = (-1)^{r(r-1)/2} \dfrac{(t+1)^{r-1} (t+r+1)^{r-1} (t+2)^{r-2} (t+3)^{r-2} \cdots (t+r)^{r-2}}{(r!)^{r-2}}.
\]
\end{lemma}
\begin{proof}
We view $t$ as a variable and work with
\[
f_{r}(x) = q_{r,t+r+1}(x-1) = \sum_{j=0}^{r} c_{j} x^{j},
\]
where
\begin{equation}\label{discreq1}
c_{j} = \dfrac{(-1)^{r-j} (t+r+1) \cdots (t+r-j+2) (t+r-j) \cdots (t+1)}{j! (r-j)!}.
\end{equation}
To clarify, for $t$ a non-negative integer, we have
\[
c_{j} = \dfrac{(-1)^{r-j} (t+r+1)!}{j! \,(r-j)! \,t! \,(t+r-j+1)}.
\]
However, from the point of view of \eqref{discreq1}, we can view $t$ as a real variable.
We are interested in the discriminant $\Delta$ of $f_{r}(x)$. Observe that
\begin{equation}\label{discreq2}
\Delta = \dfrac{(-1)^{r(r-1)/2}}{c_{r}} \,\text{Res}(f_{r}, f'_{r})
= \dfrac{(-1)^{r(r-1)/2} r!}{(t+r+1)(t+r) \cdots (t+3) (t+2)} \,\text{Res}(f_{r}, f'_{r}),
\end{equation}
where $\text{Res}(f_{r}, f'_{r})$ is the resultant of $f_{r}$ and $f'_{r}$ with respect
to the variable $x$. We express the resultant in terms of the $(2r-1) \times (2r-1)$ Sylvester determinant
\begin{equation*}
\text{Res}(f_{r}, f'_{r}) =
\begin{vmatrix}
c_{r} & c_{r-1} & c_{r-2} & \hdots & c_{1} & c_{0} & 0 & 0 & \hdots & 0 \\[3pt]
0 & c_{r} & c_{r-1} & \hdots & c_{2} & c_{1} & c_{0} & 0 & \hdots & 0 \\[3pt]
0 & 0 & c_{r} & \hdots & c_{3} & c_{2} & c_{1} & c_{0} & \hdots & 0 \\[3pt]
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots &\vdots \\[3pt]
r c_{r} & (r-1) c_{r-1} & (r-2) c_{r-2} & \hdots & c_{1} & 0 & 0 & 0 & \hdots & 0 \\[3pt]
0 & r c_{r} & (r-1) c_{r-1} & \hdots & 2 c_{2} & c_{1} & 0 & 0 & \hdots & 0 \\[3pt]
0 & 0 & r c_{r} & \hdots & 3 c_{3} & 2 c_{2} & c_{1} & 0 & \hdots & 0 \\[3pt]
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots &\vdots
\end{vmatrix}.
\end{equation*}
Observe that there are $r-1$ rows consisting of the coefficients of $f_{r}(x)$ and
$r$ rows consisting of the coefficients of $f'_{r}(x)$.
For each integer $j \in [1, r]$, we see from \eqref{discreq1} that $t+r+1$ divides $c_{j}$.
We deduce that $t+r+1$ can be factored out of each element of the first $r$ columns of
$\text{Res}(f_{r}, f'_{r})$ to show that $(t+r+1)^{r}$ is a factor of $\text{Res}(f_{r}, f'_{r})$.
For each $k \in \{ 1, 2, \ldots, r \}$, we also have from \eqref{discreq1} that $t+r-k+1$
divides $c_{j}$ for each integer $j \in [1,r]$ with $j \ne k$. In particular, each of the
$r-1$ columns not containing $k c_{k}$ have each element divisible by $t+r-k+1$.
Thus, $(t+r-k+1)^{r-1}$ divides $\text{Res}(f_{r}, f'_{r})$. Hence,
\begin{equation}\label{discreq3}
(t+r+1)^{r} (t+1)^{r-1} (t+2)^{r-1} \cdots (t+r)^{r-1}
\end{equation}
divides $\text{Res}(f_{r}, f'_{r})$. The product in \eqref{discreq3} as a polynomial in $t$
has degree $(r+1)(r-1) + 1 = r^{2}$. Hence, from \eqref{discreq2}, we see that $\Delta$
is divisible by the polynomial
\begin{equation}\label{discreq4}
(t+r+1)^{r-1} (t+1)^{r-1} (t+2)^{r-2} (t+3)^{r-2} \cdots (t+r)^{r-2}
\end{equation}
of degree $r^{2} - r$ in $t$.
We turn to making use of
\[
p_{r,t}(x) = \sum_{j=0}^{r} d_{j} \,x^{j},
\quad \text{ where }
d_{j} = \sum_{j=0}^{r} \dfrac{(t+j) (t+j-1) \cdots (t+1)}{j!} x^{j}.
\]
Observe that the discriminant of $f_{r}(x)$ and $p_{r,t}(x)$ are both polynomials in $t$
that agree at all positive integers and, hence, are identical.
We use next that $\Delta$ is the discriminant of $p_{r,t}(x)$ to show that
$\Delta$ cannot be divisible by a higher degree polynomial in $t$ than that
given by \eqref{discreq4}. Taking into account the leading coefficient of $p_{r,t}(x)$, we see that
\begin{equation}\label{discreq6}
\Delta = \dfrac{(-1)^{r(r-1)/2} r!}{(t+r)(t+r-1) \cdots (t+2) (t+1)} \,\text{Res}(p_{r,t}, p'_{r,t}),
\end{equation}
where
\begin{equation*}
\text{Res}(p_{r,t}, p'_{r,t}) =
\begin{vmatrix}
d_{r} & d_{r-1} & d_{r-2} & \hdots & d_{1} & d_{0} & 0 & 0 & \hdots & 0 \\[3pt]
0 & d_{r} & d_{r-1} & \hdots & d_{2} & d_{1} & d_{0} & 0 & \hdots & 0 \\[3pt]
0 & 0 & d_{r} & \hdots & d_{3} & d_{2} & d_{1} & d_{0} & \hdots & 0 \\[3pt]
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots &\vdots \\[3pt]
r d_{r} & (r-1) d_{r-1} & (r-2) d_{r-2} & \hdots & d_{1} & 0 & 0 & 0 & \hdots & 0 \\[3pt]
0 & r d_{r} & (r-1) d_{r-1} & \hdots & 2 d_{2} & d_{1} & 0 & 0 & \hdots & 0 \\[3pt]
0 & 0 & r d_{r} & \hdots & 3 d_{3} & 2 d_{2} & d_{1} & 0 & \hdots & 0 \\[3pt]
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots &\vdots
\end{vmatrix}.
\end{equation*}
Observe that $d_{j}$ is a polynomial of degree $j$ in $t$ for each $j \in \{ 0, 1, \ldots, r \}$.
We set $A = (a_{ij})$ to be the $(2r-1) \times (2r-1)$ matrix defining $\text{Res}(p_{r,t}, p'_{r,t})$ above, so
\[
a_{ij} =
\begin{cases}
d_{r + i - j} &\text{if } 1 \le i \le r-1 \text{ and } i \le j \le i+r \\[3pt]
(i - j + 1) \,d_{i - j + 1} &\text{if } r \le i \le 2r-1 \text{ and } i-r+1 \le j \le i \\[3pt]
0 &\text{otherwise.}
\end{cases}
\]
We make use of the definition of a determinant to obtain
\begin{equation*}
\text{Res}(p_{r,t}, p'_{r,t})
= \det A
= \sum_{\sigma \in S_{2r-1}} \bigg( \text{sgn}(\sigma) \prod_{i = 1}^{2r-1} a_{i,\sigma(i)} \bigg).
\end{equation*}
We show that independent of $\sigma \in S_{2r-1}$, the product $\prod_{i = 1}^{2r-1} a_{i,\sigma(i)}$
is a polynomial of degree at most $r^{2}$ in $t$.
In fact, more is true. If each $a_{i,\sigma(i)} \ne 0$, then we show that $\prod_{i = 1}^{2r-1} a_{i,\sigma(i)}$
is a polynomial of degree exactly $r^{2}$ in $t$. Indeed,
for such $\sigma$, we have
\begin{gather*}
i \le \sigma(i) \le i + r \quad \text{ for } 1 \le i \le r-1, \\[3pt]
i - r + 1 \le \sigma(i) \le i \quad \text{ for } r \le i \le 2r-1,
\end{gather*}
and
\begin{align*}
\deg \bigg( \prod_{i = 1}^{2r-1} a_{i,\sigma(i)} \bigg)
&= \sum_{i = 1}^{r-1} \deg( a_{i,\sigma(i)} ) + \sum_{i = r}^{2r-1} \deg( a_{i,\sigma(i)} ) \\[5pt]
&= \sum_{i = 1}^{r-1} \big(r+i - \sigma(i)\big) + \sum_{i = r}^{2r-1} \big(1+i - \sigma(i)\big) \\[5pt]
&= r^{2} + \sum_{i = 1}^{2r-1} \big(i - \sigma(i)\big) = r^{2}.
\end{align*}
We set
\[
\rho = \sum_{\sigma \in S_{2r-1}} \bigg( \text{sgn}(\sigma) \prod_{i = 1}^{2r-1} \ell(a_{i,\sigma(i)}) \bigg),
\]
where $\ell(a_{i,\sigma(i)})$ denotes the leading coefficient of $a_{i,\sigma(i)}$. Observe that if $\rho \ne 0$,
then $\det A$ is a polynomial of degree $r^{2}$ with leading coefficient $\rho$. The value of $\rho$ is the
determinant of $(2r-1) \times (2r-1)$ matrix $(\ell(a_{ij}))$. Since
\[
\ell(a_{ij}) =
\begin{cases}
1/(r+i-j)! &\text{for } 1 \le i \le r-1 \text{ and } i \le j \le i+r \\[3pt]
1/(i-j)! &\text{for } r \le i \le 2r-1 \text{ and } i-r+1 \le j \le i \\[3pt]
0 &\text{otherwise},
\end{cases}
\]
this determinant is the value of $\text{Res}(g,g')$, where
$g(x) = \sum_{j=0}^{r} x^{j}/j!$.
This polynomial truncation of $e^{x}$ has been studied by
R~F.~Coleman \cite{RFC} and I.~Schur \cite{IS, IS2}.
In particular, $g(x)$ corresponds to the generalized Laguerre polynomial $L_{r}^{(-r-1)}(x)$,
for which I.~Schur \cite{IS3} gives an explicit formula for the discriminant from which the
value of $\text{Res}(g,g')$ is easily determined (also, see \cite{RM}, Chapter~9). From these,
we see that
\[
\text{Res}(g,g') = \dfrac{1}{(r!)^{r-1}}.
\]
We deduce that $\text{Res}(p_{r,t}, p'_{r,t})$ is a polynomial of degree $r^{2}$
in $t$ with leading coefficient $1/(r!)^{r-1}$. From \eqref{discreq6}, we see that $\Delta$ is a polynomial
of degree $r^{2} - r$ in $t$ which has leading coefficient $(-1)^{r(r-1)/2} /(r!)^{r-2}$. Since \eqref{discreq4} divides $\Delta$,
the lemma follows.
\end{proof}
\begin{lemma}\label{discrimnonsquarelemma}
Let $r$ be an integer $\ge 2$. For $t$ a non-negative integer,
let $\Delta$ be the common discriminant of $p_{r,t}(x)$, $\tilde{p}_{r,t}(x)$, $\tilde{p}_{r,t}(x+1)$ and $q_{r,t+r+1}(x-1)$.
Then there is a $t_{0} = t_{0}(r)$ such that for all $t \ge t_{0}$, the value of $\Delta$ is not a square.
\end{lemma}
\begin{proof}
Suppose $r \ge 2$ and $t \ge 0$ are such that $\Delta$ is a square.
From Lemma~\ref{discrimvaluelemma}, we see that $r \ge 4$ since $\Delta < 0$ for $r \in \{ 2, 3 \}$.
We consider even and odd $r$ separately and only the case that $\Delta \ge 0$ since $\Delta < 0$
cannot be a square.
In the case that $r$ is even, Lemma~\ref{discrimvaluelemma} implies that
$(t+1)(t+r+1)$ is an integer that is a rational square.
Hence, $(t+1)(t+r+1)$ is the square of an integer. Let $\delta = \gcd(t+1,t+r+1)$.
Then $\delta$ divides the difference $(t+r+1) - (t+1) = r$, so $\delta \le r$.
Also $(t+1)/\delta$ and $(t+r+1)/\delta$ are relatively prime numbers whose
product is a square, so each of them is a square.
As $(t+r+1)/\delta - (t+1)/\delta = r/\delta \le r$ and the difference of
two consecutive squares $(n+1)^{2}-n^{2} = 2n+1$ tends to infinity with $n$,
we deduce that $(t+1)/\delta$ is bounded. In fact, taking $n^{2} = (t+1)/\delta$,
we see that
\begin{align*}
2 \sqrt{\dfrac{t+1}{r}} + 1 \le 2 \sqrt{\dfrac{t+1}{\delta}} + 1 \le \dfrac{r}{\delta} \le r
&\implies
\dfrac{4(t+1)}{r} \le (r-1)^{2} \\[5pt]
&\implies
t < t+1 \le \dfrac{r \,(r-1)^{2}}{4}.
\end{align*}
Thus, for $r$ even and $t \ge r (r-1)^{2}/4$, we have that $\Delta$ is not a square.
In the case that $r$ is odd,
Lemma~\ref{discrimvaluelemma} implies that
the largest factor of the product
\[
(t+2)(t+3) \cdots (t+r)
\]
relatively prime to $r!$ is a square.
As $(t+r) - (t+2) = r-2$, we see also that for every prime $p > r$
dividing the product $(t+2)(t+3) \cdots (t+r)$, there is a unique $j \in \{ 2, 3, \ldots, r \}$
for which $p|(t+j)$. Thus, for such a $p$ and $j$, there is a positive integer $e$
for which $p^{2e} \Vert (t+j)$. As $r \ge 5$, we deduce that there are positive integers
$a$, $b$ and $c$ each dividing the product of the primes up to $r$ and satisfying
\[
t+2 = a u^{2}, \quad
t+3 = b v^{2} \quad \text{and} \quad
t+4 = c w^{2},
\]
for some positive integers $u$, $v$ and $w$. We deduce that
\[
b^{2} v^{4} - 1 = (t+3)^{2} -1 = (t+2)(t+4) = ac (uw)^{2}.
\]
As $a$, $b$ and $c$ divide the product of the primes up to $r$,
there are finitely many equations of the form $ac y^{2} = b^{2} x^{4} - 1$ possible
for a given $r$. Each such equation is an elliptic curve containing finitely many
integral points by a theorem of Siegel \cite{LM, ST, LW}. Hence, for a fixed $r$,
the value of $x = v = \sqrt{(t+3)/b}$ is bounded from above for every possible $b$.
Thus, $t_{0}$ exists in
the case of $r$ odd, completing the proof.
\end{proof}
\section{Proof of Theorem~\ref{mainthm}}
As noted in the introduction, from \cite{BFLT} and \cite{FKP},
for $t$ sufficiently large, the polynomial
$p_{r,t}(x)$ and, hence, the polynomials $\tilde{p}_{r,t}(x)$, $\tilde{p}_{r,t}(x+1)$ and $q_{r,t+r+1}(x-1)$ are irreducible.
We begin by considering the case that $r$ is a fixed integer $\ge 8$.
From Lemma~\ref{hajirlemma} and Lemma~\ref{discrimnonsquarelemma},
it suffices to show that there is a prime $q$ in the interval $(r/2,r-2)$
such that the Newton polygon of one of
$p_{r,t}(x)$, $\tilde{p}_{r,t}(x)$, $\tilde{p}_{r,t}(x+1)$ and $q_{r,t+r+1}(x-1)$
with respect to some prime $p$ has an edge with slope $a/b$ where
$a$ and $b$ are relatively prime integers and $q|b$.
Since $r \ge 8$, one can show using explicit results on the distribution of primes (cf.~\cite{RS})
that there is a prime $q \in (r/2,r-2)$. Alternatively, from \cite{SR}, one has that there are
$3$ primes in the interval $(r/2,r]$ for $r \ge 17$ so that there must be at least $1$ prime
in the interval $(r/2,r-2)$ for $r \ge 17$. Then a simple check leads to such a prime
for $r \ge 8$.
With $q$ a prime in $(r/2,r-2)$, we consider $t \in \mathbb Z^{+}$ sufficiently large.
Note that the numbers $t+r+1-q$ and $t+1+q$ are distinct positive integers.
Let $p$ be a prime $> r$, and suppose $p^{e} \Vert (t+r+1-q)(t+1+q)$ where $e \in \mathbb Z^{+}$.
Observe that $p > r$ implies either $p^{e} \Vert (t+r+1-q)$ or $p^{e} \Vert (t+1+q)$.
We use that in fact $p$ can divide at most one of $t+r+1, t+r, \ldots, t+1$.
Suppose $p^{e} \Vert (t+r+1-q)$.
Then the Newton polygon of $q_{r,t+r+1}(x-1)$ with respect to $p$
consists of two edges, one joining $(0,e)$ to $(r-q,0)$ and
the other joining $(r-q,0)$ to $(r,e)$.
In the case that $p^{e} \Vert (t+1+q)$, the Newton polygon of $q_{r,t+r+1}(x-1)$ with respect to $p$
also consists of two edges, one joining $(0,e)$ to $(q,0)$ and
the other joining $(q,0)$ to $(r,e)$.
In either case, we see that the Newton polygon of $q_{r,t+r+1}(x-1)$ with respect to $p$
has an edge of slope $\pm e/q$. From Lemma~\ref{hajirlemma}, we can therefore
deduce for sufficiently large $t$, the Galois group of $q_{r,t+r+1}(x-1)$ is $S_{r}$
unless $q \mid e$. This is true for each prime $p > r$ with $p|(t+r+1-q)(t+1+q)$.
So suppose then that for every prime $p > r$ dividing $(t+r+1-q)(t+1+q)$, we have
$p^{e} \Vert (t+r+1-q)$ or $p^{e} \Vert (t+1+q)$ for some $e$ divisible by $q$.
We deduce that we can write
\[
t+1+q = a u^{q}
\quad \text{and} \quad
t+r+1-q = b v^{q},
\]
where $a$, $b$, $u$ and $v$ are positive integers with both $a$ and $b$ dividing
\[
\mathcal P = \prod_{\substack{p \le r \\ p \text{ prime}}} p^{q-1}.
\]
Note that $q \in (r/2,r-2)$ and $r \ge 8$, so that $q \ge 5$.
For fixed $a$ and $b$ dividing $\mathcal P$, we have $u$ and $v$ must be solutions
to the Diophantine equation
\[
a u^{q} - b v^{q} = 2q - r > 0.
\]
As this is a Thue equation, we deduce that there are finitely many integral solutions in
$u$ and $v$ (cf.~\cite{ShT}). This is true for each fixed $a$ and $b$ dividing $\mathcal P$.
As $\mathcal P$ and $q$ only depend on $r$ and $r$ is fixed, we deduce that there are finitely
many possibilities for $t+1+q = a u^{q}$. Hence, for sufficiently large $t$, we deduce that
$q \nmid e$ for some prime $p > r$ with $p^{e} \Vert (t+r+1-q)$ or $p^{e} \Vert (t+1+q)$.
Consequently, in the case that $r \ge 8$, we can conclude the Galois group of
$q_{r,t+r+1}(x-1)$ is $S_{r}$, from which the same follows for the polynomials
$p_{r,t}(x)$, $\tilde{p}_{r,t}(x)$ and $\tilde{p}_{r,t}(x+1)$.
Now, we consider the case that $r \le 7$.
We consider $t$ sufficiently large so that in particular the polynomials in Theorem~\ref{mainthm} are irreducible.
In the case that $r = 2$ , the only possibility then is that the Galois group is $S_{2}$.
For $r=3$, we use also that, by Lemma~\ref{discrimnonsquarelemma},
the discriminant of $p_{3,t}(x)$ is not a square, and this is enough to imply that
the Galois group of $p_{3,t}(x)$ over $\mathbb Q$ is $S_{3}$.
For $r = 4$, suppose $p$ is a prime $> 3$ dividing $(t+2)(t+4)$. If $p^{e}\Vert (t+2)(t+4)$, then
$p^{e}\Vert (t+2)$ or $p^{e} \Vert (t+4)$. In either case, we see that the Newton polygon of
$q_{r,t+r+1}(x-1)$ with respect to $p$ consists of an edge with slope $e/3$.
If $3 \nmid e$, then the Galois group will have a $3$-cycle so that Lemma~\ref{kconradlemma}
and Lemma~\ref{discrimnonsquarelemma} imply that the Galois group is $S_{4}$.
Otherwise, $3 \mid e$ for each prime $p > 3$ dividing
$(t+2)(t+4)$. We deduce that $t+4 = a u^{3}$ and $t+2 = b v^{3}$ where $a$ and $b$ divide $36$.
Observe that $a u^{3} - b v^{3} = 2$.
This is a Thue equation, and as before this equation has no solutions for $t$ sufficiently large. Thus,
since $t$ is sufficiently large, the Galois group of $p_{4,t}(x)$ over $\mathbb Q$ is $S_{4}$.
For $r = 5$ and $r = 7$, one can give similar arguments. Specifically, for $r = 5$ and
for a prime $p > 3$ such that $p^{e}\Vert (t+3)(t+4)$, we deduce that
either $3 \mid e$ or else there is a $\sigma$ in the Galois group of $p_{5,t}(x)$ over $\mathbb Q$
which is a $3$-cycle or is a product of two disjoint cycles, one a $3$-cycle and one a $2$-cycle.
In the case that $3 \mid e$ for every such prime $p > 3$,
we have $t+4 = a u^{3}$, $t+3 = b v^{3}$ and $a u^{3} - b v^{3} = 1$,
where $a$ and $b$ divide $36$. Since $t$ is sufficiently large, this does not occur.
In the case that $\sigma$ is a product of a $3$-cycle and a $2$-cycle, we see that
$\sigma^{2}$ is a $3$-cycle. Thus, regardless of $\sigma$, we can apply Lemma~\ref{kconradlemma}
and Lemma~\ref{discrimnonsquarelemma} to deduce that the Galois group of $p_{5,t}(x)$ over $\mathbb Q$ is $S_{5}$.
For $r = 7$ and for a prime $p > 3$ such that $p^{e}\Vert (t+4)(t+5)$, one similarly argues
that either $3 \mid e$ for every prime $p > 3$ and a Thue equation shows that this impossible
since $t$ is sufficiently large or there is a $\sigma$ in the Galois group of $p_{7,t}(x)$ over $\mathbb Q$
such that $\sigma^{4}$ is a $3$-cycle. Also, for $r = 7$ and for a prime
$p > 7$ such that $p^{e}\Vert (t+1)(t+8)$, one similarly argues
that either $7 \mid e$ for every prime $p > 7$ and a Thue equation shows that this impossible
since $t$ is sufficiently large or there is a $7$-cycle in the Galois group of $p_{7,t}(x)$ over $\mathbb Q$.
Thus, the Galois group of $p_{7,t}(x)$ over $\mathbb Q$ contains a $3$-cycle and a $7$-cycle,
and Lemma~\ref{kconradlemma} and Lemma~\ref{discrimnonsquarelemma} imply this Galois group is $S_{7}$.
We are left with the case that $r = 6$.
There are $16$ transitive subgroups of $S_{6}$ (cf.~\cite{DM}).
We can eliminate all but two of these as possibilities for
the Galois group $G$ of $p_{6,t}(x)$ over $\mathbb Q$ as follows.
Using an argument similar to the above, we consider a prime $p > 5$
such that $p^{e}\Vert (t+2)(t+6)$ to show that
either $5 \mid e$ for every prime $p > 5$ and a Thue equation shows that this impossible
since $t$ is sufficiently large or there is a $5$-cycle in $G$.
Since $t$ is sufficiently large, we deduce that $p_{6,t}(x)$ is irreducible over $\mathbb Q$,
$p_{6,t}(x)$ has a non-square discriminant in $\mathbb Q$, and $G$ contains a $5$-cycle.
The latter implies that $5$ divides $|G|$.
Of the $16$ transitive subgroups of $S_{6}$, only $4$ have size divisible by $5$, and of those
exactly $2$ are contained in $A_{5}$. Since the discriminant of $p_{6,t}(x)$ is not a square,
this leaves then just $2$ possibilities for $G$,
one is $S_{6}$ and the other is $PGL_{2}(5)$, which is a subgroup of $S_{6}$ that is
isomorphic to $S_{5}$.
For the purposes of the proof of Theorem~\ref{mainthm}, we can distinguish between
cases where $G = S_{6}$ and cases where $G = PGL_{2}(5)$ by observing that
$S_{6}$ has an element which is the product of two disjoint cycles, one a $2$-cycle
and the other a $4$-cycle, whereas $PGL_{2}(5)$ has no such element.
We consider a prime $p > 3$ such that $p^{e}\Vert (t+3)(t+5)$ for some $e \in \mathbb Z^{+}$.
If $2 \nmid e$ then ${p}_{6,t}(x)=g(x)h(x)$ where $g(x)$ and $h(x)$ are irreducible polynomials over $\mathbb{Q}_p$ of degrees $2$ and $4$ respectively. Let $F_g$ and $F_h$ denote the splitting fields of $g$ and $h$ over $\mathbb{Q}_p$ and observe that they are tamely ramified since $p>3$. Using Newton polygons, one deduces that $F_g$ is totally ramified and that the ramification index of $F_h$ is divisible by $4$. We know that $F_h$ is tamely ramified and therefore the tame inertia group is cyclic with order divisible by $4$ \cite[Corollary 1, p. 31]{CF}. Since $S_4$ has no larger cyclic subgroups, we deduce that the ramification index of $F_h$ is exactly $4$ and the tame inertia subgroup is generated by a $4$-cycle (the only possible form of an element with order $4$ in $S_4$).
Now, let $K$ be the compositum of $F_g$ and $F_h$. If $F_h\subsetneq K$, then $F_h\cap F_g = \mathbb{Q}_p$. Therefore, there is an element of the Galois group of ${p}_{6,t}(x)$ that permutes the 2 roots of $g$ and cyclicly permutes the 4 roots of $h$. That is, the Galois group of ${p}_{6,t}(x)$ contains an element which is the disjoint product of a 4-cycle and a 2-cycle.
If $K=F_h$, then $F_g\subset F_h$. Let $\tau$ be a generator of the tame inertia group of $F_h$. If $\tau$ permutes the roots $g$, then the Galois group of ${p}_{6,t}(x)$ contains an element which is the disjoint product of a 4-cycle and a 2-cycle. If $\tau$ fixes the roots of $g$, then the roots of $g$ lie in $K^\tau$, the fixed field of $\tau$. However, since $\tau$ generates the inertia subgroup of $K$, we know that $K^\tau$ is an unramified extension of $\mathbb{Q}_p$ \cite[Proposition 9.11, p. 173]{JN}. Therefore, $F_g\subset K^\tau$ is unramified, which is a contradiction with our previous deduction that $F_g$ is a totally ramified quadratic extension of $\mathbb{Q}_p$.
Since the Galois group $G$ has an element that is the product of two disjoint cycles, one a $2$-cycle
and the other a $4$-cycle, we have shown that $G = S_{6}$. In the case that $2 \mid e$
for every prime $p > 3$, we have $t+5 = a u^{2}$ and $t+3 = b v^{2}$ where $a$ and
$b$ are divisors of $6$. In this case we have, for fixed $a$ and $b$, the Diophantine equation
\begin{equation}\label{pelleq}
a u^{2} - b v^{2} = 2
\end{equation}
in the variables $u$ and $v$.
\begin{table}[h]
\centering
\renewcommand{\tabcolsep}{8pt}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|}
\hline
Pairs $(a,b)$ & All Solutions \\ \hline \hline
$(1,2)$ & $u = 2u'$ where $(1+\sqrt{2}\,)^{2m-1} = v + \sqrt{2}\,u'$ for $m \in \mathbb Z^{+}$ \\ \hline
$(2,1)$ & $v = 2v'$ where $(1+\sqrt{2}\,)^{2m} = u + \sqrt{2}\,v'$ for $m \in \mathbb Z^{+}$ \\ \hline
$(2,3)$ & $v = 2v'$ where $(5+2\sqrt{6}\,)^{m} = u + \sqrt{6}\,v'$ for $m \in \mathbb Z^{+}$ \\ \hline
$(2,6)$ & $(2+\sqrt{3}\,)^{m} = u + \sqrt{3}\,v$ for $m \in \mathbb Z^{+}$ \\ \hline
$(3,1)$ & $(1+\sqrt{3}\,)(2+\sqrt{3}\,)^{m-1} = v + \sqrt{3}\,u$ for $m \in \mathbb Z^{+}$ \\ \hline
$(6,1)$ & $(2+\sqrt{6}\,)(5+2\sqrt{6}\,)^{m-1} = v + \sqrt{6}\,u$ for $m \in \mathbb Z^{+}$ \\ \hline
\end{tabular}
\caption{Solutions to the Pell Equations}\label{table}
\end{table}
Of the $16$ possibilities for $(a,b)$ where $a$ and $b$ divide $6$,
there are $9$ for which \eqref{pelleq} can be shown to have no solutions modulo either $3$ or $4$.
For $(a,b) = (2,2)$, the equation \eqref{pelleq} is equivalent to $u^{2}-v^{2} = 1$. Since consecutive
positive squares differ by more than $1$ and since $t+5 = a u^{2}$, we deduce that there are no solutions
for $t \ge 1$. The remaining $6$ choices of $(a,b)$ are tabulated in Table~\ref{table}.
Here, the equation \eqref{pelleq} corresponds to a
Pell equation which has infinitely many solutions in \textit{positive} integers $u$ and $v$
given by the right column in the table.
These solutions were found using classical methods for solving Pell equations (cf.~\cite{GC}), and
we do not elaborate on the details. In each case, the solutions grow exponentially, and the total number
of solutions in pairs $(u,v)$ with $u$ and $v$ each $\le X$ is $O(\log X)$. As
$t+5 = a u^{2}$ and $t+3 = b v^{2}$, we deduce that the number of $t \le T$ such that $G = PGL_{2}(5)$
is at most $O(\log T)$, completing the proof of Theorem~\ref{mainthm}.
The inclusion of the phrase ``at most" in the theorem is to emphasize that we do not know that
these exceptional pairs that arose at the end of this proof give rise to cases where $G = PGL_{2}(5)$.
In fact, it is likely that $G = S_{6}$ for every sufficiently large $t$ when $r = 6$. For $t \in \{ 1,3 \}$,
which arise from the two smallest solutions coming from Table~\ref{table}, one checks that
$G = PGL_{2}(5)$. There are $37$ other positive integer values of $t \le 10^{10}$ coming from Table~\ref{table},
and one checks that for each of these we have:
\begin{itemize}
\item
For some prime $p_{1} \le 149$, the polynomial $p_{6,t}(x)$ is an irreducible sextic polynomial modulo $p_{1}$.
Hence, $p_{6,t}(x)$ is irreducible.
\item
The discriminant $\Delta$ of $p_{6,t}(x)$ is not a square. Hence,
$G$ is not contained in $A_{6}$. (Note that by Lemma~\ref{discrimvaluelemma}, if $r = 6$, then $\Delta < 0$
for all non-negative integers $t$; thus, $\Delta$ cannot be a square if $r = 6$.)
\item
For some prime $p_{2} \le 101$, the polynomial $p_{6,t}(x)$ factors as a linear polynomial times
an irreducible quintic modulo $p_{2}$. Hence, $G = PGL_{2}(5)$ or $G = S_{6}$.
\item
For some prime $p_{3} \le 109$ not dividing the discriminant $\Delta$ of $p_{6,t}(x)$,
the polynomial $p_{6,t}(x)$ factors as an irreducible quadratic times
an irreducible quartic modulo $p_{3}$. Hence, $G = S_{6}$ (using Dedekind's Theorem discussed in Section 2).
\end{itemize}
\noindent
It therefore is plausible that for $t > 3$ in general, the Galois group of $p_{6,t}(x)$ is in fact $S_{6}$.
Note that since $\tilde{p}_{r,t}(x+1) = x^{r} q_{r,t+r+1}(1/x)$, the comments about $q_{r,n}(x)$
after the statement of Theorem~\ref{mainthm} follow.
|
{
"timestamp": "2018-03-08T02:11:02",
"yymm": "1803",
"arxiv_id": "1803.02754",
"language": "en",
"url": "https://arxiv.org/abs/1803.02754"
}
|
\section{INTRODUCTION}
Nanoscale superconductivity, in which one or more dimensions are smaller than the coherence length, exhibits a range of interesting phenomena, such as Berezinskii-Kosterlitz-Thouless (BKT) phase transitions \cite{Kos_1973,Kos_1974}, excess conductivity induced by superconducting fluctuations \cite{Asl_1968}, and the superconductor-insulator quantum phase transition at zero temperature \cite{Gol_1998}, to name a few. In particular, when the size of the superconductor becomes comparable to the electron Fermi wavelength $\lambda_F$, the formation of discretized electronic energy levels results in the oscillations of the density of states at the Fermi level with the size, together with the reconfiguration of the pairing interaction, leading to the oscillatory behavior of superconducting critical temperature $T_c$ and other observables \cite{Bla_1963,Per_1996,Bia_1997,Guo_2004,Oze_2006,Eom_2006,Bru_2009,Qin_2009,Zha_2010,Che_2010,Che_2012,Sha_2015}, i.e. the so-called quantum size effects.
Recent progress in nanotechnology has allowed high-quality superconducting nanostructures to be fabricated with atomic-scale precision \cite{Uch_2017,Pin_2017}. Superconductivity is realized in atomically thin films even down to a single monolayer. Quantum size effects have been reported in atomically thin films \cite{Bao_2005}, superconducting nanoparticles \cite{Bos_2010} and islands \cite{Min_2015}, nanowires\cite{Cor_2006} and nanowire arrays \cite{Zha_2016}. However, the low-dimensional superconductivity is strongly influenced by the imperfections such as impurities, disorder and structural defects \cite{Bru_2014}. In nanofilms, increasing disorder results in the phase transition from a superconducting to an insulating state \cite{Gol_1998}. In addition, impurities locally suppress superconductivity, which in superconducting nanowires can promote a phase slip center, giving rise to the broad temperature transition and residual resistance \cite{Lan_1967,McC_1970}. Very recently, a step in atomically thin films was found to have a strong effect on electronic transport \cite{Kim_2016,Zha_2017} and vortex matter \cite{Bru_2014,Yos_2014,Rod_2015}, as a new paradigm in the interplay between the local defects and low-dimensional superconductivity. As an extended defect, the step does not only scatter electrons (leading to the modification on the overall electronic structure of the sample \cite{Liu_2013}), but also affects the flow of superconducting currents and the proximity-induced superconducting correlations \cite{Kim_2016,Zha_2017}. However, the effect of the lateral step (indentation) in ultrathin yet nanoscale wide superconductors (from here on referred to as nanoribbons) has not been investigated. Such nanoribbons are readily used as building blocks of superconducting quantum devices such as single-photon detectors \cite{Nat_2012}, phase-slip junctions \cite{Moo_2006}, and Josephson junction arrays \cite{Hav_2001}, and can be fabricated even from 2D materials such as graphene, NbSe$_2$, MoS$_2$, WS$_2$, and other \cite{Sen_2017, Ngu_2017, Zhe_2017, Pia_2017}.
In this paper, we address this issue, with a special attention drawn to superconducting nanoribbons with a constriction formed by two adjacent step-edges (see Fig.~\ref{sketch}). Starting from a long constriction, where two step-edges are far away from each other, we discuss the role of a step-edge and present how it modifies electronic states, the superconducting order parameter and the local density of states (LDOS) in the nanoribbon. Then, for a short constriction, effectively an extended quantum point contact, we show how the device becomes a quantum-confined Josephson junction, a novel object with quantum-tunable characteristics. Namely, such a device exhibits properties that are governed by quantum size effects, different inside and outside the constriction, hence behaving as a S-S'-S junction with performance broadly dependent on all sizes, temperature, and the Fermi energy (controllable by gating or doping).
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{layout1.eps}
\caption{(Color online) Oblique and bird view of a nanoribbon with a constriction formed by step-edges, with indicated geometrical parameters within the computational unit cell. The width variation of the nanoribbon is realized by imposing a tall potential barrier $U_0$ outside the sample (shaded region). The shown aspect ratio of the simulation region ($L_x:L_y$) greatly under-represents the actual one in the simulations (where $L_x:L_y\approx100:1$).}
\label{sketch}
\end{figure}
The paper is organized as follows. In Sec.~\ref{sec:2}, we present our theoretical approach and details of numerical simulations. In Sec.~\ref{norm_sec}, we present the normal-state electronic properties for the nanoribbon with a constriction. Next, we detail the superconducting properties, first for a plain nanoribbon (for necessary background) in Sec.~\ref{sec:4.1}, followed by nanoribbon with a long constriction in Sec.~\ref{sec:4.2}, and finally for the quantum-confined Josephson junction (nanoribbon with a short constriction) in Sec.~\ref{sec:4.3}. Our findings are summarized in Sec.~\ref{sec:5}.
\section{THEORETICAL MODEL AND NUMERICAL APPROACH}\label{sec:2}
We employ the Bogoliubov-de Gennes (BdG) equations to study the role of step-edges in a superconducting nanoribbon, in which quantum confinement is important. The BdG equations have been successfully used in the past to study the interplay between superconductivity and the quantum confinement, and have revealed many fascinating phenomena - among which the quantum size effect, unconventional vortex states, new Andreev bound states, and quasiparticle interference effect \cite{Sha_2006,Sha_2007,Sha2_2007, Cro_2007}.
The BdG equations are written as:
\begin{equation}
\begin{matrix}
\label{BdG_eq}
\mqty(\hat{K}_0-E_F &\Delta(\vec{r}) \\ \Delta^*(\vec{r})&-\hat{K}_0^*+E_F)\mqty(u_n(\vec{r})\\v_n(\vec{r}))=E_n\mqty(u_n(\vec{r})\\v_n(\vec{r})),
\end{matrix}
\end{equation}
where $u_n(\vec{r})$($v_n(\vec{r})$) are electron(hole)-like wave functions corresponding to the quasiparticle energy $E_n$, $E_F$ is the Fermi energy, and the single-particle Hamiltonian $\hat{K}_0$ reads
\begin{equation}
\hat{K}_0=-\frac{\hbar^2}{2m}\nabla^2+U(\vec{r}),
\end{equation}
with $U(\vec{r})$ the confining potential.
In order to find the quasiparticle energy spectrum $E_n$ and the corresponding wave functions $u_n(\vec{r})$ and $v_n(\vec{r})$, we need to solve the BdG equations self-consistently together with the relation for the pair potential $\Delta(\vec{r})$
\begin{equation}
\label{SC_gap}
\Delta(\vec{r}) = g\sum_{E_n<E_c}u_n(\vec{r})v^*_n(\vec{r})[1-2f(E_n)],
\end{equation}
where $g$ is the coupling constant, $E_c$ is the Debye cutoff energy, and $f(E_n) = [1+\exp(E_n/k_BT)]^{-1}$ is the Fermi distribution function at temperature $T$. The local density of states is
\begin{equation}
\label{SC_gap1}
N(\vec{r},E) = \sum_n [\delta(E_n-E)\abs{u_n(\vec{r})}^2+\delta(E_n+E)\abs{v_n(\vec{r})}^2].
\end{equation}
We consider a nanoribbon with a dent at the center, as shown in Fig.~\ref{sketch}. The length of the ribbon $L_x$ is much longer than the superconducting coherence length $\xi$, i.e. $L_x \gg \xi$, with a periodic boundary condition along the ribbon. The dent separates the ribbon into two parts, characterized by different width - the part of length $L_W$ and the width $W$, and the part of length $L_w$ and the width $w$ (being the constriction, i.e. $w\leq W$). The corresponding areas are $S_W=L_W \times W$ and $S_w=L_w \times w$, respectively. Note that $L_x = L_W+L_w$ and the step-edges are located where the width of the nanoribbon changes. The widths $W$ and $w$ are of the order of the Fermi wavelength $\lambda_F$. Since $\lambda_F<\xi$, $L_x \gg W\,,\,w$.
Two extreme cases will be taken into consideration. First we study the role of a single step-edge. For this purpose, we set $L_W= L_w= L_x/2$ so that the distance between two adjacent step-edges is the farthest. Since $L_x \gg \xi$ we have $L_W\,,\,L_w \gg \xi$, i.e. the interaction between the neighboring step-edges can be neglected. The other case is the role of a dent where $L_w \approx \lambda_F$. Due to the discrete energy levels inside the dent and the momentum mismatch at the step-edges, the transport properties through the dent will be strongly affected. In this paper, we do not present the results for the intermediate cases since they can be understood as superposition of two discussed extreme cases.
In order to perform numerical calculations, we embed the nanoribbon in a computational unit cell with area $S = L_x \times L_y$, as shown in Fig.~\ref{sketch}. The length of the unit cell is the same as that of the ribbon but its width is determined by the condition $L_y > \text{max}\{W,w\}$, so that the single-particle potential barrier
\begin{equation}
U(\vec{r})= \begin{cases}
0 & \text{ outside the ribbon;} \\
U_0 & \text{ inside the ribbon;}
\end{cases}
\end{equation}
can be applied outside the ribbon to confine the electrons. Since a large magnitude $U_0=20E_F$ is used for the potential barrier, the quasiparticle wave functions $u_n(\vec{r})$ and $v_n(\vec{r})$ decay exponentially at the edges of the ribbon.
To solve more efficiently the self-consistent BdG equations \eqref{BdG_eq}-\eqref{SC_gap}, we expand $u_n$($v_n$) in terms of the eigenstates $\Psi_l(\vec{r})$ of the single-electron Schr\"odinger equation for the normal state
\begin{equation}\label{Schr}
\hat{K_0}(\vec{r})\Psi_l(\vec{r})=E_l\Psi_l(\vec{r}).
\end{equation}
We first solve Eq.~\eqref{Schr} by expanding $\Psi_l(x,y)$ in terms of plane waves $\phi_{j_x,j_y}(x,y)$, i.e.
\begin{equation}
\label{Four_exp_Psi}
\Psi_l(x,y) = \sum_{j_x,j_y} c^{\,l}_{j_x,j_y} \phi_{j_x,j_y}(x,y),
\end{equation}
where $c^{\,l}_{j_x,j_y}$ are the coefficients for the $l$-th eigenstates and
\begin{equation}
\phi_{j_x,j_y}(x,y) = (L_xL_y)^{-1/2} \text{exp} \Big( i\frac{2\pi j_x}{L_x}x +i\frac{2\pi j_y}{L_y}y \Big),
\end{equation}
with $j_x,j_y \in \mathcal{Z}$. We define $j=j(j_x,j_y)$. Then, Eq.~\eqref{Schr} becomes
\begin{equation}
\label{Schr_mat}
T_j c^{\,l}_j + \sum_{j'} U_{jj'} c^{\,l}_{j'} = \zeta_l c^{\,l}_j,
\end{equation}
where
\begin{equation}
T_j = \frac{\hbar^2}{2m} \left[ \left(\frac{2\pi j_x}{L_x}\right)^2 +\left(\frac{2\pi j_y}{L_y}\right)^2\right],
\end{equation}
and
\begin{equation}
U_{jj'}=\int\dd{x}\dd{y}\phi_{j_xj_y}^*(x,y)\,U(x,y)\,\phi_{j'_xj'_y}(x,y).
\end{equation}
Eq.~\eqref{Schr_mat} has a matrix form. By diagonalizing the relevant matrix, the eigenvalues $\zeta_l$ and eigenfunctions $\Psi_l(\vec{r})$ can be obtained. We remark that $j_x$($j_y$) must remain finite in the numerical calculations, i.e. $j_x = 0,\pm 1,\ldots,\pm j_x^{\text{max}}$ and $j_y = 0,\pm1,\ldots,\pm j_y^{\text{max}}$. The choice of $j_x^{\text{max}}$ and $j_y^{\text{max}}$ depends on different parameters, including $E_F$, $E_c$, the size of the nanoribbon and the unit cell. However, when the wave functions are confined in a smaller area, a larger cut-off is needed in order to preserve the same accuracy. For example, if $w/L_y$ is taken smaller, the number of basis functions associated with the $y$ direction, $j_y^{\text{max}}$, has to be larger.
Next, we expand $u_n$($v_n$) in terms of $\Psi_l(\vec{r})$ as
\begin{equation}
\binom{u_n(\vec{r})}{v_n(\vec{r})}=\sum_{l}\binom{u^n_l}{v^n_l}\Psi_l(\vec{r}).
\label{Four_exp_uv}
\end{equation}
We use a parameter $\varepsilon$ to control the number of $\Psi_{l}$ in the expansion, such that only those $\Psi_{l}$ with energies $\zeta_l < E_F+\varepsilon E_c$ are included. After inserting Eq.~\eqref{Four_exp_uv} into the BdG Eqs.~\eqref{BdG_eq}, we obtain
\begin{equation}
\begin{aligned}
(\zeta_l-E_F) u^n_l + \sum_{l'} \Delta_{ll'} v^n_{l'} &= E_n u^n_l, \\
\sum_{l'} (\Delta_{l'l})^* u^n_{l'} + (E_F-\zeta_l) v^n_l &= E_n v^n_l,
\end{aligned}
\label{BdG_mat}
\end{equation}
where
\begin{equation}
\Delta_{ll'} =\int\dd{x}\dd{y}\Psi_{l}^*(x,y)\,\Delta(x,y)\,\Psi_{l'}(x,y),
\end{equation}
and $(\Delta_{l'l})^*$ is the conjugate transpose of $\Delta_{ll'}$. Similarly to Eq.~\eqref{Schr_mat}, Eq.~\eqref{BdG_mat} has a matrix form as well. The corresponding eigenvalues and eigenstates can be obtained after its diagonalization.
In this paper, we present the results for the following parameters: effective mass $m=2m_e$ ($m_e$ being the electron mass), $E_F = 40~\mathrm{meV}$, $E_c = 24~\mathrm{meV}$, and coupling constant $g$ is set such that the bulk gap at zero temperature is $\Delta_0=1.2~\mathrm{meV}$ and $T_c\approx 8.2~\mathrm{K}$, the coherence length at zero temperature $\xi_0={\hbar}v_F/\left(\pi \Delta_0\right)= 14.7~\mathrm{nm}$ and $k_F\xi_0 = 21$, where $v_F$ is the Fermi velocity and $k_F$ the Fermi wave-vector. The prototype material can be, e.g., NbSe$_2$ \cite{Gyg_1991, Vir_1999}. For this set of parameters, we take $L_x=1~\mathrm{\mu m}$, $L_y=12~\mathrm{nm}$. Then, we find that $j^{\text{max}}_x=500$, $j^{\text{max}}_y=16$ and $\varepsilon=3.5$ yield satisfactory results so that larger cut-off is not necessary. We also confirm that the features of our results are robust for other $k_F\xi_0$ values so these generic features can be applied to other superconducting materials (e.g., Pb, In, Ga, NbSe$_2$). All the results are calculated at zero temperature, unless specified otherwise.
\section{Normal-state electronic properties}
\label{norm_sec}
In this section, we examine the normal-state electronic properties of the nanoribbon, since any effect of the constriction on those may further manifest in the superconducting properties. The normal-state electronic properties can be completely obtained by solving the Schr\"odinger equation \eqref{Schr}. When there is no constriction ($w=W$), the normal-state electronic structures are well characterized by a series of one-dimensional (1D) subbands, in which the energy dependence of density of states (DOS) of each subband is proportional to $(E-E_j)^{-1/2}$, where $E_j=\frac{1}{m}(\frac{\pi j}{W})^2$ is the threshold energy at the $j$th subband. As a result, the DOS exhibits a peak each time $E_j$ is approached, which corresponds to the van Hove singularity of the standard 1D DOS. An example of this type is shown in Fig.~\ref{FigN1}(a) for the nanoribbon with $W = w= 10~\mathrm{nm}$ (dashed line). The DOS is defined as
\begin{equation}\label{DOS_N}
n(E) = \sum_l\delta(E-E_l)/(S_W+S_w),
\end{equation}
where $S_W+S_w$ is the area of the ribbon. We use this area because $|\Psi_l(\mathbf{r})|^2$, i.e. the probability density of the wavefunction, is negligible outside of the ribbon due to the very large potential barrier $U_0$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_N1_mod.eps}
\caption{(Color online) The DOS of a nanoribbon with a long constriction, for $W=10~\mathrm{nm}$ and $w=8~\mathrm{nm}$. Panel (a) shows $n(E)$ (solid line), compared to the case with no constriction (dashed line). Panel (b) shows the spatially averaged LDOS over the wider and the constricted part of the nanoribbon, $n_W(E)$ and $n_w(E)$ respectively, with peaks labelled according to the respective sequence of subbands in two parts of the nanoribbon.}
\label{FigN1}
\end{figure}
A constriction is introduced in the nanoribbon when $W>w$. We initially consider a long constriction where $L_W=L_w=L_x/2 \gg W,w$. In this case we can represent the system as two adjoined nanoribbons of different width. Fig.~\ref{FigN1}(a) shows the corresponding DOS, $n(E)$, of a nanoribbon with $W= 10~\mathrm{nm}$ and $w= 8~\mathrm{nm}$. It is characterized by the standard 1D DOS, with doubled peaks compared to the plain nanoribbon. The additional peaks in the $n(E)$ can be understood by considering $n_W(E)$ and $n_w(E)$, namely, the spatially averaged local density of states (LDOS) over the $W$-part and the $w$-part of the nanoribbon, respectively. These quantities are calculated as
\begin{equation}
\begin{aligned}
n_W(E) &= \int_W n_r(\mathbf{r},E) d\mathbf{r}/S_W, \\
n_w(E) &= \int_w n_r(\mathbf{r},E) d\mathbf{r}/S_w,
\end{aligned}
\end{equation}
where
\begin{equation}
n_r(\mathbf{r},E) = \sum_l|\Psi_l(\mathbf{r})|^2\delta(E-E_l),
\end{equation}
is the LDOS and $S_W$ ($S_w$) is the area of the $W$-part ($w$-part) of the nanoribbon. Note that
\begin{equation}\label{n3}
n(E)=[n_W(E)S_W+ n_w(E)S_w]/(S_W+S_w),
\end{equation}
with reference to the definition of $n(E)$ in Eq.~\eqref{DOS_N}. Accordingly, Fig.~\ref{FigN1}(b) shows the individual contribution of $n_W(E)$ and $n_w(E)$ when they are extracted from $n(E)$ [the solid line in Fig.~\ref{FigN1}(a)]. Both $n_W(E)$ and $n_w(E)$ exhibit the standard 1D DOS, as a consequence of the fact both $L_W$ and $L_w$ are sufficiently long. The peaks in $n_W(E)$ are at energies $E_j=\frac{1}{m}(\frac{\pi j}{W})^2$, and those in $n_w(E)$ are at energies $E_{j'}=\frac{1}{m}(\frac{\pi j'}{w})^2$ \footnote{More precisely, the characteristic energies $E_j(W) \rightarrow \frac{1}{m}(\frac{\pi j}{W})^2$ and $E_{j'}(w) \rightarrow \frac{1}{m}(\frac{\pi j'}{w})^2$ when the potential barrier $U_0 \rightarrow \infty$.}. In short, the DOS of the nanoribbon with a long constriction, which is given by $n(E)$, is featured by the standard 1D DOS with two sets of characteristic energies $E_j(W)$ and $E_{j'}(w)$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_N2_2.eps}
\caption{(Color online) (a) The $p$-resolved DOS, $n_p(p,E)$ (as defined in the text), for the nanoribbon with a long constriction, where $W=10~\mathrm{nm}$ and $w=8~\mathrm{nm}$. (b) Examples of electronic probability density $|\psi(x,y)|^2$ near $x=0$ for $p=0.03$, $0.97$, and $0.5$.}
\label{FigN2}
\end{figure}
To understand the obtained behavior of $n_W(E)$ and $n_w(E)$, we study the electronic wavefunctions of the normal state. For this purpose, we calculate the probability of a wavefunction $\psi(\mathbf{r})$ lying in the constriction, i.e. $p=\int_w|\psi(\mathbf{r})|^2d\mathbf{r}$, and construct the $p$-resolved DOS, $n_p(p,E)$ [see Fig.~\ref{FigN2}(a)], written as
\begin{equation}
n_p(p,E)= \left[\sum_l \delta(E-E_l)\delta(p-p_l)\right]/(S_W+S_w)
\end{equation}
Note that $0 \leqslant p \leqslant 1$ due to the normalization of the wavefunction. The integral of $n_p(p,E)$ over $p$ is the density of states $n(E)$. From Fig.~\ref{FigN2}(a), one sees that states mostly lie around $p=0.5$, i.e. when $|\psi(\mathbf{r})|^2$ is spread over the $W$- and $w$-part. The example of a probability density $|\psi(\mathbf{r})|^2$ for $p=0.5$ is shown in Fig.~\ref{FigN2}(b), and is indeed spread over the entire nanoribbon.
However, we find that there is a large number of states near $p\approx 0$ at $E_j$ and near $p\approx 1$ at $E_{j'}$ [see $n_p(p,E)$ in Fig.~\ref{FigN2}(a)]. States with $p\approx 0$ ($p\approx 1$) are localized in the $W$-part ($w$-part) and decay exponentially in the other part. Examples of these two types of $|\psi(\mathbf{r})|^2$ are also presented in Fig.~\ref{FigN2}(b), with $p=0.03$ and $0.97$, respectively. These two cases of $|\psi(\mathbf{r})|^2$ are in analogy with quantum-well states and are responsible for the standard 1D DOS appearing in $n_W(E)$ and in $n_w(E)$. We also note that there is an exclusion rule between the states with $p>0.5$ and those with $p<0.5$. That is, for the given energy $E$, the states with $p>0.5$ cannot coexist with the state with $p<0.5$, as shown in Fig.~\ref{FigN2}(a). Therefore, the electronic properties in the $W$-part can be very different from those in the $w$-part, especially at the characteristic energies $E_j$ and $E_{j'}$. This property will play a decisive role in the change of superconducting properties at the step-edge(s), where the width of the nanoribbon changes.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_N3_mod.eps}
\caption{(Color online) (a) The spatial integral of LDOS over $y$, $n_y(x,E)$, near $x=0$ for the nanoribbon with a long constriction, for $W=10~\mathrm{nm}$ and $w=8~\mathrm{nm}$. The evolution of the peaks near $x=0$ is marked by dots. (b) $n_y(E)$ for some selected $x$, vertically displaced for clarity.}
\label{FigN3}
\end{figure}
How the electronic structure changes near the step-edge (e.g. at $x=0$ in Fig. \ref{sketch}) is an interesting aspect to study. Ref.~\onlinecite{Che_2000} reports that the transitions of the electronic states around the step-edge should be sharp and abrupt. In contrast, Ref.~\onlinecite{Sah_2015} shows that the transition is smooth within a certain lateral extension. Here we found the abrupt transitions of the electronic states around the step-edge are accompanied by a somewhat smooth transition of the energy of the peaks in LDOS due to the localized states.
To this end, the spatial integral of LDOS over $y$, $n_y(x,E)$, is evaluated near the step-edge, as shown in Fig.~\ref{FigN3}(a). Two opposite behaviors are displayed in the vicinity and far from the step. In the former case, peaks are gradually shifted in energy, while they are significant and $x$-independent in the latter case, appearing at typical energies $E_j$ ($E_{j^\prime}$) in the $W$-part ($w$-part).
Near the step, broad and low peaks evolve by changing their position in energy. Bearing in mind that electronic states are energetically well defined, modifications of the position and the shape of the peaks can be inferred by noting that wavefunctions exponentially decay when passing through the step. The occurrence and the shape of a peak at energies $E_j$ ($E_{j^\prime}$) depends on how much the wavefunction is spread over the $W$-part ($w$-part) around the step. The larger the distance is from the step, the sharper the peak is because the wavefunction is well localized on that part. When the distance from the step is progressively reduced, the resulting position of a peak can be shifted in energy since a superposition effect may occur.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_N4_mod.eps}
\caption{(Color online) $n_w(E)$ of nanoribbons with $W=10~\mathrm{nm}$ and $w=8~\mathrm{nm}$, for different $L_w$. The lineplots of $n_w(E)$ are vertically displaced for clarity. Dashed vertical lines indicate $E_j$, the positions of peaks in $n_W(E)$.}
\label{FigN4}
\end{figure}
After understanding the properties of the nanoribbon with a long constriction, we turn focus to the case of a short constriction ($L_w\sim w$). In this case, the length of the $W$-part, $L_W$, is always kept sufficiently long so that the electronic properties in the $W$-part are independent of both $L_W$ and $L_w$, i.e. $n_W$ always exhibits the standard 1D DOS as shown in Fig.~\ref{FigN1}(b).
On the other hand, different electronic properties may occur when $L_w\approx w$ because, in contrast to the long constriction, here the short constriction can be viewed as an extended quantum point contact.
To demonstrate how the electronic properties change with decreasing $L_w$, we show in Fig.~\ref{FigN4} the $n_w(E)$ dependence for the nanoribbon with $W=10~\mathrm{nm}$ and $w=8$ nm, for different lengths of the constriction ($L_w$). For $L_w=30$ nm, $n_w(E)$ still exhibits characteristics of standard 1D DOS, but the appearance of the main peak at $E_{j^\prime}$ is accompanied by several secondary peaks, due to the discrete energy levels induced in the constriction by the quantum confinement in the $x$ direction. As $L_w$ is decreased, the number of secondary peaks decreases and the energy spacing between those peaks becomes larger. For example, two secondary peaks appear after the $j^\prime=3$rd peak for $L_w=20$ nm, while only one remains for $L_w=10$ nm. Meanwhile, the main peaks are displaced in energy from $E_{j^\prime}$, towards the closest peak of $n_W(E)$ at $E_j$. These shifts can be larger for smaller values of $L_w$ [see in particular the case of $L_w=4$ nm in Fig.~\ref{FigN4}]. In addition, peaks of $n_w(E)$ become more pronounced as approaching $E_j$ [cf. for example the $j^\prime=4$th peak to the preceding peaks for $L_w=4$ nm].
It is worth mentioning that for short $L_w$, $n(E) \rightarrow n_W(E)$, following from Eq.~\eqref{n3} due to $S_W \gg S_w$. Moreover, we find that all the electronic states are mixed and spread across the entire nanoribbon. Therefore, there are no localized states in the short constriction, in contrast to the long one.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_N5_mod.eps}
\caption{(Color online) The $n_w(E)$ characteristics in the case of short constriction ($L_w=4$ nm), for different width of the constriction ($w$, indicated in the figure) and fixed width of the nanoribbon [either $W=8$ nm (solid lines) or $W=8.8$ nm (dashed lines)]. The series of $n_w(E)$ are vertically displaced for clarity. $+$ and $\times$ indicate the first and the third peak, respectively. Vertical lines indicate the nearest peak in the corresponding $n_W(E)$ [for $W=8~\mathrm{nm}$ (solid) and $W=8.8~\mathrm{nm}$ (dashed)].}
\label{FigN5}
\end{figure}
Finally, Fig.~\ref{FigN5} shows a series of $n_w(E)$ for the shortest considered constriction $L_w=4~\mathrm{nm}$, now for different widths of the constriction ($w$). The solid lines and the dashed lines represent the case of nanoribbons with $W=8$ nm and $W=8.8$ nm, respectively. In all cases the $n_w(E)$ characteristic exhibits a series of broad and smooth peaks, whose shape depends on whether peaks of $n_W$ at $E_j$ are close to them. These peaks are nearly independent of the width of the nanoribbon $W$, and shift to higher energy with decreasing the width of the constriction $w$. Therefore, the case of a short constriction case can also be viewed as a nanoribbon coupled with a spectrally broadened quantum dot.
\section{Superconducting properties}\label{sec:4}
\subsection{Background: superconducting nanoribbons without a constriction}\label{sec:4.1}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{SC0_mod.eps}
\caption{(Color online) Superconducting properties of a homogeneous nanoribbon ($w=W$). Panel (a) shows the spatially averaged order parameter, $|\bar{\Delta}|$, as a function of width $W$. The characteristic behavior of $|\bar{\Delta}|$ is identified at widths indicated by dots, labeled according to the order of the oscillations along the $y$ direction and with the letter indicating the peak ($p$) or valley ($v$) of $|\bar{\Delta}(W)|$. Dotted line shows $\Delta_0$ in the bulk limit ($W \rightarrow \infty$). Panel (b) shows the spatial profile of the order parameter ($|\Delta(y)|$), for the five cases indicated in (a). Panels (c) - (e) show the corresponding DOS as a function of energy $E$. Note that DOS profiles in panel (e) are shifted vertically for clarity.}
\label{SC0}
\end{figure}
After comprehending the fundamental normal-state properties of the nanoribbon with a constriction, we move on to the analysis of the superconducting state.
The superconductivity in a nanoribbon with no constriction (i.e. $w=W$) has already been studied in detail elsewhere \cite{Sha_2006,Cro_2006}. Here, we repeat some relevant properties of the superconducting nanoribbon under quantum confinement, which will be used as a reference later on when considering the nanoribbon with a constriction ($W>w$).
Fig.~\ref{SC0}(a) shows the spatially-averaged superconducting order parameter, $\bar{\Delta}$, as a function of the width $W$ of the nanoribbon. It exhibits quantum size oscillations as a function of the width, due to the fact that the normal-state single-electron band splits into a series of subbands under the quantum confinement effect. These subbands shift in energy with $W$, giving rise to the variations in the DOS at $E_F$, i.e. the number of electrons which can contribute to Cooper-pairing. As $W$ is varied, when the bottom of a new subband approaches $E_F$, the DOS increases together with a substantial reconfiguration of the pairing interaction, leading to the resonant enhancement of superconductivity.
The quantum-confinement regime for the transverse direction of the electron motion results in the spatial variations of the order parameter along the $y$ direction, in reference to the sketch of the system in Fig.~\ref{sketch}. $\Delta(y)$ is shown in Fig.~\ref{SC0}(b) for characteristic five cases in Fig.~\ref{SC0}(a), i.e. two for resonance cases $3p$ and $4p$ and three for off-resonance cases $2v$, $3v$, and $4v$ [the number in these labels indicates the order of oscillations along the $y$ direction, and $p$ ($v$) stands for peak (valley) in $\bar{\Delta}(W)$]. $\Delta(y)$ of the resonance cases is stronger in amplitude and more spatially inhomogeneous than in off-resonance cases.
Due to the pronounced inhomogeneity of the order parameter under the resonance condition, a multi-gap structure can form in the DOS [see Figs.~\ref{SC0}(c) and (d)] \cite{Che_2010}, detectable in experimentally measured tunneling spectrum. In addition, new type of Andreev reflection and Tomasch oscillations are also induced due to strongly inhomogeneous order parameter. In contrast, when the bottom of any present subband is away from $E_F$, the superconductivity is in the off-resonant condition where the corresponding DOS is characterized by a conventional BCS gap structure [as in Fig.~\ref{SC0}(e)].
\subsection{Superconducting nanoribbons with a long constriction}\label{sec:4.2}
\begin{table}
\setlength{\tabcolsep}{10pt}
\begin{tabular}{ | c | c | c |}
case & conditions for $W$/$w$ side & ($W$,$w$)[$\mathrm{nm}$] \\ [.5ex]
I & RES/RES & $8.8$, $6.5$ \\ [.3ex]
II & RES/OFF-RES & $8.8$, $8$ \\ [.3ex]
III & OFF-RES/RES & $10.1$, $6.5$ \\ [.3ex]
IV & OFF-RES/OFF-RES & $10.1$, $5.8$
\end{tabular}
\caption{The conditions for the characteristic cases I-IV for a nanoribbon with a long constriction, and the corresponding widths $W$ and $w$. The resonance (RES) and off-resonance (OFF-RES) conditioning corresponds to peaks and valleys indicated in Fig.~\ref{SC0}(a), respectively.}
\label{table1}
\end{table}
Next we consider the superconducting state for a sample with a constriction, thus for $w < W$. First, we study the influence of a single step-edge, i.e. $L_W$, $L_w \gg \xi$, where the interaction between the adjacent step-edges is negligible. It is clear that the superconducting properties far away from the step-edges are the same as those of plain nanoribbons with the corresponding width (either $W$ or $w$). However, the superconducting features are essentially different depending on the resonance or off-resonance configuration selected for the pair $(W,w)$. Thus, we present the results for the four possible cases, whose parameters are given in Table~\ref{table1}.
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{OPstep_mod.eps}
\caption{(Color online) The spatial distribution of the order parameter, $\abs{\Delta(x,y)}$, near the step-edge for the cases I-IV of Table \ref{table1}.}
\label{OPxy}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{DOSstep_mod.eps}
\caption{(Color online) LDOS averaged across the width of the sample, DOS($x$,$E$), for the selected cases I-IV of Table~\ref{table1}, plotted in the vicinity of the step-edge.}
\label{DOSx}
\end{figure}
Fig.~\ref{OPxy} shows the spatial distribution of the order parameter, $\abs{\Delta(x,y)}$, for the characteristic four cases described in Table~\ref{table1}. The corresponding LDOS, averaged over width of the nanoribbon [DOS($x$,$E$)], is shown in Fig.~\ref{DOSx}. We only present the results in the vicinity of the left step-edge (with situation at the other step-edge being same, i.e. mirror-symmetric). We find that the behaviors of the order parameter and DOS are well described by the normal-state electronic structures in all cases. In case IV, off-resonant superconductivity is present in both parts of the sample (i.e. with widths $W$ and $w$). The corresponding $E_F$ is away from the bottom of every subband at energies $E_j$ and $E_{j^\prime}$, and the Cooper pairs, formed by the electronic states with $p\approx0.5$, are dominant. Since these electronic states spread over entire nanoribbon, the superconducting properties do not show significant variations when crossing the step-edge. As shown in Fig.~\ref{OPxy}, the order parameter of case IV does not have a sharp change in the vicinity of the step-edge, differently from cases II and III. In addition, the DOS [Fig.~\ref{DOSx}(d)] exhibits conventional BCS gap in both the constriction and the rest of the nanoribbon.
In case III, the superconducting state is in the resonance configuration in the constricted ($w$-part), and in the off-resonance configuration in the rest of the nanoribbon ($W$-part). Thus, $E_F$ is near $E_{j^\prime}$ but far away from $E_j$ so that the normal-state electronic states with $p\approx1$ are dominant over the $w$-part.
As a result, the superconducting properties on one side are very different from those on the other side of the step-edge. For example, as shown in Fig.~\ref{OPxy}(c), the order parameter drops dramatically when crossing the step-edge from the narrower $w$-side to the $W$-side. Generally, the spatial variation of the order parameter is defined by its characteristic length, i.e. the coherence length $\xi$, inside the vortex core and at the S-S' interfaces. However, the enhancement of the order parameter in the constricted $w$-part is here induced by the normal-state electronic states with $p\approx1$. These states decay exponentially when crossing the step-edge, and the characteristic length scale is of the order of $\lambda_F$ ($\xi\approx 10-1000\lambda_F$ in conventional superconductors \cite{Alt_2013}). Therefore, the order parameter exhibits a fast variation within distance of the order of $\lambda_F$ at the step-edge, in a similar fashion to the occurrence of Friedel-like oscillations near the surface of a superconductor. Due to this feature, a superconducting nano-structure with a step-edge behaves as a rather sharp, ideally contacted S-S' junction.
The localization of the resonant superconducting properties in the long constriction can also be seen in the DOS($x$,$E$) for case III [Fig.~\ref{DOSx}(c)]. As discussed above, the superconducting gap is larger and the coherence peaks are more pronounced in the $w$-part than in the $W$-part. However, these features suddenly disappear at the step-edge leading to a dramatic change in the gap structures. This is also due to the normal-state electronic states with $p\approx1$ which decay at the step-edge. Note that, due to the large variation of the gap amplitude near the step-edge, the inverse proximity effect is clearly visible. Its magnitude varies slowly away from the step-edge because the length scale of the variation is related to $\xi$. However, this effect is much less significant when compared to the effects related to the abrupt change in the normal-state electronic states in the vicinity of the step-edge.
The case II is inversely analogous to the case III, with superconducting state being in resonance in the $W$-part, while off-resonant superconductivity is present in the constriction ($w$-part). Therefore, the same conclusions can be deduced as done in case III, but for opposite sides of the step-edge. The change in superconductivity is dramatic when the step-edge is crossed, and this variation can be seen both in the profile of the order parameter [Fig.~\ref{OPxy}(b)] and the DOS [Fig.~\ref{DOSx}(b)].
Case I is peculiar because resonant superconductivity is attributed to both sides of the step-edge, indicating that $E_F$ approaches both $E_{j^\prime}$ and $E_j$. The variation of the order parameter near the step-edge is not large, being similar to case IV [cf. Figs.~\ref{OPxy}(a) and (d)]. However, the resonance conditions in the $W$-part and $w$-part are induced by the normal-state electronic states with $p\approx0$ and with $p\approx1$, respectively. Both of them are localized and the corresponding probability density decays exponentially at the step-edge so that the superconducting electronic structures abruptly change when the step is crossed. As shown in Fig.~\ref{DOSx}(a), the multi-gap features in the $W$-part are different from those in the $w$-part, but they all coalesce into single gap near the step-edge. In fact, coherence peaks are strongly suppressed at the step-edge, as a sign of the loss of the superconducting coherence. Therefore, the superconducting properties in the resonance configuration are more sensitive to the imperfections such as impurities, disorder, surface roughness and structural defects because of the localization of the electronic states, leading to the suppression of the superconducting coherence at the imperfections. In this case, the critical current is limited by the weakest point of the nanoribbon.
\subsection{Superconducting nanoribbons with a short constriction}\label{sec:4.3}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_Ss1_mod.eps}
\caption{(Color online) The spatially averaged order parameter in and out of the constriction, $\bar{\Delta}_w$ and $\bar{\Delta}_W$ respectively, as a function of $L_w$, for the cases I and II in panel (a) and cases III and IV in panel (b). Panels (c)-(f) show the corresponding $\Delta(x,y)$ for $L_w=4~\mathrm{nm}$.}
\label{FigSs1}
\end{figure}
As previously mentioned, a nanoribbon with a long constriction can be viewed as two adjoined nanoribbons, each with different width. Such picture is no longer valid when the constriction is sufficiently short. Namely, when $L_w$ is comparable to the superconducting coherence length $\xi$, the proximity effect plays an important role, reducing the difference in superconducting gap between the constriction and the rest of the nanoribbon. In Fig.~\ref{FigSs1}(a,b), we show spatial averages $\bar{\Delta}_w$ inside and $\bar{\Delta}_W$ outside the constriction as a function of the length of the constriction $L_w$, for the characteristic cases I-IV of Table~\ref{table1}. We note that $\bar{\Delta}_W$ is independent of $L_w$ in all cases, while $\bar{\Delta}_w$ approaches $\bar{\Delta}_W$ with decreasing $L_w$. The corresponding spatial profiles $\Delta(x,y)$ near the constriction for the cases I-IV and $L_w=4$ nm are presented in Fig.~\ref{FigSs1}(c)-(f), respectively. In this limit of short constriction, $L_w$ is comparable to the Fermi wavelength ($L_w\approx\lambda_F$) so that the superconducting gap difference at the step-edge strongly diminishes [see e.g. cases I and IV in Fig.~\ref{FigSs1}(c) and (f), respectively].
However, the spatial arrangement of $\Delta(x,y)$ is still consistent with the selected cases I-IV for the nanoribbon with a long constriction [cf. Figs.~\ref{OPxy}(a)-(d)].
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_Ss2_mod.eps}
\caption{(Color online) (a) The critical current $I_c$ of the quantum-confined Josephson junction as a function of the width of a short constriction $w$, for different width $W$ of the nanoribbon, length of the constriction $L_w=4$ nm, and considered length of the junction $L_j=7$ nm. A sketch of the system is shown in the inset. (b) $I_c(W)$ for different widths of the constriction $w$, and other parameters same as in (a). Peaks are emphasized by vertical lines. Results are limited to range $w<W$, as governed by the geometry of the considered system.}
\label{FigSs2}
\end{figure}
The short constriction can thus be viewed as a quantum point contact, which results in a point-contact Josephson junction in the nanoribbon. To analyze its transport properties, we calculate the Josephson current passing through the short constriction. For this purpose, we set up a junction link of length $L_j=7$ nm around the short constriction of length $L_w=4~\mathrm{nm}$, as shown in the inset of Fig~\ref{FigSs2}(a). Inside the link, the superconducting gap $\Delta$ is calculated self-consistently. Outside the link, we fix the phase of the order parameter as $\Delta(x<-1.5~\mathrm{nm})=|\Delta|e^{i0}$ and $\Delta(x>5.5~\mathrm{nm})=|\Delta|e^{i\theta}$, such that a phase difference $\theta$ is imposed between the two sides of the link. Then, the supercurrent density is calculated as
\begin{align*}
\label{densityJ}
\vec{J}(\vec{r}) &= \frac{e\hbar}{2mi} \sum_{E_n<E_c}\left \{ f(E_n)u_n^*(\vec{r})\nabla u_n(\vec{r}) \right.\\
&+ \left. [1-f(E_n)]v_n(\vec{r}) \nabla v_n^*(\vec{r})-\text{h.c.}\right \},
\end{align*}
and satisfies the continuity condition $\nabla \cdot \vec{J} = 0$ inside the link due to the self-consistent $\Delta$ \cite{Spu_2010,Cov_2006}, resulting in the current conservation inside the link [i.e. $I(x) \equiv I = \int\dd{y} J_x(x,y)$]. Outside the link, $\vec{J}$ is discontinued due to the fixed phase of the order parameter, but these regions are simply treated as current sources in the present approximation.
The calculated critical current $I_c$ of the junction exhibits a step-like variation as a function of the width of the constriction $w$ for different values of $W$, as shown in Fig.~\ref{FigSs2}(a). Steps in $I_c$ occur each time the Fermi energy $E_F$ is crossed by a peak of $n_w$ or, in an equivalent formulation, when a new channel of conductance takes part in the current transport. This step-like behavior bears similarities with the quantum conductance in the SNS junction \cite{Fur_1991}. By changing the width of the ribbon $W$, steps of $I_c$ occur for same $w$ because the peaks of $n_w$ are nearly independent of $W$.
Moreover, $I_c$ has large value when the nanoribbon is in the resonance condition, i.e. for $W=8.8$, $6.5$ and $4.4$ nm, in contrast to the low $I_c$ for $W=8.0$ and $5.8$ nm, when nanoribbon is in the off-resonant condition [see Fig.~\ref{FigSs2}(a)]. It is worth noting that $I_c$ for $W=4.4$ nm is even more enhanced than the one for $W=8.8$ nm. Such behavior is more clearly seen in Fig.~\ref{FigSs2}(b), where the critical current as a function of $W$, $I_c(W)$, is reported. The $I_c(W)$ characteristic exhibits the quantum-size oscillations with increasing amplitude as $W$ is smaller. This is due to the fact that $\bar{\Delta}(W)$ is more enhanced at resonance in narrower nanoribbons, as shown in Fig.~\ref{SC0}(a). Thus, for fixed $w$, high $I_c$ is obtained for smallest $W$ that corresponds to a resonance condition. Fig.~\ref{FigSs2}(b) also implies that $I_c(W)$ converges for $W \rightarrow \infty$ so that it only depends on $w$ in this limit.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_Ss4_mod.eps}
\caption{(Color online) (a) Critical current $I_c$ of the quantum-confined Josephson junction made as a constriction of width $w=4$ nm and length $L_w=4$ nm inside a nanoribbon of width $W=6$ nm, as a function of the electronic potential energies $eV_{G,W}$ and $eV_{G,w}$ stemming from gate voltages respectively applied outside the junction ($W$-part) and inside the junction ($w$-part). Zero gating voltages (marked by open dot) correspond to the reference Fermi energy of $E_F=40$ meV. The profiles of $I_c(eV_{G,W})$ along $eV_{G,W}=eV_{G,w}$ (same gating in entire sample) and $eV_{G,w}=0$ (no gating in the junction) are plotted in panel (b), with line types corresponding to those shown in (a). The profiles of $I_c(eV_{G,w})$ for $eV_{G,W}=0$ (no gating outside the junction) and $eV_{G,W}=35$ meV are plotted in panel (c), with line types corresponding to those shown in (a).}
\label{FigSs3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Luca_v2_Ss5_mod.eps}
\caption{(Color online) The normal state LDOS spatially averaged over the junction, $n_w(E_F)$, for given gating difference $\Delta U=eV_{G,W}-eV_{G,w}$ between the nanoribbon and the junction, for the sample with $W=6$ nm, $w=4$ nm and $L_w=4$ nm. Open circles highlight the shift of the peaks with $\Delta U$. Note that additional peaks occur when $\Delta U<0$, i.e. for potential well realized inside the junction.}
\label{FigSs4}
\end{figure}
Finally, we show that the critical current $I_c$ of a short constriction can also be broadly tuned by electronic gating. Concretely, we apply gate voltages $V_{G,W}$ outside the junction (in the $W$-part) and $V_{G,w}$ inside the junction (in the $w$-part), respectively. The gate voltage $V_G$ is assumed to induce a chemical potential shift in the gated part of the sample, by electronic potential energy $eV_G$ \cite{Zhe_2017, Pia_2017}, with $e$ being the electron charge. The evolution of the $I_c$ of the quantum-confined junction as a function of gate voltages $V_{G,W}$ and $V_{G,w}$ is shown in Fig.~\ref{FigSs3}(a), for the short constricted nanoribbon with $W=6$ nm, $w=4$ nm, and $L_w=4$ nm. To give a better understanding of the features of $I_c$ under applied gate voltages, we also present the profile of $I_c(eV_{G,W})$ for $eV_{G,W}=eV_{G,w}$ (same gating in entire sample) and for $eV_{G,w}=0$ (no gating in the junction) in Fig.~\ref{FigSs3}(b), and the profile of the $I_c(eV_{G,w})$ for $eV_{G,W}=0$ (gating only in the junction) and for $35$ meV (fixed gating outside the junction) in Fig.~\ref{FigSs3}(c). No applied gating [indicated by open dot in Figs.~\ref{FigSs3}(a)-(c)] corresponds to the reference sample with $E_F=40$ meV.
As a general trend, $I_c$ increases under positive gate voltages due to the introduction of more charge carriers and more channels of conductance taking part in the current transport. On the other hand, the negative gate voltages reduce the $I_c$ to zero, with the complete depletion of charge carriers reached for $eV_{G,W}$ or $eV_{G,w}$ below $-E_F$.
We note that $I_c$ can be tuned by either voltage $V_{G,W}$ or $V_{G,w}$. In the former case, as seen from Fig.~\ref{FigSs3}(a) and (b), $I_c(eV_{G,W})$ exhibits peaks associated with quantum-size oscillations, in agreement with $I_c(W)$ in Fig.~\ref{FigSs2}(b). These peaks are determined by the properties of the nanoribbon (not the junction) and are therefore independent of $eV_{G,w}$. In contrast, when only the gating inside the junction ($V_{G,w}$) is varied, the corresponding $I_c(V_{G,w})$ has a richer resulting behavior, with a resonance peak around $V_{G,w}=-18$ meV and a {\it double peak} structure between $V_{G,w}=0$ and $30$ meV [see Fig.~\ref{FigSs3}(a)]. In particular, the double peak is clearly observed in $I_c(V_{G,w})$ for $V_{G,W}=0$ [see Fig.~\ref{FigSs3}(c)]. However, the occurrence of these peaks in $I_c(V_{G,w})$ does depend on $V_{G,W}$, in clear opposition to the case of $I_c(V_{G,W})$ whose features are independent of $V_{G,w}$. For example, as highlighted in Fig.~\ref{FigSs3}(c), the first peak in the double peak structure in $I_c(V_{G,w})$ shifts to higher voltage when the gate voltage outside the junction $eV_{G,W}$ is increased from $0$ to $35$ meV, while the second peak entirely disappears.
To get insight into the origins of the double-peak behavior in the $I_c(eV_{G,w})$, we examine the normal-state density of states in the junction, $n_w$, as a function of the Fermi energy $E_F$, for different gate voltages applied in the junction, as plotted in Fig.~\ref{FigSs4}. The applied gate voltage results in the potential difference $\Delta U = eV_{G,W}-eV_{G,w}$ between the junction and the rest of the nanoribbon so that a potential barrier (well) is formed in the junction when $\Delta U$ is positive (negative).
When $\Delta U \geqslant 0$, only smooth and broadened peaks are found in the $n_w$, due to the potential barrier in the junction which prevents the localization of electronic states inside the junction, leading to the formation of fewer peaks. On the other hand, when $\Delta U \leqslant 0$, we find that the peaks do not only become sharper, but also additional peaks appear. This is due to the fact that the realized potential well inside the junction can support more localized states (analogously to particle in a box problem).
Therefore, we conclude that selective gating enables rich and broadly tunable behavior of the critical current of the quantum-confined Josephson junction. This rich behavior stems from a nontrivial combination of (i) the quantum resonances in either nanoribbon or constriction, determined by geometrical parameters of the sample, and shifted independently by gating outside or inside the junction, respectively, and (ii) the effects due to a potential barrier or potential well realized in the junction, depending on the applied gating inside and outside of the junction.
\section{Conclusions}\label{sec:5}
In summary, we have detailed the properties of superconducting nanoribbons with a constriction by solving the Bogoliubov-de Gennes equations self-consistently, in the regime where quantum confinement is of crucial importance. The constriction in the nanoribbon is introduced by two adjacent steps in the lateral edge.
For a long constriction, the interaction between the adjacent step-edges can be neglected as they are separated by large distance. In this case, we reported the effect of a single step-edge on the superconducting order parameter and the local density of states. We found that the shape resonances of the superconducting gap are different and spatially confined inside and outside the constricted area, separated by an abrupt change in the superconducting properties at the step-edge, on a scale of the Fermi wavelength $\lambda_F$, thereby forming a near ideal S-S' junction. This is due to the fact that the step-edge scatters the normal electronic states, especially the ones that are near the band edge, leading to a large number of localized states concentrating on either side of the step. We also note that the superconducting (inverse) proximity effect at the step-edge is featured in the LDOS, but has far less prominent role than the change in the electronic states there.
When the two step-edges are close to each other, they form a short constriction. In this case, the normal-state electronic properties of the constriction can be viewed as those of a quantum dot with spectral broadening effects. In addition, the short constriction in the nanoribbon forms a quantum point contact, leading to a quantum-confined Josephson junction, with properties tuned via quantum-size effects in and out of the constriction. The critical current of the junction exhibits a step-like behavior as a function of the width of the constriction, and can be also tuned by the width of the nanoribbon outside the junction. Finally, we demonstrated a rather effective and versatile tunability of the junction properties by local as well as global electronic gating.
Taking everything into account, and bearing in mind the number of emergent crystalline 2D superconductors whose lateral geometry can be precisely patterned, we expect that our results will generate further ideas for control of the low-dimensional superconducting condensate and quantum tailoring of much needed superconducting quantum devices such as advanced SQUID probes \cite{Vas_2013,Uri_2016, Ana_2014,Emb_2017}, novel single-photon detectors \cite{Wal_2017}, phase-slip and weak-link junctions \cite{Lom_2018}, or Josephson qubits for second generation quantum technology \cite{Wan_2015, Ste_2010, Yan_2016}.
\section*{Acknowledgments}
This work was supported by the Research Foundation-Flanders (FWO-Vlaanderen), the Special Research Funds of the University of Antwerp (TOPBOF), the Italian MIUR through the PRIN 2015 program (contract No. 2015C5SEJJ001), the MultiSuper network, and the EU-COST NANOCOHYBRI action CA16218.
|
{
"timestamp": "2018-04-23T02:03:15",
"yymm": "1803",
"arxiv_id": "1803.02709",
"language": "en",
"url": "https://arxiv.org/abs/1803.02709"
}
|
\section{Introduction}
\label{sec:intro}
Nematic liquid crystals (NLCs) are classical examples of partially ordered materials that combine the fluidity of liquids with a degree of long-range orientational order \cite{dg,virga}. There is substantial interest in pattern formation for NLCs in confinement, of which NLC-filled square chambers are popular examples \cite{tsakonas, lewissoftmatter, cleaver, luo2012}. This paper focuses on stable NLC configurations for square geometries with a square hole, referred to as an isotropic inclusion which locally destroys the surrounding nematic ordering. Such holes can be created by laser treatments or e-beam lithography techniques~\cite{e-beam}
and domains with inclusions offer new possibilities for exotic pattern formation.
This paper is primarily motivated by the numerical results in \cite{kraljmajumdar} and the analytical results in \cite{majcanevarispicer2017}, both within the celebrated Landau-de Gennes (LdG) theory for nematic liquid crystals. The LdG theory describes the nematic state by a macroscopic order parameter, the $\mathbf{Q}$-tensor order parameter which is, mathematically speaking, a symmetric traceless $3\times 3$ matrix. The eigenvectors of the $\mathbf{Q}$-tensor represent the physically preferred directions for molecular alignment or the directions of orientational ordering and the corresponding eigenvalues are a measure of the degree of the order about the eigenvectors \cite{ejam2010, dg, virga}. In \cite{kraljmajumdar}, the authors numerically discover a novel ``Well Order Reconstruction Solution'' (WORS) for square domains with tangent boundary conditions on the square edges. This WORS solution has a constant eigenframe, featured by a cross that connects the square diagonals such that the $\mathbf{Q}$-tensor has two degenerate positive eigenvalues and a distinct negative eigenvalue along the diagonal cross, referred to as negative uniaxiality which is a signature of nematic defects. The WORS is globally stable for small square domains, typically of the order of tens to hundreds of nanometers. In \cite{majcanevarispicer2017}, the authors analytically prove the existence of the WORS solution for all square sizes and at a special temperature, reduce the analysis of the WORS solution to a scalar variational problem. The authors prove the global stability of the WORS solution for small square domains, the instability of the WORS solution for larger domains and prove that the WORS solution branch undergoes a supercritical pitchfork bifurcation as the square size increases, in the reduced scalar setting. The supercritical pitchfork bifurcation result is quite specific to the scalar problem and it is not clear if it holds for the full LdG problem with five degrees of freedom.
In \cite{kraljmajumdar}, the authors numerically study the effects of square inclusions or square holes on the stability and properties of the WORS on square domains. For concentric square inclusions (i.e. square holes that have the same centre as the square domain), the WORS exists although the stability properties depend on both the square size and the domain aspect ratio (the ratio of the inclusion size to the domain size). For an off-centered square inclusion, we lose the distinctive diagonal cross and the regions of negative uniaxiality become localised near the square edges.
In this paper, we study square domains with concentric isotropic square inclusions. Mathematically, we study a boundary value problem for the LdG $\mathbf{Q}$-tensor on this domain, with $\mathbf{Q}=0$ on the inclusion boundary and with Dirichlet boundary conditions on the outer boundary consistent with the experimentally imposed tangent boundary conditions in \cite{tsakonas}.
We prove the existence of a WORS-like solution for this model problem, with a constant eigenframe and a diagonal cross (along which the LdG $\mathbf{Q}$-tensor has two equal eigenvalues) that connects the vertices of the inner and outer squares. This existence theorem is true for all square sizes and aspect ratios (ratio of the inner square size to the outer square size) and for all temperatures. Following the arguments in \cite{lamy2014} and \cite{majcanevarispicer2017}, we can also prove that the WORS is globally stable, i.e. is the global minimizer of the LdG energy for this model problem, for either squares that are sufficiently small or for aspect ratios sufficiently close to unity. In this sense, we provide some theoretical foundations for the numerical results in \cite{kraljmajumdar}.
The analysis of the WORS is inherently two-dimensional in the presence of a square inclusion by contrast with the framework in \cite{majcanevarispicer2017} where the authors could study a scalar variational problem at a special temperature. We have conflicting boundary conditions on the inner and outer squares and we need to exploit two out of the five degrees of freedom of the LdG $\mathbf{Q}$-tensor, to describe the WORS for all temperatures. We perform a $\Gamma$-convergence analysis of a reduced LdG energy, in terms of these two degrees of freedom, to deduce qualitative properties of energy minimizers in this two-dimensional setting, in the limit of the square size $\lambda \to \infty$. We are able to identify at least three competing configurations in the reduced two-dimensional setting: the WORS configuration, a BD (boundary distortion)-configuration with a pair of distinctive edge transition layers along which the LdG $\mathbf{Q}$-tensor transitions between two distinct states and an ESC-configuration around which the nematic molecules escape into the third dimension around the isotropic inclusion. We compute specific minimality criteria of the WORS in terms of the material constants, the temperature and the geometric aspect ratio, in this asymptotic limit.
The $\Gamma$-convergence analysis is complemented by a detailed numerical study of the critical points of the LdG energy for this model problem, using finite-difference based numerical methods and deflation techniques \cite{farrell2015deflation}. Numerical investigations show that the ESC-configuration cannot have lower LdG energy than the WORS or BD configurations, which is also corroborated by the minimality estimates for the WORS, BD and ESC-configurations yielded by the $\Gamma$-convergence analysis in the $\lambda \to \infty$ limit. Hence, we restrict ourselves to a detailed study of the stability of the WORS and BD configurations, both of which have constant eigenframes and have distinct defect lines or transition layers. In the case of the WORS, the transition layers are supported along the diagonals and for the BD solution, along a pair of opposite square edges. We study the second variation of the LdG energy and decompose the second variation into three components --- the second variation in the two-dimensional class of perturbations that do not distort the constant eigenframes of the WORS and BD configurations, the second variation with respect to in-plane perturbations of the eigenframe and the second variation with respect to out-of-plane perturbations of the eigenframe. We believe that this decomposition will be useful for stability analysis of general critical points for more general model problems. We numerically test the stabilities of the WORS and the BD-configurations with respect to the three different kinds of perturbations at the special temperature employed in \cite{majcanevarispicer2017}, primarily to reduce the number of variables in the problem and this temperature is a special reference point. As expected, we find that the WORS is globally stable with respect to all perturbations for aspect ratios that are sufficiently close to unity i.e. narrow square annuli. Both the WORS and BD configurations are stable with respect to out-of-plane perturbations. It is interesting that the BD-configuration is always unstable with respect to in-plane perturbations i.e. the BD-configuration is never a stable critical point of the LdG energy for this model problem.
We briefly comment on how these results relate to the numerical results in \cite{martinrobinson2017} where the authors study the solution landscape as a function of the square size in a three-dimensional LdG framework, neglecting the out-of-plane components. They numerically find that the WORS solution branch which is globally stable for small squares and loses stability as the square size increases. The WORS solution loses stability with respect to BD-like configurations in the restricted two-dimensional class of perturbations which preserve the constant eigenframe but these BD-configurations are unstable in the class of perturbations which allow for in-plane distortions of the eigenframe. Indeed, the authors numerically observe at least four bifurcating solution branches from the WORS solution branch --- two unstable BD solution branches and two stable diagonal solution branches which do not have the constant eigenframe property. For larger squares, the BD solution branches connect to the familiar stable rotated solutions, which do not have a constant eigenframe, and for which the nematic molecules rotated by $\pi$ radians in the square plane, between a pair of parallel square edges.
Finally, we comment on why the WORS and BD-configurations are stable with respect to all out-of-plane perturbations for this model problem. It is rigorously proven in \cite{sternberggolovaty2015} that for certain thin geometries (where the vertical dimension is much smaller than the lateral dimensions) and for certain surface energies consistent with tangent boundary conditions, the LdG energy minimization problem reduces to a variational problem on the two-dimensional cross-section (such as the square domain in our case) and energy minimizers indeed only have three degrees of freedom. The energy minimizers have a fixed eigenvector in the $\mathbf{z}$-direction; one degree of freedom describes the in-plane alignment of the NLC molecules and two scalar order parameters account for the in-plane ordering and the ordering about the $\mathbf{z}$-direction.
For the WORS and the BD-configurations, the in-plane alignment is fixed by the constant eigenframe and hence, they belong to a sub-class of this reduced three-dimensional setting. In light of the rigorous results in \cite{sternberggolovaty2015}, it is not surprising that the instabilities arise in the reduced three-dimensional setting.
The paper is organized as follows. In Section~\ref{sec:prelimiaries} and \ref{sec:reduction}, we set up the geometric domain and the problem definition, along with recalling the mathematical framework of the LdG theory and proving the existence and uniqueness theorems for the WORS. In Section~\ref{sec:gamma}, we perform the $\Gamma$-convergence analysis for the limit of large domains and in Section~\ref{sec:numerics}, we present and analyse our numerical results. In Section~\ref{sec:conclusion}, we briefly present our conclusions.
\section{Preliminaries}
\label{sec:prelimiaries}
We model nematic profiles on two-dimensional squares with an isotropic inclusion within the Landau-de Gennes (LdG) theory for nematic liquid crystals.
The LdG theory is one of the most powerful continuum theories for nematic liquid crystals and describes the nematic state by a macroscopic order parameter
--- the LdG $\mathbf{Q}$-tensor that is a macroscopic measure of material anisotropy.
The LdG $\Qvec$-tensor is a symmetric traceless $3\times 3$ matrix i.e.
$$\Qvec \in S_0 := \left\{ \Qvec\in \mathbb{M}^{3\times 3}\colon Q_{ij} = Q_{ji}, \ Q_{ii} = 0 \right\}.$$
A $\Qvec$-tensor is said to be (i) isotropic if~$\Qvec=0$, (ii) uniaxial if $\Qvec$
has a pair of degenerate non-zero eigenvalues and (iii) biaxial if~$\Qvec$ has three distinct eigenvalues~\cite{dg,newtonmottram}. A
uniaxial $\Qvec$-tensor can be written as $\Qvec_u = s \left(\nvec \otimes \nvec - \Ivec/3\right)$ with~$\Ivec$ the $3\times 3$ identity matrix,
$s\in\Rr$ and~$\nvec\in S^2$, a unit vector. The scalar, $s$, is an order parameter which measures the degree of orientational order.
The vector, $\nvec$, is the eigenvector with the non-degenerate eigenvalue, referred to as the ``director'' and labels the single distinguished direction of uniaxial nematic alignment~\cite{virga,dg}.
We work with a simple form of the LdG energy given by
\begin{equation} \label{eq:2}
I[\Qvec] := \int_{\Omega} \frac{L}{2} \left|\nabla\Qvec \right|^2 + f_B(\Qvec) \, \mathrm{d}A,
\end{equation}
where $\Omega\subseteq\Rr^2$ is a two-dimensional domain,
\begin{equation} \label{eq:3}
|\nabla \Qvec |^2 := \frac{\partial Q_{ij}}{\partial r_k}\frac{\partial Q_{ij}}{\partial r_k},
\qquad f_B(\Qvec) := \frac{A}{2} \tr\Qvec^2 - \frac{B}{3} \tr\Qvec^3 + \frac{C}{4}\left(\tr\Qvec^2 \right)^2.
\end{equation}
The variable~$A = \alpha (T - T^*)$ is the re-scaled temperature, $\alpha$, $L$, $B$, $C>0$ are material-dependent constants
and~$T^*$ is the characteristic nematic supercooling temperature~\cite{dg,newtonmottram}.
Further~$\rvec:=(x, \, y)$, $\tr\Qvec^2 = Q_{ij}Q_{ij}$ and $\tr\Qvec^3 = Q_{ij} Q_{jk}Q_{ki}$ for $i$, $j$, $k = 1, \, 2, \, 3$.
It is well-known that all stationary points of the thermotropic
potential, $f_B$, are either uniaxial or isotropic~\cite{dg,newtonmottram,ejam2010}.
The re-scaled temperature~$A$ has three characteristic values: (i)~$A=0$, below which the isotropic phase $\Qvec=0$ loses stability,
(ii) the nematic-isotropic transition temperature, $A={B^2}/{27 C}$, at which $f_B$ is minimized by the isotropic
phase and a continuum of uniaxial states with $s=s_+ ={B}/{3C}$ and
$\nvec$ arbitrary, and (iii) the nematic supercooling temperature, $A = {B^2}/{24 C}$, above which the isotropic state is the unique critical point of $f_B$.
We work with $A<0$ i.e. low temperatures and the numerical work in this paper focuses on a special temperature,
$A=-{B^2}/{3C}$, largely to facilitate comparison with~\cite{majcanevarispicer2017}.
Our analytical results are true for all temperatures, $A<0$. For a given $A<0$, let
$\mathscr{N} := \left\{ \Qvec \in S_0\colon \Qvec = s_+ \left(\nvec\otimes \nvec - \Ivec/3 \right) \right\}$
denote the set of minimizers of the bulk potential, $f_B$, with
\[
s_+ := \frac{B + \sqrt{B^2 + 24|A| C}}{4C}
\]
and~$\nvec \in S^2$ arbitrary. In particular, this set is relevant to our choice of Dirichlet conditions for boundary-value problems.
We non-dimensionalize the system using a change of variables, $\bar{\rvec} = \rvec/ \lambda$,
where $\lambda$ is a characteristic length scale of the system.
The re-scaled LdG energy functional is then given by
\begin{equation} \label{eq:rescaled}
\overline{I}[\Qvec] := \frac{I[\Qvec]}{L \lambda} = \int_{\overline{\Omega}}\frac{1}{2}\left| \overline{\nabla} \Qvec \right|^2
+ \frac{\lambda^2}{L} f_B\left(\Qvec \right) \, \overline{\mathrm{d}A}.
\end{equation}
In~\eqref{eq:rescaled}, $\overline{\Omega}$ is the re-scaled domain, $\overline{\nabla}$ is the gradient with respect to
the re-scaled spatial coordinates and $\overline{\mathrm{d}A}$ is the re-scaled area element. The associated Euler-Lagrange equations are
\begin{equation} \label{eq:6}
\bar{\Delta} \Qvec = \frac{\lambda^2}{L} \left\{ A\Qvec - B\left(\Qvec\Qvec - \frac{\Ivec}{3}|\Qvec|^2 \right)
+ C|\Qvec|^2 \Qvec \right\},
\end{equation}
where $(\Qvec\Qvec)_{ik} = Q_{ij}Q_{jk}$ with $i$, $j$, $k=1, \, 2, \, 3$. The system~\eqref{eq:6} comprises five coupled nonlinear
elliptic partial differential equations.
We treat $A$, $B$, $C$, $L$ as fixed constants and vary $\lambda$.
In what follows, we drop the \emph{bars} and all statements are to be understood in terms of the re-scaled variables.
\section{The Variational Problem}
\label{sec:reduction}
\begin{figure}[t]
\centering
\includegraphics[height=5 cm]{square-inclusion.pdf}
\caption{The domain~$\Omega$.}
\label{fig:domain}
\end{figure}
We take the rescaled domain~$\Omega\subseteq\Rr^2$ to be a truncated square with a
square inclusion. More precisely, for fixed~$0 < \rho < 1$, we define
\begin{equation} \label{eq:domain}
\Omega := \left\{(x, \, y)\in\Rr^2\colon |x| < 1 - \varepsilon, \ |y| < 1 - \varepsilon,
\ \rho < |x+y| < 1, \ \rho < |x-y| < 1 \right\} \! .
\end{equation}
The boundary, $\partial\Omega$, has two components, an inner boundary and an outer boundary.
The inner boundary, $\Gamma_{\mathrm{in}}$, is a square whose diagonals are parallel
to the coordinate axes, with side length~$\sqrt{2}\rho$.
The outer boundary, $\Gamma_{\mathrm{out}}$, consists of four ``long''
edges~$C_1, \, \ldots, \, C_4$, parallel to the lines~$y = x$ and~$y = -x$, and four
``short'' edges~$S_1, \, \ldots, \, S_4$, of length~$2\varepsilon$, parallel to the
$x$ and $y$-axes respectively. The long edges~$C_i$ are labeled counterclockwise
and $C_1$ is the edge contained in the first quadrant, i.e.
\[
C_1 := \left\{(x, \, y)\in\Rr^2\colon x + y = 1, \ \varepsilon \leq x \leq 1 - \varepsilon \right\} \! .
\]
The short edges~$S_i$
are also labeled counterclockwise and
\[
S_1 := \left\{(1 - \varepsilon, \, y)\in\Rr^2\colon |y|\leq \varepsilon\right\} \! .
\]
The domain is illustrated in Figure~\ref{fig:domain}.
We work with Dirichlet conditions on $\partial\Omega$. To mimic the isotropic inclusion,
we impose \emph{isotropic} boundary conditions on the inner boundary~$\Gamma_{\mathrm{in}}$,
that is, we require
\begin{equation}\label{eq:bc0}
\Qvec(\rvec) = \Qvec_{\mathrm{b}}(\rvec) := 0 \qquad \textrm{for } \rvec\in \Gamma_{\mathrm{in}}.
\end{equation}
We impose \emph{tangent} uniaxial Dirichlet conditions on the long edges, $C_1, \, \ldots, \, C_4$.
We fix $\Qvec = \Qvec_{\mathrm{b}}$ on $C_1, \, \ldots, \, C_4$ where
\begin{equation} \label{eq:bc1}
\Qvec_{\mathrm{b}}(\rvec) := \begin{cases}
s_+\left( \nvec_1 \otimes \nvec_1 - \dfrac{\Ivec}{3} \right) & \textrm{for } \rvec \in C_1 \cup C_3 \\
s_+\left( \nvec_2 \otimes \nvec_2 - \dfrac{\Ivec}{3} \right) & \textrm{for } \rvec \in C_2 \cup C_4;
\end{cases}
\end{equation}
and
\[
\nvec_1 := \frac{1}{\sqrt{2}}\left(-1, \, 1\right), \qquad \nvec_2 := \frac{1}{\sqrt{2}}\left(1, \, 1 \right).
\]
The Dirichlet condition on the short edges is defined in terms of a
function~$g\colon [-\varepsilon, \, \varepsilon]\to [-s_+/2, s_+/2]$.
We assume that~$g$ is smooth (at least of class~$C^1$), odd
(i.e. $g(-s) = -g(s)$ for any~$s$), and satisfies $g(\varepsilon) = s_+/2$;
for instance, an admissible choice for~$g$ is
\[
g(s) := \frac{s_+}{2\varepsilon} s \qquad
\textrm{for } -\varepsilon \leq s \leq\varepsilon.
\]
We fix $\Qvec = \Qvec_{\mathrm{b}}$ on $S_1, \, \ldots, \, S_4$ where
\begin{equation} \label{eq:bc2}
\Qvec_{\mathrm{b}} := \begin{cases}
g(y) \left(\nvec_1 \otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right) - \dfrac{s_+}{6}\left(2 \hat{\mathbf{z}}\otimes\hat{\mathbf{z}}
- \nvec_1\otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right) & \textrm{on } S_1\cup S_3, \\
g(x)\left(\nvec_1 \otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right) - \dfrac{s_+}{6}\left(2 \hat{\mathbf{z}}\otimes\hat{\mathbf{z}}
- \nvec_1\otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right) & \textrm{on } S_2\cup S_4.
\end{cases}
\end{equation}
Given the Dirichlet conditions~\eqref{eq:bc0},~\eqref{eq:bc1} and~\eqref{eq:bc2},
we define our admissible space to be
\begin{equation} \label{eq:admissible}
\mathscr{A} := \left\{ \Qvec \in W^{1,2}\left(\Omega, \, S_0 \right)\!\colon \Qvec = \Qvec_{\mathrm{b}}~\textrm{on} ~\partial \Omega \right\}.
\end{equation}
We look for critical points of the re-scaled functional~\eqref{eq:rescaled} of the form
\begin{equation} \label{eq:d1}
\begin{split}
\Qvec(x, \, y) &= q_1(x, \, y) \left(\nvec_1 \otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right) +
q_3(x, \, y) \left(2 \hat{\mathbf{z}}\otimes\hat{\mathbf{z}} - \nvec_1\otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right)
\end{split}
\end{equation}
subject to the boundary conditions
\begin{equation} \label{eq:d2}
q_1(x, \, y) = q_{1,\mathrm{b}} (x, \, y) :=
\begin{cases}
0 & \textrm{on } \Gamma_{\mathrm{in}} \\
{s_+}/{2} & \textrm{on } C_1\cup C_3 \\
- {s_+}/{2} & \textrm{on } C_2\cup C_4 \\
g(y) & \textrm{on } S_1\cup S_3 \\
g(x) & \textrm{on } S_2\cup S_4;
\end{cases}
\end{equation}
and
\begin{equation} \label{eq:d3}
q_3(x, \, y) = q_{3,\mathrm{b}} (x, \, y) :=
\begin{cases}
0 & \textrm{on } \Gamma_{\mathrm{in}} \\
-{s_+}/{6} & \textrm{on } \Gamma_{\mathrm{out}}.
\end{cases}
\end{equation}
For solutions of the form~\eqref{eq:d1}, the LdG Euler-Lagrange system (\ref{eq:6}) reduces to
\begin{equation} \label{eq:d4}
\begin{aligned}
\Delta q_1 &= \frac{\lambda^2}{L}\left\{A q_1 + 2B q_1 q_3 +
2C\left(q_1^2 + 3q_3^2\right) q_1 \right\} \\
\Delta q_3 &= \frac{\lambda^2}{L}\left\{A q_3 + B \left(\frac{1}{3}q_1^2 - q_3^2\right) +
2C\left(q_1^2 + 3q_3^2\right) q_3\right\} \! .
\end{aligned}
\end{equation}
The partial differential equations \eqref{eq:d4} are precisely the Euler-Lagrange
equations associated with the functional
\begin{equation} \label{eq:J}
J_\lambda[q_1, q_3]: = \int_{\Omega} \left( |\nabla q_1|^2 + 3 |\nabla q_3|^2
+ \frac{\lambda^2}{L} F(q_1,\, q_3)\right) \d A,
\end{equation}
where~$F$ is the polynomial potential given by
\begin{equation} \label{F}
F(q_1, \,q_3) := A (q_1^2 + 3 q_3^2) + 2 B \, q_3 \,(q_1^2 - 2 q_3^2)
+ C(q_1^2 + 3 q_3^2)^2 - F_{\min}
\end{equation}
and~$F_{\min} := As_+^2/3 - 2Bs_+^3/27 + Cs_+^4/9$ is a constant chosen so
that $\inf F=0$. By solving the criticality conditions~$\nabla_{(q_1, q_3)} F = 0$,
we find that~$F$ has exactly four critical points in the $(q_1, \, q_3)$-plane:
the origin~$(0, \, 0)$, which is a local maximum,
and the points
\begin{equation} \label{critical_points}
\pvec_1 := (-s_+/2, \, -s_+/6), \qquad \pvec_2 := (s_+/2, \, -s_+/6),
\qquad \pvec_3 := (0, \, s_+/3),
\end{equation}
which are global minima. These critical points are illustrated in Figure~\ref{fig:criticalpoints}.
\begin{figure}[t]
\centering
\includegraphics[height=4.4cm]{criticalpoints.pdf}
\caption{The four critical points of the
potential~$F$, which is defined by Eq.~\ref{F}. The dashed lines indicate the
``transition costs'' that are defined by Eq.~\ref{costs}}
\label{fig:criticalpoints}
\end{figure}
\begin{proposition} \label{prop:1}
We have a critical point $(q_1^s, q_3^s)$, of the functional (\ref{eq:J}) in the admissible space (\ref{eq:admissible}), subject to the boundary conditions (\ref{eq:d2}) and (\ref{eq:d3}), such that $q_1 = 0$ on $x = 0$ and $y = 0$. This in turn defines a LdG critical point of the form (\ref{eq:d1}), referred to as a Well Order Reconstruction ``WORS'' critical point for a square with an isotropic inclusion.
\end{proposition}
\begin{proof}
We follow the ideas in \cite{majcanevarispicer2017} and minimize the functional $J[q_1, \, q_3 ]$ on a quadrant of the rotated rescaled square with an isotropic inclusion, as defined in (\ref{eq:domain}).
For the minimization problem on the quadrant, we need additional boundary conditions on the square diagonals. We impose the additional boundary condition that $q_1 = 0$ on the square diagonals $x = 0$ and $y = 0$. Further, we impose $\frac{\partial q_3}{\partial \mathbf{n}} = 0$ on $x = 0$ and $y = 0$, where $\mathbf{n}$ is the unit normal to the diagonals. We can prove the existence of a minimizer $\left(q_1^*, q_3^* \right)$ of $J[q_1, \, q_3 ]$ on the quadrant in $W^{1,2}$, subject to these boundary conditions, from the direct method in the calculus of variations \cite{evans}. We define $q_1^s$ on the square by an odd reflection of $q_1^*$ about the square diagonals and $q_3^s$ by an even reflection of $q_3^*$ about the square diagonals. By using the same arguments as in \cite{majcanevarispicer2017}, we can check that $\left(q_1^s, q_3^s \right)$ is a critical point of $J[q_1, \, q_3 ]$ on the square with an isotropic inclusion, with the property $q_1 =0$ on $x = 0$ and $y=0$. We label this as the ``Well Order Reconstruction Solution''.
\end{proof}
We define the WORS as being a LdG critical point given by
\begin{equation}
\label{eq:wors}
\begin{split}
\Qvec_s = q_1^s(x, \, y) \left(\nvec_1 \otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right) +
q_3^s(x, \, y) \left(2 \hat{\mathbf{z}}\otimes\hat{\mathbf{z}} - \nvec_1\otimes \nvec_1 - \nvec_2\otimes \nvec_2 \right)
\end{split}
\end{equation}
where the pair $(q_1^s, q_3^s)$ is defined above in Proposition~(\ref{prop:1}).
There is an important distinction between the WORS for a square domain with and without an isotropic inclusion. In \cite{majcanevarispicer2017}, the authors study the WORS on a square domain without an isotropic inclusion and hence, only have the tangent uniaxial Dirichlet conditions (\ref{eq:bc1}) on the outer square edges in which case, we can have a WORS solution with constant $q_3$ at a special temperature defined by $A=-\frac{B^2}{3C}$. In this case, the WORS analysis reduces to a scalar variational problem as studied in \cite{majcanevarispicer2017}. In the case of a square with an isotropic inclusion, the boundary conditions for $q_3$ on the inner and outer square do not match and hence, we have inhomogeneous profiles for both $q_1$ and $q_3$ for all values of $A$, making this a harder problem. Next, we have a uniqueness result following the same arguments as in \cite{majcanevarispicer2017} and \cite{lamy2014}.
\begin{proposition} \label{prop:2}
The WORS defined in (\ref{eq:wors}) is the unique LdG critical point (and hence, globally stable) for either $\lambda$ sufficiently small or for $\rho$ sufficiently close to $1$ i.e. for either very small squares or for squares with inclusions with the aspect ratio approaching unity.
\end{proposition}
\begin{proof}
The proof follows by the arguments in Proposition~$4.2$ of \cite{lamy2014},
provided that we are able to bound the Poincar\'e constant of $\Omega$
in terms of the geometric parameter~$\rho$.
Let $u\in H^1(\Omega)$ be any scalar function such that~$u = 0$ on~$\partial\Omega$;
we extend $u$ out of~$\Omega$ by zero. We consider the set
$K : = \{(x, y)\in\Omega\colon x\geq 0, \, y\geq 0, \, \rho < x + y < 1\}$
and define new variables $(s, \, t)$ by
\[
x = ts, \qquad y = t(1-s)
\]
for each $(x, y)\in K$. The variables $(s, t)$ vary in the range $s\in (0, \, 1)$,
$t\in (\rho, \, 1)$.
We compute the integral of $|u|^2$ over~$K$ with respect to the coordinates
$(s, t)$, and apply the fundamental theorem of calculus in
$t$-direction, using that $u=0$
for $t = 1$:
\begin{align*}
\int_K |u(\xvec)|^2 \, \d\xvec &=
\int_0^{1}\int_{\rho}^{1} t |u(s, t)|^2 \,\d t\,\d s
\leq \int_0^{1}\int_{\rho}^{1} t \abs{\int_{t}^1 \partial_\xi
u(s, \xi) \, \d\xi}^2 \,\d t\,\d s \\
&\leq \int_0^{1}\int_{\rho}^{1}\int_{t}^{1}
t(1 - t) \abs{\partial_\xi u(s, \xi)}^2 \, \d\xi \,\d t\,\d s
\end{align*}
The last inequality follows by the H\"older inequality.
Now, we have $|\partial_t u|^2 = |s\partial_x u + (1-s)\partial_yu|^2
\leq s |\partial_x u|^2 + (1-s) |\partial_y u|^2 \leq |\nabla u|^2$,
where $\nabla$ denotes the gradient with respect to~$(x, y)$.
Using this with $\rho \leq t \leq \xi$,
and reverting to the original coordinates $(x, \, y)$, we obtain
\begin{equation*}
\int_K |u(\xvec)|^2 \d\xvec \leq
(1-\rho)^2 \int_0^{1}\int_{\rho}^{1}
\xi \abs{\partial_t u(s, \xi)}^2 \, \d\xi\,\d s
\leq (1-\rho)^2 \int_K |\nabla u(\xvec)|^2 \d\xvec.
\end{equation*}
By repeating the same argument on the other quadrants, and by adding the resulting inequalities, we conclude that
\begin{equation} \label{poincare}
\int_{\Omega} |u(\xvec)|^2 \,\d\xvec \leq (1-\rho)^2 \int_{\Omega} |\nabla u(\xvec)|^2 \,\d\xvec.
\end{equation}
By an application of the maximum principle, as in~\cite[Proposition~3]{amaz},
we know that any solution~$\Qvec$ of~\eqref{eq:6} in the admissible class~\eqref{eq:admissible} is bounded, i.e.
$|\Qvec(\xvec)|\leq M$ for any~$\xvec\in\Omega$ and a constant~$M$
that only depends on the coefficients~$A$, $B$, $C$. Now, by repeating
verbatim the arguments in \cite[Lemma 8.2]{lamy2014}, and using
the Poincar\'e inequality~\eqref{poincare}, we conclude that the boundary
value problem~\eqref{eq:6}, \eqref{eq:bc0}, \eqref{eq:bc1},
\eqref{eq:bc2} has a unique solution, provided that
\[
(1-\rho)^2 \lambda^2 < \kappa L
\]
for some positive constant $\kappa$ that only depends on~$M$, $A$, $B$, $C$.
\end{proof}
\section{The limit of large domains}
\label{sec:gamma}
In the following proposition, we analyze the asymptotic behavior
of minimizers of~\eqref{eq:J} in the limit as~$\lambda\to+\infty$.
To this end, we need to introduce some notation.
We denote~$\qvec := (q_1, \, q_3)$ and define a metric~$d$ on the
$\qvec$-plane in the following way: for any two points~$\qvec_0$,
$\qvec_1\in\Rr^2$, we let
\begin{equation} \label{dist}
d(\qvec_0, \, \qvec_1) :=\inf\left\{\int_0^1 F^{1/2}(\qvec(t))
|\qvec^\prime(t)|\,\d t\colon \qvec\in C^1([0, \, 1]; \, \Rr^2),
\ \qvec(0) = \qvec_0, \ \qvec(1) = \qvec_1 \right\} \! .
\end{equation}
This is the geodesic distance associated with the Riemannian
metric~$F^{1/2}$. However, this metric is degenerate, in that
$F^{1/2}(\pvec_1) = F^{1/2}(\pvec_2) = F^{1/2}(\pvec_3) = 0$ for
$\pvec_1$, $\pvec_2$, $\pvec_3$ given by~\eqref{critical_points}.
Despite the degeneracy, it can be proved that the infimum in~\eqref{dist}
is actually achieved by a minimizing geodesic, for any~$\qvec_0$,
$\qvec_1\in\Rr^2$ (this follows by the arguments in
\cite[Lemma~9]{Sternberg}).
Let~$\mathcal{H}^1(E)$ denote the length of a set~$E\subseteq\Rr^2$
(or, more formally, its $1$-dimensional Hausdorff measure).
For every measurable subset~$E\subseteq\Omega$,
we denote by $\chi_E$ the characteristic function of~$E$
(i.e., $\chi_E(x) :=1$ for~$x\in E$, and~$\chi_E(x) := 0$ otherwise)
and by $\partial^* E$ the reduced boundary of~$E$, that is, the set
of points~$x\in\partial E$ such that the limit
\[
\nu_E(x) := \lim_{\rho\searrow 0}\frac{\D\chi_E(B_\rho(x))}{|\D\chi_E|(B_\rho(x))}
\]
exists and~$|\nu_E(x)|=1$. (Here $\D\chi_E$ stands for the
distributional derivative of~$\chi_E$, which is a measure, and
$|\D\chi_E|$ is the total variation measure; see, e.g., the book~\cite{evans}
for a detailed discussion on the distributional derivative.)
The reduced boundary is a subset $\partial^* E \subseteq\partial E$
with the following property:
\[
\mathcal{H}^1(\partial^* E\cap\Omega) = \sup\left\{
\int_E \mathrm{div} \, \varphi \,\d A \colon
\varphi\in C^1_{\mathrm{c}}(\Omega),
\ |\varphi| \leq 1 \textrm{ on } \Omega\right\} \! .
\]
(see, e.g., \cite[Section~14]{Simon-GMT}).
If~$E$ has a regular (say, piecewise~$C^1$) boundary,
by the Gauss-Green theorem the right-hand side of this formula reduces
to~$\mathcal{H}^1(\partial E\cap\Omega)$, and
indeed~$\partial^* E = \partial E$ in this case; however,
for a generic set~$E$ with non-regular boundary, we might
have~$\partial^* E \subsetneq \partial E$.
Finally, we set~$\qvec_{\mathrm{b}} := (q_{1,\mathrm{b}}, \, q_{3,\mathrm{b}})$,
where~$q_{1,\mathrm{b}}$, $q_{3,\mathrm{b}}$ are defined by
\eqref{eq:d2}, \eqref{eq:d3} respectively.
We let~$\qvec_\lambda := (q_1, \, q_3)$ be a minimizer of the
functional~\eqref{eq:J}, for~$\lambda >0$.
\begin{proposition} \label{prop:Gamma}
There exists a subsequence~$\lambda_j\nearrow +\infty$ such that
$\qvec_{\lambda_j}$ converges, in~$L^1(\Omega)$ and a.e., to a map of
the form
\[
\qvec_\infty = \sum_{k=1}^3 \pvec_k \, \chi_{E^*_k}.
\]
Here~$\pvec_1$, $\pvec_2$, $\pvec_3$ are defined
by~\eqref{critical_points}, and~$E^*_1$, $E^*_2$, $E^*_3$ are
measurable, pairwise disjoint sets such that
$\Omega = E^*_1\cup E^*_2 \cup E^*_3$. Moreover,
$E^*_1$, $E^*_2$, $E^*_3$ minimize the following functional:
\begin{equation} \label{opt_partition}
J_\infty[E_1, \, E_2, \, E_3] :=
\sum_{i, j=1}^3 d(\pvec_i, \, \pvec_j) \,
\mathcal{H}^1(\partial^* E_i\cap \partial^* E_j\cap\Omega)
+ \int_{\partial\Omega} d(\qvec_\infty(\rvec), \, \qvec_{\mathrm{b}}(\rvec))
\, \d\mathcal{H}^1(\rvec)
\end{equation}
among all possible choices of measurable, pairwise disjoint sets $E_1$,
$E_2$, $E_3$ such that $\Omega = E_1\cup E_2\cup E_3$.
\end{proposition}
The sets~$E^*_1$, $E^*_2$, $E^*_3$ give a partition of the domain $\Omega$,
and they are optimal, in the sense that they minimise
the functional~\eqref{opt_partition}. This functional
depends on the length of the transition
layers~$\partial^* E_i\cap\partial^* E_j$ and on~$d(\pvec_i, \, \pvec_j)$,
which represents the energy cost of a transition from the
state~$\pvec_i$ to~$\pvec_j$. The functional~\eqref{eq:J} also
contains a boundary term, which accounts for the possible presence
of boundary layers.
\begin{proof}[Proof of Proposition~\ref{prop:Gamma}]
This result can be shown using classical arguments in the theory of $\Gamma$-convergence.
More precisely, Proposition~\ref{prop:Gamma} follows by the main result in~\cite{Baldo}
(see also~\cite[Theorem~3.9]{FonsecaTartar} or~\cite[Theorem~7.20]{Braides}
for similar results). The analysis in~\cite{Baldo, FonsecaTartar} does not take into account
the presence of boundary conditions, such as~\eqref{eq:d2}--\eqref{eq:d3}.
However, these can be included by straightforward modifications of the arguments,
as indicated in~\cite[Section~4.2.1 and Theorem~7.10]{Braides}.
\end{proof}
Let us introduce the transition costs
\begin{equation}\label{costs}
\begin{array}{l l}
c_1 := d(\mathbf{o}, \, \pvec_3), & \qquad c_2 := d(\mathbf{o}, \, \pvec_1) = d(\mathbf{o}, \, \pvec_2), \\
c_3 := d(\pvec_1, \, \pvec_3) = d(\pvec_2, \, \pvec_3), & \qquad c_4 := d(\pvec_1, \pvec_2), \\
\end{array}
\end{equation}
where $\mathbf{o} :=(0, \, 0)$ is the origin in the $(q_1, \, q_3)$-plane
and~$d$ is the intrinsic distance defined by~\eqref{dist}.
These costs $c_1$, $c_2$, $c_3$, $c_4$ are functions of~$A$, $B$, $C$.
We have used the symmetry of the function~$F$, given by~\eqref{F}, to deduce that
$d(\mathbf{o}, \, \pvec_1) = d(\mathbf{o}, \, \pvec_2)$ and
$d(\pvec_3, \, \pvec_1) = d(\pvec_3, \, \pvec_2)$.
By analyzing the possible configurations of~$(E_1, \, E_2, \, E_3)$,
we can identify three candidate minimizers for ~\eqref{opt_partition}
and compute their energy as a function of the transition costs~\eqref{costs}.
For the sake of simplicity, in what follows we assume that~$\varepsilon =0$, i.e.
no truncation of the domain~$\Omega$ has been made. This is acceptable because,
when~$\varepsilon$ is small, the contribution of the truncated edges to
the boundary integral in~\eqref{opt_partition} is negligible.
\begin{itemize}
\item A configuration with $\qvec = \pvec_1$ on~the first and third quadrant, and~$\qvec = \pvec_2$
on the second and fourth quadrant (i.e., $E_1 = \{(x, \, y)\in\Omega\colon xy\geq 0\}$,
$E_2 = \{(x, \, y)\in\Omega\colon xy< 0\}$, $E_3 =\emptyset$).
This configuration corresponds to \emph{the $\lambda\to+\infty$ limit of the WORS}.
It has transition layers from the isotropic state to~$\pvec_1$ or~$\pvec_2$
over the whole of the inner boundary~$\Gamma_1$,
and transition layers~$\pvec_1\to\pvec_2$ on the diagonals.
Using (\ref{opt_partition}), the energy of this WORS-like configuration in the $\lambda\to+\infty$ limit is given by
\[
J_\infty(\mathrm{WORS}) = 4\sqrt{2} \rho c_2 +
4\left(1 - \rho \right)c_4.
\]
\item Two configurations related by symmetry, with $\qvec=\pvec_1$,
respectively $\qvec=\pvec_2$ over almost the entire domain ~$\Omega$. Equivalently, in terms of the~$E_k$'s,
these configurations are given by $E_1=\Omega$, $E_2=E_3=\emptyset$
and $E_2=\Omega$, $E_1=E_3=\emptyset$, respectively. These configuration have a transition layer
at the inner boundary, from the isotropic to a uniaxial state (either $\mathbf{p}_1$ or $\mathbf{p}_2$), and two
\emph{boundary transition layers} on the edges~$C_2$, $C_4$ respectively (or $C_1$, $C_3$),
to account for the boundary conditions~\eqref{eq:d2}--\eqref{eq:d3}. We refer to these states as \emph{BD} states, to abbreviate for boundary distortion, since they have two distinctive edge transition layers along a pair of parallel outer square edges.
These two BD configurations have the same energy given by
\[
J_\infty(\mathrm{BD}) = 4\sqrt{2}\, \rho c_2 + 2\sqrt{2}c_4.
\]
\item A configuration with~$\qvec=\pvec_3$ in a neighbourhood of the inner boundary,
surrounded by the same cross structure as in the WORS, that is,
\begin{align*}
E_3 &= \left\{(x, \, y)\in\Omega\colon |x + y|\leq \rho + \eta, \
|x - y|\leq \rho + \eta \right\} \! , \\
E_1 &= \left\{(x, \, y)\in\Omega\setminus E_3\colon xy\geq 0 \right\} \! , \\
E_2 &= \left\{(x, \, y)\in\Omega\setminus E_3\colon xy< 0 \right\} \! .
\end{align*}
In this \emph{``escaped''} configuration, the isotropic core is surrounded by a uniaxial region,
$E_3$, with positive order parameter or equivalently $q_3>0$. This may be energetically convenient, if
the transition $\pvec_1\to\pvec_2$ is energetically very expensive,
compared to the transitions $\mathbf{o}\to\pvec_3$ and
$\pvec_3\to\pvec_1$, $\pvec_3\to\pvec_2$. If this is the case, the escaped configuration
reduces the length of the (very expensive) transition layer along the diagonals,
at the price of introducing a new transition layer near the core.
The overall cost of this configuration is given by
\[
J_\infty(\mathrm{ESC}) = 4\sqrt{2} \rho c_1 +
4\sqrt{2}\left( \rho + \eta\right) c_3 +
4\left(1 - \rho - \eta\right)c_4.
\]
\end{itemize}
A configuration that has an island of the state $\mathbf{p}_3$ around the core,
surrounded by a constant state~$\pvec_1$ or~$\pvec_2$,
always has greater energy than the competing BD-configuration. This follows
from the triangle inequality for the metric~$d$, which gives~$c_1 + c_3 \geq c_2$.
Therefore, we will not consider this configuration here. Other configurations, that have
``two-steps transition layers'' e.g. a transition of the form $\pvec_1\to\pvec_3\to\pvec_2$
occurring along the diagonals, can be ruled out for the same reason.
Configurations with non-straight transition layers can also be ruled out,
as the energy per transition layer is proportional to the length of the transition layer
and a non-straight transition layer between two points has greater length than a straight layer.
Now, we can compare the energy costs of these configurations.
\begin{itemize}
\item $J_\infty(\mathrm{WORS})<J_\infty(\mathrm{BD})$ if and only if $\rho > 1 - \sqrt{2}/2$.
\item We compare $J_\infty(\mathrm{ESC})$ with $J_\infty(\mathrm{WORS})$.
By substituting the explicit expressions for the two energies, we see that the inequality
$J_\infty(\mathrm{ESC})<J_\infty(\mathrm{WORS})$ is equivalent to
\[
(\sqrt{2}c_3 - c_4)\eta < \sqrt{2}(c_2 - c_1 - c_3) \rho
\]
and the right-hand side is always non-positive, due to the triangle inequality.
Thus, for this inequality to be satisfied we must have $c_4 > \sqrt{2}c_3$.
By imposing the geometric constraint that $\eta \leq 1 - \rho$,
we obtain
\[
\frac{\sqrt{2}(c_1 - c_2 + c_3)}{(c_4 - \sqrt{2}c_3)} \rho
\leq 1 - \rho.
\]
By straightforward algebraic manipulations, we conclude that
the inequality $J_\infty(\mathrm{ESC})<J_\infty(\mathrm{WORS})$
holds if and only if
\begin{equation*}
\begin{cases}
c_4 > \sqrt{2} c_3 \\
0 < \rho < R_1
\end{cases}
\quad \textrm{or} \quad
\begin{cases}
c_4 > \sqrt{2} c_3 \\
R_1 < 0,
\end{cases}
\end{equation*}
where
\begin{equation*}
R_1 := \frac{c_4 - \sqrt{2}c_3}{\sqrt{2}c_1 - \sqrt{2}c_2 + c_4}.
\end{equation*}
\item Arguing in a similar way, we conclude that
$J_\infty(\mathrm{ESC}) < J_\infty(\mathrm{BD})$ if and only if
\begin{equation*}
\begin{cases}
c_4 > \sqrt{2} c_3 \\
c_2 > c_1 \\
R_2 < \rho < 1 \\
\end{cases}
\qquad \textrm{or} \qquad
\begin{cases}
c_4 > \sqrt{2} c_3 \\
c_1 > c_2 \\
0 < \rho < R_2, \\
\end{cases}
\end{equation*}
where
\begin{equation*}
R_2 := \frac{c_4 - 2 c_3}{2 c_1 - 2 c_2}.
\end{equation*}
\end{itemize}
\section{Numerics}
\label{sec:numerics}
Let $\bar{\lambda}^2 = \dfrac{2C \lambda^2}{L}$, and we take
\begin{equation}
B = 0.64 \times 10^4 \mathrm{Nm}^{-2}, \quad C = 0.35 \times 10^4 \mathrm{Nm}^{-2}, \quad A = - \frac{B^2}{3C}, \quad \bar{\lambda}^2 = 200
\end{equation}
throughout this section if not stated differently. We choose this special value of $A$ because in the absence of a square inclusion, the WORS has a particularly simple parametrization in terms of a single variable $q_1$ and constant $q_3$ (see (\ref{eq:d1})) at this temperature \cite{majcanevarispicer2017}. In fact, this is the only temperature for which the system (\ref{eq:d4}) has a solution with constant $q_3$. Of course, we cannot have solutions with constant $q_3$ for this model problem because of the inhomogeneous boundary conditions but we still regard this temperature as a special reference point which allows for easy comparison with the results in \cite{majcanevarispicer2017}. We assume that $\bar{\lambda}^2 = 200$ is large enough for the asymptotic estimates in Section \ref{sec:gamma} to hold; we have also checked the trends with larger values of $\bar{\lambda}^2$ and they are qualitatively the same.
\subsection{Transition Costs}
First, we compute the transition costs defined in (\ref{costs}).
According to standard arguments in Riemannian geometry,
the intrinsic distance $d(\qvec_0, \, \qvec_1)$ defined in (\ref{dist}) can be calculated alternatively as
\begin{equation*} \label{dist_1}
d(\qvec_0, \, \qvec_1) =\inf\left\{ \left( \int_0^1 F(\qvec(t))|\qvec^\prime(t)|^2 \,\d t \right)^{1/2} ~\bigg|~ \qvec\in C^1([0, \, 1]; \, \Rr^2), \ \qvec(0) = \qvec_0, \ \qvec(1) = \qvec_1 \right\} \, .
\end{equation*}
The profiles of geodesic $\qvec(t) = (q_1(t), q_3(t))$ in each case of (\ref{costs}) are shown in Fig. \ref{F1}; these are the optimal profiles which minimise the intrinsic distance between the four critical points $\mathbf{o}, \mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3$ and the associated costs are given below:
\begin{equation}
c_1 = 22.3067, \quad c_2 = 34.7378, \quad c_3 = 41.6817, \quad c_4 = 60.2955.
\end{equation}
\begin{figure}[!h]
\centering
\includegraphics[width = 0.7 \linewidth]{./Fig_c1234.pdf}
\caption{The profiles of geodesic $\qvec(t) = (q_1(t), q_3(t))$ for different $\qvec_0$ and $\qvec_1$ ($A = - \frac{B^2}{3C}$).}\label{F1}
\end{figure}
Hence,
\begin{equation}
c_2 > c_1, \quad c_4 > \sqrt{2} c_3, \quad R_2 > R_1.
\end{equation}
In view of the discussion in the previous section, $$\min\{J_{\infty}({\rm WORS}), \, J_{\infty}({\rm BD})\} < J_{\infty}({\rm ESC})$$ requires that
\begin{equation}\label{ESC_no_min_1}
R_2 < \rho < R_1, ~~\text{if}~ c_4 > \sqrt{2} c_3 ~\text{and}~ c_2 > c_1,
\end{equation}
which cannot hold since $R_2 > R_1$. Therefore, ESC cannot be energetically preferred to either the WORS or BD for this choice of parameters.
Next, we perform a systematic search of the parameter space in terms of $A$, for fixed $B$ and $C$; the transition costs $c_i$ and the
quantities $R_i$, as a function of the reduced temperature $t = \frac{27 AC}{B^2}$, are numerically computed and plotted in Fig. \ref{F2}. We note that $c_2 > c_1$ and $R_2 > R_1$ hold true in all the numerical simulations. If $c_4 > \sqrt{2}c_3$, then the same arguments as above apply to exclude the ESC as a competitor for an energy minimizer; if $c_4 <\sqrt{2}c_3$, then the ESC cannot be energetically preferred to the WORS or BD according to the estimates in the previous section.
\begin{figure}[!htb]
\centering
\includegraphics[width = 0.75 \linewidth]{./Fig_ciRi.pdf}
\caption{The value of $c_i$ and $R_i$ as a function of $t = \frac{27 AC}{B^2}$ }\label{F2}
\end{figure}
\subsection{WORS and BD on a square with an isotropic core}
For the following simulations, we take
\begin{equation}
\Omega = \{(x, y) \in \mathbb{R}^2\colon \rho < \max\{|x|, \, |y|\} < 1 \},
\end{equation}
as the computational domain and seek numerical solutions of the form
\begin{equation}\label{q13}
\Qvec(x,y) = q_1(x, y)(\e_x \otimes \e_x - \e_y \otimes \e_y) + q_3(x, y) (2 \e_z \otimes \e_z - \e_x \otimes \e_x - \e_y \otimes \e_y),
\end{equation}
where $\e_x$, $\e_y$ and $\e_z$ are unit-vectors in the $x$-, $y$- and $z$-directions respectively, subject to the boundary conditions
\begin{equation}\label{BC-general}
\begin{aligned}
& \Qvec(x, y) = 0 \qquad \quad \text{on}~\Gamma_{\rm in}, \\
& \Qvec(x, \pm 1) = ~~\frac{s_{+}}{2} (\e_x \otimes \e_x - \e_y \otimes \e_y) - \frac{s_{+}}{6} (2 \e_z \otimes \e_z - \e_x \otimes \e_x - \e_y \otimes \e_y), \\
& \Qvec(\pm 1, y) = - \frac{s_{+}}{2} (\e_x \otimes \e_x - \e_y \otimes \e_y) - \frac{s_{+}}{6} (2 \e_z \otimes \e_z - \e_x \otimes \e_x - \e_y \otimes \e_y). \\
\end{aligned}
\end{equation}
For LdG critical points of the form (\ref{q13}), the Euler-Lagrange equations (\ref{eq:6}) reduce to
\begin{equation}\label{eq_q13}
\begin{cases}
\Delta q_1 = \bar{\lambda}^2 \left( \dfrac{A}{2C} q_1 + \dfrac{B}{C} q_1 q_3 + (q_1^2 + 3 q_3^2) q_1 \right) \\
\Delta q_3 = \bar{\lambda}^2 \left( \dfrac{A}{2C} q_3 + \dfrac{B}{C} (\frac{1}{3}q_1^2 - q_3^2) + (q_1^2 + 3 q_3^2) q_3 \right). \\
\end{cases}
\end{equation}
We use a standard finite-difference method and Newton's Method to solve the system of coupled partial differential equations (\ref{eq_q13}). We plot the profiles for $q_1$, $q_3$ and
biaxiality parameter $$\beta^2 = 1 - 6 \frac{\left( \tr(\Qvec^3) \right)^2}{\left( \tr(\Qvec^2) \right)^3}$$ in the WORS and BD for $\bar{\lambda}^2 = 200$ and $\rho = 0.2$ in Fig. \ref{BD_OR}.
The biaxiality parameter $\beta^2 \in \left[0, 1 \right]$ for $\Qvec \neq \mathbf{0}$, $\beta^2$ vanishes when $\Qvec$ has two degenerate non-zero eigenvalues, and $\beta^2$ is unity when one of the eigenvalues vanishes and the corresponding $\Qvec$ is maximally biaxial \cite{majcanevarispicer2017}.
\begin{figure}[!hbt]
\centering
\includegraphics[width = 0.8\textwidth]{Fig_ORBD.pdf}
\caption{(a) WORS for $\bar{\lambda}^2 = 200$ and $\rho = 0.2$. Left to right: plot and contour plot of $q_1$, plot of $q_3$, and plot of biaxiality parameter $\beta^2$.
(b) BD for $\bar{\lambda}^2 = 200$ and $\rho = 0.2$. Left to right: plot and contour plot of $q_1$, plot of $q_3$, and plot of biaxiality parameter $\beta^2$.}\label{BD_OR}
\end{figure}
The WORS has a uniaxial cross with negative order parameter, connecting the vertices of the inner square and the outer square. The BD solution is distinguished by a pair of edge transition layers, localized near $x = \pm 1$ (or $y = \pm 1$). In both cases, $q_3$ decreases monotonically from zero on the inner boundary to $q_3 = -\frac{B}{6C}$ on the outer boundary.
We compare the free energies of BD and WORS for $\bar{\lambda}^2 = 200$ and various $\rho$ in Fig. \ref{BD_OR-energy}(a), which shows that WORS is energetically preferred for relatively large $\rho$. Indeed, the $\Gamma$-convergence argument in the previous section shows that, in the limit $\bar{\lambda}^2 \rightarrow \infty$, we have $J_{\infty}({\rm WORS}) < J_{\infty}({\rm BD})$ if and only if $\rho > 1 - \sqrt{2}/2$. Numerically, we compute the critical value $\rho_0(\bar{\lambda}^2)$, such that $J_{\bar{\lambda}^2}({\rm BD}) = J_{\bar{\lambda}^2}({\rm WORS})$ when $\rho = \rho_0(\bar{\lambda}^2)$, as a function of $\bar{\lambda}^2$ in Fig. \ref{BD_OR-energy}(b).
Qualitatively, we see that $\rho_0(\bar{\lambda}^2) \rightarrow 1 - \sqrt{2}/2$ when $\bar{\lambda}^2 \rightarrow \infty$, in agreement with the $\Gamma$-convergence results in the previous section.
Since WORS is the unique LdG critical point for either $\lambda$ sufficiently small or for $\rho$ sufficiently close to $1$, BD cannot be a critical point of the functional \eqref{eq:J} for either large $\rho$ or small $\bar{\lambda}^2$. Numerically, we find that for each $\bar{\lambda}^2$, there exists a critical value $\rho_1(\bar{\lambda}^2)$, for which BD is no longer a critical point of the functional \eqref{eq:J} when $\rho \geq \rho_1(\bar{\lambda}^2)$. This critical value $\rho_1(\bar{\lambda}^2)$ is found by increasing $\rho$ gradually till we cannot numerically obtain a BD solution with a BD-like initial guess, even with the deflation technique \cite{farrell2015deflation}.
For $\bar{\lambda}^2 = 100$, $\rho_1 \approx 0.28$, whilst for the $\bar{\lambda}^2 = 200$, $\rho_1 \approx 0.42$. $\rho_1(\bar{\lambda}^2)$ as a function of $\bar{\lambda}^2$ is shown in Fig. \ref{BD_OR-energy}(c). By adapting the arguments in \cite{lamy2014} to a truncated square annulus such as ours (see also the proof of Proposition~\ref{prop:2} for more details), we can show that
the LdG energy (\ref{eq:2}) is strictly convex for
$$ 1 - C_1\bar{\lambda}^{-1} < \rho < 1, $$
where $C_1$ is a positive constant independent of $\rho$ and $\bar{\lambda}^2$.
Therefore, the LdG energy has a unique critical point, which is the WORS, for $\rho$ in this range and as $\bar{\lambda}^2$ increases, this range becomes narrower as illustrated by the estimate above
\begin{figure}[!htb]
\centering
\includegraphics[width = \linewidth]{./Fig_BD_OR_lambda.pdf}
\caption{(a) the free energy of {\rm WORS} and {\rm BD} for various $\rho$ for $\bar{\lambda}^2 = 200$. (b) the critical value $\rho_0$, for which $J_{\bar{\lambda}^2}({\rm BD}) = J_{\bar{\lambda}^2}({\rm WORS})$ when $\rho > \rho_0$. (c)the critical value $\rho_1(\bar{\lambda}^2)$, for which BD is no longer a critical point of the functional \eqref{eq:J} when $\rho \geq \rho_1(\bar{\lambda}^2)$.}\label{BD_OR-energy}
\end{figure}
We test the stabilities of the WORS and BD by solving the gradient flow equations for $q_1$ and $q_3$ in $\Omega$ as shown below:
\begin{equation}\label{grad_q13}
\begin{cases}
\pp_t q_1 = \Delta q_1 - \bar{\lambda}^2 \left( \dfrac{A}{2C} q_1 + \dfrac{B}{C} q_1 q_3 + (q_1^2 + 3 q_3^2) q_1 \right) \\
\pp_t q_3 = \Delta q_3 - \bar{\lambda}^2 \left( \dfrac{A}{2C} q_3 + \dfrac{B}{C} (\dfrac{1}{3}q_1^2 - q_3^2) + (q_1^2 + 3 q_3^2) q_3 \right), \\
\end{cases}
\end{equation}
for $\bar{\lambda}^2 = 200$, subject to the Dirichlet boundary conditions~\eqref{eq:bc0}, \eqref{eq:bc1} and \eqref{eq:bc2} and different initial conditions. We use a standard finite-difference method for the spatial derivatives and the Crank-Nicolson scheme~\cite{iserles2009first} for time-stepping in the numerical simulations.
In Fig. \ref{OR}, we solve (\ref{grad_q13}) with a WORS-like initial condition as described below
\begin{equation}\label{eq:wors_ic}
q_1(x, y) =
\begin{cases}
~~ s_{+}/2 & \textrm{for } -|y| < x < |y| \\
- s_{+}/2 & \textrm{for } -|x| < y < |x|, \\
\end{cases}
\quad q_3(x, y) = -s_{+}/6, \quad \forall (x, y) \in \Omega
\end{equation}
for $\bar{\lambda}^2 = 200$ with $\rho = 0.02$ and $\rho = 0.1$, respectively. The dynamic evolutions of $q_1$ in both cases are shown in Fig. \ref{OR}. For both cases, $\rho$ is in the range for which $J({\rm WORS}) > J({\rm BD})$ according to Fig. \ref{BD_OR-energy}(a) and yet the dynamic evolutions are different for $\rho = 0.02$ and $\rho = 0.1$. For $\rho = 0.02$, the initial condition with the diagonal cross (see \eqref{eq:wors_ic}), evolves to BD, which indicates that WORS is unstable when $\rho$ is very small. However, for $\rho = 0.1$, the solution converges to the WORS although WORS has higher free energy than BD, which indicates that the WORS is metastable with a basin of attraction.
\begin{figure}[!htb]
\centering
\includegraphics[width = 0.8\linewidth]{Fig_OR_ini.pdf}
\caption{(a) The profiles of $q_1$ for t = 0, t = 0.2, t = 0.5 and t = 2($\rho = 0.02$, $\bar{\lambda}^2 = 200$).
(b) The profiles of $q_1$ for t = 0, t = 0.2, t = 0.5 and t = 2($\rho = 0.1$, $\bar{\lambda}^2 = 200$).
}\label{OR}
\end{figure}
We also solve (\ref{grad_q13}) with a BD-like initial condition for $\bar{\lambda}^2 = 200$ with $\rho = 0.4$ and $\rho = 0.44$. The dynamic evolutions of $q_1$ are displayed in Fig. \ref{BD}, for both cases. The previous discussions illustrate that BD ceases to be a critical point of the functional \eqref{eq:J} for $\rho \gtrsim 0.42$. We choose two values of $\rho$ that are at either end of this critical value. For $\rho = 0.4$, the numerical solution converges to BD, although BD has higher free energy than WORS, which indicates that the BD state is metastable. For $\rho = 0.44$, for which there is no BD-type critical point, the solution converges to WORS as expected.
\begin{figure}[!htb]
\centering
\begin{overpic}[width = 0.8 \linewidth]{Fig_BD_ini.pdf}
\end{overpic}
\caption{(a) The profiles of $q_1$ for t = 0, t = 0.3, t = 0.4 and t = 2
($\rho = 0.4$, $\bar{\lambda}^2 = 200$).
(b) The profiles of $q_1$ for t = 0, t = 0.3, t = 0.4 and t = 2
($\rho = 0.44$, $\bar{\lambda}^2 = 200$).
}\label{BD}
\end{figure}
\subsection{Decomposition of the Second Variation of the LdG energy}
The gradient flow simulations give us some information about the stabilities
of WORS and BD in the restricted class of $\Qvec$ that have the form (\ref{q13}). In the following, we consider the second variation of the LdG energy (\ref{eq:rescaled})
about the WORS and BD-solutions, for arbitrary perturbations with five degrees of freedom. As is standard in variational problems in the calculus of variations, a solution is locally stable if the second variation of the LdG energy is positive for all admissible perturbations and a solution is unstable if we can find a perturbation for which the second variation is negative.
Consider a perturbation about the WORS or BD solutions of the form $\mathbf{W} = \Qvec + \epsilon \mathbf{V}$, where $\mathbf{V}$ vanishes at the boundary. The second variation of the LdG energy is given by:
\begin{equation}
\delta^2 F(\mathbf{V}) = \int_{\Omega} \dfrac{\lambda^2}{L} \left( A |\mathbf{V}|^2 - 2 B Q_{ij}V_{jk}V_{ki} + C |\Qvec|^2|\mathbf{V}|^2 + 2 C (\Qvec \cdot \mathbf{V})^2 \right) + |\nabla \mathbf{V}|^2 \dd \x.
\end{equation}
We write $\mathbf{V}$ as (see \cite{majcanevarispicer2017})
\begin{equation}
\begin{aligned}
\mathbf{V}(x,y) & = v_1(x, y)(\e_x \otimes \e_x - \e_y \otimes \e_y) + v_2(x, y) (\e_x \otimes \e_y + \e_y \otimes \e_x) \\
& + v_3(x, y) (2 \e_z \otimes \e_z - \e_x \otimes \e_x - \e_y \otimes \e_y) \\
& + v_4(x, y)(\e_x \otimes \e_z + \e_z \otimes \e_x) + v_5(x, y)(\e_y \otimes \e_z + \e_z \otimes \e_y ), \\
\end{aligned}
\end{equation}
where we treat the functions, $v_1 \ldots v_5$, as perturbations in the five independent basis directions.
For LdG critical points with $q_2 = q_4 = q_5 = 0$, such as the WORS and BD solutions with a constant eigenframe, we have
\begin{equation}\label{Per_V}
\begin{aligned}
\delta^2 F(\mathbf{V}) = \int_{\Omega} & \bar{\lambda}^2 \Biggl( \frac{A}{C} (v_1^2 + v_2^2 + 3 v_3^2 + v_4^2 + v_5^2) \\
& - \frac{B}{C} \bigl( q_1 (v_4^2 - v_5^2) - 2 q_3 (v_1^2 + v_2^2) + 6q_3v_3^2 + q_3(v_4^2 + v_5^2) - 4q_1v_1v_3 \bigr) \\
& + 2 \left( q_1^2 + 3 q_3^2 \right)(v_1^2 + v_2^2 + 3 v_3^2 + v_4^2 + v_5^2) + 4 (q_1v_1 + 3 q_3 v_3)^2 \Biggr) \\
& + \left( 2 |\nabla v_1|^2 + 2 |\nabla v_2|^2 + 6 |\nabla v_3|^2 + 2 |\nabla v_4|^2 + 2 |\nabla v_5|^2 \right) \dd \x. \\
\end{aligned}
\end{equation}
where $v_i \in W_0^{1,2} \left(\Omega\right)$.
We can write (\ref{Per_V}) as
\begin{equation}
\delta^2 F(\mathbf{V}) = \delta^2 F(v_1, v_3) + \delta^2 F (v_2) + \delta^2 F(v_4) + \delta^2 F(v_5),
\end{equation}
where
\begin{equation}
\begin{aligned}
& \delta^2 F(v_1, v_3) = \int_{\Omega} \bar{\lambda}^2 \Biggl( \left( \frac{A}{C} + \frac{2B}{C} q_3 + 6 (q_1^2 + q_3^2) \right) v_1^2 + \left( \frac{3A}{C} - \frac{6B}{C}q_3 + 6 q_1^2 + 54 q_3^2 \right) v_3^2 \\
& \qquad \qquad \qquad \qquad \qquad + \left( \frac{4B}{C} q_1 + 24 q_1 q_3 \right) v_1v_3 \Biggr) + \left( 2 |\nabla v_1|^2 + 6 |\nabla v_3|^2 \right) \dd \x, \\
& \delta^2 F(v_2) = \int_{\Omega} \bar{\lambda}^2 \Biggl( \frac{A}{C} + \frac{2 B}{C} q_3 + 2 \left( q_1^2 + 3 q_3^2 \right) \Biggr) v_2^2 + 2 |\nabla v_2|^2 \dd \x, \\
& \delta^2 F(v_4) = \int_{\Omega} \bar{\lambda}^2 \Biggl( \frac{A}{C} - \frac{B}{C} (q_1 + q_3) + 2 \left( q_1^2 + 3 q_3^2 \right) \Biggr) v_4^2 + 2 |\nabla v_4|^2 \dd \x, \\
& \delta^2 F(v_5) = \int_{\Omega} \bar{\lambda}^2 \Biggl( \frac{A}{C} - \frac{B}{C} (q_3 - q_1) + 2 \left( q_1^2 + 3 q_3^2 \right) \Biggr) v_5^2 + 2 |\nabla v_5|^2 \dd \x. \\
\end{aligned}
\end{equation}
Define
\begin{equation*}
\begin{aligned}
& \mathcal{V}_{13} = \left \{ v_1 \left( \e_x \otimes \e_x - \e_y \otimes \e_y \right) + v_3 \left( 2 \e_z \otimes \e_z - \e_x \otimes \e_x - \e_y \otimes \e_y \right) \right \}, \\
& \mathcal{V}_2 = \left \{v_2 \left( \e_x \otimes \e_y + \e_y \otimes \e_x \right) \right \},~~\mathcal{V}_4 = \left \{v_4 \left( \e_x \otimes \e_z + \e_z \otimes \e_x \right) \right \},~~\mathcal{V}_5 = \left \{v_5 \left( \e_y \otimes \e_z + \e_z \otimes \e_y \right) \right \}, \\
\end{aligned}
\end{equation*}
which are subspaces of $S_0$. We can consider perturbations in each subspace respectively. The perturbations in $\mathcal{V}_{13}$ do not distort the constant eigenframes of the WORS or BD solutions, the perturbations in $\mathcal{V}_{2}$ are in-plane perturbations of the eigenframe and the perturbations in $\mathcal{V}_4$ and $\mathcal{V}_5$ are out-of-plane perturbations of the eigenframe.
Firstly, we consider $\delta^2 F(v_1, v_3)$, which can be regarded as a functional of $v_1$ and $v_3$, for given $q_1$ and $q_3$. We can minimize $\delta^2 F(v_1, v_3)$ by solving the gradient flow equations
\begin{equation}\label{grad_v13}
\begin{cases}
\dfrac{\pp v_1}{\pp t} = \Delta v_1 - \bar{\lambda}^2 \left( \dfrac{1}{2}C_{11}(x, y) v_1 + \dfrac{1}{4}C_{13}(x, y) v_3 \right) \\
\dfrac{\pp v_3}{\pp t} = \Delta v_3 - \bar{\lambda}^2 \left( \dfrac{1}{24} C_{13}(x, y) v_1 + \dfrac{1}{12} C_{33}(x, y) v_3 \right), \\
\end{cases}
\end{equation}
where
\begin{equation}
\begin{aligned}
& C_{11}(x, y) = \frac{A}{C} + \frac{2B}{C} q_3 + 6 q_1^2 + 6 q_3^2, \quad C_{13}(x, y) = \frac{4 B}{C} q_1 + 24 q_1 q_3, \\
& C_{33}(x, y) = \frac{3A}{C} - \frac{6B}{C}q_3 + 6 q_1^2 + 54q_3^2. \\
\end{aligned}
\end{equation}
For $\bar{\lambda}^2 = 200$, WORS is a critical point for
$0 \leq \rho < 1$, but is unstable for small-$\rho$.
In Fig.~\ref{C123_v13}(a), we plot $C_{11}(x, y), C_{13}(x, y)$ and $C_{33}(x, y)$ for the WORS solution, using the numerically computed $q_1$ and $q_3$ corresponding to the WORS with $\rho = 0.02$. It is relatively straightforward to find $v_1$ and $v_3$ such that
$\delta F(v_1, v_3) < 0$. An example is shown in Fig. \ref{C123_v13}(b).
\begin{figure}[!h]
\centering
\begin{overpic}[width = \linewidth]{Fig_v13_OR.pdf}
\end{overpic}
\caption{(a) The profiles of $C_{11}(x, y)$, $C_{13}(x, y)$ and $C_{33}(x, y)$ for WORS with $\rho = 0.02$. (b) The profiles of $v_1$ and $v_3$ in a perturbation s.t. $\delta^2 F(v_1, v_3) < 0$ for WORS with $\rho = 0.02$.
(c) The profiles of $C_{11}(x, y)$, $C_{13}(x, y)$ and $C_{33}(x, y)$ for WORS with $\rho = 0.2$. (d) The profiles of $v_1$ and $v_3$ in numerical solution of (\ref{grad_v13}) for WORS with $\rho = 0.2$.}\label{C123_v13}
\end{figure}
Indeed, in this case, $\delta^2 F(v_1, v_3)$ is not bounded from below, because if we have $\delta^2 F(v_1^{*}, v_3^{*}) < 0$ for a particular choice of $v_1^*$ and $v_3^*$, then $\delta^2 F(c v_1^{*}, c v_3^{*}) = c^2 \delta^2 F(v_1^{*}, v_3^{*})$ for every constant $c$, which can be arbitrarily negative by choosing $c$ to be sufficiently large. As expected, the optimal profiles are localised near the diagonals, as the WORS loses stability by losing the diagonal cross and hence, the optimal perturbations have $q_1 \neq 0$ on the square diagonals to reduce the LdG energy of the perturbed state compared to the WORS.
Next we consider the WORS with $\rho = 0.2$; the corresponding profiles of $C_{11}(x, y), C_{13}(x, y)$ and $C_{33}(x, y)$ are shown in Fig. \ref{C123_v13}.(c).We solve the the gradient flow equations (\ref{grad_v13}) with random initial data and the numerical solutions of (\ref{grad_v13}), shown in Fig. \ref{C123_v13}(d), converge to $v_1 = v_3 = 0$. This indicates that $\delta^2 F(v_1, v_3) \geq 0$. We find that $\delta^2 F(v_1, v_3) \geq 0$ for the WORS with $\rho \geq 0.05$, for
$\bar{\lambda}^2 = 200$. This is consistent with the numerical simulations in \cite{martinrobinson2017} and \cite{majcanevarispicer2017} which suggest that the WORS solution loses stability with respect to BD-like solutions in the restricted class of solutions (\ref{eq:d1}) as either $\bar{\lambda}^2$ increases or $\rho$ decreases.
Similarly, we consider $\delta^2 F(v_1, v_3)$ for the BD solution, which is a critical point of the system (\ref{eq:d4}) for small-$\rho$. The numerical profiles of $C_{11}(x, y), C_{13}(x, y)$ and $C_{33}(x, y)$ for the BD-solution, with $\rho = 0.02$ and $\rho = 0.2$, are shown in Fig. \ref{BD_C123_v13}(a) and (c). In both cases, the numerical solutions of (\ref{grad_v13}), as displayed in Fig. \ref{BD_C123_v13}(b) and (d), converge to $v_1 = v_3 = 0$, which indicates that $\delta^2 F(v_1, v_3) \geq 0$ for the BD-solution, if BD is a critical point of the system. However, this is not a reflection on the stability of the BD solution with respect to arbitrary perturbations.
\begin{figure}[!ht]
\centering
\begin{overpic}[width = \linewidth]{Fig_v13_BD.pdf}
\end{overpic}
\caption{(a) The profiles of $C_{11}(x, y)$, $C_{13}(x, y)$ and $C_{33}(x, y)$ for BD with $\rho = 0.02$. (b) The profiles of $v_1$ and $v_3$ in numerical solution of (\ref{grad_v13}) for BD with $\rho = 0.02$.
(c) The profiles of $C_{11}(x, y)$, $C_{13}(x, y)$ and $C_{33}(x, y)$ for BD with $\rho = 0.2$. (d) The profiles of $v_1$ and $v_3$ in numerical solution of (\ref{grad_v13}) for BD with $\rho = 0.2$.}\label{BD_C123_v13}
\end{figure}
Next, we consider $\delta^2 F(v_2)$. According to our numerical results,
\begin{equation}
C_2(x, y) = \dfrac{A}{C} + \dfrac{2 B}{C} q_3 + 2 ( q_1^2 + 3 q_3^2) \leq 0
\end{equation}
for both the WORS and BD solutions, for $\forall \rho$. The profiles of $C_2(x, y)$ for WORS and BD with $\rho = 0.2$ are shown in Fig. \ref{Coe_v2}(a) and (c). Indeed, we can check that for $s_{+} = B/C$, $C_2(x, y) = 0$ if $(q_1, q_3) = (\pm s_{+}/2, -s_{+}/6)$. Since $C_2(x, y) \leq 0$, it is relatively straightforward to find $v_2$ for which $\delta^2 F(v_2) < 0$, for both WORS and BD-solutions when $\rho$ is small. Fig. \ref{Coe_v2}(b) is an example of $v_2$ s.t. $\delta^2 F(v_2) < 0$ for WORS with $\rho = 0.2$, and Fig. \ref{Coe_v2}(d) is an example of $v_2$ s.t. $\delta^2 F(v_2) < 0$ for BD with $\rho = 0.2$.
It turns out that we can find a $v_2$ such that $\delta^2 F(v_2) < 0$ for the BD-solution, $\forall\rho$ for which the BD-solution exists i.e. the BD-solution is always unstable with respect to perturbations of this kind. This is intuitively easy to understand since $C_2$ is numerically found to have the maximum magnitude along the transition layers featured by $q_1 = 0$. The BD-solution is distinguished by transition layers along a pair of parallel square edges which have a constant length independent of $\rho$. Consequently, we can always find an instability that manifests along the edge transition layer for the BD-solution, for all values of $\rho \leq \rho_1(\bar{\lambda}^2)$. This instability perturbs the constant eigenframe of the BD solution.
\begin{figure}[!ht]
\centering
\begin{overpic}[width = 0.75 \linewidth]{Fig_C2_rho_0_2.pdf}
\end{overpic}
\caption{(a) $C_2(x, y)$ for WORS with $\rho = 0.2$. (b) The profile of $v_2$ in a perturbation s.t. $\delta^2 F(v_2) < 0$ for WORS with $\rho = 0.2$. (c) $C_2(x, y)$ for BD with $\rho = 0.2$. (d) The profile of $v_2$ in a perturbation s.t. $\delta^2 F(v_2) < 0$ for BD with $\rho = 0.2$. }\label{Coe_v2}
\end{figure}
For $\rho$ sufficiently close to $1$, WORS is the unique LdG critical point. Hence, $\delta^2 F(v_2) \geq 0$ for the WORS, when $\rho$ is large enough. Numerically, we find that for $\bar{\lambda}^2 = 200$, $\delta^2 F(v_2) \geq 0$ when $\rho \geq 0.74$. Fig. \ref{Coe_v2_OR}(a) and (c) show the numerically computed profiles of $C_2(x, y)$ for the WORS solution, with $\rho = 0.7$ and $\rho = 0.8$ respectively. Fig. \ref{Coe_v2_OR}(b) illustrates a perturbation for which $\delta^2 F(v_2) < 0$ for the WORS with $\rho = 0.7$. For WORS with $\rho = 0.8$, we solve the gradient flow equation
\begin{equation}\label{grad_per_v2}
\frac{\pp v_2}{\pp t} = 2 \Delta v_2 - \bar{\lambda}^2 C_2(x, y) v_2,
\end{equation}
with random initial data and find that the numerical solution converges to $v_2 = 0$, as shown in Fig. \ref{Coe_v2_OR}(d), which indicates that $\delta^2 F(v_2) \geq 0$ in this case. The WORS has transition layers along the diagonals of length $(1 - \rho )$. Hence, these transition layers get shorter as $\rho$ increases and we cannot find a $v_2$ such that $\delta^2 F(v_2) < 0$ for the WORS when $\rho$ is large enough.
\begin{figure}[!hbt]
\centering
\begin{overpic}[width = 0.8 \linewidth]{Fig_C2_OR_rho_0_7.pdf}
\end{overpic}
\caption{(a) $C_2(x, y)$ for WORS with $\rho = 0.7$. (b) The profile of $v_2$ in a perturbation s.t. $\delta^2 F(v_2) < 0$ for WORS with $\rho = 0.7$. (c) $C_2(x, y)$ for WORS with $\rho = 0.8$. (d) The profile of $v_2$ in numerical solution of (\ref{grad_per_v2}) for WORS with $\rho = 0.8$.}\label{Coe_v2_OR}
\end{figure}
Similarly, we can minimize $\delta^2 F(v_4)$ and $\delta^2 F (v_5)$ by solving the gradient flow equations for $v_4$ and $v_5$
\begin{equation}\label{grad_per_v45}
\begin{aligned}
& \frac{\pp v_4}{\pp t} = 2 \Delta v_4 - \bar{\lambda}^2 C_4(x, y) v_4, \\
& \frac{\pp v_5}{\pp t} = 2 \Delta v_5 - \bar{\lambda}^2 C_5(x, y) v_5, \\
\end{aligned}
\end{equation}
with random initial data, where
\begin{equation}
C_4(x, y) = \frac{A}{C} - \frac{B}{C} (q_1 + q_3) + 2 \left( q_1^2 + 3 q_3^2 \right), \quad C_5(x, y) = \frac{A}{C} - \frac{B}{C} (q_3 - q_1) + 2 \left( q_1^2 + 3 q_3^2 \right).
\end{equation}
The profiles of $C_4(x, y)$ and $C_5(x, y)$ for the WORS-solution with $\rho = 0.02$, are shown in Fig. \ref{Coe_v45}(a), and the profiles of numerical solutions of (\ref{grad_per_v45}) are shown in Fig. \ref{Coe_v45}(b), which converge to $v_4 = v_5 = 0$. For BD with $\rho = 0.02$, the profiles of $C_4(x, y)$ and $C_5(x, y)$ are shown in Fig. \ref{Coe_v45}(c), and the numerical solutions of (\ref{grad_per_v45}) also converge to $v_4 = v_5 = 0$, as shown in Fig. \ref{Coe_v45}(d).
This can be informally understood since the numerical results show that $C_4$ and $C_5$ are negative in a small region around the isotropic inclusion.
\begin{figure}[!ht]
\centering
\begin{overpic}[width = 0.8 \linewidth]{Fig_C45_rho_0_02.pdf}
\end{overpic}
\caption{(a) $C_4(x, y)$ and $C_5(x, y)$ for WORS with $\rho = 0.02$. (b) The profiles of $v_4$ and $v_5$ in numerical solution of (\ref{grad_per_v45}) for WORS with $\rho = 0.02$.
(c) $C_4(x, y)$ and $C_5(x, y)$ for BD with $\rho = 0.02$. (d) The profiles of $v_4$ and $v_5$ in numerical solution of (\ref{grad_per_v45}) for BD with $\rho = 0.02$.}\label{Coe_v45}
\end{figure}
Numerically, we find that for $A = - \dfrac{B^2}{3C}$ and $\bar{\lambda}^2 = 200$,
\begin{itemize}
\item WORS is unstable over subspace $\mathcal{V}_{13}$ for small-$\rho$, but is stable over subspace $\mathcal{V}_{13}$ for large-$\rho$ ($\rho \geq 0.05$). BD is stable over $\mathcal{V}_{13}$ if BD is a critical point of the system ($\rho \leq 0.4$).
\item BD is unstable over subspace $\mathcal{V}_2$, WORS is unstable over $V_2$ for small-$\rho$, but is stable over $\mathcal{V}_2$ when $\rho$ is large enough ($\rho \geq 0.74$).
\item Both WORS and BD are stable over subspaces $\mathcal{V}_4 $ and $ \mathcal{V}_5$.
\end{itemize}
Hence, WORS is globally stable for $\rho$ sufficiently large, which is in accordance with Proposition \ref{prop:2}. Further, the BD-solution is always an unstable LdG critical point for this choice of parameters and we speculate that these stability results hold for $A<0$ and moderately large values of $\bar{\lambda}^2$.
\subsection{Non-existence of ESC}
We consider an ESC-like initial condition to investigate the existence/non-existence of LdG critical points with $q_3>0$ around the isotropic inclusion; the ESC-like initial condition has the form
\begin{equation}
q_1(x, y) = 0, ~~q_3(x, y) = s_{+}/3, \quad
\textrm{for } \ \rho < \max\{|x|, \, |y|\} < \rho + \eta.
\end{equation}
We choose $\rho + \eta = 0.96$ with $\rho=0.02$ and $\rho=0.2$ respectively.
The numerical results are shown in Fig.~\ref{ES}.
In both cases, we find that $q_3 \leq 0$ everywhere for the final states, and
the gradient flow solutions (see equations (\ref{eq_q13})) evolve to a BD solution and to the WORS respectively.
\begin{figure}[!htb]
\centering
\includegraphics[width = 0.8\linewidth]{Fig_ESC_ini.pdf}
\caption{(a) The profiles of $q_1$ and $q_3$ for t = 0, t = 1, t = 2 and t = 4
($\rho = 0.02$, $\bar{\lambda}^2 = 200$).
(b) The profiles of $q_1$ and $q_3$ for t = 0, t = 1, t = 2 and t = 4
($\rho = 0.2$, $\bar{\lambda}^2 = 200$).}\label{ES}
\end{figure}
For a small value of $\rho$, the numerical solution will evolve to the BD-solution by crossing WORS,
as shown in Fig. \ref{ES}(a).
By the $\Gamma$-convergence argument, we know that
$J_{\infty}({\rm ESC}) < J_{\infty}({\rm WORS})$ requires
\begin{equation}\label{Ineq_rho_eta}
\sqrt{2}(c_1 - c_2 + c_3 )\rho < (c_4 - \sqrt{2} c_3) \eta.
\end{equation}
However, during the dynamic evolution of the numerical solution, the value of $\eta$ decreases as time increases and the inequality (\ref{Ineq_rho_eta}) no longer holds.
The non-existence of ESC, at least within the restricted class of $\mathbf{Q}$-tensors of the form (\ref{q13}), is also supported by solving Euler-Lagrange equation (\ref{eq_q13}) using the deflation technique~\cite{farrell2015deflation}.
The deflation technique enables us to discover multiple distinct solutions of (\ref{eq_q13}) with one initial guess. However, we haven't observed any ESC-like solutions for several different choices of the initial conditions. For $\rho = 0.2$, we find 17 critical points. Six of them remain after discarding the the rotational symmetries, as shown in Fig. \ref{CP_rho_0_2}(a)--(f) by the profiles of $q_1$. The profiles of $q_3$ are almost the same for all cases, as shown in Fig. \ref{CP_rho_0_2}(g). Besides the WORS and BD, we find another type of metastable configuration in the restricted class, shown in Fig. \ref{CP_rho_0_2}(c), which is between the WORS and BD (retains half the diagonal cross and one edge transition layer). The critical points shown in Fig. \ref{CP_rho_0_2}(d)-(f) are saddle points even in the restricted two-dimensional class.
\begin{figure}[!htb]
\centering
\begin{overpic}[width = 0.8 \linewidth]{./Fig_q13_CP.pdf}
\end{overpic}
\caption{(a)--(f) Critical Points for $\rho = 0.2$ ($\bar{\lambda}^2 = 200$), (g) the profile of $q_3$ in all the critical points.}\label{CP_rho_0_2}
\end{figure}
For $\rho = 0.02$, we only find 3 critical points (WORS and 2 BD solutions),
shown in Fig.~\ref{CP_rho_0_02}. Here, the WORS is no longer a metastable state but acts as a saddle point of the system connecting two stable BD equilibria in the restricted class.
\begin{figure}[!htb]
\centering
\begin{minipage}{0.6\textwidth}
\begin{overpic}[width = \linewidth]{Fig_q13_cp_rho_0_02.pdf}
\end{overpic}
\end{minipage}
\caption{Critical Points for $\rho = 0.02$ ($\bar{\lambda}^2 = 200$), shown by $q_1$.}\label{CP_rho_0_02}
\end{figure}
\subsection{General Case}
The critical points of the form (\ref{q13}) are a two-dimensional subset of LdG critical points.
We have also calculated critical points of the general form
\begin{equation}\label{q12345}
\begin{aligned}
\Qvec(x,y) & = q_1(x, y)(\e_x \otimes \e_x - \e_y \otimes \e_y) + q_2(x, y) (\e_x \otimes \e_y + \e_y \otimes \e_x) \\
& + q_3(x, y) (2 \e_z \otimes \e_z - \e_x \otimes \e_x - \e_y \otimes \e_y) \\
& + q_4(x, y)(\e_x \otimes \e_z + \e_z \otimes \e_x) + q_5(x, y)(\e_y \otimes \e_z + \e_z \otimes \e_y ), \\
\end{aligned}
\end{equation}
which exploit all five degrees of freedom of LdG $\Qvec$-tensor, subject to the boundary condition (\ref{BC-general}).
\begin{figure}[!htb]
\begin{center}
\includegraphics[width = 0.9 \linewidth]{./q123_CP.pdf}
\end{center}
\caption{Critical Points found by deflation techniques for $\rho = 0.2$ ($\bar{\lambda}^2 = 200$), which are shown by the largest eigenvalue of $\Qvec$ (blue at 0, increasing to red) and the director profiles(transparent white lines).}\label{CP_q123}
\end{figure}
The critical points of the form (\ref{q12345}) satisfy the Euler-Lagrange equation
\begin{equation}
\begin{cases}
& \Delta q_1 = \bar{\lambda}^2 \left( \dfrac{A}{2C} q_1 + \dfrac{B}{2C} \Bigl( 2 q_1 q_3 - \dfrac{1}{2}(q_4^2 - q_5^2) \Bigr) + \Bigl( \frac{1}{2} \tr (\Qvec^2) \Bigr) q_1 \right) \\
& \Delta q_2 = \bar{\lambda}^2 \left( \dfrac{A}{2C} q_2 + \dfrac{B}{2C}\Bigl( 2 q_2 q_3 - q_4 q_5 \Bigr) + \Bigl( \frac{1}{2} \tr (\Qvec^2) \Bigr) q_2 \right) \\
& \Delta q_3 = \bar{\lambda}^2 \left( \dfrac{A}{2C} q_3 + \dfrac{B}{2C} \Bigl( \dfrac{1}{3}(q_1^2 + q_2^2) - q_3^2 - \dfrac{1}{6}(q_4^2 + q_5^2) \Bigr) + \Bigl( \frac{1}{2} \tr (\Qvec^2) \Bigr) q_3 \right) \\
& \Delta q_4 = \bar{\lambda}^2 \left( \dfrac{A}{2C} q_4 - \dfrac{B}{2C} \Bigl( q_3 q_4 + q_1 q_4 + q_2 q_5 \Bigr) + \Bigl( \frac{1}{2} \tr (\Qvec^2) \Bigr) q_4 \right) \\
& \Delta q_5 = \bar{\lambda}^2 \left( \dfrac{A}{2C} q_3 - \dfrac{B}{2C} \Bigl( q_3 q_5 - q_1 q_5 + q_2 q_4 \Bigl) + \Bigl( \frac{1}{2} \tr (\Qvec^2) \Bigr) q_5 \right), \\
\end{cases}
\end{equation}
where $\tr(\Qvec^2) = 2q_1^2 + 2q_2^2 + 6q_3^2 + 2q_4^2 + 2q_5^2$.
For $\rho = 0.2$, we find 28 critical points after discarding the rotational symmetries, which are shown in Fig. \ref{CP_q123}. They all satisfy $q_4 = q_5 = 0$, have two or three degrees of freedom and have $\e_z$ as a fixed eigenvector. Further, we haven't found any ESC-like configurations with $q_3 > 0$ around the isotropic inclusion. Actually, the profile of $q_3$ is almost the same for all the numerically computed critical points, as shown in Fig. \ref{CP_rho_0_2}(g).
For small $\rho$, we find two critical points with $q_4 \neq 0$ and $q_5 \neq 0$, as shown in Fig. \ref{ES_rho_0_02}(a) and (b) for $\rho = 0.02$, by using special initial guesses. The initial condition is uniaxial around the isotropic inclusion and the leading eigenvector escapes into the third dimension around the isotropic core with winding number $\pm 1$. The profiles of $q_i$ and biaxiality parameter $\beta^2$ in configuration \ref{ES_rho_0_02}(a) are shown in Fig. \ref{ES_rho_0_02}(c)-(h). We note that for such critical points, $\mathbf{Q}$ is almost uniaxial around the isotropic inclusion with $q_3>0$, so that we have a positively ordered uniaxial state with $\e_z$ as the director around the isotropic core. These two types of critical points do not exist for relatively large $\rho$ ($\rho > 0.052$ as indicated by \ref{ES_rho_0_02}(i)). We do not analyse this further in this paper, largely because these escaped critical points seem rare for this model problem. We expect these escaped critical points to occur more frequently for three-dimensional systems and not for severely confined systems such as the ones considered in this manuscript.
\begin{figure}[!htp]
\centering
\begin{overpic}[width = \linewidth]{./Fig_q45_CP.pdf}
\end{overpic}
\caption{(a)-(b) Two critical points with $q_4 \neq 0$ and $q_5 \neq 0$ for $\rho = 0.02$, shown by the largest eigenvalue of $\Qvec$ (blue at 0, increasing to red) and the director profile. (c)-(h) The profiles of $q_i$ and $\beta$ in configuration(a). (i) $L^{\infty}(q_3)$ as the function of $\rho$ in configuration (a). }\label{ES_rho_0_02}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We study LdG critical points on a square domain with an isotropic square inclusion, with tangent boundary conditions on the outer square edges. We prove the existence of a WORS-type critical point, featured by a distinctive negatively ordered uniaxial cross along the diagonals, connecting the vertices of the inner and outer squares. We partition the LdG critical points into three categories: critical points with two degrees of freedom which have a constant eigenframe (to which the WORS and BD solutions belong), critical points with three degrees of freedom which have $\e_z$ as a fixed eigenvector and critical points which exploit all five degrees of freedom.
In the two-dimensional sub-class, there are effectively three competitors: the WORS configuration, the BD configuration with negatively ordered uniaxial transition layers along a pair of opposite square edges and a third configuration somewhere in between the WORS and the BD (retains half the diagonal cross and one edge transition layer). The WORS typically loses stability with respect to BD-type solutions in the two-dimensional setting as the square size increases or as the aspect ratio of the domain decreases. It is interesting that whilst the WORS is globally stable with respect to all perturbations in certain parameter regimes, the BD solution is never a stable critical point with respect to in-plane perturbations. In fact, the in-plane perturbations are the most effective in de-stabilizing either the WORS or BD solutions, which can be intuitively understood since these perturbations distort the eigenvectors in the square plane to reduce the elastic energy (the Dirichlet energy density term in (\ref{eq:rescaled})).
We carry out a fairly exhaustive study of the LdG critical points in the reduced three-dimensional setting and recover up to twenty eight critical points for $\bar{\lambda}^2 = 200$ and $\rho = 0.2$. For moderately large values of the square size and small aspect ratios, we expect the stable solutions to have either the diagonal or rotated profiles, without any negatively ordered uniaxial defects in the domain interior. The diagonal and rotated solutions have been studied extensively in a batch of papers \cite{luo2012, tsakonas, lewissoftmatter}; informally speaking, the corresponding LdG $\mathbf{Q}$ tensor can be written as
$$ \mathbf{Q} = q \left(\mathbf{a}(x,y)\otimes \mathbf{a}(x,y) - \mathbf{I}_2/2 \right) + q_3 \left(2 \e_z \otimes\e_z - \e_x \otimes \e_x - \e_y \otimes \e_y \right),$$
where $\mathbf{a}$ is an inhomogeneous two-dimensional unit-vector in the square plane (e.g. roughly pointing along one of the square diagonals for the diagonal state) and $\mathbf{I}_2$ is the $2\times 2$ identity matrix.
These solutions necessarily have three degrees of freedom. For the model problem considered here, as heuristically explained by the analysis in \cite{sternberggolovaty2015}, we do not expect to have stable critical points with full five degrees of freedom, with the exception of perhaps very small isotropic square inclusions. It would be interesting to study the LdG critical points on a three-dimensional rectangular box, where the vertical dimension is much smaller than the cross-sectional dimension, and then gradually increase the vertical dimension to check when the out-of-plane perturbations destabilise the WORS or BD solutions. This would elucidate the existence and stability of truly five-dimensional LdG critical points and we will investigate this further in future work.
\section{Acknowledgments}
Part of this work was carried out when Y.W. was visiting the University of Bath, he would like to thank the University of Bath
and Keble College for their hospitality. He also would like to thank the Elite Program of Computational and Applied Mathematics for PhD Candidates in Peking University and his Ph.D. advisor Professor Pingwen Zhang, for his constant
support and helpful advice.
G.C.'s research was supported by the European Research Council under the
European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n° 291053;
by the Basque Government through the BERC 2014-2017 program; and by the Spanish
Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.
A.M. is supported by an EPSRC Career Acceleration Fellowship EP/J001686/1 and EP/J001686/2 and an OCIAM Visiting Fellowship, the Keble Advanced Studies Centre. She would also like to thank the Chinese Academy of Sciences where this collaboration was initiated and the Banff International Research Station where the three authors met in November 2017. The authors would like to thank Professor Paul Milewski for helpful discussions about the numerical simulations.
|
{
"timestamp": "2018-03-08T02:06:49",
"yymm": "1803",
"arxiv_id": "1803.02597",
"language": "en",
"url": "https://arxiv.org/abs/1803.02597"
}
|
\section{Introduction}
The potential of mean force (PMF) is one of the most topical issues
when facing the problem of determining the stability
of nanoparticle (NP) systems and nanocomposites~\cite{Akcora:09,Kawada:17,Baran:17}. The possibility to
calculate the net interactions between a couple of nanoparticles, hence gaining knowledge
on the overall behavior of the system, can open the way to systematic studies of macroscopic
properties of potential tecnological interest. This is particularly true for polymer
nanocomposites, where the addition of nanoparticles can sensibly improve their physico-chemical
properties (see, {\it e.g.}, Refs.~\cite{Akcora:10,Han:11,Kim:12}). For instance,
it is now well established that a specific NP dispersion state in a polymer matrix is
crucial to improve a given property of the system~\cite{Leibler:02,Kumar:13}; the knowledge of
such a dispersion state
can be gained by means of the PMF between the NPs belonging to the composite. In this context
it is worth noting that the study of interactions between nanoparticles dispersed
in polymer matrices or in solvents
is currently object of rather extensive studies, by means of experimental~\cite{Chevigny:11,You:17},
theoretical~\cite{Martin:13,Ganesan:14} and simulations~\cite{Meng:12,Karatrantos:17}
approaches.
As far as theoretical approaches are concerned, they are generally
based on the Polymer Reference Interaction Site Model (PRISM) theory
developed by Curro and Schweizer in the late '80s~\cite{Curro1,Curro2}.
This theory has
generally provided good results when facing the study of PMF, obaining a good agreement with
experimental data
(see, {\it e.g.} Refs.~\cite{Hooper:04,Jayaraman:09} and~\cite{Raos:08,Ganesan:10} for two
detailed reviews); however,
according to this theory, only generic models can be investigated and the
chemical details characterizing a given compound is lost. In order to fully
recover its chemical structure, computer-simulations based
calculations are needed. Many efforts have been dedicated to shed light on
the complex properties of polymers-nanoparticles interface, going from
atomistic to mesoscale representations (see, for instance,
Refs.~\cite{Karatrantos:16,Kumar:17} for two recent reviews).
To quote some examples, previous numerical studies have highlighted the
effects of the NP curvature on the behavior of PMF~\cite{Cerda:03},
the role played by attractive dispersions interactions between polymer and
nanoparticles~\cite{Smith:03,Marla:06} and the importance of the ratio between
coated and free polymer chain lengths~\cite{Smith:09} and of the NP
radius~\cite{Loverso:11}.
\begin{figure*}
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=5.5cm,angle=0]{silica.png} \qquad
\includegraphics[width=3.0cm,angle=0]{au-naked.png} \qquad
\includegraphics[width=7.0cm,angle=0]{au-grafted.png}
\end{tabular}
\caption{Representative snapshots of nanoparticles investigated in this
work: silica NP (left), bare gold NP (center) and coated gold NP (right).
The color legend is the following: O=red, Si=green, H=white, Au=orange, S=yellow, C=cyan.}
\label{fig:snap}
\end{center}
\end{figure*}
In all the above mentioned studies, nanoparticles have been generally represented
by means of simple coarse grained models, like bead-spring or pearl-necklace representations;
this is an unavoidable choice when tackling
the issue of investigating nanoparticles dispersed in polymer matrices. In fact,
under such conditions, an atomistic representation should require the calculation
of pairwise interactions over hundreds of thousands of atoms for long times,
making the calculation practically unaffordable. More details on the chemical structure
of the coarse-grained models can be gained by adopting a mean field representation
of the non-bonded interactions, hence giving rise to an hybrid particle-field
approach~\cite{Milano:09,Milano:10}. However, the atomistic details of the interactions
equally need to be addressed in order to provide the overall behavior of the potential
of mean force.
In this work we plan to address this point by calculating the PMF between
couples of silica and gold nanoparticles by using atomistic models.
In particular, we focus on the nature and a suitable procedure to obtain
realistic NP-NP interaction potentials.
Our investigation is focused on the effects due to the silica particle size
and to the presence (or absence)
of polyethylene (PE) chains coated onto the surface of gold nanoparticles.
Such models have been chosen in order to
consider systems with an increasing degree of complexity, from bare
nanoparticles to coated systems with increasing grafting densities.
In order to
make the calculation affordable, and aiming to clarify the microscopic
details of NP-NP interactions,
we simulate our systems in the vacuum, {\it i.e.} without any surrounding polymer matrix or solvent.
The present work is framed in the broader perspective of obtaining atomistic potentials that
will be used in subsequent studies concerning high-density coarse-grained polymer nanocomposites.
For such an aim, we perform molecular dynamics
simulations in the canonical ensemble (NVT) by means of the GROMACS
package~\cite{Gromacs:08}.
As stated in Ref.~\cite{Meng:12}, when compared to
other techniques frequently adopted for calculating the PMF, like the
umbrella sampling, this approach shows some advantages: in fact, even if the results obtained
by implementing the two techniques are basically the same, NVT simulations
allow to
gain information also on the net forces experienced by the nanoparticles,
and to discriminate between their different contributions.
For the case of silica nanoparticles we also compare simulation
results with theoretical predictions based on the Hamaker theory~\cite{Hamaker:37},
specifically developed to deal with spherical systems comprised by several Lennard-Jones interacting
particles. Such a comparison is not performed for the gold NPs, due to the presence of
coating chains and the non-spherical shape of the particles.
In addition, in order to gain further insigth into the local structure
of coated PE chains, gyration radii and end-to-end distances have been also computed.
The paper is organized as follow: in the next Section we provide details on models, theory
and simulation approaches. In
Section III we present and discuss the obtained results, finally drawing the
conclusions in the last Section.
\section{Models and methods}
\subsection{Silica and gold NPs}
\begin{table}
\begin{center}
\caption{NP systems investigated in this work. Diameters are in nm and grafting densities $\rho_g$ in
chains/nm$^2$. In the case of gold nanoparticles, the diameter is referred to the circumscribed
sphere of the cuboctahedron.}\label{tab:NP}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}cccccccccccccc}
\hline
\hline
& NP & \qquad & Diameter & \qquad & Coated & \qquad & $N_a$
& \qquad & $\rho_g$ \qquad & $N_c$ \qquad & $L_c$ \\
\hline
& Silica & \qquad & 2.5 & \qquad & No & \qquad & 1401 & \qquad & - \qquad & - \qquad & -\\
& Silica & \qquad & 4.0 & \qquad & No & \qquad & 3189 & \qquad & - \qquad & - \qquad & -\\
& Gold & \qquad & 1.6 & \qquad & No & \qquad & 79 & \qquad & - \qquad & - \qquad & - \\
& Gold & \qquad & 1.6 & \qquad & Yes & \qquad & 801 & \qquad & 2.36 \qquad & 19 \qquad & 38\\
& Gold & \qquad & 1.6 & \qquad & Yes & \qquad & 1143 & \qquad & 3.48 \qquad & 28 \qquad & 38 \\
& Gold & \qquad & 1.6 & \qquad & Yes & \qquad & 1523 & \qquad & 4.72 \qquad & 38 \qquad & 38 \\
\hline
\end{tabular*}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{Parameters of non-bonded potential
$V_{nb}(r_{ij})\equiv V_{LJ}(r_{ij})+V_{Coul}(r_{ij})+V_{rf}(r_{ij})$.}\label{tab:pot}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}cccccccc}
\hline
\hline
& Atom & \qquad & $\sigma$ (nm) & \qquad & $\epsilon$(kJ mol$^{-1})$ & \qquad
& $q$(e) \\
\hline
& Silica NP & \qquad & & \qquad & \\
& Si & \qquad & 0.392000 & \qquad & 2.510400 & \qquad & 1.020\\
& O & \qquad & 0.315400 & \qquad & 0.636840 & \qquad & -0.510\\
& H & \qquad & 0.235200 & \qquad & 0.092000 & \qquad & 0.255\\
& Gold NP & \qquad & & \qquad & \\
& Au & \qquad & 0.293373 & \qquad & 0.163176 & \qquad & 0.000\\
& S & \qquad & 0.355000 & \qquad & 1.066000 & \qquad & -0.180\\
& C & \qquad & 0.350000 & \qquad & 0.276000 & \qquad & -0.120\\
& H & \qquad & 0.250000 & \qquad & 0.138000 & \qquad & 0.060\\
\hline
\end{tabular*}
\end{center}
\end{table}
A representative picture of the nanoparticle models investigated in this
work is given in Fig.~\ref{fig:snap}: as visible, different colors
label different atom types.
As anticipated in the Introduction, we first
investigate the behavior of the PMF between a couple of
bare spherical silica nanoparticles (Fig.~\ref{fig:snap}, left panel) with
diameters of 2.5 and 4 nm; the model for these NPs has been developed
by the M{\"u}eller-Plathe group and employed for studying the interface between
silica nanoparticles and polymer matrices~\cite{Ndoro:11,Eslami:13}.
Then, we study the behavior of gold NPs (middle panel
of Fig.~\ref{fig:snap}): for such an aim we adopt a model already investigated
in Refs.~\cite{Rai:04,Milano-Gold}, constituted
by a core made of 79 Au atoms organized in a cuboctahedral
geometry with a Au-Au bond length of 0.292 nm. Finally, gold NPs are
considered coated with a variable number of PE chains, each one containing
38 monomers (right panel of Fig.~\ref{fig:snap})
and connected to the inner part of the NP through one sulfur atom.
Following the prescription of Ref.~\cite{Rai:04},
sulfur atoms are covalently bonded
to a gold atom by using a harmonic potential with an Au-S bond
length of 0.24 nm (see Tab~\ref{tab:bond}):
this method of attachment restricts the position of the S atom
to a position directly above the Au atom to which it is attached.
Further details on the model parameters can be found in Refs.~\cite{Rai:04,Milano-Gold}.
\begin{table}
\begin{center}
\caption{Parameters of bond stretching potential
$V_{b}(r)\equiv (k_r/2)(r-r_0)^2$.}
\label{tab:bond}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}cccccc}
\hline
\hline
& Bond & \qquad & $r_0$ (nm) & \qquad & k$_r$(kJ mol$^{-1}$ nm$^{-2})$ \\
\hline
& Silica NP & \qquad & & \qquad & \\
& Si-O & \qquad & 0.1630 & \qquad & $1 \cdot 10^7$ \\
& O-H & \qquad & 0.0950 & \qquad & $1 \cdot 10^7$ \\
& Gold NP & \qquad & & \qquad & \\
& Au-Au & \qquad & 0.2920 & \qquad & 400000 \\
& Au-S & \qquad & 0.2400 & \qquad & 400000 \\
& C-S & \qquad & 0.1810 & \qquad & 400000 \\
& C-H & \qquad & 0.1090 & \qquad & 400000 \\
& C-C & \qquad & 0.1552 & \qquad & 265265 \\
\hline
\end{tabular*}
\end{center}
\end{table}
The complete collection of the investigated systems
is reported in Tab.~\ref{tab:NP}, where $N_a$ is the total number of atoms,
$\rho_g$ the grafting density, $N_c$ the number of coated chains and $L_c$ the number
of monomers belonging to a single chain.
Tables~\ref{tab:pot}-\ref{tab:dih}
provide a summary of the potential energy parameters.
In particular in Tab.~\ref{tab:pot} we report all the non-bonded interactions,
i.e. the interactions between atom pairs whose distance $r_{ij}$
is not fixed by the
connectivity. In this Table we have defined:
\begin{equation}
V_{LJ}(r_{ij}) = 4\epsilon[(\sigma/r_{ij})^{12}-(\sigma/r_{ij})^{6}] \,,
\end{equation}
\begin{equation}
V_{Coul}(r_{ij}) = q_iq_j/4\pi\epsilon_0r_{ij}^2 \,,
\end{equation}
\begin{equation}
V_{rf}(r_{ij}) = V_{Coul}(r_{ij}) r_{ij} (\epsilon_{rf}-1)
(2\epsilon_{rf}+1) (r_{ij}^2/r_{cut}^3) \,.
\end{equation}
The first contribution is the standard Lennard-Jones potential, determined
by the interaction energy $\epsilon$ and the close-contact distance $\sigma$.
$V_{Coul}(r_{ij})$ is the Coulombic interaction between two atoms with
charges $q_i$ and $q_j$, $\epsilon_0$ being the vacuum permittivity.
The third contribution takes into account the effect of a reaction
field~\cite{Tironi:95} with a dielectric constant $\epsilon_{rf}$ and, after the cutoff length
$r_{cut}$, is modeled by using the Kirkwood approximation~\cite{Kirkwood:35}.
In Tab~\ref{tab:bond} we define the bond stretching potential $V_{b}(r)$,
dependent on the elastic constant $k_r$ of the material and on the
elongation $r$ respect to the equilibrium position $r_0$. Analogous
expressions hold for the bond angle potential $V_a(\theta)$ reported in
Tab~\ref{tab:ang}, where $\theta$ and $\theta_0$ are the angular
counterparts of $r$ and $r_0$ and $k_{\theta}$ is a constant
still dependent on the material properties.
Finally, parameters of the dihedral angle potentials
$V_d(\theta)$ for the polyethylene chains are
reported in Tab~\ref{tab:dih}, $\phi$ and $\phi_0$ being the analogous
of $\theta$ and $\theta_0$ for dihedral angles, $k_{\phi}$ an other
material-dependent constant and $f$ being the multiplicity
of $\phi_0$.
\begin{table}
\begin{center}
\caption{Parameters of bond angle potential $V_d(\theta) \equiv (k_{\theta}/2)
(\theta-\theta_0)^2$.}\label{tab:ang}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}cccccc}
\hline
\hline
& Bond angle& \qquad & $\theta_0$(degrees) & \qquad & k$_\theta$(kJ mol$^{-1}$ rad$^{-2})$ \\
\hline
& Silica NP & \qquad & & \qquad & \\
& O-Si-O & \qquad & 109.47 & \qquad & 469.716 \\
& Si-O-Si & \qquad & 144.00 & \qquad & 209.598 \\
& Si-O-H & \qquad & 119.52 & \qquad & 228.836 \\
& Gold NP & \qquad & & \qquad & \\
& H-C-H & \qquad & 107.8 & \qquad & 276.144 \\
& H-C-C & \qquad & 110.7 & \qquad & 292.88 \\
& C-C-C & \qquad & 112.7 & \qquad & 527.184 \\
& C-C-S & \qquad & 108.6 & \qquad & 418.4 \\
& S-C-H & \qquad & 113.4 & \qquad & 292.88 \\
\hline
\end{tabular*}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{Parameters of dihedral angle potentials
$V_a(\phi) \equiv (k_{\phi}/2)
[1-\cos f(\phi - \phi_0)]$.}\label{tab:dih}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}cccccc}
\hline
\hline
& Bond angle& \qquad & $\phi_0$(degrees) & \qquad & k$_\phi$(kJ mol$^{-1}$ rad$^{-2})$ \\
\hline
& Polyethylene & \qquad & & \qquad & \\
& H-C-C-C & \qquad & 60 & \qquad & 5.86 \\
& C-C-C-C & \qquad & 60 & \qquad & 5.86 \\
& C-C-C-S & \qquad & 60 & \qquad & 5.86 \\
\hline
\end{tabular*}
\end{center}
\end{table}
\subsection{Simulation details}
\begin{table}
\begin{center}
\caption{Minimum and maximum values of interparticle distances as functions of particle size
and grafting density. The number of investigated simulation points $N_p$ is also reported.
All distances are in nm. As stated in Tab.~\ref{tab:NP}, for
gold nanoparticles, the diameter is referred to the circumscribed sphere of the
cuboctahedron.}\label{tab:sim}
\begin{tabular*}{0.45\textwidth}{@{\extracolsep{\fill}}cccccccccccccc}
\hline
\hline
& NP & \qquad & Diameter & \qquad & Coated & \qquad & $\rho_g$
& \qquad & $r_{min}$ \qquad & $r_{max}$ \qquad & $N_p$ \\
\hline
& Silica & \qquad & 2.5 & \qquad & No & \qquad & - & \qquad & 2.5 \qquad & 8.5 \qquad & 31 \\
& Silica & \qquad & 4.0 & \qquad & No & \qquad & - & \qquad & 4.0 \qquad & 10 \qquad & 36\\
& Gold & \qquad & 1.6 & \qquad & No & \qquad & - & \qquad & 1.6 \qquad & 5.6 \qquad & 21 \\
& Gold & \qquad & 1.6 & \qquad & Yes & \qquad & 2.36 & \qquad & 1.6 \qquad & 6.6 \qquad & 26\\
& Gold & \qquad & 1.6 & \qquad & Yes & \qquad & 3.48 & \qquad & 1.6 \qquad & 7.6 \qquad & 31 \\
& Gold & \qquad & 1.6 & \qquad & Yes & \qquad & 4.72 & \qquad & 1.6 \qquad & 7.6 \qquad & 31 \\
\hline
\end{tabular*}
\end{center}
\end{table}
In the present work all simulations have been performed by using GROMACS
4.6.3.~\cite{Gromacs:08},
employing a cubic simulation box of side $L_{box}=26$ nm with periodic boundary
conditions.
In the case of silica NPs, a time step of 1 fs has
been used for all simulations. For the nonbonded interactions, a
cutoff of 1.0 nm has been used, while the coulomb long-range
electrostatic interactions have been treated by means of a generalized
reaction field~\cite{Tironi:95} with a dielectric constant $\epsilon_{rf}=6.23$ and a cutoff
of 1.0 nm.
For gold NPs we have analogously proceeded,
the only differences being the values of cutoff for nonbonded and electrostatic
interactions, both fixed to 1.35 nm.
Simulation parameters have been fixed
by following a similar procedure described in Ref.~\cite{DeNicola-JPCB}.
In all systems the temperature has been kept
constant at $T= 590 K$ by using a Berendsen thermostat~\cite{Berendsen:84} with a
time coupling $\tau = 0.1$ ps.
We have verified that results do not change if the Nose-Hoover thermostat
is used after the equilibration in place of the Berendsen one.
The temperature has been chosen high enough to
allow for a proper relaxation of the considered systems and for the subsequent PMF
calculations. The specific value of 590 $K$ has been fixed in order to
simulate silica nanoparticles in conditions similar to those reported in previous numerical
investigations of the same NPs~\cite{Ndoro:11,Eslami:13}. With the aim to investigate,
for the sake of clarity, silica and gold nanoparticles at the same temperature,
we have set $T=590 K$ for studying gold NPs also.
In order to calculate the PMF between the above said nanoparticles, we have
preliminarly built a collection of independent initial configurations with
particles placed at progressively increasing mutual distances.
In the case of gold coated NPs, we have first prepared configurations with
the higher grafting density considered in this work (i.e. $\rho_g=4.72$) and
taken from Ref.~\cite{Milano-Gold}: in such configurations, chains are
stretched and uniformly distributed over the NP surface. Configurations with
lower $\rho_g$ have been obtained by deleting some chains, in order to get
a final $N_c$ equal to 50\% ($N_c=19$) or to $\simeq$ 75\% ($N_c=28$) of the
fully coated case.
All initial systems
have been built by using the Packmol program~\cite{Packmol}, which allows one
to put the desired number of particles in a given position
inside the simulation box avoiding
overlaps. Then, we have
computed the forces experienced by the nanoparticles, whose centers of mass
are kept fixed, finally
evaluating the PMF by integrating the obtained forces:
\begin{equation}\label{eq:PMF}
U(r)=-\int_{r_{min}}^{r_{max}} F(r) dr
\end{equation}
where $U(r)$ is the PMF, $F(r)$ is the force and $r_{max}$ and $r_{min}$
are the maxim and minimum distances between the NPs, respectively.
In all simulations $r_{min}$ corresponds to the NP diameter (see Tab.~\ref{tab:NP}), while $r_{max}$
indicates a NP-NP distance where the potential can be confidently assumed equal to zero;
distances are sampled with a step of 0.2 nm. Values of $r_{min}$ and $r_{max}$, along with
the number $N_p$ of points simulated in a single run, are collectively reported
in Tab.~\ref{tab:sim}.
After a minimization procedure of 15 ps, equilibration runs of 20 ns have
been preliminarly produced, then averaging the forces over the next 10 ns.
The convergence has been ensured by verifying that the average values of the
forces do not change anymore up to the first significant figure.
Standard deviations have been calculated in the production run
according to the formula:
\begin{equation}
s=\sqrt\frac{\sum_{i=1}^N (F_i-\bar{F})^2}{(N-1)}
\end{equation}
where $F_i$ and $\bar{F}$ are respectively the instantaneous and the average
value of the force experienced by the NPs and $N$ is the number of
simulation time steps. An analogous procedure has been implemented in order to calculate
standard deviations for gyration radii and end-to-end distances.
In what follows, if not explicitly reported in the figures, error bars
corresponding to standard deviations are smaller than symbol sizes
of the corresponding curves.
\subsection{Hamaker theory}
It is known that for molecules containing a large number of atoms experiencing
pair interactions, the
evaluation of the overall potential has an high computational cost,
since it amounts to calculate a double summation over all the interaction
sites. For a couple of NPs, such a summation $U_{sum}$
is written as~\cite{Everaers:03}:
\begin{equation}
U_{sum}=\sum_{i\in NP_1} \sum_{j\in NP_2} U(r_{ij})
\end{equation}
where $U(r_{ij})$ is the pairwise potential. For particles with simple
geometrical shapes and number density $\rho_i(r)$ of interaction sites, this
relation can be generalized to a continuum approximation as:
\begin{equation}\label{eq:int}
U_{sum}=\int_{NP_1} \int_{ NP_2} \rho_1(r) \rho_2(r') U(r-r') dV dV'
\end{equation}
where $V$ is the volume of the NP. For two spheres of radius $r_1 \leq r_2$,
volume $V_i=(4\pi/3)r_i^3$, placed at a distance $r_{12} > r_1 + r_2$ and
containing particles interacting via a Lennard-Jones potential,
eq.~\ref{eq:int} can be solved by the Hamaker theory~\cite{Hamaker:37}.
Within this approach, the attractive part of the interaction can be written as:
\begin{eqnarray}\label{eq:att}
U_A & = & -\frac{A_{12}}{6}\Biggl[\frac{2r_1r_2}{r_{12}^2-(r_1+r_2)^2}
+ \frac{2r_1r_2}{r_{12}^2-(r_1-r_2)^2}
\nonumber\\[4pt]
& & \quad +{\rm ln} \left(\frac{r_{12}^2-(r_1+r_2)^2}{r_{12}^2-(r_1-r_2)^2}\right)
\Biggr]
\end{eqnarray}
where $A_{12}$ is the Hamaker constant and is takes the value
$A_{12}=4\pi^2\epsilon(\rho\sigma^3)^2$, $\epsilon$ and $\sigma$ being the Lennard-Jones parameters
and $\rho$ being the density. The repulsive part of the interaction
can be written as:
\begin{eqnarray}\label{eq:rep}
U_R & = & \frac{A_{12}}{37800}\frac{\sigma^6}{r_{12}} \Biggl [\frac
{r_{12}^2-7r_{12}(r_1+r_2)+6(a_1^2+7a_1a_2+a_2^2)}{(r_{12}-r_1-r_2)^7}
\nonumber\\[4pt]
& & \quad +\frac
{r_{12}^2+7r_{12}(r_1+r_2)+6(a_1^2+7a_1a_2+a_2^2)}{(r_{12}+r_1+r_2)^7}
\nonumber\\[4pt]
& & \quad -\frac
{r_{12}^2+7r_{12}(r_1-r_2)+6(a_1^2-7a_1a_2+a_2^2)}{(r_{12}+r_1-r_2)^7}
\nonumber\\[4pt]
& & \quad -\frac
{r_{12}^2-7r_{12}(r_1-r_2)+6(a_1^2-7a_1a_2+a_2^2)}{(r_{12}-r_1+r_2)^7}
\end{eqnarray}
By combining Eqs.~\ref{eq:att}-\ref{eq:rep} one can obtain the total
interaction. Even if the Hamaker theory strictly holds for spherical particles
interacting via a Lennard-Jones potential only, it can provide a useful
benchmark against which simulation results can be assessed.
\section{Results and discussion}
\subsection{Silica nanoparticles}
We first calculate forces and PMF between silica nanoparticles:
the absence of coated chains and the contemporary presence of many atoms
in a single spherical nanoparticle (see Tab.~\ref{tab:NP})
make these NPs ideal candidates in order to make a comparison with the Hamaker theory.
In order to perform the summation of the bead-bead interactions required by
the theory, we have employed Lennard-Jones parameters for silicon and oxygen atoms taken
from Tab.~\ref{tab:pot}, then using the Lorentz-Berthelot mixing rules.
Forces and PMF between a pair of silica nanoparticles with radius 2.5 nm are
respectively reported in panels (a) and (b) of Fig.~\ref{fig:Hamaker-r1},
along with the predictions due to the Hamaker theory. A pictorial view of two silica NPs
whose mutual distance corresponds to the minimum of the PMF is reported in the snapshot of
panel (a). We first note that the force is strongly negative
for very short interparticle distances; then,
the force shows a steep increase, first attaining positive values and then going to
zero for interparticle distances of $\simeq$ 1.5 nm.
As for the PMF (panel b), we note that
simulation results closely match the theoretical datum in providing a Lennard-Jones
behavior: in particular the potential shows a well defined minimum of $\simeq$ -500 kJ/mol
observed for a NP surface-surface distance of $\simeq$ 0.3 nm.
The attractive well is followed by a quick rise,
with the potential going to zero for interparticle separations of $\simeq$ 3 nm. The
theory slightly anticipates this trend, proving a steeper shape of the potential.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{PMF-sio2-r25-1.png}
\caption{Force (a) and PMF (b) between a pair of silica nanoparticles of diameter 2.5 nm
obtained from atomistic simulations (symbols). In panel (b) a comparison
with the Hamaker theory (full line) is reported.
}
\label{fig:Hamaker-r1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{PMF-sio2-r40-1.png}
\caption{Force (a) and PMF (b) between a pair of silica nanoparticles of diameter 4.0 nm
obtained from atomistic simulations (symbols). In panel (b) a comparison
with the Hamaker theory (full line) is reported.
}
\label{fig:Hamaker-r2}
\end{center}
\end{figure}
Such a scenario
is not significantly changed upon increasing the size of silica nanoparticles: in
Fig.~\ref{fig:Hamaker-r2} we report the same comparison with the NP diameter fixed to 4.0 nm.
In a first
instance, we note that the minimum deep is less enhanced than the previous case: such a result
is apparently counterintuitive, since the increase of the particle size (and of the number of
beads) is expected to promote the attraction. But in this case the increase of the particle volume
is not fully compensated by the increase of the beads, this circumstance making the silica
nanoparticles less dense than before. Therefore, each bead experiments a weaker interaction and
the attractive well is less enhanced. In addition, it is worth noting that a repulsive contribution to
the total interaction comes from the coulombic potential: due to the presence of hydrogen atoms on the
most external shells, two NPs coming in close contact experience a significant electrostatic
repulsion. This effect is partially offset by the hydrogen bonds between oxygen and hydrogen atoms
belonging to different NPs. However, for interparticle distances higher than 0.3 nm (see Ref.~\cite{Denicola:15}),
the hydrogen bond can not take place, this giving rise to the shoulder observed in the PMF.
These effects are not observed for smaller silica NPs, since they are given by surface interactions and
therefore are unfavoured if the number of atoms lying on the particles surface decreases.
By comparing simulation data with the Hamaker theory, we observe that
also in this case all essential features of the theoretical predictions
(and, in particular, the
depth and the position of the attractive well) are nicely catched by simulated PMF,
except for the presence of the above said repulsive shoulder.
Such an agreement constitutes a further validation of the numerical procedure implemented
for calculating the PMF; at the same time, this finding is
also indicative of a good trasferibility of the theory, since the latter appears to
accurately work regardless of the specific value of the NP size.
\subsection{Gold nanoparticles}
Once assessed simulation results for silica NPs against Hamaker predictions, we
now investigate the behavior of PMF between gold nanoparticles. In this case the
comparison with the theory can not be performed due to both the
non-spherical shape of the NPs and the presence of coated chains.
We first report the behaviors of force and PMF for a pair of bare gold NPs
in panels (a) and (b) of Fig.~\ref{fig:gold-nak}, along with a pictorial
view of the two NPs. The force appears quite noisy and lies in the range [-10 kJ/mol
$\cdot$ nm; +20 kJ/mol $\cdot$ nm]; hence, it appears that the NPs are rather low
interacting, as can be expected given the low number of atoms constituting a single
nanoparticle. Moreover, since a large part of such atoms lie on the surface of the NP
rather than in the core, the surface effects play a significant role, causing the
irregular behavior of the resulting force. By looking at the potential of mean force
(panel b) we note that is shows a smoother trend and that it is repulsive in all the
interparticle distance range, even if two very shallow minima are observed for NP surfaces placed at
0.53 and 1.90 nm. The overall behavior of the PMF is rather flat, this confirming that
the net interaction between the two NPs is low. Hence,
in comparison with silica NPs, the emerging picture is quite different, since the
strong attraction previously observed for low interparticle distances has now disappeared.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{pmf-gold-nak-1.png} \\
\caption{Force (a) and PMF (b) between a pair of bare gold
nanoparticles as a function of their mutual distance.}
\label{fig:gold-nak}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{PMF-gold-19-1.png} \\
\caption{Force (a) and PMF (b) between a pair of gold
nanoparticles coated with 19 PE chains as a function of their mutual distance.
}
\label{fig:gold-19}
\end{center}
\end{figure}
Upon coating the gold nanoparticles with PE chains the behavior of the PMF is expected to be
modified, as a consequence of the interactions between chains. In Fig.~\ref{fig:gold-19} we report
force (panel a) and PMF (panel b) between a pair of gold NPs coated with 19 PE chains,
corresponding to a grafting density of $\rho_g=2.36$: in
comparison with the bare case, the force shows now a more regular and smooth behavior, the main
feature being the quick fall towards negative values in the range of short interparticle
distances. The potential of mean force exhibits a minimum placed at a distance of $\simeq$ 0.4 nm
and, interestingly, is still negative even if the two nanoparticles surfaces are in close contact.
This is likely due to the possibility for the chains to interpenetrate, since the relatively low
number (19) of chains coated on a particle leaves enough available space for the chains
belonging to the other particle. Such an effect is strongly dependent on the number of chains,
{\it i.e.} on the grafting density $\rho_g$ and also on the distance between the nanoparticles.
A proper combination of these two parameters gives rise to the minimum observed in the PMF.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{PMF-GOLD-28-1.png} \\
\caption{Force (a) and PMF (b) between a pair of gold
nanoparticles coated with 28 PE chains as a function of their mutual distance.}
\label{fig:gold-28}
\end{center}
\end{figure}
Results for gold nanoparticles coated with 28 PE chains, corresponding to a grafting
density of 3.48, are reported in Fig.~\ref{fig:gold-28}. The force (panel a)
is now strongly
negative for close-contact configurations, then showing a small and broad shoulder before
getting the asymptotic value for interparticle separations of $\simeq$ 2 nm.
As a consequence, the PMF (panel b) is now significantly repulsive for
close-contact configurations and
shows a well-defined attractive part with a minimum for interparticle distances of $\simeq$
0.8 nm. The overall behavior of the PMF could resemble to a Lennard-Jones potential, but
in the present case the attractive well is quite broad, extending for $\simeq$ 1 nm.
The emerging picture suggests that with the increase of the grafting density there is also
an increase of the repulsion for very short ranges: this is due to the higher number of
chains that can not overlap, hence giving rise to repulsive contributions. When the two
gold nanoparticles are sligthly more distant, chains have enough space to interpenetrate,
thus generating an attractive interaction. Since there is a rather wide range of distances
where this interpenetration is possible, the minimum in the PMF appears broad and well-defined.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{PMF-gold-38-1.png}
\caption{Force (a) and PMF (b) between a pair of gold
nanoparticles coated with 38 PE chains as a function of their mutual distance.}
\label{fig:gold-38}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{Rg.png} \\
\includegraphics[width=8.0cm,angle=0]{E2e.png}
\caption{Average gyration radius (a) and average end-to-end distance (b) for PE chains with
different $N_g$ as a function of the distance between NPs surfaces. Values of $N_g$ are in the
legends.}
\label{fig:rg}
\end{center}
\end{figure}
The case corresponding to the higher grafting density investigated in the present study
($\rho_g=4.72$) is reported in Fig.~\ref{fig:gold-38}. A comparison with the previous case
shows that the force (panel a) is now remarkably more negative for very low interparticle separations;
on the other hand, the force
attains its asymptotic value for surface distances of $\simeq 2$ nm, as observed for $\rho_g=3.48$.
Overall, the behavior of the force is
similar to that previously observed, but for the strength of the force when the two NPs come in close
contact. As a consequence, the PMF (panel b) is now much repulsive for low interparticle separations and
shows a minimum of $\simeq$ -200 kJ/mol placed at a distance of $\simeq$ 1.2 nm. Upon comparing this
behavior with the cases of lower grafting densities, we note that the close-contact value of the PMF
remarkably increases with $\rho_g$; the position of the minimum is also affected by the grafting density, going
from 0.4 nm for $\rho_g=2.36$ till to 1.2 nm for $\rho_g=4.72$. The depth of the minimum is indeed
unchanged when going from $\rho_g=3.48$ to $\rho_g=4.72$. All these features can be explained in terms
of chain interpenetration: upon increasing $\rho_g$, chains belonging to different NPs are progressively
repelled from each other, this giving rise to the increasing repulsion observed in the PMF; on the other
hand, there is a preferred NP-NP distance where the chains can be more easily arranged, hence giving rise
to a minimum in the PMF, whose position is in turn dependent on $\rho_g$.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm,angle=0]{PMF-gold-all.png}
\caption{Comparison between PMF of a pair of gold
nanoparticles for different values of $N_g$ (in the legend) as a function of their
mutual distance. The behavior for low interparticle distances is highlighted in the inset.}
\label{fig:gold-all}
\end{center}
\end{figure}
More insight into the extension of PE chains and their dependence on the grafting density
can be gained by investigating their local structure.
In Fig.~\ref{fig:rg} we report the average gyration
radius $\langle R_g \rangle$ (panel a) and the average end-to-end distance $\langle EtE \rangle$
(panel b) for all the investigated values of $N_g$.
As a first instance, we note that
both $\langle R_g \rangle$ and $\langle EtE \rangle$ shows a clear dependence on $N_g$: as visible,
upon increasing the number of coating chains, values of
$\langle R_g \rangle$ and $\langle EtE \rangle$ increase in turn, this being particularly
apparent for $N_g=38$. This finding is a clear indication that for high $N_g$ (and hence for
high grafting densities) the chains are more stretched, since the available space is reduced and
they are forced to assume elongated configurations.
On the other hand, $\langle R_g \rangle$ and $\langle EtE \rangle$ shows also a slight dependence
on the interparticle separation, even if such a dependence is less enhanced than that on $N_g$: in
fact, for $N_g=28$ and, in particular, for $N_g=38$, we note the presence of a shift for
a NP-NP distance of $\simeq 2.5$ nm. In correspondence of this distance, both $\langle R_g \rangle$ and
$\langle EtE \rangle$ jump to higher values, thus signalling a point where the coated chains belonging to
a NP stop to be compressed by the chains belonging to the other NP, hence returning to their
unperturbed size. By comparing this behavior with the PMF reported in Fig.~\ref{fig:gold-28} and
Fig.~\ref{fig:gold-38}, one can note that the shift appears for the same interparticle separation
where the PMF vanishes, in agreement with the recovering of the unperturbed state of the chains.
A summarizing comparison between all the PMF calculated for gold NPs is given in Fig.~\ref{fig:gold-all}:
in particular, the increase of the repulsion strength for low interparticle distance is clearly visible.
In the inset, the region around PMF minima is magnified: here, the progressive shift of the minimum
towards higher values of the interparticle separation is enhanced, in agreement with the increase
of $\langle R_g \rangle$ and $\langle EtE \rangle$ previously discussed.
\section{Conclusions}
In the present work we have investigated the behavior of the potential of mean
force (PMF) between atomistic silica or gold nanoparticles (NPs). By performing
GROMACS molecular simulations in the canonical ensemble, we have calculated the net forces between a
pair of such particles, hence obtaining the PMF through an integration over
their mutual distances.
In the case of silica nanoparticles, the effects
due to the particle size have been taken into account upon calculating the
PMF for particles of 2.5 and 4 nm of diameter. No significant discrepancies
between the two cases have been observed, the only remarkable difference being the
appearance of a peak in the PMF between the two larger particles. Such an
effect is likely due to surface interactions given by the formation and
breaking of hydrogen bonds. In addition, we have performed a comparison between
simulation data and an analytical theory
due to Hamaker and obtained by employing the Lennard-Jones parameters of
oxygen and silicon atoms. As a result, a good agreement between theory and simulations
has been found, this suggesting
that for bare spherical particles made by a large number of atoms the overall behavior
of the PMF can be well approximated by means of a combination of site-site
Lennard-Jones potentials.
We have then investigated the behavior of PMF between
gold nanoparticles, which have been
considered in a first instance bare and then coated with an increasing number
of polyethilene chains; in such a way it is possible to detect the effect of the grafting density
$\rho_g$ on the PMF. We have found that bare gold NPs experience little
interactions and surface effects are dominant; upon coating the particles
with polyethilene chains, the profile of the PMF is deeply modified, with the
appearance of a short-range attraction characterized by a large attractive
well. Overall, the resulting PMF appears quite similar to a Lennard-Jones
potential, but for the attractive well that appears broader, extending for
distances of $\simeq$ 1 nm. An high repulsive interaction is detected for low
interparticle distances and high grafting densities.
The present work can be considered a preliminary and necessary study suited
to fully characterize effective interactions in high-density coarse-grained polymer nanocomposites,
where the knowledge of atomistic potentials constitute a crucial issue that need to be
properly taken into account. Such studies will constitute the main target of a forthcoming
investigation.
\section{Acknowledgements}
The computing resources and the related technical support used for this work
have been provided by CRESCO/ENEAGRID High Performance Computing infrastructure
and its staff~\cite{Cresco}. CRESCO/ENEAGRID High Performance Computing
infrastructure is funded by ENEA, the Italian National Agency for New
Technologies, Energy and Sustainable Economic Development and by Italian and
European research programmes, see http://www.cresco.enea.it/english for
information. G.~Muna\`o and G.~Milano acknowledge financial support from
the project PRIN-MIUR 2015-2016.
\section{Author contribution statement}
Giuseppe Milano suggested the research topic and coordinated the preparation
of the manuscript. Gianmarco Muna\`o wrote the paper and performed numerical
simulations along with Andrea Correa. Gianmarco Muna\`o and Antonio Pizzirusso
performed the structural analysis. All authors contributed to the
critical reading of the manuscript.
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|
{
"timestamp": "2018-03-08T02:10:10",
"yymm": "1803",
"arxiv_id": "1803.02727",
"language": "en",
"url": "https://arxiv.org/abs/1803.02727"
}
|
\section{Introduction}
In the first-past-the-post (FPTP) voting mechanism, each voter indicates his/her candidate of choice on a ballot, and the candidate that received the most votes wins. This is a popular voting mechanism that for instance is used in the United States. FPTP voting is related to consensus or Byzantine agreement in the sense that each voter/node has an input, and the voters need to decide on a single output. If Byzantine voters are present, it is natural to try to agree on a candidate with many votes to be robust against Byzantine behavior, e.g., \cite{BenOr,BrachaRB,QueenAlgorithm}.
FPTP does however face a lot of criticism since it has several issues, in particular the problem of wasted votes to minority parties, but also tactical voting or gerrymandering \cite{orvis2013introducing}. Moreover, one may argue that FPTP will eventually lead to a two-party system, e.g., \cite{sachs2011price}.
Preferential voting is a powerful alternative: Each voter \textit{ranks} the candidates first, second, third, etc. The collection of all rankings forms a \textit{preference profile} from which a winner (or even a ranking) is determined. Preferential voting is more expressive, harder to manipulate, and solves many of FPTP's problems. In particular there is no problem with wasted votes to minority candidates. If inputs are not just binary, preferential voting will lead to much better decisions. It is therefore remarkable that byzantine agreement research has not given preferential voting any attention.
In this paper we want to investigate how robust preferential voting is in a Byzantine environment.
In Section \ref{sec:motivation}, we first focus on some basic properties for voting rules, and see that not all of them can be satisfied if the nodes should reach agreement. This is because Byzantine voters are manipulators that modify the result to make it more favorable to themselves. In the main part of the paper (Section \ref{sec:kemeny}) we study how well the voting result intended by the correct (non-Byzantine) voters can be approximated. For this purpose we introduce the Kemeny rule which picks the most central ranking as the voting result. We will provide an algorithm that approximates the solution of the Kemeny rule in the presence of Byzantine voters and prove that this algorithm computes the best possible approximation. We believe that our paper will help to get a deeper understanding of both fault-tolerant distributed systems as well as social choice theory.
\section{Background and Motivation}\label{sec:motivation}
In search of a \emph{fair} rule to elect candidates, philosophers and mathematicians started developing various voting mechanisms and rules already in the beginning of the 18th century. In the middle of the 20th century, Kenneth Arrow \cite{Arrow1951,Arrow1963} was one of the first to formalize existing rules and analyze possibility and impossibility results in an axiomatic fashion, thereby introducing the field of Computational Social Choice. In this section we use this formalism in order to show how well Byzantine agreement connects to voting theory.
We start by considering the special case of $n$ voters voting on only two candidates $c_1$ and $c_2$.
In this setting, each voter (node) ranks the two candidates such that its preferred candidate (input value) is ranked first. A vote for a candidate $c_1$ means that the voter strictly prefers $c_1$ to $c_2$, here denoted $c_1 \succ c_2$. A central authority then applies a \textit{social choice function (SCF)} to a given preference profile in order to determine the winner (decision value), or set of winners in case of a tie. An SCF $f$ can be qualified based on the following properties:
\begin{itemize}[noitemsep]
\item $f$ is \textit{anonymous} if interchanging two \textit{voters} (swapping their names) does not change the result
\item $f$ is \textit{neutral} if renaming the \textit{candidates} (changing their names) does not change the result
\item $f$ is \textit{positively responsive} if in a case where the decision is a tie ($c_1$ is among the winners) and a voter changes its ranking from $c_2 \succ c_1$ to $c_1 \succ c_2$, candidate $c_1$ becomes the unique winner
\end{itemize}
\noindent One example of an SCF is the \textit{majority rule}. It chooses the candidate that wins most pairwise comparisons against every other candidate. Note that such a winner always exists in elections with two candidates, but not necessarily in the general case with any number of candidates.
Social choice theory shows that the majority rule satisfies all desirable properties for the special case of voting on two candidates:
\begin{theorem}[May's Theorem \cite{May}]
For two candidates and any number of voters, the majority rule is the unique SCF that satisfies anonymity, neutrality and positive responsiveness.
\end{theorem}
Interestingly, most known algorithms for binary Byzantine agreement indirectly exploit the properties of May's theorem.
Some of them make use of leaders who suggest their decision value to all nodes \cite{KingAlgorithm, QueenAlgorithm}.
The leader in these algorithms temporarily plays what is known as a dictator in voting theory.
Another type of algorithm, e.g., the shared coin algorithm in \cite{BrachaRB}, is biased towards one of the outcomes and thus violates neutrality.
In general we can say that most of the proposed algorithms try to use the majority value as the decision value if a majority exists, or an arbitrary input value otherwise, see for example \cite{BenOr,BrachaRB}. Such settings may satisfy anonymity and neutrality, but in cases where the correct nodes are undecided, i.e. there is a tie between the two input values, Byzantine nodes have a large influence on the majority value. Thus, if a correct node decides to swap two candidates in its ranking in order to make one of the candidates win, a Byzantine node can perform an opposite swap in its own ranking and return the profile to the previous state. This shows that positive responsiveness cannot be satisfied for these algorithms in the presence of Byzantine nodes.
May's theorem does not apply to the general case with more than two candidates. In fact, the majority rule gives surprisingly bad results for three or more candidates. To illustrate this, let $m$ denote the number of candidates. Assume that $n/2+1$ voters rank the candidates as $c_1 \succ c_2 \ldots \succ c_{m}$, and all other voters rank the candidates as $c_2 \succ c_3 \succ \ldots \succ c_1$. In this case candidate $c_1$ wins every pairwise comparison according to the majority rule, even though $c_2$ seems to be the candidate that is approved by more voters.
Moreover, a lot of information is lost when a single winner is sought. When it comes to preferential voting, social choice theory therefore often wants not only the input to be rankings but also the output. More formally:
\begin{definition}[Social Welfare Function]
A Social Welfare Function (SWF) is a map from a preference profile to a set of consensus rankings.
\end{definition}
\noindent For an SWF $g$, the following three properties are usually considered:
\begin{itemize}[noitemsep]
\item $g$ is \textit{dictatorial} if there is one distinguished voter whose input ranking is chosen as the single consensus ranking
\item $g$ is \textit{independent of irrelevant alternatives (IIA)} if the consensus ranking of two candidates $c_i$ and $c_j$ only depends on the relative preference of these candidates in each voter's ranking, and not on the ranking of some third candidate $c_k$
\item $g$ is \textit{weakly Paretian} if it satisfies the weak Pareto condition \cite{pareto1919}: for two candidates $c_i$ and $c_j$ which are ranked $c_i \succ c_j$ by all voters, consensus ranking has to rank $c_i \succ c_j$ as well
\end{itemize}
\noindent In contrast to IIA and weak Pareto, dictatorship is a highly undesirable property in voting theory. Unfortunately, a corresponding result to May's theorem for SWF's on three or more candidates is the famous impossibility result by Arrow:
\begin{theorem}[Arrow's Impossibility Theorem \cite{Arrow1951}]\label{thm:Arrow}
If there are at least three candidates which the members of the society are free to order in any way, then every SWF that is weakly Paretian and IIA must be dictatorial.
\end{theorem}
From the viewpoint of Byzantine agreement, an SWF should not be dictatorial since one does not want a dictator
to be a Byzantine node and choosing more than one dictator may also result in different decision values. Consequently, any reasonable Byzantine agreement protocol must either violate IIA or weak Pareto.
The IIA condition implies that the consensus ranking should remain the same if the input of every correct node does not change, no matter what the Byzantine nodes do.
However, a Byzantine node can pretend to be a correct node but change its ranking in two executions in which the correct nodes have the same inputs. This change may lead to a different consensus ranking which would violate IIA.
For the weak Pareto condition consider the case with two candidates: if every non-Byzantine voter ranks $c_1 \succ c_2$, the consensus ranking should also rank $c_1 \succ c_2$. This corresponds to a well-known validity condition in Byzantine agreement -- the \textit{All\hspace{0.1em}-\hspace{0.05em}Same\hspace{0.1em}-\?Validity}: If all correct nodes have the same input value, all correct nodes have to decide on this value. We use the weak Pareto condition to impose a validity rule on Byzantine Agreement with rankings:
\begin{description
\item[Pareto\,-\?Validity] for any pair of candidates $c_i$ and $c_j$: if all correct nodes rank $c_i\succ c_j$, then the consensus ranking should rank $c_i\succ c_j$ as well.
\end{description}
Given $m$ candidates, Pareto\,-\?Validity can be viewed as All\hspace{0.1em}-\hspace{0.05em}Same\hspace{0.1em}-\?Validity applied on each of the $\binom{m}{2}$ pairs of candidates in a ranking. Note that Byzantine agreement on a ranking is at least as hard as binary Byzantine agreement: Consider a case where the nodes agree on the ranking of the candidates $c_{3}, \ldots c_{m}$ which they rank last, but not on the two first candidates $c_1$ and $c_2$. The Pareto condition is then satisfied for every binary relation which contains at least one of the candidates $c_{3}, \ldots c_{m}$. Agreement in this case is reduced to binary Byzantine agreement on the two candidates $c_1$ and $c_2$, under the All\hspace{0.1em}-\hspace{0.05em}Same\hspace{0.1em}-\?Validity condition.
Unfortunately, there is no straightforward way to apply a binary Byzantine agreement protocol to solve Byzantine agreement on rankings. Other than binary relations on two candidates, preference profiles can form cycles, e.g., they can contain all three relations $c_i\succ c_j$, $c_j\succ c_k$ and $c_k\succ c_i$ which are each preferred by a majority of nodes.
The smallest preference profile which produces such a cycle of binary relations is called a \textit{Condorcet cycle}. It contains three rankings $c_i\succ c_j\succ c_k$, $c_j\succ c_k\succ c_i$ and $c_k\succ c_i\succ c_j$ which induce the three relations from above. Simply agreeing on each pair of candidates can thus lead to a circular decision which does not form a ranking. In order to get rid of cycles one could think of applying the quicksort algorithm on the candidates sorted with respect to the majority. This procedure however violates Pareto\,-\?Validity: Consider a candidate $c_i$ that Pareto dominates candidate $c_j$. Assume that the quicksort algorithm compares both candidates to some third candidate $c_k$ first. Then $c_j$ might win against $c_k$ and $c_i$ might lose, thus swapping $c_i$ and $c_j$ in the consensus ranking.
This consideration makes the problem of finding a consensus ranking in the presence of Byzantine nodes rather an instance of multi-valued agreement, as we discuss in Section \ref{sec:king}, which makes the problem both interesting and challenging.
\section{Related Work}
Byzantine agreement was first proposed as the Byzantine Generals problem by Pease, Shostak and Lamport \cite{ByzantineGeneralsPease,ByzantineGeneralsLamport}.
In these papers the authors showed that three nodes cannot establish agreement in the presence of one Byzantine node even if the communication system is synchronous. Given $n$ nodes, it was shown for the synchronous model that at least $t+1$ rounds are required to establish agreement \cite{FischerLynchMinRounds}, where $t<n/3$ is the number of Byzantine nodes in the system; the corresponding upper bound was provided in \cite{KingAlgorithm,QueenAlgorithm}. For the asynchronous model, the FLP impossibility result \cite{FLPimpossibility} states that there is no deterministic agreement protocol which can tolerate even one Byzantine node. The first randomized algorithm for solving Byzantine agreement proposed in \cite{BenOr} had expected exponential running time for a constant fraction of Byzantine nodes. This result was recently improved in \cite{KingSaia2016}, where the authors showed that it is possible to establish agreement within expected polynomial running time using spectral methods.
Byzantine agreement with more than two input values has mostly been considered in approximate agreement \cite{ApproximateAgreement,ApproximateAgreement2}, where the input values of the nodes converge towards some value over rounds. More recent results seek to establish agreement on a value that makes sense for applications. In \cite{PowerOfTwoChoices}, the values converge towards a value at most $\sqrt{n\log{n}}$ positions away from the median. In \cite{MedianValidity,IntervalValidity} an exact algorithm to establish agreement on a value that is at most $t/2$ positions away from the median or $t$ positions away from a minimum or a maximum was proposed. In \cite{VectorConsensus,VectorConsensusAsynch,VectorConsensusJointWork}, Byzantine agreement was further generalized to several dimensions and the nodes converge to a vector inside the convex hull of all correct input vectors. While the one-dimensional case has been investigated in depth, all previous approaches for multiple dimensions struggle to derive an algorithm which either can tolerate a constant fraction of Byzantine nodes independent on the number of dimensions, or find a solution that is not trivial.
In social choice theory, Byzantine behavior can be interpreted as manipulation of a ballot in an election, in which the manipulating party has full knowledge about all votes. Bartholdi et al.\ \cite{BartholdiManipulation} defined manipulation as a preference profile where one single voter can change its ranking such that this voter's most preferred candidate wins the election. Groups of voters have also been considered in this context, but mostly from the perspective of how hard it is for a group of nodes to manipulate the voting result given a certain voting rule \cite{manipulationHard1,manipulationHard2}. Other types of Byzantine behavior have been considered with respect to robustness of proposed voting rules. In \cite{robustVoting}, the authors investigate robustness of Borda's mean and median in the presence of outlier ballots. In \cite{robustVotingNoise}, robustness of scoring rules is considered under arbitrary noise which is described in terms of pairwise swaps of candidates in the ranking of one voter.
In this paper we will consider the Kemeny rule which was first proposed in \cite{Kemeny1959,KemenySnell}. The corresponding Kemeny median satisfies additional properties to those presented in Section \ref{sec:motivation}, but it was shown to be NP-hard to compute for an increasing number of candidates and already for four voters in \cite{BartholdiNPhard,DworkNPhard}.
At least three different $2$-approximation algorithms for the Kemeny median have been proposed in \cite{Ailon} and \cite{DiaconisGraham} respectively. In \citep{Ailon}, the approximation ratio was improved to $4/3$ using randomization, and later derandomized in \cite{vanZuylen2008}. A good overview over the Kemeny rule and an extended introduction into social choice theory can be found in \cite{Brandt2016}.
\section{A Deterministic Algorithm for Pareto\,-\?Validity} \label{sec:king}
This section focuses on Byzantine agreement protocols for rankings that satisfy Pareto\,-\?Validity. By using single transferable voting and a multi-valued Byzantine agreement algorithm, a ranking satisfying Pareto\,-\?Validity can be obtained in $(m-1)\cdot (t+1)$ rounds: In the first $t+1$ rounds, we let the voters apply the King algorithm in order to agree on the top candidate. Then every node removes this candidate from its ranking. In the next step, they will agree on the top candidate from the reduced rankings, and so on. While this procedure is simple, the number of rounds depends not only on the number of nodes, but also on the number of candidates.
In the following we present a deterministic algorithm which solves this problem in only $t+1$ rounds using the same number of messages. We do this by modifying the King algorithm to broadcast rankings instead of single candidates. In the proposed algorithm we select $t+1$ different nodes and assign each of them to one of the $t+1$ rounds of the algorithm. Such a node is called a dictator of the corresponding round. This dictator then suggest its own, possibly adjusted, ranking to all nodes, which will always be accepted if the dictator is a correct node. This way, dictators decide on the ranking of all pairs of candidates which do not satisfy the Pareto\,-\?Validity. Algorithm \ref{alg:king} presents this procedure in pseudocode.
\algblockdefx{MyIf}{EndMyIf}[1]{\textbf{if} #1 \textbf{fix $\mathbf{c_k \succ c_l}$}}{\textbf{end if}}
\begin{algorithm}
\begin{algorithmic}[1]
\Statex Every node $v$ executes the following algorithm
\For{ round $1$ to $t+1$}
\Statex \hspace{\algorithmicindent}\textit{Communication Phase:}
\Indent
\State Broadcast own input ranking $r_v$
\For{all pairs of candidates $c_i$ and $c_j$}
\If{$c_i$ is ranked above $c_j$ in at least $n-t$ rankings}\label{step:propose}
\State Broadcast ``propose $c_i \succ c_j$''
\EndIf
\EndFor
\MyIf{some ``propose $c_k\succ c_l$'' received at least $t+1$ times}\label{step:adjustRanking}
\State Adjust own ranking $r_v$ such that $c_k$ appears before $c_l$
\EndMyIf
\EndIndent
\Statex \hspace{\algorithmicindent}\textit{Dictator Phase:}
\Indent
\State Let node $w$ be the predefined dictator of the current round
\State The dictator broadcasts its ranking $r_{dictator} \coloneqq r_w$
\EndIndent
\Statex \hspace{\algorithmicindent}\textit{Decision Phase:}
\Indent
\If{$r_{dictator}$ agrees with $r_v$ in all fixed pairs $c_i \succ c_j$ from step \ref{step:adjustRanking}}\label{step:adaptDictator}
\State $r_v \coloneqq r_{dictator}$
\EndIf
\EndIndent
\EndFor
\State Return $r_v$
\end{algorithmic}
\caption{Byzantine agreement protocol on rankings (for $t < n/3$)}
\label{alg:king}
\end{algorithm}
Since we are dealing with rankings, it is not trivial to see that the nodes will always be able to agree on a proper ranking at the end of the algorithm. In the following lemmas we will prove that the nodes can adjust their rankings in Step \ref{step:adjustRanking} of Algorithm \ref{alg:king} in order to guarantee Pareto\,-\?Validity and that the outcome of the algorithm will be a proper ranking. It is easy to see that the algorithm is correct for $t < n/4$ Byzantine nodes, since the correct nodes will not be able to propose binary relations which form a Condorcet cycle in this case.
In order to show that the algorithm can tolerate $t < n/3$ Byzantine nodes, we need to exploit the fact that no Byzantine node can propose relations that form a Condorcet cycle at any point of the algorithm.
\begin{lemma}\label{lemma:noCondorcet}
There is no Condorcet cycle that can be proposed by the correct nodes if $t < n/3$.
\end{lemma}
\begin{proof}
Assume by means of contradiction that the three relations $c_i\succ c_j$, $c_j\succ c_k$ and $c_k\succ c_i$ were each proposed by at least $t+1$ nodes in Step \ref{step:propose} of Algorithm \ref{alg:king}. Each binary relation was proposed by at least one correct node who must have seen $n-t$ nodes having a ranking with such a pair.
Let $t_1$ be the number of all Byzantine nodes who proposed $c_i \succ c_j$, $t_2$ the number of those who proposed $c_j \succ c_k$ and $t_3$ those nodes who proposed $c_k \succ c_i$. Further, let $t_{1\cap 2}$ denote the number of Byzantine nodes who proposed $c_i \succ c_j \succ c_k$. The following inequality then holds: $t_1 + t_2 -t_{1\cap 2} \leq t$.
The number of correct nodes who proposed $c_i \succ c_j \succ c_k$ is then $(n-t-t_1)+(n-t-t_2)+t_{1\cap 2} - (n-t) = n-t-t_1-t_2+t_{1\cap 2}$. The number of correct nodes who proposed $c_k\succ c_i$ is $n-t-t_3\geq n-2t$. However, the two sets must have a nonempty intersection, since $$n-t-t_1-t_2+t_{1\cap 2} + n-2t -(n-t) = n-2t -t_1-t_2+t_{1\cap 2} \geq n-3t > 1.$$ Therefore, there must be at least one correct node who proposed $c_i\succ c_j\succ c_k$ and $c_k\succ c_i$ simultaneously. This is a contradiction.
\end{proof}
Note that by the properties of the King algorithm, no two opposite binary relations can be proposed in Step \ref{step:propose} simultaneously. Lemma \ref{lemma:noCondorcet} additionally shows that a Condorcet cycle cannot be proposed in Step \ref{step:propose} and thus all proposed pairs can form a ranking. It remains to show that the nodes will always be able to adjust their rankings to incorporate the proposed pairs.
\begin{lemma}\label{lem:adjustRanking}
In Step \ref{step:adjustRanking} a correct node will always be able to incorporate the proposed pairs into its own ranking.
\end{lemma}
\begin{proof}
This can be achieved by the following strategy: Divide the candidates into two sets. The first set contains all candidates which are in at least one of the pairs proposed by the $t+1$ nodes in Step \ref{step:adjustRanking}. This set of nodes will be ranked first. The second set will contain all candidates for which the node has not received any propose message. The candidates will be ranked second and will be dominated by all candidates from the first set. Next, we can rank all candidates in the first set according to the proposed relations, possibly leaving some pairs of the candidates not ranked. In the last step, all candidates which have not been ranked in each of the sets can be ranked by choosing binary relations from the local ranking of the node. This strategy outputs a ranking of candidates in which all proposed binary relations are satisfied.
\end{proof}
The next theorem summarizes the correctness results of Algorithm \ref{alg:king} and states that the consensus ranking will be valid, which can be derived with help of previous lemmas. The corresponding proof can be found in Appendix \ref{app:kingAlgo}.
\begin{theorem}\label{thm:kingAlgo}
At the end of Algorithm \ref{alg:king} all nodes will have agreed on the same ranking which additionally satisfies Pareto\,-\?Validity.
\end{theorem}
\section{Kemeny Median with Byzantine Nodes}\label{sec:kemeny}
Weakly Paretian voting rules are often not sufficient to pick the most fair ranking from a set of individual preference rankings. In search of the best possible consensus ranking we have to add restrictions on the voting rules without violating the known impossibility results of \cite{Arrow1951}. This leads us to majoritarian SWFs, one of which is the Kemeny rule.
In the following we will introduce the Kemeny rule and use it to derive a better consensus ranking in the presence of Byzantine nodes. Since Byzantine nodes have influence on the final ranking, the corresponding solutions can be qualified with respect to their approximation ratio which we define in Section \ref{sec:approx}. In Section \ref{sec:kemenyLB}, we will derive a lower bound on the approximation ratio of the Kemeny median in the presence of Byzantine nodes and further provide a matching upper bound in Section \ref{sec:kemenyAlgo}.
\begin{definition}[Kendall's $\tau$ distance \cite{Kendall}]
The Kendall's $\tau$ distance measures the distance between two rankings $r$ and $p$ on candidates $c_1,\ldots,c_m$ by counting pairs of candidates on which they disagree:
$$\tau(r, p) \triangleq | \{ (c_i,c_j) \mid c_i \succ_r c_j \hbox{ and } c_j \succ_p c_i \} | .$$
\end{definition}
\noindent This metric $\tau$ on ballots can be extended to a distance function between a ranking $r$ and a profile $\mathcal{P}$:
$$\tau(r, \mathcal{P}) \triangleq \sum_{p\in \mathcal{P}} \tau(r, p).$$
\begin{definition}[Kemeny median]
For a given profile $\mathcal{P}$, the Kemeny median is the ranking $r$ which minimizes $\tau(r,\mathcal{P}).$
\end{definition}
The Kemeny median satisfies many nice properties and to some extent guarantees that the chosen ranking is ``fair''. The most prominent quality is probably \textit{monotonicity}: if voters increase a candidate's preference level, the ranking result either does not change or the promoted choice increases in overall popularity. This quality makes the median solution more robust to Byzantine behavior. The Kemeny rule is also a so called Condorcet method because if there exists a Condorcet winner, i.e., a candidate that wins all pairwise majority comparisons, it will always be ranked as the most popular choice. Besides, it only depends on the number of voters who prefer one alternative over the other and it is reinforcing, meaning that winners which were chosen independently by two different sets of voters will also become winners if the two groups are joined.
\subsection{Byzantine Setting}\label{sec:approx}
The Kemeny median cannot be computed exactly in the presence of Byzantine nodes since they might suggest rankings which have a large distance to the Kemeny median of the correct nodes thus moving the median preference away from the actual median. A notion for approximate median rankings is therefore introduced as follows:
\begin{definition}[$\alpha$-approximation of Kemeny median]
Let $m$ be a Kemeny median of a preference profile $\mathcal{P}$. An $\alpha$-approximation of $m$ is a preference ranking $m_\alpha$ satisfying $$\tau(m_\alpha,\mathcal{P})\leq \alpha\cdot \tau(m,\mathcal{P})$$
\end{definition}
As an example consider binary agreement ($m=2$): Here $\tau$ counts the number of correct nodes who disagree with the consensus value. Any binary Byzantine agreement algorithm that satisfies All-\hspace{0.05em}Same\hspace{0.1em}-\?Validity will also satisfy $\alpha < n-t-1$.
Unlike binary agreement, it is not straightforward to see what a Byzantine node would choose as its ranking when the Kemeny rule determines the consensus ranking. Since the input vectors of nodes are rankings, each voter has to propose a strict order between candidates and the corresponding preference relation is transitive. A possible strategy for the Byzantine nodes would then be to choose exactly the opposite ranking of the Kemeny median of all correct nodes. We show in Appendix \ref{app:oppositeRanking} that this strategy works, but such a solution is not unique for most preference profiles. It is therefore difficult for the correct nodes to find out which of the rankings might have been Byzantine.
\subsection{Lower Bounds on the Approximation Ratio}
\label{sec:kemenyLB}
In this section we discuss preference profiles that are vulnerable to Byzantine nodes. The first case is based on the reduction of rankings to binary agreement and gives the highest approximation ratio for $t<n/3$.
Binary agreement does however assume that there are two groups of voters who completely disagree in their preferences. This is somewhat unlikely in practice when $m$ is sufficiently large.
In the second case we therefore exclude such binary instances and provide a lower bound based on Condorcet cycles within a preference profile which converges to the same value for large $m$. The approximation ratio usually depends on the ratio $n/t$, which will be denoted $k$ for the sake of simplicity.
For our analysis, we represent the preference profile $\mathcal{P}$ as a weighted \emph{tournament graph}, i.e., a graph where the nodes represent the candidates and weighted edges represent how many voters prefer one candidate to the other. The sum of the forward and the backward edge should be equal to the total number of rankings in the corresponding preference profile.
The ranking of a node is a directed Hamiltonian path following the order of the ranking, and all other edges are derived from the transitivity.
For any two candidates we call the edge between these candidates a \emph{majority edge} if its backward edge has a smaller weight. The backward edge we then call a \emph{minority edge}. A Kemeny median of a weighted tournament graph is the ranking that minimizes the sum of the weights of all backward edges of the graph.
Note that rankings restrict the power of Byzantine nodes in the sense that Byzantine nodes can only send transitive tournament graphs where every edge has weight $1$. Using tournament graphs, we can derive a lower bound for the binary case:
\begin{theorem}\label{thm:noCycles}
There is a tournament graph corresponding to a preference profile for which the Byzantine nodes may change the edge weights such that the median of the resulting preference profile is a $\frac{k}{k-2}$-approximation of the optimal median, where $k = n/t$. For $t$ close to $n/3$, this gives a $3$-approximation.
\end{theorem}
\begin{proof}
This tournament graph is equivalent to binary agreement. Consider therefore one pair of candidates: $t$ Byzantine nodes are only able to change the median, i.e., the majority edge, between two candidates if they can swap the majority and minority edge by supporting the minority edge with their ranking.
Assume the worst case, where the forward and the backward edge both have the same weight $n/2$ after the Byzantine nodes have added their preferences.
In the worst case the tournament graph of correct nodes had the weight $n/2$ for the majority edge. Since the correct nodes will not be able to determine the actual majority edge, they might agree on a minority edge with weight $n/2-t$ instead. The corresponding approximation ratio is then $\frac{n/2}{n/2-t} = \frac{k}{k-2}$.
Figure \ref{fig:worstCaseExample} shows a simple generalization of this argument to $m$ candidates and proves that the lower bound of $\frac{k}{k-2}$ holds, for all $m$.
\end{proof}
\def 0.45 {0.45}
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~\textbf{+}~
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~\textbf{=}~
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\node[state,minimum size=0pt] (FC2) at (0,1) {$c_2$};
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\node[state, fill=black!50, scale=0.1] at (0,-0.5) {};
\node[state, fill=black!50, scale=0.1] at (0,-1) {};
\node[state, fill=black!50, scale=0.1] at (0,-1.5) {};
\path (FC1) edge [->,thick,>=latex, bend left, blue] node [right] {} (FC2)
edge [->,thick,>=latex, bend left, blue] node [right] {$\frac{n}{2}$} (FCm)
(FC2) edge [->,thick,>=latex, bend left, blue] node [right] {} (FCm);
\path (FCm) edge [->,thick,>=latex, bend left, red] node [left] {} (FC2)
edge [->,thick,>=latex, bend left, red] node [left] {$\frac{n}{2}$} (FC1)
(FC2) edge [->,thick,>=latex, bend left, red] node [left] {} (FC1);
\end{tikzpicture}
~\textbf{=}~
\begin{tikzpicture}[baseline={(0,0)},scale=0.45]
\node[state,minimum size=0pt] (C11) at (0,3.8) {$c_1$};
\node[state,minimum size=0pt] (C12) at (0,1) {$c_2$};
\node[state,minimum size=0pt] (C1m) at (0,-3) {$c_m$};
\node[state, fill=black!50, scale=0.1] at (0,-0.5) {};
\node[state, fill=black!50, scale=0.1] at (0,-1) {};
\node[state, fill=black!50, scale=0.1] at (0,-1.5) {};
\path (C11) edge [->,thick,>=latex, bend left, blue] node [right] {} (C12)
edge [->,thick,>=latex, bend left, blue] node [right] {$\frac{n}{2}-t$} (C1m)
(C12) edge [->,thick,>=latex, bend left, blue] node [right] {} (C1m);
\path (C1m) edge [->,thick,>=latex, bend left, red] node [left] {} (C12)
edge [->,thick,>=latex, bend left, red] node [left] {$\frac{n}{2}$} (C11)
(C12) edge [->,thick,>=latex, bend left, red] node [left] {} (C11);
\end{tikzpicture}
~\textbf{+}~
\begin{tikzpicture}[baseline={(0,0)},scale=0.45]
\node[state,minimum size=0pt] (F11) at (0,3.8) {$c_1$};
\node[state,minimum size=0pt] (F12) at (0,1) {$c_2$};
\node[state,minimum size=0pt] (F1m) at (0,-3) {$c_m$};
\node[state, fill=black!50, scale=0.1] at (0,-0.5) {};
\node[state, fill=black!50, scale=0.1] at (0,-1) {};
\node[state, fill=black!50, scale=0.1] at (0,-1.5) {};
\path (F11) edge [->,thick,>=latex, bend left, blue] node [right] {} (F12)
edge [->,thick,>=latex, bend left, blue] node [right] {$t$} (F1m)
(F12) edge [->,thick,>=latex, bend left, blue] node [right] {} (F1m);
\end{tikzpicture}
\end{center}
\label{fig:worstCaseExample}
\caption{Two indistinguishable views on $m$ candidates for binary relations}
\medskip
\small
Note that the labels of the edges correspond to all edges in the same color. The left tournament graph is reached if $n/2$ nodes choose the order $c_1 \succ c_2 \succ \ldots\succ c_m$ and $n/2-t$ nodes choose $c_m \succ c_{m-1} \succ \ldots \succ c_1$. The right profile can be reached from a profile where $n/2$ nodes choose the order $c_m \succ c_{m-1} \succ \ldots \succ c_1$ and $n/2-t$ nodes choose $c_1 \succ c_2 \succ \ldots\succ c_m$. The Byzantine nodes can make all correct nodes see the tournament graph in the center by adding $t$ preference vectors $c_m \succ c_{m-1} \succ \ldots \succ c_1$ or $c_1 \succ c_2 \succ \ldots\succ c_m$ respectively.
\end{figure}
Now we present another lower bound using Condorcet cycles which can result in ambiguous views as well. Here we assume that every majority edge has a weight of more than $n/2$, thus discarding the possibility to reduce any pair of forward and backward edges in the tournament graph to binary agreement. The main difficulty in finding a good example comes from the fact that not every tournament graph has an underlying preference profile. In Appendix \ref{app:LBcycles} we discuss the necessary properties that preference profiles induce on tournament graphs and show how the best lower bound can be derived. The next theorem present the best lower bound for cycles and its generalization to $m$ candidates.
\begin{theorem} \label{thm:cycles}
By modifying directed majority cycles in the tournament graph, Byzantine nodes can increase the approximation ratio by a factor of at most $5/4$.
\end{theorem}
\begin{proof}
We start by considering a tournament graph formed by one directed cycle of candidates $c_1,\ c_2,\ c_3$, i.e., one directed cycle formed by majority edges.
Assume all correct nodes receive a view where $n-2t$ nodes prefer $c_1$ to $c_2$, i.e., $(c_1, c_2)$ is a majority edge; $n/2 + t$ nodes prefer $c_2$ to $c_3$ and $n/2+t$ nodes prefer $c_3$ to $c_1$.
For $n>8f$ or equivalently $k>8$, the edge $(c_1,c_2)$ is in the median ranking of all nodes. Since the edges $(c_2,c_3)$ and $(c_3,c_1)$ cannot be both in the median ranking, the nodes have to decide for one of the rankings. In the worst case, one of these two edges was supported by all $t$ Byzantine nodes while the other edge was not supported by any Byzantine node. This leads to two views which are not distinguishable for the correct nodes, as shown in Figure \ref{fig:worstCaseExampleCircle}.
The approximation ratio for these views is $\frac{2t+n/2-t+n/2+t}{2t+n/2-2t+n/2}= \frac{n+2t}{n} = \frac{k+2}{k} < \frac{5}{4}$.
An extension to $m$ candidates gives an approximation ratio of
$\frac{2t + (m-2)\cdot(n/2-t) + (m-2)\cdot(n/2+t)}{2t + (m-2)\cdot n/2 + (m-2)\cdot(n/2-2t)} = \frac{2t+(m-2)\cdot n}{2t+(m-2)\cdot(n-2t)} = \frac{2+(m-2)k}{2+(m-2)\cdot(k-2)} \approx \frac{k}{k-2}$ for large $m$.
\end{proof}
\def 0.45 {0.45}
\begin{figure}
\begin{center}
\begin{tikzpicture}[baseline={(0,2)},scale=0.45]
\node[state,minimum size=0pt] (C1) at (4,-2) {$c_1$};
\node[state,minimum size=0pt] (C2) at (-4,-2) {$c_2$};
\node[state,minimum size=0pt] (C3) at (0,2) {$c_3$};
\node[state,minimum size=0pt] (C4) at (0,5.2) {$c_4$};
\node[state,minimum size=0pt] (Cm) at (0,8.4) {$c_m$};
\node[state, shape=ellipse,minimum height =3.9cm, minimum width=3.1cm ] (x) at (-0.05,5.2) {};
\node[state, fill=black!50, scale=0.1] at (0,6.4) {};
\node[state, fill=black!50, scale=0.1] at (0,6.8) {};
\node[state, fill=black!50, scale=0.1] at (0,7.2) {};
\path (C1) edge [->,thick,>=latex, bend left] node [below] {$n-3t$} (C2)
(C2.west) edge [->,thick,>=latex, bend left] node[left] {$\frac{n}{2}+t$} (x.west)
(x.east) edge [->,thick,>=latex, bend left] node [right] {$\frac{n}{2}$} (C1.east)
(C1) edge [->,thick,>=latex, bend right] node [left,yshift=-6pt] {$\frac{n}{2}-t$} (x)
(x) edge [->,thick,>=latex, bend right] node [right, xshift= -1pt,yshift=-6pt] {$\frac{n}{2}-2t$} (C2)
(C2) edge [->,thick,>=latex, bend left] node [below] {$2t$} (C1)
(Cm) edge [->,thick,>=latex, bend left] node [right] {$n-t$} (C4)
(Cm) edge [->,thick,>=latex, bend right] node [left] {$n-t$} (C3)
(C4) edge [->,thick,>=latex, bend left] node [right] {$n-t$} (C3);
\end{tikzpicture}
~$\mathbf{\longleftrightarrow}$~
\begin{tikzpicture}[baseline={(0,2)},scale=0.45]
\node[state,minimum size=0pt] (C1) at (4,-2) {$c_1$};
\node[state,minimum size=0pt] (C2) at (-4,-2) {$c_2$};
\node[state,minimum size=0pt] (C3) at (0,2) {$c_3$};
\node[state,minimum size=0pt] (C4) at (0,5.2) {$c_4$};
\node[state,minimum size=0pt] (Cm) at (0,8.4) {$c_m$};
\node[state, shape=ellipse,minimum height =3.9cm, minimum width=3.1cm ] (x) at (-0.05,5.2) {};
\node[state, fill=black!50, scale=0.1] at (0,6.4) {};
\node[state, fill=black!50, scale=0.1] at (0,6.8) {};
\node[state, fill=black!50, scale=0.1] at (0,7.2) {};
\path (C1) edge [->,thick,>=latex, bend left] node [below] {$n-3t$} (C2)
(C2.west) edge [->,thick,>=latex, bend left] node[left] {$\frac{n}{2}$} (x.west)
(x.east) edge [->,thick,>=latex, bend left] node [right] {$\frac{n}{2}+t$} (C1.east)
(C1) edge [->,thick,>=latex, bend right] node [left,yshift=-6pt] {$\frac{n}{2}-2t$} (x)
(x) edge [->,thick,>=latex, bend right] node [right,xshift=-1pt,yshift=-6pt] {$\frac{n}{2}-t$} (C2)
(C2) edge [->,thick,>=latex, bend left] node [below] {$2t$} (C1)
(Cm) edge [->,thick,>=latex, bend left] node [right] {$n-t$} (C4)
(Cm) edge [->,thick,>=latex, bend right] node [left] {$n-t$} (C3)
(C4) edge [->,thick,>=latex, bend left] node [right] {$n-t$} (C3);
\end{tikzpicture}
\end{center}
\label{fig:worstCaseExampleCircle}
\caption{Two indistinguishable views on $m$ candidates for directed cycles}
\medskip
\small
We have two views which show the profiles of correct nodes only. The left tournament graph results from a profile where $n/2-t$ nodes choose $c_1 \succ c_2 \succ c_m \succ \ldots \succ c_3$, $n/2-2t$ nodes choose $c_m \succ \ldots \succ c_3 \succ c_1 \succ c_2$ and $2t$ nodes choose $c_2 \succ c_m \succ \ldots \succ c_3 \succ c_1$.
The right tournament graph results from $n/2-2t$ nodes choosing $c_1 \succ c_2 \succ c_m \succ \ldots \succ c_3$, $n/2-t$ nodes choosing $c_m \succ \ldots \succ c_3 \succ c_1 \succ c_2$ and $2t$ nodes choosing $c_2 \succ c_m \succ \ldots \succ c_3 \succ c_1$.
If the Byzantine nodes add $t$ profiles $c_m \succ \ldots \succ c_3 \succ c_1 \succ c_2$ to the left view, and $t$ profiles $c_1 \succ c_2 \succ c_m \succ \ldots \succ c_3$ to the right view, the resulting profiles become indistinguishable to the correct nodes.
\end{figure}
The received approximation ratio converges to the same approximation ratio as in the binary case for large $m$.
\subsection{Algorithm for Kemeny Median Approximation}\label{sec:kemenyAlgo}
In this section we present a synchronous algorithm for computing a consensus median which matches the lower bound on the approximation ratio presented in the previous section.
A simple idea is to use interactive consistency \cite{BrachaRB,TouegRB}: For $t+1$ rounds, the nodes exchange all information they have received until that round and after the $(t+1)$-st round they compute the Kemeny median from a set of rankings which they have received often enough. This algorithm guarantees that the set of rankings will be the same for each node and therefore all nodes will decide on the same ranking. The main drawback of interactive consistency is that it has a large message complexity. Since each of the correct rankings will be forwarded by each of the correct nodes, this message complexity is in $\Theta(mn^t)$.
Instead of exchanging large amounts of information, we can directly exploit the fact that the Byzantine nodes cannot change a Kemeny median of the preference profile of the correct nodes by more than a transitive tournament graph with edge weights $t$. The corresponding strategy is presented in Algorithm \ref{alg:kemenyMedianOpt}.
\begin{algorithm}
\begin{algorithmic}[1]
\Statex Every node $v$ executes the following algorithm
\State broadcast own ranking $r_v$
\State compute the Kemeny median of the received preference profile, call it $m_v$ \label{step:computeMedian}
\State apply Algorithm \ref{alg:king} with $m_v$ as an input value \label{step:paretoOnMedian}
\end{algorithmic}
\caption{Byzantine agreement protocol for the Kemeny median (for $t < n/3$)}
\label{alg:kemenyMedianOpt}
\end{algorithm}
The presented algorithm has the same order of round and message complexity as Algorithm \ref{alg:king}.
\begin{theorem}\label{thm:correctnessAlg2}
Algorithm \ref{alg:kemenyMedianOpt} terminates within $t+3$ rounds exchanging $O(tn^2m\log{m})$ messages. The computed consensus ranking satisfies the lower bounds from Section \ref{sec:kemenyLB} and Pareto\,-\?Validity.
\end{theorem}
In the following we give two lemmas proving the correctness of this algorithm. A full proof of Theorem \ref{thm:correctnessAlg2} is provided in Appendix \ref{app:paretoOptAlgo}.
\begin{lemma}\label{lem:localMedians}
In Step \ref{step:computeMedian} of Algorithm \ref{alg:kemenyMedianOpt}, every correct node chooses a median ranking that matches the bounds from Section \ref{sec:kemenyLB}.
\end{lemma}
\begin{proof}
Instead of all nodes in the previous section, we can consider that the Byzantine nodes change just one node's view. Since the number of Byzantine nodes remains the same and the rankings of all correct nodes are received by every node in the synchronous communication model, the Byzantine nodes can in the worst case only reach the lower bound for any correct node, but not exceed it.
\end{proof}
\begin{lemma}\label{lem:optApprox}
The computed median ranking by Step \ref{step:paretoOnMedian} of the algorithm satisfies the approximation ratios from Section \ref{sec:kemenyLB}.
\end{lemma}
\begin{proof}
Observe that the consensus ranking is derived from a preference profile formed by the medians $m_v$. Unless some correct node disagrees on an edge in this profile, the edge will be inside the consensus median since Algorithm \ref{alg:king} satisfies Pareto\,-\?Validity. This edge may not be inside the median ranking of all correct nodes, but the approximation ratio still satisfies the bounds due to Lemma \ref{lem:localMedians}.
If the correct nodes disagree on an edge in the preference profile of medians, there was at least one correct node who either chose the opposite edge (binary case) for its median ranking or a different edge in a directed cycle (non-binary case).
Consider the binary case first. There, the forward and the backward edge chosen as the Kemeny median $m_v$ will both satisfy the lower bound, since there is a correct node choosing either of the cases.
For the non-binary case we need to consider directed cycles formed by the median rankings. Every directed cycle in a tournament graph implies that there is a directed sub-cycle formed by three candidates $c_i, c_j, c_k$. The corresponding preference profile of correct median rankings must contain the three opposite rankings $c_i \succ c_j \succ c_k$, $c_j \succ c_k \succ c_i$ and $c_k \succ c_i \succ c_j$. This, however, implies that all three median rankings could have been derived from the preference profile of the rankings $r_v$ by modifying edge weights by $t$. The only case from which such a situation can result is when the forward and backward edge weights of each of the three pairs of candidates differed by at most $t$ in the preference profile of rankings $r_v$. This case, again, is equivalent to the binary case and satisfies the lower bound.
\end{proof}
Note that the computed median can have a larger Kendall's $\tau$ distance to the preference ranking of all correct nodes than any $m_v$ has in the algorithm, since the Byzantine nodes can propose their own rankings as dictators in Algorithm \ref{alg:king}. Such a ranking would still satisfy the lower bound.
\section{Discussion and Future Work}
In this paper we introduced a new Byzantine agreement problem which extends binary Byzantine agreement to rankings. We showed that rules for choosing a consensus ranking in voting theory fit well with requirements from Byzantine agreement.
We further considered a special voting rule, the Kemeny median, for which we provided an optimal Byzantine agreement protocol that can tolerate up to $t<n/3$ Byzantine nodes. We do not claim to have chosen the best voting rule at this point, since such a rule simply does not exist due to impossibility results in voting theory. Instead, we think of our results as an inspiration to consider a larger pool of voting rules, such as approval voting, the Godgson's rule, and many others.
Byzantine agreement on rankings can also be of interest in several applications. Consider distributed machine learning as an example. Training data is usually collected by different (potentially Byzantine) parties and is stored in different places. When the amount of data is very large or the data is too sensitive to share,
it is impossible to transmit the data and then train a global model. Instead, these parties could train their own models locally and then vote to predict the labels for new data points. A data point could be a picture, and every trained model outputs a ranking on the possible labels (panda $\succ$ gibbon $\succ \ldots \succ$ cat).
The different parties could then use our Byzantine preferential voting protocols to find the best ranking.
|
{
"timestamp": "2018-03-08T02:10:04",
"yymm": "1803",
"arxiv_id": "1803.02720",
"language": "en",
"url": "https://arxiv.org/abs/1803.02720"
}
|
\section{Introduction and Main Results}
Let give some basic notation and terminology that will be used in this paper. Let $E$ be real vector space. Then $E$ is called ordered vector space if it has an order relation $\leq$ that means it is reflexive, antisymmetric and transitive, and also it satisfies the axioms; if $y\leq x$ then $y+z\leq x+z$ for all $z\in E$, and if $y\leq x$ then $\lambda y\leq \lambda x$ for all $0\leq \lambda$.
An ordered vector $E$ is said to be {\em vector lattice} (or, {\em Riesz space}) if, for each pair of vectors $x,y\in E$, the supremum $x\vee y=\sup\{x,y\}$ and the infimum $x\wedge y=\inf\{x,y\}$
both exist in E. Also, $x^+:=x\vee 0$, $x^-:=(-x)\vee0$, and $\lvert x\rvert:=x\vee(-x)$ are called the {\em positive} part, the {\em negative} part, and the {\em absolute value} of $x$, respectively. In a vector lattice $E$, a subset $A$ is called as {\em solid} if, for $y\in A$ and $x\in E$ with $\lvert x\rvert\leq\lvert y\rvert$, we have $x\in A$. Also, two vector $x$, $y$ in a vector lattice is said to be {\em disjoint} whenever $\lvert x\rvert\wedge\lvert y\rvert=0$; for more details see \cite{ABur,ABPO}.
Let $E$ be vector lattice $E$ and $\tau$ be a linear topology on it. Then $(E,\tau)$ is said a {\em locally solid vector lattice} (or, {\em locally solid Riesz space}) if $\tau$ has a base which takes place with solid sets, for more details on these notions see \cite{ABur,ABPO,AB}. It is known that every linear topology $\tau$ on a vector space $E$ has a base $\mathcal{N}$ for the zero neighborhoods satisfying the followings; for each $V\in \mathcal{N}$, we have $\lambda V\subseteq V$ for all scalar $\lvert \lambda\rvert\leq 1$; for any $V_1,V_2\in \mathcal{N}$ there is another $V\in \mathcal{N}$ such that $V\subseteq V_1\cap V_2$; for each $V\in \mathcal{N}$ there exists another $U\in \mathcal{N}$ with $U+U\subseteq V$; for any scalar $\lambda$ and for each $V\in \mathcal{N}$, the set $\lambda V$ is also in $\mathcal{N}$; see for example \cite{AA,Tr}. Hence, every locally solid vector lattice satisfies these properties. Also, it follows from \cite[Thm.2.28]{ABur} that a linear topology $\tau$ on a vector lattice $E$ is a locally solid iff it is generated by a family of Riesz pseudonorms $\{\rho_j\}_{j\in J}$, where {\em Riesz pseudonorm} is a realvalued map $\rho$ on a vector space $E$ if it satisfies the following conditions; $\rho(x)\geq0$ for all $x\in X$; if $x=0$ then $\rho(x)=0$; $\rho(x+y)\leq\rho(x)+\rho(y)$ for all $x,\ y\in X$; if $\lim\limits_{n\to\infty}\lambda_n=0$ in $\mathbb{R}$ then $\rho(\lambda_nx)\to 0$ in $\mathbb{R}$ for all $x\in X$; if $\lvert x\rvert\leq \lvert y\rvert$ then $\rho(x)\leq \rho(y)$. Moreover, if a family of Riesz pseudonorms generates a locally solid topology $\tau$ on a vector lattice $E$ then $x_\alpha \tc x$ iff $\rho_j(x_\alpha-x)\to 0$ in $\mathbb{R}$ for each $j\in J$. In this article, unless otherwise, the pair $(E,\tau)$ refers to as a locally solid vector lattice, and the topologies in locally solid vector lattices are generated by families of Riesz pseudonorms $\{\rho_j\}_{j\in J}$. In this paper, unless otherwise, when we mention a zero neighborhood, it means that it always belongs to a base that holds the above properties. Recently, there are some studies on locally solid Riesz spaces; see for example \cite{AA,L,Z}.
Next, we give the concept of filters. Let $X$ be a set. A subset $\mathcal{F}$ of the power set of $X$ is said to be {\em filter} on $X$ if the followings hold; $\emptyset\notin \mathcal{F}$; if $A\in \mathcal{F}$ and $A\subseteq B$ then $B\in \mathcal{F}$; $\mathcal{F}$ is closed under finite intersections; see \cite{AB}. The second condition says that the set $X$ belongs to the filter on it. The filter can be defined tanks to its base. A nonempty subset $\mathcal{B}\subseteq\mathcal{F}$ is called a {\em filter base} for a filter $\mathcal{F}$, if $\mathcal{F}=\{F\subseteq X:\exists B\in\mathcal{B}, B\subseteq F\}$. A base $\mathcal{B}$ satisfies the following properties; $\mathcal{B}$ is nonempty; each $B\in\mathcal{B}$ is nonempty; for each $B_1, B_2\in\mathcal{B}$, there is another $B\in\mathcal{B}$ such that $B\subseteq B_1\cap B_2$.
Filter can be defined with nets. A given partially ordered set $I$ is called {\em directed} if, for each $a_1,a_2\in I$, there is another $a\in I$ such that $a\geq a_1$ and $a\geq a_2$. A function from a directed set $I$ into a set $E$ is called a {\em net} in $E$. Now, we give a relation between net and filter convergence. Let $(x_\alpha)_{(\alpha\in I)}$ be a net in the set $E$. The filter $\mathcal{F}$ which is associated of $(x_\alpha)_{(\alpha\in I)}$ is defined as follows; let $\hat{x}_\alpha=\{x_\alpha:\alpha_0\in I, \alpha\geq\alpha_0\}$, and so the collection $\mathcal{B}=\{\hat{x}_\alpha:\alpha\in I\}$ is a filter base and the filter that is generated by $\mathcal{B}$ is the associated filter of $(x_\alpha)_{(\alpha\in A)}$. So, we can give the following natural example.
\begin{exam}
Let $E$ be a set and $(x_\alpha)_{(\alpha\in I)}$ be a net in $E$. The elementary filter associated to $(x_\alpha)_{(\alpha\in I)}$ is
$$
\mathcal{F}_{(x_\alpha)}=\{F\subseteq E:\exists \alpha_0, \ x_\alpha\in F\ \text{for all} \ \alpha\geq \alpha_0\}.
$$
\end{exam}
The filter convergence is defined on topological spaces with respect to the neighborhood of limit points; see \cite[Def.3.7]{J}. In here, we define this concept on locally solid vector lattices.
\begin{defn}
Let $(E,\tau)$ be a locally solid vector lattice and $\mathcal{F}$ be a filter on the set $E$. A vector $e\in E$ is said to be a {\em limit of $\mathcal{F}$} (or, $\mathcal{F}$ is said to {\em converge to $e$}) if each zero neighborhood set containing $e$ belongs to $\mathcal{F}$, abbreviated as $\mathcal{F}\tc e$. Also, a vector $x\in E$ is said to be a {\em cluster} point of $\mathcal{F}$ if each zero neighborhood set containing $x$ intersects every member of $\mathcal{F}$.
\end{defn}
\begin{exam}
Suppose $(E,\tau)$ is a locally solid vector lattice. For a vector $e\in E$, the filter generated by $e$ is
$$
\mathcal{F}_e=\{U\subseteq E:U \ \text{is zero neighborhood and contains}\ e\}.
$$
So, we can see $\mathcal{F}\tc e$
\end{exam}
Consider a topological convergence net $(x_\alpha)_{(\alpha\in I)}\tc e\in E$. Then the elementary filter associated to $(x_\alpha)_{(\alpha\in I)}$ is also convergent to $e$ since every zero neighborhood containing $e$ belongs to that filter.
\begin{rem}
Let $(E,\tau)$ be a locally solid vector lattice. Then
$(i)$ If a filter $\mathcal{F}\tc e$ then $e$ is a cluster point of $\mathcal{F}$. Indeed, let $U$ be zero neighborhood set and it contains $e$, and $F\in\mathcal{F}$. Since $U$ and $F$ in $\mathcal{F}$, we have $U\cap F\in \mathcal{F}$. Thus, we get $U\cap F\neq\emptyset$ since $\emptyset$ is not in the filter $\mathcal{F}$.
$(ii)$ Let $\mathcal{F}_1$, $\mathcal{F}_2$ be filters on $E$ with $\mathcal{F}_1\subseteq\mathcal{F}_2$. So, if $\mathcal{F}_1\tc e$ then $\mathcal{F}_2\tc e$ for $e\in E$. Indeed, every zero neighborhood containing $e$ is in $\mathcal{F}_1$, and so is in $\mathcal{F}_2$. Hence, $\mathcal{F}_2\tc e$.
\end{rem}
In the filter convergence, the limit point may not be unique. That means a filter $\mathcal{F}$ on a locally solid vector lattice $(E,\tau)$ can be convergence both $e_1, \ e_2\in E$.
\begin{prop}
Let $(E,\tau)$ be a locally solid vector lattice, and $\mathcal{F}$ be a filter on $E$. Then the followings hold;
\item[(i)] If $\mathcal{F}\tc e$ for some $e\in E$ then $\mathcal{F}\tc e+x$ for all $x \in E$ whenever $e$ and $x$ are positive, or disjoint;
\item[(ii)] If $\mathcal{F}\tc e$ for some $e\in E$ then $\mathcal{F}\tc \lvert e\rvert$
\item[(iii)] If $\mathcal{F}\tc e$ for some $e\in E$ then $\mathcal{F}\tc e^+$ and $\mathcal{F}\tc e^-$
\end{prop}
\begin{proof}
$(i)$ Let $U$ be a zero neighborhood that contains $e+x$. So, we show $U\in \mathcal{F}$. Under the condition of positivity of $e$ and $x$, we have $e, x\in U$ since $U$ is a solid set and $e,x\leq e+x$. Therefore, by using the convergence $\mathcal{F}$ to $e$, we get $U\in \mathcal{F}$. On the other hand, if $e$ and $x$ are disjoint then we have $\lvert e+x\rvert=\lvert e\rvert+\lvert x\rvert$; see \cite[Thm.1.7(2-7)]{ABPO}. So, by solidness of $U$, we get $e, x\in U$. Therefore, $U\in \mathcal{F}$ since $\mathcal{F}\tc e$.
$(ii)$ Suppose $\mathcal{F}\tc e$ and $U$ is a zero neighborhood containing $\lvert e\rvert$. By the formula $\lvert e\rvert\leq \big\lvert \lvert e\rvert\big\rvert=\lvert e\rvert$ and by the solidness of $U$, we have $e\in U$. Therefore, $U\in \mathcal{F}$ since $\mathcal{F}\tc e$.
$(iii)$ Assume $\mathcal{F}\tc e$ and $U$ is a zero neighborhood which contains $e^+$. By the formula $\lvert e\rvert=e^++e^-$; see \cite[Thm.1.5(2)]{ABPO}, and by using the solidness of $U$, we get $e^+\in U$. Therefore, we have $U\in \mathcal{F}$ since $\mathcal{F}\tc e$. The other part of the proof is analog.
\end{proof}
In the next three results, we give some basic properties of filter convergence on locally solid vector lattices.
\begin{thm}\label{filter conv with addition}
Let $(E,\tau)$ be a locally solid vector lattice. If $\mathcal{F}_1$ and $\mathcal{F}_2$ are filters on $E$ such that $\mathcal{F}_1\tc e$ and $\mathcal{F}_2\tc x$ for some $e, x\in E$ then the set $\mathcal{F}=\{F_1\cup F_2:F_1\in\mathcal{F}_1\ \text{and}\ F_2\in\mathcal{F}_2\}$ is also a filter on the set $E$, and $\mathcal{F}\tc e+x$ whenever $e,\ x$ are positive, or disjoint.
\end{thm}
\begin{proof}
Firstly, we show that $\mathcal{F}$ is a filter. $(i)$ $\emptyset \notin \mathcal{F}$ since $\emptyset \notin \mathcal{F}_1$ and $\emptyset \notin \mathcal{F}_2$; if $F_1\in\mathcal{F}_1$ and $F_2\in\mathcal{F}_2$, and $F_1\cup F_2\subseteq A\subseteq E$ then $A\in \mathcal{F}_1$ and $A\in \mathcal{F}_2$, so $A\in \mathcal{F}$; if $F_1,G_1\in\mathcal{F}_1$ and $F_2,G_2\in\mathcal{F}_2$ then $F_1\cup F_2$, $G_1\cup G_2$ and $(F_1\cup F_2)\cap(G_1\cup G_2)$ in both $\mathcal{F}_1$ and $\mathcal{F}_2$, and so $(F_1\cup F_2)\cap(G_1\cup G_2)\in \mathcal{F}$.
Next, for the first case, we show $\mathcal{F}\tc e+x$. Let $U$ be a zero neighborhood and it contains $e+x$. By the properties of zero neighborhoods in locally solid vector lattice, we have $e,\ x\in U$ since they are positive and so that $e,x\leq e+x$. Therefore, since $\mathcal{F}_1$ and $\mathcal{F}_1$ are filters on the set $E$, and also $\mathcal{F}_1\tc e$ and $\mathcal{F}_2\tc x$, we get $U\in \mathcal{F}_1$ and $U\in \mathcal{F}_2$. So, we get $U\in \mathcal{F}$.
Now, suppose $e$ and $x$ are disjoint in $E$. Then we have $\rvert e+x\rvert=\lvert e\rvert+\lvert x\rvert$; see \cite[Thm.1.7(2-7)]{ABPO}. So, for given any zero neighborhood $U$ containing $e+x$, we have $e, \ x \in U$. Then the proof follows like first case.
\end{proof}
\begin{thm}
Suppose $(E,\tau)$ is a locally solid vector lattice, and $\mathcal{F}_1$ and $\mathcal{F}_1$ are filters on the set $E$ such that $\mathcal{F}_1\tc e$ and $\mathcal{F}_2\tc x$ for some $e,x\in E$. Then the class $\mathcal{F}=\{F_1\cap F_2:F_1\in\mathcal{F}_1, F_2\in\mathcal{F}_2, \ \text{and} \ F_1\cap F_2\neq \emptyset \}$ is also a filter on $E$, and also $\mathcal{F}\tc e+x$ whenever $e,\ x$ are positive, or disjoint.
\end{thm}
\begin{proof}
We show $\mathcal{F}$ is a filter. $(i)$ $\emptyset \notin \mathcal{F}$; if $F_1\in\mathcal{F}_1$ and $F_2\in\mathcal{F}_2$, and $F_1\cap F_2\subseteq A\subseteq E$ then $A\in \mathcal{F}_1$ and $A\in \mathcal{F}_2$, so we get $A\in \mathcal{F}$; if $F_1,G_1\in\mathcal{F}_1$ and $F_2,G_2\in\mathcal{F}_2$ such that intersection of for each pair of them is non empty then $F_1\cap G_1\in\mathcal{F}_1$, $F_2\cap G_2\in\mathcal{F}_2$, so $(F_1\cap F_2)\cap(G_1\cap G_2)=(F_1\cap G_1)\cap(F_2\cap G_2)$ is non empty and it is also in $\mathcal{F}$. Then $\mathcal{F}$ is a filter on $E$, and by using the similar way in the proof of Theorem \ref{filter conv with addition} we get the desired result.
\end{proof}
Let $(x_\alpha)_{\alpha\in I}$ be a net in any topological space. A point $x$ is called {\em cluster point} of $(x_\alpha)_{\alpha\in I}$ if, for each neighborhood $U$ of $x$, $(x_\alpha)_{\alpha\in I}$ is frequently in $U$, or equivalently, for each neighborhood $U$ of $x$, we have $(U\setminus \{x\})\cap \{x_\alpha\}\neq \emptyset$). On the other hand, let $\mathcal{F}$ be a filter on a locally solid vector lattice $(E,\tau)$. A vector $e\in E$ is said to be {\em cluster point of $\mathcal{F}$} (with respect to $\mathcal{F}$) if every zero neighborhood which contains $e$ intersects with each member of $\mathcal{F}$; see \cite[Def.3.7]{J}. So, we give the following result which has a relation between filter convergence and cluster point.
\begin{thm}
Let $(E,\tau)$ be a locally solid vector lattice and $\mathcal{F}$ be a filter on the set $E$. Then the vector $e\in E$ is a cluster point of $\mathcal{F}$ iff there exists another filter $\mathcal{F}_1$ containing $\mathcal{F}$ such that $\mathcal{F}_1\tc e$.
\end{thm}
\begin{proof}
Suppose $e\in E$ is a cluster point of $\mathcal{F}$. Then the set $\mathcal{B}=\{U\cap F:F\in\mathcal{F}, U\ \text{is a zero neighborhood and contains}\ e\}$ is a filter base. Indeed, we show the properties of filter base. $(i)$ $\mathcal{B}$ is non empty since, for each zero neighborhood $U$ containing $e$ intersects with every member of $\mathcal{F}$; $(ii)$ for $U\cap F\in \mathcal{B}$, we have $U\cap F\neq \emptyset$ since $U\cap F\in \mathcal{F}$ and $\emptyset\notin \mathcal{F}$; $(iii)$ for $U_1\cap F_1$ and $U_2\cap F_2$ in $\mathcal{B}$, we can take $B=(U_1\cap U_2)\cap( F_1\cap F_2)\in \mathcal{B}$. So, we assume it generates the filter $\mathcal{F}_1$. For each $F \in \mathcal{F}$, we have $F=E\cap F\in \mathcal{B}$, and so we get $\mathcal{F}_1\subseteq\mathcal{F}_2$. Therefore, for given a zero neighborhood $U$ containing $e$, we have $U=U\cap E\in \mathcal{B}$, and so we get $\mathcal{F}_1\tc e$.
Conversely, assume such filter $\mathcal{F}_1$ exists it means that $\mathcal{F}_1$ is a filter on the set $E$ with $\mathcal{F}\subseteq\mathcal{F}_1$ and $\mathcal{F}_1\tc e$. So, all zero neighborhoods containing $e$ is in $\mathcal{F}_1$. So, by the definition of filter, each zero neighborhood containing $e$ intersects with member of $\mathcal{F}_1$, otherwise $\emptyset \in \mathcal{F}_1$. Therefore, in particular, intersects each set in $\mathcal{F}$, and so we get $e$ is a cluster point of $\mathcal{F}$.
\end{proof}
Now, we use net to define filter on locally solid vector lattice. Let $(x_\alpha)_{\alpha\in I}$ be a net in the locally solid vector lattice $(E,\tau)$. Then we define its associated filter $\mathcal{F}$ on the set $E$ as follow; consider the tail $\hat{x}_\beta=\{x_\alpha:\alpha\in I,\alpha\geq\beta\}$ and $\mathcal{B}=\{\hat{x}_\beta:\beta\in I\}$. So, $\mathcal{B}$ is a filter base. Indeed, $(i)$ $\mathcal{B}$ is not empty; $(ii)$ every $\hat{x}_\beta\in \mathcal{B}$ is not empty since $I$ is directed set; $(iii)$ for any $\hat{x}_{\beta_1},\ \hat{x}_{\beta_2}\in \mathcal{B}$, consider the index $\beta=\max\{\beta_1,\beta_2\}$ so that $\hat{x}_\beta\subseteq \hat{x}_{\beta_1}\cap\hat{x}_{\beta_2}$ and $\hat{x}_\beta \in \mathcal{B}$. Thus, the filter which is generated by $\mathcal{B}$ is the associated filter of $(x_\alpha)_{\alpha\in I}$.
\begin{thm}\label{net convergence and filter convergence}
Let $(E,\tau)$ be a locally solid vector lattice and $(x_\alpha)_{(\alpha\in I)}$ be a net in $E$. Assume $\mathcal{F}$ is the associated filter of $(x_\alpha)_{(\alpha\in I)}$ and $e\in E$. Then $x_\alpha\tc e$ as a net iff $\mathcal{F}\tc e$ as a filter. Moreover, $e$ is a cluster point of $(x_\alpha)_{(\alpha\in A)}$ iff $e$ is a cluster point of $\mathcal{F}$.
\end{thm}
\begin{proof}
We show only the convergence part of proof, the cluster point case is analogous. Suppose $x_\alpha\tc e$ as a net. Since every locally solid vector lattice has a base of zero neighborhoods, we can consider any zero neighborhood $U$ which contains $e$. So, by definition of $\mathcal{F}$, we get $U\in \mathcal{F}$. Thus, we have $\mathcal{F}\tc e$ as a filter.
Conversely, assume the filter $\mathcal{F}$ converges to $e$. Let $V$ be a zero neighborhood in $E$ and contains $e$. Then $V\in \mathcal{F}$. By definition of $\mathcal{F}$, there exists a index $\alpha_0$ such that $\hat{x}_{\alpha_0}\subseteq V$. Therefore, we get $x_\alpha\in V$ for every $\alpha\geq \alpha_0$, and so $x_\alpha\tc e$.
\end{proof}
\begin{cor}
Let $(E,\tau)$ be a locally solid vector lattice which is generated by a family of Riesz pseudonorms $\{\rho_j\}_{j\in J}$, $(x_\alpha)_{(\alpha\in A)}$ be a net in $E$ and $\mathcal{F}$ be associated filter of it, and $e\in E$. Then $\mathcal{F}\tc e$ iff $\rho_j(x_\alpha-e)\to 0$ for all $j\in J$.
\end{cor}
\begin{proof}
It follows from \cite[Thm.2.28]{ABur} and Theorem \ref{net convergence and filter convergence}.
\end{proof}
We can give convergence of filters with respect to continuous function. Let $(E,\tau)$ and $(F,\acute{\tau})$ be locally solid vector lattices, and $\mathcal{F}$ be a filter on $E$. For a function $f:E\to F$, the set
$$
\acute{f}\mathcal{F}=\{B\subset F:f^{-1}(B)\in \mathcal{F}\}
$$
is the filter on $E$ generated by $\{f(A):A\subseteq \mathcal{F}\}$.
\begin{prop}
Suppose $(E,\tau)$ and $(F,\acute{\tau})$ are locally solid vector lattices. Then a function $f:E\to F$ is continuous iff $\acute{f}\mathcal{F}\tcc f(e)$ in $F$ for each $\mathcal{F}\tc e$ in $E$.
\end{prop}
\begin{proof}
Suppose $f$ is continuous. For fixed $e\in E$, consider the following set
$$
\mathcal{N}_e=\{A\subseteq E:\exists U \ \text{zero neighborhood s.t.}, e\in U\subseteq A\}.
$$
Thus, we get $\mathcal{N}_e\tc e$. Note also that a filter $\mathcal{F}$ converges to $e$ iff $\mathcal{N}_e\subset \mathcal{F}$. Take a filter $\mathcal{F}$ such that $\mathcal{F}\tc e$. The continuity of $f$ implies that $\mathcal{N}_{f(e)}\subseteq \acute{f}\mathcal{N}_{e}$. Therefore, if $\mathcal{F}\tc e$ then $\mathcal{N}_e\subseteq \mathcal{F}$, and so $\mathcal{N}_{f(e)}\subseteq \acute{f}\mathcal{N}_{e}\subseteq \acute{f}\mathcal{F}$ so that $\acute{f}\mathcal{F}\tcc f(e)$.
Assume $\acute{f}\mathcal{N}_e$ is the set of all subsets of $E$ whose preimage is a neighborhood of $e$. Since $\mathcal{N}\tc e$, we conclude that the preimage of any neighborhood of $f(e)$ is a neighborhood of $e$. Hence, $f$ is continuous.
\end{proof}
|
{
"timestamp": "2018-03-08T02:04:39",
"yymm": "1803",
"arxiv_id": "1803.02534",
"language": "en",
"url": "https://arxiv.org/abs/1803.02534"
}
|
\section{Introduction} \label{sec:intro}
In the present note we complete a line of research about the phase-transition behaviour of a nearest-neighbor model on Cayley trees with arbitrary degree $k\ge 2$. As first described in~\cite{re}, for a given consistent family of finite-volume Gibbs measures, the existence and multiplicity of a certain class of infinite-volume measures which are consistent with the prescribed finite-volume Gibbs measures, can be reduced to the analysis of fixed points of some non-linear integral equation of Hammerstein type. Every positive solution of the fixed point equation here corresponds to a measures which is called a splitting Gibbs measure. Every splitting Gibbs measure is also a Gibbs measure in the sense of the DLR formalism; see~\cite{B}. This approach has been successfully applied in the analysis of a variety of different models on Cayley trees with respect to their phase-transition properties; see~\cite{rb} for a comprehensive overview. In particular, starting with~\cite{ehr1}, a phase-transition of multiple splitting Gibbs measures has been detected in a model with uncountable local state space $[0,1]$ and nearest-neighbor interactions. This has motivated the subsequent analysis in~\cite{erb, bb, b1}, to further understand critical behavior of this model for all degrees of the underlying tree, where also new parameters are introduced. It is the purpose of this note to complete the analysis of this model.
\medskip
For nearest neighbors $x,y$ on the {\it Cayley tree} $\Gamma^k$ with degree $k \geq 2$ with local states ${\sigma}(x),{\sigma}(y)\in[0,1]$, we consider the potential
\begin{equation}\label{Model}
\xi_{{\sigma}(x),{\sigma}(y)}=\log\left(1+\theta \sqrt[2m+1]{4({\sigma}(x)-\frac{1}{2})({\sigma}(y)-\frac{1}{2})}\right)
\end{equation}
where $m\in\mathbb N\cup\{0\}$, $0 \leq \theta <1$ are the system parameters. It can be interpreted as a certain symmetric pair-interaction with values in $[\log(1-\theta),\log(1+\theta)]$, admitting two distinct ground states given by the all-$0$ and the all-$1$ configuration. The main result is the existence of a sharp threshold
$$\theta_{\rm c}=\frac{2m+3}{k(2m+1)}$$
such that if $\theta_{\rm c}< \theta <1$, there are exactly three translation-invariant splitting Gibbs measures and otherwise there is only one.
\section{Setup}\label{Setup}
\subsection{Gibbs measures on Cayley trees}
The {\it Cayley tree} $\Gamma^k$ of order $k \geq 1$ is an infinite tree, i.e., a graph
without cycles, such that exactly $k+1$ edges originate from each vertex. Let $\Gamma^k=(V,L)$ where
$V$ is the set of {\it vertices} and $L$ is a symmetric subset of $V \times V$, called the {\it edge set}.
The word "symmetric" means that $(x, y) \in L$ iff $(y, x) \in L$. Here, $x$ and $y$ are called the {\it endpoints} of the edge $\langle x, y\rangle$. Two vertices $x$ and $y$ are called {\it nearest neighbors}
if there exists an edge $l \in L$ connecting them and we denote $l=\langle x,y\rangle$.
For a fixed $x^0 \in V$, called the {\it root}, we defines $n$-spheres and $n$-disks in the graph distance $d(x,y)$ by
$$W_n=\{x \in V| d(x,x^0)=n\}, \ \ \ \ V_n=\bigcup \limits_{i=0}^n W_i$$
and denote for any $x \in W_n$ the set of {\it direct successors} of $x$ by
$$S(x)=\{y \in W_{n+1}: d(x,y)=1\}.$$
For $A \subset V$ let $\Omega_A=[0,1]^A$ denote the set of all configurations ${\sigma}_A$ on $A$. In particular, a configuration $\sigma$ on $V$ is then defined as a function $V\ni x \mapsto \sigma(x) \in [0,1]$.
According to the usual setup for Gibbs measure, we consider a (formal) Hamiltonian of the form
\begin{equation}\label{m}
H(\sigma)=-\sum \limits_{\langle x,y\rangle \in L}\xi_{\sigma(x),\sigma(y)},
\end{equation}
where $\xi: (u,v) \in [0,1]^2 \mapsto \xi_{u,v} \in\mathbb{R}$ is the interaction~\eqref{Model} which assigns energy only to neighboring sites. Since $\xi$ does not depend on the locations $x$ and $y$, $H$ is invariant under tree translations.
Let $\lambda$ be the Lebesgue measure on $[0,1]$ then, on the set of all configurations on $A$ the a priori measure $\lambda_A$ is introduced as the $|A|$-fold product of the measure $\lambda$. Here and in the sequel, $|A|$ denotes the cardinality of $A$. We equip $\Omega=\Omega_V$ with the standard sigma-algebra $\mathcal{B}$ generated by the cylindrical subsets. A probability measure $\mu$ on $(\Omega, \mathcal{B})$ is called a {\it Gibbs measure} (with Hamiltonian $H$) if it satisfies the DLR equation. That is, for any $n=1,2,...$ and bounded measurable test function $f$, we have that
\begin{equation}\label{DLR}
\int\mu(d{\sigma})f({\sigma})=\int \mu(d {\sigma})\int\gamma_{V_n}(d\tilde{\sigma}_{V_n}|{\sigma}_{W_{n+1}})f(\tilde{\sigma}_{V_n}{\sigma}_{\Gamma^k\setminus V_n}),
\end{equation}
where $\gamma_{V_n}(d \sigma_{V_n}|{\sigma}_{\Gamma^k\setminus V_n})$ is the Gibbsian specification
$$\gamma_{V_n}(d\tilde\sigma_{V_n}|{\sigma}_{\Gamma^k\setminus V_n})=\frac{1}{Z_{V_n}({\sigma}_{W_{n+1}})}e^{-\beta H(\tilde\sigma_{V_n}{\sigma}_{W_{n+1}})}\lambda_{V_n}(d\tilde{\sigma}_{V_n}),$$
with normalization $Z_{V_n}$ and temperature parameter $\beta \geq 0$. Such a specification is also sometimes referred to as a Markov specification; see~\cite{Ge11}.
\subsection{Representation via Hammerstein operators}
A subset of the infinite-volume Gibbs measures defined via the DLR equation~\eqref{DLR}, called the {\it splitting Gibbs measures} or {\it Markov chains}, can be represented in terms of the fixed points of some nonlinear integral operator of Hammerstein type; see~\cite{re} for details. More precisely, for every $k\in\mathbb{N}$ consider the integral operator
$H_{k}$ acting on the cone $C^{+}[0,1]=\{f\in C[0,1]: f(x)\geq 0\}$ given by
\begin{equation}\label{Ham_k}
\begin{split}
(H_{k}f)(t)=\int^{1}_{0}K(t,u)f^{k}(u)du.
\end{split}
\end{equation}
Then, the translation-invariant splitting Gibbs measures for the Hamiltonian~\eqref{m} correspond to fixed points of $H_k$ with $K(t,u)=\exp(\beta\xi_{t,u})$, often called {\it boundary laws}. Note that $H_k$ in general might generate ill-posed problems; see~\cite{Kr64,KrZa84}.
\section{Main results}
The main result of this note is the following characterization of phase-transition regimes of the model~\eqref{Model} with $\beta=1$.
\begin{thm}\label{t}
For all $n\in\mathbb N\cup\{0\}$ and $k\ge 2$ let $\theta_{\rm c}=(2n+3)/(k(2n+1))$, then the model~\eqref{Model} has
\begin{enumerate}
\item a unique translation-invariant splitting Gibbs measure if $0 \leq \theta \leq \theta_{\rm c}$ and
\item exactly three translation-invariant splitting Gibbs measures if $\theta_{\rm c} < \theta <1$.
\end{enumerate}
\end{thm}
The proof is based on a characterization of solutions to the fixed point equation for the associated Hammerstein integral operator~\eqref{Ham_k} as given in Proposition~\ref{Prop1} below. In case of the model at hand, then the analysis can be reduced to finding the fixed points of the following $2$-dimensional operator $V_k:(x,y) \in \mathbb{R}^2 \rightarrow
(x', y') \in \mathbb{R}^2$
\begin{equation}\label{V_k}
\begin{split}
V_{k,n}(x,y)=\left\{
\begin{array}{lllllll}
x'=\sum \limits_{j=0}^{\lfloor\frac{k}{2}\rfloor} \binom{k}{2j}
\frac{2n+1}{2n+1+2j} \cdot 2^{\frac{2j}{2n+1}} \cdot x^{k-2j} (\theta y)^{2j}$$
\\ [6 mm]$$y'=\sum \limits_{j=0}^{\lfloor\frac{k}{2}\rfloor} \binom{k}{2j+1}
\frac{2n+1}{2n+2+2j+1} \cdot 2^{\frac{2j}{2n+1}} \cdot x^{k-(2j+1)} (\theta y)^{2j+1}
\end{array}\right.
\end{split}
\end{equation}
with $k \geq 2$, which is then the content of Proposition~\ref{Prop2}.
\begin{pro}\label{Prop1} A function $\varphi \in C[0,1]$ is a solution of the Hammerstein equation
\begin{equation}\label{H_k}
\begin{split}
H_kf=f\end{split}
\end{equation}
with $H_k$ defined in~\eqref{Ham_k} for our model~\eqref{Model}, iff $\varphi$ has the following form
$$\varphi(t)=C_1 + C_2
\theta \sqrt[2n+1]{4(t-\frac{1}{2})},$$
where $(C_1, C_2) \in \mathbb{R}^2$ is
a fixed point of the operator $V_{k,n}$ as defined in~\eqref{V_k}.
\end{pro}
In the following proposition we characterize the fixed points of $V_{k,n}$ which readily implies Theorem~\ref{t} using Proposition~\ref{Prop1}.
\begin{pro}\label{Prop2}
Let $\theta_{\rm c}=(2n+3)/(k(2n+1))$, then there exist $x_o,y_o\in(0,\infty)$ such that the number and form of the fixed points of the operator $V_{k,n}$ are as presented in the following Table~\ref{Table1}.
\begin{table}[h]
\caption{Set of $2$-dimensional fixed points of $V_{k,n}$}
\begin{tabular}{cV{3} c | c | c | c | c | c | c | r |}
& \multicolumn{3}{c|}{fixed points if $ 0 \leq \theta \leq \theta_{\rm c}$}&\multicolumn{4}{c|}{additional fixed points if $\theta_{\rm c}< \theta <1$}\\
\hlineB{3}
$k$ even & $(0,0)$ & $(1,0)$ & &$(x_o,y_o)$ && $(x_o,-y_o)$ &\\ \hline
$k$ odd & $(0,0)$ & $(1,0)$ & $(-1,0)$& $(x_o,y_o)$ & $(-x_o,-y_o)$ & $(x_o,-y_o)$& $(-x_o,y_o)$\\ \hline
\end{tabular}
\label{Table1}
\end{table}
\noindent
Only the fixed points $(1,0)$, $(x_o,y_o)$ and $(x_o,-y_o)$ give rise to positive solutions for the Hammerstein equation~\eqref{H_k}.
\end{pro}
Let us finally give the references to the special cases considered prior to this work. \cite[Theorem 4.2 and Theorem 5.2]{erb} proves the cases $k=2,3$ with $n=1$ of~\eqref{Model} whereas
in \cite[Theorem 3.2.]{bb} the cases $k\geq 2$ with $n=1$ are given.
Finally, in \cite[Theorem 2.3]{b1} the cases $k=2$ with general $n \geq 1$ is provided.
\section{Proofs}
Note that for the model~\eqref{Model} with $\beta=1$, the kernel $K(t,u)$ of the Hammerstein operator $H_k$ is given by
$$K(t,u)=1+\theta \sqrt[2n+1]{4(t-\frac{1}{2})(u-\frac{1}{2})}.$$
\begin{proof}[Proof of Proposition~\ref{Prop1}]
Let us start with necessity. Assume $\varphi \in C[0,1]$ to be a solution of the equation~\eqref{H_k}. Then we have
\begin{equation}\label{fi2} \varphi(t)=C_1 + C_2
\theta \sqrt[2n+1]{4(t-\frac{1}{2})},
\end{equation}
where
\begin{equation}\label{c21}
C_1=\int \limits_0^1 \varphi^k(u)du \quad\text{ and }\quad C_2=\int \limits_0^1 \sqrt[2n+1]{u-\frac{1}{2}}\cdot \varphi^k(u)du.
\end{equation}
Substituting $\varphi(t)$ into the first equation of~\eqref{c21} we get
\begin{equation*}
\begin{split}
C_1&=
\int \limits_0^1 \left(C_1+ C_2 \theta\sqrt[2n+1]{4 (u-\frac{1}{2})}\right)^k du\\
&=\int \limits_0^1 \sum \limits_{i=0}^k \binom{k}{i} C_1^{k-i} \left( C_2 \theta\sqrt[2n+1]{4} \sqrt[2n+1]{u-\frac{1}{2}}\right)^i du\\
&= \sum \limits_{i=0}^k \binom{k}{i}C_1^{k-i}(\theta C_2 )^i2^{\frac{2i}{2n+1}}\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{i}{2n+1}} du.
\end{split}
\end{equation*}
Now, we use the following equality
\begin{equation}\label{1}\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{i}{2n+1}} du =\left\{
\begin{array}{ll}
0, & \hbox{if $i$ is odd and} \\
\frac{2n+1}{2n+1+i} \cdot 2^{- \frac{i}{2n+1}}, & \hbox{if $i$ is even.}
\end{array}
\right.
\end{equation}
Then we get
$$ C_1=\sum \limits_{j=0}^{\lfloor\frac{k}{2}\rfloor} \binom{k}{2j}
\frac{2n+1}{2n+1+2j}2^{\frac{2j}{2n+1}}C_1^{k-2j} (\theta C_2)^{2j}$$
and substituting the function $\varphi$ into the second equation of~\eqref{c21} we have
\begin{equation*}
\begin{split}
C_2&=\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{1}{2n+1}}\left(C_1+\theta C_2 \sqrt[2n+1]{4 (u-\frac{1}{2})}\right)^k du\cr
&=\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{1}{2n+1}} \sum \limits_{i=0}^k \binom{k}{i}C_1^{k-i} \left(\theta C_2 \sqrt[2n+1]{4} \sqrt[2n+1]{u-\frac{1}{2}}\right)^i du\cr
&= \sum \limits_{i=0}^k \binom{k}{i}C_1^{k-i}(C_2 \theta)^i2^{\frac{2i}{2n+1}}\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{i+1}{2n+1}} du.
\end{split}
\end{equation*}
Now, using the following equality
\begin{equation}\label{2}\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{i+1}{2n+1}} du =\left\{
\begin{array}{ll}
0, & \hbox{if $i$ is even and } \\
\frac{2n+1}{2n+2+i} \cdot 2^{- \frac{i+1}{2n+1}}, & \hbox{if $i$ is odd}
\end{array}
\right.
\end{equation}
we arrive at the equation
$$ C_2= \sum \limits_{j=0}^{\lfloor\frac{k}{2}\rfloor} \binom{k}{2j+1}
\frac{2n+1}{2n+2+2j+1}2^{\frac{2j}{2n+1}}C_1^{k-2j-1} (\theta C_2)^{2j+1}.$$
In particular, the point $(C_1,C_2) \in \mathbb{R}^2$ must be a fixed point of the operator $V_{k,n}$ from~\eqref{V_k}.
\medskip
For the sufficiency, assume that, a point $(C_1,C_2) \in \mathbb{R}^2$ is a fixed point of the operator $V_{k,n}$ and define the function $\varphi \in C[0,1]$ by the equality $$\varphi(t)=C_1 + C_2
\theta \sqrt[2n+1]{4(t-\frac{1}{2})}.$$
Then, we can calculate
\begin{equation}\label{k}
\begin{split}
&(H_k \varphi)(t)=\int \limits_0^1 \left(1+\sqrt[2n+1]{4} \theta \sqrt[2n+1]{(t-\frac{1}{2})(u-\frac{1}{2})}\right)\varphi^k(u)du\cr
&=\int \limits_0^1 \varphi^k(u)du+\sqrt[2n+1]{4} \theta \sqrt[2n+1]{t-\frac{1}{2}} \int \limits_0^1 \sqrt[2n+1]{u-\frac{1}{2}}\varphi^k(u) du\cr
&=\int \limits_0^1 \left(C_1 + C_2\theta \sqrt[2n+1]{4(u-\frac{1}{2})}\right)^k du\cr
&\hspace{1cm}+ \sqrt[2n+1]{4} \theta \sqrt[2n+1]{t-\frac{1}{2}} \int \limits_0^1 \sqrt[2n+1]{u-\frac{1}{2}} \left(C_1+ C_2 \theta \sqrt[2n+1]{4(u-\frac{1}{2})}\right)^k du\cr
&=\sum \limits_{i=0}^k \binom{k}{i}C_1^{k-i}(\theta C_2 )^i2^{\frac{2i}{2n+1}}\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{i}{2n+1}} du+ \sqrt[2n+1]{4} \theta \sqrt[2n+1]{t-\frac{1}{2}} \cr
&\hspace{1cm}\times \sum \limits_{i=0}^k \binom{k}{i}C_1^{k-i}(C_2 \theta)^i 2^{\frac{2i}{2n+1}}\int \limits_0^1 \left(u-\frac{1}{2}\right)^{\frac{i+1}{2n+1}} du.
\end{split}
\end{equation}
Now, we using~\eqref{1} and~\eqref{2}, from~ \eqref{k} we get
\begin{equation*}
\begin{split}
(H_k \varphi)(t)&=\sum \limits_{j=0}^{\lfloor\frac{k}{2}\rfloor} \binom{k}{2j}
\frac{2n+1}{2n+1+2j}2^{\frac{2j}{2n+1}}x^{k-2j} (\theta y)^{2j} \cr
&\hspace{0.5cm}+ \theta \sqrt[2n+1]{4(t-\frac{1}{2})} \sum \limits_{j=0}^{\lfloor\frac{k}{2}\rfloor} \binom{k}{2j+1}
\frac{2n+1}{2n+2+2j+1}2^{\frac{2j}{2n+1}} x^{k-2j-1} (\theta y)^{2j+1}\cr
&=C_1+C_2 \theta \sqrt[2n+1]{4(t-\frac{1}{2})}=\varphi(t).
\end{split}
\end{equation*}
Thus, $\varphi$ is a solution of the equation (\ref{H_k}).
\end{proof}
\begin{proof}[Proof of Proposition~\ref{Prop2}]
Let us start by assuming $k$ to be even. We determine the number and form of solutions to $V_{k,n}$ in equation~\eqref{V_k}.
If $\theta\ge 0$, then for $k$ even $(0,0)$ and $(1,0)$ are fixed points.
If $\theta>0$ then there are potentially more fixed points. Indeed, let $\theta>0$ and assume $y>0$ then, writing $z=\theta y/x$, the fixed point equation for~\eqref{V_k} becomes
\begin{equation*}\label{V_z}
\begin{split}
z=\theta\frac{\sum_{i=1,3, \dots, k-1} \binom{k}{i}\frac{2n+1}{2n+2+i} \cdot 2^{\frac{i-1}{2n+1}} \cdot z^{i}}{\sum_{i=0,2, \dots, k}\binom{k}{i}\frac{2n+1}{2n+1+i}2^{\frac{i}{2n+1}}z^{i}}=\theta\frac{F_1(z)}{F_2(z)}=f(z).
\end{split}
\end{equation*}
Hence, in order to find solutions, we have to find roots of the polynomial
\begin{equation}\label{Poly}
\begin{split}
P(z)&=\sum_{i=1,3, \dots, k+1}\binom{k}{i-1}\frac{2n+1}{2n+i}2^{\frac{i-1}{2n+1}}z^{i}-\theta\sum_{i=1,3, \dots, k-1}\binom{k}{i}\frac{2n+1}{2n+2+i}2^{\frac{i-1}{2n+1}}z^{i}\\
&=r_\theta(k,k+1)z^{k+1}+\sum_{i=1,3, \dots, k-1}r_\theta(k,i)z^{i}
\end{split}
\end{equation}
where $r_\theta(k,k+1)=\frac{2n+1}{2n+k+1}2^{\frac{k}{2n+1}}$ and
\begin{equation*}\label{V_t}
\begin{split}
r_\theta(k,i)&=\binom{k}{i}\frac{2n+1}{2n+2+i}2^{\frac{i-1}{2n+1}}\Big[\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}-\theta\Big].
\end{split}
\end{equation*}
Moreover,
\begin{equation*}
\begin{split}
r_\theta(k,i)\left\{
\begin{array}{lllllll}
<0\qquad \text{if}\quad \theta>\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}\\
=0\qquad \text{if}\quad \theta=\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}\\
>0\qquad \text{if}\quad \theta<\frac{i}{k-i+1}\frac{2n+2+i}{2n+i}
\end{array}\right.
\end{split}
\end{equation*}
and we denote the critical $\theta$ by $\theta_{k,i}$. Further note that $i\mapsto\theta_{k,i}$ is increasing. Indeed, the derivative of the continuous version is given by
\begin{equation*}
\begin{split}
\frac{4(1 + k) n^2 + (3 + k) i^2 + 4 (1 + k) n (1 + i))}{(1 + k - i)^2 (2 n + i)^2}
\end{split}
\end{equation*}
which is non-negative.
Hence, for $\theta$ below the lowest critical value, $\theta_{k,1}=\frac{2n+3}{k(2n+1)}$, all coefficients are positive and hence there is no positive real root by Descartes' rule of sign.
Further, again by Descartes' rule of sign, if we increase $\theta>\theta_{k,1}$, then there is exactly one sign change and hence, exactly one non-trivial positive real root which we denote $z_0$.
Since only odd term appear in the polynomial, with $z_0$ also $-z_0$ is a root.
In order to recover a solution $(x,y)$ from the positive non-trivial solution $z_0$, note that
\begin{equation*}\label{V_kn}
\begin{split}
V_{k,n}(x,y)=\left\{
\begin{array}{lllllll}
x=x^k\sum \limits_{i=0,2, \dots, k} \binom{k}{i}
\frac{2n+1}{2n+1+i} \cdot 2^{\frac{i}{2n+1}} \cdot (\frac{\theta y}{x})^i=x^kF_2(\frac{\theta y}{x})$$
\\ [6 mm]$$y=x^k\sum \limits_{i=1,3, \dots, k-1}\binom{k}{i}
\frac{2n+1}{2n+2+i} \cdot 2^{\frac{i-1}{2n+1}} (\frac{\theta y}{x})^i=x^kF_1(\frac{\theta y}{x})
\end{array}
\right.
\end{split}
\end{equation*}
and hence $x_0=F_2(z_0)^{1/(1-k)}>0$ and $y_0=F_1(z_0)F_2(z_0)^{k/(1-k)}>0$ solve the $2$-dimensional equation. Note that $(x_0,y_0)$ is the only solution with $\theta y_0/x_0=z_0$. Indeed, any other such solution would be $x_1=c x_0$ and $y_1=cy_0$ for some $c\in\mathbb{R}\setminus\{0\}$, but plugging this into the first line of the above equation gives $c=c^k$ which is true iff $c=1$ for even $k$.
Further, note that $F_2(-z_0)^{1/(1-k)}=F_2(z_0)^{1/(1-k)}=x_0$ and $F_1(-z_0)F_2(-z_0)^{k/(1-k)}=-F_1(z_0)F_2(z_0)^{k/(1-k)}=-y_0$ and hence also $(x_0,-y_0)$ is a solution to the $2$-dimensional fixed point equation. Using similar arguments one can show that this is the only fixed point with $-\theta y_0/x_0=-z_0$.
\medskip
For odd $k$ and $\theta\ge 0$ we have fixed points $(0,0)$, $(1,0)$ and $(-1,0)$.
For the additional fixed point, the calculations are analogous, but without the leading term $z^{k+1}$, yielding again to fixed point $z_0$ and $-z_0$ for $\theta>\theta_{k,1}$. In contrast to the case for even $k$, for odd $k$, both $(x_0,y_0)$ and $(-x_0,-y_0)$ are $2$-dimensional fixed points corresponding to $z_0$. Finally, the fixed points $(x_0,-y_0)$ and $(-x_0,y_0)$ corresponds to $-z_0$. The complete list of fixed points is recorded in Table~\ref{Table1}.
\medskip
For $(\pm x_0,\pm y_0)$ to give rise to a positive solution, by the form of solutions $\varphi$ we must have that for all $t\in[0,1]$
\begin{equation*}
\pm x_o\pm y_o\theta\sqrt[2n+1]{4(t-1/2)}>0.
\end{equation*}
Clearly, for $-x_o$, in $t=1/2$, the inequality is violated and it suffices to consider the points $(x_0,\pm y_0)$. By monotonicity in $t$, it suffices to show that
\begin{equation}\label{Criterium}
2^{-1/(2n+1)}>\theta y_o/x_o=z_o
\end{equation}
for the positive solution of the polynomial $P$ from~\eqref{Poly}. Since, the sign change in the polynomial must be from minus to plus, we need to determine its sign in $2^{-1/m}$ where we put $m=2n+1$. We show that indeed $P(2^{-1/m})>0$ which implies that~\eqref{Criterium} is satisfied and thus $(x_o,\pm y_o)$ correspond to positive solutions.
Note that
\begin{equation*}
\begin{split}
P(2^{-1/m})&=2^{-1/m}\sum_{i=1,3, \dots, k+1}\binom{k}{i-1}\frac{m}{m+i-1}-\theta\sum_{i=1,3, \dots, k-1}\binom{k}{i}\frac{m}{m+1+i}>0
\end{split}
\end{equation*}
is implied by
\begin{equation*}
\begin{split}
\sum_{i=1,3, \dots, k+1}\binom{k}{i-1}\frac{1}{m+i-1}-\sum_{i=1,3, \dots, k-1}\binom{k}{i}\frac{1}{m+1+i}>0.
\end{split}
\end{equation*}
We can further bound the left hand side from below by
\begin{equation*}
\begin{split}
\sum_{i=0}^k(-1)^i\binom{k}{i}\frac{1}{m+1+i}=\frac{k!m!}{(m+1+k)!}
\end{split}
\end{equation*}
which is positive for all $m, k$. This completes the proof.
\end{proof}
\section{Acknowledgement}
Golibjon Botirov thanks the DAAD program for the financial support and the Weierstrass Institute Berlin for its hospitality. Benedikt Jahnel thanks the Leibniz program 'Probabilistic methods for mobile ad-hoc networks' for the support.
|
{
"timestamp": "2018-03-09T02:01:13",
"yymm": "1803",
"arxiv_id": "1803.02867",
"language": "en",
"url": "https://arxiv.org/abs/1803.02867"
}
|
\section{Introduction}
The space-time evolution of nonlinear plasma oscillations/waves and their breaking is a fascinating field of study due to its far reaching applications in nonlinear plasma physics \cite{a, b, 7}. The breaking of relativistically intense plasma waves generated by a high intensity laser/particle beam pulse plays a crucial role in particle acceleration processes \cite{e, f, g, i, j, c} and inertial confinement fusion experiments \cite{a1, a2, ssnf}. Dawson \cite{1} was the first to elucidate the fact that for a nonrelativistic cold plasma, oscillations in slab geometry would be stable below a critical amplitude, called wave breaking limit which is given by $eE/m\omega _pv_\phi \approx 1$ ($e$ and $m$ respectively being the charge and mass of an electron; $\omega _p$ is the nonrelativistic plasma frequency, $v_\phi$ is the phase velocity of the plasma wave and $E$ is the self-consistent electric field amplitude). Beyond this limit multistream flow or fine scale mixing develops which destroys the collective motion of the plasma electrons and hence the oscillations break within a plasma period.
Even without approaching the above ``wave breaking limit'', Dawson pictured a novel
phenomenon where plasma oscillations start losing it's periodicity gradually with
time, provided that the characteristic frequency of oscillation, due to some physical reasons, acquires a spatial dependency. In this case, oscillations break at a particular time decided by the initial amplitude which is far below the corresponding wave breaking limit. This phenomenon is called ``phase mixing'' \cite{4,5,6,8,k,16, 3, arsu}. Due to spatial dependency of the characteristic frequency, neighbouring electrons gradually move out of phase and eventually cross each other causing the wave to break at arbitrarily small initial amplitude \cite{4, 5, 6, 8, k, 16, 3, arsu}. Thus due to phase mixing, oscillations/waves break at arbitrarily small initial amplitude far below the corresponding ``wave breaking limit''. Dawson derived a general expression for phase mixing time scale (wave breaking time) by using a physical reasoning which is based on out of phase motion of neighbouring oscillators separated by a distance equal to twice the amplitude of the oscillation/wave, and demonstrated that for ``nonrelativistic oscillations'' phase mixing can occur if there is a density inhomogeneity (either fixed \cite{ref244} or self-generated \cite{ref233}) or the geometry of the oscillation changes from planar to cylindrical/spherical \cite{1}.
Later it has been shown by several authors that plasma oscillations/waves in a slab geometry can also break in the similar fashion, if the electron's quiver velocity becomes relativistic \cite{drake, 3, 4, 5, 6, 8, k, 16, 3, arsu}. In this case phase mixing time scale crucially depends on the amplitude of the initialised oscillation/wave. By using Dawson Sheet Model, the authors in Refs. \cite{4, 5} derived an analytical expression for phase mixing time scale of a relativistically intense oscillation/wave in a slab geometry as a function of the initial amplitude. These authors \cite{4, 5} also confirmed their theoretical predictions by using a code based on Dawson Sheet Model and observed that the phase phase time scale of a relativistically intense oscillation/wave in a slab geometry is inversely proportional to the cube of the initial amplitude.
In the last few years much attention has been paid to the study of relativistically intense cylindrical and spherical plasma oscillations/waves, both theoretically \cite{17, 18, 19, 20} and experimentally \cite{21, 22}. For example, Gorbunov et. al. \cite{17,18} and Bulanov et. al. \cite{19} respectively studied the evolution of cylindrical and spherical plasma waves analytically by using Lagrange coordinates \cite{a}. These authors respectively derived the equation of motion of an electron oscillating along the radius of a cylinder \cite{17, 18} and sphere \cite{19}. The frequency of oscillation correct upto second order in oscillation amplitude were derived for cylindrical and spherical case, which turned out to be same. In the former case \cite{17, 18}, the authors observed trajectory crossing of the neighbouring electrons which leads to wave breaking via phase mixing. In the latter case \cite{19}, the authors observed that after some plasma period, the wave changes it's direction of propagation which also occurs due to spatial dependency of the characteristic frequency of the wave. This time was termed as
``turn-around time'' \cite{19}. But, an analytical expression for phase mixing time as a
function of the amplitude of the applied perturbation for cylindrical and spherical
plasma oscillations/waves were not presented. Though Gorbunov et. al. \cite{17, 18} attempted to predict a phase mixing time by using Dawson's argument \cite{1}, still the verification of their prediction was never shown explicitly.
In this paper, we extend Dawson's earlier work on cylindrical and spherical oscillations by including relativistic mass variation effect of the electrons.
In section -\ref{sec:2} we extend Dawson Sheet Model from planar to cylindrical and spherical geometry and also include the relativistic mass variation effects. We first derive the expressions for the fluid variables viz. density ($n$), electric field ($E$) and velocity ($v$) by respectively using the principle of conservation of number of particles (continuity equation), Gauss's Law and the relativistic equation of motion. Then we solve the equation of motion in respective coordinate system by using Lindstedt-Poincar\'{e} perturbation method \cite{23} and derive an approximate expression for frequency of oscillation in the weakly relativistic limit upto fourth order in oscillation amplitude. We observe that in general the expressions for frequencies acquire spatial dependency which ultimately lead to breaking via phase mixing. In section -\ref{sec:3} we study the dynamics of cylindrical and spherical plasma oscillation and derive analytical expressions for (cylindrical and spherical) phase mixing time scales as a function of initial amplitude by using Dawson's argument \cite{1}. Further, we verify this scaling by performing numerical simulations, using a code based on Dawson Sheet Model \cite{1, 4, 5, 6, 8, 16, 25, arsu} by extending it to cylindrical and spherical geometry. Finally in section -\ref{sec:4} we summarize this work.
\section{Governing Equations} \label{sec:2}
According to Dawson Sheet Model \cite{4, 5, 6, 8, 16, 25, arsu}, cylindrical and spherical oscillations arise respectively due to cylindrical and spherical sheet of charges oscillating about their equilibrium positions. These sheet of charges are embedded in a homogeneous background of immobile ions. Unlike the slab geometry, in these cases the electric field becomes space dependent due to geometrical effects {\it i.e.} for same displacement amplitude the electric field amplitude becomes different for different equilibrium positions of the sheets, which in turn makes the frequency of oscillation space dependent. This is the basic reason why phase mixing and wave breaking in these cases is an inherent phenomena, arising due to geometry of the problem \cite{1, 17, 18, 19, 20}.
In the following subsections we respectively derive the expressions for the fluid variables and derive equation of motion for cylindrical $\&$ spherical oscillations considering relativistic mass variation effect of the electrons. Let $r_0$ and $R(r_0,t)$ respectively be the equilibrium positions and displacement from the equilibrium positions of the electron sheets. So the Euler positions of the sheets can be written as $r(r_0,t) = r_0 + R(r_0,t)$.
\subsection{Fluid Variables for Cylindrical Oscillations}
In cylindrical geometry, the conservation of the number of particle yields
\begin{equation}
2\pi n_0r_{0}dr_{0} = 2\pi nrdr \nonumber \\
\end{equation}
where, $n_0$ and $n$ respectively are the equilibrium and instantaneous number density. Using the expression for $r(r_{0},t)$ gives number density as
\begin{equation}
n(r_0,t) = \frac{n_0r_0}{(r_0 + R)(1+\frac{\partial R}{\partial r_0})} \label{eq:1}
\end{equation}
Now using Gauss's Law in cylindrical geometry, we have
\begin{equation}
\frac{1}{r}\frac{\partial}{\partial r}(rE) = 4\pi e(n_0 - n) \nonumber\\
\end{equation}
Substituting the value of $n$, from Eq.(\ref{eq:1}), the electric field stands as
\begin{equation}
E(r_0,t) = 2\pi en_0\left[\frac{(r_0 + R)^2 - r_0^2}{(r_0 + R)}\right] \label{eq:2}
\end{equation}
Now the relativistically correct equation of motion of electron sheet can be written as
\begin{equation}
m\frac{d}{dt}\left[\frac{\dot{R}}{(1-\frac{\dot{R}^2}{c^2})^{1/2}}\right] = -eE \nonumber \\
\end{equation}
putting the value of $E$ from Eq.(\ref{eq:2}), we get
\begin{equation}
\frac{\ddot{R}}{(1-\frac{\dot{R}^2}{c^2})^{3/2}} = -\frac{\omega_p^2}{2}\left[\frac{(r_0 + R)^2 - r_0^2}{(r_0 + R)}\right] \label{eq:3}
\end{equation}
Here $\omega_p = 4\pi n_0e^2/m$ is the nonrelativistic plasma frequency. Here $dot$ sign represents derivative w.r.t time.
Taking $R/r_0 = \rho$, Eq.(\ref{eq:3}) modifies as
\begin{equation}
\frac{\ddot{\rho}}{(1 - \frac{r_0^2\dot{\rho}^2}{c^2})^{3/2}} + \frac{\omega_p^2}{2}\left[\frac{(1 + \rho)^2 - 1}{(1+\rho)}\right] = 0 \label{eq:4}
\end{equation}
In nonrelativistic limit ($c \rightarrow \infty$), we get the same equation as obtained by Dawson \cite{1}
\begin{equation}
\ddot{\rho} + \frac{\omega_p^2}{2}\left[\frac{(1 + \rho)^2 - 1}{(1+\rho)}\right] = 0 \nonumber
\end{equation}
In the weakly relativistic limit and small amplitude oscillations Eq.(\ref{eq:4}) can be simplified as \cite{17, 18} (expanded upto the third order of $\rho$ and second order of $\dot{\rho}$)
\begin{equation}
\ddot{\rho} - \frac{3}{2}\frac{r_0^2\omega_p ^2}{c^2}\rho\dot{\rho}^2 + \omega_p^2 \rho - \frac{\omega_p^2}{2}\rho ^2 + \frac{\omega_p^2}{2}\rho ^3 = 0 \label{eq:5}
\end{equation}
Now, by using Lindstedt - Poincar\'{e} perturbation method \cite{23}, the expression for frequency correct upto the fourth order of oscillation amplitude can be written as
\begin{equation}
\Omega_{cy}(rel) = \omega_p\left[1 + \frac{\rho_0(r_0)^2}{12} + \frac{\rho_0(r_0)^4}{512} - \frac{3\omega_p^2}{16}\frac{r_0^2\rho_0(r_0)^2}{c^2} \right] \label{eq:6}
\end{equation}
Here $\rho_0(r_0)$ is the displacement amplitude of the electrons, which in general depends on their equilibrium positions $r_0$. In nonrelativistic limit $c\rightarrow \infty$, Eqs.(\ref{eq:5}) and (\ref{eq:6}) respectively become
\begin{equation}
\ddot{\rho} + \frac{\omega_p^2}{2}\left[2\rho - \rho ^2 + \rho ^3\right] = 0 \label{eq:7}
\end{equation}
and
\begin{equation}
\Omega_{cy}(nonrel) = \omega_p\left[1 + \frac{\rho_0(r_0)^2}{12} + \frac{\rho_0(r_0)^4}{512}\right] \label{eq:8}
\end{equation}
\subsection{Fluid Variables for Spherical Oscillations}
Now following the same procedure in spherical geometry the fluid variables can be written as
\begin{equation}
n(r_0,t) = \frac{n_0r_0^2}{(r_0 + R)^2(1 + \frac{\partial R}{\partial r_0})} \label{eq:9}
\end{equation}
\begin{equation}
E(r_0,t) = \frac{4\pi en_0}{3}\left[\frac{(r_0 + R)^3 - r_0^3}{(r_0 + R)^2} \right] \label{eq:10}
\end{equation}
Using the above expression or electric field, the relativistically correct equation of motion of an electron sheet oscillating along the radius of a sphere is given by,
\begin{equation}
\frac{\ddot{R}}{(1-\frac{\dot{R}^2}{c^2})^{3/2}} + \frac{\omega_p^2}{3}\left[\frac{(r_0 + R)^3 - r_0^3}{(r_0 + R)^2}\right] = 0 \label{eq:11}
\end{equation}
In nonrelativistic limit, the above Eq. transforms to \cite{1}
\begin{equation}
\ddot{R} + \frac{\omega_p^2}{3}\left[\frac{(r_0 + R)^3 - r_0^3}{(r_0 + R)^2}\right] = 0 \nonumber
\end{equation}
In terms of $\rho$, Eq.(\ref{eq:11}) becomes
\begin{equation}
\frac{\ddot{\rho}}{(1 - \frac{r_0^2\dot{\rho}^2}{c^2})^{3/2}} + \frac{\omega_p^2}{3}\left[\frac{(1 + \rho)^3 - 1}{(1+\rho)^2}\right] = 0 \label{eq:12}
\end{equation}
In weakly relativistic limit and small amplitude oscillation, the above Eq. takes the form \cite{19}
\begin{equation}
\ddot{\rho} - \frac{3}{2}\frac{r_0^2\omega_p ^2}{c^2}\rho \dot{\rho}^2 + \omega_p^2 \rho - \omega_p^2 \rho ^2 + \frac{4\omega_p^2}{3}\rho ^3 = 0 \label{eq:13}
\end{equation}
And frequency of oscillation stands as (using Lindstedt - Poincar\'{e} perturbation method \cite{23})
\begin{equation}
\Omega_{sph}(rel) = \omega_p\left[1 + \frac{\rho_0(r_0)^2}{12} + \frac{\rho_0(r_0)^4}{72}- \frac{3\omega_p^2}{16}\frac{r_0^2\rho_0(r_0)^2}{c^2} \right] \label{eq:14}
\end{equation}
In non-relativistic limit, the equation of motion for small amplitude oscillation becomes \cite{1}
\begin{equation}
\ddot{\rho} + \frac{\omega_p^2}{3}\left[3\rho - 3\rho ^2 + 4\rho ^3\right] = 0 \label{eq:15}
\end{equation}
And the frequency of oscillation can be written as
\begin{equation}
\Omega_{sph}(nonrel) = \omega_p\left[1 + \frac{\rho_0(r_0)^2}{12} + \frac{\rho_0(r_0)^4}{72}\right] \label{eq:16}
\end{equation}
Here we want to note that Eqs.(\ref{eq:5}) and (\ref{eq:13}) respectively are the same equation derived by Gorbunov \cite{17, 18} and Bulanov \cite{19}. In the expressions of $\Omega_{cy}(rel)$ and $\Omega_{sph}(rel)$ the second term represents correction due to the relativistic effects and other terms represent additional anharmonicity introduced by cylindrical and spherical geometry respectively. In addition to this, it should be noted from the expressions of the electric field [Eqs.(\ref{eq:2}) and (\ref{eq:10})] that, even for the same displacement amplitude, the electric field depends explicitly on the equilibrium positions of the electrons (unlike the planar geometry case) which results in a position dependent restoring force. Therefore the frequency of oscillation depends on the equilibrium positions of the sheets which leads to wave breaking via phase mixing. In the next section we present a scaling law for this phase mixing time.
\section{Dynamics of Cylindrical $\&$ Spherical plasma oscillations and Calculation of Phase Mixing Time} \label{sec:3}
In this section we describe the dynamics of relativistically intense cylindrical and spherical plasma oscillations and calculate their phase mixing time. In order to study the space-time evolution and breaking of cylindrical and spherical oscillations, we first load the initial conditions needed to excite axisymmetric cylindrical and spherical oscillations respectively in an one dimensional (along the radius) sheet code (based on Dawson Sheet Model) containing $\sim$ 10000 cylindrical and spherical surfaces of charges. For cylindrical and spherical oscillations, its parameters depend only on the radial coordinate of the oscillating species (here the electron sheets) \textit{i.e.} they are azimuthally symmetric in nature. As Bessel functions and Spherical Bessel functions \cite{14,15} respectively form a complete orthogonal set (basis functions) in cylindrical and spherical coordinate systems, then any arbitrary radial perturbation imposed in these systems can be written as a superposition of these basis functions (Fourier Bessel Series) in their respective coordinate system \cite{14,15}. Therefore to excite an oscillation in cylindrically and spherically symmetric system we respectively use Bessel functions and Spherical Bessel functions as an initial perturbation. Here, we note that, the structure and propagation of nonrelativistic axisymmetrical waves in linear and nonlinear regime using such type of initial conditions has been studied first by Travelpiece $\&$ Gould \cite{10} for a cylindrical plasma column and later continued by several authors \cite{11, 12, 13}. Using these type of initial conditions the equation of motion for each electron sheet is solved using fourth order Runge-Kutta scheme. At each time step, ordering of the sheets is checked for sheet crossing. Phase mixing time is measured as the time taken by any two of the adjacent sheets to cross over \cite{1, 4, 5, 6, 8, k, 16, 17, 18, 22, arsu}.
In the following subsections we study the space-time evolution $\&$ derive the expressions for phase mixing time as a function of the amplitude of the applied perturbation for cylindrical and spherical plasma oscillations respectively.
\subsection{Phase Mixing of Relativistically Intense Cylindrical Plasma Oscillations}
The relativistic equation of motion of an electron sheet along the radius of a cylinder is given by Eq.(\ref{eq:3}). This equation can be solved numerically with the help of two initial conditions $R(r_0,t = 0)$ and $\dot{R}(r_0,t = 0)$. Here we consider plasma oscillations localized in space in the vicinity of the axis $r = 0$ \cite{17, 18}. We assume that electron velocity at $t = 0$ is zero : $\dot{R}(r_0,t = 0) = 0$. We also assume that at $t = 0$ the oscillations are excited by an electric field of the form \cite{10, 11, 12}
\begin{equation}
\bar{E}(r_0,t = 0) = \frac{eE(r_0,t = 0)}{m\omega_p v_\phi} = \Delta J_n\left[\frac{\alpha _{n m} r(r_0, 0)}{R_0}\right] \label{eq:17}
\end{equation}
For relativistically intense oscillations, $v_\phi \rightarrow c$. $J_n$ is $n$-th order Bessel function. $\alpha_{n m}$ is the $m$-th zero of $n$-th order Bessel function \cite{14, 15}. $\Delta$ is the amplitude of applied electric field perturbation and $R_0$ is the maximum value of the radius of the cylindrical system under consideration. We consider that at initial time, the electric field $E(r_0,0)$ at the boundaries of the simulation are zero i.e $E(r_0 = 0, t = 0) = 0$ and $E(r_0 = R_0, t = 0) = 0$.
In the above scenario all initial conditions are satisfied by the lowest order mode is $n = 1$ and $m = 1$ (As E is zero on the axis at $t = 0$ so $n = 0$ cannot be taken as lowest order mode as $J_0(\alpha_{01}r/R_0) \neq 0$ at $r = 0$). Therefore, here we study the space-time evolution of the lowest order mode $\Delta J_1(\alpha_{11}r/R_0)$. The length of the system $R_0$ is taken upto the first zero of the Bessel function.
To solve Eq.(\ref{eq:3}) we first numerically calculate the initial profile of $R(r_0, t = 0)$ in the following way: At initial instant of time $t = 0$, sheets are at their equilibrium position $r_0$ and radially displaced from their equilibrium positions by an amount $R(r_0, 0)$ such that they produce an electric field perturbation given by Eq.(\ref{eq:17}) {\it i.e.} $\bar{E}(r_0,t = 0) = \Delta J_1[\alpha _{11} r(r_0,0)/R_0]$. ( The Euler positions of the electrons at $t = 0$ become $r(r_0, 0) = r_0 + R(r_0, 0)$. )
On the other hand from Gauss's Law the electric field $E(r_0, 0)$ is given by
\begin{equation}
E(r_0, 0) = 2\pi en_0\frac{[r_0 + R(r_0, 0)]^2 - r_0^2}{[r_0 + R(r_0, 0)]} \label{eq:18}
\end{equation}
Comparing Eqs.(\ref{eq:17}) and (\ref{eq:18}) we find $R(r_0, 0)$ for given values of $r_0$.
Using the above initial conditions, numerical computations have been carried out and the spatial variation of electric field and electron density with time have been shown. The results of relevant simulations are illustrated in Figs-(\ref{fig:fig1}) and (\ref{fig:fig2}). Figs-(\ref{fig:fig1}) and (\ref{fig:fig2}) respectively show the show the snapshots of the electron density and electric field profiles for the value $\Delta = 0.5$. Fig-(\ref{fig:fig1}) shows that, as time progresses, the density maxima increases gradually and shows a high spike at $\omega _pt = 314.2221$ which is a signature of wave breaking via the process of phase mixing \cite{1, 4, 5, 6, 8, k, 16, 17, 18, 22, arsu}. From Fig-(\ref{fig:fig2}) we observe that, as time goes on, the radial profile of electric field becomes steeper and acquires a jump at the off-axis radial point, where electron density spikes.
Now we calculate the phase mixing time scale in the following way: Equating the electric field given by Eq.(\ref{eq:17}) with Eq.(\ref{eq:18}) and in the small amplitude limit $\Delta << 1$, we can write
\begin{equation}
\frac{\omega _pr_0}{v_\phi}\rho _0(r_0) \sim \Delta J_1 \left( \alpha _{11}\frac{r_0}{R_0}\right) \label{eq:19}
\end{equation}
Substituting the value of $\rho_0(r_0)$ from above in the expression for frequency of relativistic cylindrical plasma oscillation [Eq.(\ref{eq:6})] we get (correct upto $\Delta ^2$)
\begin{equation}
\Omega_{cy}(rel) \sim \omega_p\left[1 - \frac{3}{16}\frac{v_\phi ^2\Delta ^2}{c^2}J_1 ^2\left( \alpha _{11}\frac{r_0}{R_0}\right) + \frac{v_\phi ^2\Delta ^2}{12\omega _p^2r_0^2}J_1 ^2\left( \alpha _{11}\frac{r_0}{R_0}\right) \right] \label{eq:20}
\end{equation}
According to Dawson's argument \cite{1}, the phase mixing time $(\omega_p \tau_{mix})$ depends on the spatial derivative of frequency as $\omega_p \tau _{mix} \sim \pi/[2R_{max}(d\Omega/dr_{0})]$. Differentiating Eq.(\ref{eq:20}) w.r.t $r_0$ and noting $R_{max}$ is proportional to $\Delta$, the phase mixing time scale can be written as
\begin{equation}
\omega _p\tau_{mix} \sim \frac{1}{\Delta ^3} \label{eq:21}
\end{equation}
Here, we have omitted the proportionality constant which is a function of $r_0$. This proportionality constant is related to the position of breaking, which generally depends on the profile of the initial perturbation and is not of general interest. In the similar manner using Eq.(\ref{eq:8}), we can find that, for nonrelativistic cylindrical oscillations also, phase mixing time scale follows the same scaling law as given by Eq.(\ref{eq:21}).
In order to verify this scaling expressed by Eq.(\ref{eq:21}) we have repeated our numerical experiment for different values of $\Delta$. The variation of phase mixing time as function of the amplitude of applied perturbation is shown in Fig-(\ref{fig:fig3}) for both relativistic and nonrelativistic oscillations. In the figure, points represent the simulation results and the solid lines represent our theoretical scaling obtained from Eq.(\ref{eq:21}). By comparing the results, we observe that for a fixed value of the applied perturbation, relativistic effect reduces the phase mixing time.
\subsection{Phase Mixing of Relativistically Intense Spherical Plasma Oscillations}
In this subsection we present the space-time evolution of spherical plasma oscillations and estimate the phase mixing time in a similar fashion as described in the above subsection.
Here the spherical oscillations have been excited by electric field $E(r_0,t)$ of the form
\begin{equation}
\bar{E}(r_0,t = 0) = \frac{eE(r_0,t = 0)}{m\omega_p v_\phi} = \Delta j_\nu\left[\frac{\beta _{\nu m} r(r_0, 0)}{R_0}\right] \label{eq:23}
\end{equation}
where $j_\nu(x)$ is the Spherical Bessel function of order $\nu$ and defined as $j_\nu(x) = \sqrt{\pi /2x} J_{\nu + 1/2}(x)$. $\beta_{\nu m}$ is the $m$-th zero of $\nu$-th order Spherical Bessel function \cite{14, 15}. We have computed the displacement of the sheets from their equilibrium positions {\it i.e.} the value of $R(r_0,0)$ by comparing Eq.(\ref{eq:10}) and (\ref{eq:23}) in the similar fashion as described in the previous subsection. Here we have taken the lowest order mode $\Delta j_1(\beta_{11}r/R_0)$ and the maximum radius of the system ($R_0$) is taken upto the first zero of the Spherical Bessel function. We have solved Eq.(\ref{eq:11}) numerically and calculated the density and electric field profile respectively from the expressions (\ref{eq:9}) and (\ref{eq:10}). The snapshots of density and electric field profile at different time steps are shown in Figs-(\ref{fig:fig5}) and (\ref{fig:fig6}) respectively. Like cylindrical waves here also it is observed that phase mixing leading to wave breaking is manifested by a density burst and a sharp gradient in the electric field profile.
Now to obtain a scaling law for phase mixing of spherical oscillations, we again follow the same procedure. For a small amplitude perturbation $\Delta << 1$, $\frac{\omega _pr_0}{v_\phi}\rho _0(r_0) \sim \Delta j_1 (\beta _{11}\frac{r_0}{R_0})$ and the frequency of spherical oscillation correct upto second order in $\Delta$ for relativistic $\&$ non-relativistic case respectively can be written as,
\begin{equation}
\Omega_{sph}(rel) \sim \omega_p\left[1 - \frac{3}{16}\frac{v_\phi ^2\Delta ^2}{c^2}j_1 ^2\left(\beta _{11}\frac{r_0}{R_0}\right) + \frac{v_\phi ^2\Delta ^2}{12\omega _p^2r_0^2}j_1 ^2\left(\beta _{11}\frac{r_0}{R_0}\right) \right] \label{eq:24}
\end{equation}
and
\begin{equation}
\Omega_{sph}(non-rel) \sim \omega_p\left[1 + \frac{v_\phi ^2\Delta ^2}{12\omega _p^2r_0^2}j_1 ^2\left(\beta _{11}\frac{r_0}{R_0}\right) \right] \label{eq:25}
\end{equation}
Again following Dawson's argument \cite{1}, one can easily arrive at the same scaling law given by Eq.(\ref{eq:21}) for both relativistic and nonrelativistic case. The variation of phase mixing time $(\tau_{mix})$ as a function of applied amplitude $\Delta$ for relativistic and nonrelativistic spherical oscillations are shown in Fig-(\ref{fig:fig7}). In the figure numerical results have been displayed by points and the solid lines represent the scaling given by Eq.(\ref{eq:21}).
In all these cases we observe that, the analytical scaling law given by Eq.(\ref{eq:21}) shows a very good fit to the observed numerical results, thus vindicating our weakly relativistic calculations.
\section{Summary} \label{sec:4}
In this paper, we have demonstrated analytically and numerically the behaviour
of relativistically intense cylindrical and spherical plasma oscillations
using Dawson sheet model. Initial conditions are taken in terms of Bessel functions and Spherical Bessel functions to excite cylindrical and spherical oscillations respectively. This is because, as Bessel functions and Spherical Bessel functions are respectively the normal modes in cylindrical and spherical coordinate systems, thus any arbitrary perturbation in these systems can be written as a superposition of these basis functions in their respective coordinate systems. The expressions for frequencies have been given for both the cases and is found to be an explicit function of the equilibrium positions of the electron sheets. By performing numerical simulations it has been shown that the electron number density associated with cylindrical and spherical plasma oscillations grows sharply with time and after few plasma periods the density shows explosive behaviour due to the crossing of neighbouring electron sheets which is a signature of wave breaking. Analytical expression for the phase mixing time scale has been derived and it is observed that for both cases (cylindrical and spherical) phase mixing time scales with the cube of the oscillation amplitude, which indicates that in general scaling of phase mixing time with amplitude is independent of geometry of oscillation. Further we have found that inclusion of relativistic effects does not change the scaling of phase mixing time with amplitude of perturbation; it only hastens the process as compared to the non-relativistic case as depicted in Figs.(\ref{fig:fig3}) and (\ref{fig:fig7}). Thus cylindrical and spherical oscillations initiated by an arbitrary density perturbation (or electric field perturbation) will always phase mix and the phase mixing time scale can be estimated from the scaling law given by Eq.(\ref{eq:21}).
|
{
"timestamp": "2018-04-12T02:07:35",
"yymm": "1803",
"arxiv_id": "1803.02773",
"language": "en",
"url": "https://arxiv.org/abs/1803.02773"
}
|
\section{Introduction}
Many systems exhibit abrupt changes, or tipping, e.g. population extinction \cite{drake2010early,dodorico2005noise}, emergence of infectious diseases \cite{dibble2016waiting}, \txb{financial systems crisis \cite{may2008complex}}, compression buckling of mechanical structures \cite{vella2009macroscopic}, and climate transitions \cite{lenton2008tipping,ditlevsen2010tipping,turney2017rapid}.
Tipping is dangerous if some states of the system are associated with extreme or catastrophic events, \txb{and this explains the interest this subject has received in the last decades. Recently, different studies demonstrated that economical or environmental disasters can be modelled as dynamical systems incurring a tipping. Therefore, the development of tipping forecasting techniques with early indicators is an active research area \cite{scheffer2009early,kuehn2011mathematical,scheffer2012anticipating}}.\\
A key aspect in this context is the distinction between three types of tipping, rooted in different causes \cite{ashwin2012tipping}.\\
\textit{B-tipping} is induced by a \textit{Bifurcation} where the system state changes drastically for a small change of a control parameter. \txb{In this case, the tipping can be often predicted with techniques that rely on the popular concept of critical slowing down \cite{nazarimehr2017can,dakos2008slowing,dakos2015resilience,lenton2012early,meisel2015critical}, or that make use of other properties of the attractor \cite{karnatak2017early,jiang2018predicting}}.\\
In \textit{N-tipping}, dynamic \textit{Noise} induces jumps between several coexisting attractors \txb{(e.g. \cite{sutera1981stochastic,sura2002noise,semenov2017noise,nikolaou2015detection}); in this case, the analysis of the time series statistic can help in detecting precursor of critical transitions \cite{carpenter2006rising,chen2018rising}}.\\
\textit{R-tipping} is induced by the \textit{Rate} at which a control parameter is varied\txb{, if several possible attractors are present in the range of parameter variation. Inertial effects play a central role in R-tipping. In the case of standard R-tipping, the system starts from an attractor but, if the parameter rate of change is larger than a critical value, it cannot follow this attractor and tips to another one. \cite{ashwin2017parameter,siteur2016ecosystems,chen2015patterned,wieczorek2011excitability}. In the case of ``preconditioned R-tipping'', the system starts from an unstable condition and, depending on the rate of change, it evolves towards one of the possible attractors \cite{tony2017experimental}}.\\
\txb{Inertial effects can also delay the bifurcation, moving the tipping point to higher/lower values of the bifurcation parameter as observed, for example, by Baer and Gaekel in \cite{baer2008slow} for the FitzHugh-Nagumo model. This delay is in general a function of the parameter rate of change. Therefore we will refer to this effect as \emph{``Rate-delayed tipping''}}.\\
\begin{figure}[]
\centering
\includegraphics[width=0.75\columnwidth]{04.png}
\caption{Illustration of the various types of tipping encountered in the vicinity of the bi-stable region of a sub-critical bifurcation according to the classification proposed in \cite{ashwin2012tipping}. Solid and dashed black lines: deterministic attractor and repeller, respectively. Light to dark hues: low to high probability density. a) to c) B-tipping, \txb{Rate-delayed tipping} and N-tipping. d) Present work where B-, N- and \txb{Rate-delayed} tipping mechanisms occur simultaneously (see \cref{fig:03_ramp_exp} for the experimental data).}
\label{fig:04_summary}
\end{figure}
\txb{All those} mechanisms can manifest independently, or, like in the present study, simultaneously.
In this case, the evolution of the system results from the interplay of different time scales set by the ramp rate, the noise intensity and the system relaxation time \cite{shi2016towards}. \txb{Several examples can be found in the recent literature. Ashwin et al. \cite{ashwin2017fast} study the regimes of transition and the escape time in a network of bistable nodes as a function of the coupling and noise strengths. Sun and coworkers \cite{sun2015delay} assess the possibility of tipping for a Duffing-Van der Pol oscillator with time-delayed feedback, as a function of forcing noise intensity, feedback time delay and feedback intensity. The work from Clements and Ozgul \cite{clements2016rate} deals with two stochastic models for population dynamics, and studies the effect of the rate of change of one governing parameter on the system dynamics and on the predictability of tipping}.
\txb{Berglund and Gentz \cite{berglund2002pathwise} provide theoretical and numerical analyses for rate-delayed tipping in presence of noise in supercritical pitchfork bifurcations. An analogous study is carried out by Ritchie and Sieber in \cite{ritchie2016early} for a rate-dependent tipping in a saddle-node bifurcation.}
\txb{Kwasniok \cite{kwasniok2015forecasting} introduces a method to predict a fold and a Hopf bifurcation in presence of noise. Kuehn \cite{kuehn2017uncertainty} studies the delay in a Hopf bifurcation with a random initial condition}.\\
In this study, we show experimental evidence of simultaneous B-, N- and Rate-delayed tipping mechanisms at a Hopf subcritical bifurcation, in a lab-scale combustor subject to thermoacoustic instabilities in presence of turbulence-induced noise.\\
The four panels in figure \ref{fig:04_summary} illustrate how the three types of tipping combine in our system. In these diagrams, the amplitude $A$ is plotted as a function of the bifurcation parameter $\nu$.
In the absence of noise and for a quasi-steady change of the bifurcation parameter (figure \ref{fig:04_summary}a), the system state evolves on the deterministic attractor, leading to B-tipping and hysteresis (blue and red for increasing and decreasing $\nu$). This quasi-steady picture changes if the bifurcation parameter varies at a finite rate (figure \ref{fig:04_summary}b): bifurcation delay occurs, and it is a function of the rate (e.g. \cite{premraj2016experimental,holden1993slow,bergeot2014response}). For a quasi-steady variation of $\nu$ in the presence of stochastic forcing (\cref{fig:04_summary}c), the hysteresis is suppressed in a statistical sense. For each value of the bifurcation parameter, the state is defined by a probability density distribution. In this case, N-tipping occurs in the bistable region (e.g. \cite{samoilov2005stochastic,lenton2008tipping}). Finally, when the bifurcation parameter is varied at a finite rate in presence of stochastic forcing (\cref{fig:04_summary}d), the highest probability of state transition is delayed. This is the case discussed in this work. Our scenario therefore results from the combination of a finite-rate ramping through a stochastic subcritical bifurcation. \\
\txb{This paper is organised as follows. In \cref{sec:TA_inst}, we introduce the physical problem of thermoacoustic instabilities. In \cref{ss:stat_exp} and \cref{ss:ramp_exp} we} show experimental results where the average tipping point is delayed when the control parameter is ramped at a finite rate.
In \cref{ss:stat_mod} and \cref{ss:ramp_mod} we develop a low-order stochastic model of the system and demonstrate with a quantitative first-passage time analysis how the bifurcation delay statistic varies with the ramping rate.
Finally, in \cref{s:FPA} we consider a situation where a control parameter is ramped up and, if tipping is detected, ramped down in order to come back to the initial safe state.
In this situation, the system may suffer irreversible damage if the ramp up is too fast, which applies to many industrial applications or to natural systems like, for instance, climate transitions.
\begin{figure*}[h!]
\centering
\includegraphics[trim= 0 0 0 0, clip, width=\textwidth]{01.png}
\caption{Thermoacoustic instabilities occur in combustion chambers for aeronautic and power-generation applications. a) Schematic of our lab-scale swirled combustor. b) Illustration of this unstable coupling. Under a certain phase difference relationship, well known as the Rayleigh criterion, a constructive feedback establishes between the unsteady heat release rate $q(\boldsymbol{x},t)$ and the acoustic field $p(\boldsymbol{x},t)$. c) Thermoacoustic limit cycle in the lab-scale combustor used for this work. In the left loop, four snapshots of the flame showing the coherent motion of the flame leading to $q$ and originating from the thermoacoustic feedback. Time traces of acoustic pressure and heat release rate are shown on the right.}
\label{fig:01_TA}
\end{figure*}
\section{Thermoacoustic instabilities}
\label{sec:TA_inst}
Thermoacoustic coupling is a phenomenon that has fascinated scientists for over two centuries. In 1777, Dr. William Higgins reported, with surprise, on a hydrogen flame emitting ``sweet tones'' if placed inside a glass tube \cite{higgins1802}. In 1894, Lord Rayleigh provided an explanation to this observation: the gas in the tube resonates if the flame (or any other source) provides heat at the moment of maximum gas compression \cite{rayleigh1896}.\\
Many years after, during the Cold War, thermoacoustic instabilities became a very critical issue for one of the most challenging project ever realised by humankind: the Apollo program to take man to the Moon. As detailed in \cite{oefelein1993}, the F-1 engines propelling the Saturn V had destructive combustion instabilities that required 2000 full-scale tests, with empirical modifications of the chamber geometry before the rocket was ready for take off.\\
More recently, thermoacoustic instabilities became a recurrent issue in the development phase of heavy-duty gas turbines for power generation and turbofans for air transportation. This is because the resulting intense acoustic fields induce high-cycle fatigue of the combustion chambers \cite{lieuwen2012book}. For heavy-duty gas turbines, the pressing demand for machines with high power density and ultra low emissions, which are capable of compensating the production intermittency of the wind and solar sources, led to the use of lean premixed flames. Unfortunately, these flames are more prone to thermoacoustic instabilities. In the case of airplane turbofans, these instabilities constitute an increasingly serious obstacle to the development of new aeroengines complying with more stringent emission regulations \cite{icao2016}.\\
The suppression of these instabilities is very challenging due to the uniqueness and complexity of engines in real life application \cite{poinsot2017}.
Despite the achievements attained over the past decades in terms of passive mitigation implementation, development engineers cannot predict if a combustor prototype will have a sufficiently large pulsation-free operating window, over which the acoustic level is acceptable for the mechanical integrity of the components.
\Cref{fig:01_TA}a shows a schematic of our lab-scale combustor\footnote{Additional details about the combustor and the experimental apparatus are provided in the appendix.}. The air pre-mixed with methane enters the plenum, a volume that, in practice, evens the flow delivered by the compressor and guides it to the inlet of the burner. Then, the mixture passes through the swirler, a set of curved blades that rotate the flow. This rotational motion is essential to achieve a stable anchoring of the flame. Then, the flow enters the combustion chamber, where combustion takes place.
At any operating point, the fluctuating component $q$ of the heat release rate $Q=\bar{Q}+q$ acts as a source term in the wave equation:
\begin{equation}
\label{eq:wave}
\dfrac{\partial^2 p}{\partial t^2} - c^2 \nabla^2 p = (\gamma-1)\dfrac{\partial q}{\partial t},
\end{equation}
where $p$ is the acoustic pressure, $c$ the speed of sound and $\gamma$ the specific heat ratio. In practice, the unsteady heat release of the flame $q$ is influenced by the acoustic field $p$, via, for instance, acoustically-triggered fuel supply modulation or coherent vortex shedding, which leads to a thermoacoustic feedback loop \cite{boujo2016quantifying}.
As illustrated in \cref{fig:01_TA}b, the geometry of the combustor and the temperature distribution define a set of acoustic modes in the chamber.
Each mode is characterised by a shape and an eigenvalue. The latter determines whether the thermoacoustic mode is linearly stable or unstable. The system stability depends on several operating parameters, such as the mass flows of fuel $\dot{m}_\text{CH4}$ or air $\dot{m}_\text{air}$. The transition from linearly stable to linearly unstable regime occurs at Hopf bifurcations, where the sign \txb{of the growth rate of the mode} changes. If unstable, the thermoacoustic dynamics is characterised by a limit cycle, with amplitudes $p_\text{rms}$ and $q_\text{rms}$ being defined by the natural acoustic damping of the chamber, and by the linear and nonlinear components of the flame response to acoustic perturbations \cite{boujo2016quantifying,noiray2017method}. The non-coherent component of the heat release rate fluctuations, which is induced by turbulence, acts as a broadband forcing on this self-sustained thermoacoustic oscillation.\\
A typical operating condition for which we observe a thermoacoustic limit cycle is presented in \cref{fig:01_TA}c (see also the movie in the supplementary material). The four panels in the loop show instantaneous flame pictures and the corresponding phase-averaged flame shapes. The right plot displays the time traces of the acoustic pressure signal $p$ (in red) and the heat release rate $q$ (in blue) (note the symbols on the time trace corresponding to the four flame snapshot in the left loop). The flame exhibits a periodic motion at the frequency of the first acoustic mode (150~Hz), with sound intensity at the anti-nodes exceeding 150 dB, which is considerable for a burner operated at atmospheric pressure. This dynamic state would not be acceptable in a commercial aeronautical engine or in a heavy-duty gas turbine, because the acoustic loading, which scales with the engine operating pressure, would be highly detrimental for the mechanical components.\\
In this work, we focus on the transient thermoacoustic dynamics associated with the passage through the Hopf bifurcation when one of the critical operating parameters -- the equivalence ratio -- is ramped. We show experimental evidence of a bifurcation delay and explain the phenomenon using a surrogate low-order model. This is particularly relevant for the development of new aeronautical and land-based gas turbines, which require fast loading or deloading, and which may be at risk due to such \txb{rate-delayed} tipping points.
\section{Subcritical bifurcation}
\txb{This section presents two main results. In the first part, the results of the experimental mapping of the combustor dynamics as a function of the equivalence ratio are shown. In the second part, a low-order model of the system is derived.}
\begin{figure*}[h!
\centering
\includegraphics[width=\textwidth]{02.png}
\caption{a) Experimental records of the thermoacoustic subcritical Hopf bifurcation investigated in this work. According to the methane/air mixture equivalence ratio $\phi=(\dot{m}_\text{CH4}/\dot{m}_\text{air})/(\dot{m}_\text{CH4}/\dot{m}_\text{air})_\text{stoich.}$, the acoustic pressure recorded in the combustor has three different signatures, reflected in the different shapes of the PDFs $P(p)$ and $P(A)$. From top to bottom (increasing $\phi$): small amplitude acoustic pressure resulting from the forcing of the linearly stable thermoacoustic mode by turbulence-induced noise; bistable thermoacoustic dynamics with two intermittently visited attractors; high amplitude limit cycle. These three possible regimes are also presented by the joint PDFs of the oscillation phase portrait $P(p,\dot{p}/\omega)$ at three exemplary $\phi$. b) Surrogate oscillator model \eqref{eq:oscillator} that mimics the subcritical Hopf bifurcation when the parameter $\nu$ is increased. In the 3D plot, $P_\infty(p,\dot{p}/\omega\, ; \nu)$ and three cuts, resembling the experimental $P(p,\dot{p}/\omega)$. c) The stationary PDF $P_\infty(A\,;\,\nu)$ for the slow-varying oscillation amplitude $A$, obtained with the transformation of variables $A^2=p^2+(\dot{p}/\omega)^2$. On top of it, the deterministic pitchfork and saddle-node bifurcation diagram (in blue), and the stationary PDF $P_\infty(A)$ with the corresponding potential $V(A)$ for an overdamped particle at the three selected $\nu$.}
\label{fig:02_bif}
\end{figure*}
\subsection{Stationary experiment}
\label{ss:stat_exp}
The combustor was operated selecting one equivalence ratio $\phi$ at a time. The stationary operation was reached and a long acoustic pressure signal $p(t)$ was recorded using a microphone placed inside the chamber. The oscillation amplitude $A(t)$ was then extracted by applying the Hilbert transform to $p(t)$. The procedure was repeated for different equivalence ratios $\phi$ in the range [0.580; 0.635].
The results for five selected $\phi$ are presented in \cref{fig:02_bif}a. On the left, the measured acoustic pressure and amplitude signals are plotted,
together with their probability density functions (PDFs) $P(p)$ and $P(A)$. On the right, the joint PDFs $P(p,\dot{p}/\omega)$ for three of the presented operative points show the statistic of the phase portraits.\\
These results demonstrate how the system undergoes a subcritical Hopf bifurcation when the control parameter is varied. For low equivalence ratio $\phi$, the system state is attracted towards zero. The small fluctuations of the acoustic signal envelope are due to the stochastic forcing exerted by the intense turbulence in the combustor. For intermediate values of $\phi$, two states are possible: small amplitude acoustic pressure and high amplitude limit cycle. The intermittency between the two states is triggered by the turbulence-induced stochastic forcing (N-tipping, as in \cref{fig:04_summary}c). For higher equivalence ratio $\phi$, the stochastically-forced limit cycle is the only stable state. The reader can refer to the supplementary material for the movies of the three regimes.\\
\subsection{Non-linear oscillator model}
\label{ss:stat_mod}
\txb{
The thermoacoustic behaviour described in the previous section can be reproduced by a low-order model derived from first principles. The Helmholtz equation (\ref{eq:wave}) (hereafter rewritten in Laplace space) defines the acoustic pressure field in the combustor, given an unsteady source of heat in the volume and impedance conditions at the boundaries::
\begin{equation}
\label{helmholtz}
\nabla^{2}\widehat{p}(s,x)-\left(\frac{s}{c}\right)^2\widehat{p}(s,x)=-s\frac{(\gamma-1)}{c^{2}}\widehat{Q}(s,x) \,\, \text{in the domain},
\end{equation}
\begin{equation}
\label{helmholtz_BC}
\frac{\widehat{p}(s,x)}{\mathbf{\widehat{u}}(s,x)\cdot \mathbf{n}}=Z(s,x) \,\,\,\,\, \text{on the boundaries},
\end{equation}
where $s$ is the Laplace variable, $\widehat{p}$ and $\mathbf{\widehat{u}}$ are the transforms of acoustic pressure and velocity fluctuations, $x$ the position, $c$ the local speed of sound, $\gamma$ the specific heat ratio, $\widehat{Q}$ the heat release rate source term, $\mathbf{n}$ the outward normal to the boundary and $Z$ the acoustic impedance. This equation is valid under low Mach number conditions.\\
Although non-linear coupling among different thermoacoustic modes can occur in some practical configurations, we focus on situations where, like in the present case, one mode is dominant in the thermoacoustic dynamics. Therefore, it is possible to project the acoustic field on an orthogonal basis $\mathbf{\Psi}$ and approximate the system's dynamics with the one of the dominant mode only, which will be denoted by $\psi$ \cite{lieuwen2003statistical,culick2006unsteady}. This yields the approximation $\widehat{p}(s,x)\approx\widehat{\eta}(s)\psi(x)$, $\widehat{\eta}$ being the mode amplitude:
\begin{equation}
\label{eq:mode}
\widehat{\eta}=\dfrac{s\rho c^2}{s^2+\omega^2}\dfrac{1}{V\Lambda}\left(\dfrac{\gamma-1}{\rho c^2}\int_V\widehat{Q}(s,x)\psi^*(x)\text{d}V-\widehat{\eta}\int_{\partial V}\dfrac{|\psi(x)|^2}{Z(s,x)}\text{d}\sigma\right),
\end{equation}
where $\rho$ is the gas density and $\Lambda$ the mode normalisation coefficient. This equation can be rewritten as:
\begin{equation}
\label{eq:mode_osc}
(s^2+\alpha s+\omega_0^2)\widehat{\eta}=s\widehat{q},
\end{equation}
\begin{equation}
\label{eq:mode_alpha}
\text{with} \quad \alpha= \dfrac{\rho c^2}{V\Lambda}\int_{\partial V}\dfrac{|\psi(x)|^2}{Z(s,x)}\text{d}\sigma,
\end{equation}
\begin{equation}
\label{eq:mode_q}
\text{and} \quad \widehat{q}= \dfrac{\gamma-1}{V\Lambda}\int_V\widehat{Q}(s,x)\psi^*(x)\text{d}V.
\end{equation}
Therefore, the system dynamics can be approximated by a forced damped harmonic oscillator \eqref{eq:mode_osc} of resonance frequency $\omega_0$. The term $\alpha>0$ represents the damping mechanisms, and it is assumed to be constant, since the impedance at the boundaries is generally a smooth function of $s$ and therefore is not expected to vary significantly around $\omega_0$. The term $\widehat{q}$ is the result of the weighting on the mode shape of the volumetric heat release rate and can be decomposed into two contributions: $\widehat{q}=\widehat{q}_n+\widehat{q}_c$. The first component $\widehat{q}_n$ represents the non-coherent part of the heat release rate oscillations, induced by the intense turbulence present in practical combustors. The term $\widehat{q}_c$ refers to the coherent heat release rate fluctuations, which result from a feedback interaction with the acoustic field established in the combustor. Hence, this term can be expressed as a non-linear function of the modal amplitude $\eta$. It is customary to simplify this function by replacing it with its truncated Taylor expansion \cite{lieuwen2003statistical,culick2006unsteady}. The linear term coefficient $\beta$ of this expansion defines, together with the linear damping $\alpha$, the linear stability of the system. Absorbing in the constants the mode shape $\psi(x_p)$ at the pressure probe location $x_p$ and considering only the odd terms of the series expansion up to the fifth order leads to the following oscillator model for the thermoacoustic system:
}
\begin{equation}
\label{eq:oscillator}
\ddot{p}+\omega_0^2p=[2\nu +\kappa p^2-\mu p^4]\dot{p}+\xi(t),
\end{equation}
where $\nu=(\beta-\alpha)/2$ is the oscillation linear growth rate in $\mathrm{rad.s^{-1}}$ and $\kappa$ and $\mu$ the two positive constants that define the non-linear component of the oscillator response. The term $\xi(t)$ is a white noise forcing of intensity $\Gamma$ that models the non-coherent turbulence-induced heat release rate fluctuations.
In \cref{fig:02_bif}b, the plot shows the stochastic Hopf bifurcation featured by this oscillator, as a function of the bifurcation parameter $\nu$.
This three-dimensional representation of the stationary joint-probability density $P_\infty(p, \dot{p}/\omega\, ; \nu)$ is depicted together with 3 orthogonal cuts resembling the ones obtained from the experiments and showing that the bifurcation parameter $\nu$ of the surrogate model \eqref{eq:oscillator} corresponds to the equivalence ratio $\phi$ in the experiments.\\
It is convenient to describe the system evolution in terms of the slowly varying amplitude $A$ and phase $\varphi$, with $p(t)=A(t)\cos(\omega_0t+\varphi(t))$. By inserting this ansatz for $p$ into the second order stochastic differential equation \eqref{eq:oscillator} and by performing deterministic and stochastic averaging (e.g. \cite{noiray2017linear}), one gets first order Langevin equations for $A$ and $\varphi$. The equation for $A$ is $\mathrm{d}{A}/\mathrm{dt}=\mathcal{F}(A)+\zeta$, where $\mathcal{F}(A)=A\left[\nu+(\kappa/8)A^{2}-(\mu/16)A^4\right]+\Gamma/(4\omega_{0}^{2}A)$ and $\zeta$ is a white noise forcing of intensity $\Gamma/2\omega_0^2$. The deterministic dynamics derives from a potential with $\mathcal{F}(A)=-\mathrm{d}V/\mathrm{d}A$, and the equation does not depend on $\varphi$, which leads to the corresponding Fokker-Planck equation (FPE) for the variation in time of the amplitude PDF $P(A;t)$:
\begin{equation}
\label{eq:fp}
\dfrac{\partial P}{\partial t}=-\dfrac{\partial}{\partial A}[\mathcal{F}(A)P]+\dfrac{\Gamma}{4\omega_0^2}\dfrac{\partial^2P}{\partial A^2}
\end{equation}
Setting ${\partial P}/{\partial t}=0$, one obtains the stationary PDF $P_\infty(A\,;\,\nu)$, plotted in \cref{fig:02_bif}c as a function of the linear growth rate $\nu$, in a pitchfork bifurcation diagram fashion. To provide a visual reference, the bifurcation diagram of the deterministic case (i.e. in absence of noise, $\Gamma=0$) is superimposed in blue. This diagram shows the subcritical pitchfork and the saddle-node bifurcations governing the system.
In the bottom insets, the PDFs $P_\infty(A\,;\,\nu_i)$ for three selected values of the bifurcation parameter $\nu$ are presented. In the upper insets, the corresponding potentials are plotted. The linearly stable and stable limit cycle conditions feature a single potential well at low or high amplitude, while the bistable case has two potential wells.
The stochastic forcing causes the jumps from one basin of attraction to the other, and hence the intermittency between low-amplitude noisy fluctuations and high-amplitude limit cycle oscillations.
\section{Ramping}
\txb{In this section, the dynamics of the system under transient operation is analysed. In the first part, experimental results obtained by ramping the bifurcation parameter are provided. They highlight the presence in the system dynamics of B- and N-tipping mechanisms combined with inertial and hysteresis effects. In the second part, the model introduced in \cref{ss:stat_mod} is used to study the influence of the ramp rate on the system dynamics.}
\begin{figure*}[h!
\centering
\includegraphics[width=\textwidth]{03.png}
\caption{Experimental evidence of the bifurcation delay and of the dynamic hysteresis in the ramped $\phi$ experiment. The panels are divided in ramp up (top row) and ramp down (bottom row). The stationary probability density function $P_\infty(A;\,\phi_i)$ at seven equivalence ratios $\phi$ (grey) is given as a reference and compared to the evolution in time of the ensemble PDF $P(A;\,t_i)$, where $t_i$ is the time at which $\phi(t_i)=\phi_i$ with $\phi(t)=\phi_0+Rt$ for the ramp up and $\phi(t)=\phi_\text{E}-Rt$ for the ramp down.}
\label{fig:03_ramp_exp}
\end{figure*}
\subsection{Ramp experiment}
\label{ss:ramp_exp}
\txb{The following} test was performed on the test rig to highlight the peculiar dynamics of this combustor. The methane and air mass flows were controlled such that the equivalence ratio $\phi$ repeated 100 times the following four-step cycle: 1) linear increase for 4s from $\phi_0$ = 0.580 to $\phi_\text{E}$ = 0.635; 2) idle for 10s at $\phi_\text{E}$; 3) linear decrease for 4s back to $\phi_0$; 4) idle for 10s at the lowest equivalence ratio. \Cref{fig:03_ramp_exp} presents the results of this experiment. The panels are grouped in two rows: the upper row corresponds to the statistic of the 100 ramps up, the lower row to the one of the 100 ramps down. Each column corresponds to an equivalence ratio. The PDFs of this ramp experiment were obtained via a kernel density estimation (KDE) applied over the 100 realisations, and they are plotted in color (blue for the ramp up, red for the ramp down). In all the panels, the stationary PDF for the corresponding $\phi$ (no ramping, already presented in \cref{fig:02_bif}a) is given in grey as a reference.\\
The system experiences dynamic hysteresis: in the bistable region, even though the stationary PDF features two maxima, the system stays in the low-amplitude (resp. high-amplitude) range when $\phi$ is ramped up (resp. down). Another feature is the delay in transition, easily observable in the bottom row: the dynamic PDF peak is at an amplitude that is higher than the one of the stationary PDF at the same $\phi$. This means that the system experiences inertial effects, remaining close to the initial state longer: a bifurcation delay is observed. This observation corresponds to the case depicted in \cref{fig:04_summary}d.
\begin{figure}[h!
\centering
\includegraphics[width=0.75\columnwidth]{05.png}
\caption{Ramping cycle of $\nu$ between the values $\nu_0=-4.5$ and $\nu_\text{E}=5.5$ for the oscillator model \eqref{eq:oscillator} for two different ramp rates $R$. The contour plot represents the PDF $P(A;t)$ computed by time-marching the FPE \eqref{eq:fp}. The time is normalised with the ramp time $t_\text{ramp}=(\nu_\text{E}-\nu_0)/R$. In the insets, the ramp cycle in dimensional time. The other parameters of the model are: $\omega_0/2\pi=120$s\textsuperscript{-1}, $\kappa=8$s\textsuperscript{-1}, $\mu=2$s\textsuperscript{-1}, $\Gamma/4\omega_0^2=0.44$.}
\label{fig:05_ramp_sim}
\end{figure}
\\
\subsection{Rate-dependent bifurcation delay}
\label{ss:ramp_mod}
The ramp rate, together with the ramp profile, is expected to influence the bifurcation delay, as shown in \cite{baer2008slow} for a deterministic system. We therefore used the surrogate oscillator model to investigate this aspect in more detail. The parameter $\nu$ was varied linearly in time between two values $\nu_0$ and $\nu_\text{E}$ at different rates $R$.
Two approaches were used. The first is to simulate \eqref{eq:oscillator} in Simulink, varying the initial condition and running different realisations of the process. Extracting the envelope for each realisation and considering the ensemble statistic, it is possible to draw maps of the evolution in time of the amplitude PDF $P(A;t)$. The other approach is to integrate numerically the FPE \eqref{eq:fp} and obtain directly $P(A;t)$. The two methods closely agree, as shown in the appendix. In \cref{fig:05_ramp_sim} the results of the FPE integration are presented. A ramp up/idle/ramp down/idle cycle is solved, for two different ramp rates $R=$50 rad/s\textsuperscript{2} and $R=$10 rad/s\textsuperscript{2}. The dynamic stochastic bifurcation delay is captured and it is observed that a faster ramp leads to a more pronounced delayed transition from one stable point to the other.\\
An important aspect of the phenomenon depicted in this figure is that the state transitions are delayed with respect to the bifurcation point, but not time delayed (the horizontal axis in these figures is normalised by the physical duration of the cycle). In other words,
a higher rate of change of the time-varying potential induces a faster transition into the neighbouring basin of attraction, but the transition occurs for a delayed value of the bifurcation parameter compared to the quasi-steady picture of the system.
\begin{figure}[h!]
\centering
\includegraphics[trim= 0 0 0 0, clip, width=0.5\textwidth]{06.png}
\caption{$P_{A_\text{th}}(\nu_\text{th};\,R)$ is the probability density of the instantaneous linear growth rate $\nu$ at the first passage over the threshold amplitude $A_\text{th}$, as function of the ramp rate $R$. It is obtained from simulations of the unsteady FPE with absorbing boundary at $A=A_\text{th}$. In blue $\langle\nu_\text{th}(R)\rangle$, the linear growth rate of the system at the mean first passage time.}
\label{fig:06_fpt}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.75\textwidth]{07.png}
\caption{Two exemplary cases (square $R$=10, circle $R$=50) are simulated in Simulink, implementing a control system that ramps down $\nu$ if the danger level is reached.
Top row: different realisations of the process (thin lines, grey and red in the safety and danger zones, respectively)
and the two extreme realisations in terms of first passage time (thick grey lines) with their associated quasi-steady deterministic bifurcation diagrams (blue lines).
Bottom row: KDE of the PDFs. The mean residence and mean released energy in the danger zone are indicated too.
}
\label{fig:07_control}
\end{figure}
\section{First Passage Analysis}
\label{s:FPA}
In this section, we imagine that a tipping point is feared due to the monotonous change of a key parameter of the system, and that one wants to ramp back this parameter sufficiently early to avoid the critical transition. In that situation, the underlying time-varying potential landscape is unknown and a controller monitors the state of the system while the parameter varies.
As is usually done when new prototype engines are tested, we will take the current state of the system to feed the controller. Indeed, gas turbines and aeronautical engine combustors are equipped with a controller that constantly monitors the acoustic pressure level in the chamber. In case the measured acoustic pressure is too high, the control system intervenes, either changing the parameters to bring the operating condition back to a safe point, or in extreme cases, shutting off the flame by closing the fuel supply valve.\\
If the combustor features a subcritical bifurcation on the varied parameter (e.g. $\phi$), the system inertia is a factor that has to be taken into account. In this case, the transition from low to high amplitudes happens suddenly and, if the bifurcation delay is long, the reached acoustic pressure level can be considerable. In this situation, the control system detects the danger late and might be ineffective in avoiding damages to the system.
A way to estimate the hazard represented by the delayed bifurcation is to compute, using the surrogate model, the statistic of the time $t_\text{fp}$ needed to reach a certain danger level. This is similar to the classical problem of first passage time, often addressed in the context of bifurcation theory for stochastic dynamics in steady double-well potential \cite{torrent1988stochastic,kuske1999probability,miller2012escape,hu2010first,dibble2016waiting}. A major difference in the present situation is that the potential evolves with time. Ramp rate and noise intensity are expected to influence this escape problem as theoretically shown for other types of bifurcation in \cite{ritchie2017probability} or \cite{berglund2002beyond}. The statistic of the first passage time can be computed either performing an ensemble average over many time-domain simulations of the process, or solving the unsteady Fokker-Planck equation and imposing an absorbing boundary condition at that threshold level. Details about the two methods, with results in close agreement, are provided in the appendix.
The value $\nu(t_\text{th})=\nu_\text{th}$ of the control parameter $\nu(t)$ at the first passage time $t_\text{th}$ is of particular interest: this quantity is proportional to the danger of the delayed transition, as it determines the limit cycle amplitude when the transition occurs. This $\nu_\text{th}$ statistic can be determined as $\nu_\text{th}= \nu_0+Rt_\text{th}$.
The results are presented in \cref{fig:06_fpt}. The contour levels represent the probability density of $\nu_{th}$ as a function of the ramp rate $R$. The mean value of $\nu_{th}$ (plotted in blue, $\langle\nu_{th}\rangle$) increases with the ramp rate $R$, while the time needed to reach the danger level is shorter (see the iso-time lines). This finding indicates that a fast ramp of the control parameter is dangerous if a subcritical bifurcation is present, as exemplified in the two test cases presented in \cref{fig:07_control}. Here the process was simulated in Simulink: the parameter $\nu$ was ramped up at two different rates $R$ (10 rad/s\textsuperscript{2} and 50 rad/s\textsuperscript{2}) and when the danger level was reached, ramped back down at the maximum rate $R$=-50 rad/s\textsuperscript{2}. In the top row, many realisations of this process are presented. As a function of the initial condition and of the random excitation, each realisation has a different evolution and, therefore, a different first passage time.
The two extreme realisations (shortest and longest first passage times) are highlighted with thick lines. The respective deterministic bifurcation diagrams are superimposed to provide a visual reference. The PDFs obtained with a KDE over the realisations are plotted in the bottom row. The control system effectively brings the oscillations back to a safe level in both cases. However, the combined action of the finite ramp-down rate, dynamic hysteresis and inertia causes the system to stay in the danger zone for a certain time. The faster case $R$=50 rad/s\textsuperscript{2} is more critical: as discussed before, the crossing of the threshold level happens on average when the target $\nu$ is already high. As a result, the system abruptly reaches high-amplitude oscillations and has to travel a long distance on the bifurcation diagram upper branch before reaching the safety zone.
This effect can be gauged by comparing two quantities for the two cases $R=10$ and $50$~rad/s\textsuperscript{2}: in the latter case, the mean residence time over the safety threshold $\Delta t_\text{th}$ is twice larger and the mean released energy $\langle E_\text{th}\rangle \propto (1/ \Delta t_{th})\int_{\Delta t_\text{th}}A^2\,\mathrm{d}t$ is nine times larger.
\section{Conclusions}
A subcritical Hopf bifurcation of a thermoacoustic system was investigated in this work. A lab-scale combustor was operated under different values of methane/air equivalence ratio, which serves as bifurcation parameter: depending on its value, acoustic pressure amplitude in the chamber is either damped, intermittently switching between low and high amplitudes, or attracted towards high-amplitude, which corresponds to a stable limit cycle. The main focus of the work was on the transient dynamics: the equivalence ratio was ramped in time and dynamic hysteresis and delayed bifurcation were observed.
A non-linear oscillator surrogate model was used to investigate the effect of the ramp rate on the bifurcation delay. It was shown that when the control parameter is ramped faster, the transition from the damped regime to the limit cycle occurs for higher values of the bifurcation parameter. The corresponding first passage problem in a time-varying potential was solved with the unsteady Fokker-Planck equation and with Monte Carlo simulations of the process.
This study primarily addresses a major problem of practical combustion systems. Operating conditions of gas turbines are often varied in time, for matching power grid requirement, and similar rapid changes of the combustion regimes also occur in aeronautical engines, especially at take-off. If a subcritical thermoacoustic bifurcation is present, a delayed bifurcation results in a sudden and unexpected acoustic pressure level rise, which is detrimental to the machine integrity. Therefore a slow variation of the machine parameters is advisable, especially when mapping the operating points of a new combustor. More broadly, this study is relevant for the countless systems, which exhibit critical transitions. This work highlights the importance of carefully considering the rate of change of the bifurcation parameter, when investigating tipping points.\\
\emph{Data accessibility.} The datasets supporting this article are available at doi:10.5061/dryad.4cj4k.\\
\emph{Author contributions.} N.N., G.B. and E.B. designed research; G.B. performed the numerical simulations and analysed all the data; D.E. made the experiments and supported in the analysis of experimental data; N.N. and E.B. provided scientific advises and helped in analysing data. G.B. and N.N. wrote the paper. All the authors proofread and made suggestions about the manuscript.\\
\emph{Competing interests.} We declare we have no competing interests.\\
\emph{Funding.} This research is supported by the Swiss National Science Foundation under Grant 160579.
\bibliographystyle{apsrev_mod}
|
{
"timestamp": "2018-03-08T02:09:00",
"yymm": "1803",
"arxiv_id": "1803.02676",
"language": "en",
"url": "https://arxiv.org/abs/1803.02676"
}
|
\section{Introduction}
Supervised learning has been highly effective in solving challenging tasks in sentiment analysis over the last few years. However, the success of supervised learning for the domain in recent years has been premised on the availability of large amounts of data to effectively train models. Obtaining a large labeled dataset is time-consuming, expensive, and sometimes infeasible; and this has often been the bottleneck in translating the success of machine learning models to newer problems in the domain.
An approach that has been used to solve this problem is to crowdsource the annotation of data, and then aggregate the crowdsourced labels to obtain ground truths. Online platforms such as Amazon Mechanical Turk and CrowdFlower provide a friendly interface where data can be uploaded, and workers can annotate labels in return for a small payment. With the ever-growing need for large labeled datasets and the prohibitive costs of seeking experts to label large datasets, crowdsourcing has been used as a viable option for a variety of tasks, including sentiment scoring \cite{CSsentimentscoring}, opinion mining \cite{CScommodityreview}, general text processing \cite{Snow:2008:CFG:1613715.1613751}, taxonomy creation \cite{Bragg2013CrowdsourcingMC}, or domain-specific problems, such as in the biomedical field \cite{DBLP:journals/corr/GuanGDH17,Albarqouni2016AggNetDL}, among many others.
In recent times, there is a growing need for a fast and real-time solution for judging the sentiment of various kinds of data, such as speech, text articles, and social media posts. Given the ubiquitous use of the internet and social media today, and the wide reach of any information disseminated on these platforms, it is critical to have a efficient vetting process to ensure prevention of the usage of these platforms for anti-social and malicious activities. Sentiment data is one such parameter that could be used to identify potentially harmful content. A very useful source for identifying harmful content is other users of these internet services, that report such content to the service administrators. Often, these services are set up such that on receiving such a flag, they ask other users interacting with the same content to classify whether the content is harmful or not. Then, based on these votes, a final decision can be made, without the need for any human intervention. Some such works include: crowdsourcing the sentiment associated with words \cite{CSsentimenttoword}, crowdsourcing sentiment scoring for online media \cite{CSsentimentscoring}, crowdsourcing the classification of words to be used as a part of lexicon for sentiment analysis \cite{CSlexicon}, crowdsourcing sentiment judgment for video review \cite{CSvideoreview}, crowdsourcing for commodity review \cite{CScommodityreview}, and crowdsourcing for the production of word level annotation for opinion mining tasks \cite{CSsyntacticrelatedness}. However, with millions of users creating and adding new content every second, it is necessary that this decision be quick, so as to keep up with and effectively address all flags being raised. This indicates a need for fast vote aggregation schemes that can provide results for a stream of data in real time.
The use of crowdsourced annotations requires a check on the reliability of the workers and the accuracy of the annotations. While the platforms provide basic quality checks, it is still possible for workers to provide incorrect labels due to misunderstanding, ambiguity in the data, carelessness, lack of domain knowledge, or malicious intent. This can be countered by obtaining labels for the same question from a large number of annotators, and then aggregating their responses using an appropriate scheme. A simple approach is to use majority voting, where the answer which the majority of annotators choose is taken to be the true label, and is often effective. However, many other methods have been proposed that perform significantly better than majority voting, and these methods are summarized further in Section \ref{related}.
Despite the various recent methods proposed, one of the most popular, robust and oft-used method to date for aggregating annotations is the Dawid-Skene algorithm, proposed by \cite{dawid1979maximum}, based on the Expectation Maximization (EM) algorithm. This method uses the M-step to compute error rates, which are the probabilities of a worker providing an incorrect class label to a question with a given true label, and the class marginals, which are the probabilities of a randomly selected question to have a particular true label. These are then used to update the proposed set of true labels in the E-step, and the process continues till the algorithm converges on a proposed set of true labels (further described in Section \ref{dawidskenealgo}).
In this work, we propose a new simple, yet effective, EM-based algorithm for aggregation of crowdsourced responses. Although formulated differently, the proposed algorithm can be interpreted as a `hard' version of Dawid-Skene (DS) \cite{dawid1979maximum}, similar to Classification EM \cite{celeux1992classification} being a hard version of the original EM. The proposed method converges upto 7.84x faster than DS, while maintaining similar accuracy. We also propose a hybrid approach, a combination of our algorithm with the Dawid-Skene algorithm, that combines the high rate of convergence of our algorithm and the better likelihood estimation of the Dawid-Skene algorithm as part of this work.
\section{Related Work}
\label{related}
The Expectation-Maximization algorithm for maximizing likelihood was first formalized by \cite{10.2307/2984875}. Soon after, Dawid and Skene \cite{dawid1979maximum} proposed an EM-based algorithm for estimating maximum likelihood of observer error rates, which became very popular for crowdsourced aggregation and is still considered by many as a baseline for performance. Many researchers, to this day, have worked on analyzing and extending the Dawid-Skene methodology (henceforth, called DS), of which we summarize the more recent efforts below. The work on crowdsourced data aggregation have not been confined only for sentiment analysis or opinion mining tasks, instead most of the methods are generic and can easily used for sentiment analysis and opinion mining tasks.
A new model, GLAD, was proposed in \cite{NIPS2009_3644}, that could simultaneously infer the true label, the expertise of the worker, and the difficulty of the problem, and use this to improve on the labeling scheme. \cite{Raykar:2010:LC:1756006.1859894} improved upon DS by jointly learning the classifier while aggregating the crowdsourced labels. However, the efforts of \cite{NIPS2009_3644} were restricted to binary choice settings; and in the case of \cite{Raykar:2010:LC:1756006.1859894}, they focused on classification performance, which is however not the focus of this work.
\cite{ipeirotis2010quality} presented improvements over DS to recover from biases in labels provided by the crowd, such as cases where a worker always provides a higher label than the true label when labels are ordinal. More recently, \cite{NIPS2016_6124} analyzed and characterized the tradeoff between the cost of obtaining labels from a large group of people per data point, and the improved accuracy on doing so, as well as the differences in adaptive vs non-adaptive DS schemes.
In addition to these efforts, there has also been a renewed interest in recent years to understand the rates of convergence of the Dawid-Skene method. \cite{minimax-optimal-convergence-rates-for-estimating-ground-truth-from-crowdsourced-labels} obtained the convergence rates of a projected EM algorithm under the homogeneous DS model, which however is a constrained version of the general DS model. \cite{NIPS2014_5431} proposed a two-stage algorithm which uses spectral methods to offset the limitations of DS to achieve near-optimal rate convergence. \cite{article} recently proposed a permutation-based generalization of the DS model, and derived optimal rates of convergence for these models. However, none of these efforts have explicitly focused on increasing the speed of convergence, or making Dawid-Skene more efficient in practice. The work in \cite{IWMV} is the closest in this regard, where they proposed an EM-based Iterative Weighted Majority Voting (IWMV) algorithm which experimentally leads to fast convergence. We use this method for comparison in our experiments.
In addition to methods based on Dawid-Skene, other methods for vote aggregation have been developed, such as using Gaussian processes \cite{Rodrigues:2014:GPC:3044805.3044941} and online learning methods \cite{Welinder2010OnlineCR}. The scope of the problem addressed by Dawid-Skene has also been broadened, to allow cases such as when a data point may have multiple true labels \cite{DUAN20145723}. (In this work, we show how our method can be extended to this setting too.) For ensuring reliability of the aggregated label, a common approach is to use a large number of annotators, which may however increase the cost. To mitigate this, work has also been done to intelligently assign questions to particular annotators \cite{0768fc60fef84637864e13671a981243}, reduce the number of labels needed for the same accuracy \cite{Welinder2010OnlineCR}, consider the biases in annotators \cite{NIPS2011_4311} and so on.
Recent work on vote aggregation also includes deep learning-based approaches, such as \cite{Albarqouni2016AggNetDL,training-deep-neural-nets-aggregate-crowdsourced-responses,DBLP:journals/corr/abs-1709-01779}. A survey of many earlier methods related to vote aggregation can be found in the work of \cite{10.1007/978-3-642-41154-0_1} and \cite{sheshadri2013square}. Moreover, a benchmark collection of methods and datasets for vote aggregation is defined in \cite{sheshadri2013square}, which we use for evaluating the performance of our method.
While many new methods have been developed, the DS algorithm still remains relevant as being one of the most robust techniques, and is used as a baseline for nearly every new method. Inspired by \cite{celeux1992classification}, our work proposes a simple EM-based algorithm for vote aggregation, that provides a similar performance as Dawid-Skene but with a much faster convergence rate. We now describe our method.
\section{Proposed Algorithm}
\label{algos}
We propose an Expectation-Maximization (EM) based algorithm for efficient vote aggregation. The E-step estimates the dataset annotation based on the current parameters, and the M-step estimates the parameters which maximize the likelihood of the dataset. Starting from a set of initial estimates, the algorithm alternates between the M-step and the E-step till the estimates converge. Although formulated using a different approach to the aggregation problem, we call our algorithm Fast Dawid-Skene (FDS), because of its similarity to the DS algorithm (described in Section \ref{dawidskenealgo}).
\subsection{Preliminaries}
\label{subsec_preliminaries}
For convenience, we use the analogy of a question-answer setting to model the crowdsourcing of labels. The data shown to the crowd is viewed as a question, and the possible labels as choices of answers from the crowd worker/participant. Let the questions (data points, problems) that need to be answered be $q = \{1,2,3,\dots,Q\}$ and the annotators (participants, workers) labeling them be $a = \{1,2,3,\dots,A\}$. The task requires the participants to label each question by selecting one of the predefined set of choices (options), $c = \{1,2,3,\dots, C\}$, which has the same length across all questions. A participant is said to answer a given question when s/he chooses an option as the answer for that question. A participant need not answer all the questions, and in fact, for a large pool of questions, it is reasonable to assume that a participant might be invited to answer only a small subset of all the questions. Each question is assumed to be answered by at least one participant (ideally, more). We also assume that the choice selected by a participant for a question is independent of the choice selected by any other participant. This assumption holds for real-world applications that use contemporary crowdsourcing methods, where participants generally do not know each other, and are often physically and geographically separated, and thus do not influence each other. Besides, while answering a question, the participants have no knowledge of the choices chosen by previous participants in these settings.
\subsection{The Fast Dawid-Skene Algorithm}
\label{ouralgo}
We now derive the proposed Fast Dawid-Skene (FDS) algorithm under the assumption that each question has only one correct choice, and that a participant can select only one choice for each question. (In Section \ref{discussions}, we show how our method can be extended to relax this assumption.)
Our goal is to aggregate the choices of the crowd for a question and to approximate the correct choice. Consider the question $q$. Let the $K$ participants that answered this question be $\{q_1, q_2, \dots, q_K\}$. The value of $K$ may vary for different questions. Let the choices chosen by these $K$ participants for question $q$ be $\{c_{q_1}, c_{q_2}, \dots, c_{q_K}\}$, and the correct (or aggregated) answer to be estimated for the question $q$ be $Y_q$. We define the answer to the question $q$ to be the choice $c \in \{1,2,\dots,C\}$ for which $P\left(Y_{q} = c | c_{q_1}, c_{q_2}, \dots, c_{q_K}\right)$ is maximum. Using Bayes' theorem and the independence assumption among participants' answers, we obtain:
\begin{align}\label{e1}
P&(Y_{q} = c | c_{q_1},c_{q_2},\dots, c_{q_K})\nonumber \\
&= \frac{P(c_{q_1}, c_{q_2}, \dots, c_{q_K} | Y_{q} = c)P(Y_{q} = c)}{\sum\limits_{c=1}^{C} P(c_{q_1}, c_{q_2}, \dots, c_{q_K} | Y_{q} = c)P(Y_{q} = c)}\nonumber\\
&= \frac{\left(\prod\limits_{k = 1}^{K} P(c_{q_k} | Y_{q} = c)\right)P(Y_{q} = c)}{\sum\limits_{c = 1}^{C} \left(\prod\limits_{k = 1}^{K} P(c_{q_k} | Y_{q} = c)\right)P(Y_{q} = c) }
\end{align}
Let $T_{qc}$ be the indicator that the answer to question $q$ is choice $c$. Using our formulation:
\begin{equation}\label{e2}
T_{qc} = \begin{cases} 1 &c = \underset{j \in \{1,2,\dots,C\}}{\arg\max} P(Y_{q} = c | c_{q_1}, c_{q_2}, \dots, c_{q_K}) \\ 0 & \text{otherwise}
\end{cases}
\end{equation}
These $T_{qc}$s serve as the proposed answer sheet.
To determine the correct (or aggregated) choice for a question $q$, we need the values of $P(c_{q_k} | Y_{q} = c)$ for all $k$ and $c$, which however is not known given only the choices from the crowd annotators. However, if the correct choices are known for all the questions, we can compute these parameters. Let $q_k$ be the annotator $a$. To compute the parameters, we first define the following sets:
\begin{equation*}S_{a}^{(c)} = \left\{ i\, |\, Y_i = c \wedge a \text{ has answered question } i \right\}
\end{equation*}
and
\begin{equation*}T_{c_a}^{(c)} = \left\{ i \,|\, Y_i = c \wedge a \text{ has answered } c_a \text{ on question } i \right\}
\end{equation*}
Then, we have:
\begin{equation}\label{e3}
P(c_a | Y_{q} = c) = \frac{ \left| T_{c_a}^{(c)} \right|}{ \left| S_a^{(c)} \right|}
\end{equation}
where $\left| \cdot \right| $ denotes the cardinality of the set.
Also, $P(Y_{q} = c)$ can be defined as:
\begin{equation}\label{e4}
P(Y_{q} = c) = \frac{\text{Number of questions having answer as }c}{\text{Total number of questions}}
\end{equation}
The above quantities can be estimated if we have the correct choices, and conversely, the correct choices can be obtained using the above quantities. We hence use an Expectation-Maximization (EM) strategy, where the E-step calculates the correct answer for each question, while the M-step determines the maximum likelihood parameters using equations \ref{e3} and \ref{e4}. There are no pre-calculated values of parameters to begin with, and so in the first E-step, we estimate the correct choices using majority voting. We continue applying the EM steps until convergence. We use the total difference between two consecutive class marginals being under a fixed threshold as the convergence criterion. We discuss the convergence criterion in more detail in Section \ref{experiments}. The proposed algorithm is summarized below in Algorithm \ref{fdsalgorithm}.
\begin{algorithm}
\caption{The Fast Dawid-Skene Algorithm}\label{fdsalgorithm}
\begin{algorithmic}[1]
\Input Crowdsourced choices of $Q$ questions by $A$ participants (annotators) from $C$ choices
\Output Proposed true choices - $T_{qc}$
\State Estimate $T$s using majority voting.
\Repeat
\State \textit{M-step:} Obtain the parameters, $P(c_a | Y_{q} = c)$ and $P(Y_{q} = c)$ using Equations \ref{e3} and \ref{e4}
\State \textit{E-step:} Estimate $T$s using the parameters, $P(c_a | Y_{q} = c)$ and $P(Y_{q} = c)$, and with the help of Equations \ref{e2} and \ref{e1}.
\Until convergence
\end{algorithmic}
\end{algorithm}
\subsection{Connection to Dawid-Skene Algorithm}
\label{dawidskenealgo}
The Dawid-Skene algorithm \cite{dawid1979maximum} was one of the earliest EM-based methods for aggregation, and still remains popular and competitive to newer approaches. In this subsection, we briefly describe the Dawid-Skene methodology, and show the connection of our approach to this method.
As defined in \cite{dawid1979maximum}, the maximum likelihood estimators for the DS method are given by:
\begin{footnotesize}
\begin{align*}
\hat{\pi}_{cl}^{(a)} &= \frac{\text{number of times participant $a$ chooses $l$ when $c$ is correct}}{\text{number of questions seen by participant $a$ when $c$ is correct}}
\end{align*}
\end{footnotesize}
\noindent and $\hat{p_c}$, which is the probability that a question drawn at random has a correct label of $c$. Let $n_{ql}^{(a)}$ be the number of times participant $a$ chooses $l$ for question $q$. Let $\{T_{qc} : q = 1,2,\dots, Q\}$ be the indicator variables for question $q$. If choice $m$ is true, for question $q$, $T_{qm} = 1$ and $\forall j \ne m,\,T_{qj} = 0$. Given the assumptions made in Section \ref{subsec_preliminaries}, when the true responses of all questions are available, the likelihood is given by:
\begin{equation}\label{e8}
\prod_{q=1}^{Q} \prod_{c=1}^{C} \left\{ p_c \prod_{a=1}^{A} \prod_{l=1}^{C} \left(\pi_{cl}^{(a)}\right)^{n_{ql}^{(a)}}\right\}^{T_{qc}}
\end{equation}
where $n_{ql}^{(a)}$ and $T_{qc}$ are known. Using equation \ref{e8}, we obtain the maximum likelihood estimators as:
\begin{equation}\label{e9}
\hat{\pi}_{cl}^{(a)} = \frac{\sum_q T_{qc} n_{ql}^{(a)}}{\sum_l \sum_q T_{qc} n_{ql}^{(a)}}
\end{equation}
\begin{equation}\label{e10}
\hat{p}_c = \frac{\sum_q T_{qc}}{Q}
\end{equation}
We then obtain using Bayes' theorem:
\begin{equation}\label{e11}
p(T_{qc} = 1 | \text{data}) = \frac{\prod_{a=1}^{A} \prod_{l=1}^{C} (\pi_{cl}^{(a)})^{n_{ql}^{(a)}} p_c }{ \sum_{r=1}^{C} \prod_{a=1}^{A} \prod_{l=1}^{C} (\pi_{rl}^{(a)})^{n_{ql}^{(a)}} p_r}
\end{equation}
The DS algorithm is then defined by using equations \ref{e9} and \ref{e10} to obtain the estimates of $p$s and $\pi$s in the M-step, followed by using equation \ref{e11} and the estimates of $p$s and $\pi$s to calculate the new estimates of $T$s in the E-step. These two steps are repeated until convergence (when the values don't change over an iteration).
A close examination of the DS and proposed FDS algorithms shows that our algorithm can be perceived as a `hard' version of DS. The DS algorithm derives the likelihood assuming that the correct answers (which are ideally binary-valued) are known, but uses the values for $T_{qc}$ (which form a probability distribution over the choices) directly as obtained from equation \ref{e11}. Instead, in our formulation, we always have $T_{qc}$ as either $0$ or $1$ after each E-step. Our method is similar to the well-known Classification EM proposed in \cite{celeux1992classification}, which shows that a `hard' version of EM significantly helps fast convergence and helps scale to large datasets \cite{jollois2007speed}. We show empirically in Section \ref{experiments} that this subtle difference between DS and FDS ensures that changes in the answer sheet dampens down quickly, and allows our method to converge much faster than DS with comparable performance.
A careful implementation for both FDS and DS provides a solution in $O(QACn)$ time under the assumption that there is only one correct choice for each question, where $n$ is the number of iterations required by the algorithm to converge. As the cost per iteration of FDS would be similar to DS by the nature of its formulation, this implies that the speedup of our algorithm is proportional to the ratio of the number of iterations required to converge by the two algorithms, which we also confirm experimentally.
\subsection{Theoretical Guarantees for Convergence}
In this subsection, we establish guarantees for convergence. We prove that if we start from an area close to a local maximum of the likelihood, we are guaranteed to converge to the maximum at a linear rate. For the analysis of our algorithm's convergence, we first frame it in a way similar to the Classification EM algorithm as proposed by \cite{celeux1992classification}. Classification EM introduces an extra C-step (Classification step) after the E-step. This is the step that assigns each question a single answer, thus doing a `hard' clustering of questions based on options instead of the `soft' clustering by DS.
To continue with the proof we will use the notation used for DS. The term
$ P(c_{q_k} | Y_{q} = c)$ for FDS is replaced by $\pi_{cc_{qk}}^{q_k}$ and the term
$ P(Y_q = c) $ for FDS is replaced by $p_c$. $n_{ql}^{(a)}$ used by DS would be either $1$ or $0$ for the setting considered.
Having established the analogy, we restate the algorithm in CEM form (Algorithm \ref{cemalgorithm}).
\begin{algorithm}
\caption{The Fast Dawid-Skene Algorithm}\label{cemalgorithm}
\begin{algorithmic}[1]
\Input Crowdsourced choices of $Q$ questions by $A$ participants (annotators) from $C$ choices
\Output Proposed true choices - $T_{qc}$
\State Estimate $T$s using majority voting. This essentially does the first E and C step.
\Repeat
\State \textit{M-step:} Obtain the parameters, $\pi$s and $p$s using Equations \ref{e3} and \ref{e4}
\State \textit{E-step:} Estimate $T$s using the parameters, $\pi$ and $p$, and with the help of Equation \ref{e1}.
\State \textit{C-step:} Assign $T$s using the values obtained in the E-step and Equation \ref{e2}.
\Until convergence
\end{algorithmic}
\end{algorithm}
We prove the convergence of the CEM algorithm similar to \cite{celeux1992classification}. For the proof, let us first form partitions. We form $C$ partitions out of all the questions based on their correct answer in a step.
\begin{equation}
P_c = \{q | Y_q = c\}
\end{equation}
In the CEM approach, each question can belong to only one partition.
Now, we define the CML (Classification Maximum Likelihood) criterion:
\begin{equation}
C_2(P,p,\pi) = \sum_{c=1}^{C} \sum_{q \in P_c} \log \left({ p_c f(q, \pi_c)}\right)
\end{equation}
In the above equation, $\pi_c = \{\pi_{cj}^{(a)} | \forall j \in \{1\dots C\} \text{ and a } \in \{1\dots A\} \}$ and
\begin{equation}
f(q,\pi_c) = \prod_{a=1}^{A} \prod_{l=1}^{C} \left(\pi_{cl}^{(a)}\right)^{n_{ql}^{(a)}}
\end{equation}
To prove convergence, we define a few more notations. Note that we begin the algorithm by first doing a majority vote. This assigns each question to a class and forms the first partition. We denote this partition as $P^0$. We then proceed to the M-step and estimate $\pi$ and $p$. Let us denote this first set of parameters by $\pi^1$ and $p^1$. The next EC step gives the next partition, $P^1$. Thus, the algorithm continues to calculate $(P^{m}, p^{m+1}, \pi^{m+1})$ from $(P^{m}, p^{m}, \pi^{m})$ in the M step. Then, in the EC step, it calculates $(P^{m+1}, p^{m+1}, \pi^{m+1})$ from $(P^{m}, p^{m+1}, \pi^{m+1})$.
\begin{theorem}
For the sequence $(P^{m}, p^{m}, \pi^{m})$ obtained by FDS, the value of $C_2(P^{m}, p^{m}, \pi^{m})$ increases and converges to a stationary value. Under the assumption that $p$s and $\pi$s are well defined, the sequence $(P^{m}, p^{m}, \pi^{m})$ converges to a stationary point.
\end{theorem}
\begin{proof}
To prove the above theorem we prove that \\$C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m}, \pi^{m}) \, \forall m > 1$.\\
Note that equations \ref{e3} and \ref{e4} maximize the likelihood given the values of $T$ and $n$ (as shown by \cite{dawid1979maximum}), i. e. $T$ is known, and so $\pi$s and $p$s obtained by the M-step maximize the likelihood. We need to show that maximizing the likelihood is the same as maximizing the CML criterion, $C_2$. In the case of hard clustering, for each $q$, only one class, $c$, can have $T_{qc}$ as $1$; all other classes will have $T_{qc}$ as 0. With this observation, we can rewrite the CML criterion as:
\begin{align}
C_2(P,p,\pi) &= \sum_{c=1}^{C} \sum_{q \in P_c} \log (p_c f(q, \pi_c))\\
&= \log \left\{\prod_{q=1}^{Q} \prod_{c=1}^{C} \left( p_c f(q, \pi_c) \right)^{T_{qc}} \right\}\\
&= \log \left\{ \prod_{q=1}^{Q} \prod_{c=1}^{C} \left( p_c \prod_{a=1}^{A} \prod_{l=1}^{C} \left(\pi_{cl}^{(a)}\right)^{n_{ql}^{(a)}} \right)^{T_{qc}} \right\}
\end{align}
Thus, maximizing maximum likelihood is equivalent to maximizing $C_2$. So, we have that after the M step, $C_2(P^{m}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m}, \pi^{m})$.\\
Now, we consider the EC step. Observe that for each question $q$, we choose the answer as the option $c'$ for which $p_c' f(q,\pi_c') \ge p_c f(q,\pi_c)$ for all $c$ (By definition of the criterion for the C-step). Thus, $\log { p_c f(q, \pi_c)}$ increases individually for each question, and so cumulatively,
$C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m+1}, \pi^{m+1})$.\\
Combining the two inequalities, we obtain,
\begin{equation}
C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m}, \pi^{m})
\end{equation}
This proves that $C_2$ increases at each step. Since the number of questions are finite and so the number of partitions as well are finite; the value of $C_2$ must converge after a finite number of iterations.\\
On convergence, we obtain $ C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) = \\C_2(P^{m}, p^{m+1}, \pi^{m+1}) = C_2(P^{m}, p^{m}, \pi^{m})$ for some $m$. By definition of the C-step, the first equality implies that $P^{m+1} = P^{m}$. Also under the assumption that $p$s and $\pi$s are well defined, we have that $p^m = p^{m+1}$ and $\pi^{m+1} = \pi^m$. This proves the convergence to a stationary point.
\end{proof}
To prove the rate of convergence, we define $M$ to be the set of matrices $U \in \mathbb{R}^{C \times Q}$ of nonnegative values. The matrices are defined such that the summation of values in each column is 1 and the summation along each row is nonzero.\\
Consider the criterion to be maximized as:
\begin{equation}
C_2'(U,p,\pi) = \sum_{c=1}^{C} \sum_{q=1}^{Q} u_{qc} \log (p_c f(q, \pi_c))
\end{equation}
With the above definitions, proposition 3 of \cite{celeux1992classification} guarantees a linear rate of convergence for FDS to a local maximum from a neighborhood around the maximum.
\subsection{Hybrid Algorithm}
\label{hybridalgo}
While the proposed FDS method is quick and effective, by using the softer marginals, DS can obtain better likelihood values (which we found in some of our experiments too). A comparison of the likelihood values over multiple datasets (described in Section 4) is provided in Table 2. To bring the best of both DS and FDS, we propose a hybrid version, where we begin with DS, and at each step, we keep track of sum of the absolute values of the difference in class marginals ($p_c$s). When this sum falls below a certain threshold, we switch to the FDS algorithm and continue (Algorithm \ref{hybalgorithm}). Our empirical studies showed that this hybrid algorithm can maintain high levels of accuracy along with faster convergence (Section \ref{experiments}). We however observe that a similar likelihood to DS does not necessarily translate to better accuracy, and in fact FDS outperforms Hybrid on some datasets.
\begin{algorithm}
\caption{The Hybrid Algorithm}\label{hybalgorithm}
\begin{algorithmic}[1]
\Input Crowdsourced choices for $Q$ questions by $A$ participants given $C$ choices per question, threshold $\gamma$
\Output Aggregated choices: $T_{qc}$
\State Estimate $T$s using majority voting.
\Repeat
\State \textit{M-step:} Obtain parameters, $\hat{\pi}_{cl}^{(a)}$ and $\hat{p}_c$ using equations \ref{e9} and \ref{e10}
\State \textit{E-step:} Estimate $T$s using parameters, $\hat{\pi}_{cl}^{(a)}$ and $\hat{p}_c$ using equation \ref{e11}.
\Until $\sum_c | p_c^t - p_c^{t-1} | < \gamma$
\Repeat
\State EM steps of Algorithm \ref{fdsalgorithm} (FDS)
\Until convergence
\end{algorithmic}
\end{algorithm}
\section{Experimental Results}
\label{experiments}
We validated the proposed method on several publicly available datasets for vote aggregation, and the results are presented in this section. We first describe the datasets, competing methods used for comparison and the performance metrics used before presenting the results.
\paragraph{Datasets:} We used seven real-world datasets to compare the performance of the proposed method against other methods. These include
\textit{LabelMe} \cite{Russell2008,R7807338}, \textit{SentimentPolarity (SP)} \cite{Pang:2005:SSE:1219840.1219855,Rodrigues:2014:GPC:3044805.3044941}, \textit{DAiSEE} \cite{d2016daisee,kamath2016crowdsourced}, and four datasets from the SQUARE benchmark \cite{sheshadri2013square}: \textit{Adult2} \cite{ipeirotis2010quality}, \textit{BM} \cite{DBLP:journals/corr/abs-1209-3686}, \textit{TREC2010} \cite{Buckley10-notebook}, and \textit{RTE} \cite{Snow:2008:CFG:1613715.1613751}.
Many of the datasets had varying number of annotators per data point. For uniformity, we set a threshold for each dataset, and all data points with fewer annotators than the threshold were removed. In our experiments, we studied the performance of all the methods by varying the number of annotators from one till the threshold, by taking a random subset of all annotators for a data point at each step (We maintained the same random seed across the methods, and conducted multiple trials to verify the results presented herewith). Also, the \textit{TREC2010} dataset has an `unknown' class, which we removed for our experiments. Table 1 lists the size, the number of classes, and the number of annotators in each dataset.
\begin{table}[t]
\begin{center}
\begin{scriptsize}
\setlength\tabcolsep{3pt}
\begin{tabular}{|P{1.1cm}||P{0.5cm}|P{0.8cm}|P{1cm}||P{1.1cm}|P{1.2cm}|P{1.1cm}|}
\hline
& \# qns &\# options (per qn)& Maximum \# of annotators (per qn) & Speedup of FDS over DS in Time (Iterations) & Speedup of FDS over IWMV in Time (Iterations) & Speedup of Hybrid over DS in Time (Iterations)\\
\hhline{|=||=|=|=||=|=|=|}
Adult2 & 305 & 4 & 9 & 6.61(7.87) & 1.32(1.15) & 2.30(2.43)\\
\hline
BM & 1000 & 2 & 5 & 2.69(4.51) & 1.70(1.02) & 1.49(2.03)\\
\hline
TREC2010 & 3670 & 4 & 5 & 7.84(8.64) & 6.09(2.93) & 4.39(4.59) \\
\hline
DAiSEE & 4628 & 4 & 10 & 6.57(7.37) & 4.40(2.04) & 4.11(4.37)\\
\hline
LabelMe & 589 & 8 & 3 & 7.55(8.59) & 0.54(1.14) & 5.15(5.47)\\
\hline
RTE & 800 & 2 & 10 & 3.14(4.95) & 2.63(1.24) & 1.88(2.24)\\
\hline
SP & 4968 & 2 & 5 & 3.00(3.95) & 2.78(0.94) & 2.40(2.54)\\
\hline
\end{tabular}
\captionof{table}{Datasets Used and Speedup of FDS and Hybrid}\label{datasettable}
\end{scriptsize}
\end{center}
\end{table}
\vspace{-10pt}
\paragraph{Baseline Methods:} A total of six aggregation algorithms were used in our experiments for evaluation - Majority Voting (MV), Dawid-Skene (DS) \cite{dawid1979maximum}, IWMV \cite{IWMV}, GLAD \cite{NIPS2009_3644}, proposed Fast Dawid-Skene (FDS), and the proposed hybrid algorithm. IWMV is among the fastest methods using EM for aggregation under general settings. \cite{IWMV} compared IWMV against other well-known aggregation methods, including \cite{Raykar:2010:LC:1756006.1859894}, \cite{Karger} and \cite{LPI}, and showed that IWMV gives an accuracy comparable to these algorithms but does so in a much lesser time. We hence compare our performance to IWMV in this work. GLAD \cite{NIPS2009_3644}, another popular method, was proposed only for questions with two choices, and we hence use this method for comparison only on the binary label datasets in our experiments.
\paragraph{Performance Metrics:} For each experiment, the following metrics were observed: the accuracy of the aggregated results (against provided ground truth), time taken and number of iterations needed for empirical convergence. For DS, FDS, and Hybrid, the negative log likelihood after each iteration was also observed. For MV, only the accuracy was observed. The experiments were conducted on a 4-core system with Intel Core i5-5200U 2.20GHz processors with 8GB RAM.
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\begin{figure*}[t]
\label{fig_result_graphs}
\setlength\tabcolsep{0pt}
\begin{tabular}{@{}ccccccc@{}}
\centering
\begin{footnotesize}Adult2\end{footnotesize} & \begin{footnotesize}BM\end{footnotesize} & \begin{footnotesize}TREC2010\end{footnotesize} & \begin{footnotesize}DAiSEE\end{footnotesize} & \begin{footnotesize}LabelMe\end{footnotesize} & \begin{footnotesize}RTE\end{footnotesize} & \begin{footnotesize}SP\end{footnotesize} \\
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\subfigure
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\begin{tikzpicture}[scale=0.3]
\begin{axis}[
legend pos=outer north east,
ymajorgrids=true,
grid style=dashed,
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\addplot[
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\begin{tikzpicture}
\begin{customlegend}[legend columns=-1,
legend style={
draw=none,
column sep=1ex,
},legend entries={\begin{footnotesize}DS\end{footnotesize}}]
\csname pgfplots@addlegendimage\endcsname{blue,mark=*,sharp plot}
\end{customlegend}
\end{tikzpicture} &
\begin{tikzpicture}
\begin{customlegend}[legend columns=-1,
legend style={
draw=none,
column sep=1ex,
},legend entries={\begin{footnotesize}IWMV\end{footnotesize}}]
\csname pgfplots@addlegendimage\endcsname{black,mark=x,sharp plot}
\end{customlegend}
\end{tikzpicture} &
\begin{tikzpicture}
\begin{customlegend}[legend columns=-1,
legend style={
draw=none,
column sep=1ex,
},legend entries={\begin{footnotesize}MV\end{footnotesize}}]
\csname pgfplots@addlegendimage\endcsname{purple,mark=o,sharp plot}
\end{customlegend}
\end{tikzpicture} &
\begin{tikzpicture}
\begin{customlegend}[legend columns=-1,
legend style={
draw=none,
column sep=1ex,
},legend entries={\begin{footnotesize}GLAD\end{footnotesize}}]
\csname pgfplots@addlegendimage\endcsname{orange,sharp plot}
\end{customlegend}
\end{tikzpicture} &
\begin{tikzpicture}
\begin{customlegend}[legend columns=-1,
legend style={
draw=none,
column sep=1ex,
},legend entries={\begin{footnotesize}FDS\end{footnotesize}}]
\csname pgfplots@addlegendimage\endcsname{green,mark=square,sharp plot}
\end{customlegend}
\end{tikzpicture} &
\begin{tikzpicture}
\begin{customlegend}[legend columns=-1,
legend style={
draw=none,
column sep=1ex,
},legend entries={\begin{footnotesize}Hybrid\end{footnotesize}}]
\csname pgfplots@addlegendimage\endcsname{red,mark=triangle,sharp plot}
\end{customlegend}
\end{tikzpicture}
\end{tabular}
\caption{Experimental results: \textit{(Row 1:)} Accuracy of different methods across the considered datasets; \textit{(Row 2:)} Time taken in seconds to converge; and \textit{(Row 3:)} Number of iterations to converge. X-axis denotes the varying number of annotators studied for each dataset.}
\end{figure*}
\paragraph{Results:} The results of our experiments are presented in Figure 1 and Table \ref{logltable}. Table \ref{datasettable} shows the speedup in time and number of iterations needed to converge of FDS over DS and IWMV and of Hybrid over DS, averaged over all observations with varying number of annotators.
\begin{table}[h]
\begin{center}
\begin{scriptsize}
\begin{tabular}{ |c||c|c|c|}
\hline
& FDS & DS & Hybrid \\
\hhline{|=||=|=|=|}
Adult2 & 1283.75 & 1153.09 & 1154.97 \\
\hline
BM & 2110.16 & 2094.76 & 2100.32 \\
\hline
TREC2010 & 13109.26 & 12180.84 & 12346.91 \\
\hline
DAiSEE & 39968.08 & 36178.16 & 36350.61 \\
\hline
LabelMe & 1714.50 & 1655.94 & 1660.06 \\
\hline
RTE & 3741.61 & 3679.63 & 3680.32 \\
\hline
SP & 12472.00 & 12433.70 & 12440.70 \\
\hline
\end{tabular}
\end{scriptsize}
\captionof{table}{Negative Log Likelihood at convergence of FDS, DS and Hybrid methods}\label{logltable}
\end{center}
\end{table}
\paragraph{Performance Analysis of Fast Dawid-Skene:} The results show that FDS gives similar accuracies when compared to DS, Hybrid, GLAD, and IWMV, and a significant improvement over MV, on most datasets except for the BM and LabelMe datasets. In LabelMe, the aggregation accuracy is not at par with DS or Hybrid but is still significantly higher than MV and comparable to IWMV. In the BM dataset, the accuracies of FDS and IWMV are slightly lower than MV but both are comparable to each other. In terms of time taken, we notice that apart from the LabelMe dataset, FDS performs much better than DS, Hybrid, IWMV and GLAD all through. In the case of LabelMe, IWMV outperforms in terms of speed but the margin is very small (around 0.1 sec). This leads us to infer that in general, FDS gives comparable accuracies to other methods while taking significantly lesser time.
\paragraph{Performance Analysis of the Hybrid Method:} The goal of the Hybrid algorithm is to converge to a similar likelihood as DS in much lesser time. From the experiments (especially Table 2), we see that this is indeed the case - the log likelihood of the Hybrid algorithm is close to that of DS and consistently better than FDS. This naturally leads to accuracies almost similar to those obtained by DS, as is confirmed in the results. The total time taken for convergence is much lower for Hybrid as compared to DS. Moreover, the time taken for convergence by Hybrid is consistently low and does not deviate as much as IWMV. While IWMV outperforms Hybrid with respect to time in a few datasets, the proposed Hybrid outperforms IWMV on accuracy on those datasets. These observations support Hybrid to be an algorithm which performs with accuracies similar to DS in a much lesser time consistently over datasets.
\paragraph{Implementation Details:}
We discuss two important implementation details of the proposed methods in this section: \textit{initialization} and \textit{stopping conditions}.
As argued in \cite{dawid1979maximum}, a symmetric initialization of the parameters (all $P(Y_q = c)$s to be $1 / C$) corresponds to a start from a saddle point, from where the EM algorithm faces difficulty in converging. Instead, a good initialization is to start with the majority voting estimate. While performing majority voting, it could often happen that there is a tie between two or more options with the highest number of votes. In such situations, we randomly choose an option among those which received the highest votes\footnote{We also tried a variant, in which the option with the highest running class marginal was used to break ties.
But this variant did not perform as well as the randomized majority voting across all methods. We also ran many trials with different random seeds, and found the results to almost the same as those presented.}. We maintained the same random seed for all methods which required this decision.
The ideal convergence criterion would be when the answer sheet proposed by an algorithm stops changing. This condition is met within a few iterations for FDS and Hybrid, but DS does not converge using this criterion in a reasonable number of steps. For example, in case of the \textit{DAiSEE} dataset, DS did not converge even after 100 iterations (as compared to $\le 10$ for FDS). To address this issue,
we set the convergence criterion as the point when the difference in class marginals is less than $10^{-4}$.
We do not include the changes in participant error rates in the final convergence criterion because we observed that its fluctuations could lead to stopping prematurely. Similarly, the criterion for switching from DS to FDS in the Hybrid algorithm is the point when the change in class marginals is less than 0.005 (which happened approximately between 45-75\% of total iterations across the datasets).
\vspace{-5pt}
\section{Online Vote Aggregation}
Online aggregation of crowdsourced responses is an important setting in today's applications, where data points may be streaming in large data applications.
We consider a setting in which we have access to an initial set of questions and have obtained the proposed answer key using FDS. We also have $P(Y = c)$ and $P(c_a| Y = a) \,\forall\, c, a$ at this time. When we receive a new question and the answers from multiple participants for this new question, we first estimate the answer for this question directly using majority voting. We then update the parameters using the M-step in Algorithm \ref{fdsalgorithm}. After the M-step, we run the E-step only for this question to re-obtain the aggregated choice. To update the new knowledge which we have regarding the new participants, we run the M-step for one last time. We conducted experiments on the \textit{SP} dataset\footnote{More results, including on other datasets, on \url{https://sites.google.com/view/fast-dawid-skene/}},
and observed almost the same accuracy for online FDS as offline FDS (Table 4) for different number of annotators. Table 3 shows the results for the max number of annotators (= 5).
\begin{center}
\label{onlinetable}
\begin{scriptsize}
\begin{tabular}{ |c||c|c|c|}
\hline
& DS & FDS & Hybrid\\
\hhline{|=||=|=|=|}
Accuracy & 90.94\% & 90.60\% & 90.64\%\\
\hline
Time taken to converge (s) & 4.40 & 3.76 & 4.09 \\
\hline
\# Iterations to converge & 26 & 4 & 5\\
\hline
\end{tabular}
\captionof{table}{Online Vote Aggregation on \textit{SP} dataset.
}
\end{scriptsize}
\end{center}
\vspace{-10pt}
\begin{center}
\label{onvsofftable}
\begin{scriptsize}
\begin{tabular}{ |c||c|c|c|c|}
\hline
Accuracy & 2 & 3 & 4 & 5\\
\hhline{|=||=|=|=|=|}
FDS & 85.59\% & 88.41\% & 90.02\% & 90.74\% \\
\hline
Online FDS & 83.57\% & 88.06\% & 89.90\% & 90.60\%\\
\hline
\end{tabular}
\captionof{table}{Online FDS vs FDS for varying number of annotators.}
\end{scriptsize}
\end{center}
\section{Extension to Multiple Correct Options}
\label{discussions}
The proposed FDS method can be extended to solve the aggregation problem under different settings. We describe an extension below, using the same notations as in Section \ref{subsec_preliminaries}.
In real-world machine learning settings such as multi-label learning, a data point might belong to multiple classes, which would result in more than one true choice per question. For such cases, we now assume that participants are allowed to choose more than one choice for each question. Our Algorithm \ref{fdsalgorithm} originally assumes that every question has exactly one correct choice. To overcome this limitation, we can make a simple modification in how we interpret questions when multiple options are correct. We assume that every (question, option) pair is a separate binary classification problem, where the label is true if the option is chosen for that question, and false otherwise. This transforms a task with $Q$ questions and $C$ options each to a task with $QC$ questions and two options each. This is valid because the correctness of an option is independent of the correctness of all other options for that question in this setting. We ran experiments using this model on the Affect Annotation Love dataset \textit{(AffectAnnotation)} used in \cite{DUAN20145723} (which was specifically developed for this setting) on FDS, and compared our performance with DS and Hybrid. Our results are summarized in Table 5 (annotators=5, averaged over five subsets), showing the significantly improved results of FDS over DS. Hybrid attempts to follow DS in the likelihood estimation, and thus does not perform as well as FDS in this case. Besides, our results for FDS also performed better than the methods proposed in \cite{DUAN20145723}, which showed a best accuracy of $\approx92\%$ on this dataset.
\begin{center}
\label{multtable}
\begin{scriptsize}
\begin{tabular}{ |c||c|c|c|}
\hline
& DS & FDS & Hybrid \\
\hhline{|=||=|=|=|}
Accuracy & 88.66\% & 94.14\% & 89.26\% \\
\hline
Time taken to converge (s) & 0.44 & 0.057 & 0.14 \\
\hline
\# Iterations to converge & 29.6 & 2 & 5.8 \\
\hline
\end{tabular}
\captionof{table}{Multiple Correct Options setting on \textit{AffectAnnotation} data.
\end{scriptsize}
\end{center}
\vspace{-20pt}
\section{Conclusion}
\label{conclusion}
In this paper we introduced a new EM-based method for vote aggregation in crowdsourced data settings. Our method, Fast Dawid-Skene (FDS), turns out to be a `hard' version of the popular Dawid-Skene (DS) algorithm, and shows up to 7.84x speedup over DS and up to 6.09x speedup over IWMV in time taken for convergence. We also propose a hybrid variant that can switch between DS and FDS to provide the best in terms of accuracy and speed. We compared the performance of the proposed methods against other state-of-the-art EM algorithms including DS, IWMV and GLAD, and our results showed that FDS and the Hybrid approach indeed provide very fast convergence at comparable accuracies to DS, IWMV and GLAD. We proved that our algorithm converges to the estimated labels at a linear rate. We also showed how the proposed methods can be used for online vote aggregation, and extended to the setting where there are multiple correct answers, showing the generalizability of the methods.
\bibliographystyle{ACM-Reference-Format}
|
{
"timestamp": "2018-09-11T02:01:56",
"yymm": "1803",
"arxiv_id": "1803.02781",
"language": "en",
"url": "https://arxiv.org/abs/1803.02781"
}
|
\section{Introduction}
\label{sec:intro}
The cosmic-ray positron excess has been discovered for nearly a decade
\citep{2009Natur.458..607A,2012PhRvL.108a1103A,2013PhRvL.110n1102A},
but its origin is still a mystery. These extra positrons may originate
either from astrophysical sources like pulsars or the dark matter
annihilation/decay. Among various kinds of astrophysical sources, the
Geminga pulsar (PSR J0633+1746) has been widely believed to be a very
promising candidate to produce the positron excess
\citep{2009PhRvL.103e1101Y,2009JCAP...01..025H,2013PhRvD..88b3001Y,
2017PhRvD..96j3013H}. Geminga is one of the nearest pulsars with a
distance of $250^{+120}_{-62}$ pc \citep{2007Ap&SS.308..225F}.
Its age is estimated to be about $3.42\times10^5$ years, and the
derived spin-down energy is $1.23\times10^{49}$ erg
\citep{2005AJ....129.1993M,2018ApJ...854...57F}. All these parameters
suggest that Geminga can probably dominate the high energy positron
flux observed on the Earth. The extended TeV $\gamma$-ray halo around
Geminga pulsar observed by Milagro \citep{2007ApJ...664L..91A} and HAWC
\citep{2017ApJ...843...40A} gives straightforward evidence supporting
that Geminga can indeed generate very high energy electrons and
positrons ($e^\pm$).
However, the detailed morphological study of the very high-energy
$\gamma$-ray emission from Geminga and PSR B0656+14 (Monogem) by
HAWC suggests that the $e^\pm$ produced by these pulsars diffuse out
significantly slower than that in the average interstellar medium (ISM)
as inferred from the Boron-to-Carbon ratio (B/C) measurements
\citep{2017Sci...358..911A}. In such a case, the $e^\pm$ produced by
Geminga or Monogem can hardly reach the Earth, and thus these two
pulsars may be unlikely to account for the positron excess.
On the other hand, \citet{2017arXiv171107482H} pointed out that this
slow-diffusion scenario should not be representative even in the local
environment. Since H.E.S.S. has detected high-energy $e^\pm$ up to
$\sim$ 20 TeV\footnote{
https://indico.snu.ac.kr/indico/event/15/session/5/contribution/694},
these $e^\pm$ can only travel for $10\sim20$ pc within the cooling time
given such a slow-diffusion condition. We can hardly find any high-energy
$e^\pm$ sources within such a small distance around the solar system.
Since the HAWC data can only probe a region of $\sim30$ pc around
Geminga and Monogem, it is very likely that the slow-diffusion region
is actually limited in a small region around the sources, beyond
which particles diffuse faster as typical Galactic cosmic rays.
This scenario would be consistent with the B/C data and the H.E.S.S
$e^\pm$ spectrum \citep{2017PhRvD..96j3013H,2017arXiv171107482H}.
In this work, we investigate whether this two-zone diffusion model can
explain both the HAWC $\gamma$-ray data of Geminga and the positron
excess. The two-zone diffusion model is solved with a numerical method.
Because the diffusion coefficient has a jump at the boundary between
the two zones, the differencing scheme should be carefully dealt with.
In Section \ref{sec:method}, we introduce the two-zone diffusion model and
the numerical treatment to the propagation equation. In Section \ref{sec:rst},
we calculate the positron spectrum of Geminga in the two-zone scenario, and
compare the result with the AMS-02 data \citep{2014PhRvL.113l1102A}. Then we
conclude our work with some discussion in the last section.
\section{Method}
\label{sec:method}
The propagation process of $e^\pm$ can be described by the diffusion-cooling
equation
\begin{equation}
\frac{\partial N}{\partial t} - \nabla(D\nabla N) - \frac{\partial}{\partial
E}(bN) = Q \,,
\label{eq:prop}
\end{equation}
where $N$ is the differential number density of $e^\pm$, $D$ denotes the
diffusion coefficient, $b$ is the energy-loss rate, and $Q$ is the source
term. In the present work, we are interested in the energy range higher
than 10 GeV, so the convection and reacceleration terms which affect the
low energy spectrum are neglected \citep{dela09}. The energy-loss rate has the
form of $b(E)=b_0(E)E^2$, which describes the synchrotron and inverse Compton
radiation cooling of $e^\pm$. The interstellar magnetic field in the Galaxy is
set to be 3 $\mu$G to get the synchrotron term \citep{1996ApJ...458..194M}.
For the inverse Compton scattering term, it is necessary to consider a
relativistic correction to the scattering cross section. We follow the
calculation of \citet{schli10}, where $b_0$ is energy-dependent.
The diffusion coefficient is usually assumed to be $D(E)=\beta^\eta
D_0{(R/\rm 1\,GV)}^{\delta}$, where $D_0$ and $\delta$ are both constants,
$\beta$ is the velocity of particles in unit of light speed, $\eta$ is
a low energy correction parameter of the velocity dependence, and $R$ is
the rigidity. Assuming $D$ is spatially constant, \citet{2017PhRvD..95h3007Y}
constrained the propagation parameters with the B/C data of AMS-02
\citep{2016PhRvL.117w1102A}, and found that the best-fit model is the
diffusion model with reacceleration. The obtained propagation parameters
are $D_0=(2.08\pm0.28)\times10^{28}$ cm$^2$ s$^{-1}$ and
$\delta=0.500\pm0.012$ (hereafter Y17 model). The thickness of the propagation
halo is $z_h=5.02\pm0.86$ kpc, and the low energy correction parameter $\eta$
is not relevant for this study.
For the $e^\pm$ produced by Geminga, the propagation distance can be
estimated as $2\sqrt{D(E)t_E}$, where $t_E={\rm min}\{t_{\rm g},1/(b_0E)\}$,
i.e., the smaller one of the age of Geminga $t_{\rm g}$ and the cooling
time \citep{2017Sci...358..911A}. If we adopt the propagation parameters
of Y17, $e^\pm$ of $\simeq 1$ TeV energies can diffuse to a distance of
$\simeq 1.7$ kpc. This scale is much smaller than $z_h$ obtained in Y17.
If the particles diffuse slower in the region around Geminga, the
total diffusion distance is even less. Therefore, it should be fine
to assume a spherically symmetrical geometry of the propagation of
$e^\pm$ from Geminga. The diffusion coefficient for the two-zone model
is then
\begin{equation}
D(E, r)=\left\{
\begin{aligned}
D_1(E), & & r< r_\star \\
D_2(E), & & r\geq r_\star\\
\end{aligned}
\right.\,,
\label{eq:diff}
\end{equation}
where $r$ is the distance from Geminga, $r_\star$ is the discontinuity
shell of the diffusion coefficient, $D_1$ is the diffusion coefficient
around Geminga which is inferred by the HAWC $\gamma$-ray data
\citep{2017Sci...358..911A}, and $D_2$ is the average diffusion
coefficient in the Milky Way given by Y17. In a more realistic picture the
diffusion coefficient may change gradually. However, no observations can
constrain the transitional zone. Further as $r_\star$ is a free parameter we
can always get a equivalent result for the local $e^\pm$ flux by adjusting
$r_\star$ in this simplified picture.
The spatially-dependent diffusion equation is difficult to be solved
analytically, and we adopt a numerical method instead in this work.
We assume that Geminga is a burst-like point source, then Equation
(\ref{eq:prop}) can be rewritten as following, along with the initial condition
and the boundary conditions:
\begin{equation}
\left\{
\begin{aligned}
& \frac{\partial N}{\partial t} = \mathcal{L}N\,, \\
& N(0, E, r)=Q(E)\delta(r)\,, \\
& N(t, E_{\rm max}, r)=0\,, \\
& \left.\frac{\partial N}{\partial r}\right|_{r=0}=0\,, \\
& N(t, E, r_{\rm max})=0\,, \\
\end{aligned}
\right.
\label{eq:prop_set}
\end{equation}
where $E_{\rm max}=500$ TeV, and $r_{\rm max}=4$ kpc. The operator
$\mathcal{L}$ is the sum of the diffusion operator $\mathcal{L}_r$ and the
energy-loss operator $\mathcal{L}_E$, which are
\begin{equation}
\begin{aligned}
& \mathcal{L}_r=\frac{1}{r^2}\frac{\partial}{\partial
r}\left[r^2D(r)\frac{\partial}{\partial r}\right]\,, \\
& \mathcal{L}_E= b\frac{\partial}{\partial E}+\frac{\partial b}
{\partial E}\,. \\
\end{aligned}
\label{eq:operator}
\end{equation}
We apply the {\it operator splitting method} to deal with the two operators
separately. For the discretization of the energy-loss scheme, we follow the
method given by \citet{2014APh....55...37K}, while for the diffusion
operator, we use two different discretizations depending on if $r_i$
lies on the discontinuity surface ($i$ is the spatial step index). If not, the
well-known Crank-Nicolson scheme for constant $D$ is a good choice. For the
case of $r_i=r_\star$, the difference equation should be re-written to ensure
the conservation of the flux at both sides of the discontinuity surface. We
adopt the {\it finite volume method} to derive the differencing scheme for
this case. All the details of the discretization are presented in Appendix
\ref{app}. We should point out the differencing scheme derived by the {\it
finite volume method} (Equation [\ref{eq:Lr2}]) is different from those used
in GALPROP \citep{galp} or DRAGON \citep{2008JCAP...10..018E}; the numerical
schemes of GALPROP and DRAGON are basically applicable for continuously and
slowly changed diffusion coefficient.
Then we describe the parameter settings. In the difference equation, the time
step $\Delta t$ is set to be 1000 years, which is much smaller than the age of
Geminga. The spatial step $\Delta r$ is 1 pc, then the initial condition given
in Equation (\ref{eq:ini}) describes a spherical source with a radius of 1 pc.
For an old source like Geminga, the spectrum at the Earth can be very close to
that of a point source, especially when $E\gtrsim100$ GeV
\citep{2012MNRAS.419..624T}. We use a logarithmic scale with a ratio of 1.2 for
the energy grids, that is, $E_{l+1}/E_l=1.2$, where $l$ is the energy step
index. The $e^\pm$ injection spectrum takes the form of $Q(E)=Q_0E^{-\gamma}$,
where $\gamma\simeq2.2$, as indicated by the HAWC observation
\citep{2017Sci...358..911A}.
\section{Result}\label{sec:rst}
If $D$ is spatially uniform, Equation (\ref{eq:prop}) can be solved
analytically using the Green's function method \citep{1964ocr..book.....G}.
We verify that our numerical solution for the one-zone case matches well
with the analytical solution. For the two-zone diffusion case, there is no
analytical way to test the results. To check that there is no ``swallowing'' or
``spitting'' of particles in the discontinuity surface, we integrate the number
of particles, $N(r_i)$, for the two-zone diffusion cases with different
$r_\star$. We find that the number of particles is always the same as that of
the one-zone cases. The present radial distributions of 1 TeV $e^\pm$
are shown in Figure \ref{fig:test} for the two-zone diffusion scenarios and
also the one-zone cases. Here we assume that all the spin-down energy of
Geminga pulsar is converted to the energy of injected $e^\pm$ (conversion
efficiency), to determine the normalization.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.55\textwidth]{test2.eps}
\caption{Radial distributions of 1 TeV $e^\pm$ from Geminga at the present
age. Three different $r_\star$ are adopted for the two-zone diffusion models.
The distributions of the one-zone diffusion cases are also shown for
comparison, the black line is for the one-zone diffusion model with Y17
diffusion coefficient, and the red line is calculated with parameters inferred
by HAWC data \cite{2017Sci...358..911A}.}
\label{fig:test}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.48\textwidth]{spectra.eps}
\includegraphics[width=0.48\textwidth]{positron.eps}
\caption{Left: expected electron (or positron) spectra of Geminga at the
Earth. The conversion efficiency of Geminga pulsar is set to be 100\%.
Three different values of $r_\star$ are adopted for the two-zone diffusion
scenario. The spectra of the one-zone diffusion cases with $D_1$ (HAWC) and
$D_2$ (Y17) are also presented. Note the former is 300 times of its original
spectrum. Right: the model predicted positron spectrum compared with the AMS-02
data \citep{2014PhRvL.113l1102A}. The green dashed line is the contribution
of Geminga, with $r_\star=50$ pc and a conversion efficiency of 75\%.}
\label{fig:spec}
\end{figure}
The positron (or electron) spectra at the Earth generated by Geminga are
shown in the left panel of Figure \ref{fig:spec} for different $r_\star$.
The conversion efficiency of Geminga pulsar is also set to be 100\%.
The spectrum becomes harder for a larger $r_\star$, since fewer
low energy $e^\pm$ can reach the Earth when the slow-diffusion zone is
larger. The extreme cases of one-zone diffusion with Y17 or HAWC diffusion
coefficients are also shown for comparison. We find that at high energies,
the fluxes of the two-zone diffusion models are considerably higher than
that of the one-zone diffusion case with Y17 parameters. For a fast-diffusion
model like Y17, the propagation scale of 1 TeV $e^\pm$ of Geminga is
$\simeq1.7$ kpc. This scale is
about 7 times larger than the distance between Geminga and the Earth,
which implies that the particles have diffused to a considerably large
region. For the two-zone diffusion scenario, the propagation of particles
is hindered within the slow-diffusion zone. After being confined for
some time in the inner region, these particles enter the fast propagation
region and diffuse to the Earth. Since they spend less time in the
fast-diffusion region, they can be effectively accumulated in a relatively
smaller region and result in a higher flux at the Earth's location.
We find that the two-zone model with $r_\star=50$ pc can well reproduce
the AMS-02 data of the positron flux. The results for $r_\star>50$ pc
are too hard to explain the data, while the spectrum is too soft for the case
with a smaller $r_\star$ or the one-zone fast-diffusion model. Note that the
final spectra depend on the injection spectrum of $e^\pm$ assumed, which is
$\gamma \simeq 2.2$ in this work. The comparison between the model and the data
is shown in the right panel of Figure \ref{fig:spec}. Here the conversion
efficiency to $e^\pm$ is assumed to be 75\% in order to match the data. For the
secondary contribution from $pp$ collisions, one can refer to \citet{dela09}
and \citet{2018ApJ...854...57F}.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.55\textwidth]{spectra_m.eps}
\caption{Positron spectra of Monogem with different diffusion models,
compared with the AMS-02 data. The conversion efficiency is 100\% for all the
cases.}
\label{fig:spec_m}
\end{figure}
The $e^\pm$ spectrum of Monogem under the two-zone diffusion can be obtained in
the same way. The distance, age, and spin-down energy of Monogem is 290 pc,
$1.1\times10^5$ years, and $1.58\times10^{48}$ erg, respectively
\citep{2005AJ....129.1993M}. The observation of HAWC indicates a spectral index
of $\sim$ 2.0 for the injection spectrum of Monogem
\citep{2017Sci...358..911A}. As can be seen from Figure \ref{fig:spec_m}, the
positron spectra of Monogem is too hard to fit the AMS-02 data in the two-zone
diffusion cases, due to its farther distance, younger age, and harder injection
spectrum compared with Geminga. Thus Monogem cannot make a major contribution
to the local positron fluxes even in the two-zone diffusion model.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.55\textwidth]{test3.eps}
\caption{Density profiles of 100 TeV $e^\pm$ with diffusion of 7000
years. The red line describes the slow-diffusion case proposed by
\cite{2017Sci...358..911A}, while the green line represents the two-zone
diffusion case with $r_\star=50$ pc.}
\label{fig:comp}
\end{figure}
Finally we would like to give a simple self-consistency check if the
two-zone region can reproduce the $\gamma$-ray profile observed by HAWC.
The HAWC $\gamma$-ray data around 20 TeV are produced by $\sim100$ TeV
electrons, whose cooling time is about 7000 years. This indicates that the
parent $e^\pm$ of the $\gamma$-ray observed by HAWC should be younger than 7000
years. We show in Figure \ref{fig:comp} the density profiles of 100 TeV
$e^\pm$ from Geminga with diffusion of 7000 years. Within 30 pc from Geminga,
we find very good consistency between the one-zone slow-diffusion scenario of
\cite{2017Sci...358..911A} and the two-zone diffusion scenario with
$r_{\star}=50$ pc, which means the later proposed in this work can also
accommodate the HAWC $\gamma$-ray data. The reason for this consistency is that
7000 years is too short for $e^\pm$ to escape from the slow-diffusion region
of the two-zone model.
\section{Conclusion and Discussion}
\label{sec:discuss}
In this work we propose that a two-zone propagation scenario of cosmic
ray $e^\pm$ to account for the HAWC observations of extended halo
around Geminga pulsar and the locally observed positron flux by AMS-02.
The diffusion of $e^\pm$ is assumed to be significantly slow within a
distance of $r_\star$ from the pulsar, as inferred by the spatial
brightness profile of $\gamma$-ray emission observed by HAWC
\citep{2017Sci...358..911A}. The particles diffuse beyond
$r_\star$ with a diffusion coefficient inferred from the B/C data.
A numerical method is adopted to solve the propagation equation, and the
differencing scheme of the diffusion operator is derived by the {\it finite
volume method}, which is different from those used in GALPROP or DRAGON.
The exact value of $r_\star$ to fit the AMS-02 positron flux degenerates
with the injection spectrum of $e^\pm$, and hence can neither be too big
nor too small. For larger (smaller) value of $r_\star$, low energy
particles would be confined in the slow-diffusion region longer
(shorter), and the final positron spectrum is harder (softer).
For an injection spectral index of $\sim 2.2$ and $r_\star \sim 50$ pc,
we find that the $e^\pm$ from Geminga can reasonably account for the
positron excess, with about 75\% of the spin-down energy converted
into $e^\pm$. Actually such a result is even more natural than that
in the one-zone diffusion scenario which requires an efficiency even
slightly larger than 100\% \citep{2013PhRvD..88b3001Y}.
The origin of the slow-diffusion zone around Geminga is still unclear. It is
possible that the zone is pre-existed. For example, the shock of the parent
supernova remnant (SNR) of Geminga, which is non-observable today, may have
swept the ISM and made it more turbulent. Considering the age of $3\times10^5$
years, the shocked region can reach a size of $\sim$100 pc \citep{yamazaki06}.
We note that Geminga has a transverse velocity of $205^{+90}_{-47}$ km s$^{-1}$
\citep{2007Ap&SS.308..225F}, which means a 70 pc offset from its birth place. So
Geminga may still be within the shocked region of its SNR, where the diffusion
coefficient is smaller. Meanwhile, if most of $e^\pm$ are injected in
the early age of Geminga, when the offset of Geminga is small and the scale of
its SNR is not so large as today, our assumption of the symmetrical
slow-diffusion zone may not be impacted.
Alternatively, the slow diffusion zone can be generated intrinsically by the
$e^\pm$ injected from Geminga. Near the cosmic-ray sources, the spatial gradient
of particle density is significantly larger than that of the average ISM, which
can lead to the growth of the streaming instability
\citep{2008AdSpR..42..486P,malkov13,blasi16}. In this case, the
magnetohydrodynamic turbulence may be considerably stronger than that in the
ISM. Therefore, it is plausible that the $e^\pm$ leaving Geminga are confined in
the nearby zone for a longer time by the waves induced by themselves.
The slow-diffusion regions around pulsars may be common, which may result
in extended $\gamma$-ray halos of pulsars and can probably explain the
diffuse $\gamma$-ray excess observed in the Galactic plane
\citep{2008ApJ...688.1078A,2017arXiv170701905L,2016ChPhC..40k5001G}.
\acknowledgments{This work is supported by the National Key Program for Research
and Development (No.~2016YFA0400200) and by the National Natural Science
Foundation of China under Grants No.~U1738209,~11475189,~11475191,~11722328. Q.
Yuan acknowledges the 100 Talents Program of Chinese Academy of Science.}
|
{
"timestamp": "2018-07-26T02:07:43",
"yymm": "1803",
"arxiv_id": "1803.02640",
"language": "en",
"url": "https://arxiv.org/abs/1803.02640"
}
|
\section{\label{sec1}Introduction}
Motivated by the black hole (BH) information paradox and cosmological constant problems, it has been suggested that non-perturbative quantum gravitational effects may lead to Planck-scale modifications of BH horizons. Proposals to solve the BH information paradox include gravastars \cite{Mazur:2004fk}, fuzzballs \cite{Lunin:2001jy, Lunin:2002qf, Mathur:2005zp, Mathur:2008nj, Mathur:2012jk}, and firewalls \cite{Braunstein:2009my, Almheiri:2012rt}, amongst others \cite{Barcelo:2015noa, Kawai:2017txu}.
These Exotic Compact Objects (ECOs) all modify the standard structure of BH horizons, and should form by Page time $\sim M^3$, but can emerge as early as the ``scrambling time'' $\sim M \log M$ \cite{Hayden:2007cs, Sekino:2008he}.
Gravitational aether theory \cite{Afshordi:2008xu,PrescodWeinstein:2009mp} which modifies the Einstein field equations by adding an incompressible fluid (aether) is a possible solution to the cosmological constant problem(s). BH solutions in this theory link the BH mass with the aether pressure at infinity, and yield a comparable pressure to the observed dark energy pressure for stellar BH masses of 10-100 $M_{\odot}$. The solution of the modified Einstein field equation deviates from the GR within the order of Planck length proper distance outside the (would-be) horizon. It is also suggested that replacing the horizon with a ``wall'' could be a source of high energy astrophysical neutrino flux \cite{Afshordi:2015foa} which is a possible source for the PeV neutrinos recently detected by IceCube observatory.
A concrete physical model for replacing event horizon due to quantum gravitational effects is provided in \cite{Saravani:2012is}. The spacetime ends at about the order of Planck length proper distance outside the (would-be) horizon with a wall containing a surface fluid. It is then shown that Israel junction conditions imply that the fluid has the thermodynamic entropy matching the Bekenstein-Hawking area law, for charged rotating BHs (Also see \cite{Holdom:2016nek} for a similar horizonless spacetime solution).
Recent detections of gravitational waves from binary BH mergers by the LIGO-Virgo collaboration \cite{TheLIGOScientific:2016agk, TheLIGOScientific:2016pea, Giddings:2016tla, Abbott:2016blz, Abbott:2016nmj, Abbott:2017vtc, Abbott:2017oio, TheLIGOScientific:2017qsa, Abbott:2017gyy} provide a way to test the structure around the horizon scale.
Shortly after LIGO's first detection, GW150914, \cite{Cardoso:2016rao, Cardoso:2016oxy} argued that introducing a wall to replace horizon might yield a similar ringdown waveform as GR BHs, but produce delayed echoes (see \cite{Cardoso:2017njb, Cardoso:2017cqb} for a review) in the gravitational wave signal. Using a phenomenological template by truncating the GR merger waveforms, \cite{abedi2016echoes} carried out the first search for echoes and claimed a 2.5$\sigma$ tentative evidence for them in the the first three (candidate) events in the LIGO public data (but see \cite{Ashton:2016xff,Westerweck:2017hus} and \cite{Abedi:2017isz} for a critique/rebuttal).
An independent search \cite{Conklin:2017lwb}, using a different methodology, has recently found evidence for echoes in each of LIGO's merger events (with the notable exception of GW150914) at $\sim 3\sigma$ significance level. However, we should note that the echoes reported in \cite{abedi2016echoes} and \cite{Conklin:2017lwb} are for different events, even though they are both broadly consistent with the hypothesis of near-horizon Planck-scale structure. In particular, \cite{Abedi:2017isz,Westerweck:2017hus} fail to find echoes in GW151226, which has the most significant evidence for echoes in \cite{Conklin:2017lwb}, suggesting that the two methods capture different parts of the echo waveform.
Most recently, \cite{BNS} claim a tentative detection of (lower harmonics of) echoes, at $4.2\sigma$ level, from a ``black hole'' remnant in the aftermath GW170817 binary neutron star merger.
While one may consider other phenomenological echo templates (e.g., \cite{Maselli:2017tfq}), more realistic templates for fitting data may be found by solving (linearized) Einstein equations with modified boundary conditions near the horizon. Along this direction, most studies have so far focused on Schwarzschild BHs (e.g., \cite{Cardoso:2016rao,Cardoso:2016oxy, Price:2017cjr, Mark:2017dnq, Volkel:2018hwb, Volkel:2017kfj}). In this paper, we extend this to Kerr metric as realistic BHs have spin. \cite{Nakano:2017fvh} also presented echo templates by modelling the reflectivity of the angular momentum barrier in the Kerr spacetime. We, however, model the propagation in the full spacetime which provides a more realistic treatment at lower frequencies.
Another related work is \cite{Bueno:2017hyj} which studies the echoes of scalar gaussian wavepackets in Kerr-like wormholes. In contrast, we study generic propagation in Kerr spacetime, with arbitrary boundary conditions, which can be applied not only to scalar fields (s=0), but also massless Dirac (s = $\pm$1/2), electromagnetic (s = $\pm$1), or gravitational (s = $\pm$2) fields. Interestingly (but not surprisingly), we come to some similar conclusions, e.g., {\it i)} Spinning ECOs give rise to unstable modes which, however, do not affect the echoes till very late times (depending on whether the initial frequency range is within the superradiance regime). {\it ii)} It is hard to make a model-independent prediction for the first echo.
A related phenomenological issue that arises when we replace the horizon with a wall is the emergence of superradiant instability for horizonless ergoregions \cite{1978CMaPh..63..243F, Cardoso:2007az,Cunha:2017qtt}. While this might suggest long-term instability of spinning ECOs, which may be in conflict with astrophysical spin measurements for BHs \cite{Narayan:2013gca}, it was suggested by \cite{Maggio:2017ivp} that an absorption rate of the wall as small as 0.4\% is sufficient to quench the instability completely
We organize this paper as follows: Sec.\ref{sec2} provides the linear Einstein equations and boundary conditions used. Instead of normal boundary condition with no outgoing wave on the horizon, we put a wall standing just outside the would-be horizon. The reflection rate of the wall depends on the specific model of quantum BHs. Sec.\ref{sec3} presents echo solutions for different positions of a perfect wall and time-delays of a geometric formula given in \cite{abedi2016echoes}, while Sec. \ref{seca1} discusses how superradiance of Kerr geometry is manifested in echo templates. In Sec. \ref{sec5}, we provide an analytic fit to the echo templates, based on solutions in Sec. \ref{sec3}. We explore a soft wall with frequency-dependent reflection, as well as nonlinear corrections to initial conditions in Sec.\ref{sec4} for a more realistic picture. In Appendix \ref{a1}, we briefly discusses ergoregion instability developed in the presence of a perfect wall. While in principle the instability is significant at high spins, we show that these instabilities do not affect the first several echoes of typical binary merger events. Finally, Sec.\ref{sec7} concludes our work.
If not specified, we use units with $G=\hbar = k_B = c=1$. For concreteness, we use the best fit properties and waveforms resulting from the GW150914 merger event, provided by the LIGO-Virgo collaboration \cite{TheLIGOScientific:2016agk, TheLIGOScientific:2016pea} \footnote{https://losc.ligo.org/events/GW150914/}. In particular, the detector frame mass and reduced spin parameter of the remnant used for the echo calculation are $M_{\rm fin} = 67.6~ M_{\odot}$ and $a = 0.67$. Echo templates for other final masses can be found by rescaling our analytic templates, as long as the dimensionless binary properties are not too different from those of GW150914.
\section{\label{sec2}Propagation and Boundary Conditions in Kerr spacetime}
\begin{table
\caption{\label{master}%
Corresponding field $\psi$ for different spin weight $s$ in Master equation. Here $\rho^{-1}=-(r-i a \cos\theta)$
}
\begin{ruledtabular}
\begin{tabular}{c|cccc}
\textrm{s}&
\textrm{0}&
\textrm{-1/2, 1/2}&
\textrm{-1, 1}&-2, 2\\
\colrule
$\psi$ & $\Phi$ & $\chi_0, \rho^{-1} \chi_1$ & $\phi_0, \rho^{-2} \phi_2$ & $\Psi_0, \rho^{-4} \Psi_4$\\
\end{tabular}
\end{ruledtabular}
\end{table}
We study the propagation of gravitational waves using linearized Einstein equations in Kerr geometry which describes the spacetime of a spinning BH. In order to model an exotic compact object (ECO), we simply replace the Kerr event horizon with a wall, where boundary conditions for linear perturbations are modified. The initial condition here is an incoming wavepacket $h_{\rm in}$ from infinity, and we calculate the outgoing wavepacket $h_{\rm out}$ by solving the linear Einstein equations.
As usual, we use the Newman-Penrose (NP) Formalism which greatly simplifies perturbation in Kerr metric, reducing to only a single master equation (known as the Teukolsky equation) which describes propagation of all scalar ($s=0$), massless Dirac ($s=\pm1/2$), electromagnetic ($s=\pm 1$) and gravitational ($s=\pm 2$) fields (see \citeauthor{teukolsky1973perturbations} \cite{teukolsky1973perturbations} for details):
\begin{widetext}
\begin{eqnarray}\label{eq:teuk}
\left[\frac{(r^2+a^2)^2}{\Delta}-a^2 \sin ^2 \theta\right] \frac{\partial^2 \psi}{\partial t^2}+ \frac{4Mar}{\Delta}\frac{\partial^2 \psi}{\partial t \partial \varphi}+\left(\frac{a^2}{\Delta}-\frac{1}{\sin^2 \theta}\right)\frac{\partial^2 \psi}{\partial \varphi^2}-\Delta^{-s} \frac{\partial}{\partial r} \left(\Delta^{s+1} \frac{\partial \psi}{\partial r}\right)-\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)\nonumber\\-2s\left[\frac{a(r-M)}{\Delta}+\frac{i \cos \theta}{\sin^2\theta}\right]\frac{\partial\psi}{\partial \varphi}-2s\left[\frac{M(r^2-a^2)}{\Delta}-r-ia\cos\theta\right]\frac{\partial \psi}{\partial t}+(s^2\cos^2\theta-s)\psi= 0, ~
\end{eqnarray}
\end{widetext}
where the field $\psi$ for each spin weight $s$ corresponds to NP quantities presented in Table \ref{master}.
The Teukolsky equation (\ref{eq:teuk}) is separable in coordinates in the frequency domain and can be decomposed into 4 ODEs. Furthermore, the symmetries in time and azimuth, allows for Fourier space decomposition in $t$ and $\varphi$:
\begin{widetext}
\begin{eqnarray}
&&\psi=\frac{1}{2\pi} \int d\omega e^{i(-\omega t +m \varphi)} S[\theta] R[r],\label{R}\\
&&\Delta^{-s} \frac{d}{dr} \left( \Delta^{s+1} \frac{dR}{dr} \right)+\left[ \frac{K^2-2is(r-M)K}{\Delta}+4is\omega r -\lambda \right]R=0,\label{r}\\
&&\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\frac{dS}{d\theta}\right)+\left(a^2 \omega^2 \cos^2\theta-\frac{m^2}{\sin^2\theta}-2a\omega s \cos\theta-\frac{2ms\cos\theta}{\sin^2\theta}-s^2 \cot^2\theta+s+A_{slm}\right)S=0,\label{s}
\end{eqnarray}
\end{widetext}
where $K=(r^2+a^2)\omega -am$ and $\lambda=A_{slm}+a^2 \omega ^2 -2am\omega$. The solution for the angular mode is spin-weighted spheroidal harmonic (full discussion can be found in \cite{Berti:2005gp}). We solve the radial mode numerically based on \citeauthor{Brito:2015oca} \cite{Brito:2015oca}, with publicly available Mathematica code, which was developed to study superradiance in Kerr metric \footnote{https://centra.tecnico.ulisboa.pt/network/grit/files/amplification-factors/}. Eq \ref{r} has the following asymptotic solutions
\begin{eqnarray}
&&R=\mathcal{T} \Delta^{-s} e^{-ik_{\rm h}r^*}+\mathcal{O}e^{ik_{\rm h}r^*}, r \rightarrow r_+ , \\
&&R=\mathcal{I} \frac{e^{-i \omega r^*}}{r}+\mathcal{R}\frac{e^{i\omega r^*}}{r^{2s+1}}, r \rightarrow \infty,
\end{eqnarray}
where $r^*$ is tortoise coordinate (defined as $r^*=\int \frac{r^2+a^2}{r^2-2Mr +a^2} dr$ that approaches -$\infty$ at horizon), $k_{\rm h}=\omega-\frac{a m}{2Mr_{+}}$ and $r_+ = M+\sqrt{M^2-a^2}$.
In classical General Relativity, everything that reaches the horizon will fall into the BH, and thus theres is no outgoing wave at $ r \rightarrow r_+ $, i.e. $\mathcal{O}=0$. However, for ECOs we assume that quantum gravity effects replace the horizon with (partially) reflective wall standing the order of Planck length proper distance outside the (would-be) horizon. We shall assume that this modifies the boundary condition, so that the wall reflects the incoming energy flux (see \cite{Nakano:2017fvh} for definition of energy near horizon) with a rate $R$ but does not change the phase:
\begin{widetext}
\begin{eqnarray}
&&|\mathcal{O}|^2=R_{\rm wall} \left | \frac{C}{D}\right| ^{s/2} |\mathcal{T}|^2, \qquad \arg[\mathcal{T} \Delta^{-s} e^{-ik_{\rm h}r^*}]=\arg[\mathcal{O}e^{ik_{\rm h}r^*}]\qquad \text{when } r\rightarrow r_{\rm wall},\\
&&C=B \left\{-36 a^2 \omega ^2+36 a m \omega +[\lambda +(s+1) s-2]^2\right\}+\{ 2 [\lambda +(s+1) s]-1\} \left(96 a^2 \omega ^2-48 a m \omega \right)+144 \omega ^2 \left(M^2-a^2\right),\nonumber\\
&& \\
&&B = [\lambda +s (s + 1)]^2 + 4 m a \omega- 4 a^2 \omega^2, \quad D= 256 k_{\rm h}^2 (2 M r_+)^8 [k_{\rm h}^2 + \frac{4(M^2-a^2)}{(4 M r_+)^2}]^2 [k_{\rm h}^2 +
\frac{16(M^2-a^2)}{(4 M r_+)^2}].
\end{eqnarray}
\end{widetext}
$R_{\rm wall}=1$ would correspond to a perfectly reflective wall, but the actual reflectivity and phase change depend on the specific quantum gravity model for ECOs. In the rest of the paper, we will present solutions to these equations with different choices of the reflectivity and discuss the important properties of solutions, such as echo templates, time-delays and superradiant instability.
\begin{figure}
\includegraphics[width=0.2\textwidth]{3_1_1}
\includegraphics[width=0.2\textwidth]{3_1_2}
\caption{\label{31} Black Holes (BHs) and Exotic Compact Objects (ECOs) with an ingoing wavepacket. For BHs, angular momentum barrier reflects low frequency modes but higher frequencies cross the barrier and fall through the horizon. For ECOs with a wall standing the order of Planck length proper distance outside the (would-be) horizon, modes with intermediate frequencies can be trapped between the wall and the angular momentum barrier, slowly leaking out as repeating echoes.}
\end{figure}
\begin{figure*}
\raggedright
\includegraphics[width=\textwidth]{3}
\caption{\label{32} Echoes with different wall positions. Changing the positions of wall doesn't influence the shape of echoes a lot, but when putting wall closer to the would-be horizon and away from angular momentum barrier, the time-delay becomes bigger.}
\end{figure*}
\section{\label{sec3}Making Echoes}
Realistic predictions for Echo waveforms requires nonlinear simulations of the mergers of binary ECOs in full general relativity. As a consistent covariant formulation for dynamics of ECOs is yet non-existent, we have to rely on approximate methods to produce realistic echo templates. In order to do this using linear theory, we instead custom-design an ingoing wavepacket $\hat{h}_{\rm in}$ at infinity, so that the outgoing waveform matches the LIGO best-fit template $\hat{h}_{\rm LIGO}$ (without a wall).
The higher frequencies will go across the barrier and fall into BH, as shown in Fig \ref{31} (left), while the lower frequencies are reflected. We thus assume
\begin{eqnarray}
\hat{h}_{\rm LIGO}(\omega)= R_{\rm BH}(\omega) \hat{h}_{\rm in}(\omega),
\end{eqnarray}
where $R_{\rm BH}(\omega)$ is the reflectivity of the Kerr angular momentum barrier.
For an ECO, however, we have one more barrier near the would-be horizon as shown in Fig \ref{31} (right). Wavepackets with intermediate frequencies can now be trapped between two barriers and leak slowly every time when they hit the angular momentum barrier. Therefore, ECOs would have a similar ringdown waveform as classical BHs, but they are followed by delayed slowly decaying echoes.
\begin{eqnarray}\label{eq:hout}
\hat{h}_{\rm out}(\omega)= R_{\rm ECO}(\omega) \hat{h}_{\rm in}(\omega)=R_{\rm ECO}(\omega) \frac{\hat{h}_{\rm LIGO}(\omega)}{R_{\rm BH}(\omega)} f_{\text{cutoff}}(\omega), \nonumber\\
\end{eqnarray}
where $f_{\text{cutoff}}(\omega)$ is a low-pass filter introduced to suppress numerical noise at high frequencies, as
the reflectivity of the Kerr angular momentum barrier $R_{\rm BH}(\omega)$, in the denominator, vanishes at high frequencies. Luckily, high frequencies leak out quickly in the first echo, and have small effect on the subsequent echoes. Our choice of $f_{\text{cutoff}}$ does not affect the second and later echoes, but it changes the first echo slightly by cutting the high frequency noise:
\begin{eqnarray}
&\hat{h}_{\rm out, fin}= \hat{h}_{\rm out} f_{\text{cutoff}},\\
&f_{\text{cutoff}}=\exp\left[ -\frac{1}{2}\left(\frac{2 \pi f({\rm Hz})-299.495}{1347.73}\right)^{16}\right], \label{cutoff}
\end{eqnarray}
where $\omega = 2\pi f$.
With the equations and boundary conditions given in the last section, we can numerically solve for $R_{\rm BH}$ and $R_{\rm ECO}$ as a function of frequency. We use LIGO event GW150914 with $a=0.67$, $M=62~M_{\odot}$ and $z=0.09$. The mass is measured in the source frame and the finial mass used in our calculation is the mass in the detector frame $M_{\rm fin}=(1+z) M$. The waveform is dominated by the $(l,m) =(2,2)$ mode, which we shall focus on for the rest of the paper \footnote{Given the symmetries of Eqs. (\ref{r}-\ref{s}), we can easily extend the solution to $m=-2$ case using $R_{slm}[\omega]=R^*_{sl-m}[-\omega]$.}
The time dependence of the waveform can then be obtained by Fourier transforming $\hat{h}_{\rm out}(\omega)$, and is shown in Fig \ref{32}. We see that changing the position of the wall changes the time-delay between the echoes, but does not affect the individual echo waveforms significantly (as long as the wall is close the would-be horizon).
As we see in Fig. \ref{31}, in the geometric optics approximation, the time delay between echoes, $\Delta t_{\rm echo,geom}$ is given by the travel time from the angular momentum barrier to the wall and back \cite{abedi2016echoes}:
\begin{eqnarray}
&&\Delta t_{\rm echo,geom}=2r_*|^{r_{\rm barrier}}_{r_{\rm wall}}=2\int^{r_{\rm barrier}}_{r_{\rm wall}}dr \frac{r^2+a^2M^2}{r^2-2Mr+a^2M^2}\nonumber\\
&&=2r_{\rm barrier}-2r_{\rm wall}+2\frac{r_+^2+a^2M^2}{r_+-r_-}\ln\frac{r_{\rm barrier}-r_+}{r_{\rm wall}-r_+}\nonumber\\
&&-2\frac{r_-^2+a^2M^2}{r_+-r_-}\ln\frac{r_{\rm barrier}-r_+}{r_{\rm wall}-r_-}.
\end{eqnarray}
This can be well approximated by the following fitting function:
\begin{eqnarray}
&&\Delta t_{\rm echo,geom}
=2\frac{r_+^2+a^2M^2}{r_+-r_-}\ln\frac{M}{r_{\rm wall}-r_+}+M G(a), \label{t_geom}\\
&&G(a)\simeq \frac{0.335}{a^2-1}+4.77+7.42 (a^2-1)+4.69(a^2-1)^2,\nonumber\\ &&\\
&&r_{\rm wall}-r_{+}= \frac{\sqrt{1-a^2} d_{\rm wall}^2}{4M(1+\sqrt{1-a^2})},
\end{eqnarray}
where we find the fit of $G(a)$ for the angular momentum barrier of $l=m=2$ mode, while $d_{\rm wall}$ is the proper distance from the wall to the would-be horizon. The latter is expected to be comparable to Planck length for ECOs of quantum gravitational nature, but $\Delta t_{\rm echo}$ only depends on the exact value of $d_{\rm wall}$ logarithmically (see Fig. \ref{32}).
The echoes in both time and frequency domain for the LIGO event GW150914 are shown in Fig. \ref{51} and \ref{511} with perfect wall standing a Planck length proper distance outside the (would-be) horizon. Here, we show the Amplitude Spectral Density (ASD), which is the square root of the power spectral density. The latter is the average of the square of the fast Fourier transforms (FFTs) of the model. In the next section, we will study the structure of the echo in the frequency domain and present how superradiance affect the structure of echo.
\begin{figure*}
\includegraphics[]{5_1_2}
\caption{\label{51} Predicted echoes for LIGO event GW150914 in the time domain with different resolution, assuming a prefect wall at a Planck length proper distance outside the horizon}.
\end{figure*}
\begin{figure}
\includegraphics[width=0.43\textwidth]{5_1_1}
\caption{\label{511} Predicted echoes for LIGO event GW150914 in the frequency domains, assuming a prefect wall at a Planck length proper distance outside the horizon}.
\end{figure}
\section{\label{seca1}Superradiance}
Scattering off Kerr BH can lead to superradiance of modes with frequency $0<\omega<m \Omega_{\rm H}$, which can extract energy from a spinning background \cite{1972Natur.238..211P}. Adding a (partially) reflective wall near horizon could turn this amplification to an instability, since modes trapped between the wall and the angular momentum barrier can extract the spin energy repeatedly \cite{1978CMaPh..63..243F, Cardoso:2007az}. In this section, we study this effect for the echoes in frequency domains.
There is an odd looking spike in Fig. \ref{511} frequency domain around 183 Hz (see top panel in Fig. \ref{81} for a zoom-in). Indeed, this is exactly the threshold frequency for the superradiance. This is demonstaretd in the middle panel of Fig. \ref{81}, which shows the scattering amplification with the horizon, perfect wall and soft wall around that frequency. The vertical axis is the relative energy, extracted from around black hole by scattered gravitational waves. The blue dashed line shows superradiance slowly turning off with increasing the frequency, and we confirm that it ends exactly at frequency $f _{\rm max}=a m/[2 \pi(r_+^2+a^2)]=183~ \textrm{Hz}$, for $m=2$ as shown in the plot. In contrast, superradiance by soft wall (grey and thin curve) occurs at resonance peaks, corresponding to the ergoregion trapped mode (for more details, see Appendix \ref{a1} ). Since superradiance ends at 183Hz, the resonance peaks shift the direction, which is the reason we have an odd spike in the Fig. \ref{511} and \ref{81} top panel.
\begin{figure}
\includegraphics[width=0.43\textwidth]{8_1}
\includegraphics[width=0.43\textwidth]{8_2}
\includegraphics[width=0.43\textwidth]{8_3}
\caption{\label{81} Superradiance in frequency domain for GR Kerr BH and ECO with a wall.}
\end{figure}
The perfect wall (the red thick curve) in Fig. \ref{81} middle panel is a constant zero without any resonance peaks, since a perfect reflective wall kills superradiance, as all the energy that goes in, comes out eventually (see Appendix \ref{a1} for a subtlety in this argument). However, the odd spike structure remain in the amplitudes, as shown in Fig. \ref{81} bottom panel, where we change the vertical axis to real part of outgoing to ingoing wave at infinity. We still see the sign flip in resonance structure at 183 Hz.
In the next section, we study the echo templates resulting from solving the linearized Einstein equations, which improves the simplistic geometric picture in Fig. \ref{31}.
\section{\label{sec5}Minimal Echo templates}
Now that we have numerical predictions for echoes, we would like to provide simple fitting functions that could be used for quick visualization and data-fitting purposes. We call these fitting functions, templates. In order to find our templates, we define echoes in the time domain by the regions that surround the peaks of $|h(t)|$ and exceed a limit: $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1, -1.5$ or $-2$. $|h|_{\rm max, n}$ is the height of the $n^{\rm th}$ peak of $|h(t)|$, which we call the $n^{\rm th}$ echo. Then we fit the $n^{\rm th}$ echo to a complex gaussian
\begin{eqnarray}
h_n(t)&=& \exp [\Psi_n(t)+I \Phi_n(t)], \\ \label{gaussian}
\Psi_n(t)&=&a_0+a_1t+a_2 t^2, \\
\Phi_n(t)&=&b_0+b_1t,
\end{eqnarray}
where $a_0$, $a_1$, $a_2$, $b_0$ and $b_1$ are real numbers. This form is same as fitting the $n^{\rm th}$ echo to $ A \exp [ \frac{(t-t_0)^2}{2\sigma^2} ]$, where $A$ and $t_0$ are complex, while the width $\sigma$ is real.
\begin{figure}
\includegraphics[width=0.22\textwidth]{5_2_0_1}
\includegraphics[width=0.22\textwidth]{5_2_0_2}
\includegraphics[width=0.22\textwidth]{5_2_0_3}
\includegraphics[width=0.233\textwidth]{5_2_0_4}
\includegraphics[width=0.23\textwidth]{5_2_0_5}
\includegraphics[width=0.23\textwidth]{5_2_0_6}
\caption{\label{520} Best fit gaussians to the $2^{\rm nd}$, $5^{\rm th}$, $10^{\rm th}$ and $30^{\rm th}$ echoes within $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. We see that as high frequency modes leak out faster, later echoes decay in amplitude and become wider in time domain, and high frequency is cuted in the frequency domain.}
\end{figure}
As an example, Fig \ref{520} compares the numerical solutions and gaussian fits for the 2$^{nd}$, $10^{th}$, and $30^{th}$ echoes, with time origin shifted to center of each echo, and fitting the region with $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$.
Within this approximation, there are five real parameters for every echo that quantify its amplitude, width and center, both in time and frequency domain, as well as the overall phase at the center of the echo, as shown in Table \ref{t2}.
\begin{table*
\caption{\label{t2} Some physical quantities of a single echo defined by the five parameters from the gaussian echo template (Eq. \ref{gaussian})
}
\begin{ruledtabular}
\begin{tabular}{c|ccc}
&\textrm{width}&\textrm{center}&\textrm{peak amplitude}\\
\colrule
\textrm{time}&
$\sqrt{-1/ (2a_2})$&
$-a_1/2 a_2$&
$\exp[a_0-a_1^2/4 a_2]$\\
\colrule
\textrm{frequency} & $ \sqrt{-2 a_2}/(2 \pi)$ & $b_1/(2\pi)$ & $\exp[a_0-a_1^2/4 a_2-1/2 \log[2 \sqrt{a_2^2}]]$\\
\colrule
\colrule
\textrm{overall phase} & $ b_0-b_1 a_1/(2 a_2)$ \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table
\caption{\label{t3} Best fit gaussian echo template quantities (see Table \ref{t2} and Fig. \ref{52}) , for our minimal model of GW150914}
\begin{tabular}{|l|l|}
\cline{1-2}
\textrm{peak amplitude in time / strain}&
$2.91 \times 10^{-19}/n^{1.32}$\\
\colrule
\textrm{width in time / msec} & $4.29+ 0.883 n$ \\
\colrule
\textrm{correction to} $\Delta t_{\rm echo,geom}$ \textrm{/ msec}& $1.52+1.71/(1+n)$ \\
\colrule
\textrm{peak frequency / Hz} & $177+102/ n^{0.3}$ \\
\colrule
\textrm{Overall phase}&
$-7.26+27.1 n^{0.945}+22.6 n$\\
\cline{1-2}
\end{tabular}
\end{table}
Table \ref{t3} provides the best fit parameters of our echo templates for all echoes, based on the LIGO event GW150914 and averaging over the best fit functions with different echo domains $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1, -1.5$ or $-2$.
The best fits for each echo domain is also provided in Fig. \ref{52}. For correction to $\Delta t_{\rm echo, geom}$, we define time-delay as $\Delta t_{n}= t_{n} -t_{n-1} $. For all other plots, first echo is not included since it is very sensitive to the properties of the wall, as well as nonlinear effects from early stage of merger (see details in Sec. \ref{sec5}). The top three panels in Fig. \ref{52} show the time domain properties as a function of the echo number. Starting from the left, peak echo amplitudes in time are all well fit by decaying power laws(\cite{Correia:2018apm} argue that the decay of echoes at early stages is polynomial). Middle are the width of the echoes, becoming wider for later echoes in the time domains, as the high frequency modes leak out more quickly. The top right panel gives correction to $\Delta t_{\rm echo,geom}$ (\ref{t_geom}), while the bottom left panel shows the decay of the mean echo frequency. The bottom middle and right provide overall phase at $t_{\rm center}$ and the residuals of the best fit for the phase. We only show the residuals for the phase, as the numerical error for the phase is relatively big.
\begin{figure*}
\minipage{0.33\textwidth}
\includegraphics[width=1.151\linewidth]{5_2_1}
\includegraphics[width=1.154\linewidth]{5_2_4}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[width=1.18\linewidth]{5_2_2}
\includegraphics[width=1.175\linewidth]{5_2_5}
\endminipage\hfill
\minipage{0.32\textwidth}%
\includegraphics[width=1.18\linewidth]{5_2_3}
\includegraphics[width=1.18\linewidth]{5_2_6}
\endminipage
\caption{\label{52} Best fit gaussian template parameters (for $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1, -1.5$ or $-2$), in our minimal model of LIGO event GW150914, showing second and later echoes. The top three panels are in the time domains. Starting from left, peak amplitudes of echos in time are well fit by power laws. Middle panel is the width of the echoes, which become wider in time, as the high frequencies leak out more quickly. For the same reason, the peak frequency (bottom left) also decays with time. The top right panel gives corrections to $\Delta t_{\rm echo,geom}$ (Eq. \ref{t_geom}). Finally, the bottom middle and right provide the overall phase at $t_{\rm center}$ of each echo and the residuals of the best fit. This is the only plots we show the residuals since the numerical error for the phase is relatively big. }
\end{figure*}
To visualize the quality of the template to fit data, Fig. \ref{53} shows the ${\rm SNR}_{\rm temp}/{\rm SNR}_{\rm model}$, where ${\rm SNR}_{\rm model}$ is the predicted signal-to-noise ratio for our numerical solution of echoes (assuming white noise), while ${\rm SNR}_{\rm temp}$ is a reduced value, if we use our Gaussian approximations of Fig. \ref{520} (gray circles in Fig. \ref{53}). Using a second fit for how properties (i.e. width, center and amplitude) of $\Psi_{\rm n}(t)= \log |h_n(t)| $ depend on $n$ (Table \ref{t3}) further reduces ${\rm SNR}_{\rm temp}$ (red triangles in Fig. \ref{53}). We notice that the quality of Gaussian fit drops for later echoes, which could be either due to build-up of numerical error or systematic deviations from a single gaussian fit. The secondary fit for $\Psi_{\rm n}$ vs $n$ further reduces SNR as the width in time and time delay, shown as Fig. \ref{52}, do not have a simple behavior. However, the power law fit to the peak amplitude in time $\propto n^{-4/3}$ is surprisingly good. Also, as we discussed before, since the shapes of first few echoes are much more dependent on the initial conditions, it might be better to use independent Gaussians to fit them in data. Finding a reasonable fit for phase information $\Phi_{\rm n}$ vs $n$ proves even more challenging, as a small change in phase leads to a significant change in echo profiles. Fortunately, model-agnostic searches (e.g., \cite{Conklin:2017lwb}) based on cross-correlating different detectors can be done independent of the phase information.
\begin{figure}
\includegraphics[width=0.5\textwidth]{5_2_0}
\caption{\label{53} ${\rm SNR}_{\rm temp}$ compared to ${\rm SNR}_{\rm model}$, showing the quality of gaussian templates.}.
\end{figure}
\section{\label{sec4}Beyond the minimal model}
While our minimal model for echoes has only one free parameter (wall distance to the horizon, $d_{\rm wall}$) in addition to those of GR, the reality can be more complicated. Here, we explore the two main deviations expected from the minimal model due to nonlinear effects in GR and quantum gravity.
\subsection{Nonlinear Mergers Effects}
Our assumption of a custom-designed incoming wavepacket, as a placeholder for black hole binary merger, is almost certainly too naive to provide a realistic echo template, as it misses the nonlinear nature of the merger. While numerical simulations can now provide realistic waveforms for black hole mergers in GR, a covariant formulation of ECOs that could produce realistic echo waveforms is currently missing. However, we can get an idea about the extent of nonlinear corrections to linear results by noticing that the Kerr background for Teukolsky equation (\ref{eq:teuk}) is dynamical during the merger event, and thus the frequencies can be shifted by ${\cal O}(30\%)$, between the ingoing and outgoing waves at merger \footnote{Fort example, the best-fit for the dominant quasinormal mode frequency for GW150914 is 10-20\% offset from the linear theory predictions for the best-fit Kerr metric (Fig. 5 in \cite{TheLIGOScientific:2016src}). } . We shall explore the extent of this effect on echoes by introducing a blueshift parameter $s$, in the ingoing linear initial conditions:
\begin{eqnarray}\label{s_shift}
\hat{h}_{\rm LIGO, shifted} [f]=\hat{h}_{\rm LIGO} [ f/s ].
\end{eqnarray}
As shown in Fig. \ref{7}, redshifted (blueshifted) initial conditions give echoes which damp more slowly (quickly), since low frequencies leak more slowly through the angular momentum barrier. This also dramatically changes the amplitude of first few echoes. Blueshift parameter $s$ can be a free parameters for data fitting purposes.
\begin{figure*}
\includegraphics[width=0.3\linewidth]{7}
\includegraphics[width=0.3\linewidth]{7_3}
\includegraphics[width=0.3\linewidth]{7_4}
\caption{\label{7} Echoes predicted for GW150914, expected for redshifted (blueshifted) initial conditions with respect to our minimal model. We see that lower frequency initial conditions lead to lower amplitude, but more persistent, echoes as they cannot penetrate the angular momentum barrier efficiently.}
\end{figure*}
The effect is clearer if we compared SNR of echoes to first echo, as shown in Fig. \ref{8}. $\rm SNR^2_{\rm n}$ is $\rm SNR^2$ of our numerical solution of $\rm n^{th}$ echo and we trimmed a single echo with $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. We assume white gaussian noise ${\sigma_{\omega}}=1$ so that
\begin{eqnarray}\label{SNR}
\rm SNR^2_{\rm n}= \sum_{\omega} \frac{ | \hat{h}_{\rm n, \omega}| ^2}{{\sigma_{\omega}}^2} = \sum_{t} | h_{\rm n}| ^2. \\
\end{eqnarray}
Fig \ref{8} (right panel) shows that later echoes contain more (less) information in redshifted (blueshifted) templates, since they decay more slowly (quickly). The left panel also shows the relative SNR of 1st echo compared to the trimmed main event in our model. The fact that this number can change by more than 1.5 orders of magnitude suggests that the amplitude of 1st echo is very sensitive to the nonlinear merger physics and cannot be reliably predicted. \cite{Gupta:2018znn} simulates a binary black hole merger and finds the ratio of the energy falling into the black hole to the energy out is around 1:1, which can be used as a normalization of amplitude of echoes.
\begin{figure*}
\includegraphics[width=0.45\linewidth]{10_1}
\includegraphics[width=0.45\linewidth]{11_1}
\caption{\label{8} signal-to-noise ratios(SNR) and energy for blueshifted echoes compared with the first echo. We see that there is more (less) information in subsequent echoes for lower (higher) frequency initial conditions. Furthermore, the amplitude of first echo is hard to predict and can change by more than 1.5 orders of magnitude.We also list SNRs and energy for blueshifted first echoes compared with the event. Since we assume white noise to calculate the SNR in time domain, we trim the merger template at around 0.076 seconds before the peak (similar to the LIGO noise whitening for GW150914 template). }
\end{figure*}
Table \ref{t5} and Fig. \ref{72} compare the best fit echo parameters for different blueshift factors. We see in the left panels that the blueshifted initial condition ($s=1.2$) has a transient excess in amplitude that decays quickly and falls in line the minimal model. In contrast, the redshifted model ($s=0.8$) has a significantly smaller but more persistent amplitude. Surprisingly, the middle panels show that the redshifted echoes remain narrower in time. Even more puzzling is that the redshifted initial conditions have higher frequency echoes as shown in Fig. \ref{72} the bottom left panel. This is due to the fact that the echo peak frequency depends on the slope (and not the amplitude) of the spectral density $\hat{h}_{\rm out}(\omega)=R_{\rm ECO}(\omega) \frac{\hat{h}_{\rm LIGO}(\omega)}{R_{\rm BH}(\omega)} f_{\text{cutoff}}(\omega)$ from Eqn. \ref{eq:hout}, which involve several complicated components. As we see in the middle panel of Fig. (\ref{7}), this slope is not monotonic which leads to the counterintuitive behavior, even though the amplitude of the redshifted model is smaller compared to the blueshifted.
\begin{widetext}
\begin{table
\caption{\label{t5} Same as Table \ref{t3}, but contrasting with redshifted/blueshifted initial conditions, fitted within $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. }
\begin{tabular}{|l|l|l|l|}
\cline{1-4}
\textrm{blueshift factor $s$}&\textrm{ 0.8}&\textrm{$1$}&\textrm{$1.2$}\\
\colrule
\textrm{peak amplitude in time / strain}&$5.91 \times 10^{-20}/n^{1.14}$&
$2.92\times 10^{-19}/n^{1.33}$&
$5.31\times 10^{-19}/n^{1.54}$\\
\colrule
\textrm{width in time / msec} & $3.91+0.678 n$& $5.5+0.808 n$ & $9.48+0.711 n$ \\
\colrule
\textrm{correction to} $\Delta t_{\rm echo,geom} \textrm{/ msec}$ & $-47.8-57.0/(1+n)$& $15.4+1.64/(1+n)$& $76.2+60.4/(1+n)$ \\
\colrule
\textrm{peak frequency / Hz}& $227+95.2/ n^{0.3}$ & $175+104/ n^{0.3}$& $144+97.8/ n^{0.3}$ \\
\colrule
\textrm{Overall phase} & $-3.06+30.2n^{0.945}-25.9n$ & $-6.65+28.5n^{0.945}-23.8n$ & $-12.7+35.2n^{0.945}-29.4n$ \\
\cline{1-4}
\end{tabular}
\end{table}
\end{widetext}
\begin{figure*}
\minipage{0.33\textwidth}
\includegraphics[width=1.151\linewidth]{7_2_1}
\includegraphics[width=1.151\linewidth]{7_2_4}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[width=1.16\linewidth]{7_2_2}
\includegraphics[width=1.17\linewidth]{7_2_5}
\endminipage\hfill
\minipage{0.32\textwidth}%
\includegraphics[width=1.18\linewidth]{7_2_3}
\includegraphics[width=1.18\linewidth]{7_2_6}
\endminipage
\caption{\label{72} Same is Fig. (\ref{52}), but using the different blueshift factors $s$ (Eq. \ref{s_shift}) for echo initial conditions (fitted for $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$). We see that redshifted initial conditions yield weaker, but more persistent echoes (see text for details).
}
\end{figure*}
\subsection{Soft Wall}
Motivated by quantum models of black holes, the wall must at least partially absorb the energy incident on the wall \cite{abedi2016echoes}. For example, in fuzzball models \cite{Mathur:2012jk} high energy particles (with $\hbar \omega \gg kT_{\rm H}$, where $T_{\rm H}$ is the Hawking temperature) excite the fuzzball microstates and thus will be absorbed by the wall. On the other hand, particles with $\hbar\omega \leq kT_{\rm H} $ may be (at least partially) reflected (but see \cite{Guo:2017jmi} for recent counter-arguments). Ringdown phase of mergers of two BHs is in the intermediate range ($\sim 100$ Hz for GW150914). Therefore, a realistic quantum gravity model for the echoes is expected to involve a {\it soft} wall. For example, frequency of electromagnetic emissions from accretion into BHs is much higher, which is expected to be absorbed by the wall \cite{Broderick:2009ph, Broderick:2015tda}. However, possible loopholes that could lead to astrophysical observables from quantum effects have been exploited in \cite{Pen:2013qva, Afshordi:2015foa}.
A wall that absorbs high frequency modes will dramatically decrease the amplitude of the first echo, since these modes leak out quickly every time the wavepacket hits the angular momentum barrier. Therefore, the first echo contains most of the high frequency modes which, as shown in the top left panel in Fig \ref{43}, would be absorbed for a soft wall.
Of course, the actual frequency-dependent reflection of the wall depends on the specific quantum theory of black holes. We explore a phenomenological model for the wall with a Gaussian-like energy reflection rate
\begin{equation}
R_{\rm wall}(\omega) \simeq \exp\left[-\left( \alpha\frac{\omega}{T_{\rm H}} \right)^q\right],
\end{equation}
where $T_{\rm H}=\frac{r_+^2-a^2}{4 \pi r_+(r_+^2+a^2)}$ is the Hawking temperature for Kerr BH. While smooth $R_{\rm wall}$'s, such as gaussian or Boltzmann reflectivity ($q=2$ or $1$, respectively) may appear natural, they do tend to essentially wipe out the echoes, unless $\alpha \ll 1$, which is inconsistent with the tentative echoes found in \cite{abedi2016echoes}. In contrast, a sharper function with, e.g., $q=12$ then can damp the first echo, but not significantly influence later echoes, as shown in Fig \ref{42} \footnote{Fig \ref{42} also shows that if the wall absorbs too much, the late echoes will stop decaying. This is due to superradiant instability which we shall discuss in the next section.}. We can also compare these reflectivity functions with that of the angular momentum barrier of the Kerr BH, for the same spin and mass, as shown in Fig \ref{44}, which provides another motivation for sharper $R_{\rm wall}$'s.
\begin{figure}
\includegraphics[width=0.5\textwidth]{4_4}
\caption{\label{44} Comparison of soft wall reflectivity coefficients that we use, with that of the Kerr angular momentum barrier \cite{Nakano:2017fvh}. The thin and dashed lines are the two reflectivity rates used in Fig \ref{42}.}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{4_2}
\caption{\label{42} Echoes for GW15014, for soft vs. perfect walls. The top (gray) curve assumes a perfect wall/mirror, while the lower curves show soft walls with different energy reflectivity coefficients.
}
\end{figure}
\begin{widetext}
\begin{table
\caption{\label{t4} Same as Table \ref{t3}, but contrasting perfect ($R_{\rm wall}=1$) and soft ($R_{\rm wall}= \exp[-(0.055 \frac{\omega}{T_{\rm H}})^{12}]$) walls, fitted within $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. }
\begin{tabular}{|l|l|l|}
\cline{1-3}
\textrm{wall type}&\textrm{perfect}&\textrm{soft}\\
\colrule
\textrm{peak amplitude in time / strain}&
$2.78 \times10^{-19}/n^{1.31}$&
$2.33 \times 10^{-19}/n^{1.36}$\\
\colrule
\textrm{width in time / msec} & $5.5+0.808 n$ & $8.17+0.659 n$ \\
\colrule
\textrm{correction to} $\Delta t_{\rm echo,geom}$ \textrm{msec} & $15.4+1.64/(1+n)$& $14.3+1.52/(1+n)$ \\
\colrule
\textrm{peak frequency / Hz} & $175+104/ n^{0.3}$& $177+96.8/ n^{0.3}$ \\
\colrule
\textrm{Overall phase} & $-6.65+28.5 n^{0.945} -23.8 n$& $-5.2+26.8 n^{0.945} -22.6 n$ \\
\cline{1-3}
\end{tabular}
\end{table}
\end{widetext}
\begin{figure*}
\minipage{0.33\textwidth}
\includegraphics[width=1.151\linewidth]{4_3_1}
\includegraphics[width=1.151\linewidth]{4_3_4}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[width=1.16\linewidth]{4_3_2}
\includegraphics[width=1.17\linewidth]{4_3_5}
\endminipage\hfill
\minipage{0.32\textwidth}%
\includegraphics[width=1.18\linewidth]{4_3_3}
\includegraphics[width=1.18\linewidth]{4_3_6}
\endminipage
\caption{\label{43} Same is Fig. (\ref{52}), comparing walls with different energy reflectivity coefficients (fitted for $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$). We see that echo amplitudes and peak frequencies decay more quickly for softer walls (see text for details).
}
\end{figure*}
Table \ref{t4} and Fig. \ref{43} compare the template for the perfect and soft walls, similar to Figs. (\ref{52}) and (\ref{72}). We see in the top left panel that, due to absorption of high frequency modes, the power law fit to the amplitudes could be extended to first echo for the soft walls. More generally, echoes decay faster for a softer wall.
As echoes for a wall with $R_{\rm wall}= \exp[- (0.06\frac{ \omega}{T_{\rm H}})^8]$ decay too fast, we only focus on the $R_{\rm wall}= \exp[-(0.055 \frac{\omega}{T_{\rm H}})^{12}]$ case in subsequent panels of Fig. \ref{43}, and provide numerical fits for echo properties in Table \ref{t4}. With this choice, the evolution of echo properties is similar to those in Figs. (\ref{52}) and (\ref{72}), with the notable difference that peak frequency decays more rapidly as the soft wall absorbs high frequencies.
\section{\label{sec7}Conclusions}
We have provided realistic templates for echoes of BH mergers by numerically solving the linearized Einstein equation (or Teukolsky equation) in Kerr spacetime with boundary conditions at a Planck length proper distance outside the (would-be) event horizon. We obtain analytic approximations for the echo waveforms and time-delays, and explore their dependence on the softness of the wall (or frequency-dependence of the reflection rate), as well as nonlinear effects during merger event. These analytic templates should be useful in echo searches in current and future gravitational wave data. Finally, we studied the occurrence of superradiant instability and showed that it has negligible effect, for the first few dozen echoes of in typical BH mergers such as GW150914.
Let us close with some open questions and future directions:
\begin{itemize}
\item The strain is dominated by mode $l=2, m=\pm{2}$. We only show mode $m=2$ here and solution of $m=-2$ can easily be found by $R_{slm}[\omega]=R^*_{sl-m}[-\omega]$. More realistic templates should combine all other modes by appropriate weight.
\item We cannot provide a reliable waveform for the first echo as it is too sensitive to the {\it ad hoc} cutoff function (\ref{cutoff}) that we use to set up our initial conditions. This highlights the need for a covariant numerical implementation of ECOs within a dynamical spacetime, which could provide realistic nonlinear initial conditions for echoes.
\item Another big uncertainty is the expected softness of the wall. While this is ultimately a question for the quantum models of black holes, it highlights the need for a covariant and causal description of the wall dynamics. It might be possible to describe this dynamics in terms of the properties of a surface (2+1d) fluid and Israel junction conditions (e.g., see \cite{Saravani:2012is}).
\item The computation of the echo phase beyond $\sim 20$ echoes is limited by numerical precision and frequency resolution. This can be improved in the future, by either brute force or novel numerical/analytic methods.
\end{itemize}
\nocite{*}
\begin{acknowledgements}
We thank Vitor Cardoso, Rafael Sorkin, Huan Yang, Aaron Zimmerman, Bob Holdom, Randy S. Conklin, Ren Jing, Ofek Birnholtz, William East and Connor Adair for helpful comments and discussions. We also thank all the participants in our weekly group meetings for their patience during our discussions. This work was supported by the University of Waterloo, Natural Sciences and Engineering Research Council of Canada (NSERC), and the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Innovation.
\end{acknowledgements}
\section{\label{sec1}Introduction}
Motivated by the black hole (BH) information paradox and cosmological constant problems, it has been suggested that non-perturbative quantum gravitational effects may lead to Planck-scale modifications of BH horizons. Proposals to solve the BH information paradox include gravastars \cite{Mazur:2004fk}, fuzzballs \cite{Lunin:2001jy, Lunin:2002qf, Mathur:2005zp, Mathur:2008nj, Mathur:2012jk}, and firewalls \cite{Braunstein:2009my, Almheiri:2012rt}, amongst others \cite{Barcelo:2015noa, Kawai:2017txu}.
These Exotic Compact Objects (ECOs) all modify the standard structure of BH horizons, and should form by Page time $\sim M^3$, but can emerge as early as the ``scrambling time'' $\sim M \log M$ \cite{Hayden:2007cs, Sekino:2008he}.
Gravitational aether theory \cite{Afshordi:2008xu,PrescodWeinstein:2009mp} which modifies the Einstein field equations by adding an incompressible fluid (aether) is a possible solution to the cosmological constant problem(s). BH solutions in this theory link the BH mass with the aether pressure at infinity, and yield a comparable pressure to the observed dark energy pressure for stellar BH masses of 10-100 $M_{\odot}$. The solution of the modified Einstein field equation deviates from the GR within the order of Planck length proper distance outside the (would-be) horizon. It is also suggested that replacing the horizon with a ``wall'' could be a source of high energy astrophysical neutrino flux \cite{Afshordi:2015foa} which is a possible source for the PeV neutrinos recently detected by IceCube observatory.
A concrete physical model for replacing event horizon due to quantum gravitational effects is provided in \cite{Saravani:2012is}. The spacetime ends at about the order of Planck length proper distance outside the (would-be) horizon with a wall containing a surface fluid. It is then shown that Israel junction conditions imply that the fluid has the thermodynamic entropy matching the Bekenstein-Hawking area law, for charged rotating BHs (Also see \cite{Holdom:2016nek} for a similar horizonless spacetime solution).
Recent detections of gravitational waves from binary BH mergers by the LIGO-Virgo collaboration \cite{TheLIGOScientific:2016agk, TheLIGOScientific:2016pea, Giddings:2016tla, Abbott:2016blz, Abbott:2016nmj, Abbott:2017vtc, Abbott:2017oio, TheLIGOScientific:2017qsa, Abbott:2017gyy} provide a way to test the structure around the horizon scale.
Shortly after LIGO's first detection, GW150914, \cite{Cardoso:2016rao, Cardoso:2016oxy} argued that introducing a wall to replace horizon might yield a similar ringdown waveform as GR BHs, but produce delayed echoes (see \cite{Cardoso:2017njb, Cardoso:2017cqb} for a review) in the gravitational wave signal. Using a phenomenological template by truncating the GR merger waveforms, \cite{abedi2016echoes} carried out the first search for echoes and claimed a 2.5$\sigma$ tentative evidence for them in the the first three (candidate) events in the LIGO public data (but see \cite{Ashton:2016xff,Westerweck:2017hus} and \cite{Abedi:2017isz} for a critique/rebuttal).
An independent search \cite{Conklin:2017lwb}, using a different methodology, has recently found evidence for echoes in each of LIGO's merger events (with the notable exception of GW150914) at $\sim 3\sigma$ significance level. However, we should note that the echoes reported in \cite{abedi2016echoes} and \cite{Conklin:2017lwb} are for different events, even though they are both broadly consistent with the hypothesis of near-horizon Planck-scale structure. In particular, \cite{Abedi:2017isz,Westerweck:2017hus} fail to find echoes in GW151226, which has the most significant evidence for echoes in \cite{Conklin:2017lwb}, suggesting that the two methods capture different parts of the echo waveform.
Most recently, \cite{BNS} claim a tentative detection of (lower harmonics of) echoes, at $4.2\sigma$ level, from a ``black hole'' remnant in the aftermath GW170817 binary neutron star merger.
While one may consider other phenomenological echo templates (e.g., \cite{Maselli:2017tfq}), more realistic templates for fitting data may be found by solving (linearized) Einstein equations with modified boundary conditions near the horizon. Along this direction, most studies have so far focused on Schwarzschild BHs (e.g., \cite{Cardoso:2016rao,Cardoso:2016oxy, Price:2017cjr, Mark:2017dnq, Volkel:2018hwb, Volkel:2017kfj}). In this paper, we extend this to Kerr metric as realistic BHs have spin. \cite{Nakano:2017fvh} also presented echo templates by modelling the reflectivity of the angular momentum barrier in the Kerr spacetime. We, however, model the propagation in the full spacetime which provides a more realistic treatment at lower frequencies.
Another related work is \cite{Bueno:2017hyj} which studies the echoes of scalar gaussian wavepackets in Kerr-like wormholes. In contrast, we study generic propagation in Kerr spacetime, with arbitrary boundary conditions, which can be applied not only to scalar fields (s=0), but also massless Dirac (s = $\pm$1/2), electromagnetic (s = $\pm$1), or gravitational (s = $\pm$2) fields. Interestingly (but not surprisingly), we come to some similar conclusions, e.g., {\it i)} Spinning ECOs give rise to unstable modes which, however, do not affect the echoes till very late times (depending on whether the initial frequency range is within the superradiance regime). {\it ii)} It is hard to make a model-independent prediction for the first echo.
A related phenomenological issue that arises when we replace the horizon with a wall is the emergence of superradiant instability for horizonless ergoregions \cite{1978CMaPh..63..243F, Cardoso:2007az,Cunha:2017qtt}. While this might suggest long-term instability of spinning ECOs, which may be in conflict with astrophysical spin measurements for BHs \cite{Narayan:2013gca}, it was suggested by \cite{Maggio:2017ivp} that an absorption rate of the wall as small as 0.4\% is sufficient to quench the instability completely
We organize this paper as follows: Sec.\ref{sec2} provides the linear Einstein equations and boundary conditions used. Instead of normal boundary condition with no outgoing wave on the horizon, we put a wall standing just outside the would-be horizon. The reflection rate of the wall depends on the specific model of quantum BHs. Sec.\ref{sec3} presents echo solutions for different positions of a perfect wall and time-delays of a geometric formula given in \cite{abedi2016echoes}, while Sec. \ref{seca1} discusses how superradiance of Kerr geometry is manifested in echo templates. In Sec. \ref{sec5}, we provide an analytic fit to the echo templates, based on solutions in Sec. \ref{sec3}. We explore a soft wall with frequency-dependent reflection, as well as nonlinear corrections to initial conditions in Sec.\ref{sec4} for a more realistic picture. In Appendix \ref{a1}, we briefly discusses ergoregion instability developed in the presence of a perfect wall. While in principle the instability is significant at high spins, we show that these instabilities do not affect the first several echoes of typical binary merger events. Finally, Sec.\ref{sec7} concludes our work.
If not specified, we use units with $G=\hbar = k_B = c=1$. For concreteness, we use the best fit properties and waveforms resulting from the GW150914 merger event, provided by the LIGO-Virgo collaboration \cite{TheLIGOScientific:2016agk, TheLIGOScientific:2016pea} \footnote{https://losc.ligo.org/events/GW150914/}. In particular, the detector frame mass and reduced spin parameter of the remnant used for the echo calculation are $M_{\rm fin} = 67.6~ M_{\odot}$ and $a = 0.67$. Echo templates for other final masses can be found by rescaling our analytic templates, as long as the dimensionless binary properties are not too different from those of GW150914.
\section{\label{sec2}Propagation and Boundary Conditions in Kerr spacetime}
\begin{table
\caption{\label{master}%
Corresponding field $\psi$ for different spin weight $s$ in Master equation. Here $\rho^{-1}=-(r-i a \cos\theta)$
}
\begin{ruledtabular}
\begin{tabular}{c|cccc}
\textrm{s}&
\textrm{0}&
\textrm{-1/2, 1/2}&
\textrm{-1, 1}&-2, 2\\
\colrule
$\psi$ & $\Phi$ & $\chi_0, \rho^{-1} \chi_1$ & $\phi_0, \rho^{-2} \phi_2$ & $\Psi_0, \rho^{-4} \Psi_4$\\
\end{tabular}
\end{ruledtabular}
\end{table}
We study the propagation of gravitational waves using linearized Einstein equations in Kerr geometry which describes the spacetime of a spinning BH. In order to model an exotic compact object (ECO), we simply replace the Kerr event horizon with a wall, where boundary conditions for linear perturbations are modified. The initial condition here is an incoming wavepacket $h_{\rm in}$ from infinity, and we calculate the outgoing wavepacket $h_{\rm out}$ by solving the linear Einstein equations.
As usual, we use the Newman-Penrose (NP) Formalism which greatly simplifies perturbation in Kerr metric, reducing to only a single master equation (known as the Teukolsky equation) which describes propagation of all scalar ($s=0$), massless Dirac ($s=\pm1/2$), electromagnetic ($s=\pm 1$) and gravitational ($s=\pm 2$) fields (see \citeauthor{teukolsky1973perturbations} \cite{teukolsky1973perturbations} for details):
\begin{widetext}
\begin{eqnarray}\label{eq:teuk}
\left[\frac{(r^2+a^2)^2}{\Delta}-a^2 \sin ^2 \theta\right] \frac{\partial^2 \psi}{\partial t^2}+ \frac{4Mar}{\Delta}\frac{\partial^2 \psi}{\partial t \partial \varphi}+\left(\frac{a^2}{\Delta}-\frac{1}{\sin^2 \theta}\right)\frac{\partial^2 \psi}{\partial \varphi^2}-\Delta^{-s} \frac{\partial}{\partial r} \left(\Delta^{s+1} \frac{\partial \psi}{\partial r}\right)-\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)\nonumber\\-2s\left[\frac{a(r-M)}{\Delta}+\frac{i \cos \theta}{\sin^2\theta}\right]\frac{\partial\psi}{\partial \varphi}-2s\left[\frac{M(r^2-a^2)}{\Delta}-r-ia\cos\theta\right]\frac{\partial \psi}{\partial t}+(s^2\cos^2\theta-s)\psi= 0, ~
\end{eqnarray}
\end{widetext}
where the field $\psi$ for each spin weight $s$ corresponds to NP quantities presented in Table \ref{master}.
The Teukolsky equation (\ref{eq:teuk}) is separable in coordinates in the frequency domain and can be decomposed into 4 ODEs. Furthermore, the symmetries in time and azimuth, allows for Fourier space decomposition in $t$ and $\varphi$:
\begin{widetext}
\begin{eqnarray}
&&\psi=\frac{1}{2\pi} \int d\omega e^{i(-\omega t +m \varphi)} S[\theta] R[r],\label{R}\\
&&\Delta^{-s} \frac{d}{dr} \left( \Delta^{s+1} \frac{dR}{dr} \right)+\left[ \frac{K^2-2is(r-M)K}{\Delta}+4is\omega r -\lambda \right]R=0,\label{r}\\
&&\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\frac{dS}{d\theta}\right)+\left(a^2 \omega^2 \cos^2\theta-\frac{m^2}{\sin^2\theta}-2a\omega s \cos\theta-\frac{2ms\cos\theta}{\sin^2\theta}-s^2 \cot^2\theta+s+A_{slm}\right)S=0,\label{s}
\end{eqnarray}
\end{widetext}
where $K=(r^2+a^2)\omega -am$ and $\lambda=A_{slm}+a^2 \omega ^2 -2am\omega$. The solution for the angular mode is spin-weighted spheroidal harmonic (full discussion can be found in \cite{Berti:2005gp}). We solve the radial mode numerically based on \citeauthor{Brito:2015oca} \cite{Brito:2015oca}, with publicly available Mathematica code, which was developed to study superradiance in Kerr metric \footnote{https://centra.tecnico.ulisboa.pt/network/grit/files/amplification-factors/}. Eq \ref{r} has the following asymptotic solutions
\begin{eqnarray}
&&R=\mathcal{T} \Delta^{-s} e^{-ik_{\rm h}r^*}+\mathcal{O}e^{ik_{\rm h}r^*}, r \rightarrow r_+ , \\
&&R=\mathcal{I} \frac{e^{-i \omega r^*}}{r}+\mathcal{R}\frac{e^{i\omega r^*}}{r^{2s+1}}, r \rightarrow \infty,
\end{eqnarray}
where $r^*$ is tortoise coordinate (defined as $r^*=\int \frac{r^2+a^2}{r^2-2Mr +a^2} dr$ that approaches -$\infty$ at horizon), $k_{\rm h}=\omega-\frac{a m}{2Mr_{+}}$ and $r_+ = M+\sqrt{M^2-a^2}$.
In classical General Relativity, everything that reaches the horizon will fall into the BH, and thus theres is no outgoing wave at $ r \rightarrow r_+ $, i.e. $\mathcal{O}=0$. However, for ECOs we assume that quantum gravity effects replace the horizon with (partially) reflective wall standing the order of Planck length proper distance outside the (would-be) horizon. We shall assume that this modifies the boundary condition, so that the wall reflects the incoming energy flux (see \cite{Nakano:2017fvh} for definition of energy near horizon) with a rate $R$ but does not change the phase:
\begin{widetext}
\begin{eqnarray}
&&|\mathcal{O}|^2=R_{\rm wall} \left | \frac{C}{D}\right| ^{s/2} |\mathcal{T}|^2, \qquad \arg[\mathcal{T} \Delta^{-s} e^{-ik_{\rm h}r^*}]=\arg[\mathcal{O}e^{ik_{\rm h}r^*}]\qquad \text{when } r\rightarrow r_{\rm wall},\\
&&C=B \left\{-36 a^2 \omega ^2+36 a m \omega +[\lambda +(s+1) s-2]^2\right\}+\{ 2 [\lambda +(s+1) s]-1\} \left(96 a^2 \omega ^2-48 a m \omega \right)+144 \omega ^2 \left(M^2-a^2\right),\nonumber\\
&& \\
&&B = [\lambda +s (s + 1)]^2 + 4 m a \omega- 4 a^2 \omega^2, \quad D= 256 k_{\rm h}^2 (2 M r_+)^8 [k_{\rm h}^2 + \frac{4(M^2-a^2)}{(4 M r_+)^2}]^2 [k_{\rm h}^2 +
\frac{16(M^2-a^2)}{(4 M r_+)^2}].
\end{eqnarray}
\end{widetext}
$R_{\rm wall}=1$ would correspond to a perfectly reflective wall, but the actual reflectivity and phase change depend on the specific quantum gravity model for ECOs. In the rest of the paper, we will present solutions to these equations with different choices of the reflectivity and discuss the important properties of solutions, such as echo templates, time-delays and superradiant instability.
\begin{figure}
\includegraphics[width=0.2\textwidth]{3_1_1}
\includegraphics[width=0.2\textwidth]{3_1_2}
\caption{\label{31} Black Holes (BHs) and Exotic Compact Objects (ECOs) with an ingoing wavepacket. For BHs, angular momentum barrier reflects low frequency modes but higher frequencies cross the barrier and fall through the horizon. For ECOs with a wall standing the order of Planck length proper distance outside the (would-be) horizon, modes with intermediate frequencies can be trapped between the wall and the angular momentum barrier, slowly leaking out as repeating echoes.}
\end{figure}
\begin{figure*}
\raggedright
\includegraphics[width=\textwidth]{3}
\caption{\label{32} Echoes with different wall positions. Changing the positions of wall doesn't influence the shape of echoes a lot, but when putting wall closer to the would-be horizon and away from angular momentum barrier, the time-delay becomes bigger.}
\end{figure*}
\section{\label{sec3}Making Echoes}
Realistic predictions for Echo waveforms requires nonlinear simulations of the mergers of binary ECOs in full general relativity. As a consistent covariant formulation for dynamics of ECOs is yet non-existent, we have to rely on approximate methods to produce realistic echo templates. In order to do this using linear theory, we instead custom-design an ingoing wavepacket $\hat{h}_{\rm in}$ at infinity, so that the outgoing waveform matches the LIGO best-fit template $\hat{h}_{\rm LIGO}$ (without a wall).
The higher frequencies will go across the barrier and fall into BH, as shown in Fig \ref{31} (left), while the lower frequencies are reflected. We thus assume
\begin{eqnarray}
\hat{h}_{\rm LIGO}(\omega)= R_{\rm BH}(\omega) \hat{h}_{\rm in}(\omega),
\end{eqnarray}
where $R_{\rm BH}(\omega)$ is the reflectivity of the Kerr angular momentum barrier.
For an ECO, however, we have one more barrier near the would-be horizon as shown in Fig \ref{31} (right). Wavepackets with intermediate frequencies can now be trapped between two barriers and leak slowly every time when they hit the angular momentum barrier. Therefore, ECOs would have a similar ringdown waveform as classical BHs, but they are followed by delayed slowly decaying echoes.
\begin{eqnarray}\label{eq:hout}
\hat{h}_{\rm out}(\omega)= R_{\rm ECO}(\omega) \hat{h}_{\rm in}(\omega)=R_{\rm ECO}(\omega) \frac{\hat{h}_{\rm LIGO}(\omega)}{R_{\rm BH}(\omega)} f_{\text{cutoff}}(\omega), \nonumber\\
\end{eqnarray}
where $f_{\text{cutoff}}(\omega)$ is a low-pass filter introduced to suppress numerical noise at high frequencies, as
the reflectivity of the Kerr angular momentum barrier $R_{\rm BH}(\omega)$, in the denominator, vanishes at high frequencies. Luckily, high frequencies leak out quickly in the first echo, and have small effect on the subsequent echoes. Our choice of $f_{\text{cutoff}}$ does not affect the second and later echoes, but it changes the first echo slightly by cutting the high frequency noise:
\begin{eqnarray}
&\hat{h}_{\rm out, fin}= \hat{h}_{\rm out} f_{\text{cutoff}},\\
&f_{\text{cutoff}}=\exp\left[ -\frac{1}{2}\left(\frac{2 \pi f({\rm Hz})-299.495}{1347.73}\right)^{16}\right], \label{cutoff}
\end{eqnarray}
where $\omega = 2\pi f$.
With the equations and boundary conditions given in the last section, we can numerically solve for $R_{\rm BH}$ and $R_{\rm ECO}$ as a function of frequency. We use LIGO event GW150914 with $a=0.67$, $M=62~M_{\odot}$ and $z=0.09$. The mass is measured in the source frame and the finial mass used in our calculation is the mass in the detector frame $M_{\rm fin}=(1+z) M$. The waveform is dominated by the $(l,m) =(2,2)$ mode, which we shall focus on for the rest of the paper \footnote{Given the symmetries of Eqs. (\ref{r}-\ref{s}), we can easily extend the solution to $m=-2$ case using $R_{slm}[\omega]=R^*_{sl-m}[-\omega]$.}
The time dependence of the waveform can then be obtained by Fourier transforming $\hat{h}_{\rm out}(\omega)$, and is shown in Fig \ref{32}. We see that changing the position of the wall changes the time-delay between the echoes, but does not affect the individual echo waveforms significantly (as long as the wall is close the would-be horizon).
As we see in Fig. \ref{31}, in the geometric optics approximation, the time delay between echoes, $\Delta t_{\rm echo,geom}$ is given by the travel time from the angular momentum barrier to the wall and back \cite{abedi2016echoes}:
\begin{eqnarray}
&&\Delta t_{\rm echo,geom}=2r_*|^{r_{\rm barrier}}_{r_{\rm wall}}=2\int^{r_{\rm barrier}}_{r_{\rm wall}}dr \frac{r^2+a^2M^2}{r^2-2Mr+a^2M^2}\nonumber\\
&&=2r_{\rm barrier}-2r_{\rm wall}+2\frac{r_+^2+a^2M^2}{r_+-r_-}\ln\frac{r_{\rm barrier}-r_+}{r_{\rm wall}-r_+}\nonumber\\
&&-2\frac{r_-^2+a^2M^2}{r_+-r_-}\ln\frac{r_{\rm barrier}-r_+}{r_{\rm wall}-r_-}.
\end{eqnarray}
This can be well approximated by the following fitting function:
\begin{eqnarray}
&&\Delta t_{\rm echo,geom}
=2\frac{r_+^2+a^2M^2}{r_+-r_-}\ln\frac{M}{r_{\rm wall}-r_+}+M G(a), \label{t_geom}\\
&&G(a)\simeq \frac{0.335}{a^2-1}+4.77+7.42 (a^2-1)+4.69(a^2-1)^2,\nonumber\\ &&\\
&&r_{\rm wall}-r_{+}= \frac{\sqrt{1-a^2} d_{\rm wall}^2}{4M(1+\sqrt{1-a^2})},
\end{eqnarray}
where we find the fit of $G(a)$ for the angular momentum barrier of $l=m=2$ mode, while $d_{\rm wall}$ is the proper distance from the wall to the would-be horizon. The latter is expected to be comparable to Planck length for ECOs of quantum gravitational nature, but $\Delta t_{\rm echo}$ only depends on the exact value of $d_{\rm wall}$ logarithmically (see Fig. \ref{32}).
The echoes in both time and frequency domain for the LIGO event GW150914 are shown in Fig. \ref{51} and \ref{511} with perfect wall standing a Planck length proper distance outside the (would-be) horizon. Here, we show the Amplitude Spectral Density (ASD), which is the square root of the power spectral density. The latter is the average of the square of the fast Fourier transforms (FFTs) of the model. In the next section, we will study the structure of the echo in the frequency domain and present how superradiance affect the structure of echo.
\begin{figure*}
\includegraphics[]{5_1_2}
\caption{\label{51} Predicted echoes for LIGO event GW150914 in the time domain with different resolution, assuming a prefect wall at a Planck length proper distance outside the horizon}.
\end{figure*}
\begin{figure}
\includegraphics[width=0.43\textwidth]{5_1_1}
\caption{\label{511} Predicted echoes for LIGO event GW150914 in the frequency domains, assuming a prefect wall at a Planck length proper distance outside the horizon}.
\end{figure}
\section{\label{seca1}Superradiance}
Scattering off Kerr BH can lead to superradiance of modes with frequency $0<\omega<m \Omega_{\rm H}$, which can extract energy from a spinning background \cite{1972Natur.238..211P}. Adding a (partially) reflective wall near horizon could turn this amplification to an instability, since modes trapped between the wall and the angular momentum barrier can extract the spin energy repeatedly \cite{1978CMaPh..63..243F, Cardoso:2007az}. In this section, we study this effect for the echoes in frequency domains.
There is an odd looking spike in Fig. \ref{511} frequency domain around 183 Hz (see top panel in Fig. \ref{81} for a zoom-in). Indeed, this is exactly the threshold frequency for the superradiance. This is demonstaretd in the middle panel of Fig. \ref{81}, which shows the scattering amplification with the horizon, perfect wall and soft wall around that frequency. The vertical axis is the relative energy, extracted from around black hole by scattered gravitational waves. The blue dashed line shows superradiance slowly turning off with increasing the frequency, and we confirm that it ends exactly at frequency $f _{\rm max}=a m/[2 \pi(r_+^2+a^2)]=183~ \textrm{Hz}$, for $m=2$ as shown in the plot. In contrast, superradiance by soft wall (grey and thin curve) occurs at resonance peaks, corresponding to the ergoregion trapped mode (for more details, see Appendix \ref{a1} ). Since superradiance ends at 183Hz, the resonance peaks shift the direction, which is the reason we have an odd spike in the Fig. \ref{511} and \ref{81} top panel.
\begin{figure}
\includegraphics[width=0.43\textwidth]{8_1}
\includegraphics[width=0.43\textwidth]{8_2}
\includegraphics[width=0.43\textwidth]{8_3}
\caption{\label{81} Superradiance in frequency domain for GR Kerr BH and ECO with a wall.}
\end{figure}
The perfect wall (the red thick curve) in Fig. \ref{81} middle panel is a constant zero without any resonance peaks, since a perfect reflective wall kills superradiance, as all the energy that goes in, comes out eventually (see Appendix \ref{a1} for a subtlety in this argument). However, the odd spike structure remain in the amplitudes, as shown in Fig. \ref{81} bottom panel, where we change the vertical axis to real part of outgoing to ingoing wave at infinity. We still see the sign flip in resonance structure at 183 Hz.
In the next section, we study the echo templates resulting from solving the linearized Einstein equations, which improves the simplistic geometric picture in Fig. \ref{31}.
\section{\label{sec5}Minimal Echo templates}
Now that we have numerical predictions for echoes, we would like to provide simple fitting functions that could be used for quick visualization and data-fitting purposes. We call these fitting functions, templates. In order to find our templates, we define echoes in the time domain by the regions that surround the peaks of $|h(t)|$ and exceed a limit: $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1, -1.5$ or $-2$. $|h|_{\rm max, n}$ is the height of the $n^{\rm th}$ peak of $|h(t)|$, which we call the $n^{\rm th}$ echo. Then we fit the $n^{\rm th}$ echo to a complex gaussian
\begin{eqnarray}
h_n(t)&=& \exp [\Psi_n(t)+I \Phi_n(t)], \\ \label{gaussian}
\Psi_n(t)&=&a_0+a_1t+a_2 t^2, \\
\Phi_n(t)&=&b_0+b_1t,
\end{eqnarray}
where $a_0$, $a_1$, $a_2$, $b_0$ and $b_1$ are real numbers. This form is same as fitting the $n^{\rm th}$ echo to $ A \exp [ \frac{(t-t_0)^2}{2\sigma^2} ]$, where $A$ and $t_0$ are complex, while the width $\sigma$ is real.
\begin{figure}
\includegraphics[width=0.22\textwidth]{5_2_0_1}
\includegraphics[width=0.22\textwidth]{5_2_0_2}
\includegraphics[width=0.22\textwidth]{5_2_0_3}
\includegraphics[width=0.233\textwidth]{5_2_0_4}
\includegraphics[width=0.23\textwidth]{5_2_0_5}
\includegraphics[width=0.23\textwidth]{5_2_0_6}
\caption{\label{520} Best fit gaussians to the $2^{\rm nd}$, $5^{\rm th}$, $10^{\rm th}$ and $30^{\rm th}$ echoes within $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. We see that as high frequency modes leak out faster, later echoes decay in amplitude and become wider in time domain, and high frequency is cuted in the frequency domain.}
\end{figure}
As an example, Fig \ref{520} compares the numerical solutions and gaussian fits for the 2$^{nd}$, $10^{th}$, and $30^{th}$ echoes, with time origin shifted to center of each echo, and fitting the region with $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$.
Within this approximation, there are five real parameters for every echo that quantify its amplitude, width and center, both in time and frequency domain, as well as the overall phase at the center of the echo, as shown in Table \ref{t2}.
\begin{table*
\caption{\label{t2} Some physical quantities of a single echo defined by the five parameters from the gaussian echo template (Eq. \ref{gaussian})
}
\begin{ruledtabular}
\begin{tabular}{c|ccc}
&\textrm{width}&\textrm{center}&\textrm{peak amplitude}\\
\colrule
\textrm{time}&
$\sqrt{-1/ (2a_2})$&
$-a_1/2 a_2$&
$\exp[a_0-a_1^2/4 a_2]$\\
\colrule
\textrm{frequency} & $ \sqrt{-2 a_2}/(2 \pi)$ & $b_1/(2\pi)$ & $\exp[a_0-a_1^2/4 a_2-1/2 \log[2 \sqrt{a_2^2}]]$\\
\colrule
\colrule
\textrm{overall phase} & $ b_0-b_1 a_1/(2 a_2)$ \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table
\caption{\label{t3} Best fit gaussian echo template quantities (see Table \ref{t2} and Fig. \ref{52}) , for our minimal model of GW150914}
\begin{tabular}{|l|l|}
\cline{1-2}
\textrm{peak amplitude in time / strain}&
$2.91 \times 10^{-19}/n^{1.32}$\\
\colrule
\textrm{width in time / msec} & $4.29+ 0.883 n$ \\
\colrule
\textrm{correction to} $\Delta t_{\rm echo,geom}$ \textrm{/ msec}& $1.52+1.71/(1+n)$ \\
\colrule
\textrm{peak frequency / Hz} & $177+102/ n^{0.3}$ \\
\colrule
\textrm{Overall phase}&
$-7.26+27.1 n^{0.945}+22.6 n$\\
\cline{1-2}
\end{tabular}
\end{table}
Table \ref{t3} provides the best fit parameters of our echo templates for all echoes, based on the LIGO event GW150914 and averaging over the best fit functions with different echo domains $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1, -1.5$ or $-2$.
The best fits for each echo domain is also provided in Fig. \ref{52}. For correction to $\Delta t_{\rm echo, geom}$, we define time-delay as $\Delta t_{n}= t_{n} -t_{n-1} $. For all other plots, first echo is not included since it is very sensitive to the properties of the wall, as well as nonlinear effects from early stage of merger (see details in Sec. \ref{sec5}). The top three panels in Fig. \ref{52} show the time domain properties as a function of the echo number. Starting from the left, peak echo amplitudes in time are all well fit by decaying power laws(\cite{Correia:2018apm} argue that the decay of echoes at early stages is polynomial). Middle are the width of the echoes, becoming wider for later echoes in the time domains, as the high frequency modes leak out more quickly. The top right panel gives correction to $\Delta t_{\rm echo,geom}$ (\ref{t_geom}), while the bottom left panel shows the decay of the mean echo frequency. The bottom middle and right provide overall phase at $t_{\rm center}$ and the residuals of the best fit for the phase. We only show the residuals for the phase, as the numerical error for the phase is relatively big.
\begin{figure*}
\minipage{0.33\textwidth}
\includegraphics[width=1.151\linewidth]{5_2_1}
\includegraphics[width=1.154\linewidth]{5_2_4}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[width=1.18\linewidth]{5_2_2}
\includegraphics[width=1.175\linewidth]{5_2_5}
\endminipage\hfill
\minipage{0.32\textwidth}%
\includegraphics[width=1.18\linewidth]{5_2_3}
\includegraphics[width=1.18\linewidth]{5_2_6}
\endminipage
\caption{\label{52} Best fit gaussian template parameters (for $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1, -1.5$ or $-2$), in our minimal model of LIGO event GW150914, showing second and later echoes. The top three panels are in the time domains. Starting from left, peak amplitudes of echos in time are well fit by power laws. Middle panel is the width of the echoes, which become wider in time, as the high frequencies leak out more quickly. For the same reason, the peak frequency (bottom left) also decays with time. The top right panel gives corrections to $\Delta t_{\rm echo,geom}$ (Eq. \ref{t_geom}). Finally, the bottom middle and right provide the overall phase at $t_{\rm center}$ of each echo and the residuals of the best fit. This is the only plots we show the residuals since the numerical error for the phase is relatively big. }
\end{figure*}
To visualize the quality of the template to fit data, Fig. \ref{53} shows the ${\rm SNR}_{\rm temp}/{\rm SNR}_{\rm model}$, where ${\rm SNR}_{\rm model}$ is the predicted signal-to-noise ratio for our numerical solution of echoes (assuming white noise), while ${\rm SNR}_{\rm temp}$ is a reduced value, if we use our Gaussian approximations of Fig. \ref{520} (gray circles in Fig. \ref{53}). Using a second fit for how properties (i.e. width, center and amplitude) of $\Psi_{\rm n}(t)= \log |h_n(t)| $ depend on $n$ (Table \ref{t3}) further reduces ${\rm SNR}_{\rm temp}$ (red triangles in Fig. \ref{53}). We notice that the quality of Gaussian fit drops for later echoes, which could be either due to build-up of numerical error or systematic deviations from a single gaussian fit. The secondary fit for $\Psi_{\rm n}$ vs $n$ further reduces SNR as the width in time and time delay, shown as Fig. \ref{52}, do not have a simple behavior. However, the power law fit to the peak amplitude in time $\propto n^{-4/3}$ is surprisingly good. Also, as we discussed before, since the shapes of first few echoes are much more dependent on the initial conditions, it might be better to use independent Gaussians to fit them in data. Finding a reasonable fit for phase information $\Phi_{\rm n}$ vs $n$ proves even more challenging, as a small change in phase leads to a significant change in echo profiles. Fortunately, model-agnostic searches (e.g., \cite{Conklin:2017lwb}) based on cross-correlating different detectors can be done independent of the phase information.
\begin{figure}
\includegraphics[width=0.5\textwidth]{5_2_0}
\caption{\label{53} ${\rm SNR}_{\rm temp}$ compared to ${\rm SNR}_{\rm model}$, showing the quality of gaussian templates.}.
\end{figure}
\section{\label{sec4}Beyond the minimal model}
While our minimal model for echoes has only one free parameter (wall distance to the horizon, $d_{\rm wall}$) in addition to those of GR, the reality can be more complicated. Here, we explore the two main deviations expected from the minimal model due to nonlinear effects in GR and quantum gravity.
\subsection{Nonlinear Mergers Effects}
Our assumption of a custom-designed incoming wavepacket, as a placeholder for black hole binary merger, is almost certainly too naive to provide a realistic echo template, as it misses the nonlinear nature of the merger. While numerical simulations can now provide realistic waveforms for black hole mergers in GR, a covariant formulation of ECOs that could produce realistic echo waveforms is currently missing. However, we can get an idea about the extent of nonlinear corrections to linear results by noticing that the Kerr background for Teukolsky equation (\ref{eq:teuk}) is dynamical during the merger event, and thus the frequencies can be shifted by ${\cal O}(30\%)$, between the ingoing and outgoing waves at merger \footnote{Fort example, the best-fit for the dominant quasinormal mode frequency for GW150914 is 10-20\% offset from the linear theory predictions for the best-fit Kerr metric (Fig. 5 in \cite{TheLIGOScientific:2016src}). } . We shall explore the extent of this effect on echoes by introducing a blueshift parameter $s$, in the ingoing linear initial conditions:
\begin{eqnarray}\label{s_shift}
\hat{h}_{\rm LIGO, shifted} [f]=\hat{h}_{\rm LIGO} [ f/s ].
\end{eqnarray}
As shown in Fig. \ref{7}, redshifted (blueshifted) initial conditions give echoes which damp more slowly (quickly), since low frequencies leak more slowly through the angular momentum barrier. This also dramatically changes the amplitude of first few echoes. Blueshift parameter $s$ can be a free parameters for data fitting purposes.
\begin{figure*}
\includegraphics[width=0.3\linewidth]{7}
\includegraphics[width=0.3\linewidth]{7_3}
\includegraphics[width=0.3\linewidth]{7_4}
\caption{\label{7} Echoes predicted for GW150914, expected for redshifted (blueshifted) initial conditions with respect to our minimal model. We see that lower frequency initial conditions lead to lower amplitude, but more persistent, echoes as they cannot penetrate the angular momentum barrier efficiently.}
\end{figure*}
The effect is clearer if we compared SNR of echoes to first echo, as shown in Fig. \ref{8}. $\rm SNR^2_{\rm n}$ is $\rm SNR^2$ of our numerical solution of $\rm n^{th}$ echo and we trimmed a single echo with $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. We assume white gaussian noise ${\sigma_{\omega}}=1$ so that
\begin{eqnarray}\label{SNR}
\rm SNR^2_{\rm n}= \sum_{\omega} \frac{ | \hat{h}_{\rm n, \omega}| ^2}{{\sigma_{\omega}}^2} = \sum_{t} | h_{\rm n}| ^2. \\
\end{eqnarray}
Fig \ref{8} (right panel) shows that later echoes contain more (less) information in redshifted (blueshifted) templates, since they decay more slowly (quickly). The left panel also shows the relative SNR of 1st echo compared to the trimmed main event in our model. The fact that this number can change by more than 1.5 orders of magnitude suggests that the amplitude of 1st echo is very sensitive to the nonlinear merger physics and cannot be reliably predicted. \cite{Gupta:2018znn} simulates a binary black hole merger and finds the ratio of the energy falling into the black hole to the energy out is around 1:1, which can be used as a normalization of amplitude of echoes.
\begin{figure*}
\includegraphics[width=0.45\linewidth]{10_1}
\includegraphics[width=0.45\linewidth]{11_1}
\caption{\label{8} signal-to-noise ratios(SNR) and energy for blueshifted echoes compared with the first echo. We see that there is more (less) information in subsequent echoes for lower (higher) frequency initial conditions. Furthermore, the amplitude of first echo is hard to predict and can change by more than 1.5 orders of magnitude.We also list SNRs and energy for blueshifted first echoes compared with the event. Since we assume white noise to calculate the SNR in time domain, we trim the merger template at around 0.076 seconds before the peak (similar to the LIGO noise whitening for GW150914 template). }
\end{figure*}
Table \ref{t5} and Fig. \ref{72} compare the best fit echo parameters for different blueshift factors. We see in the left panels that the blueshifted initial condition ($s=1.2$) has a transient excess in amplitude that decays quickly and falls in line the minimal model. In contrast, the redshifted model ($s=0.8$) has a significantly smaller but more persistent amplitude. Surprisingly, the middle panels show that the redshifted echoes remain narrower in time. Even more puzzling is that the redshifted initial conditions have higher frequency echoes as shown in Fig. \ref{72} the bottom left panel. This is due to the fact that the echo peak frequency depends on the slope (and not the amplitude) of the spectral density $\hat{h}_{\rm out}(\omega)=R_{\rm ECO}(\omega) \frac{\hat{h}_{\rm LIGO}(\omega)}{R_{\rm BH}(\omega)} f_{\text{cutoff}}(\omega)$ from Eqn. \ref{eq:hout}, which involve several complicated components. As we see in the middle panel of Fig. (\ref{7}), this slope is not monotonic which leads to the counterintuitive behavior, even though the amplitude of the redshifted model is smaller compared to the blueshifted.
\begin{widetext}
\begin{table
\caption{\label{t5} Same as Table \ref{t3}, but contrasting with redshifted/blueshifted initial conditions, fitted within $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. }
\begin{tabular}{|l|l|l|l|}
\cline{1-4}
\textrm{blueshift factor $s$}&\textrm{ 0.8}&\textrm{$1$}&\textrm{$1.2$}\\
\colrule
\textrm{peak amplitude in time / strain}&$5.91 \times 10^{-20}/n^{1.14}$&
$2.92\times 10^{-19}/n^{1.33}$&
$5.31\times 10^{-19}/n^{1.54}$\\
\colrule
\textrm{width in time / msec} & $3.91+0.678 n$& $5.5+0.808 n$ & $9.48+0.711 n$ \\
\colrule
\textrm{correction to} $\Delta t_{\rm echo,geom} \textrm{/ msec}$ & $-47.8-57.0/(1+n)$& $15.4+1.64/(1+n)$& $76.2+60.4/(1+n)$ \\
\colrule
\textrm{peak frequency / Hz}& $227+95.2/ n^{0.3}$ & $175+104/ n^{0.3}$& $144+97.8/ n^{0.3}$ \\
\colrule
\textrm{Overall phase} & $-3.06+30.2n^{0.945}-25.9n$ & $-6.65+28.5n^{0.945}-23.8n$ & $-12.7+35.2n^{0.945}-29.4n$ \\
\cline{1-4}
\end{tabular}
\end{table}
\end{widetext}
\begin{figure*}
\minipage{0.33\textwidth}
\includegraphics[width=1.151\linewidth]{7_2_1}
\includegraphics[width=1.151\linewidth]{7_2_4}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[width=1.16\linewidth]{7_2_2}
\includegraphics[width=1.17\linewidth]{7_2_5}
\endminipage\hfill
\minipage{0.32\textwidth}%
\includegraphics[width=1.18\linewidth]{7_2_3}
\includegraphics[width=1.18\linewidth]{7_2_6}
\endminipage
\caption{\label{72} Same is Fig. (\ref{52}), but using the different blueshift factors $s$ (Eq. \ref{s_shift}) for echo initial conditions (fitted for $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$). We see that redshifted initial conditions yield weaker, but more persistent echoes (see text for details).
}
\end{figure*}
\subsection{Soft Wall}
Motivated by quantum models of black holes, the wall must at least partially absorb the energy incident on the wall \cite{abedi2016echoes}. For example, in fuzzball models \cite{Mathur:2012jk} high energy particles (with $\hbar \omega \gg kT_{\rm H}$, where $T_{\rm H}$ is the Hawking temperature) excite the fuzzball microstates and thus will be absorbed by the wall. On the other hand, particles with $\hbar\omega \leq kT_{\rm H} $ may be (at least partially) reflected (but see \cite{Guo:2017jmi} for recent counter-arguments). Ringdown phase of mergers of two BHs is in the intermediate range ($\sim 100$ Hz for GW150914). Therefore, a realistic quantum gravity model for the echoes is expected to involve a {\it soft} wall. For example, frequency of electromagnetic emissions from accretion into BHs is much higher, which is expected to be absorbed by the wall \cite{Broderick:2009ph, Broderick:2015tda}. However, possible loopholes that could lead to astrophysical observables from quantum effects have been exploited in \cite{Pen:2013qva, Afshordi:2015foa}.
A wall that absorbs high frequency modes will dramatically decrease the amplitude of the first echo, since these modes leak out quickly every time the wavepacket hits the angular momentum barrier. Therefore, the first echo contains most of the high frequency modes which, as shown in the top left panel in Fig \ref{43}, would be absorbed for a soft wall.
Of course, the actual frequency-dependent reflection of the wall depends on the specific quantum theory of black holes. We explore a phenomenological model for the wall with a Gaussian-like energy reflection rate
\begin{equation}
R_{\rm wall}(\omega) \simeq \exp\left[-\left( \alpha\frac{\omega}{T_{\rm H}} \right)^q\right],
\end{equation}
where $T_{\rm H}=\frac{r_+^2-a^2}{4 \pi r_+(r_+^2+a^2)}$ is the Hawking temperature for Kerr BH. While smooth $R_{\rm wall}$'s, such as gaussian or Boltzmann reflectivity ($q=2$ or $1$, respectively) may appear natural, they do tend to essentially wipe out the echoes, unless $\alpha \ll 1$, which is inconsistent with the tentative echoes found in \cite{abedi2016echoes}. In contrast, a sharper function with, e.g., $q=12$ then can damp the first echo, but not significantly influence later echoes, as shown in Fig \ref{42} \footnote{Fig \ref{42} also shows that if the wall absorbs too much, the late echoes will stop decaying. This is due to superradiant instability which we shall discuss in the next section.}. We can also compare these reflectivity functions with that of the angular momentum barrier of the Kerr BH, for the same spin and mass, as shown in Fig \ref{44}, which provides another motivation for sharper $R_{\rm wall}$'s.
\begin{figure}
\includegraphics[width=0.5\textwidth]{4_4}
\caption{\label{44} Comparison of soft wall reflectivity coefficients that we use, with that of the Kerr angular momentum barrier \cite{Nakano:2017fvh}. The thin and dashed lines are the two reflectivity rates used in Fig \ref{42}.}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth]{4_2}
\caption{\label{42} Echoes for GW15014, for soft vs. perfect walls. The top (gray) curve assumes a perfect wall/mirror, while the lower curves show soft walls with different energy reflectivity coefficients.
}
\end{figure}
\begin{widetext}
\begin{table
\caption{\label{t4} Same as Table \ref{t3}, but contrasting perfect ($R_{\rm wall}=1$) and soft ($R_{\rm wall}= \exp[-(0.055 \frac{\omega}{T_{\rm H}})^{12}]$) walls, fitted within $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$. }
\begin{tabular}{|l|l|l|}
\cline{1-3}
\textrm{wall type}&\textrm{perfect}&\textrm{soft}\\
\colrule
\textrm{peak amplitude in time / strain}&
$2.78 \times10^{-19}/n^{1.31}$&
$2.33 \times 10^{-19}/n^{1.36}$\\
\colrule
\textrm{width in time / msec} & $5.5+0.808 n$ & $8.17+0.659 n$ \\
\colrule
\textrm{correction to} $\Delta t_{\rm echo,geom}$ \textrm{msec} & $15.4+1.64/(1+n)$& $14.3+1.52/(1+n)$ \\
\colrule
\textrm{peak frequency / Hz} & $175+104/ n^{0.3}$& $177+96.8/ n^{0.3}$ \\
\colrule
\textrm{Overall phase} & $-6.65+28.5 n^{0.945} -23.8 n$& $-5.2+26.8 n^{0.945} -22.6 n$ \\
\cline{1-3}
\end{tabular}
\end{table}
\end{widetext}
\begin{figure*}
\minipage{0.33\textwidth}
\includegraphics[width=1.151\linewidth]{4_3_1}
\includegraphics[width=1.151\linewidth]{4_3_4}
\endminipage\hfill
\minipage{0.32\textwidth}
\includegraphics[width=1.16\linewidth]{4_3_2}
\includegraphics[width=1.17\linewidth]{4_3_5}
\endminipage\hfill
\minipage{0.32\textwidth}%
\includegraphics[width=1.18\linewidth]{4_3_3}
\includegraphics[width=1.18\linewidth]{4_3_6}
\endminipage
\caption{\label{43} Same is Fig. (\ref{52}), comparing walls with different energy reflectivity coefficients (fitted for $\ln\left[|h(t)|/|h|_{\rm max, n}\right] > -1.5$). We see that echo amplitudes and peak frequencies decay more quickly for softer walls (see text for details).
}
\end{figure*}
Table \ref{t4} and Fig. \ref{43} compare the template for the perfect and soft walls, similar to Figs. (\ref{52}) and (\ref{72}). We see in the top left panel that, due to absorption of high frequency modes, the power law fit to the amplitudes could be extended to first echo for the soft walls. More generally, echoes decay faster for a softer wall.
As echoes for a wall with $R_{\rm wall}= \exp[- (0.06\frac{ \omega}{T_{\rm H}})^8]$ decay too fast, we only focus on the $R_{\rm wall}= \exp[-(0.055 \frac{\omega}{T_{\rm H}})^{12}]$ case in subsequent panels of Fig. \ref{43}, and provide numerical fits for echo properties in Table \ref{t4}. With this choice, the evolution of echo properties is similar to those in Figs. (\ref{52}) and (\ref{72}), with the notable difference that peak frequency decays more rapidly as the soft wall absorbs high frequencies.
\section{\label{sec7}Conclusions}
We have provided realistic templates for echoes of BH mergers by numerically solving the linearized Einstein equation (or Teukolsky equation) in Kerr spacetime with boundary conditions at a Planck length proper distance outside the (would-be) event horizon. We obtain analytic approximations for the echo waveforms and time-delays, and explore their dependence on the softness of the wall (or frequency-dependence of the reflection rate), as well as nonlinear effects during merger event. These analytic templates should be useful in echo searches in current and future gravitational wave data. Finally, we studied the occurrence of superradiant instability and showed that it has negligible effect, for the first few dozen echoes of in typical BH mergers such as GW150914.
Let us close with some open questions and future directions:
\begin{itemize}
\item The strain is dominated by mode $l=2, m=\pm{2}$. We only show mode $m=2$ here and solution of $m=-2$ can easily be found by $R_{slm}[\omega]=R^*_{sl-m}[-\omega]$. More realistic templates should combine all other modes by appropriate weight.
\item We cannot provide a reliable waveform for the first echo as it is too sensitive to the {\it ad hoc} cutoff function (\ref{cutoff}) that we use to set up our initial conditions. This highlights the need for a covariant numerical implementation of ECOs within a dynamical spacetime, which could provide realistic nonlinear initial conditions for echoes.
\item Another big uncertainty is the expected softness of the wall. While this is ultimately a question for the quantum models of black holes, it highlights the need for a covariant and causal description of the wall dynamics. It might be possible to describe this dynamics in terms of the properties of a surface (2+1d) fluid and Israel junction conditions (e.g., see \cite{Saravani:2012is}).
\item The computation of the echo phase beyond $\sim 20$ echoes is limited by numerical precision and frequency resolution. This can be improved in the future, by either brute force or novel numerical/analytic methods.
\end{itemize}
\nocite{*}
\begin{acknowledgements}
We thank Vitor Cardoso, Rafael Sorkin, Huan Yang, Aaron Zimmerman, Bob Holdom, Randy S. Conklin, Ren Jing, Ofek Birnholtz, William East and Connor Adair for helpful comments and discussions. We also thank all the participants in our weekly group meetings for their patience during our discussions. This work was supported by the University of Waterloo, Natural Sciences and Engineering Research Council of Canada (NSERC), and the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Innovation.
\end{acknowledgements}
|
{
"timestamp": "2018-06-21T02:12:23",
"yymm": "1803",
"arxiv_id": "1803.02845",
"language": "en",
"url": "https://arxiv.org/abs/1803.02845"
}
|
\section{Introduction}
Why the tenuous solar outer atmosphere, or corona, is much hotter than the underlying layers remains one of the greatest challenges for solar theory and modeling. Detailed diagnostics of the coronal thermal structure come from extreme ultraviolet (EUV) emission. This EUV emission is produced by heavy ions in various ionization states and, depends on the amount of these ions and on plasma temperature and density.
Since 2010, the Atmospheric Imaging Assembly (AIA) aboard the \textit{Solar Dynamics Observatory (SDO)} has dazzled scientists and the public alike with its high-resolution, full-Sun images of myriad coronal loops at the Sun in several extreme ultraviolet (EUV) bandpasses. Yet it has long been suspected that AIA reveals only a fraction of the multitude of coronal loops that may actually be present. The observability of these coronal loops depends on their temperature and a sufficient abundance of ionized elements, particularly iron. Should some mechanism drain the coronal loops of their highly ionized iron, the loops might become invisible in EUV.
In fact, any non-uniformity of the elemental distribution in space or variability in time affects thermal diagnostics of the corona \citep{2014ApJ...786L...2W, 2015ApJ...802L...2C, 2017ApJ...844...52D}.
Here, we propose a mechanism by which solar coronal loops can be depleted of heavy ions via the migration of these ions to one end of the loop in the transition region between the corona and chromosphere. Basic plasma physics arguments show that electron drag forces due to electric currents can sweep ions to and trap them at one end of the loop. This mechanism offers an explanation why coronal loops---often highly twisted current-carrying loops---suddenly seem to appear ``out of nowhere'' at flaring times, and implies that the beautiful and rich images returned by AIA may actually reveal only a subset of the hot coronal loops present at the Sun. In particular, based on analysis of many time frames of many ARs observed with AIA, \citet{2014ApJ...797...50A}
noted that: ``…not all twisted and current-carrying loop structures are illuminated before the flare and thus part of the free energy is invisible before the flare.''
Our mechanism predicts ionized chemical element concentrations in some areas of the solar atmosphere, where the electric current is directed upward. We detected these areas observationally, by comparing the electric current density at the photosphere with the EUV brightness in a few active regions. We found a significant excess in EUV brightness in the areas with positive current density rather than negative one\footnote{This trend has also been recently noted by \cite{2017ApJ...847..143H, 2017ApJ...847..113H} in the \citet{janvier_et_al_2014} and \citet{musset_et_al_2015} data set.}. In this way, we found important evidence in favor of substantial concentrations of heavy ions in current-carrying magnetic flux tubes. For short, we call such areas of the heavy ion concentration the ``ion traps'' that hold enhanced ion levels until the trap is disrupted by a flare, whether large or small.
\section{Ion Traps}
The microscopic picture of the electric current in a multicomponent plasma is interesting and far from trivial at any scale---from a lab circuit to stellar atmospheres. In the steady conditions the conventional Ohm's law applies. This implies that the mean velocities of various plasma components are determined by balancing the electric force (that tends to infinitely accelerate charged particles) by the counter-acting drag force. The drag force originates from collisions between the various plasma components (protons, electrons, and multiply charged ions). In the simplest case of the hydrogen plasma, the behavior of the plasma components is relatively simple: the electric current is mainly carried by electrons, which move in the direction opposite to the electric field vector with a velocity larger than the proton velocity by a factor of the proton-to-electron mass ratio; the protons move slowly along the electric field. This picture is valid for any singly ionized ion ($Z=1$). However, the situation changes drastically \citep{Gurevich_1961, Holman_1995, FT_2013} if more highly ionized ions ($Z>1$) are present in addition to the singly ionized ones, which is the case in the hot solar corona. While the electric force ($F=eZE$) does drive those ions in the direction of the electric field vector, the counter-acting dynamic drag force from collisions with the moving electrons, proportional to $Z^2$, is stronger provided that $Z>1$; see the details of the derivation in section 2.1. This implies that the more highly charged ions move in the same direction as the electrons, opposite the electric field direction as shown in the loop-top inset in Figure~\ref{f_CDIS_cartoon}. This, apparently anomalous behavior of the multiply ionized ions turns out to produce ion concentrations at certain areas of the solar atmosphere, which we call ``ion traps.''
Let us track the destiny of these high-$Z$ ions as they move along the current-carrying magnetic flux tube towards the righthand footpoint (see Figure~\ref{f_CDIS_cartoon}). As soon as a given ion enters the transition region between the hot corona and relatively colder chromosphere, it begins to accumulate electrons due to the cooler environment at a ``recombination'' region, indicated by the circled R in Figure~\ref{f_CDIS_cartoon}. When the ion reaches the ionization state of $Z=1$, the force balance reverts and the singly ionized atom is drawn upward toward the hot corona, where it soon becomes doubly ionized, and starts to move down again.
This means that the right-hand foot point of our idealized current-carrying coronal flux tube will start accumulating heavy elements with time as soon as the loop emerges from the photosphere, and thus it will become an ‘ion trap’ at a height range where heavy ions oscillate between ionization stages $Z=1$ and $Z=2$. This enhancement is ultimately counterbalanced by the depletion of these heavy ions from the left foot point and their redistribution over the coronal portion of the loop; which requires, according to our estimates, from half an hour to a few days, depending on the loop length, density, and electric current; see below.
In addition, the ion trap is supplied by the low-FIP ions from below. Indeed, the chromosphere is cool, $T \sim 5-10$~kK; the low-FIP elements are mainly singly-ionized there, while the high-FIP elements are mainly neutral. Thus, there is a substantial reservoir of the singly-ionized low-FIP ions to take part in the charge flow forming the electric current. By drifting up within this current flow, these ions are filtrated up to the coronal portion of the electric circuit as compared with the high-FIP atoms, which are mainly neutral and do not take part in the electric current flow. In contrast, no heavy ion supply is available at the left foot point of our flux tube because the singly ionized ions move down there and practically no supply of more highly ionized ions, which would move upward, is available in the cool chromosphere. This implies that the right footpoint (ion trap) is enriched with low-FIP ions. Any mechanism capable of sporadically lifting the ion trap plasma up (due to turbulence-enhanced diffusion, plasma flows, or evaporation episodes) will accordingly enrich the coronal portion of the loop with low-FIP ions leading to the FIP effect.
\begin{figure}[!t]
\epsscale{1.035}
\plotone{cartoon_v2c.eps}
\caption{A schematic representation of a current-carrying (twisted) loop in the solar atmosphere. The inset at the top of the loop shows a few plasma components (electrons, $Z=-1$, protons, $Z= 1$, and He III ions, $Z=2$) and the directions in which they move under the action of an external electric field $\vE$ (adopted here to be directed from the right to the left end of the loop, called footpoint) driving the electric current. The convention is that the vertical component of electric current is positive if it is directed upward and negative if it is directed downward. The transition region between the hot corona and much cooler chromosphere is marked by a circled R, which indicates the region where the more highly charged ions recombine into lower ionization states, while moving down through progressively cooler plasma. \label{f_CDIS_cartoon}}
\end{figure}
This idealized picture, although rather simplistic, is relevant to the real corona. Indeed, the current-carrying (twisted) loops are both implied and observed in the corona: some coronal electric current must necessarily be present to support non-potential coronal magnetic field required to drive the solar activity such as flares and eruptions. In addition, twisted coronal structures, presumably, loops, are often observed in EUV. Another idealization of our cartoon is that only the steady-state electric current is explicitly taken into the picture.
In reality, there are episodes of energy release leading to random or regular motions of the coronal plasma and chromospheric evaporation, which tends to fill the coronal part of the magnetic loop with heated chromospheric plasma and, thus, reduce or entirely smooth out any non-uniformity in the ion distribution along the loop. It is the plasma mixing / evaporation episodes that are likely responsible for distributing this ion trap plasma enriched with the low-FIP ions over a larger portion of the corona. Eventually, the net heavy ion distribution over the flux tube will be set up by balancing between these (and, likely, other) counter-acting effects. Taking into account the probable complicated coronal dynamics, it would be unrealistic to expect that all heavy ions drained down to the trap to form a purely hydrogen coronal loop; however, an appreciable non-uniformity of the spatial distribution of the heavy ions along the coronal flux tube and a noticeable asymmetry in distribution of heavy ions between the two opposite footpoints, with respectively upward and downward electric current, are expected to exist.
\subsection{Microscopic picture of the multi-component electric current}
Here we address the composition of the electric current in terms of its microscopic components---electrons, protons, and heavier ions---and how these components move relative to each other within the plasma. The result will depend on both the ion charge and number density of the given plasma component \citep{FT_2013}.
For simplification we consider the case of a three-component plasma consisting of electrons ($e$), protons ($p$) (and, perhaps, other singly-ionized atoms), and one more type of heavier ion ($i$) with a charge $Z|e|$ ($Z>1$), mass $m_i$, and number density $n_i$ so that $n_e=n_p+Zn_i$. Next, we assume that a relatively weak (compared with the Dreicer field, $E_{De}$) constant and uniform DC electric field $\vE$ is applied to this multi-component plasma, so a constant electric current develops under conditions when the conventional Ohm's law applies. To find the mean velocities of the plasma components we compute the balance between the electric force acting on the given plasma component and dynamic friction forces acting on this component from all other plasma species.
The dynamic frictional force $\mbox{\boldmath{$\cal F$}}_a$ produced by a plasma component $a$ on a test particle can be computed using the averaged momentum exchange between that plasma component and the test particle \citep{Trubnikov_1965}:
\begin{equation}\label{Eq_fric_F_def}
\mbox{\boldmath{$\cal F$}}_a(\vU)=-\frac{Z^2Q}{\mu_a}\int\frac{\vU-\vV}{|\vU-\vV|^3}f_a(\vV)d^{\,3}V,
\end{equation}
where $\vU$ is the velocity of the test particle, $\mu_a=Mm_a/(M+m_a)$ is the reduced mass defined by the test particle mass $M$ and $a$-component particle mass $m_a$, $f_a(\vV)$ is the distribution function over velocity $V$ of the component $a$, and $Q=4{\pi}e^4\ln\Lambda_C$, where $\ln\Lambda_C$ is the Coulomb logarithm. To find the mean velocity of each plasma component, we compute the frictional force acting on a `mean' test particle of the `$b$' component. This is done by averaging the force~(\ref{Eq_fric_F_def}) over the distribution function $f_b(\vU)$, as $\overline{\mbox{\boldmath{$\cal F$}}}_{ab}=\int\mbox{\boldmath{$\cal F$}}_a(\vU)f_b(\vU)d^{\,3}U$.
It is convenient to perform these integrations denoting $\vU=\vu_T+\vu$ and $\vV=\vv_T+ \vv$, where $\vu_T$ and $\vv_T$ are the thermal velocities of the corresponding plasma components, while $\vu$ and $\vv$ are velocities of the components due to the external electric field. For the case of a sub-Dreicer field, $E/E_{De}\ll1$, the thermal electron velocity is larger than the electron drift velocity or any ion velocity; however, the drift ion velocity is not necessarily smaller than the thermal proton velocity. Accordingly, for the collisions between electrons and ions, the thermal electron velocity always dominates the denominator of Equation~(\ref{Eq_fric_F_def}), but it does not contribute to the numerators at all because of the isotropy of the thermal distribution, so the numerators are solely determined by the mean velocities $\vu$ and $\vv$.
Now we can compute the balance of forces for the three-component plasma. The balance of forces acting on the electron component is
\begin{equation}\label{G_1b}
-|e|\vE-\frac{Qn_p}{m_ev_{Te}^3}(\vv_e-\vv_p)-\frac{Z^2Qn_i}{m_ev_{Te}^3}(\vv_e-\vv_i)=0
\end{equation}
where we neglected $v_{Tp}$ and $v_{Ti}$ in the denominators, because $v_{Te}{\gg}v_{Tp}$, $v_{Te}{\gg}v_{Ti}$ and adopted $\mu_a{\approx}m_e$ because $m_p{\gg}m_e$.
Unlike this case, for the collisions between the heavy ions and protons, the denominator may or may not be dominated by the thermal proton velocity depending on parameters. If $|\vv_i|{\ll}v_{Tp}$, then we have for protons
\begin{equation}\label{G_1protons}
|e|\vE-\frac{Qn_e}{m_ev_{Te}^3}(\vv_p-\vv_e)-\frac{Z^2Qn_i}{m_{ip}v_{Tp}^3}(\vv_p-\vv_i)=0,
\end{equation}
where we adopted for simplicity $v_{Tp}{\gg}v_{Ti}$;
with the same accuracy we use below the proton mass $m_p$ for the reduced ion-proton masses $m_{ip}$.
Then, adding up Eqns~(\ref{G_1b}) and (\ref{G_1protons}) and discarding a small term, we eliminate the electric field:
\begin{equation}\label{G_1sum}
-\frac{QZ(Z-1)n_i}{m_ev_{Te}^3}\vv_e-\frac{Z^2Qn_i}{m_{ip}v_{Tp}^3}\vv_p+\frac{Z^2Qn_i}{m_{ip}v_{Tp}^3}\vv_i=0
\end{equation}
and then, using the momentum conservation law in the reference frame, where the plasma does not move as a whole, we eliminate the electron velocity $\vv_e=-(n_pm_p\vv_p+n_im_i\vv_i)/(n_em_e)$.
Substitution of this expression in Eq~(\ref{G_1sum}) yields:
\begin{equation}\label{G_1p4}
\left[1+\frac{m_{ip}v_{Tp}^3}{m_ev_{Te}^3}\frac{Z-1}{Z}\frac{n_im_i}{m_en_e}\right]\vv_i= $$$$ \left[1-\frac{m_{ip}v_{Tp}^3}{m_ev_{Te}^3}\frac{Z-1}{Z}\frac{n_pm_p}{m_en_e}\right]\vv_p.
\end{equation}
Next, we make an assumption about the elemental abundances $n_p$ and $n_i$. Here we consider a common (coronal or chromospheric) plasma, where hydrogen is the dominant element, thus $n_p{\gg}n_i$. In this case we can safely neglect the first term (``1'') compared with the second one in the brackets in the second line in Eq~(\ref{G_1p4}), which yields:
\begin{equation}\label{G_1p4a}
\left[1+\frac{m_{ip}v_{Tp}^3}{m_ev_{Te}^3}\frac{Z-1}{Z}\frac{n_im_i}{m_en_e}\right]\vv_i=-\frac{m_{ip}v_{Tp}^3}{m_ev_{Te}^3}\frac{Z-1}{Z}\frac{n_pm_p}{m_en_e}\vv_p.
\end{equation}
This equality is remarkable as it cleanly demonstrates
that, in the presence of external electric field, \textit{positive} admixture ions move in the direction \textit{opposite} to proton motion. Given that the protons move along the external electric field, the \textit{positive} admixture ions (with $Z>1$) move oppositely to the electric field, in the \textit{same} direction as the \textit{negative} electrons. This apparently anomalous behavior \citep{Gurevich_1961, Furth_Rutherford_1972, Holman_1995} is driven by the dynamic frictional force produced by the relatively fast electron component on the ions with $Z>1$. This dynamic frictional force is proportional to $Z^2$, which is stronger than the electric force (proportional to $Z$). Eqn~(\ref{G_1p4a}) also tells us that a small fraction of ions can achieve a relatively large mean velocity if the second term in the lefthand brackets is much less than unity; for more abundant ions like helium and oxygen the second term may be larger than or comparable to 1, so their net velocities are lower than that of minor ions.
This holds, as has been mentioned, for a reasonably weak electric field that yields a relatively slow ion drift velocity, $|\vv_i|{\ll}v_{Tp}$. If this condition does not hold, the cross-section of the proton-ion collisions goes down as ${\propto}|\vv_i|^{-2}$, so the third terms drops out from Eq.~(\ref{G_1p4a}). This means that the proton-ion collisions become inefficient, so the ions can be picked up by the flow of electrons and accelerated almost to the electron drift velocity \citep{Gurevich_1961}, $v_i{\approx}(Z-1)v_e/Z$; these runaway ions start to appear when the electric field exceeds a critical field \citep{Gurevich_1961, 2015PhPl...22e2122E} $E_{ci}$, which is yet much smaller than the electron Dreicer field $E_{De}$; $E_{ci}{\sim}(m_e/m_p)^{1/3}E_{De}$. The result of this analytical consideration was fully confirmed by numerical simulations \citep{2015PhPl...22e2122E}. Such ``runaway'' ions are routinely detected in laboratory experimens \citep{2002PhRvL..89w5002H, 2013PhRvL.111a5006F, 2013NucFu..53h3017Z, 2015PhPl...22b0702E}.
\subsection{Ion Trap Formation Time}
Assuming the Spitzer conductivity $\sigma_{\rm S}$ we find the electric field that corresponds to the given electric current density:
\begin{equation}\label{Eq_E_vs_j}
E=\frac{j}{\sigma_{\rm S}} =
3.88\cdot 10^{-5}\times $$$$\left(\frac{j}{160~{\rm mA~m^{-2}}}\right)\times\left(\frac{T}{10^6~{\rm K}}\right)^{-3/2} \left(\frac{\ln\Lambda_C}{20}\right)~{\rm V~m^{-1}},
\end{equation}
which is to be compared with the electron Dreicer field
\begin{equation}
\label{Eq_Accel_Dreicer_field}
E_{De}\approx6\cdot10^{-3}\left(\frac{n_e}{10^{9}~{\rm cm}^{-3}}\right)\left(\frac{T}{10^6~{\rm K}}\right)^{-1}\left(\frac{\ln\Lambda_C}{20}\right)~~{\rm V/m}
\end{equation}
Let us estimate an expected time needed for the ion trap to develop. The electric current density is $j=|e|n_ev_e$; thus, the drift electron velocity is $v_e = j/(|e|n_e)$, which is convenient to write in the form:
\begin{equation}\label{Eq_v_driftVSj}
v_e= 10^5 \left(\frac{10^9{\rm cm^{-3}}}{n_e}\right) \left(\frac{j}{160~{\rm mA~m^{-2}}}\right)~{\rm cm~s^{-1}}.
\end{equation}
The time of flight $\tau_e=L/v_e$ along a flux tube with a length $L$ is
\begin{equation}\label{Eq_tau_e_drift}
\tau_e= 10^4 \left(\frac{L}{10^9{\rm cm}}\right) \left(\frac{n_e}{10^9{\rm cm^{-3}}}\right) \left(\frac{160~{\rm mA~m^{-2}}}{j}\right)~{\rm s}.
\end{equation}
Thus, a thermal electron, as well as the runaway heavy ions having $v_i \approx (Z-1)v_e/Z$ (provided $E> E_{ci}$), drifts along the length of a flux tube with $L=10^9$~cm filled with a plasma with density $n_e\sim10^8{\rm cm^{-3}}$ and electric current density $j\sim 200~{\rm mA~m^{-2}}$ in less than 20 minutes. If the condition $E> E_{ci}$ does not hold, the ion drift velocity is roughly an order of magnitude smaller than the electron drift velocity {(but the multiply charged ions still drift in the same direction as electrons)}; thus, the ion travel time along the loop is accordingly longer. Given the plausible range of the coronal loop parameters, the time needed to establish a ion trap ranges from less than one hour up to a few days.
{In principle, this drift time has to be compared with the ion recombination time needed for a multiply charged ion to recombine to the singly ionized state. The recombination rate $\Gamma_R$ is $\Gamma_R = \alpha_R n_e$, where $\alpha_R$ is the (total) recombination rate coefficient
and $n_e$ is the electron number density.
Although the recombination rate coefficient depends on the temperature and is different for various ions, we can use a typical range of $\alpha_R=10^{-12}-10^{-11}$~cm$^3$s$^{-1}$ \citep{1981ApJ...249..399W, 1997ApJ...479..497N} and $n_e=10^{11}-10^{12}$~cm$^{-3}$ in the TR and upper chromosphere to estimate the typical recombination time $\tau_r=1/\Gamma_R$ to be of the order a second; thus, the recombination is a much faster process than the ion drift. Recombination of multiply charged ions to the singly ionized stage will need the corresponding multiple number of the recombination steps, whose total time could sum up to a few dozens of seconds, which is still much shorter than the drift time. Thus, the trap formation time is specified by the ion drift time rather than the recombination time.}
\section{Observational Manifestations of the Ion Traps}
\label{S_Obs_Ion_Traps}
Observationally, the detection of the heavy ion distribution non-uniformity associated with the direction of the electric current vector is challenging for several reasons. The first of them is the already mentioned dynamics of the coronal plasma: for a given flux tube the non-uniformity of the ion distribution is expected to be minimal following transient energy depositions, when the plasma evaporated from the chromosphere and has filled the coronal part of the loop in response to an energy release episode. The current-driven concentration of the heavy ions in one footpoint discussed here will then redevelop during a relaxation phase that also includes plasma cooling. The longer the relaxation lasts, the stronger the concentration of the heavy ions is expected in one of the footpoints, which is favorable for detecting the effect. However, if the relaxation lasts too long, the plasma gets too cold and too tenuous to produce significant EUV emission. Thus, even though this is the state in which we expect the most concentrated ion population at the TR, it is also the state in which it is difficult to observe in EUV.
The ions expected in the traps are mostly singly and doubly ionized, and thus do not emit radiation in the EUV wavelength range covered by most AIA channels \citep{lemen_et_al_2012} with the exception of He~II. Their presence could be detectable in the 304 Å channel, dominated by the strong He II 303.7 Å line, as well as in the 1600 Å channel, which includes a few low-ionization species, as well as the recombination continuum from singly ionized Mg and Si to the neutral species. Because of the optically thin character of the EUV emission, it would be very difficult to isolate a single twisted flux tube and study the distribution of heavy ions along this given loop without a considerable bias. Here, to evaluate the significance of this effect, we analyze the EUV brightness from a relatively large field-of-view (FOV) assuming that even if the non-uniformity of the heavy ion distribution is present in only a small subset of the coronal twisted loops, this non-uniformity will yield a measurable asymmetry in the EUV brightness. Here we take advantage of the fact that a narrow transition region (between the chromosphere and corona) emits strongly in some UV / EUV lines; thus, the UV / EUV brightness of the transition region is expected to be stronger at the areas corresponding to upward electric current.
Ideally, we would check this expectation by comparing distributions of the vertical component of the electric current density and of the EUV brightness at the transition region level. In practice, the electric current density can be computed only at the photospheric level, where vector magnetic measurements are available \citep{scherrer_et_al_2012} which is roughly 2~Mm below the transition region. Nevertheless, for a solar active region observed from approximatively above, the direction of the electric current vector at the photospheric level will be a good proxy for that at the transition region height. Although the EUV emission from the transition region alone cannot be isolated in line-of-sight integrated images, the transition region contribution is significant for the cooler channels (such as 1600~\AA, 304~\AA, or 171~\AA).
Therefore, our strategy of searching for the ion traps is (1) to compute the vertical component of the electric current density for each photospheric pixel of an active region observed near the disk center; (2) separate all the pixels into two groups with either positive or negative current; and (3) compare the EUV brightness corresponding to positive current with that corresponding to negative current.
For the analysis we selected active regions located near the disk center before and during X-class flares: AR11156 on February 15 2011, AR11166 on March 9 2011, AR11283 on September 6 2011 and AR11520 on July 12 2012.
\subsection{Computation of the electric current density at the photosphere. }
\label{S_j_compute}
We derive the magnetic field and electric current density maps from the ``level 1p IQUV'' data of HMI on \textit{SDO} \citep{scherrer_et_al_2012, schou_et_al_2012}. This instrument provides images of the entire Sun in six narrow spectral bands in a single iron line (FeI 617.33 nm) and in four different states of polarization. This set of $6\times4=24$ images provides spectropolarimetric data to enable the calculation of the full vector magnetic field at the altitude of formation of the spectral line, which is at the photosphere. The ``level 1p IQUV'' HMI data and, thus, the magnetic field vector maps are available every 12 minutes.
The electric current density can be calculated from the magnetic field using Ampere's law; but since only the magnetic field vector distribution in a plane parallel to the surface of the Sun, rather than in a 3D volume, is measured, then only the vertical component of the electric current density $j_z$ can be calculated:
\begin{equation}
j_z=\frac{c}{4\pi}\left(\frac{{\partial}B_y}{{\partial}x}-\frac{{\partial}B_x}{{\partial}y}\right)
\label{eqn_ampere}
\end{equation}
Where $z$ is the spatial coordinate perpendicular to the solar surface and $c$ is the speed of light.
The calculation of the magnetic field and vertical density \citep[described in][]{janvier_et_al_2014,musset_et_al_2015,bommier_2016} employs here the inversion code UNNOFIT \citep{bommier_et_al_2007,bommier_2016} based on the Milne-Eddington model of the solar atmosphere. The advantage of this code for our purpose is its specificity to take into account a magnetic filling factor as a free parameter of the Levenberg-Marquardt algorithm \citep{bommier_et_al_2007}. As a result, the method better determines the field inclination, which is of great importance when studying the current density, determined from the horizontal components of the magnetic field, as described by equation~\ref{eqn_ampere}.
To quantify the level above which a current density is
significantly non-zero, we estimated the standard deviation in a region of the current density map located outside the active region and where no clear current patterns appear. The standard deviation calculated on several maps for the active region studied in this paper was found
to be $\sigma=15\pm2$~mA$/$m$^2$. Therefore, locations where the calculated current density is smaller than 15~mA$/$m$^2$ are considered current-free.
\subsection{Combination of electric current density maps and EUV images.}
The AIA instrument \citep{lemen_et_al_2012} on \textit{SDO} provides high resolution images of the entire Sun in 10 narrow spectral bands corresponding to 8 nominal ion lines, mainly iron and helium, and 2 continuum bands. Each band is sensitive to one or more different plasma temperatures, see Figure~13 in \citet{lemen_et_al_2012}, and therefore are emitted from characteristic heights in the solar chromosphere, transition region and corona; see Table~1 in \citet{lemen_et_al_2012}. In addition, in the loop transition region and chromosphere the emission in some AIA bands has strong contributions from cooler components. For example, the 1600 Å channel includes a few low-ionization species, as well as the recombination continuum from singly ionized Mg and Si to the neutral species.
Following the magnetic field cadence from HMI, the vertical electric current density maps are calculated every 12 minutes. In order to compare the AIA images with these maps, we selected the AIA images closest in time with a given magnetic field map. Moreover, the magnetic field and current density maps are calculated in the heliographic frame, whereas the AIA images are taken in the local reference frame. As described in more details in \citet{musset_et_al_2015}, magnetic field and current density maps can be traced in the local reference frame at a chosen time; we therefore apply a required coordinate transformation and the correction for the rotation of the Sun between the map and the EUV image at the same time. We note that no pointing error is expected since both instruments are on the same spacecraft, take images of the entire Sun, and the data are already well co-aligned. In addition, we select active regions near the center of the solar disk to avoid any noticeable projection effect. The combination of the current density map and the map of EUV brightness at 304~\AA\ is shown in Figure~\ref{f_CDIS_map}.
\begin{figure}[!t]
\epsscale{1.}
\plotone{aia_and_currents75_304.pdf}
\caption
Top: Image of the 304~\AA\ emission from AIA. Current densities above 75~mA~m$^{-2}$ (in red) and below -75~mA~m$^{-2}$ (in blue), are over plotted. Bottom: Brightest 304~\AA\ areas on top of the vertical electric current map. The strongest electric currents show up as ``ribbons'' that appear as elongated dark blue (negative) or dark red (positive) patterns. The brightest 304~\AA\ EUV emission tends to follow the red patterns, but clearly avoids the blue ones, demonstrating a striking asymmetry of the EUV brightness associated with positive or negative electric currents at the photosphere.
\label{f_CDIS_map}}
\end{figure}
\subsection{AIA intensities in positive and negative vertical current densities.}
\label{S_AIA_asymmetry}
The starting point for the analysis is a set of scatter plots representing the distribution of AIA intensity with respect to the magnitude of current density for both positive and negative currents. These scatter plots are shown in Figure~\ref{f_CDIS_regress} for two EUV channels, 304~\AA\ and 1600~\AA.
To aid the eye, filled contours of the binned 2D density estimates of the distribution of AIA intensities have been displayed on top of the scatter plots. The binned density has been calculated with the R function bkde2D of the KernSmooth package.
In these examples, a clear excess of high AIA intensities is apparent for positive current densities in comparison to negative current densities. In order to better quantify this excess, we extract the mean trend out of the scattered data points by computing a non-parametric local linear regression of the EUV brightness values in regard to the current density magnitudes, separately for positive and negative current densities. The non-parametric regression is employed to avoid making assumptions on the relation between the AIA intensity and the current density. A local regression is performed on each data point, taking into account only the neighboring data points (i.e. a fraction of data points closest to the data point considered). Here, we performed a locally linear regression with 70\% of the data points neighboring the data point considered. This non-parametric regression was performed with the locfit function of the locfit package in R.
The results of the non-parametric local linear regression along with 95\% confidence intervals are displayed on top of the scatter plots and 2D histograms for 304~\AA\ and 1600~\AA\ AIA channels; see Figure~\ref{f_CDIS_regress}.
\begin{figure*}[!t]
\epsscale{.5}
\plotone{scatterplot304new.eps}
\plotone{scatterplot1600new.eps}
\caption
Scatter plots of AIA intensity versus the amplitude of the vertical component of the current density (in mA m$^{-2}$) for each pixel, for negative (blue dots) and positive (yellow dots) current densities, for (a) 304~\AA\ and (b) 1600~\AA\ passbands. The contours of the binned 2D density estimates for each distribution are overlapped in the same colors with semi-transparency. The non-parametric local linear regressions of the scattered distributions are also shown as solid lines in red and blue for the positive and negative current distributions, respectively. The 95\% confidence intervals of the regressions are bounded by dashed lines in the corresponding color. The departure between red and blue lines increases and become larger than the confidences intervals for stronger currents.
\label{f_CDIS_regress}}
\end{figure*}
\begin{figure*}[!t]
\includegraphics[width=0.99\linewidth]{histograms}
\caption
Histograms of 304~\AA\ (top) 1600~\AA\ (second line), 211~\AA\ (third lien) and 171~\AA\ (bottom) EUV brightness for the pixels associated with positive (yellow) and negative (blue) currents. The histograms are plotted separately for no, weak, moderate and strong currents. The asymmetry clearly increases towards stronger currents, although it is more visible for some EUV channels than others. \label{f_CDIS_hist_others}}
\end{figure*}
A complementary way to visualize the EUV brightness asymmetry is to plot EUV brightness histograms for pixels associated with the positive and negative currents separately. Figure~\ref{f_CDIS_hist_others} displays such histograms for four EUV channels in the four domains of electric current density magnitude (no, weak, moderate, and strong currents). The asymmetry between the two distributions clearly increases towards stronger currents.
\begin{figure*}[!t]
\epsscale{.25}
\plotone{20110215_0024_scatterplot_304.eps}
\plotone{20110215_0124_scatterplot_304.eps}
\plotone{20110215_0205_scatterplot_304.eps}
\plotone{20110215_0212_on_0148_scatterplot_304.eps}
\plotone{20110309_2300_scatterplot_304.eps}
\plotone{20110309_2312_scatterplot_304.eps}
\plotone{20110309_2324_scatterplot_304.eps}
\plotone{20110906_2112_scatterplot_304.eps}
\plotone{20120712_1512_scatterplot_304.eps}
\plotone{20120712_1612_scatterplot_304.eps}
\plotone{20120712_1624_scatterplot_304.eps}
\plotone{20110906_2212_scatterplot_304.eps}
\caption
Scatter plots of AIA intensity versus the amplitude of the vertical component of the current density (in mA m$^{-2}$) for each pixel, for negative (blue dots) and positive (yellow dots) current densities, for 304~\AA\ for 12 time frames for four different analyzed active regions. The layout of each plot is the same as in Figure~\ref{f_CDIS_regress}(a). {In the top right corner of each panel is indicated the time of the plot $t$ in regards to the time of the peak of the flare $P$ in minutes. }Out of 12 cases shown in the Figure (out of 13, which includes the time frame discussed in the main text) we see a statistically significant (95\% confidence interval) brightness excess above positive current in 5 panels (6 cases out of 13 total); no statistically significant excess is identified in 7 remaining panels (in only one of these seven cases we see an insignificant excess above the negative currents; in all other cases we either see ``positive'' excess, in three cases, or do not see any, in three remaining cases). We never see a statistically significant excess above the negative currents.
Note that some cases evidence saturation of the brightest AIA pixels. This saturation changes the correlation between EUV intensity and current density, but cannot in itself explain the observational effect we find.
\label{f_CDIS_regress_many}}
\end{figure*}
\section{Discussion}
We analyzed several time instances of four different ARs (the total of 13 time instances have been considered). Although in some instances the effect is not or only barely seen, in other instances it is strikingly prominent. To be specific, we discuss in some detail the case of the distribution of EUV brightness vs positive and negative currents at the very beginning of the X2.2 class flare on February 15 2011.
Figure~\ref{f_CDIS_map} shows the vertical current density map derived {at the onset of the impulsive phase} at 01:48 UT, which is compared with an almost simultaneous AIA image.
The positive vertical electric currents at the photosphere are shown in red and the negative ones are shown in blue, while the areas of the strongest 304~\AA\ EUV emission are shown by semitransparent grey shades. {We note that these ribbons are in fact similar to the brightest ribbons in other AIA passbands at this onset of the flare impulsive phase.} The asymmetry of the bright EUV emission vs electric current distribution is striking: the bright areas tend to project onto positive photospheric currents (in red), while they clearly avoid the negative currents (in blue); this is particularly true for the parallel red and blue ``current ribbons,'' highlighting the strongest currents in the area ($x=200-240''$; $y\sim-230''$) slightly to the right and upward from the map center. The opposite correlation is neither expected nor observed: an equally bright EUV area in the middle of the map projects on a weak-current region that confirms that the electric current is not the cause of the brightness itself (in agreement with the currently accepted paradigm adopting that the Joule heating does not play a dominant role in the coronal heating). The top panel of Figure~\ref{f_CDIS_map} shows the 304~\AA\ EUV image, from which the distinction between the transition region ribbons and coronal loops is well seen, so we confidently conclude that a significant fraction of the strongest EUV brightness comes from transition region ribbons with positive electric current, but no EUV brightness enhancement is observed from the other ribbon, which has a comparably strong but negative electric current.
To quantify this finding we inspected the entire map statistically.
Specifically, for each of no, weak, moderate, and strong current subgroups, we compared histograms of the EUV intensities corresponding to positive and negative current densities in the same plot. These histograms are shown in Figure~\ref{f_CDIS_hist_others} for four EUV channels {at the same time frame (01:48 UT) at the onset of the impulsive phase}.
This analysis reveals a strong asymmetry in EUV brightness between the areas associated with positive and negative electric currents for all subgroups with a significant electric current---weak, moderate, and strong. The asymmetry is absent in the case of no significant current. A similar behavior is observed for 1600~\AA, 171~\AA, and 211~\AA\ EUV channels. Even though no contribution of singly or doubly ionized ions is expected in 171~\AA\ channel, a significantly larger amount of Fe~IX--X ions is expected to be found above the ion trap footpoint just after the flare-initiated disruption. A similar excess can be expected at a later phase, due to the electric current-induced migration of heavy ions towards such footpoint, before they fall back in and recombine, which is also true for ``hotter'' ions forming emission in other AIA channels.
Another way of examining the relationship between the vertical component of the electric current density and the associated EUV brightness is to compute a non-parametric local linear regression of the EUV brightness values in regard to the current density magnitudes, separately for positive and negative current densities (see sec.~\ref{S_AIA_asymmetry} and Figure~\ref{f_CDIS_regress}). It has to be pointed out that the non-parametric local linear regression we used is different and more general than the linear least squares fit regression. The non-parametric regression as we applied is the best adapted to our case where the relationship between the current density and the EUV intensity is unknown and, in the general case, nonlinear. Figure~\ref{f_CDIS_regress} displays these regressions for the 304 and 1600~\AA\ EUV channels, where the departure between the regressions for the positive and negative currents becomes significant for current densities greater than $\sim50$~mA/m$^2$.
Finally, Figure~\ref{f_CDIS_regress_many} shows the 2D scatter plots, {binned 2D density estimates}, and non-parametric local linear regression curves along with 95\% confidence intervals for 12 time frames studied for four different ARs. {The time frames have been selected such as to cover different stages relative to the corresponding flare: a relatively quiet pre-flare phase, impulsive phase onset, and a post-impulsive relaxation phase.} This set of data confidently shows that we either see a statistically significant asymmetry in favor of positive currents, or do not see the asymmetry at all (which is not surprising given the many competing factors capable of hiding the effect as we have explained above). {Although it might be premature to firmly conclude about dynamics of this effect, we note that the positive current excess becomes more pronounced in the last two panels of 2011-Feb-15 flare (top row) and the third panel of the 2011-Mar-09 flare, which are all obtained from the flare relaxation phase. This evolutionary pattern is consistent with our model prediction as explained in Section~\ref{S_Obs_Ion_Traps}.}
From our cartoon (Fig.~\ref{f_CDIS_cartoon}) one could expect EUV brightness ``holes'' from the areas with the negative electric current.
Interestingly, \citet{2015ApJ...808L...7D} used the \textit{Hinode}/EIS data and found that at certain footpoints of post-flare loops, a so-called inverse FIP effect developed over roughly a half an hour, where the Ar/Ca ratio was greatly enhanced relative to both coronal and photospheric ratios. Although they did not consider electric currents, the creation of such anomaly might naturally be created by forming a low-FIP ion ``hole'' due to depletion of the low-FIP Ca ions from the footpoint; the reported time scale is perfectly consistent with our estimate; see Equation~(\ref{Eq_tau_e_drift}).
Recently, \cite{2017ApJ...847..113H} noted the coincidence of higher EUV brightness and upward electric currents in the observations of the 2011 February 15 flare of \cite{janvier_et_al_2014, musset_et_al_2015} as well as a preferred coincidence of the maxima of the electron X-ray emission seen by \textit{RHESSI} with the upward current ribbons \citep{musset_et_al_2015}. \cite{2017ApJ...847..113H} suggested that these observations show that accelerated electrons precipitated and deposited energy preferentially in the upward current ribbon. The author interpreted this observational result as an evidence for the existence of field-parallel acceleration during solar flares.
That proposed model, however, requires a rather extreme (though not impossible) parameter regime; in particular the electric current density would need to be orders of magnitude larger than that which is typically observed at the photosphere. Our work offers an interpretation of the observed EUV asymmetry that does not require an electric current density larger than the observed photospheric values, {although more theoretical work and data analysis is needed to deeply understand the exact origin and firmly quantify the magnitudes of the EUV brightness asymmetries and their role in solar physics}.
\section{Conclusions}
We found evidence in favor of a
striking asymmetry in EUV brightness distributions associated with positive and negative electric currents, which is a natural outcome of our predicted current-driven concentration of heavy ions.
The presented evidence calls for a more systematic and detailed study of this novel fundamental effect and its potential role in FIP fractionation, temporal variation, and spatial non-uniformity of the elemental abundances.
In particular, the FIP effect, which is an enhancement of coronal abundances of the elements with low FIP relative to their photospheric abundances, is a likely and natural consequence of the electron drag at the loop footpoint, as this force acts on the low-FIP atoms which are singly ionized in the photosphere and chromosphere, and leaves the neutral high-FIP elements behind, leading to an enhanced abundance of low-FIP ions in the ion trap and eventually in the loop coronal body as observed.
\vspace{-0.25cm}
\acknowledgements
This work was supported in part by NSF grant AGS-1262772, NASA grant NNX14AC87G and 80NSSC18K0015 to New Jersey
Institute of Technology, and by an NSF Faculty Development Grant (AGS-1429512) to the University of Minnesota. This work was granted access to the HPC resources of MesoPSL financed by the Region Ile de France and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme Investissements d'Avenir supervised by the French Agence Nationale pour la Recherche
\clearpage
\bibliographystyle{apj}
|
{
"timestamp": "2018-03-09T02:00:53",
"yymm": "1803",
"arxiv_id": "1803.02851",
"language": "en",
"url": "https://arxiv.org/abs/1803.02851"
}
|
\section{Introduction}
When a high-intensity laser pulse is irradiated on a material, its surface is instantaneously ionized, and the electrons in the ionized material, i.e., plasma, are then accelerated close to the speed of light by the ponderomotive force of the laser light.
These energetic electrons are often called relativistic electrons (REs).
The energy distribution of REs is approximated by a Maxwell-Boltzmann distribution function with slope temperature $T_\textrm{RE}$ as $dN/dE \propto \exp \left( -E/T_\textrm{RE} \right)$ where $N$ and $E$ denote the number and energy, respectively.
The scaling laws of $T_\textrm{RE}$ on laser intensity have been investigated experimentally \cite{Malka1996, Beg1997, Tanimoto2009}, theoretically, and computationally \cite{Wilks1992a, Haines2009, Kluge2011, Pukhov1999}.
These scaling laws are useful to determine laser parameters for high-intensity short pulse laser experiments and to design applications.
The effect of pulse duration on $T_\textrm{RE}$ is not considered explicitly in the reported scaling laws; however, recent computational and theoretical studies \cite{Kemp2012, Sorokovikova2016} have revealed that $T_\textrm{RE}$ generated by multi-picosecond (multi-ps) laser pulses could be several times higher than that predicted by the reported scaling laws.
With the development of kilojoule-class high-power lasers such as LFEX \cite{Miyanaga2006}, NIF-ARC \cite{Crane2010}, LMJ-PETAL \cite{Batani2014}, and OMEGA-EP \cite{Maywar2008}, it has become possible to irradiate relativistic laser pulses continuously over multi-ps.
In this study, we have clarified the generation of super-ponderomotive RE (SP-RE) in multi-ps laser-plasma interaction using ultra-high-contrast LFEX laser pulses realized using a plasma mirror (PM).
The slope temperature of REs was increased more than twice by extending the laser pulse duration from 1.2 ps to 4.0 ps.
The following two acceleration mechanisms were identified as essential for the generation of SP-REs in multi-ps laser-plasma interaction with the help of particle-in-cell (PIC) simulations.
One mechanism is the generation of SP-REs by the combination of a laser field and a quasi-static electric field reported by Sorokovikova \textit{et al.} \cite{Sorokovikova2016}.
In a laser-heated plasma, a quasi-static electric field is generated spontaneously by charge separation at the forward edge of the plasma expansion, and the direction of this field is generally parallel to the direction of laser propagation.
Such a quasi-static electric field is able to push electrons along the laser propagation direction; therefore, electrons can stay in the acceleration phase longer than that without a quasi-static electric field, i.e., electrons undergo higher energy gain.
The other mechanism is multiple electron injection in the region where the laser field and quasi-static electric field coexist due to the cyclotron motion of REs in a self-generated quasi-static azimuthal magnetic field \cite{Krygier2014, Nakamura2010}.
This distinctive injection mechanism is referred to as loop-injected direct acceleration (LIDA).
A tens of megagauss (MG) quasi-static magnetic field also develops within the expanding plasma in multi-ps laser-plasma interaction and LIDA plays a significant role in the generation of SP-REs in multi-ps laser-plasma interaction.
The LIDA is triggered by the transition from the hole boring phase to the blowout phase in a laser-heated plasma.
Here, we obtained the equation of transition timing for arbitrary laser pulses.
\section{Experimental observation of super-ponderomotive electrons}
We have experimentally investigated the dependence of RE energy distributions on the pulse durations under conditions free from pre-plasma formation.
The experiment was conducted using the LFEX laser system at the Institute of Laser Engineering, Osaka University.
The LFEX laser consists of four beams, where the spot diameter of the spatially overlapped LFEX beams on a target was 70 $\mathrm{\mu m}$ of the full width at half maximum (FWHM), and 30\% of the laser energy was contained in this spot.
One LFEX beam delivered 300 J of 1.053 $\mu$m wavelength laser light with a 1.2 ps duration (FWHM), and the peak intensity of one beam was 2.5$\times$10$^{18}$ $\rm{W/cm^2}$.
It is well known that SP-REs can be accelerated in a long-scale-length pre-plasma; therefore, a PM \cite{Doumy2004} was implemented to realize the pre-plasma-free condition to exclude the other known mechanisms from this experiment.
The contrast ratio of the LFEX laser pulse was improved by two orders of magnitude through implementation of the PM \cite{Arikawa2016} down to 10$^{11}$ at 150 ps before the main pulse (as shown in Fig.\,\ref{fig:experimental_setup}(a)).
These clean intense laser pulses create the ideal situation where the REs are accelerated predominantly in the inherent plasma formed by the main laser pulse itself during the picosecond time range.
The density scale length of the preformed plasma was calculated to be 1.5 $\mu$m at 10 ps before the intensity peak from a 2D radiation hydrodynamics simulation with the PINOCO-2D code \cite{Nagatomo2007}.
These ``clean" pulses were focused on a 1 mm$^3$ gold cube.
The thickness of the gold cube is also an important parameter to investigate RE acceleration by multi-ps laser-plasma interactions.
The REs generate a sheath electric field at the rear surface of the target.
This sheath field refluxes especially low energy REs and the refluxed REs are re-injected to the acceleration region.
This recirculation process also generates SP-REs, which was investigated by Yogo and Iwata \textit{et al.} \cite{Yogo2017, Iwata2017}.
One cycle of the recirculation process takes at least 6.7 ps in the 1 mm-thick gold cube, which is longer than the pulse durations (1.2 or 4.0 ps) in this experiment; therefore, the recirculation process can be eliminated from the SP-RE mechanisms in this study.
LFEX laser pulses can be stacked temporally with arbitrary delays between the beams, as shown in Fig.\,\ref{fig:experimental_setup}(b).
In this study, a single beam (case A: 1.2 ps FWHM pulse duration and peak intensity of $2.5\times10^{18}$ $\rm{W/cm^2}$) was used and two types of four-stacked beams (case B: 4.0 ps FWHM pulse envelope and peak intensity of $3.0\times10^{18}$ $\rm{W/cm^2}$, and case C: 1.2 ps FWHM pulse duration and peak intensity of $1.0\times10^{19}$ W/cm$^2$).
We emphasize here that the leading edge of the stacked pulse remains similar to that of the single beam. If the pulse duration is extended by adjusting the pulse compressor of the laser system, the leading edge would inevitably be modified into a more gradual shape.
The energy distribution of REs emanated from the target to the vacuum was measured with an electron energy analyzer located 20.9$^{\circ}$ from the incident axis of the LFEX laser.
Figure \ref{fig:experiment}(a) shows the experimental results of the time-integrated energy distribution.
The slope temperatures were 0.65 MeV for case A (red circles) and 1.7 MeV for case B (green triangles).
The slope temperature for case B was more than twice that for case A, even though the peak intensities were very close.
The energy distributions of REs obtained for case B (green triangles) and case C (blue squires) were almost identical, even though the peak intensities were different by a factor of four.
These slope temperatures cannot be explained using the reported scaling laws, whereby the dependence of the slope temperature on the pulse duration is not considered.
\section{Two-dimensional (2D) PIC simulations with experimental conditions}
\subsection{Electron acceleration dynamics in multi-picosecond laser-plasma interaction}
The experimental results were compared with those computed using the 2D PIC simulation code (PICLS-2D \cite{Sentoku2008}).
Calculations were performed with temporal and spatial scales that were comparable to the experimental scales.
The gold cube was replaced with a 20 $\mu$m planar plasma with a peak density of 40$n_{c}$, where $n_{c}$ = 1.0$\times$10$^{21}$ cm$^{-3}$ is the critical electron density for 1.053 $\mu$m wavelength light.
The bulk plasma has an exponential density profile from 0.1 to 40$n_{c}$ and a scale length of 1 $\mu$m.
Due to computational limitations, the ionization degree was fixed to be +40 in the PICLS-2D simulation, which was determined based on the result of a one-dimensional PICLS simulation with the dynamic ionization model of gold described by field ionization \cite{Kato1998} and a fast electron collisional ionization \cite{Lotz1970}. The ionization degree rose from +10 (given by the radiation hydrodynamic code PINOCO-2D) to around +40 for first several picoseconds.
In the 2D-PIC simulation with dynamic ionization, it was reported that ionizing defocusing counteracting laser filamentation and self-focusing occurs
However, it does not significantly affect the short-scale-length pre-plasma in the order of the laser wavelength.
\begin{figure*}
\begin{center}
\includegraphics*[width=160mm]{Figure1.png}
\end{center}
\caption{(Color online) (a) Experimental setup. The geometrical positions of the target, PM and the diagnostics instruments, and the ray trace are illustrated. (b) Temporal intensity profiles of LFEX laser pulses. Pulses temporally stacked to generate various pulse shapes.
} \label{fig:experimental_setup}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics*[width=160mm]{Figure2.png}
\end{center}
\caption{(Color online) RE energy distributions measured experimentally and computationally by changing the intensity and duration of laser pulses.
(a) Comparison of experimental data for cases A (red circles) and B (green triangles), and computational data for cases A (grey line) and B (black line).
(b) Comparison of experimental data for cases A (red circles) and C (blue squares), and computational data for cases A (grey line) and C (black line).
Comparison between laser pulse shapes (red lines) and temporal evolution of the maximum energy of REs (lines between circles) for cases (c) A, (d) B, and (e) C.
} \label{fig:experiment}
\end{figure*}
The slope temperatures of the REs in the simulation were 0.7, 2.0, and 2.0 MeV for cases A, B, and C, respectively.
Thus, the PIC simulation reproduces well the experimentally observed dependence of the slope temperature on the laser intensity and pulse duration \cite{Kojima2016a}, as shown in Fig.\,\ref{fig:experiment}.
Figures \ref{fig:experiment} (c)--(e) show a comparison of the pulse shapes (red lines) and the temporal evolution of maximum energy of REs (blue lines between circles) for cases A, B, and C.
The temporal evolution of the maximum energy of the REs is similar to the laser pulse shapes for the cases of 1.2 ps pulse duration (cases A and C).
In contrast, the situation for the 4.0 ps pulse duration (case B) is completely different.
For case B, the maximum energy increases, even after the timing when the laser intensity reaches the plateau at 2.0 ps.
The most energetic REs were produced near the end of the intensity plateau (5.5 ps).
The time-integrated energy distributions of the REs for cases B and C seem to be identical; however, the temporal behavior of RE acceleration in case B is completely different from that in case C.
Figures \ref{fig:track}(a)--(f) show three selected RE trajectories at two different periods ($t$ = 3.0--3.5 and 5.0--5.5 ps) overlaid on the electron densities [Figs.\,\ref{fig:track}(a) and (b)], self-generated azimuthal magnetic fields [Figs.\,\ref{fig:track}(c) and (d)], and self-generated electric fields [Figs.\,\ref{fig:track}(e) and (f)].
Figures \ref{fig:track}(a) and (b) are colored using the lookup table of electron density logarithm normalized with the critical density ($n_\textrm{c}$).
When a high-intensity laser is irradiated on a target, quasi-static electric and magnetic fields are spontaneously generated on the target surface.
The quasi-static term indicates that the time variation of the fields is slower than that of the laser field.
The electric field is formed with plasma expansion and its direction is perpendicular to the target. The magnetic field is in the azimuthal direction of the laser axis.
These self-generated electric and magnetic fields assist RE acceleration as discussed below.
\begin{figure*}
\begin{center}
\includegraphics[width=160mm]{Figure3.png}
\end{center}
\caption{Three examples of RE trajectories at two different periods ($t$ = 3.0--3.5 and 5.0--5.5 ps) overlaid on the electron densities [(a) and (b)], self-generated azimuthal magnetic fields [(c) and (d)], and self-generated electric fields [(e) and (f)].
The electron density maps [(a) and (b)] are colored using the lookup table of electron density logarithm normalized according to the critical density ($n_\textrm{cr}$).
(g,h) Kinetic energies of REs along the longitudinal position for the two different periods.
\\} \label{fig:track}
\end{figure*}
In the earlier period (the top panels of Fig.~\ref{fig:track}), the REs move around the near-critical density region.
The energetic RE source is initially accelerated to 3--4 MeV by the reflected laser field in the near-critical density region.
3--4 MeV is close to the kinetic energy (3.5 MeV) of a RE obtained by the ponderomotive force from the reflected laser field ($a_0 = 1.7$ and $I = 4.0\times10^{18} \mathrm{W/cm^2}$) without absorption of the incident laser field.
The electron travels outwardly (the opposite direction of laser propagation) through the magnetic and electric fields that are generated by the Biermann battery effect and charge separation.
In this period, the self-generated magnetic field strength is not sufficient to change the RE motion.
The self-generated electric field decelerates the outwardly moving RE, and the RE eventually stops and is then accelerated again inwardly.
The effect of the quasi-static electric field not only directly imparts additional energy to the electrons but also reduces the dephasing rate of the RE from the acceleration phase of the laser field \cite{Sorokovikova2016, Robinson2013, Robinson2015, Paradkar2012a, Arefiev2016c, Arefiev2012, Kemp2009}. The RE continues to ride on the acceleration phase, whereby the RE gains energy from the laser field.
i.e., the RE obtains more energy when it is accelerated by the incident laser field.
In this simulation, the RE is accelerated up to 15 MeV by the combination of the quasi-static electric field and the laser field, as shown in Fig.\,\ref{fig:track}(g).
In the later period (bottom panels of Fig.\,\ref{fig:track}), the self-generated magnetic field is sufficiently strong that some of the REs (blue and green trajectories) are reflected outwardly by the $\bm{v} \times \bm{B}$ force and they are re-injected to the region where both the self-generated electric field and laser field coexist (Fig.\,\ref{fig:track}(b), loop(ii)).
In loop (ii), the turning point of the RE is farther from the near-critical density region than that in loop (i) because the RE receives more kinetic energy in loop (i).
After loop (ii), the kinetic energy of the REs reaches beyond 15 MeV, as shown in Fig.\,\ref{fig:track}(h).
This re-injection mechanism is the LIDA \cite{Krygier2014}.
The solid lines in Figs.\,\ref{fig:histogram}(a) and (b) show energy distributions of REs accelerated in the two periods.
The histograms show the ratio of the RE numbers between the two groups: one group (red bars) consists of REs that experienced single loop-injection and the other (green bars) consists of REs that experienced multiple loop-injection.
The correlation between multiple loop-injections and energetic RE generation is clearly evident; namely, the highest energy component of REs in Fig.\,\ref{fig:histogram}(b) above 20 MeV is generated predominantly by multiple loop-injection.
\begin{figure*}
\begin{center}
\includegraphics*[width=160mm]{Figure4.png}
\end{center}
\caption{(a,b) Energy distributions (solid lines) of REs accelerated in the two periods (3.0--3.5 and 5.0--5.5 ps).
The histograms show the ratio of the RE numbers between the two groups, where one group (red bars) consists of REs that experienced single loop-injection and another group (green bars) consists of REs that experience multiple loop-injections due to LIDA.
A correlation between multiple loop-injections and energetic electron generation is clearly evident.
\\} \label{fig:histogram}
\end{figure*}
\subsection{Generation of a giant quasi-static magnetic field during multi-ps laser-plasma interaction}
The PIC simulation shows that the quasi-static magnetic field is generated by three different mechanisms in case B, which are dependent on the time during the multi-ps laser-plasma interaction.
At the leading edge of the 4 ps flat-top pulse ($<$2 ps), the ponderomotive force of the incident laser pushes the relativistic critical density surface ($\gamma n_c$) into the overdense region, and the heated underdense plasma expands into the vacuum.
An electric field is generated at the outer boundary of the expanding plasma (which is referred to as the first electric field.).
An azimuthal magnetic field is generated in the overdense plasma due to the $\nabla n \times \nabla I$ effect \cite{Sudan1993, Mason1998, Wilks1992a, Tripathi1994}, where $n$ and $I$ are the plasma electron density and laser intensity, respectively.
When the laser intensity reaches the plateau at 2.0 ps, plasma evacuation by the laser field is eventually halted by the charge separation due to depletion of the local electron density.
The $\nabla n \times \nabla I$ mechanism becomes relatively small, whereas the $\nabla T \times \nabla n$ (Biermann battery) effect \cite{Stamper1971, Borghesi1998, Max1978, Kolodner1979, Sandhu2002a, Schumaker2013a} becomes the dominant mechanism for generation of the magnetic field.
Here, $T$ is the plasma electron temperature.
Along with a change of the generation mechanism, the generation region also moves from the overdense region to the underdense region.
The strongest magnetic field is generated at the edge of the laser spot in the underdense plasma (Fig.\,\ref{fig:Ex_Jy_Bz}(f)).
This magnetic field influences the motion of REs around the near-critical density region.
Some of the REs are moved transversely from the laser spot by the $\bm{E} \times \bm{B}$ drift.
The drift current heats the surface of the bulk plasma via the two-stream instability.
Enhancement of the energy transfer to the transverse direction due to the surface magnetic field is discussed in Refs.\,\cite{Jaanimagi1981,Paradkar2010a,Forslund1982}.
The electric field that contributes to the $\bm{E} \times \bm{B}$ drift is a weak electric field generated in a limited region near the critical density surface.
The heated bulk plasma begins to expand at the edge of the laser spot, while the expansion is suppressed at the inside of the laser spot by the laser ponderomotive pressure.
The heated bulk plasma surface, which has been flat so far, deforms into a bow shape (which is referred to as a bow-shaped bulk plasma surface).
The first electric field is carried out by the plasma expansion far away from the critical density surface and no longer contributes to the drift.
When the thermal pressure of the heated bulk plasma exceeds the ponderomotive pressure of the incident laser at 3.8 ps, the bulk plasma begins to expand at the inside of the laser spot, and the strong quasi-static electric field (the second electric field) is then generated at the near-critical density region.
Figure \ref{fig:Ex_Jy_Bz}(a) shows the electric fields in the longitudinal direction ($E_\mathrm{x}$) of the two regions.
The second electric field is generated at the expansion front of the heated bulk plasma at the inside of the laser spot.
The newly generated strong electric field contributes to the $\bm{E} \times \bm{B}$ drift by combination with the magnetic field (Fig.\,\ref{fig:Ex_Jy_Bz}(b)).
REs move along the bow-shaped bulk plasma surface by the $\bm{E} \times \bm{B}$ drift.
When the REs flow in the plasma, the return-current is driven to maintain current neutrality in the plasma.
Figure \ref{fig:Ex_Jy_Bz}(c) shows the RE drift current in the lower density region and the return-current flow in the higher density region.
The current loop produced by the spatial separation between the RE drift current and the return current generates a magnetic field along the outer edge of the bow-shaped bulk plasma surface (Fig.\,\ref{fig:Ex_Jy_Bz}(c)).
This third magnetic field (30--50 $\mathrm{MG}$) is stronger than the magnetic field generated by the $\nabla T \times \nabla n$ effect ($<$10 $\mathrm{MG}$).
In the plasma region where RE current terminates, the electric field is enhanced by the inflow of electrons (Fig.\,\ref{fig:Ex_Jy_Bz}(d)).
The positive feedback between the growth of the fields and the field-driven drift current results in the rapid growth of the quasi-static electric and magnetic fields with time (Figs.\,\ref{fig:Ex_Jy_Bz}(c)--(h)).
The maximum energy of the REs increases from 3.5 ps until 5.5 ps, which corresponds to the timing of rapid growth of the self-generated fields.
The SP-RE are accelerated by a laser field under a quasi-static self-generated electric field.
In addition, when positive feedback starts, the strength of the self-generated magnetic field grows by several tens of MG approximately several picoseconds after the beginning of the laser-plasma interaction, and the strong magnetic field begins the LIDA.
Thus, SP-RE acceleration is not a process that gradually progresses with time but a process that proceeds in a threshold manner.
This has not been pointed out in previous studies on REs acceleration by multi-ps laser pulse. \cite{Sorokovikova2016,Kemp2012,Peebles2017,Yogo2017,Iwata2017}
\begin{figure*}
\begin{center}
\includegraphics*[width=160mm]{Figure5.png}
\end{center}
\vspace*{-0.5cm}
\caption{Spatial maps of longitudinal electric field $E_x$ [((a), (d), and (g)], transverse current density $J_y$ [(c) and (f)], and azimuthal magnetic field $B_\theta$ [(b), (e), and (h)] at 3.8, 4.0, and 4.5 ps.
Loop current by the REs and return current rapidly enhance the strength of the electric and magnetic fields.
\\} \label{fig:Ex_Jy_Bz}
\end{figure*}
\subsection{Transition timing to super-ponderomotive electron acceleration}
The SP-RE acceleration is started when the plasma thermal pressure exceeds the laser ponderomotive pressure.
Figure \ref{fig:Plasma_compression}(a) shows the evolution of an initially exponential plasma profile during the interaction with a high-intensity laser pulse. The color map shows the electron density ($\log_{10}(n_e/n_c)$) and the red solid line shows the temporal intensity profile of the laser.
At $t=$3.8\,ps, the motion of the relativistic critical interface stops even though the laser pulse is still irradiated, and the state of the laser-plasma interaction transits from the hole boring phase to the blowout phase.
The position of the interface that interacts with the laser pulse having an arbitrary intensity temporal profile is obtained by integrating the velocity of the interface with respect to time \cite{Kemp2008},
\begin{widetext}
\begin{equation}
\begin{split}
x_i (t)= x_c(0) + 2l_s \ln \biggl[1+\frac{c}{2l_s} \sqrt{\frac{R \cos \theta}{(1+R)} \frac{Zm_e}{M_i }} \int_{t_0}^{t} \biggl( \frac{\gamma(t)^2 -1}{\gamma(t) } \biggr)^{1/2} dt \biggr]. \\
\label{eq: momentum conservation at the laser-plasma interaction surface_low}
\end{split}
\end{equation}
\end{widetext}
Here, $I(t)/c=m_ec^2n_c a_0^2(t)/2$ is used and the variables are explained in the Methods section. $t_0$ is the time when the normalized laser amplitude $a_0$ reaches 1.
Note that the position of the interface $x_{c}$ should vary with time.
However, here the initial position of the critical density, i.e., $x_{c}=x_{c}(0)=$constant, was substituted considering that the temporal profiles of realistic lasers increase from 0 to the peak intensity.
The transition timing can be obtained by coupling Eq.\,\eqref{eq: momentum conservation at the laser-plasma interaction surface_low} with the hole boring limit density, which is derived from the momentum transfer equation for the stationary state of the interface \cite{iwata}:
\begin{equation}
\begin{split}
\frac{n_s}{n_c} = 8 \epsilon^2 a_0^2 \biggl[ \frac{1+R-(1-R)\beta_h^{-1}\alpha^{-1}}{2} \biggr],
\label{eq: electron_number_density_plateau_hot_by_nc}
\end{split}
\end{equation}
where $\epsilon$ is the polarization factor ($\epsilon$=1 and $\sqrt{2}$ for linear and circular polarization, respectively), the plasma is assumed to be composed of REs ($n_h$) and bulk electrons ($n_b$) as $n_e=n_h+n_b$, and the momentum flux of the bulk electron component is negligible compared to that of the RE component (i.e., $ n_e T_e c \beta_e \approx n_h T_h c\beta_h$). $\beta_h$ is the ratio of the drift velocity of REs ($v_h$) to the speed of light, $c$. $\alpha \equiv ir/2$ is the geometrical factor, where $r = 1$ for the non-relativistic Maxwell momentum distribution and $r = 2$ for the relativistic Maxwell (Maxwell-J\"uttner) momentum distribution. Here, $i =$ 1, 2, or 3 represents the dimension of the momentum distribution.
When a 1D relativistic Maxwell distribution $\alpha=1$ is assumed, the relativistic limit for the RE velocity $\beta_h=1$, and linear polarization $\epsilon=1$, Eq.\,(\ref{eq: electron_number_density_plateau_hot_by_nc}) reduces to $n_s/n_c = 8 R a_0^2$.
By substituting $a_{0}=1.79$ and $R=0.7$, the electron density threshold $n_{s}$ for the experimental condition of case B in Fig.\,\ref{fig:Plasma_compression} is obtained as $n_{s}=17.9\,n_c$. This density is almost identical to the electron density threshold in which the plasma compression terminates in the PIC simulation.
Substituting Eq.\,(\ref{eq: electron_number_density_plateau_hot_by_nc}) and the initial electron density profile ($n_e(x)=n_c \exp[(x-x_c)/l_s]$) into Eq.\,(\ref{eq: momentum conservation at the laser-plasma interaction surface_low}) yields
\begin{widetext}
\begin{equation}
\begin{split}
8 \epsilon^2 a_0^2 \biggl[ \frac{1+R-(1-R)\beta_h^{-1}\alpha^{-1}}{2} \biggr] = \exp\Biggl\{ \frac{x_c +2l_s \ln \biggl[1+\frac{c}{2l_s}\sqrt{\frac{R\cos \theta}{(1+R)}\frac{Zm_e}{M_i}} \int_{t_0}^{t_s} \sqrt{\frac{\gamma(t)^2-1}{\gamma(t)}} dt \biggr]-x_c }{l_s} \Biggr\},
\label{eq: transition_timing_1}
\end{split}
\end{equation}
\end{widetext}
where $a_0$ is the normalized laser intensity. When the laser intensity is constant in time, the transition timing is then obtained as
\begin{widetext}
\begin{equation}
\begin{split}
t_s= F_c \biggl[ \frac{2l_s}{c} \biggl\{ \sqrt{4 \epsilon^2 a_0^2 [ 1+R-(1-R)\beta_h^{-1}\alpha^{-1} ]}-1 \biggr\} \sqrt{\frac{(1+R)}{R\cos \theta}\frac{\alpha m_p Z^*}{Zm_e} \frac{\gamma}{\gamma^2-1}} \biggr] + t_0,
\label{eq: transition_timing_4}
\end{split}
\end{equation}
\end{widetext}
where $M_{i}=\alpha^* m_p Z^*$ represents the ion mass, $m_p$ is the proton mass, $Z^*$ is the ion charge number for the fully ionized state, and $\alpha^*=1$ for hydrogen and $\alpha^*=2$ for other species.
The correction factor $F_c$ is added to take the laser pulse profile into account.
For cases where the laser intensity is constant in time, $F_{c}=1$.
The lines in Fig.\,\ref{fig:Plasma_compression}(b) show the transition timing calculated from Eq.\,(\ref{eq: transition_timing_4}) with $F_{c}=1$ for various reflectivities.
Here, we assumed that a gold plasma ($Z^*=197$) with the preformed plasma scale length $l_s=$1 ${\rm \mu m}$ in the charge state of $Z=40$ interacts with a linearly polarized laser ($\epsilon=1$). The spot size of the LFEX laser is large; therefore, it is assumed that electron acceleration occurs one-dimensionally ($\alpha=1$).
As an approximate trend, when the normalized laser intensity $a_0$ or reflectivity $R$ increases, the transition timing is delayed. Low reflectivity reduces the effective laser intensity at the interface and reduces the hole boring limit density.
When the normalized laser intensity is $a_0$=1.79 (intensity is $I\lambda^{2}=4.0\times 10^{18}\,{\rm W\,\mu m^{2}/cm}^{2}$), the transition timing is estimated to be $t_{s}=$2.8\,ps, which is in good agreement with the simulation result.
\begin{figure*}
\begin{center}
\includegraphics*[width=140mm]{Figure6.png}
\end{center}
\vspace*{-0cm}
\caption{(a) Evolution of the initially exponential plasma profile during interaction with a laser pulse having a normalized laser intensity of $a_{0}=1.79$ ($I\lambda^{2}=4.0\times 10^{18}\,{\rm W\,\mu m^{2}/cm}^{2}$). The motion of the interface calculated from Eq.\,(\ref{eq: momentum conservation at the laser-plasma interaction surface_low}) (red dotted line) is in good agreement with the motion obtained by the PIC simulation until $t=$3.8\,ps (black solid line). (b) Transition timing for constant intensity temporal profile calculated from Eq.\,(\ref{eq: transition_timing_4}) for a gold target with a preformed plasma scale length of $\ell_{s}=1\,{\rm \mu m}$ in the charge state of $Z=40$.
Correction factor of the pulse temporal profile $F_{c}$ for cases of (c) low reflectivity $R=0.3$ and (d) high reflectivity $R=0.7$.
The normalized laser intensity $a_0$ or half width at half maximum (HWHM) of the Gaussian leading edge increases; therefore, a larger correction factor is required.
} \label{fig:Plasma_compression}
\end{figure*}
In an actual case, the laser intensity increases in time with the Gaussian profile, so that the transition timing is delayed compared with that obtained for $F_{c}=1$.
The color maps in Figs.\,\ref{fig:Plasma_compression}(c) and (d) show the dependence of the correction factor $F_c$ on the normalized laser intensity $a_{0}$ and the half width at half maximum (HWHM) of the Gaussian leading edge for the low reflectivity case ($R$=0.3) and high-reflectivity case ($R$=0.7).
As the normalized laser intensity or HWHM increases, a larger correction factor is required. In the present range of $1 \le a_{0} \le 6$ and 0\,ps$\:\le$ HWHM $\le\:$1.8\,ps, the correction factor increases only approximately 1.2 times at the maximum; therefore, it is sufficient to use Eq.\,\eqref{eq: transition_timing_4} with $F_{c}=1$ for a rough estimation of the transition timing.
\section{Summary}
In summary, with the development of kilojoule-class high-power lasers, it has become possible to continuously irradiate relativistic laser pulses on matter over multi-ps.
Although electron acceleration using a conventional sub-ps laser pulse has been explained theoretically as the interaction of a single electron with a laser field, it is necessary to consider the collective effect of electrons when the pulse duration reaches the multi-ps range.
Energetic RE generation was experimentally clarified with an average energy far beyond the ponderomotive scaling using the prepulse-free LFEX laser.
During the multi-ps interaction, a quasi-static electric field is generated by plasma expansion. In addition, a quasi-static magnetic field is gradually generated due to three different mechanisms of the $\nabla n \times \nabla I$ effect, the $\nabla T \times \nabla n$ effect, and a loop current driven by the ${\textbf E}\times{\textbf B}$ drift.
The third mechanism of the current loop rapidly amplifies the magnetic field strength by the positive feedback between the electric and magnetic fields and the field-driven drift current.
Under the quasi-static electric field, REs are accelerated efficiently above ponderomotive scaling by the laser field because the dephasing rate of the REs from the laser field is reduced.
Furthermore, when the quasi-static magnetic field becomes sufficiently strong to reflect REs back to the laser-plasma interaction region, the reflected REs gain further additional energy.
The boosting timing of electron acceleration by the LIDA mechanism is related to the transition timing of the laser-plasma interaction state from the hole boring phase to the blowout phase.
The equation for transition timing can be derived from the equations for the motion of a relativistic critical density interface and the equations for the electron density where the hole boring terminates.
In this study, the equation for transition timing was extended to a laser pulse with an arbitrary intensity temporal profile. The theoretical result was then compared with the result of PIC simulation.
The mechanism for the generation of SP-REs in the multi-ps laser-plasma interaction reported here is useful for various applications.
For instance, Yogo \textit{et al.} reported that the maximum proton energy is enhanced more than twice by extending the pulse duration to the multi-ps regime, due to the electron temperature evolution beyond the ponderomotive energy in the over picoseconds interaction.
The acceleration mechanism of REs investigated here is also important for fast-ignition inertial confinement fusion and laboratory astrophysics using high-intensity multi-ps laser pulses.
\begin{acknowledgments}
The authors thank the technical support staff of the Institute of Laser Engineering (ILE) at Osaka University and those of the Plasma Simulator at the National Institute for Fusion Science (NIFS) for assistance with laser operation, target fabrication, plasma diagnostics, and computer simulations.
We also acknowledge A. Sagisaka, K. Ogura, A. S. Pirozhkov, M. Nishikino, and K. Kondo of the Kansai Photon Science Institute, National Institutes for Quantum and Radiological Science and Technology for valuable discussions on intensity contrast improvement using the PM.
This work was supported by the Collaboration Research Program between the NIFS and ILE at Osaka University, the ILE Collaboration Research Program, and by the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) through Grants-in-Aid for Scientific Research (Nos. 24684044, 24686103, 70724326, 15K17798, 25630419, 16K13918, and 16H02245), the Bilateral Program for Supporting International Joint Research of the Japan Society for the Promotion of Science (JSPS), and Grants-in-Aid for Fellows from JSPS (Nos. 14J06592, 17J07212 and 15J00850).
\end{acknowledgments}
\section*{Author contributions}
S. K. and S. F. are the principal investigators who proposed and organized the experiment.
M. H. performed the PIC simulations in the collaboration with T. J. and H. S..
N. I., S. K., M. H. and Y. S. developed the theoretical model.
Y. A. and A. M. designed and constructed the large size plasma mirror.
H. M. and Y. O. carried out the theoretical analysis.
H. N. and A. S. performed the radiation hydrodynamic simulations.
K. M., S. S., S. L., K. F. F. L. and Y. A. measured electron energy distribution of high energy component with some help from T. O..
S. T. measured electron energy distribution of low energy component with some help from Z. Z. and A. Y..\,
S. T. and J. K. are in charge of LFEX laser facility development in ILE.
M. N., H. N., H. S. and H. A. supervised the project and provided overall guidance.
All authors participated in the discussions and contributed to the preparation of the manuscript.
\section{Methods}
\subsection{PM implementation}
The LFEX parabola cannot be focused at an offset position far from the target chamber center due to its mechanical limitations; therefore, a spherical PM was used that allows the original focal pattern to be relayed at an offset position with respect to the target chamber center.
A spherical concave mirror (2-inch diameter and 202 mm curvature length) with a 1.053 $\mu$m anti-reflection coating on both surfaces was placed after the focus point, as shown in Fig.\,\ref{fig:experimental_setup}(a). The LFEX was focused at 3 mm above the target chamber center (offset position). The image at the offset position is relayed to the target chamber center with an image magnification of 1 by the spherical mirror. According to ray-trace code calculations, the deterioration of the image due to spherical aberration and astigmatism of the spherical mirror is negligible compared to the 70 $\mu$m diameter of the LFEX spot.
The laser energy fluence on the PM surface was optimized to be 90 J/cm$^2$ to obtain acceptable reflectivity (50$\%$) and spatial uniformity of the reflected pulse \cite{Morace2017}.
\subsection{Model for the hydrodynamics of the critical surface irradiated by multi-ps laser pulse}
We have previously derived the transition timing from the hole boring to
the blowout phase $t_{s}$, under constant laser irradiation (Eq.\,(8) in Ref.\,\cite{iwata}).
Here, the equation of transition timing is extended to a laser pulse with an arbitrary intensity temporal profile and the theoretical result is compared with that obtained by PIC simulation.
At the interface, plasma is pushed into a high-density region by the laser ponderomotive pressure caused by hole boring.
The hole boring velocity is conventionally derived with assumption of the initial and terminal states of the plasma components (ions, bulk electrons) and REs \cite{Ping2012,Kemp2008,Bagnoud2017,Vincenti2014,Sentoku2006}.
It is assumed that the laser is reflected at the interface, which is moving with velocity $v_p$, and that electrons and ions are initially immobile.
The flow velocity of REs is assumed to be $v_{h}$.
In a frame moving with the interface at velocity $v_p$, ions and bulk electrons drift at $-v_p$ toward the interface, where they are reflected. $1-f_e$ is the fraction of electrons reflected elastically to the velocity $+v_p$, at the interface. The remaining fraction $f_e$ of electrons are heated by the laser and accelerated to relativistic velocities $p_h/\gamma m_e$. All ions are assumed to be reflected elastically to $+v_p$, which
is the same assumption as that made by Vincenti \textit{et al.} \cite{Vincenti2014}.
The equations of the momentum flux conservation and the energy flux conservation are given by
$(1+R)I(t)/c\cos \theta \approx 2M_i n_i(t) v_p^2 + f_e n_e(t) v_h(t) p_h(t)$ and $(1-R)I(t)\cos \theta = f_e n_e(t) v_h(t) p_h(t) c$, respectively.
Here, $R$ is the reflectivity of the incident laser on the plasma and $\theta$ is the laser incident angle. $M_i$ and $n_i$ ($m_e$ and $n_e$) are the ion (electron) mass and number density, respectively.
Reflection and density steepening occur at the relativistic critical electron density $\gamma(t) n_c$ with $\gamma(t)=\sqrt{1+(1+R)a_0^2(t)/2}$,
where $a_0$ is the normalized laser field amplitude.
The initial electron density profile is assumed to be $n_e(x)=(\gamma n_c)\exp[(x-x_c(0))/l_s]$ with scale length $l_s$.
Note that in Ref.\,\cite{iwata}, $\cos\theta =1$ and $f_{e}=1$ are assumed.
The dashed lines in Fig.\,\ref{fig:Plasma_compression}(a) indicate the motion of the interface calculated by Eq.\,(\ref{eq: momentum conservation at the laser-plasma interaction surface_low}) with various reflectivities.
The red dashed line ($R$=0.7) reproduces the motion obtained by the PIC simulation until $t=$3.8\,ps.
This result shows that the velocity of the interface at each time can be determined by the momentum and energy balance among the laser, ions, and hot electrons, and that the integration of the interface velocity, Eq.\,\eqref{eq: momentum conservation at the laser-plasma interaction surface_low}, explains the motion of the interaction surface.
|
{
"timestamp": "2018-03-08T02:03:59",
"yymm": "1803",
"arxiv_id": "1803.02514",
"language": "en",
"url": "https://arxiv.org/abs/1803.02514"
}
|
\section{Resolving mazes with electricity}
\subsection{Preliminary considerations about electrical circuits and thermography}
The maze-solving problem and the shortest path problem are inspiring problems in algorithmics
and they involve many fields of science,
such as robotics or optimization.
In addition to numerical methods, many experimental methods have been proposed
to solve these problems, including fluids~\cite{Fuerstman2003},
memristors~\cite{Pershin2011},
living organisms (ants~\cite{Stratton1973}, honey bees~\cite{Zhang2000}, amoeba or ``blobs"~\cite{Nakagaki2000}, nematodes~\cite{Qin2007}, plants~\cite{Adamatzky2014})
or plasma~\cite{Reyes2002}.
In this chapter, a solution by a simple physical method using an electric current is proposed.
\begin{figure}[h]
\includegraphics[width=\textwidth]{labys_V3.eps}
\caption{\label{fig:labys}Examples of labyrinths :
a) The labyrinth on the floor of the cathedral at Chartres (France);
b) The logo of \textit{Monuments historiques} (national heritage sites) in France;
c) A handwritten labyrinth that was designed according to the intriguing instructions ``You have two minutes to design a maze that takes one minute to solve", reproduced from the \textit{Inception} movie (real. C. Nolan, 2010).}
\end{figure}
\begin{figure}[h]
\includegraphics[width=\textwidth]{mazes_V2.eps}
\caption{\label{fig:mazes}Examples of mazes:
a) A computer generated maze;
b) A plan of the Palace of Knossos (now a ruin near the town of Heraklion in Crete), the historical location of the myth of the Minotaur;
c) Map of a city with ways (streets and avenues).}
\end{figure}
First, mazes and labyrinths should be distinguished.
Labyrinths have only one way, which is very complicated and which generally leads to the center,
as can be seen in drawings on the floor of several cathedrals (see Fig.~\ref{fig:labys}).
In contrast, mazes possess a complex branching (see Fig.~\ref{fig:mazes}).
Although labyrinths are fascinating from a symbolic point of view, mazes are more interesting
\footnote{The title of this chapter is a tribute to the American sci-fi writer Robert Silverberg and his novel ``The Man in the Maze".}.
This chapter presents a simple physical method to solve a maze, using an electric current.
The maze can be done by an electrical circuit that is constituted by, for example, copper tracks printed on an epoxy card~\cite{Ayrinhac2014}.
Basically, two points of the maze are connected with a battery:
if the entrance and the output of the maze are connected, then the electric current flows and
the maze is solved.
If they are unconnected, then the circuit is open and no current flows.
A simple ohmmeter (usually a multimeter in a particular mode) gives the answer:
if the resistance between two points is very low, then the path is continuous;
in contrast, if the resistance is very high, then the path is broken.
However, in this method, the exact path followed by the current is unknown.
Thermography is a contactless and nondestructive method that can reveal the good path.
The power $P$ dissipated by a resistor, with electrical resistance $R$, is given by Joule's law
\begin{equation}
\label{eq:jouleslaw}
P=R I^{2}.
\end{equation}
For a resistor obeying Ohm's law $U=RI$ the electrical energy provided by the battery is integrally converted into heat.
When the electric charges flow, the temperature increase in the tracks is due to Joule heating.
The temperature increase $\Delta T$ is limited by thermal losses in the circuit.
A first origin of thermal losses is conduction, which depends on the surrounding materials
and the contact areas (controlled by the size of the circuit).
A second origin is radiation produced by a hot body.
A third origin is the convective heat transfer between an object and the surrounding fluid---in this case, the atmosphere.
Given that the radiation heat transfer is negligible at low temperature,
the following simple scaling law is relevant for a standard circuit on printed circuit board (PCB)~\cite{Adam2004} :
\begin{equation}
\label{eq:scalinglaw}
\Delta T \propto I^{2}.
\end{equation}
The increase in temperature is visualized by a thermal camera that
detects infrared radiation.
\begin{figure}[h]
\begin{tabular}{cc}
\includegraphics[width=.45\linewidth]{labysmall.eps} &
\includegraphics[height=.3\linewidth]{circuit.eps} \\
\end{tabular}
\caption{
\label{fig:labysmall} (left) The maze used in the experiment;
\label{fig:circuit} (right) The maze on a printed circuit board with dimensions 15$\times$15~cm$^{2}$.
The conductive copper tracks have a thickness of 35~$\mu$m
and a width of 800~$\mu$m.
The circuit is covered by a plastic transparent sheet.}
\end{figure}
\begin{figure}[h]
\begin{tabular}{cc}
\includegraphics[width=.5\linewidth]{laby_resolu_vf.eps} &
\includegraphics[width=.5\linewidth]{fig1.eps} \\
\end{tabular}
\caption{\label{fig:fig1} (left) The studied maze without dead ends ;
(right) Thermal image (320 $\times$ 240 pixels) of the circuit connected to the battery at points labelled (A) and (B).
The infrared light captured by the camera immediately shows the correct path! The colour bar on the right of the image is the temperature scale in degrees Celsius.
This scale is calculated with an emissivity parameter of $0.95$.
The spectral range of the camera is in the long-wave (LW) region (i.e. 7.5-13~$\mu$m).
Note that the minimum temperature on the scale is not the room temperature.}
\end{figure}
Thermal cameras are often used for educational purposes~\cite{Vollmer2001, Mollman2007, Xie2011, Haglund2016, Netzell2017}
to provide a clear visualization of invisible phenomena
or to illustrate complex phenomena.
There a wide range of topics in physics~\cite{Vollmerbook} or in chemistry~\cite{Xie2011a}
where a thermal camera may come in handy.
With this kind of apparatus, qualitative as well as quantitative applications are possible.
Although prices have decreased significantly in recent years, thermal cameras are still rather expensive.
However, there are other devices suitable for thermal imaging applications: such as a
simple webcam with an IR filter~\cite{Gross2005}
or a smartphone-based device such as FLIR ONE or Seek Thermal.
The main purpose of an infrared camera is to convert IR radiation intensity into a temperature measurement and to show the spatial variations in a false color visual image.
Intensity is integrated from a spectral band, generally in the long-wave infrared (7.5-13 µm) region.
The temperature is given by a formula accounting for three phenomena~\cite{Vollmerbook}: the true thermal emissions from the object,
the thermal radiation emitted by its surroundings and reflected by the object, and the atmospheric absorption.
For a proper temperature measurement, knowledge of certain parameters (e.g., emissivity, humidity, distance, and ambient temperature) is necessary.
Thermal imaging can find a wide application in electronics. For example, electrical components in the microelectronic boards of computers produce heat that can damage the circuits. To avoid failures, processors or power transistors need to be cooled by fans or Peltier modules, for example. IR imaging is a non-contact and non-destructive technique that can be used to test and survey electronic boards, allowing a diagnostics of possible malfunctions. Given that these boards are often made of different materials, the differences in components emissivity makes quantitative temperature measurements difficult (an explanation of emissivity will be given later).
Temperature measurement depends strongly on the emissivity $\epsilon$ of materials.
Unfortunately, for metals, the emissivity is very low, and they are hard to see directly in thermography.
Emissivity is defined as the ratio of the amount of the radiation emitted
from the surface to that emitted by a blackbody at the same temperature~\cite{Vollmerbook}.
A blackbody is a perfect absorber for all incident radiation. It appears black when cooled at 0~K and
when heated up it emits light at all wavelengths and the resulting spectrum
(given by Planck's law~\cite{Vollmerbook}) depends only on the temperature of the blackbody.
Kirchhoff's law of thermal radiation states that $\epsilon=\alpha$
where $\alpha$ is the absorption coefficient~\cite{Besson2009}.
This formula means that, for an opaque body, the more a body absorbs, the more it emits light.
A metal is a good reflector, so it has a bad absorption and, therefore, a poor emissivity.
\subsection{Experiments on circuits}
A regular maze (see Fig.~\ref{fig:circuit}) is printed on a epoxy card with tracks made of copper.
A transparent plastic sheet, which possess a higher emissivity than bare metal,
is placed on top of the circuit to ensure that the temperature increase is seen by thermography.
The transparent cover sheet allows the maze to be seen in both in infrared light and visible light.
With a thermal camera, the correct track appears to be immediately illuminated,
despite the complexity of the circuit (see Fig.~\ref{fig:fig1}).
Our maze is designed to highlight the following special features (see Fig.~\ref{fig:figs23}):
\begin{itemize}
\item In case of branching with a path twice as long as the parallel branch,
the shorter path appears to be more brightly illuminated compared to the longer path.
This happens because the trace
resistance $R$ is proportional to the length of the resistor $\ell$, such as
$R = \ell / \sigma A$, where $\sigma$ is the electrical conductivity and $A$ the cross-sectional area.
The voltage $U$ is equal in the two branches and gives $I_{\ell}=2I_{2\ell}$,
so with equation~(\ref{eq:scalinglaw}),
the temperature increase in the shortest path is four times higher than in the longer path $\Delta T_{\ell}=4\Delta T_{2\ell}$.
\item If the branching is configured as a Wheatstone bridge, then
the parallel branch does not appear.
See Fig.~\ref{fig:simu9} and the associated text for explanations.
\item In case of multiple paths, the branching is complicated and the shortest path is hard to be seen.
\end{itemize}
\begin{figure}[h]
\begin{tabular}{cc}
\includegraphics[width=.45\linewidth]{fig2_V2.eps} &
\includegraphics[width=.45\linewidth]{fig3_V2.eps} \\
\end{tabular}
\caption{\label{fig:figs23}
(left) The battery is connected to the circuit by the points labelled (1) and (2).
This picture illustrates the consequences of two particular circuit configurations:
the difference between two paths with a path with double length compared to another (lower arrow)
and the Wheatstone Bridge (upper arrow).
The battery (right) is connected to the circuit by the points labelled (I) and (II).
Because there are many good branches with equal lengths, it is difficult to identify the shortest path.}
\end{figure}
This kind of demonstration is possible provided that several conditions are met: the tracks should have the same section and they should be built with the same material, the branching should not be too complex (i.e., one-solution mazes) and the correct path should exist among many dead ends. So, the ideal circuit is an intermediate between a labyrinth and a maze.
The circuit can be drawn by an ink pen on a paper sheet (see Fig.~\ref{fig:papermaze}).
However, despite the care taken in the drawing, the tracks are not perfectly regular and the paper is much more fragile than a PCB and can be torn easily.
Nevertheless, this method is cheaper and faster than printing the same maze on a PCB.
To further reduce costs, the IR camera can be replaced by a temperature sensitive liquid crystal film (around 15\$),
to obtain qualitatively the same result (see Fig.~\ref{fig:thermalsheet}).
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=.4\linewidth]{papermaze_imgcolor_V2.eps} &
\includegraphics[width=.5\linewidth]{papermaze_imgtherm.eps} \\
\end{tabular}
\caption{\label{fig:papermaze}
(left) A maze is drawn on a paper sheet with a pen delivering conductive ink (around 1~$\Omega$/cm).
It takes about 10 minutes to draw the whole maze.
This maze was previously presented in detail in Ref.~\cite{Ayrinhac2014}.
Despite the care taken in the drawing, the tracks are not perfectly regular
and the paper is much more fragile compared to a PCB, and can be torn easily
(right). The correct path appears illuminated with an infrared camera.
The temperature increase is clearly seen in this case because the paper has a higher emissivity compared to metallic conductive tracks.
Due to the sideways spreading of the heat, the correct path in the image looks ``blurred".
This method is cheaper and faster compared to printing the same maze on a PCB.}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=.4\linewidth]{thermalsheet1.eps} &
\includegraphics[width=.5\linewidth]{thermalsheet2_V2.eps} \\
\end{tabular}
\caption{\label{fig:thermalsheet}
(left) A thermochromic liquid crystal film (the size of the sheet is 15$\times$15 cm$^{2}$), sensitive to temperature,
with a hand print.
The transition from black to color occurs between 20--25$^{\circ}$C.
The colour change is reversible and quick, with a response time about 10~ms~\cite{Stasiek2014}.
(right) The correct path of the maze appears illuminated with the liquid crystal film placed on the PCB.}
\end{figure}
\subsection{Simulated circuits}
To investigate more complex topologies, various circuits were simulated
in the permanent regime using Kirchhoff's laws~\cite{Chabaybook} :
the algebraic sum of currents at a node is zero (Kirchhoff node rule),
and the directed sum of the voltages around a loop is zero (Kirchhoff loop rule).
The operation involves a solution of a linear system
involving resistances, currents and applied voltages
with $n$ equations, where $n$ is the number of branches in the electrical network.
The studied circuits are directly generated by drawing the tracks in an image file
(see the example in Fig.~\ref{fig:circuit_equiv}).
The battery voltage is 10~V and the track electrical resistance is 1~$\Omega$ by unit of length
(equal to the track width). This resistance value is arbitrary.
The nodes are not be taken into account in the calculation of resistance.
In the nodes, the current is calculated by the averaging of the surroundings currents.
The resulting picture represents the current $I$ in amperes at each point of the circuit.
Note that the resulting picture is not a thermography image rendering.
\begin{figure}[h]
\begin{tabular}{ccc}
\includegraphics[width=.2\linewidth]{maze0_V2.eps} ~~~~~~~~~~&
\includegraphics[width=.3\linewidth]{circuit_equiv.eps} ~~~~~~~~~~&
\includegraphics[width=.3\linewidth]{maze0_simu.eps} \\
\end{tabular}
\caption{\label{fig:circuit_equiv}
(left) A circuit drawn in an image file (9$\times$8 pixels), imitating copper tracks on a PCB.
(center) The equivalent electrical circuit.
(right) Distribution of the intensities obtained by the application of Kirchhoff's laws (see text).
The color bar at the right indicates the intensity values in each branch. }
\end{figure}
In a grid-like circuit (Fig.~\ref{fig:simu18}), the current is spread over the whole circuit.
In this case, all of the paths are equivalent and the current appears equal in all of the paths.
In another example with disordered tracks (Fig.~\ref{fig:simu13}), the shortest path appears clearly.
In some cases, the current can fall to zero in a part of the circuit. This is due to the creation of an ``electrical bridge", also called Wheatstone bridge,
as illustrated in the Fig.~\ref{fig:simu9}.
The process to follow the shortest path between two points connected by the battery is to choose
at each node the branch where the intensity is maximum.
Generally speaking, the shortest path is the path where the intensity is maximized.
This idea is sustained by the basic electric conception that more current follows the path of less resistance.
The resistive grid was early used to explore some physical problems, such as solution of partial differential equations~\cite{Liebmann1950},
or mobile robot path planning~\cite{Tarassenko1991}.
In robot path planning, a collision-free environment can be modelled with a resistive grid of uniform resistance,
and obstacles are represented by regions of infinite resistance.
The path planning can be evaluated in real space if the robot moves through a maze, for example,
or in the configurational space where the dimensions are the degrees of freedom of the robot, considering a robot manipulator arm, for example.
The path from start to goal is found using voltage measurements from successive nodes.
In the limit of the continuous case, if we assume that conductivity is uniform and constant,
then the electromagnetism equations imply that for steady currents (in
regions where there is no sources) the electric potential $V$ obeys Laplace's equation.
The two-dimensional Laplace's equation
\begin{equation}
\frac{\partial^{2} V(x,y)}{\partial x^{2}}+\frac{\partial^{2} V(x,y)}{\partial y^{2}}=0,
\end{equation}
may be solved to calculate the electric potential $V$ at every point $(x,y)$.
The direction of the movement is given locally by the direction of the voltage gradient $\vec{\nabla} V$.
Globally, this approach produces an optimal path solution, depending on the limit conditions to avoid spurious local minima.
\begin{figure}[h]
\begin{tabular}{cc}
\includegraphics[width=.3\linewidth]{maze20_a.eps} ~~~~~~~~~~~~~~~~&
\includegraphics[width=.4\linewidth]{maze20_b_V2.eps} \\
\end{tabular}
\caption{\label{fig:simu18}
(left) A grid-like circuit. The battery is located in the upper branch
(right). Distribution of the intensities inside the studied circuit.
The color bar at the right indicates intensity values in each branch.}
\end{figure}
\begin{figure}[h]
\begin{tabular}{cc}
\includegraphics[width=.3\linewidth]{maze13_a_V2.eps} ~~~~~~~~~~~~~~~~&
\includegraphics[width=.4\linewidth]{maze13_b_V3.eps} \\
\end{tabular}
\caption{\label{fig:simu13}
(left) A circuit maze with disordered tracks. The battery is located in the upper branch
(right). Distribution of the intensities inside the studied circuit. The color bar at the right indicates the intensity values. }
\end{figure}
\begin{figure}[h]
\begin{tabular}{ccc}
\includegraphics[width=0.25\linewidth]{maze9_a_V2.eps} ~~~&
\includegraphics[width=0.35\linewidth]{maze9_b_V3.eps} ~~~&
\includegraphics[width=0.3\linewidth]{bridge.eps} \\
\end{tabular}
\caption{\label{fig:simu9}
(left) A complex circuit. The battery is located in the upper branch
(center). In the simulated circuit, the central part does not appear.
This is due to the creation of an electrical ``bridge"
(right). Diagram of an electrical circuit containing an electrical ``bridge".
Resistors and currents are labelled $R$ and $i_{n}$, respectively.
If the circuit is well-balanced, then all $R$ are equal, and it can be demonstrated that $i_{3}=0$.
This configuration is called a Wheatstone bridge and it is often used to measure resistance.}
\end{figure}
\section{Discussion of the physical mechanisms}
After the experimental demonstration, a physical question remains:
How does the electric current choose the correct path amongst many others?
A common explanation is that the battery produces a potential difference $\Delta V$ at the two extremities of the circuit and
the resulting electric field $\vec{E}$ possesses a magnitude constant and a direction along the wire.
This is especially puzzling in a maze circuit, where the electric field must follow the multiple bends of the circuit.
In contrast, the electric field $\vec{E}$ and then the potential difference $\Delta V$ is produced by distributions of point charges.
So, where are the charges producing the electric field inside the wires?
This question is challenging because \textit{electrokinetics} and \textit{electrostatics}
are two topics that are usually treated separately in physics textbooks:
charges distributions on one side, and electrical circuits on the other side.
Consequently, for most students, the two topics are unconnected, leading to many misconceptions~\cite{Rainson1994};
that is, commonly held beliefs that have no basis in actual scientific knowledge.
This point was extensively examined in the literature~\cite{Rosser1970, Heald1984, Jackson1996, Preyer2002, Muller2012}.
The electric charges responsible for the electric field inside the conductor are located on the surface of the wire.
This fact is known from the pioneering works of Weber and Kirchhoff~\cite{Assis2007}, but it has been completely forgotten during the last 150 years.
The quantity of surface charge is very small: the order of magnitude of the charge necessary to turn an electric current 1~A around a corner
is equal to about the charge of one electron~\cite{Rosser1970}.
A quantitative estimate in a typical circuit~\cite{Muller2012} for the magnitude of the surface charge density
is $10^{-12}$-$10^{-10}$~C$\cdot$m$^{-2}$.
Comparatively, the quantity of charge moving inside the wires is much higher, about 10$^{-6}$~C$\cdot$m$^{-2}\cdot$s$^{-1}$, which corresponds to 1~A.
Although the quantity of surface charges is small, their role is essential \cite{Jackson1996}:
they ensure that the equality of the potential in the conductors is at equilibrium,
they permit the circulation of the charges
and they produce an electric field outside the wires.
Two types of surface charges can be distinguished \cite{Muller2012}:
at the boundary of two conductors with different resistivities
and at the surface of the conductors.
The free electrons in the metal are pushed by the electric force arising from the electric field.
If there is a curve, then they pile-up on the surface and their electric field changes the pathway of the incoming moving charges.
There is a \textit{feedback mechanism} between the surface charges and the charges moving inside.
Chabay and Sherwood \cite{Chabaybook} provide an excellent and very accessible overview
of this problem for undergraduate students.
This feedback mechanism explains how the charges avoid dead-ends of the circuit maze.
First, they pile-up on the extremity of the dead-end;
the build-up of negative charge pushes the arriving electrons and
then the flowing current reaches zero.
Finally, the only path left for the charges to follow is the solution path of the maze.
This phenomenon is analogous to liquid propagating in a microfluidic network~\cite{Fuerstman2003}.
Electric circuits are often compared to hydraulic circuits from a pedagogical point of view
(the most complete comparison can be found in Table~1 in Ref.~\cite{Oh2012}).
However, there are fundamental differences between electrons and water:
electrons do not interact with one another
and energy is not carried by the free electrons.
Energy is carried outside the circuit by the electromagnetic fields
forming the Poynting vector $\vec{S} = \frac{1}{\mu_{0}} \vec{E} \times \vec{B}$.
This formula combines the magnetic field $\vec{B}$ due to electric current inside the wires (moving charges)
and the electric field $\vec{E}$ due to surface charges.
Solving the maze with an electric current
reveals the existence of a transient period between the beginning of the experiment
and the moment that the current is stabilized in the solved maze
\footnote{In the following discussion, we do not consider the thermal equilibration of the system,
which requires more time than electric equilibration.}.
Simulations performed by Preyer in a simple RC circuit~\cite{Preyer2002}
can give us a better understanding of phenomena observed in the transient state.
Just after the connection of the battery at two points of the maze,
the electric field spreads through the circuit at the speed of light.
During this step, the surface charges build on the tracks.
The surface charges locally change the electric field
and the current because they influence each other.
This feedback mechanism occurs in the transient state and is at work when the uniform current flow is established,
which means that the maze is solved.
All of the changes occur at the speed of light $c$ inside the material around the circuit (usually air),
which is also the speed that the information propagates between different parts of the circuit.
The drift velocity $v$ of charges moving inside the wires is much slower than $c$,
typically a few microns a second.
The simple propagation of the electric field through the whole circuit needs a time $\tau \approx \ell / c$,
where $\ell$ is the characteristic length of the maze,
but the time $\tau'$ needed for the feedback mechanism to operate is much longer~\cite{Preyer2002} with $\tau' > 2 \ell / c$,
which can be considered as the minimum time needed to solve the maze.
With this physical method, the maze resolution is fast. This explains why
this is considered to be the fastest and the cheapest method among many other physical methods~\cite{Adamatzky2017},
especially if the circuit is drawn by ink pen on a paper sheet.
\section{Acknowledgements}
The thermal camera was provided by the ``UFR de physique" at UPMC.
The author is indebted to M. Fioc for his valuable comments on the manuscript.
|
{
"timestamp": "2018-03-08T02:06:45",
"yymm": "1803",
"arxiv_id": "1803.02594",
"language": "en",
"url": "https://arxiv.org/abs/1803.02594"
}
|
\section{Introduction}
Elastic capsules consist of a thin elastic shell enclosing
a fluid inside. Elastic microcapsules are found in nature,
for example, as red
blood cells or virus capsids. They can also be produced
artificially by various methods, for example,
interfacial polymerization at liquid-liquid interfaces or multilayer
polyelectrolyte deposition \cite{Neubauer2013}.
Artificially produced microcapsules are attractive systems
for encapsulation and transport, for example, in
delivery and release systems.
Their overall shape is often nearly spherical,
and the shell can be treated
as a two-dimensional elastic solid with a curved equilibrium shape.
In experiments and for applications,
elastic properties of capsules can be
tuned by varying size, thickness, and shell materials.
For applications involving delivery by rupture of capsules
it is necessary to understand and
characterize the mechanical properties and elastic instabilities
of capsules.
The mechanical properties of elastic capsules are governed
by the elastic shell, which is curved (typically spherical)
in its equilibrium shape.
This gives rise to
different characteristic instabilities in response to external
forces \cite{Fery04,Vinogradova2006,Neubauer2013}.
Thin elastic membranes bend much more easily than stretch.
This protects a curved equilibrium shape against deformations
changing its Gaussian curvature and that is the reason for the
stability of capsules under uniform compression.
In contrast to fluid drops, elastic capsules under uniform compression
fail in a buckling instability
below a critical volume or critical internal pressure
\cite{Gao01, Sacanna11, Datta2012, Knoche11, Knoche14, Knoche2014a}.
Buckling-type instabilities
can also be triggered by external
forces, for example, in electrostatically
driven buckling transitions of charged shells \cite{Jadhao14} or in
hydrodynamic flows \cite{Boltz15}.
Under point force loads, for example, exerted by atomic force microscopy tips,
elastic capsules indent linearly at small forces and assume
buckled shapes in the nonlinear
regime at higher forces \cite{Fery04,Zoldesi08,Vella2012}.
As opposed to fluid droplets, elastic capsules can also
develop wrinkles upon deformation
\cite{Rehage2002,Vella2011,Aumaitre2013,Knoche13}
if compressive hoop stresses arise.
Microcapsules can be manipulated and deformed in
hydrodynamic flow \cite{Pieper1998,Rehage2002,Barthes-Biesel2011},
by micromanipulation using an atomic force
microscope \cite{Fery04,Vinogradova2006}
or micropipettes or capillaries \cite{Aumaitre2013,Knoche13}.
Another promising route to exert
mechanical forces and to actuate elastic capsules in a noninvasive manner
is via magnetic or electric fields
\cite{Degen08,Karyappa2014}. For magnetic fields
this requires the presence of magnetizable material
either in the shell or in the capsule interior.
The whole capsule then acquires a magnetic dipole moment,
which can be manipulated in external magnetic fields.
For actuation by electric fields the capsule has to contain polarizable
dielectric material such that the capsule acquires an electrostatic dipole
moment, which can be manipulated by an electric field.
Homogeneous fields orient dipole moments but
also induce
capsule deformations, which increase the size of the dipole moment
after orientation.
Therefore, homogeneous fields always lead to
stretching and elongation of the capsule.
Inhomogeneous fields can also exert a net force
on the capsule and induce directed motion at
fixed magnetic dipole moment along the field gradient.
In the following we focus on
spherical elastic capsules that are filled with a
(quiescent) magnetic fluid and deformed
in homogeneous external magnetic fields.
As magnetic fluid we consider
a ferrofluid, which is a liquid that is magnetizable by external
magnetic fields because it consist of ferromagentic or ferrimagnetic
nanoparticles
suspended in a carrier fluid. Because of the small particle size, ferrofluids
are stable against phase separation and show superparamagnetic behavior
\cite{Rosensweig85}. Ferrofluids are used in technical and
medical applications \cite{Voltairas01, Holligan03, Liu07}.
All our results also apply to elastic capsules filled with a
(quiescent) dielectric fluid which are placed in a
homogeneous external electric field.
The problem of ferrofluid droplets in
uniform external magnetic fields has already been theoretically
studied in the
literature. Also, a spherical ferrofluid droplet
is elongated in the direction of the magnetic field
for increasing field strength; the resulting elongated shape
was observed to be nearly spheroidal
\cite{Arkhipenko79}. Bacri and Salin \cite{Bacri82}
used the assumption of a spheroidal shape
for a quite precise approximation of the
elongation by minimizing the total energy.
Although the droplet is only elongated by the field, an
abrupt shape transition is possible \cite{Bacri82}:
Beyond a threshold magnetic field strength
the spheroidal droplet becomes unstable and
elongates discontinuously into a shape with
conical tips.
The conical shape is stabilized by a positive feedback between
shape and magnetic field distribution:
A sharp tip gives rise to a diverging field
strength at the tip, which in turn generates strong
stretching forces stabilizing the sharp tip.
The mechanism of forming sharp tips is
reminiscent of the normal field instabilities (Rosensweig
instabilities) of free planar ferrofluid surfaces in a perpendicular
homogeneous magnetic field, which were first described by Cowley and
Rosensweig \cite{Rosensweig67} and later extended by
a nonlinear stability analysis to study
subsequent pattern formation \cite{Boudouvis1987}.
The discontinuous shape transition to a conical shape exhibits
hysteresis and only occurs above a critical susceptibility $\chi_c$
of the ferrofluid. In Refs.\ \cite{Li1994,Ramos1994} a value
$\chi_c \simeq 16.59$ was found below which no conical shape can exist;
a slender-body approximation in Ref.\ \cite{Stone99} gives
$\chi_c \simeq 14.5$. Using the approximative energy minimization for
spheroidal shapes of Bacri and Salin \cite{Bacri82} gives
$\chi_c \simeq 19.8$. The jump in droplet elongation at the transition to a
conical shape depends on the magnetic susceptibility: Large
elongation jumps are possible for high susceptibilities. This
behavior was investigated in more detail in several numerical
studies \cite{Guillaume92, Lavrova04, Afkhami10}. Apart from free
ferrofluid droplets, the deformation behavior of sessile droplets on
a plate \cite{Zhu11} or sedimenting ferrofluid drops in external
fields \cite{Korlie08} have also been investigated for homogeneous
external magnetic fields.
Dielectric droplets in a
homogeneous external electric field exhibit the same
shape transition from a spheroidal to a conical shape.
For the electric field, however, free charges exist, and
conducting droplets are easily realized experimentally. In fact, the first
experimental observations of conical droplet shapes were
made for water droplets \cite{Zeleny1917} and soap bubbles
\cite{Wilson_Taylor1925}. In Ref.\ \cite{Ramos1994} it was
shown that also a conducting liquid droplet surrounded by
an outer conducting liquid in a homogeneous electric field
exhibits conical shapes above a corresponding critical conductivity
ratio $\sigma/\sigma_{\rm out} = 1+ \chi_c \simeq 17.59$.
In the limit of an ideally conducting droplet with
infinite susceptibility or
infinite conductivity (both resulting in zero electric field
inside the droplet),
the conical solutions in Refs.\ \cite{Li1994,Ramos1994,Stone99} approach
Taylor’s cone solution with a
half opening angle $\simeq 49.3^\circ$ \cite{Taylor1964}.
Both for liquid metal (i.e., ideally conducting) and
dielectric droplets, the conic cusp formation has been studied dynamically
and dynamic self-similar solutions have been obtained
\cite{Zubarev2001,Zubarev2002}.
Fluid droplets, which are neither perfect
conductors nor perfect insulators,
disintegrate at higher external electric fields
by emitting jets of fluid at the tip, from which
small droplets pinch off. Also for this process,
scaling laws for droplet sizes
could be theoretically obtained \cite{Collins2008,Collins2013}.
In a ferrofluid-filled elastic capsule the
ferrofluid drop is enclosed by a thin elastic membrane,
which will modify the
transition from a spheroidal to a conical shape observed for droplets.
Such ferrofluid-filled capsules
have already been realized experimentally.
Neveu-Prin {\it et al.}\ \cite{NeveuPrin93}
encapsulated ferrofluids by polymerization and analyzed
the magnetization behavior of the magnetic capsules.
Degen {\it et al.}\ \cite{Degen08}
investigated experimentally elastic capsules filled with a magnetic
liquid in an external magnetic field. They used a
magnetic liquid
consisting of micrometer-sized magnetic particles
that do not show the special
properties of ferrofluids but form long chains in the presence of
external magnetic fields.
These magnetite-filled elastic capsules could be actuated
to deform in a magnetic field.
A quantitative theoretical description of their
deformation is still missing.
In Ref.\ \cite{Karyappa2014}, capsules
filled with a dielectric liquid in an external
electric field were investigated experimentally and theoretically
with a focus on small deformations.
We will describe the elastic shell by a nonlinear
elastic model based on a Hookean elastic energy density
for thin shells,
assume axisymmetric capsules, and
calculate the shape at force equilibrium by solving shape
equations as they have been derived in Refs.\
\cite{Knoche11, Knoche13}.
As stated above, homogeneous magnetic fields acting on ferrofluid-filled
capsules
give rise to stretching and elongation of the capsule in order
to increase the total dipole moment. Therefore, stretching tensions
are dominant in the elastic shell. This is why we will
consider the limiting case of vanishing
bending modulus and bending moments
for most of the present work, which is commonly called
the elastic membrane limit (as opposed to the elastic shell case).
In our numerical approach, the magnetic field inside the capsule
is calculated using a coupled finite element and boundary element method.
The capsule shape provides the geometric boundary for the
field calculation.
Vice versa, the magnetic field distribution
couples to the shape equations via the magnetic surface stresses.
We solve the full coupled problem numerically by an iterative method.
We combine this numerical approach with several analytical approaches to
investigate the capsule deformation in a homogeneous
magnetic field as a function of the magnetic field strength
and Young modulus of the capsule material.
First we characterize the linear deformation regime of spheroidal
capsules for small fields both numerically and analytically.
Then we answer the question to what extent
the elastic shell will suppress the discontinuous
spheroidal to conical
shape transition of a ferrofluid droplet and
whether elastic properties such as the Young modulus
of the shell material can be used to tune and control
the instability.
We show that conical shapes can also occur for capsules with
nonlinear Hookean membranes
but require diverging strains at the conical tips.
As a real elastic material is not able to support arbitrarily high strains,
we expect that diverging local stretch factors at the capsule poles
indicate that real capsules tend to rupture close to
the poles as soon as the conical shape is assumed.
Then the existence of a sharp discontinuous shape transition
into a conical shape provides an interesting route to trigger
capsule rupture at the poles at rather well-defined magnetic
(for ferrofluid-filled capsules)
or electric (for dielectric-filled capsules) field values.
The subsequent rupture process has some analogies to the onset of the
disintegration of droplets in electric fields
\cite{Collins2008,Collins2013}, but our
static approaches based on nonlinear Hookean material
laws are not suited to model the rupture process itself.
We find that the discontinuous shape transition
between spheroidal and conical shapes
with hysteresis effects and shape bistability
is also present for elastic ferrofluid-filled capsules.
Numerically, we obtain a complete classification of the shape transition
in the parameter
plane of dimensionless magnetic field strengths (magnetic Bond number)
and the dimensionless ratio of the Young modulus of the shell material
and the surface tension of the ferrofluid.
These findings are partly corroborated by an analytical
approximative energy minimization extending
the spheroidal shape approximation
of Bacri and Salin \cite{Bacri82}
to ferrofluid-filled capsules.
For conical shapes we generalize the slender-body approximations
of Stone {\it et al.}\ \cite{Stone99}, which allows us to
quantify the divergence of local stretch factors at the capsule poles
and to show that the same analytic formula as for ferrofluid droplets
governs the dependence of the cone angle on the magnetic
susceptibility $\chi$ (or the dielectric susceptibility
${\varepsilon}/{\varepsilon_{\rm out}}-1$ for a dielectric droplet with
dielectric constant
$\varepsilon$ in a surrounding liquid with $\varepsilon_{\rm out}$).
In particular, we predict the critical susceptibility $\chi_c$, above
which the hysteretic shape transition between spheroidal and
conical capsule shapes can be observed, to be
{\it identical} to the critical value for ferrofluid or dielectric droplets.
We also find that, for elastic capsules,
magnetic stretching can give rise to
wrinkling along the capsule equator region.
We predict the parameter range for the
appearance of wrinkles and the extent of the wrinkled
region on a spheroidal capsule
depending on its elastic properties and its elongation.
\section{Theoretical model and numerical methods}
\label{sec:theory}
We start with a ferrofluid drop suspended in an external
nonmagnetic liquid of the same density as the ferrofluid, which eliminates
gravitational forces.
Thus the drop is force-free except for the surface
tension $\gamma$, which forces the drop to be spherical and is balanced
by internal pressure.
If the drop is enclosed by an elastic shell, for example, after
a polymerization reaction at the liquid-liquid interface,
we have a spherical elastic capsule. We assume
that the relaxed rest shape of this capsule is spherical
with a rest radius $R_0$, which is given by the fixed
volume $V_0 = 4\pi R_0^3/3$ of the droplet or capsule.
After applying a uniform magnetic field $H_0\vec{e}_\text{z}$ in
the $z$ direction,
the resulting shape of the capsule becomes stretched in the
$z$ direction, but the capsule shape and magnetic field
distribution remain axisymmetric around
the $z$ axis.
A uniform external magnetic field causes mirror-symmetric forces on the
capsule, resulting in a shape with reflection symmetry with respect
to the plane $z=0$ (see Fig.\ \ref{fig:geometry}).
\subsection{Geometry}
\label{sec:geometry}
We describe the axisymmetric shell using cylindrical
coordinates $r$, $z$, and $\varphi$. The capsule's shell is thin compared to its
diameter, so we consider the shell to be a two-dimensional elastic
surface. Because of the axial symmetry,
we only need the contour line $r(z)$
to describe the whole capsule shape.
For our calculations, we parametrize the surface by the arc length $s_0$ of the
undeformed spherical contour
with $s_0 \in [0, L_0 = \pi R_0]$, starting at the lower apex
and ending at the upper apex. Using the reflection symmetry,
we only need half of
that interval, $s_0 \in [0, {L_0}/{2}]$, to describe the capsule's shape
completely. In addition to the coordinates $r(s_0)$ and $z(s_0)$, we
define a
slope angle $\psi(s_0)$ by the unit vector
$\vec{e}_s$ following the contour line via $\vec{e}_s = (\cos\psi,
\sin\psi)$.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=0.25\textwidth,clip]{parametrization.pdf}
\caption{
Illustration of the parametrization in cylindrical coordinates
($r,z,\varphi$) and the contour line with arc length $s$. The
complete capsule is obtained by revolution of the red contour line, while the
angle $\psi$ describes its slope. This contour line is calculated
numerically. The polar radius is called $a$, while $b$ denotes the
equatorial radius.
}
\label{fig:geometry}
\end{center}
\end{figure}
\subsection{Magnetostatics}
\subsubsection{Forces by the ferrofluid}
In order to calculate the shape of the capsule in an
external magnetic field, we have
to take the magnetic forces that are caused by the ferrofluid
on the capsule surface into
account.
Because we are interested in a static solution, we can
assume that the
fluid is at rest. Then the fluid can only exert hydrostatic
forces {\it normal}
to the surface, while tangential components are zero.
In order to calculate the normal
magnetic force density $f_m(r,z)$ on the surface,
we use the magnetic stress tensor by Rosensweig,
\cite{Rosensweig85}
\begin{equation}
\label{eq:stress_tensor}
f_m(r,z) = \mu_0 \int\limits_0^{H(r,z)}M(r,z)\text{d}H(r,z)
+ \frac{\mu_0}{2}M_n^2(r,z).
\end{equation}
Here $M = |\vec{M}|$ is the absolute value
of the magnetization and $M_n =\vec{M}\cdot \vec{n}$
its normal component ($\vec{n}$ is the outward unit
normal to the capsule surface).
Magnetization $M$ and magnetic field $H$ are taken
on the inside of the capsule surface.
We assume a linear magnetization law
\begin{equation}
\vec{M} = \chi\vec{H}
\label{eq:linear}
\end{equation}
with a susceptibility $\chi$ for the ferrofluid ($\chi = \mu-1$ in terms of
its magnetic permeability $\mu$), which is justified for small fields
$H\ll M_s/3\chi$, where $M_s$ is the saturation magnetization of the
ferrofluid.
References \cite{Boudouvis1988,Basaran1992}
studied the behavior of drops with a nonlinear Langevin magnetization
(polarization) law.
The saturation of the magnetization or polarization forbids sharp
tips and leads to more rounded drops.
It was shown, on the other hand, that the linear law is a very good
approximation for small and even medium fields.
This typically requires the maximum magnetic flux density
$B_{\text{max}}=\mu_0H_{\text{max}}$ to be in a range of $50-100\,{\rm
mT}$, depending on the specific fluid \cite{Chantrell1978,Zhu11}.
For a linear magnetization we can rewrite Eq.\ \eqref{eq:stress_tensor} as
\begin{equation}
\label{fm}
f_m(r,z) = \frac{\mu_0\chi}{2}\left[H^2(r,z) + \chi H_n^2(r,z)\right]
\end{equation}
(assuming $\chi_{\rm out}=0$ for the external
non-magnetic liquid or using $\chi = {\mu}/{\mu_{\rm out}}-1$
in terms of the magnetic permeabilities $\mu$ of the ferrofluid and the
$\mu_{\rm out}$ of the external liquid),
where $H = |\vec{H}|$ and $H_n$ is the
normal component of the magnetic field.
We will use this position-dependent normal magnetic
force density to modify the pressure in our elastic
equations in Sec.\ \ref{sec:shape_eqns}.
\subsubsection{Calculation of the magnetic field}
\label{fieldcalculation}
To calculate the total magnetic field, i.e., the
superposition of the external uniform
field and the field from the ferrofluid magnetization,
we use the fact that
ferrofluids are generally non-conducting \cite{Rosensweig85}.
Then Maxwell's
equations give $\nabla \times {\bf H} = 0$, which allows us to introduce
a scalar
magnetic potential $u$ with $\vec{\nabla} u = \vec{H}$.
From Maxwell's equation $\vec{\nabla} \cdot \vec{B} = \vec{\nabla}
\cdot \mu_0(\vec{H} + \vec{M})=0$
we get Poisson's equation in magnetostatics
\begin{equation}
\label{eq:Poisson}
\vec{\nabla}^2 u(r,z) = - \vec{\nabla}\cdot \vec{M}(r,z).
\end{equation}
For the linear magnetization law \eqref{eq:linear},
Poisson's equation simplifies to the Laplace equation
$\vec{\nabla}^2 u(r,z) = 0$.
For the numerical solution of this partial differential equation we use a
coupled axisymmetric
finite element -- boundary element method \cite{Costabel87,
Wendland88, Arnold83, Arnold85} with a cubic spline interpolation for the
boundary \cite{Ligget81}. This combination of methods was also
used by Lavrova {\it et al.}\ for
free ferrofluid drops \cite{Lavrova04, Lavrova05,Lavrova06}
and earlier for electric drops, e.g.,
by Harris and Basaran \cite{Harris1993}.
The finite element method (FEM) is used to solve
Eq.\ \eqref{eq:Poisson} in the magnetized domain inside the capsule
and the boundary element method (BEM) for the nonmagnetic domain outside.
Both domains are coupled by the continuity
conditions of magnetostatics for
$u$ and its normal derivative on the boundary of the capsule,
\begin{equation}
\label{eq:magnetic_boundary}
u_{\text{in}} = u_{\text{out}},~~
\mu\frac{\partial u_\text{in}}{\partial n}
= \frac{\partial u_\text{out}}{\partial n},
\end{equation}
with $\mu_{\rm out}=1$ for the external nonmagnetic liquid.
Both the FEM and BEM exploit axial symmetry and effectively
operate in the two-dimensional
$rz$ plane, where the axisymmetric capsule shape
is described by a contour line $(r(s),z(s))$.
For the FEM we
use a standard Galerkin method with linear elements on a
triangular two-dimensional
grid in the $rz$ plane that is created with a Delauney
triangulation using the Fade2D software package \cite{Fade2D}, where
we set a
fixed number of grid points on the capsule's boundary.
In the BEM we express solutions $u(\vec{r}_0)$ of the
Laplace equation $\vec{\nabla}^2 u = 0$ for $\vec{r}_0$ on the
outside or the boundary
of the capsule in terms of integrals over the boundary of the
capsule.
Using fundamental solutions with rotational
symmetry \cite{Wrobel02}, we have to solve a set of one-dimensional integrals
over the whole boundary of the capsule
\begin{equation}
\label{eq:BEM_integral_eq}
cu(\vec{r}_0) - \int\limits_0^L \left[
u(\vec{r})
\frac{\partial u_{\text{ax}}^*(\vec{r}_0, \vec{r})}{\partial {n}}
- \frac{\partial u(\vec{r})}{\partial {n}}
u_{\text{ax}}^*(\vec{r}_0, \vec{r}) \right]
r \text{d}s = z_0.
\end{equation}
Here $u_{\text{ax}}^*(\vec{r}_0, \vec{r})
\equiv \int_0^{2\pi}u^*(\vec{r}_0, \vec{r})\text{d}\varphi$
is the axially symmetric fundamental solution of Laplace's equation,
which is obtained from the
fundamental solution
$u^*(\vec{r}, \vec{r}_0) = {1}/{4\pi|\vec{r} - \vec{r}_0|}$
of Laplace's equation,
$\Delta u^*(\vec{r}, \vec{r}_0) = -\delta(\vec{r} - \vec{r}_0)$.
In the integral equation (\ref{eq:BEM_integral_eq}),
$u$ and its normal derivative
are evaluated on the outside of the capsule surface.
The point $\vec{r}_0$
is the point where $u$ is to be calculated,
while the integrals are taken over points $\vec{r}(s)$ on the capsule
contour. Both $\vec{r}$ and $\vec{r}_0$ lie in the same $rz$-plane.
For the geometric factor $c$, we have $c={1}/{2}$ for
points $\vec{r}_0$ on the boundary $\Gamma$ and $c = 1$ for points
$\vec{r}_0$ in the exterior domain.
The vector $\vec{n}$ denotes the outward unit
normal vector and $z_0$ describes the $z$-component of $\vec{r}_0$.
On the right-hand side of \eqref{eq:BEM_integral_eq},
$z_0$ can be interpreted as the potential of the external
electric field.
For numerical evaluation, the integrals in Eq.\ \eqref{eq:BEM_integral_eq}
are discretized by a point
collocation method and solved by applying Gaussian quadrature for nonsingular
integrands and a midpoint rule for weakly singular integrands.
The FEM and BEM are coupled at the boundary by the continuity conditions
(\ref{eq:magnetic_boundary}).
The FEM provides values
for $u$ on every finite element grid point inside the capsule
including values $u_{\text{in}}$ on
the inner side of the boundary; in addition, the normal derivatives
${\partial u_\text{in}}/{\partial n}$
on the inside of the discretized capsule boundary are needed
for the FEM but remain {\it a priori} unknown.
Values for these normal derivatives on the boundary points of the
FEM grid are obtained by
the BEM method.
Our BEM uses linear interpolation for $u$ between the discretized
boundary points.
We use the continuity conditions \eqref{eq:magnetic_boundary}
to write the boundary integral equation
\eqref{eq:BEM_integral_eq} in terms of quantities on the
inner capsule boundary.
Using one BEM equation \eqref{eq:BEM_integral_eq}
for each boundary point (with $c=1/2$),
we obtain a set of equations
that allows us to calculate the
unknown derivatives ${\partial u_\text{in}}/{\partial n}$
for given $u_\text{in}$ and to get a
closed system of equations for $u$ everywhere inside
the capsule.
After solving the resulting system of FEM equations
we know $u$ everywhere inside
the capsule.
For the calculation of $u$ inside the capsule and thus
for the calculation of the magnetic force density $f_m(r,z)$ acting
on the capsule using (\ref{fm}), which is also calculated
with the magnetic field on the inside, it is
not necessary to calculate $u$ in the
entire external domain explicitly. This is done implicitly by the BEM.
If needed (for example, in order to calculate the field in the exterior
regions in Fig.\ \ref{fig:fieldplot}),
$u$ can be calculated by
solving \eqref{eq:BEM_integral_eq} for
points $\vec{r}_0$ in the exterior with $c=1$.
In a ferrofluid capsule or drop with sharp edges, very high
field strengths can arise [see Fig.\ \ref{fig:fieldplot}(c)].
Also field gradients can be large, which makes pointed shapes prone
to discretization errors caused by the grid.
This effect can be countered to
some degree by placing more FEM
grid points at the tip in order to improve
the precision there, which is, however, limited by the BEM part of the
solution scheme:
The collocation points must not come too close to the symmetry axis because
the weakly singular integrals become strongly singular
on the $z$ axis \cite{Lavrova2006}. This
leads to massively increasing numerical errors near the axis and a decrease of
the overall precision.
Overall, our numerical scheme to calculate singular
BEM integrals is not the most advanced as a trade-off for simplicity.
There are more elaborated schemes for the integration of singular integrals
as, for example, developed over many years by Gray
{\it et al.} \cite{Gray1990,Gray2004}, which could
provide a more elegant way to deal with the problem.
We use the following compromise for the discretization:
We place $N=250$ elements on the boundary such that the
length $L_i$ of the $i$th boundary
element (beginning at the equator) is given by
\begin{align}
L_i = c_0\exp\left(\ln(l_0)\frac{i-1}{N} \right).
\label{eq:Li}
\end{align}
The constant $c_0$ is chosen in order to obtain the
correct total arc length $L$, which is given by
the meridional stretch factors $\lambda_s = {\text{d}s}/{\text{d}s_0}$
of the deformed capsule [see Eq.\ (\ref{eq:lambda}) below],
$\sum_{i=1}^N L_i = L/2 = \int_0^{{L_0}/{2}} \lambda_s ds_0$.
We choose $l_0 = 0.1$ ($l_0 = 1$ gives a constant element length and $l_0 <
1$ leads to a higher element density at the capsule's tip).
Increasing $N$
beyond 250 does not improve the precision
significantly. A higher density of points at the
capsule's tip (lower $l_0$) leads to stronger oscillations in the iterative
solution scheme (see Sec.\ \ref{sec:iterative_solution} below).
\begin{figure*}[htbp]
\begin{center}
\includegraphics[width=0.8\textwidth,clip]{H_fieldplot.pdf}
\caption{
Numerical results for the magnetic field distribution and capsule
shape (two-dimensional projection)
for a capsule filled with a ferrofluid with a susceptibility of $\chi
= 21$. The ratio of Young's modulus and surface tension is
$Y_{2\text{D}}/\gamma = 100$.
The external magnetic field $H_0$ is uniform and points in the upward
direction.
Arrows indicate the local direction of $H$; the color codes for
the absolute value of $H$ in units of $H_0$. The (a) spherical capsule and
the (b) spheroidal capsule have uniform fields inside, while the field in
the (c) conical-shaped capsule increases strongly in the tips.
The elongations $a/b$
(ratio of the polar radius to the equatorial radius) are
(a) $a/b = 1$, (b) $a/b = 2.26$,
and (c) $a/b = 5.38$.
The magnetic Bond numbers $B_m$
[see definition in Eq.\eqref{BmDefinition}]
are (a) $B_m = 0$ , (b) $B_m = 262.4$, and (c) $B_m = 702.2$.
}
\label{fig:fieldplot}
\end{center}
\end{figure*}
\subsubsection{Electric fields and dielectric liquid}
\label{sec:electric}
Our approach to elastic capsules filled with a ferrofluid in a
magnetic field also applies to capsules filled with a dielectric fluid
in an electric field.
The generic situation for a capsule filled with a
fluid with dielectric constant $\varepsilon$ is to be suspended
in a dielectric liquid with a different $\varepsilon_{\rm out}\neq
\varepsilon$,
which does not equal unity $\varepsilon_{\rm out} \neq 1$.
Then the dielectric force density in a linear medium is
\begin{equation}
f_e(r,z) = \frac{\varepsilon_0\varepsilon_{\rm out} \chi_\varepsilon}{2}
\left( E^2(r,z) + \chi_\varepsilon
E_n^2(r,z)\right)~,~~
\chi_\varepsilon \equiv \frac{\varepsilon}{\varepsilon_{\rm out}}-1,
\label{fmepsilon}
\end{equation}
which is completely analogous to (\ref{fm}) with
$\chi_\varepsilon$ playing the role of the susceptibility $\chi$.
For the general case, Poisson's equation becomes
\begin{equation}
\vec{\nabla}^2 \phi(r,z) = -\frac{1}{\epsilon_0}\vec{\nabla}\cdot \vec{P}(r,z),
\end{equation}
with the electric potential $\phi$ and the polarization $\vec{P}$.
For a linear polarization law, it simplifies to the Laplace equation
$\vec{\nabla}^2 \phi(r,z) = 0$.
\subsection{Equilibrium shape of the capsule}
\label{sec:elastic_modell}
\subsubsection{Elasticity and shape equations}
\label{sec:shape_eqns}
The capsule is deformed by the normal magnetic
stresses $f_m$ from the ferrofluid. We
have to calculate the resulting deformed equilibrium
shape, where all
elastic stresses, surface tension and magnetic stress are
balanced everywhere on the capsule.
Every point of the reference shape $[r_0(s_0), z_0(s_0)]$
is mapped onto a new point $[r(s_0), z(s_0)]$. The deformed
shape $[r(s_0), z(s_0)]$ is calculated by solving shape equations, which are
derived from nonlinear
theory of thin shells \cite{Libai98,Pozrikidis03, Knoche11,
Knoche13}. We use a Hookean elastic energy density with a spherical
rest shape. The Hookean
elastic energy density (defined as energy per undeformed unit area)
is given by
\begin{equation}
w_s = \frac{1}{2}\frac{Y_{2\text{D}}}{1-\nu^2}
(e_s^2 +2 \nu e_s e_\varphi+e_\varphi^2)
+\frac{1}{2}E_{\text{B}}(K_s^2+2\nu K_sK_\varphi + K_\varphi^2).
\label{eq:ws}
\end{equation}
Here $e_s$ and $e_\varphi$ are meridional and circumferential strains that
contain the stretch factors $\lambda_s$ and $\lambda_\varphi$:
\begin{equation}
e_s = \lambda_s - 1, \quad e_\varphi = \lambda_\varphi -1~,~~
\lambda_s = \frac{\text{d}s}{\text{d}s_0}, \quad
\lambda_\varphi = \frac{r}{r_0}.
\label{eq:lambda}
\end{equation}
Here and in the following,
quantities with subscript 0 refer to the undeformed
spherical reference shape and quantities without 0 describe
the deformed shape.
Analogously, the
bending strains $K_s$ and $K_\varphi$ are generated by the
curvatures $\kappa_s$ and $\kappa_\varphi$:
\begin{equation*}
K_s= \lambda_s\kappa_s - {\kappa_s}_0, \quad
K_\varphi = \lambda_\varphi \kappa_\varphi - {\kappa_\varphi}_0~,~~
\kappa_s = \frac{\text{d}\psi}{\text{d}s}, \quad
\kappa_\varphi = \frac{\sin\psi}{r}.
\end{equation*}
In the elastic energy (\ref{eq:ws}), $Y_{2\text{D}}$ is the
two-dimensional Young modulus governing stretching deformations,
$E_\text{B}$ is the bending modulus, and $\nu$ is the
two-dimensional Poisson ratio.
Elastic properties are usually only weakly
$\nu$ dependent; we use $\nu = 1/2$, which is the
typical value for an incompressible polymeric material.
The arc length of the deformed capsule's contour is given by
$ L = \int_0^{L_0} \lambda_s ds_0$,
while $L_0 = \pi R_0$ is the fixed arc length
of the undeformed spherical capsule.
In experiments, the capsule's shell is constructed by polymerization
on the surface of a drop. Therefore, the undeformed reference shape,
which is spherical in the absence of gravity,
is also a solution of the Laplace-Young equation
\begin{equation}
\label{LaplaceYoung}
\gamma(\kappa_s+\kappa_\varphi) = p,
\end{equation}
where $\gamma$ is the surface tension of the droplet.
The solution of the Laplace-Young equation
will be discussed in detail in
Sec.\ \ref{sec:DropletTheorie} below.
In the following, we will neglect the bending energy, which means we set
$E_\text{B} = 0$. The characteristic length scale of the problem is the
radius $R_0$ of the undeformed sphere, such that
the neglect of the bending energy corresponds to the limit of large
F{\"o}ppl-von K\'{a}rm\'{a}n numbers
$\gamma_{\rm FvK}\equiv Y_{2\text{D}}R_0^2/E_B$.
This is the limiting case of an elastic Hookean membrane and
is a good approximation for two reasons. First,
we will only consider capsules with thin shells as they were prepared in
experiments \cite{Knoche13, Degen08}. The shell thickness $D$ is very
small compared to the capsule size, $D \ll R_0$. With $Y_{2\text{D}}
\propto D$ and $E_\text{B} \propto D^3$ it follows that
$\gamma_{\rm FvK}\sim (R_0/D)^2 \gg 1$ and
stretching energies are typically larger than bending energies.
The second argument
is that the homogeneous magnetic field acting on the ferrofluid-fluid
capsule predominantly stretches and elongates the capsule in order
to increase its total dipole moment.
This increases stretching energies, whereas
the capsules develop high curvatures only
at the conical tips. However,
we show below that stretch factors diverge at conical tips, so
the stretching energy dominates over the
bending energy associated with these high curvatures
also in the tip regions.
Elastic tensions in the shell (defined as force per
deformed unit length) derive from the surface elastic energy density
by
\begin{align}
\begin{split}
\tau_s &= \frac{1}{\lambda_\varphi}\frac{\partial w_s}{\partial e_s}
= \frac{Y_{2\text{D}}}{(1-\nu^2)\lambda_\varphi}
\left[(\lambda_s-1) + \nu(\lambda_\varphi-1)\right],\\
\tau_\varphi &= \frac{1}{\lambda_s}\frac{\partial w_s}{\partial
e_\varphi} = \frac{Y_{2\text{D}}}{(1-\nu^2)\lambda_s}
\left[(\lambda_\varphi-1) + \nu(\lambda_s-1)\right].
\end{split}
\label{eq:taulambda}
\end{align}
Although we use a Hookean
elastic energy density, the constitutive relation (\ref{eq:taulambda})
is {\it nonlinear} because
of the additional $1/\lambda$ factors, which arise for purely geometrical
reasons: The Hookean elastic energy density is defined per
undeformed unit area such that $\partial w_s/\partial e_s$ is
the force per undeformed unit length, whereas the Cauchy stresses $\tau_s$
and $\tau_\varphi$ are defined per deformed unit length.
In addition to the elastic tensions $\tau_s$ and $\tau_\varphi$,
there is also a contribution
from an isotropic effective surface tension $\gamma$
between the outer liquid and the capsule.
Such a contribution arises
either as the sum of surface tensions of the liquid outside
with the outer capsule surface and the liquid inside with the
inner capsule surface or, if
the capsule shell is porous such that there is still contact
between the liquids outside and inside the capsule,
with additional contributions
from the surface tension between outside and inside liquids.
In the absence of elastic tensions, the surface tension $\gamma$
also gives rise to the spherical rest shape of the capsule.
For macroscopic capsules the surface tensions should be negligible,
but for microcapsules with weak walls they should not be neglected.
We expect the effective surface tension $\gamma$ to be somewhat
smaller than typical liquid-liquid surface tensions, which are
around $\gamma = 50\,{\text{mN}}/{\text{m}}$; we will use
$\gamma = 10\,{\text{mN}}/{\text{m}}$ below.
The equilibrium of forces in the deformed elastic membrane is described by
\begin{align}
0&=\tau_s\kappa_s + \tau_\varphi\kappa_\varphi +
(\kappa_s+\kappa_\varphi)\gamma -p,
\label{eq:equilibrium_eqns_norm}\\
0&=\frac{\cos \psi}{r}\tau_\varphi -
\frac{1}{r}\frac{\text{d}(r\tau_s)}{\text{d}s},
\label{eq:equilibrium_eqns_tang}
\end{align}
where Eq.\ (\ref{eq:equilibrium_eqns_norm})
describes the normal force equilibrium and Eq.\
(\ref{eq:equilibrium_eqns_tang})
tangential force equilibrium (in the $s$ direction,
equilibrium in the $\varphi$ direction is always fulfilled by axial
symmetry).
In the presence of magnetic forces, the pressure
\begin{equation}
p(s) = p_0 + f_m(s)
\label{eq:pfm}
\end{equation}
is modified by the magnetic stress $f_m$, which is
a position-dependent normal stress
pointing outwards and thus stretching the capsule and given
by the magnetic field at the capsule surface [see Eq.\ (\ref{fm})].
It is important to note that magnetic forces are always
normal to the surface such that they do no enter the tangential
force equilibrium (\ref{eq:equilibrium_eqns_tang}).
The (homogeneous) pressure $p_0$ is the Lagrange multiplier for the
volume constraint $V=V_0=4\pi R_0^3/3$.
The equations of force equilibrium and geometric relations
can be used to derive
a system of four first-order differential equations with the
arc length $s_0$ of the undeformed spherical contour as
an independent variable,
which are called shape equations in the following:
\begin{align}
\label{eq:shape_eqns}
\begin{split}
r'(s_0)&=\lambda_s\cos\psi~~,~~
z'(s_0)=\lambda_s\sin\psi,\\
\psi'(s_0)&=\frac{\lambda_s}{\tau_s + \gamma}
\left[-\kappa_\varphi(\tau_\varphi+\gamma) + p(s_0)\right],\\
\tau_s'(s_0)&=\lambda_s\frac{\cos\psi}{r}(\tau_\varphi-\tau_s).
\end{split}
\end{align}
In these shape equations, the surface tension $\gamma$ gives an
isotropic and constant
stress contribution, in addition to the elastic stresses
$\tau_s$ and $\tau_\varphi$. This is because
we assume that the undeformed rest state,
where the elastic stresses $\tau_s$ and $\tau_\varphi$ vanish, is
identical to the shape of a ferrofluid droplet of surface tension $\gamma$.
We neglect that $\gamma$ could change during capsule preparation and
during elastic deformation.
The system of shape equations
is closed by the constitutive relation
(\ref{eq:taulambda}) for $\tau_\varphi$ and the
relations
\begin{align*}
\lambda_s &= (1-\nu^2)\lambda_\varphi\frac{\tau_s}{Y_{2\text{D}}}
-\nu(\lambda_\varphi-1)+1~~
\mbox{with}~
\lambda_\varphi = \frac{r}{r_0},~~
\kappa_\varphi = \frac{\sin\psi}{r},
\end{align*}
where the first relation derives from
the constitutive relation (\ref{eq:taulambda}) for $\tau_s$
and the second relation is geometrical.
For further details on the derivation of the shape equations, we refer
the reader to
Refs.\ \cite{Libai98, Knoche13, Knoche11}.
\subsubsection{Numerical solution of the shape equations}
The system of shape equations \eqref{eq:shape_eqns} has to be solved
numerically. The integration starts at the pole with $s_0 = 0$
and runs to the capsule's equator at $s_0 = {L_0}/{2}$. To integrate the four
first-order differential equations we have three boundary conditions
at $s_0 = 0$:
\begin{equation}
r(0) = 0, \quad z(0)~{\rm arbitrary}, \quad \psi(0)=0.
\label{eq:shape_eqns_bc}
\end{equation}
The condition for $r(0)$ follows from the absence of
holes in the capsule. We can choose $z(0)$ arbitrarily because
the external magnetic field does not depend on the $z$ coordinate.
The boundary condition $\psi(0)=0$ at the pole
seems to exclude possible conical capsule shapes with
$\psi(0)>0$. We discuss this issue below in Sec.\ \ref{sec:conical}
and in Appendix \ref{app:cone_tip2}.
There we derive the boundary condition $\psi(0)=0$
for finite stretches $\lambda_s$ and $\lambda_\varphi$ at the poles.
The boundary condition $\psi(0)=0$ also arises if the magnetic forces
$f_m$ remain finite at the poles such that the normal
force equilibrium requires finite
curvatures at the poles.
Conical shapes, however, have
divergent stretches $\lambda_s$ and $\lambda_\varphi$
and divergent magnetic normal forces $f_m$ at their
conical tips. In the numerical calculation of capsule shape and
magnetic field we have to
discretize the capsule surface such that divergences are cut off
(this numerical issue is discussed in more detail also
in Appendix \ref{app:errors})
and the boundary condition $\psi(0)=0$
for finite stretches $\lambda_s$ and $\lambda_\varphi$
or finite magnetic force $f_m$ is appropriate.
Then the right-hand side
of the shape equation for $\tau_s$ in \eqref{eq:shape_eqns}
vanishes, $\tau_s'(0)=0$ for $s_0=0$ [see also Eq.\ (\ref{eq:tau1_app})],
which can be used to start the
integration at the pole.
{\it A priori}, a fourth boundary condition for
the tension $\tau_s(0)$ at the pole is unknown.
On the other hand, we have $\psi\left({L_0}/{2}\right)=\pi/2$ as a
matching condition at $s_0 = {L_0}/{2}$ to prevent kinks there. With the
help of this matching condition, we can use a shooting method to determine
$\tau_s(0)$.
To increase numerical stability, we expand the shooting
method to a multiple shooting method, where we use several integration
intervals with several matching points.
To keep the volume of the capsule constant, we have to use the internal
pressure $p_0$ as the Lagrange multiplier, which is adjusted during the
calculation. In order to do so, $p_0$ becomes another shooting parameter
with $V-V_0$ as the corresponding residual. In this work, we use a fourth
order Runge-Kutta scheme with a step size of $\Delta s_0=10^{-6}$ in the first
integration interval starting at the apex and $\Delta s_0=10^{-4}$ in all
other intervals, while there is a total of 250 integration intervals.
\subsubsection{Wrinkling}
\label{sec:WrinkleTheorie}
A ferrofluid-filled capsule is stretched
in a uniform external magnetic field in the
direction of the magnetic field.
As opposed to a ferrofluid droplet, a capsule can develop
wrinkles if circumferential compressive stresses arise as a result
of this stretching.
Because of volume conservation, the circumferential
radius of the capsule has to decrease in the equator region
giving rise to compression with $\lambda_\varphi < 1$ in this region
and a region of negative elastic stress $\tau_\varphi < 0$ develops.
In
contrast to a droplet with a liquid surface and constant
surface tension $\gamma >0$, regions of negative
total hoop stress
$\gamma+ \tau_\varphi < 0$ can develop for capsules if the
negative elastic hoop stress exceeds the surface tension.
Then the elastic shell can reduce its total energy by
developing wrinkles in the circumferential direction (see
Fig.\ \ref{fig:3Dwrinkling} for illustration).
These wrinkles cost stretching energy in the meridional $s$ direction
and bending energy,
but this is compensated by a
release of compressional stresses and a reduction of
elastic compression energy in the $\varphi$ direction.
Strictly speaking, $\gamma + \tau_\varphi<0$
is only an approximation neglecting the bending energy, which will
also increase upon wrinkling, and the negative stress has to exceed
a small Euler-like threshold value.
We expect the wrinkles to occur in a region near the capsule
equator. Thus they will be roughly parallel to the external magnetic field and
therefore we assume that they do not effect the magnetic
properties of the capsule.
In order to introduce wrinkling in the shape
equations, we will use the same approach that has been used
for pendant capsules in Ref.\ \cite{Knoche13}.
The wrinkles will break the axial symmetry.
In the wrinkled regions, where
$\gamma + \tau_\varphi<0$, we approximate the shape by
an axisymmetric pseudomidsurface
$(\overline{r}(s_0), \overline{z}(s_0))$ for which we
use modified axisymmetric shape equations, where we set
$\gamma + \tau_\varphi=0$. This condition states that
the total circumferential hoop stress is completely relaxed by
fully developed
wrinkles \cite{Davidovitch11}.
This leads us to a new set of equations (see also Ref.\ \cite{Knoche13}),
which read
\begin{align}
\label{eq:wrinkle_eqs}
\overline{r}'(s_0)&= \lambda_s\cos\overline{\psi},~~
z'(s_0) =\lambda_s\sin\overline{\psi},~~
\overline{\psi}'(s_0) =\frac{\lambda_s}{\overline{\tau}_s
+ \overline{\gamma}}p,~~
\overline{\tau}_s'(s_0) = -\lambda_s\frac{\cos\overline{\psi}}{\overline{r}}
(\overline{\tau}_s+\overline{\gamma}).
\end{align}
We also have to introduce a modified effective surface tension
$\overline{\gamma}= {\lambda_\varphi}/{\overline{\lambda}_\varphi}$,
because the real surface area exceeds the pseudosurface area, and
we have to model this increase of $E_\gamma$ by increasing $\gamma$
instead. This new system of
differential equations is closed by the relations
\begin{equation*}
\lambda_s =\frac{\overline{\tau}_s\overline{\lambda}_\varphi
+Y_{2\text{D}}}{Y_{2\text{D}}- \nu \gamma}, ~~
\overline{\lambda}_\varphi =\frac{\overline{r}}{r_0}.
\end{equation*}
In order to calculate $\overline{\gamma}$, the circumferential stretch factor
$\lambda_\varphi$ of the \textit{real, wrinkled} surface has to be calculated
via the constitutive relations \eqref{eq:taulambda}.
To calculate wrinkled capsule shapes we start to solve the
shape equations \eqref{eq:shape_eqns} as described before. As soon as the
condition $\tau_\varphi + \gamma < 0$ is valid, we continue the calculations
by solving the modified system \eqref{eq:wrinkle_eqs}.
By following the
solution of the modified system, we can calculate
the length $\L_\text{w}$ of the wrinkled region
\begin{align}
L_{\text{w}} = \int\limits_{\tau_\varphi + \gamma < 0} \text{d}s.
\label{eq:Lw}
\end{align}
At this point, it is also possible to calculate the wavelength of the
wrinkles using the same methods as in Ref.\ \cite{Knoche13}.
Here we will mainly be interested in the extent $L_\text{w}$
of the wrinkled region.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.2\textwidth,clip]{wrinkle_skizze.pdf}
\caption{
Three-dimensional illustration of a wrinkled capsule. The length
$L_{\text{w}}$ of the wrinkles is measured as the length of the
region, where $\tau_\varphi + \gamma < 0$.
The wrinkling wavelength is not determined explicitly here.
}
\label{fig:3Dwrinkling}
\end{center}
\end{figure}
\subsubsection{Ferrofluid droplet}
\label{sec:DropletTheorie}
The special case $Y_{2\text{D}}=0$ describes a ferrofluid droplet
without an
elastic shell and has been treated in the literature before.
The balance of forces on the
surface is given by the Laplace-Young equation
\eqref{LaplaceYoung}.
Using the definitions of $\kappa_s$ and $\kappa_\varphi$,
this equation can be translated into
\begin{equation*}
\frac{\text{d}\psi}{\text{d}s} = \frac{p}{\gamma} - \frac{\sin\psi}{r}.
\end{equation*}
In order to have a parametrization in the reference arc length $s_0$ and a
fixed integration interval, we introduce a constant stretch factor
$\lambda_s$,
which is adjusted as a shooting parameter. The boundary and matching
conditions are the same as in the case of the elastic shape
equations. Together with the already known geometrical relations for $r$ and
$z$ we get a system of three shape equations for a droplet:
\begin{align}
\label{dropsystem}
\begin{split}
r'(s_0)&=\lambda_s\cos\psi,~~
z'(s_0)=\lambda_s\sin\psi,~~
\psi'(s_0) = \lambda_s\left(\frac{p_0 + f_m}{\gamma}
- \frac{\sin\psi}{r}\right).
\end{split}
\end{align}
This system is solved in the same way as the shape equations
for elastic capsules in the previous
sections. The basic shooting parameters are given by $\lambda_s$ and $p_0$.
Our solution scheme for the Laplace-Young equation is chosen
such that it is
completely analogous and comparable to the elastic shape equations.
There are several other ways
to solve this equation with a volume constraint,
for example, by employing finite elements \cite{Brown1980}.
\subsection{Iterative numerical solution of the coupled problem}
\label{sec:iterative_solution}
The magnetostatic and the elastic problem are coupled: The
capsule shape determines the boundary conditions for the magnetic
field via the continuity conditions (\ref{eq:magnetic_boundary}), while
the normal magnetic force density $f_m(r,z)$
acting on the capsule surface [see Eq.\ (\ref{fm})] enters
the shape equations (\ref{eq:shape_eqns}) via the pressure [see
Eq.\ (\ref{eq:pfm})].
To find a joint solution we use an iterative numerical
solution scheme. We start
with the reference shape and calculate the corresponding magnetic field
$\vec{H}(r,z)$ for a given external field $\vec{H}_0$. Then, we
can calculate a deformed shape of the capsule using this magnetic field. Now we
recalculate the magnetic field and so on until the iteration converges.
At this fixed point, the solution of the shape equations and the
magnetic field are self-consistent. This iterative coupling of elastic shape
equations to an external field calculated by a boundary element method is
similar to the iterative scheme used in Ref.\ \cite{Boltz15} to
calculate the shape of sedimenting capsules in an external flow
field. For the problem of ferrofluid droplets, an analogous
iterative strategy has been introduced in
Refs.\ \cite{Lavrova04,Lavrova05,Lavrova06,Lavrova2006}.
The iteration can cause numerical problems in the solution of the
the nonlinear elastic shape equations. If the
capsule shape changes rapidly during the iteration,
the shooting method used to solve
the shape equations does not find a solution.
This problem can be reduced by slowing down the
iteration. To solve the elastic shape equations in the $n$th step, we
use a convex linear combination of the
updated magnetic field $\vec{H}'_n$ and
the magnetic field $\vec{H}_{n-1}$ from
the previous iteration step instead of $\vec{H}'_n$ itself
\cite{Lavrova2006,Boltz15}:
\begin{align}
\vec{H}_n = \vec{H}_{n-1}
+ \alpha(\vec{H}'_n - \vec{H}_{n-1}).
\end{align}
The parameter $\alpha$ ranges between 0 and 1 and has to be lowered in
situations of quickly changing shapes of the capsule. Finally, it is
switched back to 1 to ensure real convergence.
To track a solution as a function of the magnetic field
strength,
it is helpful to increase the external magnetic field
$\vec{H}_0$ in small steps $\Delta\vec{H}_0$ and let the
capsule's shape converge after each step. This slows down the
calculation speed drastically but increases numerical
stability and helps to track
a specific branch of stable
solutions (see Sec.\ \ref{sec:hysteresis_effects}).
A problem with the
iterative solution scheme can arise if the capsule shape becomes nearly
conical with a very sharp tip of high curvature. Then the numerical error in
the calculation of the magnetic field (see
Sec.\ \ref{fieldcalculation}), makes it difficult or even prohibitive
to reach a fixed point of the
iterative scheme. Instead the iteration gives
oscillations of the capsule shape around the required fixed point, which
worsens the quality of the results.
The iterative strategy used here directly converges to stationary
shapes without simulation of the real dynamics.
An alternative to our iterative scheme is to directly
simulate the dynamics for the
fluid from the electromagnetic, elastic, and hydrodynamic
forces. Then the fluid motion is simulated over
time until it reaches a steady state.
This method was used by Karyappa {\it et al.}\
for elastic capsules in electric fields \cite{Karyappa2014}.
For liquid droplets,
there are comparable problems with sharp tips and numerical singularities,
where the full dynamics could by solved to great accuracy,
such as the emission of fluid jets at the tip of drops
in electric fields \cite{Collins2008}, pinch-off dynamics \cite{Suryo2006},
and coalescence phenomena \cite{Anthony2017}.
The errors of the field calculation with finite elements at such sharp tips
can also be reduced by using advanced mesh algorithms,
such as the elliptic mesh generation \cite{Christodoulou1992}.
\subsection{Control parameters and non-dimensionalization}
\label{sec:control_parameters}
In order to identify the relevant control parameters
and reduce the parameter space, we introduce dimensionless quantities.
We measure lengths in units of the radius $R_0$ of the spherical
rest shape, energies in units of $\gamma R_0^2$, i.e., tensions
in units of the surface tension $\gamma$ of the ferrofluid, and
magnetic fields in units of the external field $H_0$.
The problem is then governed by essentially three dimensionless control
parameters.
The magnetic Bond number $B_m$,
\begin{equation}
\label{BmDefinition}
B_m \equiv \frac{\mu_0 R_0\chi H_0^2}{2\gamma},
\end{equation}
is the dimensionless strength of the magnetic force density.
With this dimensionless number, the Laplace-Young equation
(\ref{LaplaceYoung})
for a ferrofluid droplet can be written in dimensionless form
\begin{align*}
\tilde{\kappa}_s + \tilde{\kappa}_\varphi
&=\widetilde{p}_0 + B_m \left( \tilde{H}^2 + \chi \tilde{H}_n^2 \right),
\end{align*}
with $\tilde{H} \equiv H/H_0$, $\tilde{\kappa} \equiv R_0 \kappa$, and
$\tilde{p} \equiv pR_0/\gamma$.
The scaled droplet shape described by this Laplace-Young equation then only
depends on the two dimensionless
parameters $B_m$ and $\chi$.
The dimensionless Young modulus
${Y_{2\text{D}}}/{\gamma}$ is the control parameter for elastic properties
of the capsule shell. Another dimensionless control parameter for
elastic properties is Poisson's ratio $\nu$,
which is set to $\nu=1/2$ and thus
fixed throughout this paper.
The limit
${Y_{2\text{D}}}/{\gamma}=0$ describes a droplet without an elastic shell
while
${Y_{2\text{D}}}/{\gamma} \gg 1$ describes a system dominated by the
shell elasticity.
The three dimensionless parameters
$B_m$, ${Y_{2\text{D}}}/{\gamma}$, and the magnetic susceptibility
$\chi$ of the ferrofluid
uniquely determine the capsule shape
(apart from its overall size $R_0$).
In the following we consider Bond numbers $B_m$ between $0$ and $10^3$ (see
Sec.\ \ref{sec:results}).
For a typical ferrofluid-filled capsule with $\chi=21$, $R_0 = 1\,$mm
\cite{Karyappa2014,Zwar2018},
and
$\gamma = 0.01\,{\text{N}}/{\text{m}}$, these Bond numbers
correspond to magnetic field strengths
$H$ between $0$ and about $500\,$kA/m
(or fields $B=\mu_0H$ between $0$ and $0.5\,$T).
We consider dimensionless Young moduli ${Y_{2\text{D}}}/{\gamma}$
from $10^{-2}$ (nearly no elasticity)
to $100$ (elastically dominated)
and the purely elastic limit ${Y_{2\text{D}}}/{\gamma} = \infty$
(where the definition of $B_m$ is not useful anymore).
For the analogous problem of a dielectric droplet in an external
electric field $E_0$
we can introduce a dielectric Bond number $B_e$ by
$B_e = {\varepsilon_0\varepsilon_{\rm out} R_0\chi_\varepsilon E_0^2}/{2\gamma}$,
where $\chi_\varepsilon$ is the analog of the
magnetic susceptibility $\chi$ and has been defined in (\ref{fmepsilon}).
\section{Analytical approaches}
In this section we introduce three approximative analytical approaches
to the problem, which describe ferrofluid-filled elastic capsules
in three different deformation regimes.
The first approach is the analysis of the linear response of the
capsule to small magnetic forces.
The second approach applies to spheroidal shapes at moderate magnetic forces
and is an approximative minimization of the
total magnetic and elastic energy under the assumption of a spheroidal
shape and uniform stretch factors. This extends
the approximative energy minimization
of Bacri and Salin \cite{Bacri82} for ferrofluid droplets to capsules.
Finally, we investigate conical capsule shapes as they can arise
under strong magnetic forces. We investigate the existence of
conical shapes
and derive the governing equations
in a slender-body approximation by extending
the approach of Ref.\ \cite{Stone99} from conical droplets
to conical capsules.
\subsection{Linear shape response at small fields}
\label{sec:lin_def}
In this section we derive the linear response of the
spherical capsule shape to small magnetic forces. In particular,
we derive the
elongation $a/b$ of the capsule,
where $a$ denotes the capsule's polar radius and $b$ its
equatorial radius (see Fig.\ \ref{fig:geometry}).
Details of the derivation
are given in Appendix \ref{app:lin_def}; here we present the main
results.
At small fields displacements change linearly in the magnetic force
density $f_m$. Therefore, radial and tangential displacements
$u_R(\theta)$ and $u_\theta(\theta)$ (using spherical coordinates with a
polar angle $\theta$ and assuming axisymmetry)
are of $O(H^2)$.
In order to calculate the displacements we consider
the force equilibria in normal direction, i.e., the Laplace-Young equation
(\ref{eq:equilibrium_eqns_norm}),
and in tangential direction, i.e., Eq.\ (\ref{eq:equilibrium_eqns_tang}).
For a liquid
ferrofluid droplet with an isotropic surface tension $\gamma$
both force-equilibria give equivalent results.
Expanding to linear order
in the displacements around the spherical shape, we obtain two coupled
differential equations for the functions $u_R$ and $u_\theta$.
These linearized force-equilibrium equations can be solved exactly.
The solution takes the form
\begin{equation}
u_R= A+B\cos^2\theta,~~u_\theta = C\sin\theta\cos\theta,
\label{eq:AnsatzuRcap}
\end{equation}
where
$A$, $B$, and $C$ are determined in Appendix \ref{app:lin_def}
explicitly.
We find
$B = {\mu_0(5+\nu)\chi^2 H^2R_0^2}/{8[Y_{2\text{D}}+(5+\nu)\gamma]}$
from the normal force equilibrium, and
$C = - 2(1+\nu)B/(5+\nu)$ from the tangential force equilibrium,
and the pressure is adjusted such that
$A=-B/3$ in order to fulfill the volume constraint.
The functional form $u_R = A+B\cos^2\theta$ of the
normal displacement leads to a spheroidal shape in linear response.
For a spheroid we can use the relation $H=3H_0(3+\chi)$ and
obtain $B= R_0 B_m 9(5+\nu)\chi/4(3+\chi)^2[Y_{2\text{D}}/\gamma+(5+\nu)]$.
The linear response approach remains valid as long as $A,B,C\ll R_0$
or $B_m/[Y_{2\text{D}}/\gamma+(5+\nu)] \ll (3+\chi)^2/\chi \approx \chi$.
From the displacement $u_R(\theta)$ we can calculate its
elongation
\begin{equation*}
\frac{a}{b} \approx 1 + \frac{u_R(0)-u_R(\pi/2)}{R_0} = 1+ \frac{B}{R_0}
\end{equation*}
in linear order in the displacement.
For a ferrofluid droplet with surface tension
$\gamma$ and without any elastic tensions, i.e., $Y_{2\text{D}}/\gamma = 0$,
we get, for the elongation $a/b$
in linear order [see Eq.\ (\ref{eq:droplinear_app})],
\begin{align*}
\frac{a}{b} &= 1 + \frac{9\mu_0R_0\chi^2}{8\gamma(3+\chi)^2}H_0^2.
\end{align*}
For the general case $Y_{2\text{D}}/\gamma > 0$, we find
[see Eq.\ (\ref{eq:capdroplinear_app})]
\begin{equation}
\frac{a}{b} = 1 +\frac{9\mu_0R_0\chi^2(5+\nu)}
{8 [Y_{2\text{D}}+\gamma(5+\nu)](3+\chi)^2}
H_0^2
= 1 +\frac{9}{4}\frac{\chi}{(3+\chi)^2}
\frac{B_m}{{Y_{2\text{D}}}/{\gamma}(5+\nu) + 1},
\label{eq:lin_approx}
\end{equation}
which gives a precise prediction of the capsule's
elongation for small fields, as a comparison with the numerical results
in Fig.\ \ref{fig:lin_region} shows.
To leading order in $B_m$
Eq.\ (\ref{eq:lin_approx})
agrees with the results from a similar small deformation approach
in Ref.\ \cite{Karyappa2014} for capsules filled with a
dielectric liquid in electric fields.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.5\textwidth,clip]{lin_region.pdf}
\caption{
Elongation ${a}/{b}$ of a capsule filled with a ferrofluid with
$\chi = 21$ as a function of
$B_m /[Y_{2\text{D}}/\gamma + (5+\nu)]$ for different values of
$Y_{2\text{D}}/\gamma$ in the region of small deformations.
The solid line describes the linear
approximation from Eq.\ \eqref{eq:lin_approx}.
The best agreement between the numerical data and the linear
approximation is given for a
purely elastic system without wrinkling effects (closed purple circles).
Wrinkling effects lead to considerable deviations (squares).
}
\label{fig:lin_region}
\end{center}
\end{figure}
\subsection{Approximative energy minimization for spheroidal shapes}
\label{sec:AnalyticalApprox}
In this section we derive an analytical approximation for the
elongation $a/b$ of the capsule at moderate magnetic forces
by minimizing an
approximative total energy, which assumes a spheroidal shape
for magnetic and elastic contributions.
For ferrofluid droplets, the spheroidal approximation is based on
the experimental observation that the droplet shape in uniform
magnetic fields is very similar to a prolate spheroid \cite{Arkhipenko79,
Bacri82, Afkhami10} for sufficiently small
magnetic Bond numbers before a transition into a conical shape
can take place.
Our numerical results show that this behavior remains
qualitatively unchanged with an additional elastic shell (see
Sec.\ \ref{sec:spheroid}).
Therefore, we consider a capsule with
prolate spheroidal shape. Analogously
to Bacri and Salin \cite{Bacri82},
we use an energy argument by minimizing the total energy of
the capsule at fixed volume $V= (4\pi/3) ab^2 = V_0$.
The total energy consists of three different contributions.
First is the surface energy $E_\gamma$, which is caused by the surface tension
$\gamma$. It is proportional to the surface area $A$ and given by
\begin{equation}
E_\gamma = \gamma A = 2 \pi ab\left[\frac{b}{a} + \frac{1}{\epsilon}
\arcsin{\epsilon}\right]\gamma,
\label{Egamma}
\end{equation}
where $\epsilon \equiv \sqrt{1-{b^2}/{a^2}}$ is the eccentricity.
The
second energy contribution is the magnetic field energy
$E_\text{mag}$. According to Ref.\ \cite{Stratton41}, $E_{\text{mag}}$ can be
written as
\begin{equation}
E_{\text{mag}} = -\frac{V\mu_0}{2} \frac{\chi}{1 + n\chi} H_0^2
\label{Emag}
\end{equation}
for $\mu_{\rm out}=1$ and with
the demagnetization factor $n = ({b^2}/{2a^2\epsilon^3})\left[-2\epsilon +
\ln{\left( ({1+\epsilon})/({1-\epsilon})\right)} \right]$.
The third energy contribution is
the elastic stretching energy $E_\text{el}$, which we construct by taking the
energy density $w_s$ from Sec.\ \ref{sec:elastic_modell},
\begin{equation*}
E_\text{el} = \int w_s \text{d}A_0
= \int\frac{1}{2}\frac{Y_{2\text{D}}}{1-\nu^2}
(e_s^2 +2 \nu e_se_\varphi+e_\varphi^2)\text{d}A_0,
\end{equation*}
with $e_\text{s} = \lambda_s - 1$ and $e_\varphi = \lambda_\varphi-1$, as
defined in Sec.\ \ref{sec:shape_eqns}.
At this point,
the stretch factors $\lambda_s$ and $\lambda_\varphi$ are unknown and
we need further approximations. An acceptable approximation
for spheroidal shapes, which is checked below by comparison
with the numerics (see Fig.\ \ref{fig:lambda_s})
is constant stretch factors throughout the shell, i.e., $\lambda_s,
\lambda_\varphi = \text{const}$, which leads to
\begin{equation}
E_\text{el} =\frac{1}{2}\frac{Y_{2\text{D}}}{1-\nu^2}
(e_s^2 +2 \nu e_s e_\varphi+e_\varphi^2)A_0.
\label{Eel}
\end{equation}
We approximate the circumferential stretch factor
$\lambda_\varphi$ by the stretching
of a fiber at the capsule equator and set
\begin{equation*}
\lambda_\varphi = \frac{b}{R_0}.
\end{equation*}
In meridional direction we approximate $\lambda_s$ by taking the ratio of the
perimeter $P_\text{ellipse}$ of the corresponding ellipse, which generates the
prolate spheroid by rotation, and the perimeter
$P_\text{circle}=2\pi R_0$ of a great
circle on the initial sphere. The perimeter of the
ellipse is given by an elliptic integral. Therefore, we use Ramanujan's
approximation \cite{Ramanujan14}, which leads us to
\begin{equation*}
\lambda_s = \frac{P_\text{ellipse}}{P_\text{circle}}
\approx \frac{a+b}{2R_0}
\left(1 + \frac{3\eta^2}{10 + \sqrt{4-3\eta^2}}\right),
\end{equation*}
with $\eta \equiv {(b-a)}/{(b+a)}$.
As the last step,
we have to minimize the total energy
$E_\text{tot} = E_\gamma + E_{\text{mag}} + E_\text{el}$
with respect to the elongation ratio ${a}/{b}$ at fixed volume
$V= (4\pi/3) ab^2=V_0$
in order to get the equilibrium elongation as a function
of the magnetic Bond number $B_m$ for spheroidal shapes.
Details of the calculation are presented in Appendix \ref{app:Bm}.
We obtain a closed but quite complicated analytical expression
for the inverse relation $B_m=g(b/a)$, i.e.,
the magnetic Bond number $B_m$ as a function of the inverse
elongation $b/a<1$ for spheroidal shapes in Eq.\ (\ref{eq:BMab}).
The function $g(k)$ in Eq.\ (\ref{eq:BMab})
still depends on three dimensionless parameters:
the susceptibility $\chi$, the dimensionless
Young modulus $Y_{2\text{D}}/\gamma$, and Poisson's ratio $\nu$.
This relation reduces to the results of Bacri and Salin \cite{Bacri82}
for ferrofluid droplets in the limit $Y_{2\text{D}}= 0$.
\subsection{Conical membrane shapes with normal magnetic forces}
\label{sec:conical}
For ferrofluid-filled droplets a shape transition into a
stable conical shape with $\psi(0)> 0$
is possible above a critical susceptibility $\chi_c$ and
at high magnetic fields \cite{Bacri82,Wohlhuter92,Li1994,Stone99}.
We want to show that a conical shape
with a strictly conical tip can also exist for an elastic
capsule with spherical rest shape and normal
magnetic stretching forces if the constitutive relation
is of the nonlinear form
(\ref{eq:taulambda}).
Details of the argument are presented in
Appendix \ref{app:cone_angle}.
The existence of sharp cones in deformed membranes is an
important issue in deformations of membranes with planar rest shape
\cite{Witten2007}.
A membrane of thickness $D$
prefers bending deformations (energy proportional to $D^3$)
over stretching deformations
(energy proportional to $D$). If external forcing or constraints
are such that stretching can be avoided, the membrane responds by pure
bending.
Any deformation of such an unstretched membrane
has to preserve the metric and thus the vanishing Gaussian curvature
of a plane. This results in so-called developable cones, which
have zero Gaussian curvature everywhere except at the tip of the cone.
Cones only develop in response to external
forces or constraints, typically under compressional constraints or forcing
as in the crumpling of paper.
Then unstretched membranes develop folds or wrinkles
around the developable cones in order to accommodate the excess
area that occurs under compression \cite{BenAmar1997,Cerda1998,Witten2007}.
Our ferrofluid elastic membranes differ in several respects.
The magnetic forces are always {\it stretching}
forces and they are always
{\it normal} to the surface such that the tangential force
equilibrium (\ref{eq:equilibrium_eqns_tang})
only involves internal stresses of the membrane.
Under stretching forces the membrane cannot respond by
pure bending and changes in the metric are unavoidable.
However,
the forcing depends on the magnetic field distribution [see Eq.\ (\ref{fm})]
and becomes concentrated in points of high fields, which are typically
points of high curvature. This establishes a
positive feedback between
shape and magnetic field distribution that
can stabilize conical tips.
Moreover, we consider membranes with spherical rest shape and, thus,
non-zero Gaussian curvature $K=1/R_0^2$. This is another reason
why deformation
into a cone with $K=0$ is impossible without stretching.
Similar conditions (normal forces and spherical rest shape)
are fulfilled for spherical shells under
point forces, where conical solutions have also been obtained
\cite{Vella2012} and to which most of our results regarding the existence
of conical shapes should also apply.
The tangential force equilibrium (\ref{eq:equilibrium_eqns_tang})
has to be fulfilled in the vicinity of the conical tip
and is independent of the stretching magnetic forces,
which are always normal.
In combination with the nonlinear constitutive relations
(\ref{eq:taulambda}) this requires that the stretching tensions remain
finite and isotropic at the conical tip, i.e.,
$\tau_s(0) = \tau_\varphi(0)>0$ at $s_0=0$.
From the constitutive relations then also follows
the isotropy of the stretches
$\lambda_s(0)=\lambda_\varphi(0)$ at the tip.
However, stretches are not necessarily finite at a conical tip.
For finite isotropic
stretches $\lambda_s(0)=\lambda_\varphi(0)<\infty$ at the
pole, l'H{\^o}pital's rule applied at $s_0=0$ gives
$\lambda_\varphi(0) = \lambda_s(0)\cos[\psi(0)]$ [see
Eq.\ (\ref{eq:lhospital})].
Then isotropy requires $\psi(0)=0$ and it follows that
a sharp conical tip with $\psi(0) >0$
is impossible if stretches remain finite at the tip.
Finite isotropic stretches at the pole thus always lead to
flat tips with $\psi(0)=0$ as for the spheroidal shapes.
For diverging and asymptotically isotropic stretches
\begin{equation}
\lambda_s(s_0)\approx \lambda_\varphi(s_0)\approx {\rm const}\, s_0^{-\beta},
\label{eq:lambdadiv}
\end{equation}
with an exponent $\beta>0$;
however, l'H{\^o}pital's rule does not apply at $s_0=0$.
Then we find instead that isotropy of the diverging
stretches requires a conical tip with
the relation
\begin{equation}
\beta = \cos[\psi(0)]-1 = \sin\alpha-1
\label{eq:betaalpha}
\end{equation}
between the exponent $\beta$
and the half opening angle $\alpha = \pi/2 - \psi(0)$ of the
conical tip [see Eq.\ (\ref{eq:betaalpha_app})].
This result can be obtained from a modified l'H{\^o}pital's rule
or directly from analyzing stretches for a deformation into a conical
tip under the constraint of isotropy of the stretches at the tip
[see Eq.\ (\ref{eq:lambdaphi_d3_app})].
For the nonlinear constitutive relation (\ref{eq:taulambda})
diverging and isotropic stretches are still compatible with
finite and isotropic tensions, which approach
$\tau_s(0) = \tau_\varphi(0) =
{Y_{2\text{D}}}/{(1-\nu)}$, see Eq.\ (\ref{eq:tau0_app}), at the tip.
Moreover, $\beta>-1$ according to (\ref{eq:betaalpha}) and, therefore,
the divergence is such that the elastic energy [the energy density
(\ref{eq:ws}) integrated over the tip area] remains finite.
Any numerical approaches to capsule shell mechanics and the calculation
of the magnetic fields rely on discretization.
In the numerical solution of
axisymmetric shape equations the arc length $s_0$
is discretized. After discretization in the numerics,
stretches necessarily remain
finite at potential conical tips at the apices.
Then our results for finite stretches apply, and we have to choose
a boundary condition $\psi(0)=0$.
Also, for the calculation of the magnetic fields,
we discretize the boundary of the capsule [see Eq.\
(\ref{eq:Li})]. Therefore, also magnetic fields remain finite
at conical tips. Then also the normal magnetic forces remain
finite and can only support finite curvatures at the tip of the
conical shape.
This leads to a rounding of conical tips and, thus, also requires
$\psi(0)=0$.
This implies that, in the numerical calculations,
all shapes of ferrofluid capsules will
have rounded tips with $\psi(0)=0$; the rounding
of a conical tip for these numerical reasons will happen on the
scale of the discretization of the problem.
A boundary condition $\psi(0)$
for the numerical solution of the shape
equations [see Eq.\ (\ref{eq:shape_eqns_bc})] has also
been used
in Refs.\ \cite{Lavrova04, Lavrova05,Lavrova06,Lavrova2006}
for ferrofluid droplet shapes.
\subsection{Slender-body approximation for conical capsules}
\label{sec:Slender}
For ferrofluid droplets, the conical shape
could be investigated analytically using a slender-body approximation
\cite{Stone99}, which we want to adapt for conical
shapes of the ferrofluid-filled capsule.
We have shown that conical shapes can also exist for
ferrofluid-filled capsules but they involve
diverging isotropic stretches at the conical tip.
Tensions are isotropic,
remain finite at the conical tip and approach the
limiting values $\tau_s(0) = \tau_\varphi(0) =
{Y_{2\text{D}}}/{(1-\nu)}$ [see Eq.\ (\ref{eq:tau0_app})].
The capsule shape is described by
a function $r(z)$ in cylindrical coordinates.
In a slender-body approximation, we assume $\partial_z r \ll 1$;
for a conical tip with half opening angle $\alpha = \pi/2 - \psi(0)$,
we have $\partial_z r \approx \tan\alpha$ in the vicinity of the tip.
Then we can neglect small
radial field components and approximate
the magnetic field as parallel to the $z$ axis, ${\bf H} = H(z) \vec{e}_z$.
The field $H(z)$ is determined by
\begin{align}
H_0 &= H(z) - \frac{\ln A}{2}\chi \partial_z^2\left[r^2(z) H(z)\right],
\label{eq:Hz_slender}
\end{align}
where $A$ is the aspect ratio of the slender shape, which can be expressed
in terms of the half opening angle, $A=1/\tan \alpha$,
for a conical shape \cite{Stone99}.
This relation is unchanged as compared to fluid droplets as it is
a result of the slender shape and magnetic boundary conditions only
and independent of the surface elasticity underlying the shape.
In the slender-body approximation we also assume $\partial_z^2 r\ll 1/r$
such that the meridional curvature is small $\kappa_s\ll \kappa_\varphi
\approx 1/r(z)$.
Then the
Laplace-Young equation describing normal force equilibrium becomes
\begin{align}
\label{eq:fbal_cap_slender}
\frac{1}{r(z)} \left\{\tau_\varphi[r(z)] + \gamma\right\}
&= p_0 + f_m.
\end{align}
This relation differs from the corresponding relation for fluid
droplets by the appearance of the additional elastic tension
$\tau_\varphi=\tau_\varphi(r)$.
As shown in Appendix \ref{app:cone_tip2},
tangential force equilibrium is fulfilled in the vicinity of the
conical tip if stretches are diverging, and the resulting
circumferential tension is
\begin{align}
\tau_\varphi(r) &=
\frac{Y_{2\text{D}}}{1-\nu}\left[ 1 - 2R_0 \sin\alpha
({a \tan\alpha})^{-1/\sin\alpha}
r^{1/\sin\alpha -1} \right]
\label{eq:tau1}
\end{align}
[see Eq.\ (\ref{eq:tau1_app})]
in the vicinity of the conical tip. Note that $a$ still denotes the polar
radius.
In Appendix \ref{app:cone_eqs} we also outline how the
tension $\tau_\varphi(r)$ could be calculated for a general shape
$r(z)$, in principle.
The Laplace-Young equation (\ref{eq:fbal_cap_slender}) with an elastic tension
(\ref{eq:tau1}) and the slender-body field equation (\ref{eq:Hz_slender})
provide two coupled equations for $r(z)$ and $H(z)$.
The pressure $p_0$ has to be chosen such that the resulting shape $r(z)$
fulfills the volume constraint
\begin{align}
V_0 &= \pi \int_{-a}^{a} r^2(z) dz.
\label{eq:vol_slender}
\end{align}
The three equations (\ref{eq:Hz_slender}), (\ref{eq:fbal_cap_slender}),
and (\ref{eq:vol_slender}) governing slender (and, in particular, conical)
shapes of a
ferrofluid-filled capsule only differ in the appearance of the additional
elastic tension $\tau_\varphi=\tau_\varphi(r)$ from the corresponding equations
for ferrofluid droplets from Ref.\ \cite{Stone99}.
They can be also be
solved analogously as for ferrofluid droplets, in principle.
\section{Results}
\label{sec:results}
\subsection{Spheroidal capsule shapes}
\label{sec:spheroid}
While the capsule is spherical at $B_m = 0$, it becomes
elongated for increasing magnetic field or Bond number
$B_m$ similarly to a ferrofluid droplet. We can quantify
the elongation by the ratio of capsule length $a$ in the
$z$ direction and capsule diameter $b$ at the equator, $a/b$.
At small or moderate magnetic fields ferrofluid capsules
assume a prolate spheroidal shape to a very good approximation;
one example is shown in Fig.\ \ref{fig:fieldplot}(b).
For small fields we calculated
the linear response of the capsule exactly
in Sec.\ \ref{sec:lin_def} and Appendix \ref{app:lin_def}
and found displacements (\ref{eq:AnsatzuRcap}), which
describe a prolate spheroid
with an elongation $a/b>1$ given by Eq.\
(\ref{eq:lin_approx}). This analytical result is
in excellent agreement with numerical results for small fields
(see Fig.\ \ref{fig:lin_region}).
The linear response regime is valid as long as
$a/b-1 \ll 1$ or $B_m \ll {\left[{Y_{2\text{D}}}/{\gamma}(5+\nu) +
1\right]}(3+\chi)^2/\chi$ according to Eq.\ (\ref{eq:lin_approx}).
Small magnetic fields are easily accessible and
for many ferrofluids, susceptibilities are rather small
(for example, $\chi \simeq 0.36$
in Ref.\ \cite{Zhu11}). Therefore, spheroidal shapes in
the linear response regime are
experimentally easily accessible. Then
the linear response relation (\ref{eq:lin_approx})
can be used as experimental method
to deduce unknown capsule material properties, for example,
Young's modulus $Y_{\text{eD}}$ if the magnetic properties of the ferrofluid
are known.
At moderate magnetic fields,
the capsule shape remains very similar to a prolate spheroid
for all elongations $a/b \lesssim 3$, which was one basic assumption of
the approximative energy minimization in
Sec.\ \ref{sec:AnalyticalApprox}.
Figure \ref{fig:ellipsoidplot} demonstrates this for shapes with
${a}/{b}=2$.
The spheroidal approximation
works better for systems dominated by the surface tension,
i.e., for small ratios ${Y_{2\text{D}}}/{\gamma}$.
For fixed Bond number $B_m$ and susceptibility $\chi$ the
elongation decreases with increasing ${Y_{2\text{D}}}/{\gamma}$
because of the additional stretching energy of the shell as compared
to a droplet, so a ferrofluid
droplet (${Y_{2\text{D}}}/{\gamma}=0$)
always shows the highest elongation.
For small fields, this trend can be quantified
with the linear response relation (\ref{eq:lin_approx}).
For smaller elongations, the spheroidal approximation
tends to work better.
\begin{figure}[htbp]
\begin{center}
\includegraphics[angle=0,width=0.9\textwidth]{ellipsoid_comparison.pdf}
\caption{
Comparison of numerically calculated
$r(z)$ contour of a capsule with
${Y_{2\text{D}}}/{\gamma}=100$, and $\chi = 21$,
for a value $B_m$
chosen such that the elongation is
${a}/{b}=2$ (the inset
shows the location of the pictured shapes in the
$B_m$-${a}/{b}$ plane) with
a spheroid.
The shape calculated without wrinkling (blue solid line)
shows very good agreement with a spheroid of the same volume and
elongation (red dashed line). Taking wrinkling into account leads to
visible deviations (green dotted line).
}
\label{fig:ellipsoidplot}
\end{center}
\end{figure}
The other assumption in
the approximative energy minimization in
Sec.\ \ref{sec:AnalyticalApprox} was
constant stretch factors throughout the shell, i.e.,
$\lambda_s, \lambda_\varphi = \text{const}$ (and thus constant
elastic tensions $\tau_\varphi$ and $\tau_s$).
Also this approximation works very well for spheroidal shapes
with elongations $a/b \lesssim 3$,
as the numerical results in Fig.\ \ref{fig:lambda_s}
for $a/b=2$
(left scale, red line) show.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1\textwidth,clip]{lambda_s0.pdf}
\caption{
(a) Stretch factors in the meridional direction $\lambda_s(s_0)$ following
the whole contour line from the south pole ($s_0 = 0$) to the equator
(${s_0}/{R_0} = {\pi}/{2}$) for ${Y_{2\text{D}}}/{\gamma}=100$
and $\chi = 21$. The left scale (red dashed line) gives
almost constant
stretch factors for a spheroidal shape with
${a}/{b} = 2$. The right scale (blue solid line) gives
diverging stretch factors
for a conical shape with
${a}/{b} = 5.34$.
(b) Logarithmic plot of $\lambda_s(s_0)$ near the tip
for $s_0/R_0 < 10^{-1}$.
The function $\lambda_s(s_0) = {\rm const}\, s_0^{-\beta}$ [see Eq.\
\eqref{eq:lambdadiv}]
was fitted to the data of the conical shape,
which gave $\beta = 0.562$, corresponding
to an angle $\alpha = 25.98^{\circ}$ in Eq.\ \eqref{eq:betaalpha}.
(c) Zoom in to the tip of the contour line $z(r)$
for the conical shape; the half opening angle
is $\alpha \approx 25^{\circ}$.
}
\label{fig:lambda_s}
\end{center}
\end{figure}
As a result, the approximative energy minimization in
Sec.\ \ref{sec:AnalyticalApprox} gives very good results for
moderate magnetic fields, i.e., for all elongations
$a/b \lesssim 3$, where we always find
prolate spheroidal shapes, as the comparison with
numerical results in
Fig.\ \ref{fig:ab_Bm_plot} shows.
\subsection{Conical capsule shapes and capsule rupture}
\label{sec:cone}
For large magnetic fields or Bond numbers
$B_m$ and at sufficiently high susceptibilities $\chi$,
ferrofluid capsules can also assume
conical shapes, such as the shape in
Fig.\ \ref{fig:fieldplot}(c), which
have also been found for ferrofluid droplets
\cite{Li1994,Stone99}.
We investigated the possibility of conical shapes
for elastic capsules with normal magnetic forces
above in Sec.\ \ref{sec:conical} and found
that stretch factors have to
diverge at the conical tips,
$\lambda_s(s_0)\approx \lambda_\varphi(s_0)\approx {\rm const}\, s_0^{-\beta}$
[see Eq.\ (\ref{eq:lambdadiv}], with an exponent
$\beta = \sin\alpha-1$, which is determined by the
half opening angle $\alpha = \pi/2 - \psi(0)$ of the
conical tip [see Eqs.\ (\ref{eq:betaalpha}) and (\ref{eq:betaalpha_app})].
This behavior is confirmed by our numerical results
in Fig.\ \ref{fig:lambda_s}
(left scale, blue line).
The stretch factors diverge but are asymptotically isotropic at the
tips. The
nonlinear constitutive relations (\ref{eq:taulambda}) then
result in finite and isotropic tensions
$\tau_s(0) = \tau_\varphi(0) =
{Y_{2\text{D}}}/{(1-\nu)}$ [see Eq.\ (\ref{eq:tau0_app})].
Diverging stretch factors cannot be realized in an actual
material without rupture. Typical
alginate capsule materials can only resist stretch factors of
$\lambda\simeq 1.2$ before rupture;
highly stretchable hydrogels can resist
stretch factors up to $\lambda \sim 20$
\cite{Sun12}.
Therefore, a real capsule should rupture at the
poles at the transition into a conical shape
and we conclude
that investigations of conical shapes
are primarily of theoretical interest.
Such rupture events have actually been observed in
Ref.\ \cite{Karyappa2014} for capsules
filled with a dielectric liquid in external
electric fields.
We expect that the nonlinear Hookean material law
will become invalid at such high stretch factors prior
to rupture. Then constitutive relations which are more realistic
for high strains should be used.
Nevertheless, the appearance of large
stress factors is a robust feature of the conical
shape independently of the material law.
Conical shapes cannot be described quantitatively by the
approximative energy minimization from
Sec.\ \ref{sec:AnalyticalApprox} as
spheroidal shapes with a large elongation $a/b$,
which is clearly shown by
the deviations between
numerical results (data points)
and the approximative energy minimization from
Sec.\ \ref{sec:AnalyticalApprox} (solid lines) for the conical
shapes in
Fig.\ \ref{fig:ab_Bm_plot}.
For ferrofluid droplets, conical shapes can be described
by a slender-body theory \cite{Stone99}, which we generalized in
Sec.\ \ref{sec:Slender} to ferrofluid-filled capsules.
The three governing equations (\ref{eq:Hz_slender}),
(\ref{eq:fbal_cap_slender}),
and (\ref{eq:vol_slender}) from Sec.\ \ref{sec:Slender}
can be used to describe
conical capsule shapes quantitatively.
As pointed out above, the tensions remain
finite and isotropic at the conical tip, i.e.,
$\tau_s(r) \approx \tau_\varphi(r) \approx {Y_{2\text{D}}}/{(1-\nu)}$
for small $r$ [see Eq.\ (\ref{eq:tau0_app})].
Then the slender-body equation (\ref{eq:fbal_cap_slender})
from normal force balance actually becomes
identical to the corresponding equation for a droplet
from Ref.\ \cite{Stone99}, however, with an effectively
increased surface tensions $\gamma_{\rm eff} = \gamma + \tau_\varphi(0)$.
Also the other two equations (\ref{eq:Hz_slender})
and (\ref{eq:vol_slender}) are identical such that
we obtain very similar slender conical shapes
for capsules and droplets, which can be mapped onto each other
by a simple shift of the surface tension.
The mechanism underlying the stabilization of the conical
shape is analogous to ferrofluid droplets
because tensions remain finite and isotropic
at the conical tip.
A sharp conical tip with
curvatures $\kappa_\varphi \propto 1/r$
gives rise to diverging magnetic fields
$H\propto r^{-1/2}$ and normal magnetic forces
\begin{equation}
f_m \propto H^2 \propto r^{-1},
\label{eq:Hdivergence}
\end{equation}
{\it both} for ferrofluid droplets and capsules.
These strong magnetic stretching forces
stabilize the conical tip against high
elastic restoring forces.
The normal component of the elastic force
is mainly due to the finite circumferential tension $\gamma+\tau_\varphi(0)$
acting along the high circumferential
curvature $\kappa_\varphi \propto 1/r$ at the conical tip, resulting
in an elastic force
$f_{\rm el} \propto [\gamma+\tau_\varphi(0)]\kappa_\varphi\propto r^{-1}$
with the same divergence.
Magnetic and elastic normal forces balance in the Laplace-Young equation
(\ref{eq:fbal_cap_slender}) in the slender-body approximation.
The magnetic field exponent $H\propto r^{-1/2}$ is {\it identical} for
capsules and droplets, as long as the elastic tensions at the conical
tip are finite.
This exponent determines the critical susceptibility $\chi_c$ above which
a shape transition into conical shapes is possible and therefore
we also find the identical $\chi_c$ for capsules and droplets
as discussed in the following section.
\subsection{Spheroidal-conical shape transition of capsules}
Upon increasing the magnetic field or the magnetic Bond number $B_m$
at fixed capsule
elasticity ${Y_{2\text{D}}}/{\gamma}>0$ and
for a sufficiently large and fixed ferrofluid susceptibility $\chi$,
we find a discontinuous shape transition from
spheroidal to conical capsule shapes, similar to what has been
found for ferrofluid droplets (${Y_{2\text{D}}}/{\gamma}=0$)
\cite{Bacri82,Li1994,Stone99}.
One of our main results is the diagram of capsule
elongation $a/b$ as a function of Bond number $B_m$
in Fig.\ \ref{fig:ab_Bm_plot} for different values of
elasticity parameters ${Y_{2\text{D}}}/{\gamma}$ and for
$\chi=21$, where a lower
spheroidal branch and an upper conical branch and a
discontinuous transition between both branches
can be identified.
In the following sections we will discuss different
aspects of this shape transition in more detail.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1\textwidth,clip]{ab_Bm.pdf}
\caption{
Elongation ${a}/{b}$ of a capsule filled with a ferrofluid
with $\chi = 21$ as a function of
magnetic Bond number $B_m$ for
different values of the dimensionless
elastic parameter ${Y_{2\text{D}}}/{\gamma}$.
The magnetic Bond number is rescaled by ${Y_{2\text{D}}}/{\gamma} +
(5+\nu)$,
which is motivated by the small field behavior [see
Eq.\ (\ref{eq:lin_approx})].
The solid lines describe the theoretical results from
approximative energy minimization (see
Sec.\ \ref{sec:AnalyticalApprox}). Open (closed) symbols
denote numerical data for
increasing (decreasing) $B_m$.
The agreement is good for small
elongations; the approximation fails for higher elongations, especially
at the shape transition (close-up in the right diagram), where ${a}/{b}$
jumps for small changes of $B_m$. Hysteresis effects are clearly visible in
that area. There are two sets of numerical data for
${Y_{2\text{D}}}/{\gamma}=\infty$: Square data points are based on the
modified shape equations that take wrinkling into account, while
diamonds are calculated without wrinkling.
There are also two sets of data without elasticity:
The upper data points
(black) describe a droplet with a real conical tip with a
cone angle of $\psi(0)=70.2^{\circ}$,
as it was given in Ref.\ \cite{Stone99};
for the lower points (blue) we used the boundary
condition $\psi(0)=0$.
Dashed lines indicate the position of shape transitions.
Above these lines, shapes are conical, while they are spheroidal below.
The markers (b) and (c) correspond to the
shapes in Fig.\ \ref{fig:fieldplot}.
}
\label{fig:ab_Bm_plot}
\end{center}
\end{figure}
\subsubsection{Critical susceptibility $\chi_c$}
\label{sec:chic}
For ferrofluid droplets,
a discontinuous shape transition was observed in experiments
\cite{Bacri82, Bashtovoi87} and numerical simulations \cite{Lavrova04,
Afkhami10} only for susceptibilities $\chi>\chi_c$, i.e., above
a critical susceptibility $\chi_c$.
In Ref.\ \cite{Li1994}
a value $\chi_c =\mu_c/\mu_{\rm out}-1 \simeq 16.59$
was found below which no conical shape can exist;
the slender-shape approximation for droplets
from Ref.\ \cite{Stone99}, which we generalized to elastic capsules in
Sec.\ \ref{sec:Slender},
gives $\chi_c = 16e/3 \simeq 14.5$.
The approximative energy minimization
of Bacri and Salin \cite{Bacri82}, which we
generalized to elastic capsules in
Sec.\ \ref{sec:AnalyticalApprox}, gives $\chi_c \simeq 19.8$
for ferrofluid droplets.
Numerically, a range of $\chi_c \simeq 19$ to $\chi_c \simeq 19.5$
is observed \cite{Wohlhuter92}.
The question arises whether a critical susceptibility $\chi_c$
can also be found for the existence of a discontinuous spheroidal-conical
transition for ferrofluid-filled
elastic capsules.
For given $\chi$ and half opening angle $\alpha$ of the conical
shape electromagnetic boundary conditions determine the divergence
$H \propto r^{\mu-1}$ of the field via the equation \cite{Li1994,Ramos1994}
\begin{equation}
P_\mu(\cos\alpha) P_\mu'(-\cos\alpha) + (\chi+1)
P_\mu(-\cos\alpha) P_\mu'(\cos\alpha)=0.
\label{eq:Li1994}
\end{equation}
Because of the finite elastic tension $\tau_\varphi(0)$
at the conical tip, the magnetic field at the tip of a conical
capsule
diverges with the same $\mu=1/2$ [see Eq.\ (\ref{eq:Hdivergence})]
as for a conical droplet. Therefore, we find the same
critical susceptibility $\chi_c\simeq 16.59$, above which a
conical solution can exist, for both capsules and droplets.
In the slender-body approach,
Eq.\ (\ref{eq:Hz_slender}) determines $\chi_c$ and
applies unchanged to both
slender conical droplets and
ferrofluid-filled capsules.
Also the magnetic field divergence $H \propto r^{-1/2}$
is identical in both cases, so the
analysis of Eq.\ (\ref{eq:Hz_slender}) predicts the same
critical value $\chi_c = 16e/3 \simeq 14.5$ for ferrofluid-filled capsules
as for ferrofluid droplets.
In particular, both the analysis of Eq.\ (\ref{eq:Li1994})
and the slender-body approach predict that
the value
for $\chi_c$ to be {\it independent} of the
Young modulus $Y_{2\mathrm{D}}$ of the capsule.
This result is corroborated by our numerics for $\chi=21$, where we
{\it always} observe a spheroidal-conical shape transition, even for
$Y_{2\text{D}}/\gamma \to \infty$ [see Eq.\ (\ref{fig:ab_Bm_plot})].
This result
is in contrast, however, to what we find using the approximative
energy minimization for spheroidal shapes from
Sec.\ \ref{sec:AnalyticalApprox}. Analyzing Eq.\ (\ref{eq:BMab}),
$B_m=g(k)=g(b/a)$,
for the saddle points of the function $g(k)$ gives the critical value
of the susceptibility $\chi_c$ [the two equations $g'(k)=0$ and $g''(k)=0$
determine two critical parameter values $k=k_c$ and $\chi=\chi_c$].
Using this approach, we find a $\chi_c$,
which is strongly increasing with the Young modulus $Y_{2\text{D}}/\gamma$,
such that we find $\chi_c >21$ already for $Y_{2\text{D}}/\gamma> 0.015$,
which clearly disagrees with all our numerical and analytical results.
The reason for this disagreement is the failure of the approximative energy
minimization to correctly describe conical shapes as
discussed in Sec.\ \ref{sec:cone}.
It is interesting to consider the robustness of our result
of a $Y_{2\text{D}}$-independent $\chi_c$ that is identical
to the $\chi_c$ for ferrofluid droplets
with respect to the constitutive relation. We used the nonlinear
Hookean constitutive relation (\ref{eq:taulambda}),
which can only support {\it finite} tensions at a conical tip,
even for diverging stretches (see Sec.\ \ref{sec:conical}).
A simple linear Hookean constitutive relation [missing the $1/\lambda$-factors
in Eq.\ (\ref{eq:taulambda})] behaves differently and
exhibits diverging tensions $\tau_\varphi \sim r^{-\sigma}$ with $\sigma>0$
at a conical tip.
Then tangential force equilibrium (\ref{eq:fbalt_cap0}) also requires
$\tau_s \sim\tau_\varphi \sim r^{-\sigma}$ but with an anisotropy
$\tau_\varphi/\tau_s = 1-\sigma$.
With the linear constitutive relation
this in turn leads to stretches $\lambda_s\sim \lambda_\varphi\sim
r^{-\sigma}$ with an anisotropy
$\lambda_\varphi/\lambda_s = (1-\nu-\sigma)/(1-\nu + \nu\sigma)\equiv \delta$
or $\delta(\sigma) = (1-2\sigma)/(1+\sigma)$ for a Poisson ratio $\nu=1/2$.
Requiring this anisotropy in Eq.\ (\ref{eq:lambda_d2_app})
at a conical tip with half opening angle $\alpha$
leads to a modified differential equation (\ref{eq:ODE_d_app})
and a divergence
$\lambda_s \sim \lambda_\varphi \sim r^{1-1/\delta(\sigma)\sin\alpha}$.
Consistency with $\lambda_s\sim \lambda_\varphi\sim
r^{-\sigma}$ then requires
\begin{equation*}
\sigma = \frac{1}{\delta(\sigma) \sin\alpha} -1
= \frac{1+\sigma}{1-2\sigma} \frac{1}{\sin\alpha} -1,
\end{equation*}
which determines the divergence
$\sigma=\sigma(\alpha)$ of tensions
$\tau_s \sim\tau_\varphi \sim r^{-\sigma}$
as a function of the opening angle $\alpha$.
At the conical tip we have now curvatures
$\kappa_\varphi \propto 1/r$ in combination
with circumferential tensions $\tau_\varphi \sim r^{-\sigma}$
such that normal force balance also requires magnetic
forces $f_m \propto H^2 \propto r^{-1-\sigma}$ [cf.\
Eq.\ (\ref{eq:Hdivergence})].
Thus, we have to use
$\mu = 1-\sigma(\alpha)$ instead of $\mu=1/2$ in $H \propto r^{\mu-1}$
in Eq.\ (\ref{eq:Li1994}) and obtain
a modified equation for the cone angle $\alpha$ as a function of the
parameter $\chi$.
This equation has a solution only above $\chi_c \simeq 40.5$ and thus
the critical value $\chi_c$ is strongly increased for a
strictly linear Hookean constitutive relation.
Our numerical results corroborate this result as we find
only spheroidal capsule shapes for a strictly linear constitutive relation
at a susceptibility $\chi=21$.
This shows that the value of $\chi_c$ is very sensitive to changes
in the constitutive relation and a measurement of $\chi_c$ allows us to
draw conclusions about the constitutive relation of the
capsule material.
\subsubsection{Critical Bond numbers}
Our numerical solutions of the shape equations show
that the discontinuous spheroidal-conical shape transition
that exists for ferrofluid droplets \cite{Bacri82,Li1994,Stone99}
persists for ferrofluid-filled elastic capsules and
shows qualitatively similar features.
Both for droplets and for capsules,
the driving force of the shape transition is the
lowering of the magnetic field energy in the conical shape.
Above an upper critical Bond number $B_{m,c2}$
the spheroidal shape becomes unstable and the droplet or capsule deforms
into a much more elongated, conical shape.
This shape transition is discontinuous, i.e., the deformation
into the conical shape is associated with a
jump in $a/b$. The discontinuous transition between spheroidal to conical
shapes also exhibits hysteresis: Lowering the Bond number starting
from values $B_m>B_{m,c2}$, the conical shape becomes unstable at a
lower critical Bond number $B_{m,c1}$ with $B_{m,c1}<B_{m,c2}$.
The discontinuous spheroidal-conical
transition only exists above the critical susceptibility $\chi_c$.
In other words, both droplets and capsules exhibit a line of discontinuous
shape transitions in the
$\chi$-$B_m$ plane for $\chi>\chi_c$,
which terminates at a critical point located
at $\chi=\chi_c$. The lines $B_{m,c1}(\chi)$ and $B_{m,c2}(\chi)$
are the limits of stability (spinodals) of this shape transition
and meet in the critical point.
Figure \ref{fig:ab_Bm_plot} shows the capsule elongation with
respect to $B_m$ for different values of the dimensionless
elastic parameter ${Y_{2\text{D}}}/{\gamma}$ of the capsule. We
choose $\chi = 21$, which is only slightly above $\chi_c$.
This ensures that we
have a shape transition for a ferrofluid
droplet (corresponding to the limit ${Y_{2\text{D}}}/{\gamma}=0$),
on the one hand, and relatively small and thus
numerically more stable
elongations in the conical shape, on the other hand.
Figure \ref{fig:ab_Bm_plot} clearly shows a discontinuous jump in elongation
and hysteresis effects also for capsules with
${Y_{2\text{D}}}/{\gamma}>0$.
\subsubsection{Stretch factors as order parameter}
The discontinuous jump in the elongation
ratio $a/b$ at the spheroidal-conical transition
is difficult to localize
for larger values of ${Y_{2\text{D}}}/{\gamma}$, as
Fig.\ \ref{fig:ab_Bm_plot} shows.
More suitable order parameters for the spheroidal-conical
transition
are the stretch factors $\lambda_s$ and $\lambda_\varphi$.
Because the stretch factors diverge at the tips of the conical shape
(the divergence is only limited by numerical discretization effects),
whereas they stay finite at the poles of spheroidal shape (see
Fig.\ \ref{fig:lambda_s} and our above discussion), we can directly
employ the stretch factor $\lambda_s(s_0=0)$ at one of the poles as
a convenient order parameter.
For $\chi=21$ and ${Y_{2\text{D}}}/{\gamma}=100$,
the shape transition occurs
where ${a}/{b}$ has a rather small jump from about 5.2 to 5.35 for
increasing Bond number
$B_m$, whereas the stretch factor $\lambda_s(s_0=0)$ exhibits a much
bigger jump by a factor of more than 10, as
demonstrated in Fig.\ \ref{fig:lambda_s_ab}.
Also the shape hysteresis at the spheroidal-conical shape transition
can be clearly seen for the order parameter $\lambda_s(s_0=0)$.
Using this order parameter, we can
detect the spheroidal-conical shape transition of
ferrofluid-filled capsules by the criterion
\begin{align}
\label{eq:transition_criterion}
\lim\limits_{\Delta B_m \to 0} |\lambda_s(s_0 = 0, B_m)
- \lambda_s(s_0 = 0 ,B_m + \Delta B_m)| > 0,
\end{align}
where we use values $\Delta B_m = 0.005$ for ${Y_{2\text{D}}}/{\gamma}<1$
up to values $\Delta B_m = 0.5$ for ${Y_{2\text{D}}}/{\gamma}=100$ in practice
[$B_{m,c1}$ and $B_{m,c2}$ grow approximately linearly with
${Y_{2\text{D}}}/{\gamma}$ (see Fig.\ \ref{fig:hysteresisplot} below)
such that larger values $\Delta B_m$ can be
used for larger ${Y_{2\text{D}}}/{\gamma}$; smaller values of $\Delta B_m$
give more precise results].
For ferrofluid droplets, i.e., in the limit
${Y_{2\text{D}}}/{\gamma}\approx 0$, we still have to use jumps
in the elongation $a/b$ for small changes
$\Delta B_m$ in the magnetic Bond number to detect the
spheroidal-conical shape transition.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.5\textwidth,clip]{lambdas_bm.pdf}
\caption{
Meridional stretch factor
$\lambda_s$ at the capsule pole $s_0=0$ as a function of
Bond number $B_m$ for ${Y_{2\text{D}}}/{\gamma}=100$ and $\chi=21$.
The stretch factor clearly exhibits a jump at
the location of the discontinuous shape transition
and hysteretic behavior.
}
\label{fig:lambda_s_ab}
\end{center}
\end{figure}
We note that the discretization problem at the sharp conical tip
mentioned above causes high relative errors in the
numerical values of stretch factors in the tip area.
Therefore, our numerical results
for the diverging stretch factors at the tips of
conical capsule shapes cannot be numerically exact.
The detection of a divergence in $\lambda_s$ at the poles,
which we use to detect the transition into a conical shape, is, however,
still possible even in the presence of numerical errors.
\subsubsection{Shape hysteresis}
\label{sec:hysteresis_effects}
In order to track the range of elastic control parameters
${Y_{2\text{D}}}/{\gamma}$, where a discontinuous shape transition
with hysteresis can be observed (for fixed $\chi=21$), we
use the stretch factor $\lambda_s(s_0=0)$ as
the order parameter and
the criterion \eqref{eq:transition_criterion} to determine
$B_{m,c1}$ and $B_{m,c2}$. We determine $B_{m,c2}$ by increasing the
Bond number in small steps $\Delta B_m>0$
to locate the jump in
the stretch factor $\lambda_s(s_0=0)$ at the pole, when the
spheroidal shape becomes unstable. Analogously, we determine
$B_{m,c1}$ by decreasing the
Bond number in small steps $\Delta B_m<0$ to locate the jump in
$\lambda_s(s_0=0)$, when the
conical shape becomes unstable (see Fig.\ \ref{fig:lambda_s_ab}).
Repeating this procedure for
increasing values of the elastic control parameter
${Y_{2\text{D}}}/{\gamma}$, we obtain
the location and size of the hysteresis loop $B_{m,c1}<B_m<B_{m,c2}$
for a fixed susceptibility as a function of
${Y_{2\text{D}}}/{\gamma}$ (see Fig.\ \ref{fig:hysteresisplot}).
We see that $B_{m,c1}$ and $B_{m,c2}$ increase (approximately linear) for
increasing ${Y_{2\text{D}}}/{\gamma}$ because of the
increasing elastic energy needed for the same deformation.
Note that the absolute numerical
values of $B_{m,c1}$ and $B_{m,c2}$ cannot be considered exact as they
are depending on the
discretization of the magnetic field calculation
(see also Appendix \ref{app:errors}).
The approximative energy minimization for spheroidal shapes from
Sec.\ \ref{sec:AnalyticalApprox} can be used to
calculate approximative values for $B_{m,c1}$ and $B_{m,c2}$ from
Eq.\ (\ref{eq:BMab}), $B_m=g(k)=g(b/a)$
[the two equations $g'(k)=0$ and $B_{m} = g(k)$ determine the
critical Bond numbers $B_m=B_{m,c1/2}$ and a corresponding critical
inverse aspect ratio $k=k_c$]. We find that
the hysteresis loop closes already for $Y_{2\text{D}}/\gamma> 0.015$
for $\chi=21$ (see Fig.\ \ref{fig:hysteresisplot}),
which is equivalent to our above finding (see
Sec.\ \ref{sec:chic}) that
$\chi_c >21$ for $Y_{2\text{D}}/\gamma> 0.015$ in
the approximative energy minimization.
Comparison with our numerical results in Fig.\ \ref{fig:hysteresisplot}
shows that the
approximative energy minimization gives quite accurate results
for the upper critical Bond number $B_{m,c2}$, i.e.,
the stability limit of the spheroidal shape.
It fails completely to predict the lower critical Bond number $B_{m,c1}$,
i.e., the stability limit of the conical shape, because
it is not able to describe conical shapes quantitatively
(see Sec.\ \ref{sec:cone}).
The numerical calculation shows
hysteresis behavior for {\it all} values of ${Y_{2\text{D}}}/{\gamma}$
(see Fig.\ \ref{fig:hysteresisplot}).
Only the relative size of the hysteresis loop,
$ \Delta B_{m,c} \equiv {2(B_{m,c2} - B_{m,c1})}/{(B_{m,c2} + B_{m,c1})}$,
decreases slightly for increasing ${Y_{2\text{D}}}/{\gamma}$
in the numerical results.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1\textwidth,clip]{hysteresis_l01.pdf}
\caption{
(a) Critical Bond numbers $B_{m,c1}$ (lower data points)
and $B_{m,c2}$ (upper data points) for varying ${Y_{2\text{D}}}/{\gamma}$
with $\chi = 21$. The solid lines describe the prediction by the
approximative energy minimization for spheroidal shapes.
Both critical Bond numbers increase for increasing
${Y_{2\text{D}}}/{\gamma}$.
In the region $B_{m,c1}<B_m< B_{m,c2}$ there are
hysteresis effects in the spheroidal-conical shape transition.
(b) Relative size $\Delta B_{m,c}$ of the hysteresis
area for a wider range of ${Y_{2\text{D}}}/{\gamma}$.}
\label{fig:hysteresisplot}
\end{center}
\end{figure}
\subsection{Wrinkling}
\subsubsection{Wrinkled shapes}
As opposed to liquid droplets, elastic capsules can develop wrinkles
if a part of the shell is under compressive stress
\cite{Rehage2002,Vella2011,Aumaitre2013,Knoche13}.
Wrinkles have also been considered for the equivalent problem of
capsules filled with a dielectric liquid in an external
electric field in Ref.\ \cite{Karyappa2014}.
As it was stated in Sec.\ \ref{sec:WrinkleTheorie}, wrinkles
appear if the total hoop stress becomes compressive,
$\tau_\varphi + \gamma < 0$. Then we have to use modified
shape equations (\ref{eq:wrinkle_eqs}) in the numerical
calculation of the shape.
As can be seen in Fig.\ \ref{fig:ab_Bm_plot}, taking
wrinkling into account has
a visible effect on the capsule's elongation for higher values of
${Y_{2\text{D}}}/{\gamma}$. If wrinkling is taken into account
capsules elongate because
wrinkling reduces the compressional stretch energy, which is stored near
the equator. This elastic energy gain can be used
for a further elongation of
the capsule at the same field strength to lower the magnetic energy.
This also results in stronger deviation
from the spheroidal shape.
To visualize this effect, Fig.\ \ref{fig:ellipsoidplot} shows
the projection of the contour line of the upper right quadrant of capsules
with and without wrinkling using the same elongation ${a}/{b}=2$.
While the shape is indistinguishable from a spheroid without wrinkling,
the wrinkled shape deviates from a spheroid.
Also in the presence of wrinkling, the discontinuous
spheroidal-conical shape transition where the elongation increases
persists.
In the following, we will focus on the effect of wrinkles on
the spheroidal branch of shapes.
\subsubsection{Extent of wrinkled region}
In order to characterize the wrinkling tendency of
spheroidal capsules we calculate the extent of the wrinkled region
$L_{\text{w}}$ [cf.\ Eq.\ (\ref{eq:Lw}) and Fig.\ \ref{fig:3Dwrinkling}],
which can easily be measured in experiments.
First we use the wrinkle criterion $\tau_\varphi + \gamma < 0$
to calculate the
extent of the wrinkled region in the linear
response regime for small magnetic fields as outlined
in Sec.\ \ref{sec:lin_def} and Appendix \ref{app:lin_def}.
In the linear response regime, we calculate the deviation from
a sphere with radius $R_0$ to leading order.
We can characterize the size of the wrinkled region
in terms of the polar angle
$\theta$
as $\theta_{\text{w}} < \theta <\pi - \theta_{\text{w}}$ where
$\theta_{\text{w}}$ is the smallest polar wrinkle
where wrinkles appear,
$\tau_\varphi(\theta_{\text{w}}) + \gamma = 0$. This angle
is related to the length $L_{\text{w}}$ of the wrinkled region by
$L_{\text{w}} = R_0 (\pi-2\theta_{\text{w}})$:
An angle of $\theta_{\text{w}} = \pi/2$ implies the absence of wrinkles,
while $\theta_{\text{w}} = 0$ means that the wrinkles extend from pole
to pole.
Using Eq.\ (\ref{eq:tauphi_app}) for $\tau_\varphi$,
we find
\begin{equation}
\cos^2\theta_{\text{w}} =
\frac{5}{9}-\frac{\gamma R_0}{Y_{2\text{D}} B} \frac{5+\nu}{3}
=
\frac{5}{9}-\frac{4(3+\chi)^2}{27\chi}
\frac{1+(5+\nu)\gamma/Y_{2\text{D}}} {B_m }.
\label{eq:wrinkle_lin_approx}
\end{equation}
Interestingly, $\theta_{\text{w}}$ is universal
and given by $\cos^2\theta_{\text{w}}=5/9$ for purely elastic capsules
($\gamma/Y_{2\text{D}} = 0$), i.e., it does not depend on the magnetic field or
capsule elongation.
This is also the limiting result for large values of
$B_m/[1+(5+\nu)\gamma/Y_{2\text{D}}]$ (see Fig.\ \ref{fig:lin_region_wrinkling}).
We note, however, that linear response theory is only applicable
if $B_m/[1+(5+\nu)\gamma/Y_{2\text{D}}] \ll \chi Y_{2\text{D}}/\gamma$.
For small magnetic fields, the results for
$\theta_{\text{w}}$ from the linear response
prediction \eqref{eq:wrinkle_lin_approx} agree well
with numerical results, as Fig.\ \ref{fig:lin_region_wrinkling}
shows.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.54\textwidth,clip]{lin_def_wrinkle_X21.pdf}
\caption{
Extent of the wrinkled region
represented by the polar angle $\theta_{\text{w}}$
as a function of $B_m/[1+(5+\nu)\gamma/Y_{2\text{D}}]$.
The lines are the linear response result
\eqref{eq:wrinkle_lin_approx},
crosses and stars are numerical data points for different values
of $Y_{2\text{D}}\gamma$, which all collapse to the linear
response result. The red (dashed) line gives the asymptotic result
$\cos^2\theta_{\text{w}}=5/9$ for large values
of $B_m/[1+(5+\nu)\gamma/Y_{2\text{D}}]$ and for purely elastic capsules
($\gamma/Y_{2\text{D}} = 0$).
}
\label{fig:lin_region_wrinkling}
\end{center}
\end{figure}
Now we address the extent of the wrinkled region
beyond linear response and calculate numerically
the relative extent
of the wrinkled region, ${L_{\text{w}}}/{L}$.
A value ${L_{\text{w}}}/{L} = 0$
means that there are no wrinkles, while ${L_{\text{w}}}/{L} = 1$
describes a system where wrinkles extend from pole to pole.
In Fig.\ \ref{fig:Lw_plot}, we change $B_m$ and calculate
${L_{\text{w}}}/{L}$ for different values of the capsule elongation
${a}/{b}$ in the spheroidal shape, i.e., for $a/b<5$.
We use $\chi=21$ and consider several
values of the elastic parameter ${Y_{2\text{D}}}/{\gamma}$.
As Fig.\ \ref{fig:Lw_plot} shows, there are no wrinkles for thin
stretchable capsules, i.e., wrinkles only occur above a
critical value of the dimensionless elastic parameter for
\begin{equation}
\frac{Y_{2\text{D}}}{\gamma}>8.93 ~~\mbox{for}~\chi=21.
\label{eq:Ycrit}
\end{equation}
This result is only very weakly dependent on $\chi$: We find
${Y_{2\text{D}}}/{\gamma}> 9.03$ for $\chi=1$ and
${Y_{2\text{D}}}/{\gamma}> 8.87$ for $\chi=100$.
For small $Y_{2\text{D}}$, wrinkles are energetically unfavorable, i.e., the
reduction of stretching energy $E_{\text{el}}$ by wrinkles
is smaller than the increase of
$E_\gamma$ due to the increase of the surface area.
Slightly above the critical value (\ref{eq:Ycrit}),
wrinkles can only occur for capsules with
elongations ${a}/{b} \simeq 2.4$.
Further increasing $Y_{2\text{D}}$ (or shell thickness),
the wrinkles become longer and appear for a wider range of elongations.
The extent of wrinkling is still limited by two effects.
At the lower elongation $a/b$, where ${L_{\text{w}}}/{L}=0$,
a certain elongation is needed to create a sufficient
compressional stress at the equator to overcome the surface tension.
The upper elongation $a/b$, where ${L_{\text{w}}}/{L}=0$, is
the point where the capsule is elongated so much that
the transverse strain, which is related to Poisson's number $\nu$
and tends to shrink the capsule in the circumferential direction,
counteracts any energy gain by the
wrinkles.
The wrinkles' length ${L_{\text{w}}}/{L}$ for different elongations $a/b$ turns
out to be almost independent of the susceptibility $\chi$.
In systems completely dominated by the elasticity and with negligible
surface tension, there are wrinkles for almost all elongations. The
wrinkle length quickly rises to a maximum and then slowly decreases due to
the transverse strain.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth,clip]{wrinkling_X21.pdf}
\caption{
Relative wrinkle length ${L_{\text{w}}}/{L}$ as a function of elongation
${a}/{b}$ for spheroidal capsules with
fixed $\chi=21$ and different values of
${Y_{2\text{D}}}/{\gamma}$. There are no wrinkles
(${L_{\text{w}}}/{L}=0$)
for ${Y_{2\text{D}}}/{\gamma} \lesssim 8.93$. The range of
${L_{\text{w}}}/{L}> 0$ and the extent of wrinkles
increase with ${Y_{2\text{D}}}/{\gamma}$ until they
converge to an asymptotic curve for thick shells.
}
\label{fig:Lw_plot}
\end{center}
\end{figure}
\section{Discussion and Conclusion}
\label{sec:conclusion}
Magnetic or electric fields provide an interesting and fairly easily
realizable route to the manipulation of elastic capsules if capsules can
be filled with ferrofluids or dielectric substances.
In this work we investigated the deformation of ferrofluid-filled capsules with
thin elastic shells in uniform external magnetic fields numerically
and using several analytic approaches.
Our results apply unchanged to elastic capsules filled
with a dielectric liquid in an external uniform electric field
(see Sec.\ \ref{sec:electric}).
Numerically, we obtained
equilibrium shapes by solving the coupled elastic and the magnetostatic
problems in an iterative manner. To calculate the
magnetic field, we used a combination of the finite element
method and the boundary element method for a given capsule shape.
The elastic capsule was described
by nonlinear shell theory with a Hookean elastic law. By
neglecting the bending rigidity we had to solve a system of four shape
equations describing the force equilibrium in the absence
of wrinkling and modified shape
equations to take the effect of wrinkling into account. In addition
to the dimensionless control parameters, the magnetic Bond number
$B_m$ and susceptibility $\chi$, that characterize
ferrofluid drops, we used the dimensionless
ratio ${Y_{2\text{D}}}/{\gamma}$ as an
elastic control parameter.
As for ferrofluid droplets, we found spheroidal shapes at small and moderate
magnetic fields, conical shapes at high magnetic fields, and a
discontinuous shape transition between spheroidal and conical shapes.
The general behavior of ferrofluid-filled capsules is comparable to
drops but higher Bond numbers $B_m$ are needed to reach the same
elongation
due to the additional elastic forces.
For small fields, the capsule shape is exactly spheroidal
and its elongation is very well described by a linear response theory,
which is in good agreement with our numerical results
(see Fig.\ \ref{fig:lin_region}). The small field regime is easily
accessible in experiments and our result (\ref{eq:lin_approx})
for the elongation $a/b$ can be used to determine the
Young modulus $Y_{2\text{D}}$ of the capsule material
from elongation measurements
if the magnetic properties of the ferrofluid
are known.
Also at moderate magnetic fields, capsule shapes with elongations
$a/b \lesssim 3$
are prolate spheroids to a very good approximation and
can be well described
by an approximative energy minimization, as Fig.\ \ref{fig:ab_Bm_plot} shows.
For high fields a conical shape is possible.
Capsules in a conical shape must
have finite isotropic tensions and
diverging isotropic stretches at the conical tip [see
Eq.\ (\ref{eq:lambdadiv})]
with a divergence exponent, which is given by the
half opening angle $\alpha$ of the
conical tip [see Eqs.\ (\ref{eq:betaalpha}) and (\ref{eq:betaalpha_app})].
The finiteness of tensions at the tip is a consequence of the
nonlinear constitutive relations (\ref{eq:taulambda}).
An important consequence of the divergence of stretches
at the tips of a conical shapes is that
conical shapes are probably not observable
experimentally because the high stretch factors give rise
to rupture close to the capsule tips.
Another consequence of such high stretch factors
is that the nonlinear Hookean material law will become locally invalid.
A real elastic capsule
material will show plastic behavior for high stretches,
followed by strain hardening and, finally,
the material's destruction \cite{Mercade-Prieto2011}.
Our results can explain experimental
observations of rupture of capsules
filled with a dielectric liquid in external
electric fields, where the capsules' shells
were destroyed near the tip \cite{Karyappa2014}.
Then the existence of the sharp discontinuous shape transition
into a conical shape can
provide an interesting tool to trigger
capsule rupture at rather well-defined magnetic
(for ferrofluid-filled capsules)
or electric (for dielectric-filled capsules) field values in
future applications of such capsules as delivery systems.
Capsule rupture at the tips has some analogies with the disintegration of
droplets in electric fields by emitting fluid jets at the tip
\cite{Collins2008,Collins2013}.
Real fluid drops, which are not perfect
conductors or perfect insulators,
disintegrate at higher external electric fields
by emitting jets of fluid at the tip.
This is known from experiments \cite{Wilson_Taylor1925} as
well as quite precisely understood in theory \cite{Collins2008,Collins2013}.
In our setup of a fluid inside an elastic shell,
the emission of a fluid jet is prevented by the shell at first.
However, the tangential stresses at the tip that lead to the
formation of a fluid jet may support the destruction of the shell
near the tip.
Once the shell is broken, a jet can be emitted.
The rupture process itself cannot be described by our
numerical approach and is an interesting topic for future work.
Our elastic shape equation approach provides a very precise tool
to solve the static elastic part of the problem, as long as nonlinear Hookean
elasticity can be used. Also the generalization to other
material laws, which are more appropriate for large strains,
is possible \cite{hegemann2017}.
Breaking of axisymmetry and topology changing rupture events cannot be
easily incorporated into the shape equation approach, however.
Also the magnetic field calculation should
be improved if rupture is addressed, in particular
in the capsule's tip region by using, for example,
an elliptic mesh generation for the finite element method.
One idea for a future improved simulation method that
captures possible rupture processes at the tip
is a dynamic simulation,
where the magnetohydrodynamics of the fluid and the viscoelastic dynamics
of the capsule shell including rupture processes
can be calculated explicitly, similarly to what has been
achieved for droplets in electric fields \cite{Collins2008,Collins2013}.
We presented a complete shape diagram in
Fig.\ \ref{fig:ab_Bm_plot} and characterized the discontinuous
shape transition between spheroidal and conical shapes.
The slender-body theory predicts that this discontinuous shape transition
only exists above the same critical value $\chi_c$ as
for ferrofluid droplets, which was predicted to lie between
$\chi_c \simeq 14.5$ \cite{Stone99} and $\chi_c\simeq 16.59$ \cite{Li1994}.
It also predicts that $\chi_c$ is independent of the
Young modulus $Y_{2\mathrm{D}}$ of the capsule.
We predict that the critical $\chi_c$ will be very sensitive
to the constitutive relation of the material. A strictly linear
constitutive relation, for example, could give rise to
diverging tensions at a
conical tip, resulting in much higher values for $\chi_c$.
We used the meridional stretch factor $\lambda_s$ at the pole
as a suitable order parameter to detect the spheroidal-conical
transition, because stretches diverge at the tip
of conical shapes but remain finite for spheroidal shapes, resulting
in a pronounced jump of the stretch factor in the numerical calculations.
The spheroidal-conical
transition exhibits hysteresis effects in an interval
$B_{m,c1}<B_m< B_{m,c2}$ between two
critical Bond numbers, which are the limit of stability
of the spheroidal and conical shapes.
In the hysteresis interval
both types of shapes are metastable.
The interval
has its maximum size for ferrofluid
droplets and decreases slightly with increasing
Young's modulus of the elastic shell.
In the numerical calculations for $\chi=21$, we
observe hysteresis effects for all ${Y_{2\text{D}}}/{\gamma}$,
which shows that, indeed, $\chi_c<21$ for all the Young moduli.
It turned out that the formation of wrinkles is an important
effect in systems with low surface tension $\gamma$. It has a visible effect
on the elongation and the specific shape. Wrinkles appear for the first time
for ${Y_{2\text{D}}}/{\gamma} \gtrsim 8.93$ (for $\chi=21$), and
are almost
always present for systems with lower surface tension, even at very low
elongations. Using this knowledge, it is possible to determine, for example,
${Y_{2\text{D}}}/{\gamma}$ in experiments by a simple measurement of the
wrinkle length $L_{\text{w}}$, which should be easy to perform in
practice.
\begin{appendix}
\section{Linear response at small magnetic fields}
\label{app:lin_def}
In this appendix we derive the linear response of the capsule
elongation $a/b$ for small applied magnetic fields.
Without applied field, the capsule is spherical with a rest radius $R_0$.
In the presence of a surface tension $\gamma$, this also requires
an internal pressure $p_0 = 2\gamma/R_0$ (Laplace-Young equation).
If a small magnetic field is applied the additional position-dependent
normal magnetic
force density $f_m = O(H^2)$ [see Eq.\ (\ref{fm})] acts on the spherical
surface,
resulting in normal displacements $u_R(\theta)\vec{e}_R$ and tangential
displacements $u_\theta(\theta)\vec{e}_\theta$, where we use spherical
coordinates with the polar
angle $\theta$
(i.e., $\theta=0$ at the upper pole and $\theta=\pi/2$ at the equator)
and the spherical coordinate
unit vectors $\vec{e}_R$ and $\vec{e}_\theta$.
Because of axisymmetry the displacements do not depend on the
azimuthal angle $\varphi$ and there is no displacement in direction
$\vec{e}_\varphi$.
The deformed capsule surface is parametrized as
$\vec{r}(\theta,\varphi) =
[R_0 + u_R(\theta)]\vec{e}_R(\theta,\varphi)$ using
polar and azimuthal angles $\theta$ and $\varphi$.
The new equilibrium shape has
small displacements $u_R, u_\theta = O(H^2)$
and fulfills force equilibrium in two independent directions
on the surface. We will consider normal force equilibrium as described
by the Laplace-Young equation [see Eq.\ (\ref{eq:equilibrium_eqns_norm})]
and tangential force equilibrium [see Eq.\ (\ref{eq:equilibrium_eqns_tang})].
We start with the Laplace-Young equation
\begin{align}
\label{eq:fbal_cap}
\kappa_s (\tau_s+\gamma) +\kappa_\varphi (\tau_\varphi + \gamma)
&= p_0 + f_m,
\end{align}
where $\gamma$ is a surface tension, $\tau_s$ and $\tau_\varphi$ are
elastic tensions, and
$f_m = (\mu_0\chi/2)[H^2 + \chi (\vec{n}\cdot\vec{H})^2]$ is
the small normal magnetic force density (\ref{fm}) causing
small displacements.
The pressure will change to linear order in the
displacements
$p_0 = 2\gamma/R_0 + O(u_R, u_\theta)$
to ensure a fixed volume.
In spherical coordinates and in linear order in the displacements,
the stretch factors can be calculated using
$r = \sqrt{g_{\varphi\varphi}} =|\partial_\varphi \vec{r}|$
($r_0 = R_0\sin\theta$) and
$\text{d}s = \sqrt{g_{\theta\theta}} d\theta =
|\partial_\theta \vec{r}|d\theta$
($ds_0 = R_0 d\theta$):
\begin{align*}
\lambda_s &= \frac{\text{d}s}{\text{d}s_0} = \frac{|\partial_\theta
\vec{r}|}{R_0}
= 1 + \frac{1}{R_0} \left({u_R}+ {\partial_\theta u_\theta}\right),
\\
\lambda_\varphi &= \frac{r}{r_0} = \frac{|\partial_\varphi
\vec{r}|}{R_0\sin\theta}
= 1 + \frac{1}{R_0}\left( u_R+ {u_\theta}\cot\theta\right).
\end{align*}
In linear order in the displacements
the constitutive relations (\ref{eq:taulambda})
can then be written as \cite{LL7}
\begin{align}
\tau_\varphi - \nu \tau_s&= Y_{2\text{D}}(\lambda_\varphi-1)
= \frac{Y_{2\text{D}}}{R_0}(u_\theta \cot\theta + u_R),\\
\tau_s - \nu \tau_\varphi &= Y_{2\text{D}}(\lambda_s-1) =
\frac{Y_{2\text{D}}}{R_0}\left(\partial_\theta u_\theta + u_R\right).
\label{eq:taulambda2}
\end{align}
Elastic tensions are small for small magnetic fields,
$\tau_s, \tau_\varphi = O(u_R, u_\theta) = O(H^2)$,
whereas the fluid surface tension $\gamma$ cannot
be considered small.
Therefore, we also need to consider curvature corrections up to linear
order $O(u_R, u_\theta)$ in
Eq.\ (\ref{eq:fbal_cap}):
\begin{align*}
\kappa_s+ \kappa_\varphi \approx \frac{2}{R_0}
-\frac{1}{R_0^2} \left( 2 u_R - \partial_\theta^2 u_R + \partial_\theta u_R
\cot\theta \right).
\end{align*}
On the right-hand side of Eq.\ (\ref{eq:fbal_cap}), we can use
$\vec{n} = \vec{e}_R$ for the outward unit
normal to $O(H^2)$.
This results in the following normal force balance
to linear order in the displacements, i.e., to ${\cal O}(H^2)$:
\begin{align}
& -\gamma \left( 2 u_R - \partial_\theta^2 u_R + \partial_\theta u_R
\cot\theta \right) + R_0 (\tau_s + \tau_\varphi)
\nonumber\\
&~~~= (p_0R_0^2 - 2R_0\gamma) + \frac{\mu_0}{2}\chi H^2R_0^2
(1+\chi \cos^2\theta).
\label{eq:fbal_cap2}
\end{align}
We first solve this equation
for a ferrofluid droplet ($Y_{2\text{D}}/\gamma=0$), where
the elastic stresses and thus $u_\theta$ are zero.
Boundary conditions are $\partial_\theta u_R(0) = \partial_\theta u_R(\pi/2)=0$
and $u_\theta(0) = u_\theta(\pi/2)=0$ to avoid kinks
(we are not considering conical shapes in the linear response)
or holes in the shape.
Then $u_\theta=0$ and an ansatz
\begin{equation}
u_R= A+B\cos^2\theta
\label{eq:AnsatzuR_app}
\end{equation}
leads to a solution
\begin{align}
B &= \frac{\mu_0}{8\gamma}\chi^2 H^2R_0^2
\approx \frac{9\mu_0 \chi^2}{8\gamma(3+\chi)^2} H_0^2R_0^2
\label{eq:uRB}\\
A &= -\frac{1}{2} \left(\frac{p_0R_0}{\gamma}- 2\right)R_0
- \frac{\mu_0}{4\gamma}\chi\left(1+\frac{\chi}{2} \right) H^2R_0^2.
\label{eq:uRA}
\end{align}
To leading order in $u_R = O(H^2)$, the ansatz (\ref{eq:AnsatzuR_app})
describes a spheroid such that we can replace
the magnetic field $H$ in (\ref{eq:uRB})
by the analytically known value for a field inside a
spheroid \cite{Stratton41},
\begin{equation}
H = {H_0}/{(1+n\chi)},
\label{eq:H_app}
\end{equation}
where $n$ denotes the demagnetization factor.
To leading order $O(H^2)$ it is also correct to
use the result $n=1/3$ for a sphere (\ref{eq:H_app}).
Moreover, volume conservation requires
\begin{equation}
A= -B/3,
\label{eq:VolAB_app}
\end{equation}
which determines the pressure correction
$p_0 = 2\gamma/R_0 + O(H^2)$ from Eq.\ (\ref{eq:uRA}).
For the deformation $a/b$
we find, to leading order in $u_R = O(H^2)$,
\begin{equation}
\frac{a}{b} = \frac{R_0 + u_R(0)}{R_0+ u_R(\pi/2)}
\approx 1 + \frac{B}{R_0}
= 1+\frac{9\mu_0 R_0\chi^2}{8\gamma(3+\chi)^2}\chi H_0^2.
\label{eq:droplinear_app}
\end{equation}
For a ferrofluid-filled elastic capsule we also need to consider
the force equilibrium in the tangential
direction because the total tensions
$\gamma+\tau_s\neq \gamma+\tau_\varphi$ become anisotropic now
(for a liquid interface with $\tau_s=\tau_\varphi=0$
the force equilibrium in tangential direction
becomes exactly equivalent to the normal force equilibrium,
i.e., the Laplace-Young equation).
The tangential force equilibrium (\ref{eq:equilibrium_eqns_tang})
can be written as
\begin{align*}
\tau_\varphi &= \partial_r(r \tau_s) = \tau_s + r \partial_r \tau_s
= \tau_s+ \frac{\partial_\theta \tau_s}{\partial_\theta r}.
\end{align*}
Using
$r =|\partial_\varphi \vec{r}| = \sin\theta\left(
R_0 + u_R+ {u_\theta}\cot\theta\right)$ and Eq.\ (\ref{eq:taulambda2})
for the elastic stresses,
the tangential force equilibrium becomes
\begin{align}
& \tau_\varphi -\tau_s
= \frac{Y_{2\text{D}}}{(1+\nu)R_0}\left(u_\theta \cot\theta -
\partial_\theta u_\theta\right)
\nonumber\\
&= \partial_r \tau_s = \frac{Y_{2\text{D}}}{(1-\nu^2)R_0}\left(
\tan \theta \partial_\theta^2u_\theta + \nu \partial_\theta u_\theta
-\frac{\nu u_\theta}{\cos\theta \sin\theta} + (1-\nu)
\tan\theta \partial_\theta u_R \right).
\label{eq:fbalt_cap2}
\end{align}
For the ferrofluid capsule,
the two force equilibria (\ref{eq:fbal_cap2}),
where $\tau_s$ and $\tau_\varphi$ have to be expressed in terms of
the displacements using the constitutive relations (\ref{eq:taulambda2}),
\begin{align*}
\tau_s+ \tau_\varphi &= \frac{Y_{2\text{D}}}{(1-\nu)R_0}
\left(2u_R + u_\theta \cot\theta +
\partial_\theta u_\theta\right),
\end{align*}
and Eq.\ (\ref{eq:fbalt_cap2}) have to
be solved for the deformed capsule shape.
Boundary conditions are $\partial_\theta u_R(0) = \partial_\theta u_R(\pi/2)=0$
and $u_\theta(0) = u_\theta(\pi/2)=0$.
For the fluid limit $Y_{2\text{D}}/\gamma=0$, we derived an
exact solution above.
For the ferrofluid capsule, we make an ansatz
\begin{equation}
u_R= A+B\cos^2\theta,~~u_\theta = C\sin\theta\cos\theta,
\label{eq:AnsatzuRcap_app}
\end{equation}
which still describes a spheroid to leading order in
the displacements because $u_\theta\neq 0$ only generates an
additional tangential displacement.
Then the tangential force equilibrium gives
\begin{align}
C &= - \frac{2(1+\nu)}{5+\nu} B.
\label{eq:uRCcap}
\end{align}
For the ferrofluid capsule,
the normal force equilibrium (\ref{eq:fbal_cap2}) gives
\begin{align}
B&= \frac{\mu_0(5+\nu) }{8[Y_{2\text{D}}+(5+\nu)\gamma]}\chi^2 H^2R_0^2,
\label{eq:uRBcapdrop} \\
A&= \frac{1-\nu}{2} \left(\frac{p_0R_0}{Y_{2\text{D}}}-
2\frac{\gamma}{Y_{2\text{D}}}\right)R_0
+ \frac{\mu_0}{4(1-\nu)Y_{2\text{D}}}\chi H^2R_0^2\left(1+\frac{\chi}{2}
\right)
- \frac{C}{1+\nu}
\label{eq:uRAcapdrop}
\end{align}
and the relation $A=-B/3$ [see Eq.\ (\ref{eq:VolAB_app}]
from the fixed volume constraint determines
the pressure $p_0$.
For the deformation $a/b$
we find, to leading order in $u_R = O(H^2)$,
\begin{align}
\frac{a}{b} &= \frac{R_0 + u_R(0)}{R_0+ u_R(\pi/2)}
\approx 1 + \frac{B}{R_0}
=
1 + \frac{9\mu_0 R_0 \chi^2 (5+\nu)}
{8 [Y_{2\text{D}}+\gamma(5+\nu)](3+\chi)^2}H_0^2.
\label{eq:capdroplinear_app}
\end{align}
The criterion for wrinkling is $\tau_\varphi + \gamma <0$,
where
\begin{align}
\tau_\varphi &= \frac{Y_{2\text{D}}}{(1-\nu^2)R_0}\left[ u_\theta \cot\theta
+ (1+\nu)u_R
+ \nu \partial_\theta u_\theta \right]
\approx B\frac{1-\nu^2}{5+\nu}\left( -\frac{5}{3} +3 \cos^2\theta\right)
\label{eq:tauphi_app}
\end{align}
from Eq.\ (\ref{eq:taulambda2}) and using Eq.\ (\ref{eq:AnsatzuRcap_app})
with Eqs.\ (\ref{eq:uRCcap}) and (\ref{eq:VolAB_app}).
\section{Approximative energy minimization for spheroidal shapes}
\label{app:Bm}
In this appendix we derive an analytical approximation for the
elongation $a/b$ of the capsule at moderate magnetic forces
by minimizing an
approximative total energy, which assumes a spheroidal shape
for magnetic and elastic contributions and constant elastic
stretch factors throughout the shell.
We minimize the total energy, the sum of surface, magnetic, and
elastic energies with respect to the inverse
elongation ratio $k\equiv {b}/{a}<1$
at fixed volume $V = (4\pi/3) ab^2=V_0$ (quantities $...|_V$ are
at fixed volume $V$):
\begin{equation*}
0 = \frac{\mathrm{d} E_\gamma|_V}{\mathrm{d}k}
+\frac{\mathrm{d} E_{\text{mag}}|_V}{\mathrm{d}k}
+ \frac{\mathrm{d} E_{\text{el}}|_V}{\mathrm{d}k}.
\end{equation*}
For fixed volume $V= (4\pi/3) ab^2= (4\pi/3)R_0^2= V_0$,
we have
\begin{equation*}
a|_V = R_0 k^{-2/3} ~,~~ b|_V = R_0 k^{1/3}.
\end{equation*}
The surface energy (\ref{Egamma}), which is proportional
to the surface area $A$ at fixed volume, can then be written as
\begin{align*}
E_\gamma|_V &= \gamma A|_V ~~~\mbox{with}~~
A|_V = A_0
\frac{1}{2} k^{-{1}/{3}}\left(k+\frac{1}{\epsilon}\arcsin{\epsilon} \right),
\end{align*}
where $\epsilon = \epsilon(k) \equiv \sqrt{1-k^2}$ is the spheroid's
eccentricity and $A_0 = 4\pi R_0^2$ the area of the undeformed sphere.
The magnetic energy (\ref{Emag}) is
given as
\begin{align*}
E_{\text{mag}}|_V &= -\frac{V_0\mu_0}{2} \frac{\chi}{1+n\chi}H_0^2
= -\gamma A_0 B_m \frac{1}{3(1+n\chi)},
\end{align*}
where $n$ is the demagnetization factor
\begin{equation*}
n= n(k) = \frac{k^2}{2\epsilon^3(k)}
\left(-2\epsilon(k) + \ln{\frac{1+\epsilon(k)}{1-\epsilon(k)}} \right)
\end{equation*}
and
$B_m = {\mu_0R_0\chi H_0^2}/{2 \gamma}$ is the Bond number.
Finally, we calculate the elastic stretch energy (\ref{Eel}) via
\begin{equation*}
E_{\text{el}}|_V
= A_0 \frac{Y_{2\mathrm{D}}}{2(1-\nu^2)}
\left[(e_s|_V)^2+ 2\nu e_s|_V e_\varphi|_V + (e_\varphi|_V)^2\right]
\end{equation*}
using the approximation of constant $e_s$ and $e_\varphi$.
At fixed volume, we find
\begin{align*}
e_s &= \frac{P_\text{ellipse}}{P_\text{circle}} -1
\approx \frac{a+b}{2R_0}
\left(1 + \frac{3\eta^2}{10 + \sqrt{4-3\eta^2}}\right)-1,\\
e_s|_V &= \frac{k^{-2/3}(1+ k)}{2}
\left(1 + \frac{3\eta^2(k)}{10 + \sqrt{4-3\eta^2(k)}}\right) -1, \\
e_\varphi &= \frac{b}{R_0}-1~~,~~
e_\varphi|_V = k^{1/3}-1,
\end{align*}
with $\eta = \eta(k) \equiv ({b-a})/({b+a}) = (k-1)/(k+1)$.
Now we can find the elongation $k$ that minimizes the
total energy at fixed volume; $k$ can only be determined implicitly
as a function of the magnetic field $H_0$
by the following relation between the Bond number
$B_m = {\mu_0R_0\chi H_0^2}/{2 \gamma}$
and a complicated function $g(k)$ of the elongation $k$,
which also depends on the susceptibility $\chi$, the dimensionless
Young modulus $Y_{2\text{D}}/\gamma$, and Poisson's ratio $\nu$:
\begin{align}
B_m &=\frac{\mu_0R_0\chi H_0^2}{2 \gamma} = g(k)
~~\mbox{with}\nonumber\\
g(k) &\equiv -3\left(\frac{1}{\chi} + n(k)\right)^2\chi
\frac{c_1(k) + \frac{Y_{2\mathrm{D}}}{2\gamma(1-\nu^2)} c_2(k,\nu)}{c_3(k)},
\label{eq:BMab}
\end{align}
where
\begin{align*}
c_1(k) &\equiv \frac{1}{A_0} \frac{\text{d}A|_V}{\text{d}k}, \\
c_2(k,\nu) &\equiv \left( 2 e_s|_V \frac{\mathrm{d}e_s|_V }{\mathrm{d}k}
+ 2\nu\left( e_\varphi|_V \frac{\mathrm{d}e_s|_V }{\mathrm{d}k} +
e_s|_V \frac{\mathrm{d}e_\varphi|_V }{\mathrm{d} k} \right)
+ 2e_\varphi|_V \frac{\mathrm{d} e_\varphi|_V }{\mathrm{d} k} \right),\\
c_3(k) &\equiv \frac{\text{d}n}{\text{d}k}
= \frac{-3k}{\epsilon^4(k)} + \left(\frac{k}{\epsilon^3(k)}
+ \frac{3k^3}{2\epsilon^5(k)} \right)
\ln{\frac{1+\epsilon(k)}{1-\epsilon(k)}}.
\end{align*}
The functions $c_1(k)$ and $c_3(k)$ from surface and magnetic energies
depend on the inverse elongation ration $k=b/a<1$
only, whereas the function $c_2(k,\nu)$ from the elastic energy also
depends on Poisson's ratio $\nu$ (which is set to $\nu=1/2$ and thus
fixed throughout this paper).
This relation reduces to the results of Bacri and Salin \cite{Bacri82}
for ferrofluid droplets in the limit $Y_{2\text{D}}= 0$, where
the function $c_2(k,\nu)$ drops from Eq.\ (\ref{eq:BMab}).
The solid lines in Fig.\ \ref{fig:ab_Bm_plot} show plots of
$1/k = a/b$ versus $B_m$ as given by the relation $B_m = g(k)$.
\section{Conical shapes for
elastic membranes with spherical rest shape}
\label{app:cone_angle}
\subsection{Stretches and tensions at a conical tip with
normal magnetic forces}
\label{app:cone_tip}
In this appendix we show that a conical shape, as it is
observed for ferrofluid
drops at a critical field strength, is also possible for an elastic
capsule with a spherical rest shape and stretched by normal magnetic forces
but requires diverging and asymptotically isotropic
stretches with an exponent determined by the opening angle
of the cone, whereas elastic tension have to remain
finite and isotropic at the tip of the cone.
A sharp conical tip implies a non-zero
slope angle $\psi(s_0=0)>0$, where $\alpha = \pi/2-\psi(0)$
is half of the opening angle of the cone.
In contrast to a ferrofluid droplet with constant and isotropic
surface tension $\gamma$, an elastic capsule develops additional
elastic tensions $\tau_s$ and $\tau_\varphi$, which depend on the state
of stretching, i.e., the stretches $\lambda_s$ and $\lambda_\varphi$
with respect to the spherical rest shape
via the nonlinear constitutive relations (\ref{eq:taulambda}),
and
which have to fulfill an additional tangential force equilibrium
(\ref{eq:equilibrium_eqns_tang}) that
we rewrite as
\begin{align}
\tau_\varphi &= \partial_r(r \tau_s) = \tau_s + r\partial_r \tau_s.
\label{eq:fbalt_cap0}
\end{align}
It is important to note that the tangential force equilibrium
does not contain external magnetic forces, which are always normal to the
surface [see Eqs.\ (\ref{eq:pfm}) and (\ref{eq:equilibrium_eqns_norm})].
The internal
tangential force equilibrium has to be compatible with the deformation
into a conical tip.
First we show that $\tau_s(0)$ and $\tau_\varphi(0)$ have to remain
finite at the tip at $s_0=0$ (corresponding to $r=0$).
The reason for a divergence of one
of the tensions can only be a divergence of one or both of the
stretches. According to the nonlinear constitutive relations
(\ref{eq:taulambda}), only one of the tensions can exhibit a divergence
($\lambda_s/\lambda_\varphi$ and $\lambda_\varphi/\lambda_s$ cannot both
diverge). Then it is easy to verify that a single divergent
tension at $r=0$ contradicts the force equilibrium
(\ref{eq:fbalt_cap0}). Therefore, both tensions have to remain
finite at $s_0=0$ (or $r=0$).
Next we show that finiteness of the tensions at the conical tip
necessarily leads to tension isotropy
$\tau_s(0) = \tau_\varphi(0)$ at the
tip. Because magnetic forces are stretching forces, both tensions
are equal and stretching, $\tau_s(0) = \tau_\varphi(0)>0$.
If $\tau_s(0) \neq \tau_\varphi(0)$, the tangential
force equilibrium (\ref{eq:fbalt_cap0}) immediately leads to
$\partial_r \tau_s \approx [\tau_\varphi(0)- \tau_s(0)]/r$ for
small $r$, resulting in a logarithmically diverging $\tau_s \propto -\ln r$
for small $r$ contradicting finiteness.
The equality $\tau_s(0) = \tau_\varphi(0)$ at the
tip also leads to isotropy of the stretches
$\lambda_s(0)=\lambda_\varphi(0)$ at the tip because of the
constitutive relations
(\ref{eq:taulambda}), however,
not necessarily to finiteness of the stretches at the tip.
Therefore, we have to discuss the cases of finite and diverging
stretches $\lambda_s=\lambda_\varphi$ at the conical tip
separately.
We start with finite isotropic stretches,
$\lambda_s(0)=\lambda_\varphi(0)<\infty$.
Then we can apply l'H{\^o}pital's rule at the tip $s_0=0$:
\begin{align}
\lambda_\varphi(0) =
\lim\limits_{s_0 \to 0}\frac{r}{r_0} =
\lim\limits_{s_0 \to 0} \frac{r'}{r_0'} =
\frac{\lambda_s\cos[\psi(0)]}{\cos[\psi_0(0)]}
=\lambda_s(0)\cos[\psi(0)]
\label{eq:lhospital}
\end{align}
[where we used $\psi_0(0)=0$ for the spherical rest shape].
Equality of the stretches $\lambda_s(0)=\lambda_\varphi(0)$
then leads to the conclusion $\psi(0)=0$, i.e.,
a sharp conical tip is impossible if stretches remain finite at the tip.
L'H{\^o}pital's rule can no longer be applied if the stretches diverge
at the tip (remaining asymptotically isotropic), i.e.,
\begin{equation}
\lambda_s(s_0)\approx \lambda_\varphi(s_0)\approx {\rm const}\, s_0^{-\beta}
\label{eq:lambdadiv_app}
\end{equation}
for $s_0\approx 0$
with an exponent $\beta>0$.
Because of $\lambda_s = r'/\cos\psi$, this requires
$r(s_0) \approx {\rm const}\, s_0^{1-\beta}/(1-\beta)\cos\psi(0)$
for $s_0\approx 0$, whereas $r_0(s_0) = R_0 \sin(s_0/R_0) \approx s_0$
for the spherical rest shape.
Then Eq.\ (\ref{eq:lhospital}) is replaced by
\begin{align*}
\lambda_\varphi(s_0) =
\lim\limits_{s_0 \to 0}\frac{r}{r_0} =
\frac{{\rm const} s_0^{-\beta}}{(1-\beta)\cos\psi(0)} =
\lim\limits_{s_0 \to 0} \frac{1}{1-\beta}\frac{r'}{r_0'}
= \lambda_s(s_0)
\frac{\cos[\psi(0)]}{1-\beta}
\end{align*}
for $s_0\approx 0$. The equality $\lambda_s(s_0)\approx \lambda_\varphi(s_0)$
necessarily leads to the condition
\begin{equation}
\beta = \cos[\psi(0)]-1 = \sin\alpha-1
\label{eq:betaalpha_app}
\end{equation}
between the exponent $\beta$ of the divergent stretches
and the half opening angle $\alpha = \pi/2 - \psi(0)$ of the
conical tip.
In conclusion, a deformation of the spherical rest shape into a sharp
conical tip with $\psi(0)> 0$ is only possible if stretches
are asymptotically isotropic and diverge as
$\lambda_s(s_0)\approx \lambda_\varphi(s_0)\sim s_0^{-\beta}$
with an exponent $\beta$, which is related by
Eq.\ (\ref{eq:betaalpha_app}) to the opening angle $2\alpha$ of the cone.
Because of the nonlinear constitutive relation (\ref{eq:taulambda}),
diverging and isotropic stretches are compatible with
finite and isotropic tensions at the tip with
\begin{equation}
\tau_s(0) = \tau_\varphi(0) =
\frac{Y_{2\text{D}}}{1-\nu}.
\label{eq:tau0_app}
\end{equation}
Note that away from the tip ($s_0>0$), tensions and stretches
feature anisotropic corrections.
\subsection{Governing equations for stretches and tensions
in a conical shape with spherical rest shape}
\label{app:cone_eqs}
In this section we present how to systematically calculate
stretches and elastic tensions in a deformation from a spherical
rest shape into
a conical shape by deriving the governing equations.
This is the basis of the generalization of the slender-body theory
of Stone {\it et al.}\ from ferrofluid conical droplets to
capsules.
We assume that the conical shape is given by a function $r(z)$,
where $z$ runs from the bottom of the cone at $z=-a$ to its top at
$z=a$.
We will show that, if the
conical shape $r(z)$ is known, we can calculate
all stretches and tensions in this shape.
The rest shape is spherical and parametrized analogously
by a function $r_0(z_0) = (R_0^2 - z_0^2)^{1/2}$
with $z_0 \in [-R_0,R_0]$.
For the following it is advantageous to replace $z$ and $z_0$
by coordinates
$d=a+z$ measuring the distance from the lower conical tip
and $d_0 = R_0 + z_0$ measuring the distance from the corresponding
south pole of the sphere. This geometry is illustrated in
Fig. \ref{fig:appendix_sketch}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=0.7\textwidth,clip]{appendix_sketch.pdf}
\caption{
Illustration of the geometry at the capsule's south pole
(not true to scale) for
(a) a conical tip and (b) the spheroidal reference shape.
}
\label{fig:appendix_sketch}
\end{center}
\end{figure}
Given a conical shape $r(d)$ and the spherical rest shape
\begin{equation}
r_0(d_0) = (2R_0d_0 -d_0^2)^{1/2},
\label{eq:spherical_app}
\end{equation}
we want to show how the function $d(d_0)$ describing the
stretching in the $z$ direction
can be calculated systematically
from the tangential force equilibrium
(\ref{eq:fbalt_cap0}) or (\ref{eq:equilibrium_eqns_tang}) and
the constitutive relations (\ref{eq:taulambda}).
If the conical shape $r(d)$ and the function $d(d_0)$ are given
[and the spherical rest shape $r_0(d_0)$]
the meridional and hoop stretches can be calculated
as a function of $d_0$ by
\begin{align}
\begin{split}
\lambda_\varphi &= \frac{r(d(d_0))}{r_0(d_0)} =
\frac{r(d(d_0))}{(2R_0d_0 -d_0^2)^{1/2}},
\\
\lambda_s &= \frac{\text{d}s}{\text{d}s_0}
= \frac{\left[1+ r'(d(d_0))^2\right]^{1/2}}
{\left[1+r_0'(d_0)^2\right]^{1/2}}
d'(d_0)
= {\left[1+ r'(d(d_0))^2\right]^{1/2}}\frac{(2R_0d_0 - d_0^2)^{1/2}}{R_0}
d'(d_0),
\end{split}
\label{eq:lambda_d_app}
\end{align}
where $\lambda_z \equiv d'(d_0) = {\text{d}z}/{\text{d}z_0}$
is the stretch in the $z$ direction.
If both stretches are known then the constitutive relations
(\ref{eq:taulambda}) can be used to express tensions $\tau_s$
and $\tau_\varphi$ as algebraic functions
of the stretches $\lambda_s$ and $\lambda_\varphi$ from Eq.\
(\ref{eq:lambda_d_app}) and thus as functions of $d_0$, the
conical shape
$r(d)$, and the unknown function $d(d_0)$ and its derivative.
These tensions have to fulfill the tangential
force equilibrium (\ref{eq:fbalt_cap0}), which we rewrite
in terms of stretches using the constitutive relations
(\ref{eq:taulambda}) ,
\begin{align}
\tau_\varphi &= \tau_s + r\partial_r \tau_s,
\nonumber\\
\lambda_\varphi^3 - (1+\nu)\lambda_\varphi^2 &=
\lambda_s^2\lambda_\varphi - (1+\nu)\lambda_s\lambda_\varphi
+ r\lambda_s\left\{ \lambda_\varphi (\partial_r \lambda_s) -
(\partial_r \lambda_\varphi)\left[\lambda_s - (1+\nu)\right] \right\}.
\label{eq:fbalt_cap1}
\end{align}
Plugging in the stretches from (\ref{eq:lambda_d_app}) and using
\begin{equation*}
\partial_r = \frac{1}{\partial_{d_0}r}\partial_{d_0} =
\frac{1}{r'(d(d_0)) d'(d_0)} \partial_{d_0},
\end{equation*}
we obtain a complicated nonlinear differential equation for the
unknown function $d(d_0)$ and its derivative $\lambda_z(d_0)= d'(d_0)$.
If this differential equation can be solved, all stretches
and tensions arising from the deformation from $r_0(d_0)$ into
$r(d)$ are determined, in principle.
Unfortunately, this equation cannot be solved in general.
In the next section we obtain features of
a solution close to the conical tip.
\subsection{Stretches and tensions in the vicinity of a conical tip
for a spherical rest shape}
\label{app:cone_tip2}
In the vicinity of a the conical tip the conical shape
$r(d)$ with a half opening angle $\alpha$
becomes strictly conical, and we can use
\begin{equation}
r(d) = d \tan\alpha,
\end{equation}
resulting in stretches
\begin{align}
\begin{split}
\lambda_\varphi & =
\frac{\tan\alpha}{(2R_0d_0 -d_0^2)^{1/2}} d(d_0),
\\
\lambda_s &
= \frac{1}{\cos\alpha}\frac{(2R_0d_0 - d_0^2)^{1/2}}{R_0}
d'(d_0).
\end{split}
\label{eq:lambda_d2_app}
\end{align}
Close to the conical tip, $\lambda_s$ and $\lambda_\varphi$
are diverging and asymptotically equal according to
Appendix \ref{app:cone_tip}.
Requiring $\lambda_s = \lambda_\varphi$ for small $d_0$ gives a differential
equation
\begin{equation}
d'(d_0) = \sin\alpha \frac{R_0}{2R_0d_0 - d_0^2} d(d_0),
\label{eq:ODE_d_app}
\end{equation}
which is solved by
\begin{align*}
d(d_0) &= a \left( \frac{d_0}{2R_0-d_0} \right)^{(\sin\alpha)/2}
\approx a \left(\frac{d_0}{2R_0} \right)^{(\sin\alpha)/2}
\propto d_0^{(\sin\alpha)/2},
\end{align*}
where we use a boundary condition $d(R_0) = a$ resulting
from the conservation of the
mirror symmetry plane at $z=z_0=0$.
This results in
\begin{equation*}
r(d(d_0)) = \tan\alpha \,[d(d_0)] \approx
a \tan\alpha \left(\frac{d_0}{2R_0} \right)^{(\sin\alpha)/2}
\end{equation*}
and, using (\ref{eq:lambda_d2_app}),
\begin{align}
\lambda_s = \lambda_\varphi &\approx \frac{a}{2R_0} \tan\alpha
\left(\frac{d_0}{2R_0} \right)^{(\sin\alpha-1)/2}
= \frac{a \tan\alpha}{2R_0} \left( \frac{r}{a\tan\alpha}.
\right)^{1-1/\sin\alpha}
\label{eq:lambdaphi_d3_app}
\end{align}
Noting that $d_0 \approx R_0 [1- \cos(s_0/R_0)]\approx s_0^2/2R_0$
for the spherical rest shape, the exponent in (\ref{eq:lambdaphi_d3_app})
is exactly equivalent to our above result
(\ref{eq:betaalpha_app}), $\beta = 1- \sin\alpha$,
for the relation
between the exponent $\beta$ of the divergent stretches
$\lambda_s(s_0)\approx \lambda_\varphi(s_0)\approx s_0^{-\beta}$
and the half opening angle $\alpha$ of the
conical tip.
Away from the tip, the stretches and tensions acquire
anisotropic corrections.
Therefore, we start with an ansatz
\begin{align}
\begin{split}
\lambda_s &= b r^{-\tilde{\beta}} + b_s r^{-\gamma}
~,~~
\lambda_\varphi = b r^{-\tilde{\beta}} + b_\varphi r^{-\gamma},
\\ \tilde{\beta} &= 1/\sin\alpha-1,~~
b \approx ({a \tan\alpha})^{1+\tilde{\beta}}/{2R_0}
\end{split}
\label{eq:lambda_Ansatz_app}
\end{align}
for small $r$ in the vicinity of the conical tip,
where $\gamma <\tilde{\beta}$. We use this ansatz
in the tangential force balance relation (\ref{eq:fbalt_cap1})
derived in Appendix \ref{app:cone_eqs}.
First we obtain the tensions, which are isotropic
and in agreement with (\ref{eq:tau0_app}) to leading order
but also acquire anisotropic corrections
\begin{align*}
\tau_s \frac{1-\nu^2}{Y_{2\text{D}}} &=
1+\nu + \frac{b_s-b_\varphi}{b} r^{\tilde{\beta}-\gamma}
- \frac{1+\nu}{b} r^{\tilde{\beta} }
+ \frac{(1+\nu)b_s}{b^2} r^{2\tilde{\beta}-\gamma}
-\frac{b_\varphi(b_s+b_\varphi)}{b^2} r^{2\tilde{\beta}-2\gamma},
\\
\tau_\varphi \frac{1-\nu^2}{Y_{2\text{D}}} &=
1+\nu + \frac{b_\varphi-b_s}{b} r^{\tilde{\beta}-\gamma}
- \frac{1+\nu}{b} r^{\tilde{\beta} }
+ \frac{(1+\nu)b_\varphi}{b^2} r^{2\tilde{\beta}-\gamma}
-\frac{b_s(b_s+b_\varphi)}{b^2} r^{2\tilde{\beta}-2\gamma},
\end{align*}
neglecting terms $O(r^{3\tilde{\beta}-2\gamma})$.
These expression are used in the tangential force balance relation
(\ref{eq:fbalt_cap1}), $\tau_\varphi-\tau_s = r\partial_r \tau_s$,
in which we compare coefficients order by order in $r$ in order to
determine the exponent $\gamma$ and the coefficients $b_s$ and $b_\varphi$.
If we assume $\gamma>0$ the leading order terms are
$O(r^{\tilde{\beta}-\gamma})$, and comparing coefficients
gives a contradictory relation $2=\gamma -\tilde{\beta} <0$.
It follows that
\begin{equation*}
\gamma=0,
\end{equation*}
i.e., the leading anisotropic
corrections in the stretches
(\ref{eq:lambda_Ansatz_app}) are constant.
Continuing with $\gamma=0$, terms
$O(r^{\tilde{\beta}-\gamma})$ and $O(r^{\tilde{\beta}})$
are of equal order and comparing all coefficients gives
\begin{equation*}
b_s - b_\varphi = \frac{\tilde{\beta}(1+\nu) }{2+\tilde{\beta}} >0,
\end{equation*}
i.e., the anisotropy close to the tip is such that
$\lambda_s >\lambda_\varphi$ and $\tau_s >\tau_\varphi$.
For the tensions this results in
\begin{align}
\begin{split}
\tau_s &=
\frac{Y_{2\text{D}}}{1-\nu}\left( 1 - \frac{1}{b}
\frac{1}{1+\tilde{\beta}/2} r^{\tilde{\beta}} \right),
\\
\tau_\varphi &=
\frac{Y_{2\text{D}}}{1-\nu}\left( 1 - \frac{1}{b}
\frac{1}{1+\tilde{\beta}} r^{\tilde{\beta}} \right),
\end{split}
\label{eq:tau1_app}
\end{align}
which specifies the leading anisotropic corrections
to Eq.\ (\ref{eq:tau0_app}).
Finally, we can compare
coefficients of all terms
$O(r^{2\tilde{\beta}})$ for $\gamma=0$
to obtain
\begin{equation*}
b_\varphi^2-b_s^2 + (1+\nu)(b_\varphi-b_s) =
2\tilde{\beta}(1+\nu) b_s - 2\tilde{\beta} b_\varphi(b_s+b_\varphi)
\end{equation*}
which can be used to go on and determine both $b_s$ and $b_\varphi$
if needed.
\section{Discretization errors}
\label{app:errors}
To observe the transition to a conical shape, it is necessary
to have a high resolution for the finite element-boundary element method
in the tip of the capsule.
If we consider the number of boundary elements to be fixed to $N = 250$, we
can vary the density of elements near the tip by changing the parameter $l_0$
[see Sec.\ \ref{fieldcalculation} and Eq.\ (\ref{eq:Li})].
For different values of $l_0$, we see a
quite different numerical behavior.
Every result in the text above is calculated with $l_0 = 0.1$.
For significantly smaller values of $l_0$, we cannot calculate conical shapes.
The problem is that our shooting method for the elastic shape equations
does not find solutions anymore due to extremely high
and rapidly changing stretch factors at the tip [$\lambda_s(s_0 = 0) > 10^{4}$].
On the other hand, with constant element density ($l_0 = 1$),
a shape transition cannot be found anymore; the capsule's shape stays rounded.
This indicates that the numerical calculation of the shape transition is
prone to changes of $l_0$. An example of this phenomenon can be seen in
Fig.\ \ref{fig:hysteresisplot_l0_comparison}, which is identical to
Fig.\ \ref{fig:hysteresisplot} but with additional data for $l_0 = 0.2$.
Lowering the elements' density
at the tip leads to slightly different values for
the critical Bond numbers and lowers the relative sizes
of the hysteresis loops,
especially for higher values of $Y_{2\text{D}}/\gamma$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1\textwidth,clip]{hysteresis_l0_comparison.pdf}
\caption{
Comparison of data from Fig. \ref{fig:hysteresisplot} for $l_0=0.1$ (blue)
with data for $l_0=0.2$ (red). There is an increasing deviation for
higher values of $Y_{2\text{D}}/\gamma$.
}
\label{fig:hysteresisplot_l0_comparison}
\end{center}
\end{figure}
\end{appendix}
|
{
"timestamp": "2018-04-16T02:07:03",
"yymm": "1803",
"arxiv_id": "1803.02607",
"language": "en",
"url": "https://arxiv.org/abs/1803.02607"
}
|
\section*{Introduction}
Nearly thirty years ago, a short period in the history of modern physics, quantum algorithms were presented to the scientific community as ingenious strategies to realize tasks thought to be hard with classical approaches. Significant advances in this sense sprouted within only a few years from different areas of research in cryptography \cite{Bennett84}, algorithms \cite{Shor94}, simulation \cite{Lloyd96} and communication \cite{Bennett93}, ever since placing a \textit{quantum-} before the name of each field. This shift of paradigm required to think out of the box and to start considering quantum mechanics not only as a stage to investigate but, as much interestingly, as a resource for practical purposes.
Photons naturally fit this quantum revolution \cite{Dowling03} as an effective system to process information: they propagate fast, do not interact with the environment and can be easily manipulated. Moreover, future advances will benefit from the technological infrastructures and know-how already developed in the classical context, thus further encouraging to proceed in this direction. While this aspect is particularly relevant for instance for quantum communication, where the availability of fiber networks can play a key role, milestone achievements have been reached also in the scope of information processing, raising a bridge between the linear-optical platform and information processing \cite{Knill01}.
Encouraged by these considerations, we will review the state of the art in photonic quantum information to provide the reader with a broad perspective in a unified framework. This review article is structured in three chapters, where we attempt to present the numerous works in a convenient classification, even though most of them easily overlap. Chapter 1 focuses on the single-photon encoding schemes and on the technological state of the art for experimental implementations, namely single-photon sources, integrated circuits and single-photon detectors. Chapter 2 delves into the field of quantum communication, describing various theoretical schemes developed to this purpose, overviewing the developments of quantum repeaters and distributed blind quantum computing protocols for photonic quantum networks. This chapter also presents recent significant achievements in long-distance communication and quantum key distribution. Finally, Chapter 3 presents the latest achievements in photonic quantum simulation, discussing the potentialities of single- and multi-photon quantum walks for quantum computing and simulation in the domain of quantum chemistry.
For the sake of clarity and completeness, we will also refer the interested reader to more specialized literature and review articles on each topic.
\section{Implementing quantum information\break with single photons}
Quantum information can be encoded in a variety of physical systems, ranging from photonic states, solid state devices, atomic or nuclear spin systems to electrons, Josephson junctions, superconducting devices (see Table \ref{table:qubits}). Photonic states present several advantages with respect to other platforms, due to the lack of interaction with the external environment that thus corresponds to long decoherence times. As we discuss in the next sections, this feature is particularly relevant in applications such as long-distance quantum communications. In this section we will review the encoding of quantum information in single photons with a focus on discrete-variable approach in the visible domain. For an overview of quantum information processing in the microwave regime, the interested reader can refer to Refs. \cite{Gu17, You11, Xiang13} for specialized, comprehensive reviews on the field, while for continuous-variable quantum information we refer the reader to Ref. \cite{Braunstein14}. Section \ref{sec:encoding} reviews the main degrees of freedom employed to encode qubits or qudits in this physical platform, while Section \ref{sec:phot.tech} summarizes the main recently-developed technological platforms to generate, manipulate and detect single-photon states.
\subsection{Single-photon encoding}
\label{sec:encoding}
Photon-based quantum information uses the degrees of freedom of light (see Fig. \ref{fig:FigureEncoding}), suitable quantities related to propagation directions (\textit{path} encoding), to momentum (\textit{polarization} encoding), to light spatial distribution (\textit{orbital angular momentum} encoding) and to time (\textit{time-bin} and \textit{time-frequency} encoding). All encoding strategies present their own advantages and weaknesses, and can be combined in a hybrid configuration. In this section we will review each of them focusing on discrete-variable systems, briefly recalling their operation with examples from the latest literature.
\begin{table*}[h!]
\renewcommand*{\arraystretch}{1.2}
\footnotesize
\centering
\caption{\label{table:qubits} Physical systems for encoding a qubit. Some degrees of freedom also allow for the implementation of qudits.}
The choice for the most suitable system for implementing a qubit depends on the task: quantum computation, communication or simulation. In this review we will focus on discrete-variable photonic implementations.
\begin{center}
\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}cc
\mr
System & Quantity & Encoding \\
\mr
\multirow{4}{5cm}{ \centering{Single photon}}
& Polarization & Horizontal / Vertical \\
& Orbital angular momentum & Left / Right \\
& Number & 0/1 photons \\
& Time & Early/Late \\ \hline
\centering{Continuous-variable fields} & Quadratures & Amplitude-/Phase- squeezing \\ \hline
\multirow{3}{5cm}{\centering{Josephson junction}}
& Charge & 0/1 Cooper pair \\
& Current & Clock-/Counter- clockwise \\
& Energy & Ground/Excited state \\ \hline
\centering{Quantum dot, Optical lattice, Nuclear spin} & Spin & $\uparrow/\downarrow$ \\ \hline
\multirow{2}{5cm}{\centering Electrons}
& Charge & 0/1 electrons \\
& Spin & $\uparrow/\downarrow$\\ \hline
\centering{Non-abelian anyons} & Topology & Braiding \\
\mr
\end{tabular*}
\end{center}
\end{table*}
\subsubsection{Encoding in angular momentum --}
The angular momentum of light concerns the rotation of the electromagnetic field vector, a dynamic quantity that influences light-matter interaction.
Two forms of angular momentum exist: spin angular momentum (SAM), associated to its circular polarization, and orbital angular momentum (OAM), associated to the spatial structure of the wavefront. While the two mechanisms cannot be separated in the general case of focused or divergent light beams, their nature becomes manifest for sufficiently collimated ones \cite{MartinezHerrero10}. In the latter case, the total angular momentum $\mathbf{J}$ takes the simpler form
\begin{equation}
\label{eq:TotalAngularMomentum}
\mathbf{J} \propto \int d^3\mathbf{r}\left[ \Big(|E_L|^2 - |E_R|^2 \Big) - \sum_{j:x,y,z} \left(\imath E_j^\ast \frac{\partial E_j}{\partial \phi} \right) \right]
\end{equation}
\noindent where the first term, the SAM contribution, corresponds to the familiar left (\textit{L}) and right (\textit{R}) circular polarizations while the second, associated to the OAM, describes wavefronts with the typical helical profiles and is thus related to light spatial distribution. The paraxial approximation is the natural framework for light manipulation and encoding \cite{Kok10}, thus in the following sections we will focus on this regime. For specialized reviews, we refer the reader to Refs. \cite{Bliokh15_1, Bliokh15_2}.
\paragraph{Polarization.} Polarization qubits represent a common and practical way to encode quantum information for most applications, thanks to the ease of generation (see Section \ref{sec:sources}) and to the availability of effective tools for manipulation. Indeed, in the limit of paraxial waves of Eq.\eref{eq:TotalAngularMomentum}, the SAM component interacts with anisotropic transparent systems such as birefringent crystals, which are consequently widely employed for polarization manipulation.
A qubit encoded in polarization is usually written as $\ket{\Psi} = \alpha \ket{H}+ \beta \ket{V}$, where \textit{H} and \textit{V} stand for horizontal and vertical polarization, respectively, and
\begin{equation}
\ket{p} = \int_{-\infty}^{\infty} d\mathbf{k} \ f(\mathbf{k})\ e^{-\imath w_k t} \ \hat{a}^\dagger (\mathbf{k},p) \ket{0}
\end{equation}
\noindent where $p=(H,V)$, $f(.)$ is a wave packet mode function and $\hat{a}^\dagger (\mathbf{k},p)$ is the creation operator for a photon with momentum $\mathbf{k}$ and polarization $p$ \cite{Kok10}. Polarization qubits can be as well expressed in the diagonal basis $\ket{\pm} = \frac{1}{\sqrt{2}}\left( \ket{H}\pm \ket{V} \right)$ or with left and right circular polarizations as $\ket{^L/_R} = \frac{1}{\sqrt{2}}\left( \ket{H} \pm \imath \ket{V} \right)$. Together, the three pairs of states form a set of mutually unbiased bases encoded in polarization \cite{Hou15}, at the core of several applications that we will overview in the following sections.
Polarization encoding has always played an important role in a significant number of investigations in quantum information, ranging from quantum simulation \cite{Sansoni12, Matthews13} to quantum computation \cite{Walther05, Prevedel07, Heilmann14, Barz14, Ciampini16} and communication \cite{Kim01, Ma16}. Its ubiquitous presence in quantum information processing has been further enhanced by the numerous advances in entanglement generation, manipulation and distribution \cite{Wang16, Valles13, Sansoni10, Olislager13, Muller14, Matsuda12, Kaiser14, Hamel14, Bhatti15, Barreiro10} and to the development of suitable theoretical and experimental frameworks for its manipulation in integrated devices \cite{Crespi12, RojasRojas14, Bonneau12}. Moreover, polarization qubits are increasingly coupled to other degrees of freedom of single photons, such as orbital angular momentum \cite{Bhatti15, Barreiro10, Fickler12, Nagali09prl}, path \cite{Walther05, Prevedel07, Ciampini16, Vallone09, Orieux15, Wang16}, time-energy \cite{Chen06, Steinlechner17} and all together \cite{Barreiro05} in so-called hyper-entangled states \cite{Kwiat97}, efficient resources for quantum computation, communication and work extraction protocols.
\begin{figure}[t!]
\centering
\includegraphics[trim={0 0 0 0},clip, width=\linewidth]{FigureEncoding.pdf}
\caption{\footnotesize Encoding quantum information in a single photon exploiting different degrees of freedom. Possible choices include polarization (only a qubit of information can be carried in this case), path, time-bin and orbital angular momentum (larger dimensionalities can be riched). Legend - QWP: quarter-wave plate, HWP: half-wave plate, PBS: polarizing beam splitter, BS: beam splitter, $\phi$: phase shift, SLM: spatial light modulator.}
\label{fig:FigureEncoding}
\end{figure}
\paragraph{Orbital angular momentum.}
The OAM component, related to the spatial distribution of the electromagnetic field, can be in turn divided in two terms: one \textit{internal}, origin-independent and associated to twisted wavefronts, and one \textit{external}, given by $\mathbf{L}_e=\mathbf{r} \times \mathbf{P}$ and thus origin-dependent. Quantum information processing based on OAM refers to the first component. The OAM carried by such \textit{optical vortices} is described by the phase term $e^{-\imath q \theta}$, where $\theta$ is the angular coordinate and $q$ is an unbounded integer. For eigenstates of the orbital angular momentum we have $q=Q=\frac{1}{2\pi} \oint d \xi$, where the integral is evaluated around the vortex singularity for a field with phase $\xi$, and $Q$ is the \textit{topological charge} that counts the number of helices in the phase profile. Correspondingly, single photons will carry quantized values of OAM given by $L=q \hbar$ \cite{Cardano12}.
Different techniques have been developed so far to produce and manipulate OAM states. An efficient tool is provided by spiral wave plates \cite{Schemmel14, Bierdz13}, whose thickness increases in a transparent spiral structure so that light experiences a phase gradient during the propagation. However, their cost, the selectivity in wavelength and the hardness to generate qudits still limit their applicability \cite{DAmbrosio16}. Further widely employed tools are metamaterials or cylindrical lens pairs \cite{Willner15}, capable to convert Hermite-Gauss to Laguerre-Gauss OAM modes, diffraction gratings in the form of pitch-fork holograms \cite{Cardano15, Gruneisen11} or spatial light modulators (SLM) \cite{Lavery12, Cardano15, Cardano16}, capable to modify the intensity and/or the phase profile of a beam point-by-point. The ease and versatility of SLMs can be exploited in many applications ranging from optical communication \cite{Krenn14, Bolduc13} to holography-based optical tweezers \cite{Padgett11, Dholakia11}.
Lastly, interesting possibilities are offered by the \textit{q-plate} (QP), a device built on liquid crystals, polymers or sub-wavelength gratings that allows to manipulate OAM depending on the input polarization state \cite{Karimi09, Cardano12, Cardano15, Cardano16, Zhang10, Giovannini11, Nape17, D'Errico17, D'Ambrosio12, D'Ambrosio13, Nagali09}. In the so-called \textit{tuning} condition, a $q$-plate implements the transformations $QP \ket{^L/_R}\ket{l} = \ket{^L/_R}\ket{l \pm 2q}$, i.e. it flips the qubit polarization in the circular basis and shifts the OAM of a quantity $\Delta l=\pm 2q$. Here, $q$ is related to the topological charge of the QP, which in turn depends on its internal pattern. Applications offered by the QP \cite{D'Ambrosio12, D'Ambrosio13, D'Ambrosio12alignment, DAmbrosio16, Nagali09prl} include the generation of intra-photon entanglement between polarization and OAM degrees of freedom by producing states of the form
\begin{equation}
QP \ket{_{H}^{V}}\ket{l} = \ket{_{L}^{R}}\ket{l \pm 2q} \pm \ket{_{R}^{L}}\ket{l \mp 2q}.
\end{equation}
An important aspect for OAM encoding is to develop practical and reliable techniques to analyze it. While, in fact, polarization encoding requires to resolve only two components, which can be done easily by means of waveplates and polarizing beam splitters, the number of OAM modes is potentially unbounded and as such challenging to characterize \cite{Giovannini13}. Several methods exist so far (see Ref. \cite{D'Errico17} and references therein): the above-mentioned spiral phase plates \cite{Schemmel14, Bierdz13}, holograms and spatial light modulators \cite{Lavery12, Cardano15, Cardano16, Gruneisen11} and q-plates \cite{Karimi09, Cardano12, Cardano15, Cardano16, Zhang10, Giovannini11, Nape17, D'Errico17, D'Ambrosio12, Nagali09}, as well as diffractions through apertures, interference with uniform plane waves, Dove prism interferometers, rotational Doppler frequency shifts and spatial sorting of helical modes \cite{Bhatti15, Karimi09}. In this direction, a spectrum analyzer is a device capable to measure the instantaneous power and phase distributions of OAM components \cite{Zhou17oam, Gruneisen11, Bierdz13, Mirhosseini13, Malik14, Forbes16, Zhao17oam, Piccirillo15}. Once generated and analyzed, OAM states can be manipulated with high degree of control to encode quantum information in the infinite OAM Hilbert space spanned by $l \in Z$, i.e. in the qudits $\psi = \sum_l \alpha_l \ket{l}$ \cite{Barreiro10, Pan16oam, Zhou16prl, Beltran17, Nape17, Babazadeh17, D'Ambrosio12, Nagali09, Zhang14oam, Wang17sagnac}.
OAM states represent a fundamental resource for several applications in quantum information. At a fundamental level, OAM has enabled researches on optimal quantum cloning of OAM-encoded qubits \cite{Nagali09, Bouchard17} and photonic quantum walks in the orbital angular momentum states, with theoretical analyses \cite{Hamilton11, Innocenti17} and experimental demonstrations \cite{Cardano15, Cardano16, Zhang10}. Additional features include the capability of performing a coined quantum walk in the OAM space, providing a viable alternative to other bulk or integrated implementations (see Section \ref{sec:QW}).
Long-distance quantum communication (see Section \ref{sec:LDQC}) benefits from the availability of photonic information carriers able to distribute multiple superposition states in OAM entangled \cite{Ibrahim13, Goyal14teleport, Goyal14qkd, Bhatti15, Krenn15, Hiesmayr16, Malik16, Erhard17, Fickler16, Babazadeh17, Erhard17ghz, Leonhard17, D'Ambrosio12alignment} or hyperentangled \cite{Barreiro10, Giovannini11, Jabir17qkd, D'Ambrosio12, D'Ambrosio12alignment, Farias15, DAmbrosio16, Nagali09prl} states. Moreover, photonic 'flying' qubits encoded in OAM can be prepared in alignment-free states \cite{D'Ambrosio12alignment, Farias15}, being insensitive to rotation of the reference frame. These features have unlocked a number of advancements in free-space quantum key distribution and quantum communication \cite{Vallone14, Mirhosseini15, Bouchard17, Lei15, Wei17, Wang17oam, Pan17fibonacci, Sit17, Goyal14qkd, Mafu13, D'Ambrosio12alignment} with high bit exchange rates \cite{Bozinovic13, Wang12oam}. Related to the capability of producing alignment-free states, hybrid encoding between polarization and OAM can be exploited to enhance the sensitivity to angular rotations \cite{D'Ambrosio13, Jha11}. The confidence in its potential has led to several investigations to support OAM-based photonic quantum networks for delivering information, from sorting \cite{Wei17, Zhang14oam, Wang17oam, Karimi09, Fickler14} and routing OAM states \cite{Garcia-Escartin12, Erhard17, Lavery11} to implementing quantum repeaters with teleportation \cite{Wang15teleport, Goyal13, Goyal14teleport} and quantum memories \cite{Ding13oam, Bussieres14}. First results towards integration of OAM devices have also been reported \cite{Cai12}.
The feasibility of this approach for free-space communication has been further supported by a number of theoretical \cite{Roux11, GonzalezAlonso13, GonzalezAlonso16, Leonhard17, Padgett15} and experimental \cite{Ibrahim13, Farias15, Krenn15} investigations in non-optimal conditions, addressing the issue of beam propagation in a turbulent atmosphere.
\subsubsection{Encoding in propagation direction --}
Path encoding, or encoding in the optical modes in the case of single photons, is the representation of qubits in terms of occupied spatial modes. Path-encoded qubits, as well as qudits \cite{Rossi09}, are perfectly fit for photonic integrated circuits, since waveguide arrays inherently implement a spatial separation and coupling between modes is easily accomplished with directional couplers. Moreover, the high stability and interferometric complexity offered by integrated circuits are useful features for most applications in quantum technologies \cite{Matthews09, Shadbolt12, Ciampini16}. Indeed, accurate control over these qubits for universal manipulation is possible with relatively low technological requirements when compared to other encoding schemes \cite{Bonneau12, Carolan15}. Conversely, an integrated manipulation of polarization qubits requires ad-hoc designs that represent additional challenges, as demonstrated for instance by the fabrication of rotated waveplates in integrated femtosecond laser-written circuits \cite{Corrielli14, Heilmann14}. To this aim, several experimental demonstrations have been shown for tunable all-optical path-entanglement generation via parametric down-conversion in integrated non-linear waveguides \cite{Silverstone14, Jin14, Harris14, Titchener15}, mostly on SoI and $\textup{LiNbO}_3$ \cite{Solntsev14}. Fiber integrated sources have also been reported for the generation of high-dimensional path-encoded qubits, as interface or preliminary step towards the integration in a circuit \cite{Schaeff12}. Finally, control over non-ideal implementation has been strengthened with investigations on the effect of losses over path-entanglement \cite{Antonosyan14} and on state tomographies on tunable integrated circuits \cite{Shadbolt12}.
\subsubsection{Encoding in time --}
Using time as degree of freedom offers various advantages over other encoding schemes. We present an overview of the approach and of its latest achievements, with a distinction between time-bin and time-energy encodings.
\paragraph{Time-bin encoding.} Time is a natural and effective resource to write information on single-photon quantum states. The mechanism to encode information involves a Mach-Zehnder interferometer with one arm longer than the other. The amplitude associated to an incoming photon is split at the first beam splitter of the Mach-Zehnder and passes through the unbalanced arms: we denote with $\ket{l}$ the state of a photon that has taken the long path, while $\ket{s}$ represents a photon that has taken the short one. This path difference must be stable, i.e. any fluctuation in the temporal delay must be smaller than one wavelength, and longer than the coherence length of each photon to allow a reliable discrimination of the arrival times. A proper dynamic control of the temporal delay is thus desirable to compensate for mechanic and thermal instabilities. A qubit encoded in the photon arrival time can then be written in the superposition $\ket{\Psi} = \frac{1}{\sqrt{2}} \left( \ket{l}+ \ket{s} \right)$, where the states $\ket{p}$ are given by
\begin{equation}
\ket{p} = \int_{-\infty}^{\infty} dz\ f\left(\frac{t -\frac{z}{c} + p \tau}{\delta t}\right) e^{-\imath \omega (t-\frac{L}{c} + p \tau)} \ \hat{a}^\dagger \ket{0}
\end{equation}
\noindent being $p=(l,s)$, $f(.)$ a wave packet mode function, $\omega$ a fixed angular frequency and $\tau$ the time delay experienced between the two arms of length $L$.
Time-bin encoding presents some advantages with respect to the previous schemes. First, it is suitable for integrated photonic devices, where photons can be generated, manipulated and measured without the need for external encoding devices. Moreover, its resilience to noise acting on polarization, such as depolarizing media or decoherence and mode dispersion, makes time-bin a good candidate for applications in state teleportation \cite{Marcikic03, deRiedmatten04, Landry07}, quantum communication and quantum key distribution, both free-space and in-fiber \cite{Yu15, Marcikic04, Tang16qkd, Gundogan15}. In this direction, several experiments have been carried out for tests of non-locality \cite{Marcikic04, Donohue13, Marcikic02, Guo17timebin}. Entangled photon pairs have been reported with femtosecond pulses \cite{Marcikic02}, including deterministic and narrowband atom-cavity sources \cite{NisbetJones13}, sources integrated on type-II periodically poled lithium niobate \cite{Martin13} and silicon wire \cite{Harada08} waveguides or micro-ring resonators \cite{Wakabayashi15}. Time-bin manipulation on chip \cite{Xiong15}, storage \cite{Gundogan15} and measurement \cite{Donohue13} have been reported to provide the necessary components for linear-optical quantum networks. Besides quantum communication, time-bin encoding has been proposed as a suitable scheme also for quantum walks \cite{Schreiber10, Schreiber12, Boutari16} and for photonic \textsc{BosonSampling} \cite{He17, Motes14timebin} (see Section \ref{sec:BosonSampling}).
\paragraph{Time-frequency/energy encoding.} Starting from time-bin encoding, where the allowed states are 'early' and 'late', an extension of this scheme to multiple states consists in correlating the arrival time with single-photons' energy \cite{Zhong15, Nunn13, Steinlechner17}. Relevant applications can be found in quantum communication and quantum key distribution to take advantage of the low decoherence with qubits delivery \cite{Hayat12, Roslund14, Brendel99, Zhong15, Kaiser16, Nunn13, Steinlechner17}. Notwithstanding, frequency-encoded experimental demonstrations have been reported also in the context of quantum computation \cite{Humphreys13, Campbell14, Menicucci10, Menicucci11, Yokoyama13, Chen14, Soudagar07} often operating with cluster states. Extensions to more than two photons have also been reported \cite{Shalm13}, representing a proof-of-principle investigation for multi-photon Franson interferometry and for engineering discrete- and continuous-variable hyperentangled states.
We conclude this paragraph mentioning theoretical and experimental investigations to manipulate photonic qubits encoded in time \cite{Hosseini09, Soudagar07, Donohue13} and time-frequency \cite{Autebert16, Reddy14, Brecht14, Saglamyurek14, Hayat12, Huntington04, Olislager10}.
\begin{table}[ht!]
\renewcommand*{\arraystretch}{1.1}
\centering
\caption{\label{SourcesTable} Platforms reported in the last three years for single-photon sources in the range 750-1550 nm.}
\footnotesize
\begin{tabular*}{\linewidth}{l@{\extracolsep{\fill}}clc
\br
Process & Ref. & Platform & $\lambda (nm)$ \\
\mr
PDC & \cite{Tian16} & PPKTP & 795+795 \\
& \cite{Jabir17source} & PPKTP & 810+810 \\
& \cite{Weston16} & PPKTP & 1570+1570 \\
& \cite{Kaneda16} & PPKTP & \;\,800+1590 \\
& \cite{Slussarenko17} & PPKTP & 1550+1550 \\
& \cite{Vergyris16} & PPLN & 1310+1560 \\
& \cite{Krapick16} & PPLN & 1551+1611+1625 \\
& \cite{Montaut17} & PPLN & 1560+1560 \\
& \cite{Vergyris17} & PPLN & 1560+1560 \\
& \cite{Ding15} & PPLN & 1551+1571 \\
& \cite{Autebert16} & AlGaAs & 1566+1566 \\
& \cite{Setzpfandt15} & $\textup{LiNbO}_3$ & 1342+1342 \\
& \cite{Guo17} & AlN $\mu R$ & 1550+1550 \\ \hline FWM & \cite{Kultavewuti17} & AlGaAs & 1533+1577 \\
& \cite{Ramelow15} & $\textup{Si}_3\textup{N}_4$ $\mu R$ & 1550+1550 \\
& \cite{Cruz-Delgado16} & Few-mode fiber & 620+777 \\
& \cite{Rogers16} & Silicon microdisk & 1497+1534 \\
& \cite{Cordier17} & LF-HC-PCF & 1552+1552 \\
& \cite{Yana15} & Silica FLWw & \;\,830+1130 \\ \hline
QD exciton & \cite{Olbrich17} & InGaAs & 1550 \\
& \cite{Portalupi15} & InGaAs & 945 \\
& \cite{Somaschi16} & InGaAs & 890 \\
& \cite{Loredo16} & InGaAs & 932 \\
& \cite{Kirsanske17} & InGaAs & 907 \\
& \cite{Schlehahn18} & InAs/GaAs & 925 \\
& \cite{Snijders17} & InAs/GaAs & 933 \\
& \cite{Ding16QD} & InAs/GaAs & 897 \\
& \cite{Davanco17} & InAs/GaAs & 1130 \\
QD biexciton \; & \cite{Heindel17} & InGaAs & 905+905 \\
& \cite{Huber17} & GaAs & 786+786 \\
& \cite{Jons17} & InAsP & 930+930 \\
QD triexciton & \cite{Khoshnegar17} & InAsP & 894+940 \\ \hline
Fluorescence & \cite{Benedikter17} & Si-V center & $\sim$750 \\
\br
\end{tabular*}\\
PDC: spontaneous parametric down-conversion; FWM: four-wave mixing; FLWw: femtosecond laser written waveguide; QD: quantum dot. PPLN: periodically poled lithium niobate. PPKTP: periodically poled potassium titanyl phosphate.\\ InAs: indium arsenide; GaAs: gallium arsenide; InGaAs: indium gallium arsenide; InAsP: indium arsenic phosphide; AlGaAs: aluminium gallium arsenide. AlN: aluminum nitride; Si-V: silicon-vacancy in diamond; LF-HC-PCF: liquid-filled hollow-core photonic crystal-fiber. $\mu R$: microresonator.
\end{table}
\normalsize
\subsection{Photonic technologies}
\label{sec:phot.tech}
In this section we will overview the main technological components for photonic quantum information processing. Three main stages can be identified, as shown schematically in Fig. \ref{fig:FigureTechnology}. First, a crucial requirement is the capability of efficiently generating single-photon states (Sec.\ref{sec:sources}), requiring indistinguishability of correlated states and good control over the degrees of freedom. Then, suitable platforms should be capable of manipulating single- or multi-photon states to perform unitary transformations (Sec.\ref{sec:circuits}). Finally, photons should be efficiently measured with appropriate detection systems (Sec.\ref{sec:detectors}).
\subsubsection{Single-photon sources --}
\label{sec:sources}
Ideally, a good single-photon source should emit only one photon at a time, on demand, at high generation rates and in well-defined states in spatial, temporal and spectral modes. Moreover, different sources should be capable to generate identical photons and their implementation should allow for integration in miniaturized platforms. In parallel, it is crucial to generate correlated states of more than one photon, being entanglement a key resource in several quantum information protocols (see Section \ref{sec:entanglement}). Current sources can fullfil only a limited number of the above requirements, while keeping very high performances for specific applications.
Several approaches for single-photon sources have been developed in the last decades (see Table \ref{SourcesTable}) \cite{Wang16ten, Ramelow15,Guo17,Setzpfandt15,Weston16,Kaneda16,Vergyris16,Tian16,Montaut17,Jabir17source,Vergyris17,Krapick16,Yana15, Kultavewuti17, Cruz-Delgado16,Rogers16,Higginbottom16,Benedikter17,Peng16,Somaschi16,Loredo16,Heindel17,Olbrich17, Jons17,Khoshnegar17,Ding15,Ding16QD,Davanco17,Schlehahn18,Snijders17,Huber17,Kirsanske17,Portalupi15,Geng16,Li17,Cordier17, Orieux13, Horn13, Spring17, Kruse15, Sansoni17, Atzeni18}. Among probabilistic sources we mention (a) parametric down-conversion (PDC) in bulk crystals \cite{Wang16ten}, semiconductors \cite{Boitier14, Autebert16}, microresonators \cite{Ramelow15, Guo17} and optical waveguides \cite{Setzpfandt15, Weston16,Kaneda16, Vergyris16, Tian16, Montaut17, Jabir17source, Vergyris17, Krapick16, Hamel14, Ding15, Orieux13, Horn13, Kruse15, Sansoni17, Atzeni18} and (b) four-wave mixing (FWM) in optical waveguides \cite{Yana15, Kultavewuti17, Spring17}, few-mode fibers \cite{Cruz-Delgado16} and microdisks \cite{Rogers16}. Deterministic sources include trapped ions \cite{Higginbottom16}, colour centers \cite{Benedikter17} and quantum dots (QD) \cite{Lodahl17} with GaAs \cite{Huber18}, InGaAs \cite{Muller14,Portalupi15, Somaschi16, Loredo16, Heindel17}, InAsP NW \cite{Jons17, Khoshnegar17} or InAs/GaAs \cite{Ding16QD, Davanco17, Schlehahn18, Snijders17}.
\begin{table*}[h]
\renewcommand*{\arraystretch}{1.1}
\centering
\caption{\label{CircuitsTable} Integrated circuits for quantum information processing.}
\footnotesize
\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}ccccccc
\br
Year & Ref. & Technology & Photons & Modes & $\lambda (nm)$ & Tunable $\phi$ & Application \\
\mr
2008 & \cite{Politi08} & SoS & 2 & 6 & 804 & n.r. & CNOT \\
2009 & \cite{Politi09} & SoS & 4 & 12 & 790 & n.r. & Shor's factoring algorithm \\
& \cite{Smith09} & UVW, SoS & 2 & 2 & 830 & 1 & First reconfigurable UVW circuit \\
& \cite{Matthews09} & SoS & 4 & 2 & 780 & 1 & First reconfigurable SoS circuit \\
2010 & \cite{Laing10} & SoS & 2 & 6 & 804 & n.r. & CNOT \\
& \cite{Sansoni10} & FLW & 2 & 2 & 806 & n.r. & Polarization insensitive BS \\
2011 & \cite{Crespi11} & FLW & 2 & 4 & 806 & n.r. & CNOT with partially polarizing BS \\
2012 & \cite{Shadbolt12} & SoS & 2 & 6 & 808 & 8 & State generation/detection, Bell test \\
& \cite{Bonneau12} & Ti:LN & 2 & 2 & 1550 & 1 $^{(a)}$ & First reconfigurable Ti:LN circuit \\
& \cite{Bonneau12silicon} & SoI & 2 & 2 & 1550 & 1 & First reconfigurable SoI circuit \\
2014 & \cite{Heilmann14} & FLW & 1 & 2 & 815 & n.r. & Integrated waveplates \\
& \cite{Corrielli14} & FLW & 2 & 6 & 800 & n.r. & Integrated waveplates \\
& \cite{Silverstone14} & SoI & 2 & 2 & 1550 & 1 & On-chip interference of two sources \\
& \cite{Jin14} & LN & 2 & 4 & 1560 & 1 & On-chip interference of two sources \\
& \cite{Humphreys14} & FLW & 2 & 2 & 830 & 1 & Strain-optic active control \\
& \cite{Metcalf14} & UVW, SoS & 3 & 6 & 830 & 1 & Teleportation \\
& \cite{Peruzzo14} & SoS & 2 & 6 & n.a. & 8 & Variational eigenvalue solver \\
2015 & \cite{Flamini15} & FLW & 2 & 2 & 1550 & 1 & First reconfigurable FLW circuit \\
& \cite{Carolan15} & SoS & 3 & 6 & 808 & 30 & H, CNOT, $\sigma, R(\alpha)$\\
& \cite{Xiong15} & $\textup{Si}_3\textup{N}_4$ & 2 & 5 & 1550 & 15 & Time-bin entanglement\\
2016 & \cite{Ma16} & Si & Attenuated laser & 3 & 1550 & 4 $^{(a,b)}$ & BB84 protocol \\
& \cite{Wang16} & SoI & 2 & 4+2 $^{(d)}$ & 1550 & 5 & Entanglement distribution \\
& \cite{Poot16} & SiN & 2 & 6 & 1550 & n.r. & CNOT \\
& \cite{Sibson17} & SoI & Attenuated laser & 3 & 1550 & 5 $^{(c)}$ & QKD protocol \\
2017 & \cite{Ding17} & SoI & Attenuated laser & 4+8 $^{(d)}$ & 1550 & 8+10 $^{(d)}$ & QKD protocol \\
& \cite{Harris17} & SoI & Attenuated laser & 26 & 1570 & 176 & Quantum transport \\
\br
\end{tabular*}\\
SoS: silica on silicon; SoI: silica on insulator; FLW: femtosecond laser written; UVW: UV written;\\BS: beam splitter; Ti:LN: titanium indiffusion lithium niobate; SiN: Silicon nitride.\\ Modulation in (a) polarization, (b) intensity, (c) pulse. (d): two integrated circuits. n.r.: non-reconfigurable; n.a.: not available.
\end{table*}
\normalsize
\subsubsection{Integrated quantum circuits --}
\label{sec:circuits}
Traditional optical instruments consisted of bulk optical components, unavoidably large and unpractical, more susceptible to problems of stability, scalability and adaptability to different applications. Just like the evolution undergone by the electronic components, the last decade has seen a strong effort for an optical miniaturization in dielectric materials \cite{Tanzilli12,Orieux16}.
Integrated optical circuits are built upon architectures of directional couplers, with additional geometries to account for deformation-induced phase shifts. Furthermore, recent developments have shown the capability to introduce active reconfigurable elements, thus allowing the fabrication of multi-purpose devices \cite{Carolan15, Flamini15, Harris17}. One widely adopted material for the integration remains fused silica thanks to its numerous benefits: low propagation losses, low birefringence, operation from visible to infrared, good coupling efficiency with single mode fibers and low temperature dependence. Notwithstanding, and differently from miniaturized electronic circuits, linear optical circuits do not have unique platform and manifacturing technique (see Table \ref{CircuitsTable}) \cite{Bogdanov17}:
\paragraph{Silicon-on-Insulator (SoI).} Si-based platforms, including silicon (Si), silicon nitride (SiN) and silicon carbide (SiC), are advanced platforms whose development benefits from the know-how given by electronics technologies \cite{Vivien13}. Si-based devices present a very high refractive index that allows for reduced-size circuits and that is suitable for nonlinear processes. Limitations are the low mode-matching with optical fibers and the relatively high propagation losses. Slightly more favorable conditions can be met with devices based on III-V compound semiconductors, which include indium phosphide \cite{Abellan16}, gallium arsenide \cite{Wang14} and gallium nitride \cite{Xiong11}.
\paragraph{Silica-on-Silicon (SoS).} A crystal silicon substrate is covered by a layer of silica ($\textup{SiO}_2$), wherein waveguides are etched with rectangular cross sections. A second layer of undoped silica is then laid on top of the structure to enclose the doped silica core and protect the waveguides \cite{Politi08}. Limitations of the SoS technique are the need for a mask and the restriction to one polarization, due to the birefringence induced by the rectangular cross-sections.
\paragraph{UV writing.} Waveguides are inscribed by focusing a strong laser pulse in a photosensitive B- and Ge-doped silica layer, placed within two layers of undoped silica and on top of a third translating silicon layer \cite{Svalgaard94, Spring13, Metcalf14}. This technique does not require the adoption of masks, thus reducing the complexity of fabrication, and allows for arbitrary 3D geometries.
\paragraph{Femtosecond laser writing (FLW).} The mechanism underlying the process is the non-linear absorption of strong pulses tightly focused in a glass substrate, which causes a permanent and localized modification in the refractive index. Waveguides are drawn by translating the sample at constant speed in 3D geometries \cite{Crespi16, Chaboyer15}. The possibility of writing circular cross-sections and the low birefringence of silica allow for waveguides with low dependance on polarization \cite{RojasRojas14,Sansoni12, Corrielli14}.
\paragraph \noindent On the above platforms, photonic circuits can be implemented \cite{Burgwal17, Flamini17} according to linear-optical interferometric schemes capable to perform arbitrary unitary evolutions \cite{Reck, Clements16}, or designs optimized for Fourier and Hadamard transformations \cite{Crespi16, Flamini17}.
\subsubsection{Single-photon detectors --}
\label{sec:detectors}
Photodetectors are devices that trigger a macroscopic electric signal when stimulated by \textit{one and only one} incoming photon (photon number resolving detectors, PNR) or by \textit{at least one} photon. Detecting photons with high probability and reliability is a key requirement for most tasks, often representing a bottleneck for the overall efficiency of an apparatus. Due to the very low energy of a single photon ($\sim10^{-19} \,\textup J$), a PNR detector requires high gain and low noise to be able to discriminate the correct number.
Non-PNR detectors include single-photon avalanche photodiodes (SPAD) on InGaAs \cite{Zhang15,Comandar15} or Ge-on-Si \cite{Martinez17,Warburton13}, quantum dots \cite{Weng15}, negative feedback avalanche diodes \cite{Yan12, Korzh14, Covi15}, superconducting nanowires \cite{Li15, Zhang15nw, Yamashita16, Atikian14, Tyler16, Arpaia15, Takesue15,LeJeannic16,Zhang17nbn,Zadeh16,Wang17nbn,Vorobyov17,Miki17,Krapick17}, artificial $\Lambda$-type three-level systems \cite{Inomata16} and up-conversion detectors \cite{Ma17, Pelc11, Hu12, Pelc12,Pomarico10,Zheng16}. Si-based SPADs exhibit good performances with visible light but still suffer in the infrared window, due to the incompatibility between good IR absorption and low-noise. PNR detectors include instead transition-edge sensors \cite{Miller11, Calkins13, Gerrits11, Hopker17, Lamas-Linares13}, parallel superconducting nanowire single-photon detectors \cite{Najafi12, Heath14}, quantum dot coupled resonant tunneling diodes \cite{Weng15}, organic field-effect transistors \cite{Yuan13} and multiplexed SPADs \cite{Avenhaus10, Thomas12}. Recently, large effort has been devoted to the optical integration of superconducting detectors on waveguide structures, such as on LiNb$\textup{O}_3$ \cite{Tanner12,Hopker17}, GaAs \cite{Sprengers11, Jahanmirinejad12, Reithmaier13, Sahin15, Zhou14, Kaniber16, Najafi15, Mattioli16, Li16nw}, Si \cite{Pernice12, Akhlaghi15}, S$\textup{i}_3$N$\textup{i}_4$ \cite{Cavalier11, Ferrari15, Kahl15, Schuck13, Schuck16, Beyer15, Shainline17} and diamond \cite{Rath15, Atikian14}.
For a more detailed discussion on the state of the art, we refer the interested reader to specialized reviews on the topic \cite{Eisaman11, Natarajan12}.
\begin{figure*}[t!h]
\centering
\includegraphics[trim={0 0 0 0},clip, width=\textwidth]{FigureTechnology.pdf}
\caption{\footnotesize Technologies for photonic quantum information processing. Three main stages can be identified. (i) Generation of photonic states, either indistinguishable single photons or entangled states. (ii) Manipulation, where integrated platforms enable apparatuses of increasing complexity. (iii) Measurement of photonic states, where detectors either with photon number resolution or without are currently under development. Legend - PDC: parametric down-conversion, FWM: four-wave mixing, FLW: femtosecond laser writing, SPAD: single-photon avalanche photodiode, TES: transition edge sensor.}
\label{fig:FigureTechnology}
\end{figure*}
\section{Quantum communication}
Quantum communication aims to connect distant quantum processors with an increased level of security. In recent years, advances in all these areas have led to the implementation of first quantum networks in different locations (see Table \ref{table:qNetworks}). In this section we will briefly overview the state of the art in this research.
\begin{table*}[h!]
\renewcommand*{\arraystretch}{1.1}
\centering
\caption{\label{table:qNetworks} Worldwide in-fiber photonic quantum networks (not exhaustive list).}
\footnotesize
\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}ccccc}
\br
Network & Ref. & Launch & Location & Nodes & Distance (km) \\
\mr
DARPA &\cite{Elliott05} & 2003 & USA & 10 & 29 \\
SECOQC &\cite{Peev09} & 2003 & Austria & 6 & 200 \\
SwissQuantum &\cite{Stucki11} & 2009 & Switzerland & 3 & 35 \\
Hierarchical Quantum Network &\cite{Xu09, chineseNetwork16} & 2009 & China & 32 & 2000 \\
Tokyo QKD Network &\cite{Fujiwara16, Sasaki11} & 2010 & Japan & 5 & 45 \\
Los Alamos National Lab &\cite{Hughes13} & 2011 & USA & n.a. & 50 \\
\br
\end{tabular*}\\
DARPA: Defense Advanced Research Projects Agency; SECOQC: SEcure COmmunication based on Quantum Cryptography.
\end{table*}
\normalsize
\subsection{Protocols for quantum communication}
Ever since the intuition of Bennett and Brassard with the famous BB84 protocol \cite{Bennett84}, it is known that quantum information allows one to devise algorithms to achieve classically unparalleled results in information transfer. Quantum infrastructures for managing information and delivering entanglement are now believed to join current technologies in a number of relevant tasks. Photons are by far the most suitable physical system to implement flying qubits to deliver information: they experience negligible decoherence through free space or optical fibers, they allow a high spatial control, and the technology for linear-optical devices to manipulate qubits is accessible and at an advanced stage \cite{Nielsen_Chuang, Saleh07}.
\subsubsection{Photons as quantum information carriers --}
\label{sec:entanglement}
Quantum communication protocols require two or more channels to exchange information between the parties, at least one classical and one quantum. Proper measurement are carried out on quantum systems shared between the parties, while classical messages can be exchanged to guide or complete the transfer. Entanglement is an essential physical resource to this aim \cite{Horodecki09} and, as such, it is necessary to protect it during the transfer. In the last decades a large number of theoretical investigations was reported to rigorously define and develop the field, which is still growing and reaching now a mature age \cite{Yuan10, Krenn16}. The theoretical framework, however multifaceted and oriented to various improvements, ultimately relies on few fundamental ingredients (see Fig. \ref{fig:FigureQuantumCommunication}) introduced a couple of decades ago, which have now been shown in first demonstrations \cite{Diamanti16, Krenn16}.
\paragraph{Dense coding.}
Quantum mechanics allows to increase the capacity of a quantum communication channel by transferring two bits of classical information with only one qubit\cite{Bennett92dense}. Suppose Alice and Bob share an entangled photon pair, each keeping one photon. Alice can encode two bits on her photon by choosing and applying on her photon one between four unitary transformations: the identity operation \(\mathds{1}\) (00), a bit flip $\sigma_x$ (01), a phase flip $\sigma_z$ (10) or both a bit flip and a phase flip $\sigma_y$ (11). Alice sends her particle to Bob, who measures both particles in the Bell basis. The outcome of the Bell measurement (BM) will then correspond to the two bits of information sent by Alice.
One challenge for implementing dense conding is represented by the BM stage, since achieving a never-failing BM is impossible using only linear-optical elements such as beam splitters, phase shifters, detectors and ancillary particles \cite{Lutkenhaus99}. A solution to this issue consists in exploiting hyperentangled photons \cite{Barreiro05}, leading to complete BM demonstrations in orbital angular momentum/polarization \cite{Barreiro08}, momentum/polarization \cite{Barbieri07} and time/polarization \cite{Schuck06, Williams17}. The latter schemes have the additional advantage of being feasible to prepare and of enabling an efficient transmission through optical fibers. So far, the maximum channel capacity ($1.665 \pm 0.018$) has been reported in Ref.\cite{Williams17}, beating the limit of $\log_2 3 \sim 1.585$ bits obtainable by means of only linear optics and entanglement, due to the impossibility of perfectly discriminating all four Bell states in this case.
\paragraph{Entanglement purification.}
One requirement is the capability of delivering entanglement without having it spoilt between distant nodes of a network, as it happens with a probability increasing exponentially in the length of a noisy transmission. Entanglement purification allows to circumvent this issue by extracting high-quality entangled pairs from a given ensemble using only local transformations and classical communication \cite{Bennett96}. More specifically \cite{Yuan10}, suppose Alice and Bob share two copies of the mixed state
\begin{equation}
\rho_{A,B} = \mathcal{F} \ket{\phi^\dagger}_{A,B}\bra{\phi^\dagger}+(1-\mathcal{F}) \ket{\psi^\dagger}_{A,B}\bra{\psi^\dagger},
\end{equation}
\noindent with $\ket{\psi^\dagger} \propto \ket{0,1}+ \ket{1,0} $ and $\ket{\phi^\dagger} \propto \ket{0,0}+ \ket{1,1} $, and they want to purify it so as to increase the probability of having a maximally-entangled state $\ket{\phi^\dagger}$. The original approach was based on CNOT operations \cite{Bennett96}; few years later a simpler solution was introduced \cite{Pan01} and verified experimentally \cite{Pan03} based on only polarizing beam splitters. While the probability of success is $50\%$ lower than the one with CNOT gates, the quality of these optical elements allow for a purification with much higher precision and control. Via a proper combination of projective measurements in polarization, it is possible to increase the amplitude associated to the maximally-entangled pair $\ket{\phi^\dagger}$, which can be used to quantify the fidelity of the transmission, as $\mathcal{F} \rightarrow \mathcal{F}^2/\left(\mathcal{F}^2 + (1-\mathcal{F})^2\right)$ when $\mathcal{F}>\frac{1}{2}$.\\ Entanglement purification is a key ingredient also in quantum computation, such as for quantum error-correction protocols, in quantum cryptography and quantum teleportation. For a more complete overview of the state of the art we refer to Refs. \cite{Yuan10,Sheng15,Zhang17,Simon17,Kalb17}.
\paragraph{Quantum state teleportation.}
Ever since its introduction \cite{Bennett93}, quantum state teleportation (QST) has been one of the most notable examples of quantum communication. In terms of classical communication, QST enables the transfer of one qubit by sending two bits of classical information, i.e. a reversed dense coding. The mechanism works as follows: suppose Alice and Bob share an entangled photon pair $\ket{\psi^-}_{A,B} = \frac{1}{\sqrt{2}} \left( \ket{0,1}_{A,B} - \ket{1,0}_{A,B} \right)$ and Alice wants to teleport to Bob the -unknown- qubit $\ket{\psi}_{T} = \frac{1}{\sqrt{2}} \left( \alpha \ket{0}_{T} + \beta \ket{1}_{T} \right)$. The complete system of three photons has the form
\begin{eqnarray}
\ket{\psi}_{T} \otimes \ket{\psi^-}_{A,B} = &-& \frac{1}{2} \ket{\psi^-}_{T,A} \left( \alpha \ket{0}_{B} + \beta \ket{1}_{B} \right) \label{qT:a} \\
&-& \frac{1}{2} \ket{\psi^+}_{T,A} \left( \alpha \ket{0}_{B} - \beta \ket{1}_{B} \right) \label{qT:b} \\
&+& \frac{1}{2} \ket{\phi^-}_{T,A} \left( \alpha \ket{1}_{B} + \beta \ket{0}_{B} \right) \label{qT:c} \\
&+& \frac{1}{2} \ket{\phi^+}_{T,A} \left( \alpha \ket{1}_{B} - \beta \ket{0}_{B} \right) \label{qT:d}.
\end{eqnarray}
\noindent Alice can then perform a joint Bell measurement on her two photons $A$ (entangled with Bob's) and $T$, associate two bits to the Bell state found out of the four and send them to Bob via a classical channel. At this point, Bob can simply perform one of four transformations (identity operation for \eref{qT:a}, phase flip $\sigma_z$ for \eref{qT:b}, bit flip $\sigma_x$ for \eref{qT:c} or both bit flip and phase flip $\sigma_y$ for \eref{qT:d}) on his entangled photon $B$ to shape it as the one -unknown- possessed by Alice.
The original work by Bennett \textit{et al.} \cite{Bennett93} has rapidly triggered a large number of investigations \cite{Nielsen_Chuang} for a broad range of applications \cite{Pirandola15}. Among all, teleportation schemes were shown to enable new approaches for universal quantum computation \cite{Gottesman99, Ishizaka08}, in particular as one-way quantum computers \cite{Raussendorf01}.
From the experimental perspective, numerous achievements have been reported on photonic platforms proving the feasibility of the scheme already with state-of-the-art technology. After the first demonstrations in 1997-1998 \cite{Bouwmeester97, Boschi98}, one further proof appeared with the unconditional teleportation of optical coherent states with squeezed-state entanglement \cite{Furusawa98}. Later on, in 2001 Kim \textit{et al.} reported an experimental teleportation where all four states were distinguished in the Bell-state measurement \cite{Kim01}, while Jennewein \textit{et al.} provided a proof of the nonlocality of the process and of entanglement swapping \cite{Jennewein01}. One year later, in 2002 Pan \textit{et al.} were performing four-photon experiments for high-fidelity teleportation \cite{Pan02} and Lombardi \textit{et al.} teleported qubits encoded in vacuum--one-photon states \cite{Lombardi02}. The beginning of the new century saw a true race towards more complex implementations. In 2004, for instance, a single-mode discrete teleportation scheme using a quantum dot single-photon source has been demonstrated \cite{Fattal04}. At the same time, increasing the teleportation distances \cite{Xia17} became an interesting benchmark to assess the feasibility of practical implementations for future quantum networks \cite{Marcikic03, Ursin04, deRiedmatten04, Landry07, Jin10, Ma12, Yin12}. Today the current record for the longest distance is kept by Ren \textit{et al.} who teleported single-photon qubits from a ground observatory to a satellite 1400 km high in atmosphere \cite{Ren17}. Next to the discrete-variable schemes of the first demonstrations, teleportation was also reported on squeezed entangled states \cite{Zhang03, Bowen03, Takei05, Takei05squeezedTele, Yonezawa07, Yukawa08, Lee11}, for which a review is available \cite{Pirandola06}. Indeed, the main reasons behind the increasing interest toward continuous-variable schemes concern practical advantages, since Bell measurements can be realized by means of only passive linear-optical elements and homodyne detection with very high precision. To bridge the gap between discrete and continuous variables, a hybrid approach has been recently proposed and tested \cite{Takeda13}.
Finally, interesting achievements on photonic teleportation have been demonstrated with the first implementation on integrated circuits \cite{Metcalf14}, which may turn useful for future realizations of quantum network nodes, as well as schemes with simultaneous teleportation of multiple degrees of freedom \cite{Wang15teleport} and teleportation of qudits \cite{Goyal14teleport}, as opposed to the conventional two-level discrete approach.
In the above realizations, where photons encode information through all the stages of the protocol, the state of the qubit was transferred from one photon to another. Though the sources of single photons and photon pairs are still spatially close, the process is indeed the core of future long-distance quantum communication as we envisage it today (see Section \ref{sec:LDQC}) \cite{Kimble08, Krenn16}. One further requirement for a reliable and practical quantum communication is the capability of storing information for subsequent uses. The past decade has seen a strong effort directed towards the development of matter-light interfaces as building blocks for quantum computation and communication, where entanglement between single-photon states and atomic ensembles represents an effective solution. In nearly ten years several works have been carried out in this context \cite{Hammerer10}. Single-photon qubits have been teleported on atomic ensembles \cite{Krauter13, Barrett04} such as Caesium atoms \cite{Sherson06} or $^{87}$Rb atoms \cite{Chen08teleport, Nolleke13, Bao12}, on a pair of trapped calcium ions \cite{Riebe04} or diamond \cite{Hou16}. In this sense, single photons can also turn effective as mediators between distant matter qubits, as already reported with single trapped ytterbium ions \cite{Olmschenk09}, quantum dots \cite{Gao13}, rare-earth-ion doped crystals \cite{Bussieres14} or diamond spin qubits \cite{Pfaff14}.
\begin{figure*}[t!h]
\centering
\includegraphics[trim={0 0 0 0},clip, width=\textwidth]{FigureQuantumCommunication.pdf}
\caption{\footnotesize Main ingredients for photon-based quantum communication described in this section. Photons represent the most promising system to encode quantum information in this field for their speed, ease of manipulation and long coherence time. }
\label{fig:FigureQuantumCommunication}
\end{figure*}
\paragraph{Entanglement swapping.}
In order to increase the distance between nodes of a quantum network without decreasing the quality of the transmission, i.e. the degree of entanglement distributed, it is necessary to develop techniques to protect or restore the state of the encoding photon. Entanglement swapping is a promising solution to this issue \cite{Khalique14} and, nowadays, entanglement distribution includes it as a fundamental routine in quantum repeaters (see Section \ref{qRepeaters}) \cite{Yuan10}. Entanglement swapping is actually very similar to teleportation, the only difference being that the particle to be teleported is part of a second entangled pair. Schematically, given two pairs of entangled particles \textit{A}-\textit{B} and \textit{C}-\textit{D} shared by Alice and Bob, for instance in the Bell state $\ket{\psi^{-}}$, the goal is to convert the whole system to a new entangled pair \textit{A}-\textit{D} by performing a Bell measurement on \textit{B} and \textit{C}:
\begin{eqnarray}
\ket{\psi^{-}}_{A,B} &\ket{\psi^{-}}_{C,D}& \propto \nonumber \\ &\ket{\psi^{-}}_{A,D}& \ket{\psi^{-}}_{B,C} \;+\; \ket{\psi^{+}}_{A,D} \ket{\psi^{+}}_{B,C} \;+ \nonumber \\
&\ket{\phi^{-}}_{A,D}& \ket{\phi^{-}}_{B,C} \;+\; \ket{\phi^{+}}_{A,D} \ket{\phi^{+}}_{B,C}
\end{eqnarray}
\noindent After the Bell measurement on \textit{B}-\textit{C}, and whatever outcome Alice receives, Bob remains with an entangled pair \textit{A}-\textit{D} even though they did not share any entanglement or interaction. Similarly to teleportation, photons offer a convenient solution for an implementation of the scheme.
First demonstrations of entanglement swapping go back to 1998 by Pan \textit{et al.} \cite{Pan98}, to 2001 by Jennewein \textit{et al.} and to 2002 by Sciarrino \textit{et al.}, all based on parametric down-conversion sources. Subsequent experiments still employed probabilistic single-photon sources at near-infrared and telecom wavelengths, the telecommunication windows for optical fibers, in bulk or waveguide as in Ref.\cite{Jin15} and references therein. Entanglement swapping has also been tested with discrete \cite{Sun17, Weston18} and continuous variables \cite{Takei05}, or even with a hybrid approach \cite{Takeda15}.
Future implementations of quantum repeaters in optical networks will require active synchronization of the single-photon sources involved in the exchange of information between the various nodes. To this aim, proof-of-principle demonstrations of this scheme have been reported in the last decade, where two synchronized independent sources generated entangled photon pairs for entanglement swapping and non-locality tests \cite{Yuan10}.
\subsubsection{Decoherence-free communication --}
\label{sec:decoh.free}
So far, we have described how entanglement purification, entanglement swapping, dense coding and teleportation allow for the distribution of entanglement, the essential resource for quantum communiction, between in-principle arbitrarily distant locations. However, non-ideal operating conditions may hinder or completely prevent their concrete implementation. Two methods can be adopted to circumvent the issue of noisy channels: to increase the resilience of a protocol to losses and noise \cite{Gottesman03, Xiang10, Kocsis13} (two-way classical communications), or to decrease their influence by encoding information in immune states (decoherence-free subspaces, DFS), for both quantum computation \cite{Lidar98} and quantum communication tasks \cite{Klein06}. In the following we provide simple examples of DFS schemes, while further on we will overview some of the most recent experimental demonstrations.
In DFS protocols, qubits are encoded in states that do not experience decoherence in a given channel thanks to known symmetries of the subspace.
As an example \cite{Yuan10}, let us consider two qubits evolving through a unitary transformation $U(t)$ such that
$ U(t) \ket{0} = e^{-\imath \omega_0 t} \ket{0}$ and $ U(t) \ket{1} = e^{-\imath \omega_1 t} \ket{0}$. We can see that $\ket{\psi^{+}} = \frac{1}{\sqrt{2}} (\ket{01}+\ket{10}) $ is an invariant of $U(t)$ since $ U(t) \ket{\psi^{+}} = e^{-\imath (\omega_0+\omega_1) t} \ket{\psi^{+}}$, which is equivalent to $\ket{\psi^{+}}$ up to a global phase.
A second example is given by the unitary transformation
\begin{equation}
\cases{
U(\theta, \alpha) \ket{0} = \cos{\theta} \ket{0} + e^{\imath \alpha} \sin{\theta} \ket{1} \\
U(\theta, \alpha) \ket{1} = e^{- \imath \alpha} \sin{\theta} \ket{0} + \cos{\theta} \ket{1}
}
\end{equation}
\noindent The transformation $ U(\theta,0)$ is called \textit{collective rotation noise}. In this case, the states $\ket{\phi^{+}} = \frac{1}{\sqrt{2}} (\ket{00}+\ket{11}) $ and $\ket{\phi^{-}} = \frac{1}{\sqrt{2}} (\ket{01}-\ket{10}) $ are invariants, while $\ket{\phi^{-}}$ remains unchanged even under the more generic $U(\theta,\alpha)$. Thus, by modelling the noise in a given quantum channel, it is in principle possible to construct quantum states that are immune to all decoherence effects corresponding to that specific pattern.
The first experimental demonstration of DFS communication was reported in 2000 by Kwiat \textit{et al.} \cite{Kwiat00}, where the authors induced a controllable collective decoherence on an entangled photon pair generated by spontaneous parametric down-conversion (PDC) and observing that, as predicted, one specific entangled state was immune to decoherence. More recently, in 2005 Jiang \textit{et al.} reported a test of DFS against collective noise on polarization and phase with four two-qubit states again from PDC \cite{Jiang05}, while in 2006 Chen \textit{et al.} used single photons, entangled in polarization and time, to compensate for errors induced by a collective rotation of the polarization \cite{Chen06}. Importantly, their scheme was also alignment-free, i.e. with no need for a shared reference frame, and insensitive to phase fluctuations in the interferometer. In the same year, Zhang \textit{et al.} reported a fault-tolerant implementation of quantum key distribution (see Section \ref{sec:QKD}) with polarization encoding, capable to account for bit-flip errors and collective rotation of the polarization state, without the need for a calibration of the reference frame \cite{zhang06}. In 2008, Yamamoto \textit{et al.} demonstrated an entanglement distribution scheme with state-independent DFS from PDC, with an approach robust against fluctuations of the reference frame between distant nodes \cite{Yamamoto08}. Other significant features reported in their work were the capability to extend the scheme to multipartite states, thanks precisely to the state-independence, and its applicability also with single-mode fibers.
More recently, various proposals and experimental tests have been reported on variants or improvements of the original schemes. For instance, Ikuta \textit{et al.} in 2011 proposed and demonstrated a solution for increasing the efficiency of DFS-based entanglement distribution over lossy channels \cite{Ikuta11}. Additional features of their work were the use of backward propagation of coherent light in combination with single-photon states, and the violation of the Clauser-Horne-Shimony-Holt inequality \cite{Nielsen_Chuang} to prove non-locality in the transmission. In 2017, novel investigations on matter qubits were reported by Wang \textit{et al.}, with a room-temperature quantum memory realized with two nuclear spins coupled to the electronic spin of a single nitrogen-vacancy center in diamond \cite{Wang17dfs}, and by Zwerger \textit{et al.}, with the demonstration of a quantum repeater (see Section \ref{qRepeaters}) using ion-photon entangled states protected via DFS against collective dephasing \cite{Zwerger17}.
\subsection{Long-distance quantum communication}
\label{sec:LDQC}
Future quantum networks, engineered to deliver quantum information between distant nodes on the globe, are the sought-after goal underlying most of the investigations described above (see Fig. \ref{fig:FigureQuantumNetwork}). This challenge already counts first demonstrations in Austria \cite{Peev09}, China \cite{Xu09, chineseNetwork16}, Japan \cite{Fujiwara16, Sasaki11}, Switzerland \cite{Stucki11} and USA \cite{Elliott05, Hughes13} (see Table \ref{table:qNetworks}). We will now overview the state of the art towards their implementations.
\begin{figure*}[t!]
\centering
\includegraphics[trim={0 0 0 0},clip, width=\textwidth]{FigureQuantumNetwork.pdf}
\caption{\footnotesize Schematic view of the main nodes in a large-scale quantum network, comprising blind quantum computing stages for the end user, quantum repeaters for long-distance transmission and quantum key distribution performed either via fiber networks or via free-space links.}
\label{fig:FigureQuantumNetwork}
\end{figure*}
\subsubsection{Quantum repeaters --}
\label{qRepeaters}
Quantum communication promises to enable quantum information protocols distributed over distant locations. In the previous section we have shortly analyzed some of the main theoretical ingredients of these quantum networks, together with the first concrete demonstrations of their feasibility. The key resource for its fulfillment is provided by quantum entanglement, at the core of quantum dense coding, teleportation, entanglement purification and entanglement swapping. By using these schemes it is possible to create entanglement between two nodes of a network to transfer quantum information. We have also described how this resource deteriorates easily and exponentially fast with the length of transmission for several reasons. Photon losses are the first cause of deterioration: even though fiber attenuation at telecom wavelength can be partly reduced, even ultra-low loss optical fibers still unavoidably lead to an exponential decrease in the generation rate of entangled pairs. This is a fundamental limitation imposed by quantum mechanics: the maximum number of entanglement bits (ebits) that can be distributed over a lossy channel with transmissivity $T$ is in fact equal to $-\log_2 (1-T)$ ebits per channel use \cite{Pirandola17}. In addition, errors can occurr all along the transmission due to imperfect gates or measurements, without the possibility of duplicating quantum states deterministically \cite{Nielsen_Chuang} and with conditions much more sensitive than in the classical case.
Notwithstanding, two approaches exist to circumvent these issues: reducing the transmission loss using free-space communication \cite{Yin12, Wang12oam, Vallone14, Steinlechner17, SchmittManderbach07, Resch05, Peng05, Jin10, Aspelmeyer03, Wang13satellite, Nauerth13, Ren17, Patel14} and decoherence-free subspaces \cite{Kwiat00,Chen06,Yamamoto08,Zwerger17,Wang17dfs,Ikuta11,Jiang05,GonzalezAlonso13}, or employing quantum repeaters.
Quantum repeaters (QR), proposed in 1998 by Briegel \textit{et al.} \cite{Briegel98}, currently represent a strong candidate to circumvent these issues. In principle, QRs allow to improve the fidelity of transmission with time-overhead polynomial in the transmission length. In the original proposal, QRs operate by splitting the channel into a suitable number of intermediate segments linked by as many QRs, where active control can compensate for fiber attenuation, gate errors and possible noise. Once a sufficiently strong entanglement is established between two target nodes, routine quantum communication can start effectively. Today numerous schemes and platforms exist for long-distance quantum communication based on QRs \cite{Muralidharan16}. QRs can be grouped in three main classes, according to the technique adopted for correcting propagation errors \cite{Muralidharan16}, as summarized schematically in Table \ref{table:QR}.
\begin{table}[h!]
\renewcommand*{\arraystretch}{1.4}
\centering
\caption{\label{table:QR} Quantum repeaters can be classified in three generations according to the approach used to correct errors.\\Table adapted from Ref.\cite{Muralidharan16}.}
\footnotesize
\begin{center}
\begin{tabular*}{\linewidth}{c@{\extracolsep{\fill}}ccccc}
\mr
Error & Solution & I & II & III \\
\mr
\multirow{2}{1cm}{\centering{Loss}}
& HEG ($\leftrightarrow$) & \checkmark & \checkmark & \\
& QEC ($\rightarrow$) & & & \checkmark\\ \hline
\multirow{2}{1cm}{\centering{Gate}}
& HEP ($\leftrightarrow$) & \checkmark & & \\
& QEC ($\rightarrow$) & & \checkmark & \checkmark\\
\mr
\end{tabular*}
\end{center}
HEG: Heralded entanglement generation. HEP: Heralded entanglement purification. QEC: quantum error correction.\\
$\longrightarrow$: one-way communication; $\longleftrightarrow$: two-way communication.
\end{table}
The first generation of QRs (I) employs heralded entanglement generation (HEG) and heralded entanglement purification (HEP) to reduce the deterioration induced by losses from an exponential to a polynomial scaling (see Section \ref{sec:entanglement}) \cite{Briegel98, Sangouard11, Dur99, VanLoock06, Duan01, Zwerger12}. The idea here is to create entanglement between adjacent nodes and to use teleportation to exchange information. Entanglement swapping and purification can then be properly combined to sequentially extend the distribution of entangled photon pairs until two nodes are connected. The overall rate for quantum communication based on this scheme decreases polynomially with the distance; however, a good solution to this drawback could be multiplexing in one of the degrees of freedom (see Section \ref{sec:encoding}) to increase the transmission rate. A number of other approaches have been tested experimentally in the last decade. An efficient scheme using double-photon guns based on semiconductor quantum dots, polarizing beam splitters and probabilistic optical CNOT gates was proposed in 2003 by Kok \textit{et al.} \cite{Kok03}. Active purification of arbitrary errors using only two qubits at each QR, based on nuclear and electronic spins in nitrogen-vacancy color centers in diamond, was proposed by Childress \textit{et al.} \cite{Childress06}. Further proposals involve multi-mode memories based on photon echo in solids doped with rare-earth ions \cite{Simon07}. A combination of photonic and atomic platforms was also investigated by Sangouard \textit{et al.} \cite{Sangouard09, Sangouard11}, while combinations of quantum-dot qubits and optical microcavities have been studied by Wang \textit{et al.} \cite{Wang12repeater}. Recently, a novel measurement-based QR was shown to enable entanglement purification and entanglement swapping \cite{Zwerger12} for one- and two-dimensional networks using cluster states and photon-ion entanglement \cite{Wallnofer16, Zwerger17}. A thourough analysis under noisy conditions of measurement-based quantum network coding scheme for QRs was later provided by Matsuo \textit{et al.} \cite{Matsuo17}. Finally, for an overview on quantum memories in the context of quantum communication we refer to Ref.\cite{Simon17} and references therein.
The second generation of QRs (II) employs (i) HEG to reduce the deterioration induced by losses and (ii) quantum error correction (QEC) to correct gate errors \cite{Jiang09, Munro10}. Teleportation with CNOT gates and entanglement swapping are still sequentially applied to extend the entanglement to distant nodes; however, the use of QEC in place of HEP speeds up the process by avoiding the time delays due to non-adjacent-nodes signalling \cite{Bratzik14}. With this approach, quantum memories are employed at each side of a QR to save the state of the entangled photon pairs while waiting for the classical signals to carry out teleportation \cite{Simon07,Nolleke13,Munro12,Mazurek14,Heshami16,Gundogan15,Ding13oam,Chen08teleport, Campbell14,Bussieres14,Bao12,Laplane17, Kalb17}. Indeed, QEC with qubit-repetition codes and Calderbank-Shor-Steane codes \cite{Nielsen_Chuang} were shown to be effective against imperfect quantum memories operation, photon losses and gate errors \cite{Bernardes12}. However, starting from 2012, alternative all-photonic schemes without quantum memories were presented by Munro \textit{et al.} \cite{Munro12} and by Azuma \textit{et al.} \cite{Azuma12, Azuma15} based on multi-photon cluster-states, loss-tolerant measurements and local high-speed active feedforward controls. Recently, Ewert \textit{et al.} achieved similar results using only \textit{locally}-prepared Bell states, instead of more demanding cluster states \textit{non-locally} entangled between nodes of a segment \cite{Ewert16}. One further scheme, feasible with current technology, was proposed in 2017 by Vinay \textit{et al.} \cite{Vinay17} based on double-heralded entanglement generation and brokered Bell-state measurements.
The third generation of QRs (III) employs QEC to deterministically correct errors from both propagation losses and gate errors \cite{Pant17, Azuma15, Fowler10, Munro12, Muralidharan14, Namiki16, Muralidharan17, Li13}. The idea is to iteratively transfer a block of $\sim 200$ qubits \cite{Munro12, Muralidharan14} from one node to the next in each lossy segment and to apply QEC to recover the encoding. Differently from the first two schemes (I, II), the latter is fully fault-tolerant and it involves one-way signalling between the segments, thus being much faster than the previous ones, in particular than type I. However, schemes II and III represent more demanding solutions from a purely technological perspective, mainly for the high control required to perform reliable error correction and, consequently, for the reduced maximal distance allowed between nodes.
\subsubsection{Blind quantum computing --}
\label{BQC}
In the previous sections we reviewed the physical resources for the implementation of a linear-optical quantum network. Entanglement plays a fundamental role for quantum communication, while various quantum repeaters have been developed with increasing performances, paving the way for an effective management of quantum information in intra-city and global networks. However, building a network with hundreds of nodes and synchronized single-photon sources seems still a demanding target for the near-term future. In this section we discuss quantum computation in an optical quantum network, where a client resorts to a central server with more advanced quantum technology to perform computation in a manner as safe as possible: the server should obtain no information on client's inputs, algorithms and outputs \cite{Fitzsimons17review}.
The first model of blind quantum computation (BQC) was proposed by Childs in 2001 \cite{Childs05}, where Alice, who cannot access a quantum computer, asks Bob to help her perform quantum computation without telling him her input, output and task. Childs described a protocol where Alice succeeds in this goal, with the possibility of even detecting whether Bob is honest or introducing errors. His solution was later improved by Arrighi and Salvail in 2003 \cite{Arrighi06}, with a blind protocol for the class of functions admitting an efficient generation of random input-output pairs like factoring. However, these schemes still allocate high resources to the client, who is supposed to have access to quantum memories and SWAP quantum gates. A step ahead was achieved in 2009 by Broadbent, Fitzsimons, and Kashefi with an interactive one-way BQC model (BFK) employing measurement-based quantum computing, where Alice only needs to produce single-qubit states with no need for quantum memories \cite{Broadbent09} and where she is capable to detect malicious errors (see Section \ref{sec:validation}).
From 2009 several other BQC protocols have been proposed, which may be classified according to the number of servers employed \cite{Sheng16}: \textit{single-server} BQC protocols \cite{Childs05, Arrighi06, Barz12, Dunjko12cv, Morimae12, Giovannetti13, Sueki13, Morimae13, Morimae14, Fisher14, Morimae15, Chien15, Gheorghiu15, Takeuchi16, Fitzsimons17, Mantri13, Mantri17}, where Alice is capable to generate quantum states and distribute them to Bob, \textit{two-server} \cite{Broadbent09, Sheng16, Morimae13server2} and \textit{three-server} \cite{Li14}, where Alice can get rid of quantum capabilities. Results have been reported also for a purely classical user \cite{Mantri17}, though some limitations have been recently opposed by Aaronson \textit{et al.} \cite{Aaronson17} building upon complexity considerations analogous to those for \textsc{BosonSampling} (see Section \ref{sec:BosonSampling}).
In measurement-only BQC, the idea is for Alice to secretly hide \textit{trap} qubits in her state. If Alice discovers an unwanted change of a trap she can conclude that Bob is not honest and quit the task. The probability that Alice accepts a malicious Bob can then be made exponentially small using quantum error correcting codes \cite{Morimae14, Fitzsimons17}.
The feasibility of BQC has already been proved by various photonic experiments. The first demonstration was reported in 2011 by Barz \textit{et al.} \cite{Barz12}, where photon states were sent by the client to the quantum server, which produced four-photon blind cluster states to perform Grover search and Deutsch algorithms \cite{Nielsen_Chuang}. In 2013, Barz \textit{et al.} \cite{Barz13} proposed and demonstrated experimentally a technique to test whether a quantum computer is really quantum and whether it provides correct outputs (see Section \ref{sec:validation}). In the same year, Fisher \textit{et al.} \cite{Fisher14} demonstrated arbitrary computations on encrypted qubits, only requiring the client to prepare single qubits with limited classical communication. Finally, in 2016 Greganti \textit{et al.} \cite{Greganti16} reported photonic BQC where the client sends four-qubit states to the server to implement arbitrary two-qubit entangling gates.
The above demonstrations represent significant achievements for BQC, showing that integration of these algorithms on linear-optical networks are in principle feasible using single photons. We mention that also continuous variables schemes have been proposed \cite{Morimae12cv, Dunjko16}, while numerous works have analyzed their feasibility with respect to efficiency \cite{Mantri13, Giovannetti13, PerezDelgado15 } and resilience to errors \cite{Morimae12, Sueki13, Chien15} and to noisy implementations (see also Section \ref{sec:decoh.free}) \cite{Sheng16,Takeuchi16}. However, it was observed that hybrid systems with matter qubits for carrying out quantum computation and single photons for quantum communication could offer an even more advantageous solution in larger-size distributed protocols \cite{Fitzsimons17review}.
\subsubsection{Photonic quantum key distribution --}
\label{sec:QKD}
Quantum key distribution (QKD) is a technique to generate a shared random secret key between two parties. The advantage of QKD is the possibility to detect possible attacks from a malicious third party, since any measurement carried out by the latter influences the shared system. Once a secret key is shared, the two parties can start standard classical communication.
\begin{table}[hb!]
\renewcommand*{\arraystretch}{1.2}
\centering
\footnotesize
\caption{\label{QKDtable} }
Several experimental demonstrations of quantum communication have been reported in the last two years, from specific applications to long-distance communication.
\begin{tabular*}{\linewidth}{l@{\extracolsep{\fill}}ccc}
\br
Year & Ref. & Focus for QKD & Distance (km) \\
\mr
2016$\:$ & \cite{Schiavon16} & Renes2004 protocol & 0 \\
& \cite{Autebert16qkd} & BBM92 protocol & 50 \\
& \cite{Sun16} & WDM GPOM & 24 \\
& \cite{Collins16} & QDS & 90 \\
& \cite{Tang16} & MDI DS & 36 \\
& \cite{Yin16} & MDI DS & 404 \\
& \cite{Sun16decoy} & Phase-encoded passive-DS & 10 \\
& \cite{Dynes16} & DP-QPSK DS & 100 \\
& \cite{Lee16} & HD DS DO & 43 \\
& \cite{Nape16} & HD & 0 \\
& \cite{Canas16} & HD MCF DS & 0.3 \\
& \cite{Dynes16mcf} & MCF & 53 \\
& \cite{Liao16} & Free-space in daylight DS & 53 \\
2017$\:$ & \cite{Zhang17QSDC} & QSDC & 0 \\
& \cite{Sit17} & Free-space HD & 0.3 \\
& \cite{Sibson17} & DS on tunable circuit & 20 \\
& \cite{Ding17} & HD MCF DS & 20 \\
& \cite{Sun17} & Entanglement swapping & 12 \\
& \cite{Wang17} & WDM QAM DS & 80 \\
& \cite{Collins17} & QDS & 90/134 $^{(a)}$ \\
& \cite{Roberts17} & MDI DS QDS & 50 \\
& \cite{Yin17} & MDI DS QDS & 55 \\
& \cite{Frohlich17} & DS BB84 & 200/240 $^{(b)}$ \\
& \cite{Kiktenko17} & Polarization/phase-encoding & 45 \\
& \cite{Pugh17} & Free-space & 10 \\
& \cite{Yin17_1200} & Satellite & 1200 \\
& \cite{Liao17} & Satellite DS & 700 \\
& \cite{Takenaka17} & Satellite & 1000 \\
& \cite{Liao17satellite-to-ground} & Satellite DS & 1200 \\
& \cite{Ren17} & Satellite teleportation & 1400 \\
2018$\:$ & \cite{Liao18} & Satellite DS & 7600 \\
\br
\end{tabular*}\\
WDM: wavelength-division-multiplexing;\\GPOM: gigabit-capable passive optical network;\\MCF: multicore fiber; QDS: quantum digital signature;\\MDI: measurement-device-independent; DS: decoy-state; DP-QPSK: dual polarisation quadrature phase shift keying; QAM: quadrature amplitude modulation; HD: high-dimension;\\DO: dispersive optics; QSDC: quantum secure direct communication. $^{a}$: 134 km is simulated with additional optical attenuation. $^{b}$: with/without multiplexing.
\end{table}
\normalsize
\paragraph{Technological requirements and security.} The security of a QKD system is measured with the distance $\epsilon$ between the corresponding outcomes probability distribution and the ideal one with a perfect key, typically $\sim10^{-10}$. This difference can be made in principle arbitrarily small, at the price of requiring more stringent hardware performances and by applying suitable privacy amplification \cite{Nielsen_Chuang}; however, one should bear in mind that this threshold must include all subroutines involved in a QKD system, so that only the overall \textit{composable security} $\tilde{\epsilon} =\sum \epsilon_k$ is relevant. There are still many limitations to overcome to develop QKD systems \cite{Diamanti16}, the main being (i) the transmission rate and range, much lower than in classical communications, (ii) the high cost to produce and maintain the hardware, (iii) the need for authentication and integrity and (iv) that new classical algorithms can be designed to be immune to quantum agents. First QKD prototypes have also been hacked \cite{Lo14,Moskovich15}, though techniques exist to correct gate errors with linear optics \cite{Mazurek14, Kalamidas05, Yamamoto05, Li07}.
QKD protocols are capable to establish secure communication channels, but no protection is warranted to the single-photon sources and detectors employed in the system. Alice's source represents probably the safest stage, since it can be secured by optical isolators and verified in situ \cite{Gottesman04}, thus most quantum hacking algorithms currently exploit the receiver's detectors (e.g. detection efficiency and dead time), even though a complete list of weak points would involve also attacks based on channel/device calibration or photon wavelength \cite{Lo14}.
In principle, countermeasures could be found any time a loophole is discovered just like for classical cryptosystems. However, this is naturally not desirable for an approach that aims to build inherently secure protocols of communication. One approach to counter hacking attacks is \textit{device-independent} (DI-) QKD \cite{Masanes11, Reichardt13} where Alice's and Bob's devices are considered as black boxes and the security depends on the violation of a Bell inequality to confirm quantum correlations\cite{Hensen15}. However, hardware technological challenges still make DI-QKD impractical for current state-of-the-art.
A second approach based on time-reversed QKD is \textit{measurement-device-independent} (MDI-) QKD \cite{Braunstein12, Lo12}, which allows Alice and Bob to perform QKD protocols even in the case of untrusted devices \cite{Inamori02}, -reasonably- assuming trusted sources. One advantage of MDI-QKD is its feasibility with state-of-the-art technology, with key rates orders of magnitude higher than in DI-QKD.
\paragraph{Implementation of QKD.} Currently the most widely adopted protocols for QKD are BB84 \cite{Bennett84} and E91 \cite{Ekert91}, as well as \textit{quantum secret sharing} \cite{Hillery99} and \textit{third-man quantum cryptography} \cite{Zukowski98}, though numerous other schemes have been developed \cite{Nielsen_Chuang, Bennett92, Hatakeyama17} based on discrete variables, continuous variable or distributed phase reference coding. Single photons offer the most promising platform to encode information that can be secretly shared thanks to entanglement. In particular, both quantum secret sharing and third-man quantum cryptography rely on three-particle polarization entangled states $\ket{\psi} \propto \ket{000}+\ket{111} $ known as GHZ \cite{Zukowski98}. Other schemes based on attenuated lasers are also of importance, for instance for differential phase shift \cite{Hatakeyama17, Collins16}, coherent one-way (COW) \cite{Gisin04} and decoy-state protocols \cite{SchmittManderbach07, Sun17, Canas16, Sun16decoy}. We refer the reader to recent comprehensive reviews \cite{Lo14, Diamanti16, Moskovich15} on the topic for a thorough overview of the extensive field. For a list of the most recent implementations see instead Table \ref{QKDtable}.
Information can be transferred between distant locations either via free-space ($\lambda\sim800$ nm) or via optical fibers ($\lambda\sim 1310$ nm or $\lambda\sim 1550$ nm) using any of the degrees of freedom overviewed in Section \ref{sec:encoding}. Polarization or orbital angular momentum are indeed more fit for free-space communication, since birefringence and mechanical instabilities in fibers can affect them heavily, thus making time-bin or frequency encoding more suitable. Protocols based on entangled photon pairs could achieve the longest distances tolerating higher losses up to $\sim70$ dB, as well as requiring higher technological requirements, while distributed-phase-reference QKD is a more promising solution for shorter distances ($\sim100$ km) \cite{Hatakeyama17, Gisin04}.\\
In the context of in-fiber QKD, a new approach that was recently introduced to achieve higher key rates is offered by wavelength-division multiplexing (WDM), where two quantum signals are transferred simultaneously on the same optical fibers. Quantum information can be exchanged even with strong classical signals thanks to dense wavelength multiplexing on the same fiber, showing that the integration of QKD protocols on existing telecom networks can be a viable path. In 2009, Chapuran \textit{et al.} demonstrated WDM-QKD in a reconfigurable network with single photons at 1310 nm and classical channels at 1550 nm \cite{Chapuran09}, while Peters \textit{et al.} reported WDM-QKD at 1550 nm for both quantum and strong classical channels \cite{Peters09}. More recently, in 2014 Patel and coworkers set a new record by transferring bidirectional 10 Gb/s classical channels with a key rate of 2.38 Mbps and fiber distances up to 70 km \cite{Patel14}. One of the main sources of noise in these experiments was due to spontaneous anti-Stokes Raman scattering \cite{Peters09, Chapuran09}, whose influence was shown to be mitigated by selecting an optimal wavelength, allowing for unprecedented terabit classical data transmission up to 80 km \cite{Sun16, Wang17}.
While in-fiber QKD could achieve high transmission rates for short distances, by exploiting pre-existing telecom infrastuctures, free-space QKD represents an alternative promising way for future QKD networks on a global scale. First demonstrations of ground-based quantum communication with single photons were reported with progressively higher records in distance and key rate \cite{Aspelmeyer03, Peng05, Resch05, Ursin07, SchmittManderbach07, Jin10, Yin12, Wang12oam, Heindel12, Rau14, Vallone14, Steinlechner17}. Yet, one further promising approach would be to employ satellites as nodes of the network, an ambitious project that could allow to cover the whole globe. Preliminary tests in this direction were carried out independently in 2013 by Wang \textit{et al.}, simulating three experiments of QKD with decoy states \cite{Canas16, SchmittManderbach07, Sun16decoy} with a setup operating on moving (ground) and floating (hot-air balloon) platforms with high-loss channels \cite{Wang13satellite}, and by Nauerth \textit{et al.}, demonstrating an instance of BB84 between a high-speed airplane and a ground station for a distance of 20 km \cite{Nauerth13}. Later on, in 2015 and 2016, two papers reported the transmission of single photons using satellite corner cube retroreflectors as quantum transmitters in orbit \cite{Vallone15, Dequal16}. In 2017 six further achievements were reported: ground-to-aircraft \cite{Pugh17} and satellite-to-ground \cite{Liao17, Takenaka17, Liao17satellite-to-ground} QKD, satellite-based entanglement distribution over 1200 kilometers \cite{Yin17_1200} and ground-to-satellite teleportation of single-photon qubits for a distance up to 1400 km \cite{Ren17}. Finally, in 2018 Liao \textit{et al.} successfully performed decoy-state QKD between a low-Earth-orbit satellite and multiple ground stations separated by 7600 km on Earth \cite{Liao18}, providing landmark steps forward toward the realization of future long-distance quantum networks.
\section{Photonic quantum simulation}
\label{sec:simulation}
In this chapter we will review the field of photonic quantum simulation, starting with single-photon dynamics in quantum walks (Section \ref{sec:QW}) and extending it to multiphoton evolution (Section \ref{sec:BosonSampling}). In Section \ref{sec:validation} we will review some of the most recent results on the problem of verification, whose increasing relevance goes in parallel to the developments in quantum simulation. Finally, in Section \ref{qChemistry} we describe the first applications in quantum chemistry and condensed matter. For a comprehensive review of quantum simulation in various fields and physical platforms we refer the reader to Refs. \cite{Buluta09, Georgescu14}.
\subsection{Photonic quantum walks}
\label{sec:QW}
Quantum walks (QW), the extension of classical random walks to a quantum framework, have gained an increasing role in modelling single- and multi-particle evolutions in several scenarios \cite{Grafe16} thanks to their platform-independent formulation. Importantly, the universality of the linear-optical platform for quantum computation \cite{KLM} has further prompted several experimental tests of QWs with single photons.
Two classes of QWs exist (see Fig.\ref{fig:FigureQuantumWalk}): \textit{discrete-time} (DTQW), where the evolution is split in discrete steps and random events suddenly influence the dynamics, and \textit{continuous-time} (CTQW), where the evolution on a given lattice is described by a time-independent Hamiltonian and by the adjacency matrix of the corresponding graph. In this section we will focus on the experimental implementation of photonic QWs, reviewing the latest achievements in this area with some links to the theoretical background. The reader interested in the theory of quantum walks may refer to Ref. \cite{Venegas-Andraca12} for a comprehensive collection.
\begin{figure*}[t!h]
\centering
\includegraphics[trim={0 0 0 0},clip, width=\textwidth]{FigureQuantumWalk.pdf}
\caption{\footnotesize Quantum walks are classified in two types: \textit{discrete-time} and \textit{continuous-time}. While the former can count on various photonic implementations, the latter has been enabled by integrated waveguide lattices. In each subfigure, $Q_i$ indicates the different steps of the quantum walks. Legend - QWP: quarter-wave plate, HWP: half-wave plate, PBS: polarizing beam splitter, BS: beam splitter, GP: glass plate, POL: polarizer, FPBS: fiber polarizing beam splitter.}
\label{fig:FigureQuantumWalk}
\end{figure*}
\paragraph{Discrete-time quantum walks.}
DTQWs can be described by single photons evolving through a network of beam splitters. In this pictorial representation, each time a photon enters a beam splitter its wave function is split in two parts proceeding on separate optical modes. In terms of creation operators acting on the two modes, the evolution can be ruled for instance by the \textit{Hadamard coin}
\begin{equation}
\left(
\begin{array}{c}
a^{\dagger}_1\\
a^{\dagger}_2
\end{array} \right)_{in}
\longrightarrow \quad
\frac{1}{\sqrt{2}} \left(
\begin{array}{cc}
1& i \\
i & 1
\end{array} \right)
\left(
\begin{array}{c}
a^{\dagger}_1\\
a^{\dagger}_2
\end{array} \right)_{out}
\end{equation}
\noindent We stress that also other approaches can accomplish the same task \cite{Cardano15, Cardano16}: we only need a process \^{Q} that, given a chosen encoding, operates the transformation $Q \ket{1} \longrightarrow \alpha \ket{0} + \beta \ket{1} $. Choosing \^{Q} as a building block for the QW, after $n$ steps the walker will be in the superposition $ \sum_{j}^{n} c_j\, a^{\dagger}_j \ket{0} $, where the amplitudes $c_j$ can be balanced, symmetric or asymmetric over the optical modes according to the symmetries of the walker \cite{Sansoni12, Carolan14} and to the evolution \cite{Schreiber11, Crespi13anderson}.
First implementations of DTQWs were reported in the first years of this century, motivated by the connection found between QWs and quantum computation \cite{Childs03, Childs09} for which, just at that time, Knill, Laflamme and Milburn were showing that linear-optical platforms are universal \cite{KLM}. The very first demonstrations were carried out on bulk optics, confirming the theoretical predictions for single photons dynamics \cite{Do05, Broome10} and wave packet interference \cite{SoutoRibeiro08}. However, bulk implementations unavoidably suffer from mechanical instabilities and issues of alignment critical for larger-size experiments. A first solution was found in 2010 with novel fiber-loop designs \cite{Schreiber10,Schreiber11, Regensburger11, Schreiber12}, which provided a more stable approach suitable for further improvements in the direction of scalable schemes. Thanks to this new approach it was possible to investigate QWs with a larger number of steps, yielding a richer landscape of photon interference. A relevant step forward was given by the introduction of integrated photonic circuits (see Section \ref{sec:circuits}), which provided furter advantages in stability, alignment and compactness \cite{Sansoni12, Crespi13anderson, Carolan14, Grafe14, Harris17}. Moreover, integrated circuits are fit for realizing reconfigurable photonic architectures, allowing to dynamically tune the evolution and, thus, to explore more complex scenarios within the same quantum walk \cite{Carolan15, Flamini15, Harris17}. Relevant tests on the foundations of quantum mechanics carried out on integrated circuits include, for instance, bosonic-fermionic evolutions \cite{Sansoni12}, indistinguishable-distinguishable bosons \cite{Carolan14} and quantum trasport phenomena \cite{Harris17} such as wavefunction localization in disordered media \cite{Crespi13anderson}. Very recently, in 2015 an approach for DTQWs was reported based on the orbital angular momentum of light, so that the evolution takes place in the OAM degree of freedom without the need for an interferometer \cite{Cardano15, Cardano16}. This new approach provides indeed a flexible solution that may enable scalable investigations. Finally, in 2016 Boutari \textit{et al.} proposed and demonstrated experimentally a time-bin-encoded scheme for QWs, based on a network of optical ring cavities, which simultaneously achieves low losses, high fidelity, reconfigurability. Such scheme can be further generalized to multidimensional lattices \cite{Boutari16}.
\begin{table*}[h!]
\renewcommand*{\arraystretch}{1.2}
\centering
\footnotesize
\caption{\label{QWtableDT} Photonic discrete-time quantum walks (QW).}
\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}ccccc}
\br
Year & Ref. & Platform & Focus & Photons & Structure \\
\mr
2005 & \cite{Do05} & Bulk optics & Quantum quincunx & 1 & 8 modes \\
2008 & \cite{SoutoRibeiro08} & Bulk optics & Wave packet reshaping & 1 & 1 step \\
2010 & \cite{Schreiber10} & Fiber loop & Robust implementation & 2 & 5 steps \\
& \cite{Broome10} & Bulk optics & QW with tunable decoherence & 1 & 6 steps \\
2011 & \cite{Regensburger11} & Fiber loop & Features of the dynamics & Attenuated laser & 70 steps \\
& \cite{Schreiber11} & Fiber loop & Decoherence and disorder & 1 & 28 steps \\
2012 & \cite{Kitagawa12} & Bulk optics & Topological phases & 1 & 7 steps \\
& \cite{Sansoni12} & Integrated photonics & Bosonic-fermionic evolution & 2 & 8 modes \\
& \cite{Schreiber12} & Fiber loop & Entanglement on 2D QW & Attenuated laser & 12 steps \\
2013 & \cite{Crespi13anderson} & Integrated photonics & Anderson localization & 2 & 16 modes \\
& \cite{Jeong13} & Fiber loop & Delayed-choice 2D QW, Grover QW & 1 & 4 steps \\
2014 & \cite{Grafe14} & Integrated photonics & High-order single-photon W-states & 1 & 2,4,5,8,16 modes \\
2015 & \cite{Cardano15} & Orbital angular momentum & Wave packet dynamics & 2 & 5 steps \\
2016 & \cite{Boutari16} & Optical ring cavities & Low-loss tunable QW & Attenuated laser & 62 steps \\
& \cite{Cardano16} & Orbital angular momentum & Topological quantum transitions & 1 & 6 steps \\
2017 & \cite{Cardano17} & Orbital angular momentum & Zak phases, topological invariants & Attenuated laser & 7 steps \\
& \cite{Pitsios17} & Integrated photonics & Entanglement after spin chain quench & 2 & 5 modes \\
& \cite{Harris17} & Integrated photonics & Quantum transport & Attenuated laser & 26 modes \\
\br
\end{tabular*}\\
\end{table*}
\normalsize
\paragraph{Continuous-time quantum walks.}
\begin{table*}[th!]
\renewcommand*{\arraystretch}{1.2}
\centering
\footnotesize
\caption{\label{QWtableCT} Photonic continuous-time quantum walks implemented with integrated circuits.}
\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}ccccc}
\br
Year & Ref. & Technology & Focus & Photons & Waveguides \\
\mr
2007 & \cite{Schwartz07} & Optical induction$^{(a)}$ & Transport, Anderson localization & Attenuated laser & n.a. \\
2008 & \cite{Perets08} & SoI & Evolution on quantum walks & Attenuated laser & $\sim100$ \\
2010 & \cite{Peruzzo10} & $\textup{SiO}_x\textup{N}_y$ & Quantum correlations & 2 & 21 \\
2011 & \cite{Owens11} & FLW & Evolution in an elliptic waveguide array & 2 & 6 \\
& \cite{Martin11} & FLW & Anderson localization & Attenuated laser & 101 \\
2012 & \cite{Lahini12} & n.a. & Quantum correlations & Attenuated laser & 29 \\
& \cite{Crespi12} & FLW & Quantum Rabi model & Attenuated laser & 15 \\
2013 & \cite{Rechtsman13} & FLW & Floquet topological insulator & Attenuated laser & 49/308 $^{(b)}$ \\
& \cite{Spagnolo13} & FLW & Bosonic coalescence & 3 & 3 \\
& \cite{Matthews13} & $\textup{SiO}_x\textup{N}_y$ & Fermionic statistics & 2 & 10 \\
2014 & \cite{Solntsev14} & $\textup{LiNbO}_3$ & Tunably-entangled biphoton states & 2 & 21 \\
& \cite{Carolan14} & $\textup{SiO}_x\textup{N}_y$ & Bosonic clouding & 3,4,5 & 21 \\
& \cite{Poulios14} & FLW & Quantum correlations & 2 & 9 \\
2015 & \cite{Keil15} & FLW & Majorana dynamics & Attenuated laser & 26 \\
& \cite{Crespi15fano} & FLW & Quantum decay and Fano interference & 2 & 27 \\
& \cite{Lebugle15} & FLW & N00N state Bloch oscillations & 2 & 16 \\
2016 & \cite{Biggerstaff16} & FLW & Quantum transport & Attenuated laser & 4+30 \\
& \cite{Caruso16} & FLW & Quantum transport & Attenuated laser & 18+62 \\
& \cite{Weimann16} & FLW & Discrete fractional Fourier transform & 2 & 8 \\
\br
\end{tabular*}\\
FLW: femtosecond laser written. SoI: silicon on insulator. n.a.: not available.\\$^{(a)}$: in $\textup{Sr}_{0.6}\textup{Ba}_{0.4}\textup{Nb}_{2}\textup{O}_{6}$; (b): for two honeycomb photonic lattices.
\end{table*}
\normalsize
CTQWs represent another platform for investigating statistical features of the evolution of bosonic states \cite{Schwartz07, Perets08, Owens11, Biggerstaff16, Caruso16} and simulating complex processes \cite{Crespi12, Spagnolo13, Rechtsman13, Matthews13, Keil15, Crespi15fano, Lebugle15}. CTQWs can be implemented with arrays of $N$ evanescently-coupled waveguides tailored to produce a time-dependent or time-independent lattice Hamiltonian \cite{Bromberg09}
\begin{equation}
\check{H} = \hbar \sum_{i=1}^N \left( \beta_i \, a_i^\dagger a_i + \sum_{j=1}^N \kappa_{i,j} \, a_j^\dagger a_i \right)
\end{equation}
\noindent where $\beta_i$ is the propagation constant for the $i$th waveguide and $\kappa_{i,j}$ is the coupling between waveguides ($i$,$j$), with the usual restriction to $j=i\pm 1$.\\For an analogy with DTQWs, where the free evolution is decomposed in discrete steps, now waveguides are arranged in 1D or 2D structures in such a way that light can continuously jump between neighbor sites.
The evolution of the $k$th creation operator is given by \cite{Bromberg09}
\begin{equation}
\cases{
a^{\dagger}_k(z) = e^{i \beta z} \sum_l U_{k,l}(z) \;a^{\dagger}_l (z_0) \\ U_{k,l}(z) = (e^{i \kappa z})_{k,l}
}
\end{equation}
\noindent where $z$ is the position along the propagation direction. Waveguide arrays have been implemented so far on integrated photonic circuits, inscribed using one of the fabrication techniques described in Section \ref{sec:circuits}.
\paragraph{Quantum transport.}
Quantum transport phenomena concern the spatial evolution of the wave function during a QW. This process can occur in two regimes: in \textit{ordered} and in \textit{disordered} lattices. A photon propagating through a static-ordered lattice shows a \textit{ballistic} spread for a distance proportional to the evolution time $T$, due to the interference of the wave packet amplitudes across the CTQW. For disordered systems a further distinction can be made: \textit{dynamic disorders} are associated to \textit{diffusive} transports, where interference no longer affects the evolution and a classical-like spreading is found for a distance proportional to $\sqrt{T}$, while \textit{static disorders} lead to a complete halting in the process, a phenomenon known as Anderson localization \cite{Segev13, Abouraddy12} where the wave packet localizes on the initial sites of the lattice \cite{Schwartz07, Lahini08, Martin11, Crespi13anderson}. Several analyses show that a suitable combination of quantum coherence and environmental noise can provide an effective enhancement in quantum transport \cite{Rebentrost09, Caruso09, Caruso11, Viciani15, Novo16, Biggerstaff16, Harris17}. In this context, reconfigurable photonic circuits can provide a useful platform to characterize it, allowing to study the continuous transitions between different regimes \cite{Harris17}. Finally, we mention that lattices subject to external gradient forces exhibit a characteristic periodic oscillation between spreading and localization of the wave packet, the so-called \textit{Bloch oscillations}, which have been observed experimentally in photonic platforms \cite{Morandotti99, Pertsch99, Dreisow09, Regensburger11, Lebugle15}.
\paragraph{Multi-photon quantum walks.}
So far we have reviewed studies that focus on single-particle evolutions in discrete- or continuous-time QWs. However, the full landscape of phenomena gets enriched if we consider photonic states with $n=2$ photons \cite{Crespi13anderson, Sansoni12, Schreiber10, Cardano15, Owens11, Spagnolo13, Matthews13, Solntsev14, Peruzzo10, Poulios14, Weimann16, Carolan14, Crespi15fano, Lebugle15}, thanks to the emergence of more complex interference patterns \cite{Weimann16, Carolan14}, highlighting for instance two-photon quantum correlations \cite{Peruzzo10, Poulios14} with bosonic-fermionic transitions \cite{Sansoni12, Matthews13}, and enabling the possibility of simulating relevant physical processes \cite{Crespi15fano, Lebugle15}.
Finally, the possibility of further increasing the number of injected photons to $n=3$ \cite{Metcalf13, Spagnolo13} or $n>3$ photons is believed to disclose even larger potentialities, which we briefly discuss in the next section with the \textsc{BosonSampling} problem.
\subsection{\textsc{BosonSampling}}
\label{sec:BosonSampling}
In the previous section we reviewed the recent achievements in photonic quantum walks, which gained increasing attention after the discovery that multi-particle quantum walks with interactions allow for universal quantum computation \cite{KLM, Childs13}. A milestone result in the intertwined developments of quantum walks and quantum computation is represented by the proof, by Aaronson and Arkhipov \cite{AA} in 2010, that $n$-photon states evolving in a discrete-time quantum walk can provide a first experimental evidence of a superior quantum computational power \cite{Gard15, Harrow17}.
\textsc{BosonSampling} consists in sampling from the output distribution of an interferometer implementing a Haar-random $m$-mode transformation $U$ on $n$ indistinguishable bosons. The computational complexity of the problem is rooted in the hardness \cite{Valiant79} of evaluating the permanent in the scattering amplitudes \cite{Scheel08}
\begin{equation}
\bra {t_1...t_m} \hat U \ket{s_1...s_m} = \left( \prod_{i=1}^m s_i! \, t_i! \right)^{-\frac{1}{2}} \textup{Per\,} (U_{S,T})
\end{equation}
\noindent where the integer $s_k$ ($t_k$) is the occupation number for the input (output) mode $k$ ($\sum s_k=\sum t_k=n$) and $U_{S,T}$ is the $n \times n$ matrix obtained by repeating $s_k$ ($t_k$) times the $k$th column (row) of $U$. Assuming two highly plausible conjectures \cite{AA}, Aaronson and Arkhipov showed that, should a classical polynomial-time algorithm exist capable to solve \textsc{BosonSampling}, it would imply the collapse of the polynomial hierarchy of complexity classes to the third level, a possibility of huge consequences widely believed to be unlikely.
Despite its simple formulation, requiring only single photons with no entanglement, no adaptive measurements and no ancillary qubits, the necessity to scale up the number of photons and modes represents a technological challenge. For an overview of the state of the art in the development of single-photon sources, integrated photonic interferometers and single-photon detectors, the reader can refer to Sections \ref{sec:sources}, \ref{sec:circuits} and \ref{sec:detectors}.
State-of-the-art classical simulation of \textsc{BosonSampling} depends on the number of photons as in Table \ref{table:BShardness}.
\begin{table}[h!]
\renewcommand*{\arraystretch}{1.25}
\centering
\caption{\label{table:BShardness}
The largest classical simulation of \textsc{BosonSampling} (up to 30 photons) was reported in 2017 based on Metropolised independence sampling (MIS) \cite{Neville17}. The computational complexity for exact classical \textsc{BosonSampling} was given in the same year in Ref. \cite{Clifford18} by Clifford and Clifford (CC): state-of-the-art sampling time is equal, within a factor of 2, to computing the permanent of one single scattering matrix \cite{Wu16}.
}
\footnotesize
\begin{center}
\begin{tabular}{cccc}
\mr
Approach & $n\sim$ & Hardware & Classical technique \\
\mr
Simulated & 30 & Laptop & MIS \cite{Neville17} \\ \hline
\multirow{ 2}{*}{Exact} & 10 & Laptop & Brute force \\
& 50 & Tianhe-2 \cite{Wu16} & CC \cite{Clifford18} \\
\mr
\end{tabular}
\end{center}
\end{table}
Soon after its introduction, in 2013 four experimental demonstrations with $n=3$ photons were reported \cite{Broome13, Crespi13, Spring13, Tillmann13}. Since then, several investigations have been performed to study the scalability in imperfect conditions, such as in the presence of losses \cite{Aaronson15, Motes15, Garcia-Patron17, Oszmaniec18}, partial distinguishability \cite{Shchesnovich15, Tillmann15} and generic experimental errors \cite{Leverrier15, Motes15, Rohde12, Shchesnovich14}. Furthermore, the scalability of parametric down-conversion sources and the relevance of non-ideal detectors' efficiencies have been addressed in Ref.\cite{Motes13} and investigated experimentally in 2015 with the \textit{Scattershot} implementation \cite{Lund14, Bentivegna15}.
\begin{table*}[h!]
\renewcommand*{\arraystretch}{1.15}
\centering
\footnotesize
\caption{\label{BStable} Experimental demonstrations of photonic \textsc{BosonSampling}.}
\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}cccccc}
\br
Year & Ref. & Technology & Features & Photons & Modes &Validation\\
\mr
2013 & \cite{Broome13} & in-fiber & & 2, 3 & 6 & \\
& \cite{Crespi13} & FLW & & 3 & 5 & \\
& \cite{Spring13} & SoS & & 3 & 6 & \\
& \cite{Tillmann13} & Si & & 3 & 5 & \\
2014 & \cite{Spagnolo14} & FLW & First validation & 3 & 5, 7, 9, 13 & U, D \\
& \cite{Carolan14} & $\textup{SiO}_x\textup{N}_y$ & Bosonic clouding & 3 & 9 & U, D \\
2015 & \cite{Carolan15} & SoS & Full reconfigurability & 3 & 6 & D, MF \\
& \cite{Tillmann15} & FLW & Full interference spectrum & 3 & 5 & \\
& \cite{Bentivegna15} & FLW & Scattershot \textsc{BosonSampling} & 3 & 9, 13 & U, D \\
2016 & \cite{Crespi16} & FLW & 3D Fourier interferometer & 2 & 4, 8 & D, MF \\
2017 & \cite{Loredo17} & in-fiber & High purity and brightness & 2, 3 & 6 & U, D \\
& \cite{He17} & Bulk (time-bin) & High purity and brightness & 3, 4 & 6, 8 & U, D, MF \\
& \cite{Wang17bs} & Fused-quartz & High purity and brightness & 3, 4, 5 & 9 & U, D \\
\br
\end{tabular*}\\
U, D, MF: 'Uniform', 'Distinguishable' and 'Mean-Field' samplers.\\
SoS: silica on silicon. FLW: femtosecond laser written. \\
Some demonstations have reduced computational complexity due to bunched input states $\ket{2, 2}$ \cite{Spring13} and $\ket{3, 3}$ \cite{Carolan15}, in the case of standard \textsc{BosonSampling} \cite{AA}, or due to partial coverage of the full Hilbert space in the Scattershot version \cite{Bentivegna15}.
\end{table*}
\normalsize
Since 2015, a number of alternative schemes have been proposed: driven \textsc{BosonSampling}, where the input photons are generated within the interferometer \cite{Barkhofen17}, \textsc{BosonSampling} with microwave photons \cite{Peropadre16} or squeezed states \cite{Hamilton17, Kruse18}, one using Gaussian measurements and the symmetry of the evolution under time reversal \cite{Chakhmakhchyan17, Chabaud17}, and one time-bin loop-based \cite{Motes14timebin}. Furthermore, somehow in analogy with the link between quantum walks and quantum computation, recently \textsc{BosonSampling} was shown to be equivalent to short-time evolutions of $n$ excitations in a XY model of 2$n$ spins \cite{Peropadre16}. This feature has suggested multiport photonic interferometers as good candidates for the implementation of quantum simulators or even general-purpose quantum computers \cite{Peropadre16, GonzalezAlonso16bs, Huh15}. Nevertheless, in 2017 new classical algorithms have been proposed to solve \textsc{BosonSampling} for systems with dimensionality larger than that allowed by near-term technological advances \cite{Neville17, Clifford18, Renema17, Garcia-Patron17, Chakhmakhchyan17alg, Oszmaniec18}, thus increasing the challenge to achieve quantum supremacy, i.e. the regime with quantum advantage \cite{Gard15, Harrow17}. Experimental state of the art for photonic \textsc{BosonSampling} is shown in Table \ref{BStable} and in Fig. \ref{fig:FigureBosonSampling}.
\begin{figure}[t!]
\centering
\includegraphics[trim={0 0 0 0},clip, width=0.5\textwidth]{FigureBosonSampling.pdf}
\caption{\footnotesize Increase of the Hilbert-space dimensionality $\mathrm{dim} H$ in \textsc{BosonSampling} experimental implementations with single photons. Top inset: scheme of an integrated photonic device for \textsc{BosonSampling}, where a no-collision state is injected in the first optical modes and measured at the output after a Haar-random unitary evolution. Legend - PDC: parametric down-conversion, FLW: femtosecond laser writing. Recently the implementation of a lossy \textsc{BosonSampling} experiment has been reported \cite{Wang18}.\\Bottom legend: references to the related literature.\\ a: \cite{Broome13}. b: \cite{Spring13}. c: \cite{Tillmann13}. d: \cite{Crespi13}. e: \cite{Spagnolo14}. f: \cite{Carolan14}. g: \cite{Carolan15}. h: \cite{Tillmann15}. i: \cite{Bentivegna15}. j: \cite{Loredo17}. k: \cite{He17}. l: \cite{Wang17bs}}
\label{fig:FigureBosonSampling}
\end{figure}
\subsection{Verification of quantum simulation}
\label{sec:validation}
The problem of verification, fundamental for both classical and quantum computation, is closely related to the issue of security in information processing.
One good example is offered by \textsc{BosonSampling} \cite{AA}: a computational problem where a special-purpose quantum device is believed to outperform classical supercomputers. For similar problems that live outside the complexity class NP, it is in general not possible to verify the correctness of an outcome. Thus, in a scenario where untrusted parties can deviate calculations from their ideal flow, it is essential to develop techniques to ensure that robust computation is guaranteed to the maximum possible extent (fault-tolerance, i.e. amplification of the detection rate) or, equivalently, that malicious processes can be easily detected (e.g. using traps to detect errors) \cite{Broadbent09}. In this sense, the problem of verification in quantum computation is connected with the universal blind computation model described in Section \ref{BQC}. In the last few years, several protocols for verifiable quantum computations have been proposed \cite{Broadbent09, Reichardt13, Morimae14, Morimae14oneclean, Hajdusek15, Gheorghiu15, Fitzsimons15, Kapourniotis15, Kashefi15, Hayashi15, Morimae16verification, Gheorghiu17}, where the attention focused on schemes with either fewer experimental requirements or higher generality and security. We refer the reader to Ref.\cite{Fitzsimons17} for a comprehensive review on the problem of verification in blind quantum computing and to Refs. \cite{Kashefi15, Kapourniotis15} for a short overview of the differences between the various protocols.
We have shortly described in Section \ref{BQC} the first experimental demonstrations of verifiable blind quantum computing, performed using polarization qubits represented by multi-photon states generated via parametric down-conversion \cite{Barz12, Barz13, Fisher14, Greganti16}.
Beyond these works, the problem of verification has drawn large attention also in quantum communication and quantum simulation. In the scope of secure quantum communication, for instance, where quantum networks are employed to perform distributed quantum computations over possibly untrusted nodes, it is necessary to verify the goodness of the shared multipartite entangled states, the key resource behind these protocols \cite{Pappa12, Grafe14}.
To this aim, in 2014 Bell \textit{et al.} reported the experimental demonstration of a graph-state quantum secret sharing protocol \cite{Bell14}, whose implementation was subsequently verified using the scheme in Ref.\cite{Pappa12}. In 2016, further improvements led McCutcheon \textit{et al.} to a verification of multipartite entanglement for photonic quantum networks, adopting two single-photon sources similar to Ref.\cite{Bell14} to produce three- and four-photon polarization encoded states \cite{McCutcheon16}.
We conclude this section discussing the role of verification in quantum simulation and, in particular, in \textsc{BosonSampling}. In the latter case, as for analogous experiments involving multi-photon interference in linear-optical multiport interferometers, it is crucial to check the indistinguishability of the input photons to ensure the correctness of the overall operation. The \textit{validation} of \textsc{BosonSampling} developed rapidly in the last few years just after the first experimental demonstrations, and its applications will potentially go beyond its original purpose. Since a full certification of \textsc{BosonSampling} is believed to be not possible \cite{Aaronson14}, all current protocols aim at ruling out the most plausible unwanted scenarios, namely experiments where devices are injected with distinguishable input photons and with mean-field states \cite{Tichy14}. Currently there exist several protocols to validate \textsc{BosonSampling}, most of which have already been successfully demonstrated experimentally \cite{Spagnolo14, He17, Wang17, Crespi16, Carolan15, Viggianiello17sys, Viggianiello17tvd, Bentivegna14, Bentivegna15, Carolan14, Agresti17, Wang18, Giordani18}. An efficient certification protocol for photonic state preparation was also introduced theoretically \cite{Aolita15}, allowing to discriminate relevant classes of Fock states and Gaussian/non-Gaussian pure states. We summarize them in Table \ref{tableValidation}, with an overview of the techniques adopted and the hypotheses which are designed for.
\begin{table}[b!]
\renewcommand{\arraystretch}{1.2}
\footnotesize
\centering
\begin{center}
\begin{tabular*}{\linewidth}{c@{\extracolsep{\fill}}ccccc}
\br
Test & Ref. & U & D & MF & O \\
\mr
\ RNE & \cite{Aaronson14, Spagnolo14, He17, Wang18, Bentivegna15} & $\checkmark$ & & & \ \\
\ LR & \cite{Spagnolo14, Wang17bs, He17, Wang18, Bentivegna15} & $\checkmark$ & $\checkmark$ & & $\checkmark$ \ \\
\ ZTL & \cite{Tichy14, Dittel17, Dittel18, Crespi15, Crespi16, Carolan15, Viggianiello17sys, Viggianiello17tvd} & & $\checkmark$ & $\checkmark$ & \ \\
\ Bayesian & \cite{Bentivegna14, Wang17bs, He17, Viggianiello17tvd}& $\checkmark$ & $\checkmark$ & & $\checkmark$ \ \\
\ Bunching & \cite{Carolan14, Shchesnovich16} & & $\checkmark$ & & \ \\
\ CG & \cite{Wang16bubbles, Agresti17} & $\checkmark$ & $\checkmark$ & & $\checkmark$ \ \\
\ $n=m$ & \cite{Liu16} & & $\checkmark$ & $\checkmark$ & \ \\
\ Statistical & \cite{Walschaers16, Giordani18} & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ \ \\
\mr
\end{tabular*}
\end{center}
\caption{Validation of multi-photon interference against uniform distribution (U), experiments with distinguishable photons (D), mean-field states (MF) or other hypotheses (O). See Table \ref{BStable} for a cross reference with \textsc{BosonSampling} experiments.\\
RNE: row-norm estimator; LR: likelihood ratio; ZTL: zero-transmission law (suppression law); CG: Coarse-graining. }
\label{tableValidation}
\end{table}
\subsection{Photonic simulation in quantum chemistry and condensed matter}
\label{qChemistry}
Computational chemistry employs simulations to study properties of molecules or to predict unknown chemical phenomena. Indeed, analytical solutions for quantum many-body problems are available only for the simplest systems, thus making simulations necessary and heavily based on classical techniques such as the Born-Oppenheimer or Hartree-Fock approximations, the density-functional theory or even machine learning \cite{Olson17}. Notwithstanding, when modelling the quantum nature of highly correlated many-body systems, it became clear that the only solution to counteract the exponential increase in computational resources, the so-called \textit{curse of dimensionality}, was to exploit other quantum systems for information processing \cite{Feynman82}. The belief in this quantum approach was indeed supported in the 90's by the first efficient quantum algorithms \cite{Nielsen_Chuang}, which were rapidly followed by breakthrough discoveries applicable to quantum chemistry \cite{Lloyd96, Abrams99}. The algorithm developed in 2005 by Aspuru-Guzik \textit{et al.} \cite{AspuruGuzik05} offered an exponential speed-up in computational resources, scaling linearly in the number of qubits and polynomially in the number of gates. Today there exist numerous algorithms for simulating quantum chemistry, for which a complete review goes beyond the scope of this section. The reader interestered in a detailed and comprehensive excursus of the state of the art may find Ref. \cite{Olson17} a useful resource, while Refs. \cite{AspuruGuzik12, Noh16} may be suitable for a more experiments-oriented overview. In the following, we will briefly describe the latest achievements of quantum chemistry simulations on photonic platforms.
Photonic technologies provide an effective platform for quantum simulation, thanks to the single photons' low decoherence, speed and controllability that we discussed in the previous chapters. First experimental demonstrations in bulk optics were reported in 2009 for the simulation of anyons, fractional-statistics particles responsible for the fractional quantum Hall effect \cite{Lu09, Pachos09} and for the calculation of the energy spectrum of a hydrogen molecule \cite{Lanyon10}. The approach adopted in the latter work consisted in encoding the state of the molecule on single-photon polarization qubits, simulating its evolution in the Born-Oppenheimer approximation and estimating the energy $E$ of its eigenstates $\ket{\psi}$ using the quantum phase estimation (QPE) algorithm \cite{Abrams99}, since $ e^{i E t / \hbar} \ket{\psi} = e^{i 2 \pi \phi} \ket{\psi} $. In this case, with an appropriate choice of the basis, the Hamiltonian matrix reduces to two 2$\times$2 blocks (plus two 1$\times$1 blocks), allowing a map between subspaces and single qubit states. Since the operators act on single qubits, it was possible to perform QPE separately on the polarization qubit of the quantum register, accessing the outcome via the control entangled photon \cite{Lanyon10}. In 2010, Kaltenbaek \textit{et al.} generated a photonic valence-bond-solid state, the gapped ground state of a two-body Hamiltonian on a spin-1 chain, as a useful resource for the implementation of single-qubit quantum logic gates \cite{Kaltenbaek10}. In 2011, Ma \textit{et al.} used two polarization entangled photon pairs to study the process of frustration in a tetramer, a system with four spin-1/2 particles, observing the transition between localized and resonating-valence-bond states at varying Heisenberg interaction strength \cite{Ma11, Ma14}. The analog simulation required measuring the output of tunable beam splitter injected with single-photon states from each pair. All states have been characterized by retrieving the total energy and the pairwise quantum correlations, whose values are conditioned by the quantum monogamy. In 2012, Kitagawa \textit{et al.} employed a discrete-time bulk quantum walk to demonstrate the topological protection of bound states in both static and dynamic scenarios, a useful tool that finds application for instance in quantum computation. Topological transitions have been also the focus of a recent study carried out with orbital angular momentum encoding \cite{Cardano16}.
Recently the development of integrated photonic circuits for quantum walks (QW) has prompted the realization of various quantum simulations (see Section \ref{sec:QW}). Among discrete-time QWs, for instance, we have discussed the experimental investigation on the transition between fermionic and bosonic states \cite{Sansoni12}, enabled by polarization-encoded symmetric (bosonic) or anti-symmetric (fermionic) wave functions. Very recently, a new quantum approach based on variational methods and phase estimation was introduced \cite{Santagati18}, and experimentally tested, to approximate eigenvalues for ground and excited states. Further results were reported also with continuous-time QWs, with the observation of two-photon quantum correlations \cite{Peruzzo10, Lahini12, Matthews13, Poulios14}, quantum Rabi model \cite{Crespi12}, floquet topological insulators \cite{Rechtsman13}, growth of entanglement in spin chains \cite{Pitsios17}, Majorana dynamics \cite{Keil15}, Fano interference \cite{Crespi15fano} and Bloch oscillations \cite{Lebugle15}.
We conclude this section mentioning a very recent result in quantum simulation, which connects it to that of \textsc{BosonSampling} (see Section \ref{sec:BosonSampling}). On one hand, in fact, quantum simulations were conceived to surpass the capabilities of classical computers. On the other hand, Aaronson and Arkhipov have shown \cite{AA} that \textsc{BosonSampling}, i.e. the simulation of many-boson statistics, is indeed capable to provide a concrete evidence of this computational advantage. Within this framework, in 2015 Huh \textit{et al.} found the first application of \textsc{BosonSampling} for quantum simulation to evaluate molecular vibronic spectra \cite{Huh15, Peropadre16, GonzalezAlonso16bs}.
\section*{Discussion}
Photonic quantum technologies provide a promising platform for researches and applications in several contexts. This review attempted to gather their most recent advances, to provide the reader with a unified framework for the various ingredients. General aspects addressed in this manuscript are the generation, manipulation and detection of single-photon states from the technological perspective, as well as the fundamental theoretical tools developed for quantum communication and quantum simulation. The large effort devoted to these technologies is indeed testified also by the several achievements reported during the completion of this review, which make it challenging to stay up-to-date in this rapidly evolving field.
\section*{Acknowledgments}
This work was supported by the ERC-Starting Grant 3D-QUEST (3D-Quantum Integrated Optical Simulation; grant agreement no.307783; http://www.3dquest.eu) and by the H2020-FETPROACT-2014 Grant QUCHIP (Quantum Simulation on a Photonic Chip; grant agreement no.641039; http://www.quchip.eu).
\section*{References}
\providecommand{\newblock}{}
|
{
"timestamp": "2018-03-26T02:09:24",
"yymm": "1803",
"arxiv_id": "1803.02790",
"language": "en",
"url": "https://arxiv.org/abs/1803.02790"
}
|
\section{Introduction}
As a generalization of Sasakian-space-form, Alegre et al. \cite{ALEGRE1} introduced the notion of generalized
Sasakian-space-form as that an almost contact metric manifold
$\bar{M}(\phi,\xi,\eta,g)$
whose curvature tensor $\bar{R}$ of $\bar{M}$ satisfies
\begin{align}
\label{eqn1.3}
\bar{R}(X,Y)Z &=f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber& - g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\big\}+f_3\big\{\eta(X)\eta(Z)Y\\
\nonumber& - \eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}
\end{align}
for all vector fields $X$, $Y$, $Z$ on $\bar{M}$ and $f_1,f_2,f_3$ are certain smooth functions on
$\bar{M}$. Such a manifold of dimension
$(2n+1)$, $n>1$ (the condition $n>1$ is assumed throughout the paper),
is denoted by $\bar{M}^{2n+1}(f_1,f_2,f_3)$ \cite{ALEGRE1}. Many authors studied this space form with different aspects.
For this, we may refer (\cite{HUI1}, \cite{HUI2}, \cite{HUI3}, \cite{HUI4}, \cite{HUI5}, \cite{HUI6}, \cite{KISH} and \cite{HUI8}).
It reduces to Sasakian-space-form if $f_1 = \frac{c+3}{4}$, $f_2 = f_3 = \frac{c-1}{4}$ \cite{ALEGRE1}.
After introducing the semisymmetric linear connection by Friedman and Schouten \cite{FRID},
Hayden \cite{HAYD} gave the idea of metric connection with torsion on a Riemannian manifold.
Later, Yano \cite{YANO} and many others (see, \cite{HUI7}, \cite{SHAIKH1}, \cite{SULAR} and references therein)
studied semisymmetric metric connection in different context. The idea of semisymmetric non-metric
connection was introduced by Agashe and Chafle \cite{AGAS}.
The Schouten-van Kampen connection introduced
for the study of non-holomorphic manifolds (\cite{SCHO}, \cite{VRAN}). In $2006$, Bejancu \cite{BEJA3}
studied Schouten-van Kampen connection on foliated manifolds. Recently Olszak \cite{OLSZ}
studied Schouten-van Kampen connection on almost(para) contact metric structure.
The Tanaka-Webster connection (\cite{TANA}, \cite{WEBS}) is the canonical affine connection defined on a non-degenerate
pseudo-Hermitian CR-manifold. Tanno \cite{TANN} defined the Tanaka-Webster connection for contact metric manifolds.
The submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ are studied in (\cite{ALEGRE3}, \cite{PM1}, \cite{PM2}).
In \cite{ALEGRE3}, Alegre and Carriazo studied submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect
to Levi-Civita connection $\bar{\nabla}$. The present paper deals with study of such
submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to semisymmetric metric connection,
semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-webster connection respectively.
\section{preliminaries}
In an almost contact metric manifold $\bar{M}(\phi,\xi,\eta,g)$, we have \cite{BLAIR}
\begin{align}
\label{eqn2.1}
\phi^2(X) = -X+\eta(X)\xi,\ \phi \xi=0,
\end{align}
\begin{align}
\label{eqn2.2}
\eta(\xi) = 1,\ g(X,\xi) = \eta(X),\ \eta(\phi X) = 0,
\end{align}
\begin{align}
\label{eqn2.3}
g(\phi X,\phi Y) = g(X,Y)-\eta(X)\eta(Y),
\end{align}
\begin{align}
\label{eqn2.4} g(\phi X,Y) = -g(X,\phi Y).
\end{align}
In
$\bar{M}^{2n+1}(f_1,f_2,f_3)$, we have \cite{ALEGRE1}
\begin{align}
\label{eqn2.6f}
(\bar{\nabla}_X\phi)(Y) = (f_1-f_3)[g(X,Y)\xi - \eta(Y)X],
\end{align}
\begin{align}
\label{eqn2.7g}
\bar{\nabla}_X\xi = -(f_1-f_3) \phi X,
\end{align}
where $\bar{\nabla}$ is the Levi-Civita connection of $\bar{M}^{2n+1}(f_1,f_2,f_3)$.
\indent Let $M$ be a submanifold of
$\bar{M}^{2n+1}(f_1,f_2,f_3)$. If $\nabla$ and $\nabla^\perp$ are the induced
connections on the tangent bundle $TM$ and the normal bundle $T^\perp{M}$ of $M$, respectively
then the Gauss and Weingarten formulae are given by \cite{YANOKON1}
\begin{align}
\label{eqn2.5}
\bar{\nabla}_XY = \nabla_XY +h(X,Y),\ \bar{\nabla}_XV = -A_VX + \nabla_X^{\perp}V
\end{align}
for all $X,Y\in\Gamma(TM)$ and $V\in\Gamma(T^{\perp}M)$, where $h$ and $A_V$ are second fundamental form and shape operator (corresponding to the normal vector field V), respectively and they are related by \cite{YANOKON1} $ g(h(X,Y),V) = g(A_VX,Y)$.
For any $X\in\Gamma(TM)$, we may write
\begin{equation}
\label{eqn2.17f} \phi X=TX+FX,
\end{equation}
where $TX$ is the tangential component and $FX$ is the
normal component of $\phi X$.
In particular, if $F=0$ then $M$ is invariant \cite{BEJA} and here $\phi (TM)\subset TM$.
Also if $T=0$ then $M$ is anti-invariant \cite{BEJA} and here $\phi (TM)\subset T^\bot M$.
Also here we assume that $\xi$ is tangent to $M$.
The semisymmetric metric connection $\widetilde{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$
on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by \cite{YANO}
\begin{align}
\label{eqn2.41d}
\widetilde{\bar{\nabla}}_{X}Y= \bar{\nabla}_X Y+\eta(Y)X-g(X,Y)\xi.
\end{align}
The Riemannian curvature tensor $\widetilde{\bar{R}}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\widetilde{\bar{\nabla}}$ is
\begin{eqnarray}
\label{eqn1.3s}
&&\widetilde{\bar{R}}(X,Y)Z \\
\nonumber&&=(f_1-1)\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y- g(Y,\phi Z)\phi X\\
\nonumber&& + 2g(X,\phi Y)\phi Z\big\}+(f_3-1)\big\{\eta(X)\eta(Z)Y - \eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi\\
\nonumber&& - g(Y,Z)\eta(X)\xi\big\}+(f_1-f_3)\{g( X, \phi Z)Y-g( Y,\phi Z)X+g(Y,Z)\phi X\\
\nonumber&&-g(X,Z)\phi Y\}.
\end{eqnarray}
The semisymmetric non-metric connection ${\bar{\nabla}}^{'}$ and the Riemannian connection $\bar{\nabla}$
on ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by \cite{AGAS}
\begin{align}
\label{eqn2.41h}
\bar{\nabla}^{'}_{X}Y= \bar{\nabla}_X Y+\eta(Y)X.
\end{align}
The Riemannian curvature tensor ${\bar{R}}^{'}$ of $\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to ${\bar{\nabla}}^{'}$ is
\begin{eqnarray}
\label{eqn1.3t}
{\bar{R}}^{'}(X,Y)Z &=&f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber&-&g(Y,\phi Z)\phi X + 2g(X,\phi Y)\phi Z\big\}+f_3\big\{\eta(X)\eta(Z)Y\\
\nonumber& -&\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\\
\nonumber&+&(f_1-f_3)[g(X,\phi Z)Y-g( Y,\phi Z) X]\\
\nonumber&+&\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y.
\end{eqnarray}
The Schouten-van Kampen connection $\hat{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$
of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by \cite{OLSZ}
\begin{align}
\label{eqn2.41k}
\hat{\bar{\nabla}}_{X}Y=\bar{\nabla}_X Y+(f_1-f_3)\eta(Y)\phi X-(f_1-f_3)g(\phi X,Y)\xi.
\end{align}
The Riemannian curvature tensor $\hat{\bar{R}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\hat{\bar{\nabla}}$ is
\begin{eqnarray}
\label{eqn1.3v}
&&\hat{\bar{R}}(X,Y)Z \\
\nonumber&&=f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y\\
\nonumber&&- g(Y,\phi Z)\phi X+ 2g(X,\phi Y)\phi Z\big\}+\{f_3+(f_1-f_3)^2\}\big\{\eta(X)\eta(Z)Y \\
\nonumber&&- \eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\\
\nonumber&&+(f_1-f_3)^2\big[g(X,\phi Z)\phi Y-g(Y, \phi Z)\phi X\big].
\end{eqnarray}
The Tanaka-Webster connection $\stackrel{\ast}{\bar{\nabla}}$ and the Riemannian connection $\bar{\nabla}$
of ${\bar{M}}^{2n+1}(f_{1},f_{2},f_{3})$ are related by \cite{CHO}
\begin{align}
\label{eqn2.41e}
\stackrel{\ast}{\bar{\nabla}}_{X}Y= \bar{\nabla}_X Y+\eta(X)\phi Y+(f_1-f_3)\eta(Y)\phi X-(f_1-f_3)g(\phi X,Y)\xi.
\end{align}
The Riemannian curvature tensor $\stackrel{*}{\bar{R}}$ of
$\bar{M}^{2n+1}(f_{1},f_{2},f_{3})$ with respect to $\stackrel{*}{\bar{\nabla}}$ is
\begin{eqnarray}
\label{eqn1.3u}
&&\stackrel{*}{\bar{R}}(X,Y)Z \\
\nonumber&&=f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_2\big\{g(X,\phi Z)\phi Y- g(Y,\phi Z)\phi X\\
\nonumber&&+ 2g(X,\phi Y)\phi Z\big\}+\{f_3+{(f_1-f_3)^2}\}\big\{\eta(X)\eta(Z)Y - \eta(Y)\eta(Z)X\\
\nonumber&&+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}+{(f_1-f_3)^2}\big[g(X,\phi Z)\phi Y\\
\nonumber&&-g(Y,\phi Z)\phi X\big]+2(f_1-f_3)g(X,\phi Y)\phi Z.
\end{eqnarray}
\section{Submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with $\widetilde{\bar{\nabla}}$}
\begin{lemma}
If $M$ is invariant submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\widetilde{\bar{\nabla}}$, then
$\widetilde{\bar{R}}(X,Y)Z$ is tangent to $M$, for any $X,Y,Z\in \Gamma(TM)$.
\end{lemma}
\begin{proof}
If $M$ is invariant then from (\ref{eqn1.3s}) we say that $\widetilde{\bar{R}}(X,Y)Z$ is
tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$. This proves the lemma.
\end{proof}
\begin{lemma}
If $M$ is anti-invariant submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\widetilde{\bar{\nabla}}$, then
\begin{eqnarray}
\label{eqn7.3}
&&tan(\widetilde{\bar{R}}(X,Y)Z) \\
\nonumber&&=(f_1-1)\big\{g(Y,Z)X-g(X,Z)Y\big\}+(f_3-1)\big\{\eta(X)\eta(Z)Y\\
\nonumber&&-\eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi-g(Y,Z)\eta(X)\xi\big\},
\end{eqnarray}
\begin{eqnarray}
\label{eqn7.4}
nor(\widetilde{\bar{R}}(X,Y)Z)&=&(f_1-f_3)\{g(Y,Z)\phi X-g(X,Z)\phi Y\}
\end{eqnarray}
for any $X,Y,Z\in \Gamma(TM)$.
\end{lemma}
\begin{proof}
Since $M$ is anti-invariant, we have $\phi X,\phi Y\in \Gamma(T^\bot M)$.
Then equating tangent and normal component of (\ref{eqn1.3s}) we get the result.
\end{proof}
\begin{lemma}
If $f_1(p)=f_3(p)$ and $M$ is either invariant or anti-invariant submanifold
of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\widetilde{\bar{\nabla}}$, then
$\widetilde{\bar{R}}(X,Y)Z$ is tangent to $M$ for any $X,Y,Z\in \Gamma(TM)$.
\end{lemma}
\begin{proof}
Using Lemma $3.1$ and Lemma $3.2$ we get the result.
\end{proof}
\begin{lemma}
If $M$ is invariant or anti-invariant submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\widetilde{\bar{\nabla}}$, then
$\widetilde{\bar{R}}(X,Y)V$ is normal to $M$, for any $X,Y,\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
\end{lemma}
\begin{proof}
If $M$ is invariant from (\ref{eqn1.3s}) we have $\widetilde{\bar{R}}(X,Y)V$ normal to $M$, and if $M$ is anti-invariant then $\widetilde{\bar{R}}(X,Y)V=0$ i.e. $\widetilde{\bar{R}}(X,Y)V$ normal to $M$ for any $X,Y,\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
\end{proof}
\begin{lemma}
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_2(p)\neq0$, $f_1(p)=f_3(p)$ and $TM$ is invariant
under the action of $\widetilde{\bar{R}}(X,Y)$, $X,Y\in \Gamma(TM)$, then $M$ is either invariant or anti-invariant.
\end{lemma}
\begin{proof}
For $X,Y\in \Gamma(TM)$, we have from (\ref{eqn1.3s}) that
\begin{eqnarray}
\label{eqn3.1}
\widetilde{\bar{R}}(X,Y)X&=&(f_1-1)\big\{g(Y,X)X-g(X,X)Y\big\}+f_2\big\{g(X,\phi X)\phi Y\\
\nonumber&-&g(Y,\phi X)\phi X + 2g(X,\phi Y)\phi X\big\}+(f_3-1)\big\{\eta(X)\eta(X)Y \\
\nonumber&-&\eta(Y)\eta(X)X+g(X,X)\eta(Y)\xi - g(Y,X)\eta(X)\xi\big\}\\
\nonumber&+&(f_1-f_3)\{g(\phi Y,X)X-g(\phi X,X)Y+g(Y,X)\phi X\\
\nonumber&-&g(X,X)\phi Y\}.
\end{eqnarray}
Note that $\widetilde{\bar{R}}(X,Y)X$ should be tangent if
$[-3f_2g(Y,\phi X)\phi X+(f_1-f_3)\{g(Y,X)\phi X-g(X,X)\phi Y\}]$ is tangent.
Since $f_2(p)\neq0$, $f_1(p)=f_3(p)$ at any point $p$ then by similar way of proof of Lemma $3.2$ of \cite{ALEGRE3}, we can prove that either $M$ is invariant or anti-invariant. This proves the Lemma.
\end{proof}
\begin{remark}
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with
respect to $\widetilde{\bar{\nabla}}$. If $f_1(p)\neq f_3(p)$ and $TM$ is invariant
under the action of $\widetilde{\bar{R}}(X,Y)$, $X,Y\in \Gamma(TM)$, then $M$ is invariant.
\end{remark}
From Lemma $3.3$ and Lemma $3.5$, we have
\begin{theorem}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect
to $\widetilde{\bar{\nabla}}$. If $f_2(p)\neq0$, $f_1(p)=f_3(p)$ then $M$ is either invariant or anti-invariant if and only if $TM$ is
invariant under the action of $\widetilde{\bar{R}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.
\end{theorem}
\begin{proposition}
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to
$\widetilde{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant under
the action of $\widetilde{\bar{R}}(U,V)$ for any $U,V\in \Gamma(T^\bot M)$.
\end{proposition}
\begin{proof}
Replacing $X,Y,Z$ by $U,V,X$ in (\ref{eqn1.3s}), we get
\begin{eqnarray}
\label{eqn3.2}
\widetilde{\bar{R}}(U,V)X &=&(f_1-1)\big\{g(V,X)U-g(U,X)V\big\}+f_2\big\{g(U,\phi X)\phi V\\
\nonumber&-& g(V,\phi X)\phi U + 2g(U,\phi V)\phi X\big\}+(f_3-1)\big\{\eta(U)\eta(X)V\\
\nonumber&-& \eta(V)\eta(X)U+g(U,X)\eta(V)\xi - g(V,X)\eta(U)\xi\big\}\\
\nonumber&+&(f_1-f_3)\{g(\phi V,X)U-g(\phi U,X)V+g(V,X)\phi U \\
\nonumber&-&g(U,X)\phi V\}.
\end{eqnarray}
As $M$ is invariant, $U,V\in \Gamma(T^\bot M)$, we have
\begin{equation}\label{eqn3.3}
g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0
\end{equation}
for any $X\in \Gamma(TM)$. Using (\ref{eqn3.3}) in (\ref{eqn3.2}), we have
\begin{equation}\label{eqn3.4}
\widetilde{\bar{R}}(U,V)X=2f_2g(U,\phi V)\phi X,
\end{equation}
which is tangent as $\phi X$ is tangent.
This proves the proposition.
\end{proof}
\begin{proposition}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect
to $\widetilde{\bar{\nabla}}$. If $f_2(p)\neq0$, $f_1(p)=f_3(p)$ for each $p\in M$ and $T^\bot M$ is invariant under the action
of $\widetilde{\bar{R}}(U,V)$, $U,V\in\Gamma(T^\bot M)$, then $M$ is either invariant or anti-invariant.
\end{proposition}
\begin{proof}
The proof is similar as it is an Lemma $3.4$, just assuming that $\widetilde{\bar{R}}(U,V)U$ is normal for any $U,V\in \Gamma(T^\bot M)$.
\end{proof}
\section{Submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with ${\bar{\nabla}}^{'}$}
\begin{lemma}
If $M$ is either invariant or anti-invarint submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with
respect to ${\bar{\nabla}}^{'}$, then
${\bar{R}}^{'}(X,Y)Z$ is tangent to $M$ and ${\bar{R}}^{'}(X,Y)V$ normal to $M$ for any $X,Y,Z\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
\end{lemma}
\begin{proof}
If $M$ is invariant then from (\ref{eqn1.3t}) we say that ${\bar{R}}^{'}(X,Y)Z$ is
tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$. \\
If $M$ is anti-invariant then
\begin{equation}\label{eqn4.1}
g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.
\end{equation}
From (\ref{eqn1.3t}) and (\ref{eqn4.1}) we have
\begin{eqnarray}
\label{eqn4.2}
{\bar{R}}^{'}(X,Y)Z &=&f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}+f_3\big\{\eta(X)\eta(Z)Y \\
\nonumber&-& \eta(Y)\eta(Z)X+g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}\\
\nonumber&+&[\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y],
\end{eqnarray}
which is tangent. \\
If $M$ is invariant then from (\ref{eqn1.3t}), it follows that ${\bar{R}}^{'}(X,Y)V$ is normal to $M$, and if $M$ is anti-invariant then ${\bar{R}}^{'}(X,Y)V=0$ i.e. ${\bar{R}}^{'}(X,Y)V$ is normal to $M$ for any $X,Y\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
This proves the Lemma.
\end{proof}
\begin{lemma}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to ${\bar{\nabla}}^{'}$. If $f_2(p)\neq 0$ for each $p\in M$ and $TM$ is invariant
under the action of $\bar{R}^{'}(X,Y)$, $X,Y\in \Gamma(TM)$,
then $M$ is either invariant or anti-invariant.
\end{lemma}
\begin{proof}
For $X,Y\in \Gamma(TM)$, we have from (\ref{eqn1.3t}) that
\begin{eqnarray}
\label{eqn4.3}
{\bar{R}}^{'}(X,Y)X &=&f_1\big\{g(Y,X)X-g(X,X)Y\big\}+f_2\big\{g(X,\phi X)\phi Y\\
\nonumber&-&g(Y,\phi X)\phi X + 2g(X,\phi Y)\phi X\big\}+f_3\big\{\eta(X)\eta(X)Y\\
\nonumber&-&\eta(Y)\eta(X)X+g(X,X)\eta(Y)\xi - g(Y,X)\eta(X)\xi\big\}\\
\nonumber&-&(f_1-f_3)g(\phi X, Y) X+\{\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y\}.
\end{eqnarray}
Note that ${\bar{R}}^{'}(X,Y)X$ should be tangent if $3f_2(p)g(Y,\phi X)\phi X$ is tangent.
Since $f_2(p)\neq0$ for each $p\in M$, as similar as proof of Lemma $3.2$ of \cite{ALEGRE3},
we may conclude that either $M$ is invariant or anti-invariant. This proves the Lemma.
\end{proof}
From Lemma $4.1$ and Lemma $4.2$, we have
\begin{theorem}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect
to ${\bar{\nabla}}^{'}$. If $f_2(p)\neq0$ for each $p\in M$, then $M$ is
either invariant or anti-invariant if and only if $TM$ is invariant under the
action of ${\bar{R}}^{'}(X,Y)$ for all $X,Y\in\Gamma(TM)$.
\end{theorem}
\begin{proposition}
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to
${\bar{\nabla}}^{'}$. If $M$ is invariant, then $TM$ is invariant
under the action of ${\bar{R}}^{'}(U,V)$ for any $U,V\in \Gamma(T^\bot M)$.
\end{proposition}
\begin{proof}
Replacing $X,Y,Z$ by $U,V,X$ in (\ref{eqn1.3t}), we get
\begin{eqnarray}
\label{eqn4.4}
{\bar{R}}^{'}(U,V)X &=&f_1\big\{g(V,X)U-g(U,X)V\big\} +f_2\big\{g(U,\phi X)\phi V\\
\nonumber&-& g(V,\phi X)\phi U + 2g(U,\phi V)\phi X\big\}+f_3\big\{\eta(U)\eta(X)V\\
\nonumber&-& \eta(V)\eta(X)U+g(U,X)\eta(V)\xi - g(V,X)\eta(U)\xi\big\}\\
\nonumber&+&(f_1-f_3)\{g( U,\phi X) V-g( V,\phi X)U\}\\
\nonumber&+&\{\eta(V)\eta(X)U-\eta(U)\eta(X)V\}.
\end{eqnarray}
As $M$ is invariant, $U\in \Gamma(T^\bot M)$, we have
\begin{equation}\label{eqn4.5}
g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0
\end{equation}
for any $X\in \Gamma(TM)$. Using (\ref{eqn4.5}) in (\ref{eqn4.4}), we have
\begin{equation}\label{eqn4.6}
{\bar{R}}^{'}(U,V)X=2f_2g(U,\phi V)\phi X,
\end{equation}
which is tangent as $\phi X$ is tangent. This proves the proposition.
\end{proof}
\begin{proposition}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to
${\bar{\nabla}}^{'}$. If $f_2(p)\neq0$ for each $p\in M$ and
$T^\bot M$ is invariant under the action of ${\bar{R}}(U,V)$, $U,V\in\Gamma(TM)$,
then $M$ is either invariant or anti-invariant.
\end{proposition}
\begin{proof}
The proof is similar as the proof of Lemma $4.2$, just imposing that ${\bar{R}}^{'}(U,V)U$ is normal for any $U,V\in \Gamma(TM)$.
\end{proof}
\section{Submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with $\hat{\bar{\nabla}}$}
\begin{lemma}
If $M$ is either invariant or anti-invarint submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with
respect to $\hat{\bar{\nabla}}$, then
$\hat{\bar{R}}(X,Y)Z$ is tangent to $M$ and $\hat{\bar{R}}(X,Y)V$ is
normal to $M$ for any $X,Y,Z\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
\end{lemma}
\begin{proof}
If $M$ is invariant then from (\ref{eqn1.3v}) we say that $\hat{\bar{R}}(X,Y)Z$ is
tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.
\noindent If $M$ is anti-invariant then
\begin{equation}\label{eqn6.5.1}
g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.
\end{equation}
From (\ref{eqn1.3v}) and (\ref{eqn6.5.1}) we have
\begin{eqnarray}
\label{eqn6.5.2}
\hat{\bar{R}}(X,Y)Z &=&f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(X)\eta(Z)Y-\eta(Y)\eta(Z)X \\
\nonumber&+& g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\},
\end{eqnarray}
which is tangent.\\
If $M$ is invariant from (\ref{eqn1.3v}) we have $\hat{\bar{R}}(X,Y)V$ is normal to $M$, and if $M$ is anti-invariant then $\hat{\bar{R}}(X,Y)V=0$ i.e. $\hat{\bar{R}}(X,Y)V$ is normal to $M$ for any $X,Y\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
This proves the Lemma.
\end{proof}
\begin{lemma}
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\hat{\bar{\nabla}}$.
If $3f_2\neq(f_1-f_3)^2$ on $M$ and $TM$ is invariant
under the action of $\hat{\bar{R}}(X,Y)$, $X,Y\in \Gamma(TM)$,
then $M$ is either invariant or anti-invariant.
\end{lemma}
\begin{proof}
For $X,Y\in \Gamma(TM)$, we have from (\ref{eqn1.3v}) that
\begin{eqnarray}
\label{eqn6.5.3}
\hat{\bar{R}}(X,Y)X &=&f_1\big\{g(Y,X)X-g(X,X)Y\big\}+f_2\big\{g(X,\phi X)\phi Y\\
\nonumber&-& g(Y,\phi X)\phi X + 2g(X,\phi Y)\phi X\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(X)\eta(X)Y-\eta(Y)\eta(X)X\\
\nonumber&+& g(X,X)\eta(Y)\xi-g(Y,X)\eta(X)\xi\big\}\\
\nonumber&+&(f_1-f_3)^2\big\{g(X,\phi X)\phi Y- g(Y,\phi X)\phi X\big\}.
\end{eqnarray}
Now, we see that $\hat{\bar{R}}(X,Y)X$ should be tangent if $\{3f_2+(f_1-f_3)^2\}g(Y,\phi X)\phi X$ is tangent.
Since $3f_2\neq-(f_1-f_3)^2$ then in similar way of proof of Lemma $3.2$ of \cite{ALEGRE3} we may conclude that either $M$ is invariant or anti-invariant. This proves the Lemma.
\end{proof}
From Lemma $5.1$ and Lemma $5.2$, we can state the following:
\begin{theorem}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect
to $\hat{\bar{\nabla}}$. If $3f_2\neq-(f_1-f_3)^2$, then $M$ is
either invariant or anti-invariant if and only if $TM$ is invariant under the
action of $\hat{\bar{R}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.
\end{theorem}
\begin{proposition}
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to
$\hat{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant
under the action of $\hat{\bar{R}}(U,V)$ for any $U,V\in \Gamma(T^\bot M)$.
\end{proposition}
\begin{proof}
Replacing $X,Y,Z$ by $U,V,X$ in (\ref{eqn1.3v}), we get
\begin{eqnarray}
\label{eqn6.5.4}
\hat{\bar{R}}(U,V)X &=&f_1\big\{g(V,X)U-g(U,X)V\big\} +f_2\big\{g(U,\phi X)\phi V\\
\nonumber&-&g(V,\phi X)\phi U + 2g(U,\phi V)\phi X\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(U)\eta(X)V-\eta(V)\eta(X)U\\
\nonumber&+&g(U,X)\eta(V)\xi - g(V,X)\eta(U)\xi\big\}\\
\nonumber&+&(f_1-f_3)^2\big\{g(U,\phi X)\phi V- g(V,\phi X)\phi U\big\}.
\end{eqnarray}
As $M$ is invariant, $U\in \Gamma(T^\bot M)$, we have
\begin{equation}\label{eqn6.5.5}
g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0
\end{equation}
for any $X\in \Gamma(TM)$. Using (\ref{eqn6.5.5}) in (\ref{eqn6.5.4}), we have
\begin{equation}\label{eqn6.5.6}
\hat{\bar{R}}(U,V)X=2f_2g(U,\phi V)\phi X,
\end{equation}
which is tangent as $\phi X$ is tangent.
This proves the proposition.
\end{proof}
\begin{proposition}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to
$\hat{\bar{\nabla}}$. If $3f_2\neq-(f_1-f_3)^2$ on $ M$ and
$T^\bot M$ is invariant under the action of $\hat{\bar{R}}(U,V)$, $U,V\in\Gamma(T^\bot M)$,
then $M$ is either invariant or anti-invariant.
\end{proposition}
\begin{proof}
The proof is similar as the proof of Lemma $5.2$, just imposing that $\hat{\bar{R}}(U,V)U$ is normal for any $U,V\in \Gamma(T^\bot M)$.
\end{proof}
\section{Submanifolds of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with $\stackrel{*}{\bar{\nabla}}$}
\begin{lemma}
If $M$ is either invariant or anti-invarint submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with
respect to $\stackrel{*}{\bar{\nabla}}$, then
$\stackrel{*}{\bar{R}}(X,Y)Z$ is tangent to $M$ and $\stackrel{*}{\bar{R}}(X,Y)V$ is
normal to $M$ for any $X,Y,Z\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
\end{lemma}
\begin{proof}
If $M$ is invariant then from (\ref{eqn1.3u}) we say that $\stackrel{*}{\bar{R}}(X,Y)Z$ is
tangent to $M$ because $\phi X$ and $\phi Y$ are tangent to $M$.
\noindent If $M$ is anti-invariant then
\begin{equation}\label{eqn5.1}
g(X,\phi Z)=g(Y,\phi Z)=g(\phi X,Z)=g(\phi Y,Z)=0.
\end{equation}
From (\ref{eqn1.3u}) and (\ref{eqn5.1}) we have
\begin{eqnarray}
\label{eqn5.2}
\stackrel{*}{\bar{R}}(X,Y)Z &=&f_1\big\{g(Y,Z)X-g(X,Z)Y\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(X)\eta(Z)Y- \eta(Y)\eta(Z)X\\
\nonumber&+&g(X,Z)\eta(Y)\xi - g(Y,Z)\eta(X)\xi\big\}
\end{eqnarray}
which is tangent. \\
If $M$ is invariant from (\ref{eqn1.3u}) we have $\stackrel{*}{\bar{R}}(X,Y)V$ normal to $M$ and if $M$ is anti-invariant then $\stackrel{*}{\bar{R}}(X,Y)V=0$ i.e. $\stackrel{*}{\bar{R}}(X,Y)V$ normal to $M$ for any $X,Y\in \Gamma(TM)$ and $V\in \Gamma(T^\bot M)$.
This proves the Lemma.
\end{proof}
\begin{lemma}
let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to $\stackrel{*}{\bar{\nabla}}$.
If $\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}(p)\neq0$ for each $p\in M$ and $TM$ is invariant
under the action of $\stackrel{*}{\bar{R}}(X,Y)$, $X,Y\in \Gamma(TM)$,
then $M$ is either invariant or anti-invariant.
\end{lemma}
\begin{proof}
For $X,Y\in \Gamma(TM)$, we have from (\ref{eqn1.3u}) that
\begin{eqnarray}
\label{eqn5.3}
\stackrel{*}{\bar{R}}(X,Y)X&=&f_1\big\{g(Y,X)X-g(X,X)Y\big\}+f_2\big\{g(X,\phi X)\phi Y\\
\nonumber&-& g(Y,\phi X)\phi X + 2g(X,\phi Y)\phi X\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(X)\eta(X)Y-\eta(Y)\eta(X)X \\
\nonumber&+&g(X,X)\eta(Y)\xi- g(Y,X)\eta(X)\xi\big\}\\
\nonumber&+&(f_1-f_3)^2\big\{g(X,\phi X)\phi Y- g(Y,\phi X)\phi X\big\}\\
\nonumber&+&2(f_1-f_3)g( X, \phi Y)\phi X.
\end{eqnarray}
Now we see that $\stackrel{*}{\bar{R}}(X,Y)X$ should be tangent if $\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}(p)g(Y,\phi X)\phi X$ is tangent.
Since $\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}(p)\neq0$ then by similar way of
proof of Lemma $3.2$ of \cite{ALEGRE3} we can proved that either $M$
is invariant or anti-invariant. This proves the Lemma.
\end{proof}
From Lemma $6.1$ and Lemma $6.2$, we have
\begin{theorem}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect
to $\stackrel{*}{\bar{\nabla}}$. If $\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}(p)\neq0$, then $M$ is
either invariant or anti-invariant if and only if $TM$ is invariant under the
action of $\stackrel{*}{\bar{R}}(X,Y)$ for all $X,Y\in\Gamma(TM)$.
\end{theorem}
\begin{proposition}
Let $M$ be a submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to
$\stackrel{*}{\bar{\nabla}}$. If $M$ is invariant, then $TM$ is invariant
under the action of $\stackrel{*}{\bar{R}}(U,V)$ for any $U,V\in \Gamma(T^\bot M)$.
\end{proposition}
\begin{proof}
Replacing $X,Y,Z$ by $U,V,X$ in (\ref{eqn1.3u}), we get
\begin{eqnarray}
\label{eqn5.4}
\stackrel{*}{\bar{R}}(U,V)X &=&f_1\big\{g(V,X)U-g(U,X)V\big\}+f_2\big\{g(U,\phi X)\phi V\\
\nonumber&-&g(V,\phi X)\phi U + 2g(U,\phi V)\phi X\big\}\\
\nonumber&+&\{f_3+(f_1-f_3)^2\}\big\{\eta(U)\eta(X)V-\eta(V)\eta(X)U\\
\nonumber&+&g(U,X)\eta(V)\xi- g(V,X)\eta(U)\xi\big\}\\
\nonumber&+&(f_1-f_3)^2\big\{g(U,\phi X)\phi V- g(V,\phi X)\phi U \big\}\\
\nonumber&+&2(f_1-f_3)g(U,\phi V)\phi X.
\end{eqnarray}
As $M$ is invariant, $U\in \Gamma(T^\bot M)$, we have
\begin{equation}\label{eqn5.5}
g(X,\phi U)=-g(\phi X,U)=g(\phi V,X)=0
\end{equation}
for any $X\in \Gamma(TM)$. Using (\ref{eqn5.5}) in (\ref{eqn5.4}), we have
\begin{equation}\label{eqn5.6}
\stackrel{*}{\bar{R}}(U,V)X=\{2f_2+2(f_1-f_3)\}g(U,\phi V)\phi X,
\end{equation}
which is tangent as $\phi X$ is tangent.
This proves the proposition.
\end{proof}
\begin{proposition}
Let $M$ be a connected submanifold of $\bar{M}^{2n+1}(f_1,f_2,f_3)$ with respect to
$\stackrel{*}{\bar{\nabla}}$. If $\{3f_2+2(f_1-f_3)+(f_1-f_3)^2\}(p)\neq0$ for each $p\in M$ and
$T^\bot M$ is invariant under the action of $\stackrel{*}{\bar{R}}(U,V)$, $U,V\in\Gamma(T^\bot M)$,
then $M$ is either invariant or anti-invariant.
\end{proposition}
\begin{proof}
The proof is similar as the proof of Lemma $6.2$, just considering
that $\stackrel{*}{\bar{R}}(U,V)U$ is normal for any $U,V\in \Gamma(T^\bot M)$.
\end{proof}
\noindent{\bf Acknowledgement:} The first author (P. Mandal) gratefully acknowledges to
the CSIR(File No.:09/025(0221)/2017-EMR-I), Govt. of India for financial assistance.
The Third author (S. K. Hui) are thankful to University of Burdwan for providing administrative and technical support.
|
{
"timestamp": "2018-03-08T02:07:20",
"yymm": "1803",
"arxiv_id": "1803.02610",
"language": "en",
"url": "https://arxiv.org/abs/1803.02610"
}
|
\section{Introduction}\label{sI}
Large-scale magnetic ($B$) fields appear to be quite common in our universe (e.g.~see~\cite{CT}-\cite{V}), with a verified presence in stars, galaxies, galaxy clusters, high-redshift protogalaxies and possibly even in intergalactic voids. Nevertheless, the origin, the evolution and the role of these large-scale cosmic magnetic fields remain essentially unknown, although their widespread presence might suggest that they are of primordial nature. The case for cosmological magnetic fields got stronger when recent reports claimed the existence of coherent magnetic fields in the low density intergalactic space (where no dynamo amplification is likely to operate) with strengths around $10^{-15}~G$~\cite{AK}-\cite{CBF}. Additional support comes from the fact that galaxies (like our Milky Way), galaxy clusters and remote protogalaxies have $B$-fields of similar ($\mu G$-order) strengths, which could be a sign of a common origin for all these fields.
It has long been known that large-scale cosmological magnetic fields, if present, would have affected the evolution of (baryonic) density perturbations, during both the linear and the non-linear regime of structure formation. More specifically, the presence of the $B$-field is believed to slow down the standard growth-rate of linear density gradients by an amount proportional to the square of the Alv\'en speed. Nevertheless, the available Newtonian and relativistic studies (see~\cite{RR}-\cite{TS} and~\cite{TB1}-\cite{BMT} respectively) account only for the contribution of the magnetic pressure, namely of the field's positive pressure. The effects of the magnetic tension, that is of the negative pressure exerted along the field lines themselves, have never been accounted for. The only exception has been a recent Newtonian study, where the role of the magnetic tension on the linear evolution of density inhomogeneities in the post-recombination universe was investigated~\cite{VT}. That work indicated that the aforementioned two magnetic agents may have opposing action, but the results were not conclusive.
Here, we provide the first (to the best of our knowledge) fully relativistic study of magnetised density perturbations that incorporates the effects of the field's tension, in addition to those of its (positive) pressure. Our starting point is a perturbed, nearly flat, Friedmann-Robertson-Walker (FRW) universe permeated by a weak large-scale magnetic field. The latter could be primordial in origin, or a later addition to the phenomenology of our universe (e.g.~see~\cite{KKT,Wetal} for recent reviews). Confining to the post-recombination epoch, where structure formation starts in earnest, we set the matter pressure to zero and focus on the role and the implications of the $B$-field. The latter affects the linear evolution of density inhomogeneities through the Lorentz force, which splits into a pressure and a tension part. Not surprisingly, since we are dealing with dust, the magnetic pressure becomes the sole source of support against the gravitational pull of the matter. The linear contribution of the magnetic tension, on the other hand, is two-fold. There are pure-tension stresses, similar but not identical to those identified in the Newtonian study of~\cite{VT}, and a purely relativistic magneto-curvature stress triggered by the non-Euclidean geometry of the host space. Both of these tension stresses reflect the elasticity of the magnetic forcelines and their generic tendency to react against any agent (physical or geometrical) that distorts them from equilibrium~\cite{P}-\cite{T}.
We analyse the role of the $B$-field in a step-by-step approach, accounting for the effects of the magnetic pressure first, before gradually incorporating those of the field's tension. In the first instance, our results recover those of the earlier studies. We confirm that, when dealing with dust, there is a purely magnetic Jeans length below which density perturbations cannot grow. Instead, the density gradients oscillate with an amplitude that decays to zero. Well outside the aforementioned Jeans scale, on the other hand, the perturbations grow essentially unimpeded by the field's presence. Incorporating the effects of the magnetic tension does not seem to affect the large-scale evolution of the density gradients, since they continue to grow as if there was no $B$-field present. On wavelengths near and below the Jeans threshold, however, the standard picture changes when the tension stresses are accounted for. Although the density perturbations still oscillate with decreasing amplitude on scales well inside the Jeans length, they now decay to a finite (rather than to zero) amplitude. Moreover, close to the Jeans length, where the magnetic pressure balances out the gravitational pull of the matter and the field's tension becomes the main player, the perturbations experience a slow (logarithmic) growth. Despite the weakness of the effect, this result clearly demonstrates the opposing action of the aforementioned two magnetic agents and reveals the, as yet unknown, role of the magnetic tension.
In our final step we also incorporate the magneto-curvature (tension) stresses into the linear equations. However, our assumption of a spatially flat FRW background means that the associated effects are (by default) too weak to make a ``visible'' difference. The magneto-curvature stresses identified here could (in principle) play the dominant role during a curvature-dominated era, which could occur in the very early or in the very late evolution of the universe. In such a case the type of spatial curvature (i.e.~whether it is positive or negative) is of paramount importance, because it determines the nature of the magneto-geometrical effect. Nevertheless, to look into the possible implications of such stresses, one needs to consider cosmological backgrounds with nonzero spatial curvature, which goes beyond the scope of the present work.
\section{Relativistic magnetohydrodynamics}\label{sRMHD}
We will study the role of cosmological magnetic fields on density perturbations by applying the 1+3 covariant formalism to relativistic cosmic magnetohydrodynamics (MHD), an approach that has been proven a powerful tool in the past (e.g.~see~\cite{TB1}-\cite{BMT}).
\subsection{The 1+3 spacetime splitting}\label{ss1+3SS}
Introducing a family of timelike observers allows for the 1+3 threading of the spacetime into time and 3-dimensional space. The temporal direction is defined by the observers' 4-velocity field ($u_a$ -- normalised so that $u^{a}u_{a}=-1$), while their rest-space is associated with the projector $h_{ab}=g_{ab}+ u_{a}u_{b}$ (where $g_{ab}$ is the spacetime metric). The latter is a symmetric tensor that projects orthogonally to the 4-velocity vector (i.e.~$h_{ab}u^b=0$) and acts as the metric of the 3-space when there is no rotation. Using both $u_a$ and $h_{ab}$, we define the temporal and spatial derivatives of a general tensor field $S_{ab...}{}^{cd...}$ as
\begin{equation}
\dot{S}_{ab\cdots}{}^{cd\cdots}= u^{e}\nabla_{e}S_{ab\cdots}{}^{cd\cdots} \hspace{10mm} {\rm and} \hspace{10mm} {\rm D}_{e}S_{ab\cdots}{}^{cd\cdots}= h_e{}^{s}h_a{}^{f}h_b{}^{p}h_q{}^{c}h_r{}^{d}\cdots \nabla_{s}S_{fp\cdots}{}^{qr\cdots}\,, \label{dD}
\end{equation}
respectively (with $\nabla_a$ representing the 4-dimensional covariant derivative operator).
\subsection{Matter fields and kinematics}\label{ssMFKs}
The 1+3 formalism utilises a systematic decomposition of all physical variables and operators into their irreducible temporal and spatial components. For instance, relative to the observers introduced earlier, which in our case are always comoving with the matter, the energy-momentum tensor of a general imperfect fluid splits as
\begin{equation}\label{Tab}
T_{ab}= \rho u_au_b+ ph_{ab}+ 2q_{(a}u_{b)}+ \pi_{ab}\,,
\end{equation}
with $\rho=T_{ab}u^au^b$, $p=T_{ab}h^{ab}/3$, $q_a=-h_a{}^bT_{bc}u^c$ and $\pi_{ab}=h_{\langle a}{}^ch_{b\rangle}{}^dT_{cd}$ representing the energy density, the isotropic pressure, the energy flux and the viscosity of the matter respectively.\footnote{Square brackets indicate antisymmetrisation, round ones symmetrisation and angled brackets denote the symmetric and traceless part of second rank spacelike tensors. For instance, $\pi_{ab}=h_{(a}{}^ch_{b)}{}^dT_{cd}-ph_{ab}$. Also, we use geometrised and Heaviside-Lorentz units throughout this manuscript.} Note that the quantities on the right-hand side of the above correspond to the total matter, which may also include electromagnetic fields.
In an analogous way, the covariant derivative of the observers' 4-velocity decomposes into the irreducible kinematic variables according to
\begin{equation}\label{velocity gradient}
\nabla_{b}u_{a}= \frac{1}{3}\,\Theta h_{ab}+ \sigma_{ab}+ \omega_{ab}- A_{a}u_{b}\,,
\end{equation}
where $\Theta=\nabla^a u_a={\rm D}^au_a$ is the volume expansion/contraction scalar, $\sigma_{ab}={\rm D}_{\langle b}u_{a\rangle}$ is the shear tensor, $\omega_{ab}={\rm D}_{[b}u_{a]}$ is the vorticity tensor and $A_a=\dot{u}_a$ is the 4-acceleration vector. By construction, $\sigma_{ab}u^a=0= \omega_{ab}u^a=A_{a}u^a$, which ensures that all three of them are spacelike. The volume scalar monitors the mean separation between neighbouring observers and it is also used to introduce a characteristic length-scale along their worldlines. This is the cosmological scale-factor ($a$) defined by means of $\dot{a}/a=\Theta/3$. The shear and the vorticity describe kinematic anisotropies and rotation respectively, while the 4-acceleration implies the presence of non-gravitational forces. Note that the antisymmetry of the vorticity tensor means that it can be replaced by the vector $\omega_a=\varepsilon_{abc}\omega^{bc}/2$, where $\varepsilon_{abc}$ is the Levi-Civita tensor of the 3-space.
The evolution of the kinematic variables defined above follows after applying the Ricci identities to the 4-velocity vector, namely from the expression $2\nabla_{[a}\nabla_{b]}u_c=R_{abcd}u^d$, where $R_{abcd}$ is the Riemann tensor of the spacetime. The Ricci identities decompose into a set of three timelike formulae, monitoring the evolution of the $u_a$-field, and into an equal number of spacelike relations that act as constraints. Referring the reader to~\cite{TCM} for further discussion and details, we will only provide the propagation equation of the volume scalar, namely
\begin{equation}\label{Raychaudhuri}
\dot{\Theta}=-\frac{1}{3}\,\Theta^{2}- \frac{1}{2}\,(\rho+3p)- 2\left(\sigma^2-\omega^2\right)+ {\rm D}^{a}A_{a}+ A^{a}A_{a}+ \Lambda\,,
\end{equation}
with $\Lambda$ representing the cosmological constant, $\sigma^2= \sigma_{ab}\sigma^{ab}/2$ and $\omega^2= \omega_{ab}\omega^{ab}/2$ by definition. The above expression, which is commonly known as the Raychaudhuri equation, applies to a general spacetime filled with an imperfect fluid of arbitrary electrical conductivity.
\subsection{Magnetohydrodynamics and conservation laws}\label{ssMHDCLs}
The post-inflationary universe is treated as a very good electrical conductor, at least on subhorizon scales where causal microphysical processes readily apply (see also footnote~2 below). In such an environment, the ideal-MHD limit is believed to provide an excellent physical approximation. Mathematically speaking this means setting $\varsigma\rightarrow\infty$, where $\varsigma$ is the electrical conductivity of the cosmic medium. In a frame comoving with the matter, the covariant form of Ohm's law reads $\mathcal{J}_a=\varsigma E_a$, with $\mathcal{J}_a$ and $E_a$ representing the spatial currents and the electric field respectively (e.g.~see~\cite{J}). Then, in the presence of finite 3-currents, $E_a\rightarrow0$ as $\varsigma\rightarrow\infty$. All these ensure that at the ideal-MHD limit, Maxwell's equations reduce to one propagation formula
\begin{equation}\label{Faraday}
\dot{B}_{\langle a\rangle}= -{2\over3}\,\Theta B_a+ \left(\sigma_{ab}+\epsilon_{abc}\omega^c\right)B^b
\end{equation}
and three constraints
\begin{equation}\label{Gauss,Coulomb,Ampere}
{\rm curl}B_a= \mathcal{J}_a- \varepsilon_{abc}A^bB^c\,, \hspace{10mm} 2\omega_aB^a= \mu \hspace{10mm} {\rm and} \hspace{10mm} {\rm D}^aB_a= 0\,,
\end{equation}
where $\dot{B}_{\langle a\rangle}=h_a{}^b\dot{B}_b$, ${\rm curl}B_a=\varepsilon_{abc}{\rm D}^bB^c$ and $\mu$ is the electric charge density of the matter~\cite{TB1}-\cite{BMT}. The former of the above, namely Eq.~(\ref{Faraday}), guarantees that the magnetic forcelines always connect the same particles at all times~\cite{E1}. In other words, at the ideal MHD approximation, the $B$-field is frozen into the highly conductive matter.
In the absence of electric fields, the total energy-momentum tensor of the magnetised matter, assuming that the latter is a perfect fluid of arbitrarily high electrical conductivity, reads
\begin{equation}\label{Ttot}
T_{ab}= \left(\rho+\frac{1}{2}\,B^2\right)u_au_b+ \left(p+\frac{1}{6}\,B^2\right)h_{ab}+ \Pi_{ab}\,,
\end{equation}
where $B^2=B_aB^a$ and $\Pi_{ab}=\Pi_{\langle ab\rangle}= (B^2/3)h_{ab}-B_aB_b$. The former provides a measure of the magnetic energy density and isotropic pressure, while the latter defines the anisotropic pressure of the $B$-field~\cite{TB1}-\cite{BMT}. Following (\ref{Tab}) and (\ref{Ttot}), we deduce that our magnetised medium behaves as an imperfect fluid with effective energy density $\rho+B^2/2$, effective isotropic pressure $p+B^2/6$ and effective viscosity $\Pi_{ab}$. The latter is a symmetric and trace-free spacelike tensor, which unveils the generically anisotropic nature of the $B$-field. Note that $\Pi_{ab}$ has positive eigenvalues orthogonal to the magnetic forcelines and negative parallel to them. More specifically, it is straightforward to show that $\Pi_{ab}n^b=(B^2/3)n_a$ and that $\Pi_{ab}k^b=-(2B^2/3)k_a$, where $n_a$ and $k_a$ are the unit vectors normal and along $B_a$ respectively. The positive eigenvalues are associated with the ordinary magnetic pressure and reflect the tendency of the field lines to fend off. The negative eigenvalue, on the other hand, manifests the tension properties of the magnetic forcelines, their elasticity and their intrinsic ``preference'' to remain as straight as possible~(e.g.~see~\cite{P,M}).
The conservation laws for the energy and the momentum densities of a highly conductive magnetised fluid follow from the (twice contracted) Bianchi identities and from Maxwell's equations. In particular, assuming a perfect medium, the timelike and the spacelike parts of the aforementioned Bianchi identities lead to the energy density
\begin{equation}\label{rho}
\dot{\rho}= -\Theta(\rho+p)
\end{equation}
and to the momentum-density
\begin{equation}\label{A}
(\rho+p)A_a= -{\rm D}_ap- \varepsilon_{abc}B^b\mathcal{J}^c\,,
\end{equation}
conservation laws, namely to the continuity equation and to the Navier-Stokes equation respectively.\footnote{Following (\ref{A}), the magnetic effects on the fluid propagate via the Lorentz force and require the presence of coherent electric currents. These are generated after inflation, which means that their size cannot exceed that of the causal horizon. Therefore, the magnetic effects discussed in this work apply primarily to subhorizon scales.} At the same time, the induction equation (see relation (\ref{Faraday}) above) leads to the conservation law of the magnetic energy density, namely to~\cite{TB1}-\cite{BMT}
\begin{equation}\label{dotB2}
\left(B^2\right)^{\cdot}= -{4\over3}\,\Theta B^2- 2\sigma_{ab}\Pi^{ab}\,.
\end{equation}
Expressions (\ref{rho}) and (\ref{dotB2}) reveal that, at the ideal-MHD limit, the energy density of the magnetised matter and that of the $B$-field itself are separately conserved.
\section{Magnetised density inhomogeneities}\label{ssMDIs}
Inhomogeneities in the density distribution of the matter are affected by pressure gradients. As mentioned above, the magnetic field is an additional source of pressure, both positive and negative. In what follows we will study the implications of these two different types of pressure for the linear evolution of magnetised density perturbations in the post-recombination universe.
\subsection{The key variables}\label{ssKVs}
Following the earlier relativistic treatments of~\cite{TB1,TM} (see also~\cite{BMT} for a review), we monitor inhomogeneities in the density distribution of matter by means of the dimensionless gradient
\begin{equation}\label{Drel}
\Delta_a= \frac{a}{\rho}\,{\rm D}_a\rho\,.
\end{equation}
The above variable, which depicts spatial variations in the matter density as measured by a pair of neighbouring observers, is supplemented by the auxiliary quantities
\begin{equation}\label{ZBrel}
\mathcal{Z}_a= a{\rm D}_a\Theta \hspace{10mm} {\rm and} \hspace{10mm} \mathcal{B}_a= \frac{a}{B^2}\,{\rm D}_aB^2\,.
\end{equation}
These, in turn, monitor local inhomogeneities in the volume expansion and in the magnetic energy density respectively. Note that all of the above vanish identically in an FRW background (see \S~\ref{ssBM} next) and for this reason they are gauge-invariant linear perturbations~\cite{SW}.
\subsection{The background model}\label{ssBM}
Our aim is to study the magnetic implications for the evolution of density perturbations in a perturbed almost-FRW universe. We therefore select as our background model a spatially flat Friedmann model with zero cosmological constant. Also, to enhance the linear magnetic effects, we will allow for the presence of completely random and sufficiently weak background magnetic field. The randomness implies that $\langle B_a\rangle=0$, which preserves the isotropy of the FRW host, while $\langle B^2\rangle\neq0$. The weakness ensures that, although the $B$-field contributes to the background energy density, its input is small (i.e.~$\langle B^2\rangle\ll\rho$), leaving the standard FRW dynamics unaffected.
The symmetries of the Friedmannian spacetimes imply that the only surviving background variables are time-dependent scalars. All the rest vanish identically and they will be therefore treated as first-order (gauge-invariant) perturbations. These include, among others, the inhomogeneity variables introduced in \S~\ref{ssKVs} earlier. Then, using overbars to denote the zero-order quantities, while setting $\bar{\Theta}=3H$ (where $H=\dot{a}/a$ is the unperturbed Hubble parameter) and $\bar{B}^2=\langle B^2\rangle$, with $\bar{B}^2=\bar{B}^2(t)$, the background evolution is monitored by the set
\begin{equation}\label{bgrFr}
3H^2= \bar{\rho}\,, \hspace{10mm} \dot{H}= -H^2- \frac{1}{6}\left(\bar{\rho}+3\bar{p}\right)\,, \hspace{10mm} \dot{\bar{\rho}}= -3H\left(\bar{\rho}+\bar{p}\right)
\end{equation}
and
\begin{equation}\label{bgrB2}
(\bar{B}^2)^\cdot= -4H\bar{B}^2\,.
\end{equation}
Note that we have ignored the magnetic contribution to the zero-order Friedmann and Raychaudhuri equations (see expressions (\ref{bgrFr}a) and (\ref{bgrFr}b) above), given that $\bar{B}^2\ll\bar{\rho}$ in the background. We also remind the reader that, at the ideal MHD limit, the energy density of the matter and that of the $B$-field are separately conserved (see Eqs.~(\ref{bgrFr}c) and (\ref{bgrB2})). The latter relation also unveils the radiation-like evolution of the zero-order magnetic field, namely that $\bar{B}^2\propto a^{-4}$, which also guarantees magnetic-flux conservation.
\subsection{Linear evolution of the inhomogeneities}\label{ssLEIs}
The nonlinear formulae describing the general evolution of magnetised density inhomogeneties can be found in~\cite{TB1}-\cite{BMT}, where we refer the reader for further discussion and technical details. Here, we will linearise these relations around a spatially flat FRW background (with zero cosmological constant) permeated by a sufficiently random and weak magnetic field (see \S~\ref{ssBM} before). In doing so, we will treat the magnetic energy-density and pressure gradients as first-order perturbations, which makes the perturbed $B$-field (and its spatial gradients) half-order perturbations.\footnote{The magnetic contribution to the linear equations comes always through terms of order $B^2$ (see expressions (\ref{linD'rel})-(\ref{linB'rel})), which ensures the perturbative consistency of the adopted linearisation scheme.} On these grounds, the linear evolution of the inhomogeneities is monitored by the propagation formulae~\cite{BMT}
\begin{equation}\label{linD'rel}
\dot{\Delta}_a= 3wH\Delta_a- (1+w)\mathcal{Z}_a+ \frac{3aH}{\bar{\rho}}\,\varepsilon_{abc}B^b{\rm curl}B^c+ 2aH(1+w)c_{\rm a}^2A_a\,,
\end{equation}
\begin{equation}\label{linZ'rel}
\dot{\mathcal{Z}}_a= -2H\mathcal{Z}_a- \frac{1}{2}\,\bar{\rho}\Delta_a- \frac{1}{2}\,\bar{\rho}(1+w)c_{\rm a}^2\mathcal{B}_a+ \frac{3a}{2}\,\varepsilon_{abc}B^b{\rm curl}B^c+ a{\rm D}_a{\rm D}^bA_b
\end{equation}
and
\begin{equation}\label{linB'rel}
\dot{\mathcal{B}}_a= \frac{4}{3(1+w)}\,\dot{\Delta}_a- \frac{4wH}{1+w}\,\Delta_a- \frac{4aH}{\bar{\rho}(1+w)}\,\varepsilon_{abc}B^b{\rm curl}B^c- 4aHA_a\,.
\end{equation}
According to (\ref{linD'rel}) and \ref{linZ'rel})), the magnetic field also sources inhomogeneities, both in the density of the matter and in the volume expansion of the universe. In the above $w=\bar{p}/\bar{\rho}$ is the background barotropic index of the matter and $c_{\rm a}^2=\bar{B}^2/\bar{\rho}(1+w)$ defines the zero-order Alfv\'{e}n speed. By construction, the latter satisfies the constraint $c_{\rm a}^2\ll1$ due to the overall weakness of the $B$-field. Finally, to linear order, the 4-acceleration vector seen on the right-hand side of the above is given by the momentum conservation law (see Eq.~(\ref{A}) in \S~\ref{ssMHDCLs}), which now reads
\begin{equation}
\rho(1+w)A_a= -{\rm D}_ap- \varepsilon_{abc}B^b{\rm curl}B^c= {\rm D}_ap- {1\over2}\,{\rm D}_aB^2+ B^b{\rm D}_bB_a\,, \label{MHDAa}
\end{equation}
since $J_a={\rm curl}B_a=\varepsilon_{abc}{\rm D}^bB^c$ to linear order (see Eq.~(\ref{Gauss,Coulomb,Ampere}a)). Note that in the second equality of the above the Lorentz force splits into its pressure and tension stresses, given by ${\rm D}_aB^2/2$ and $B^b{\rm D}_bB_a$ respectively. In what follows, we will investigate the implications of these two magnetic agents for the evolution of linear perturbations in the density distribution of the matter.
\subsection{Types of inhomogeneities}\label{ssTIs}
The variables defined in \S~\ref{ssKVs}, namely $\Delta_a$, $\mathcal{Z}_a$ and $\mathcal{B}_a$, contain collective information about three types of inhomogeneities: scalar, vector and tensor. The former monitors overdensities or underdensities in the matter distribution, which we usually refer to as density perturbations. Vector inhomogeneities, on the other hand, describe rotational (vortex-like) distortions in the matter. Finally, tensor inhomogeneities describe changes in the shape of the density profile under constant volume. We may decode all this information by taking the comoving spatial gradient of $\Delta_a$ and then implementing the irreducible decomposition (e.g.~see~\cite{TCM})
\begin{equation}\label{Delab}
\Delta_{ab}= a{\rm D}_b\Delta_a= {1\over3}\,\Delta h_{ab}+ \Sigma_{ab}+ \varepsilon_{abc}W^c\,.
\end{equation}
Here $\Delta=a{\rm D}^a\Delta_a$ is the scalar describing overdensities/underdensities in the matter, $W_a=-a{\rm curl} \Delta_a/2$ is the vector monitoring density vortices and $\Sigma_{ab}=a{\rm D}_{\langle b}\Delta_{a\rangle}$ is the symmetric and trace-free tensor following changes in the shape of the density profile. Clearly, similar decompositions also apply to the expansion and the magnetic energy-density gradients~\cite{TM,BMT}.
The anisotropic nature of the $B$-field ensures its interaction with all of the aforementioned three types of inhomogeneities. Here, we will focus on the linear evolution of scalar density perturbations after recombination. This restriction means that we may set the matter pressure and the associated barotropic index to zero.
\section{Magnetised density perturbations}\label{sMDPs}
When dealing with dust, the magnetic field becomes the sole source of pressure support. However, this does not a priori guarantee that the growth of density perturbations will slow down in the magnetic presence, since the $B$-field is a source of negative pressure (tension) as well.
\subsection{Linear evolution of density perturbations}\label{ssLEDPs}
Scalar perturbations in the matter density, in the volume expansion of the universe and in the magnetic energy density are monitored by
\begin{equation}\label{scalarsD,Z,B}
\Delta= a{\rm D}^a\Delta_a\,, \hspace{10mm} \mathcal{Z}= a{\rm D}^a\mathcal{Z}_a\,, \hspace{10mm} {\rm and} \hspace{10mm} \mathcal{B}= a{\rm D}^a\mathcal{B}_a\,,
\end{equation}
respectively (see \S~\ref{ssTIs} above). Then, setting $w=0$ and $c_{\rm a}^2=\bar{B}^2/\bar{\rho}\ll1$, the comoving 3-divergences of Eqs.~(\ref{linD'rel}), (\ref{linZ'rel}) and (\ref{linB'rel}) lead to the linear propagation formulae
\begin{equation}\label{scalarD'rel}
\dot{\Delta}= -\mathcal{Z}+ \frac{3}{2}\,Hc_{\rm a}^2\mathcal{B}- Hc_{\rm a}^2\mathcal{K}- \frac{6a^2H}{\bar{\rho}} \left(\sigma_B^2-\omega_B^2\right)\,,
\end{equation}
\begin{eqnarray}\label{scalarZ'rel}
\nonumber \dot{\mathcal{Z}}&=& -2H\mathcal{Z}- \frac{1}{2}\,\bar{\rho}\Delta+ \frac{1}{4}\,\bar{\rho}c_{\rm a}^2\mathcal{B}- \frac{1}{2}\,c_{\rm a}^2{\rm D}^2\mathcal{B}- \frac{1}{2}\,\bar{\rho}c_{\rm a}^2\mathcal{K}- 3a^2\left(\sigma_B^2-\omega_B^2\right) \\&& +\frac{2a^2}{\bar{\rho}}\,{\rm D}^2\left(\sigma_B^2-\omega_B^2\right)
\end{eqnarray}
and
\begin{equation}\label{scalarB'rel}
\dot{\mathcal{B}}= \frac{4}{3}\,\dot{\Delta}\,,
\end{equation}
respectively.\footnote{In deriving Eq.~(\ref{scalarZ'rel}) we have also used the linear auxiliary relation
\begin{equation}\label{scalar A}
a^2A= -\frac{1}{2}\,c_{\rm a}^2\mathcal{B}+ \frac{1}{3}\,c_{\rm a}^2\mathcal{K}+ \frac{2a^2}{\bar{\rho}}\left(\sigma_B^2-\omega_B^2\right)\,,
\end{equation}
where $A={\rm D}_aA^a$ is the 3-divergence of the 4-acceleration. Note that the last two terms on the right-hand side of the above, together with the last two terms of (\ref{scalarD'rel}) and the last three terms of (\ref{scalarZ'rel}), represent tension stresses, the effects of which were not included in the relativistic solutions of~\cite{TB1}-\cite{BMT}. The Newtonian analogues of the magnetic shear and vorticity, on the other hand, were accounted for in~\cite{VT} (see Eqs.~(23) and (27) there).} Note that the scalars $\sigma_B^2={\rm D}_{\langle b}B_{a\rangle}{\rm D}^{\langle b}B^{a\rangle}/2$ and $\omega_B^2={\rm D}_{[b}B_{a]}{\rm D}^{[b}B^{a]}/2$ are respectively related to shape and rotational distortions in a field-line congruence, which makes the tensors $\sigma_{ab}^B= {\rm D}_{\langle b}B_{a\rangle}$ and $\omega_{ab}^B= {\rm D}_{[b}B_{a]}$ the magnetic analogues of the kinematic shear and vorticity (see \S~\ref{ssMFKs}). Also, $\mathcal{K}=a^2\mathcal{R}$ by definition, with $\mathcal{R}$ representing the perturbed 3-Ricci scalar, which means that the fifth term on the right-hand side of Eq.~(\ref{scalarZ'rel}) carries the combined effects of magnetism and spatial curvature. Note that this particular stress reflects the vector nature of the $B$-field and derives from a purely geometrical coupling between magnetism and spacetime curvature~\cite{TM,T}. The latter comes into play via the Ricci identities and adds to the standard interaction between matter and geometry that the Einstein field equations introduce. Following~\cite{BMT}, the rescaled 3-Ricci scalar evolves as
\begin{equation}
\dot{\mathcal{K}}= -{4\over3}\,Hc_{\rm a}^2\mathcal{K}+ 2Hc_{\rm a}^2\mathcal{B}\,, \label{lcK}
\end{equation}
to linear order. Finally, we should point out that the second term on the right-hand side of Eq.~(\ref{scalarD'rel}) and the third and fourth terms on the right-hand of (\ref{scalarZ'rel}) are due to the (positive) magnetic pressure. On the other hand, the last two terms of (\ref{scalarD'rel}) and the last three terms of (\ref{scalarZ'rel}) carry the effects of the field's tension.
\subsection{The wave-like equation}\label{ssW-LE}
Taking the time derivative of (\ref{scalarD'rel}), using the rest of the propagation formulae and keeping up to linear order terms, we obtain the following wave-like equation for the density perturbations
\begin{equation}\label{D''}
\ddot{\Delta}= -2H\dot{\Delta}+ \frac{1}{2}\,\bar{\rho}\Delta+ \frac{2}{3}\,c_{\rm a}^{2}{\rm D}^{2}\Delta+ \frac{2}{3}\,c_{\rm a}^{2}\rho\mathcal{K}+ 4a^{2}\left(\sigma_B^2-\omega_B^2\right)- \frac{2a^2}{\bar{\rho}}\,{\rm D}^2\left(\sigma_B^2-\omega_B^2\right)\,.
\end{equation}
with additional terms due to the universal expansion, the presence of matter (including the $B$-field) and spacetime curvature. In deriving the above we have also used the linear propagation formulae $(\sigma_B^2)^{\cdot}=-6H\sigma_B^2$ and $(\omega_B^2)^{\cdot}= -6H\omega_B^2$, which in turn follow from the linear auxiliary relation $({\rm D}_bB_a)^{\cdot}=-3H{\rm D}_bB_a$. The latter is obtained after combining the linear commutation law (A.2.2) of~\cite{BMT} with the linearised magnetic induction equation (i.e.~with $\dot{B}_a=-2HB_a$ -- see expression (\ref{Faraday}) in \S~\ref{ssMHDCLs}). Note that, in the absence of matter pressure, the Alfv\'en speed has become the wave velocity as well. Also note that the magneto-curvature effects reverse when the (rescaled) 3-curvature scalar ($\mathcal{K}$) changes from positive to negative and vice versa.
Our next step is to harmonically decompose Eq.~(\ref{D''}), by introducing the standard scalar harmonics functions $\mathcal{Q}^{(n)}$, with $\dot{\mathcal{Q}}^{(n)}=0$ and ${\rm D}^2\mathcal{Q}^{(n)}=-(n/a)^2\mathcal{Q}^{(n)}$. Then, setting $\Delta=\sum_n\Delta_{(n)}\mathcal{Q}^{(n)}$, $\mathcal{K}=\sum_n\mathcal{K}_{(n)}\mathcal{Q}^{(n)}$ and $(\sigma_B^2-\omega_B^2)= \sum_n(\sigma_B^2-\omega_B^2)_{(n)}\mathcal{Q}^{(n)}$, with ${\rm D}_a\Delta_{(n)}=0={\rm D}_a\mathcal{K}_{(n)}={\rm D}_a(\sigma_B^2-\omega_B^2)_{(n)}$, arrive at
\begin{eqnarray}\label{D harmonic}
\nonumber \ddot{\Delta}_{(n)}&=& -2H\dot{\Delta}_{(n)}+\frac{1}{2}\,\bar{\rho}\left[1-\frac{4}{9}\,c_{\rm a}^2 \left(\frac{\lambda_H}{\lambda_n}\right)^2\right]\Delta_{(n)}+ \frac{2}{3}\,\bar{\rho}c_{\rm a}^2\mathcal{K}_{(n)} \\&&
+4\left[1+\frac{1}{6}\left(\frac{\lambda_H}{\lambda_n}\right)^2\right] \left(\Sigma_B^2-\Omega_B^2\right)_{(n)}\,,
\end{eqnarray}
where $\lambda_H=1/H$ is the Hubble radius, $\lambda_n=a/n$ is the physical scale of the perturbation (with $n$ being the comoving wavenumber), while $\Sigma_B^2=a^2\sigma_B^2$ and $\Omega_B^2=a^2\omega_B^2$ define the rescaled magnetic shear and the magnetic voricity respectively. As expected, in the absence of these stresses, expressions (\ref{D''}) and (\ref{D harmonic}) reduce to the wavelike formula obtained in~\cite{BMT} (see Eq.~(7.4.5) there). Also, for a direct comparison between (\ref{D''}) and its Newtonian analogue, we refer the reader to Eq.~(27) in~\cite{VT}.
The second term on the right-hand side of (\ref{D harmonic}) conveys the opposing action of gravity and (magnetic) pressure. These effects cancel each other out (and the aforementioned term goes to zero) at a specific wavelength, which is given by
\begin{equation}\label{Jeans}
\lambda= \lambda_J= \frac{2}{3}\,c_{\rm a}\lambda_H
\end{equation}
and marks the (purely magnetic) Jeans length~\cite{TM,BMT}. On scales much larger than $\lambda_J$, gravity prevails and the perturbations grow. When $\lambda_n\ll\lambda_J$, however, the (positive) pressure of the $B$-field dominates and prevents the perturbations from growing (see \S~\ref{sSs} next).
\section{Linear solutions}\label{sSs}
We will examine the magnetic effects on the evolution of density perturbations in three steps. First, we will allow the magnetic pressure to act alone. Then, we will consider the simultaneous action of magnetic pressure and tension, leaving the role of the magneto-curvature coupling last.
\subsection{Magnetic pressure effects}
Without the magnetic tension terms, which include the magneto-curvature stresses as well, Eq.~(\ref{D harmonic}) reduces to
\begin{equation}\label{dif1.1}
\ddot{\Delta}_{(n)}=-2H\dot{\Delta}_{(n)}+\frac{1}{2}\,\bar{\rho} \left[1-\left(\frac{\lambda_J}{\lambda_n}\right)^2\right] \Delta_{(n)}\,,
\end{equation}
having substituted for the (magnetic) Jean's length from definition (\ref{Jeans}). During the dust era $a\propto t^{2/3}$, $H=2/3t$ and $\bar{\rho}=4/3t^2$. At the same time, the scale-ratio $\alpha=\lambda_J/\lambda_n$, which carries the magnetic effects, remains constant (recall that $c_{\rm a}\propto t^{-1/3}$ and $\lambda_n\propto t^{2/3}$ after equipartition, while $\lambda_H\propto t$ always and $\lambda_J=c_{\rm a}\lambda_H$). Then, the above differential equation assumes the form
\begin{equation}\label{dif1.2}
\frac{{\rm d}^2\Delta_{(n)}}{{\rm d}t^2}= -\frac{4}{3t}\,\frac{{\rm d}\Delta_{(n)}}{{\rm d}t}+ \frac{2}{3t^2}\left(1-\alpha^2\right)\Delta_{(n)}
\end{equation}
and accepts the power-law solution
\begin{equation}\label{sol1}
\Delta_{(n)}= C_1\,t^{s_1}+ C_2\,t^{s_2}\,,
\end{equation}
with $s_{1,2}=-[1\mp\sqrt{25-24\alpha^2}]/6$. Therefore, in the absence of the $B$-field (i.e.~when $\alpha=0$), we recover the standard non-magnetised solution for linear density perturbations in the dust era (i.e.~$s_{1,2}=-1, 2/3$ -- e.g.~see~\cite{TCM}). On the other hand, recalling that $\alpha^2=(4c_{\rm a}^2/9) (\lambda_H/\lambda_n)^2$, we deduce that the magnetic pressure inhibits the growth of these distortions by an amount proportional to the Alfv\'en-speed squared.\footnote{Analogous magnetic effects on the linear evolution of density perturbations were also observed during the radiation epoch, in solutions where only the pressure of the $B$-field was accounted for (see~\cite{TB2,BMT} for details).} Moreover, the impact of the aforementioned effect is scale-dependent. We will therefore consider the following three characteristic cases:
\begin{itemize}
\item $\lambda_n\gg\lambda_J$: On scales much larger than the magnetic Jean's length, we have $\alpha= \lambda_J/\lambda_n\ll1$ and therefore solution (\ref{sol1}) reduces to
\begin{equation}\label{1a}
\Delta= C_1\,t^{2/3}+ C_2\,t^{-1}
\end{equation}
We have thus recovered the standard non-magnetised solution, which implies that the magnetic pressure has no effect on large scales.
\item $\lambda_n\ll\lambda_J$: Here, $\alpha= \lambda_J/\lambda_n\gg1$, in which case solution (\ref{sol1}) takes the (imaginary) form
\begin{equation}\label{1b}
\Delta_{(n)}= t^{-1/6}\left(C_1\,t^{\imath\alpha\sqrt{2/3}}+ C_2\,t^{-\imath\alpha\sqrt{2/3}}\right)\,.
\end{equation}
Consequently, on small scales, the magnetic pressure dominates forcing the perturbations to oscillate (with amplitude that decreases as $t^{-1/6}$).
\item $\lambda_n=\lambda_J$: At the $\alpha= \lambda_J/\lambda_n=1$ threshold the last term on the right-hand side of Eq.~(\ref{dif1.2}) vanishes and solution (\ref{sol1}) reads
\begin{equation}\label{1c}
\Delta= C_1+ C_2\,t^{-1/3}\,.
\end{equation}
In other words, on wavelengths equal to the Jeans length, the magnetic pressure balances the gravitational pull of the matter and the perturbations maintain constant amplitude.
\end{itemize}
Overall, the effects of the field's pressure are only felt on scales close and below the magnetic Jeans length. On larger wavelengths, the perturbations grow as if there was no $B$-field present. These results are identical to those obtained in the Newtonian study of~\cite{VT} and very close (both qualitatively and quantitatively) to those of the earlier relativistic treatments~\cite{TB1,BMT}.
\subsection{Combined pressure and tension effects}\label{ssCPTEs}
The opposite nature of the magnetic-pressure contribution and that of the field's tension (i.e.~positive vs negative) indicate that these two agents may act against each other. Here, we will attempt to clarify the matter by considering the combined effect of pressure and tension on the evolution of linear density perturbations. It should be noted, however, that the magneto-curvature stresses (which are also due to the magnetic tension) will remain switched off. Then, the density gradients are monitored by the linear system
\begin{equation}\label{D'' p+t}
\ddot{\Delta}_{(n)}= -2H\dot{\Delta}_{(n)}+ \frac{1}{2}\,\bar{\rho}\left[1 -\left(\frac{\lambda_J}{\lambda_n}\right)^2\right]\Delta_{(n)}+ 4\left[1+{1\over6}\left(\frac{\lambda_H}{\lambda_n}\right)^2\right] \left(\Sigma_B{}^2-\Omega_B{}^2\right)_{(n)}
\end{equation}
and
\begin{equation}\label{S}
\left(\Sigma_B{}^2-\Omega_B{}^2\right)^{\cdot}= -4H\left(\Sigma_B{}^2-\Omega_B{}^2\right)\,.
\end{equation}
The latter implies that $\Sigma_B$, $\Omega_B\propto a^{-2}$ on all scales and follows from the fact that $\Sigma_B=a\sigma_B$ and $\Omega_B=a\omega_B$, with $\sigma_B$, $\omega_B\propto a^{-3}$ (see Eqs.~(\ref{D''}) and (\ref{D harmonic}) in \S~\ref{ssW-LE}). Given that $H=2/3t$ and $\bar{\rho}=4/3t^2$ after equilibrium, the above recast as
\begin{equation}\label{d2Delta}
\frac{{\rm d}^2\Delta_{(n)}}{{\rm d}t^2}= -\frac{4}{3t}\frac{{\rm d}\Delta_{(n)}}{{\rm d}t}+ \frac{2}{3t^2}\left(1-\alpha^2\right)\Delta_{(n)}+ 4\left[1+{1\over6}\,\beta^2\left({t\over t_0}\right)^{2/3}\right] \left(\Sigma_B{}^2-\Omega_B{}^2\right)_{(n)}
\end{equation}
and
\begin{equation}\label{dSigma}
\frac{{\rm d}}{{\rm d}t}\left(\Sigma_B{}^2-\Omega_B{}^2\right)= -\frac{8}{3t}\left(\Sigma_B{}^2-\Omega_B{}^2\right)\,,
\end{equation}
respectively. As before, $\alpha=\lambda_J/\lambda_n=$~constant after equipartition, while $\beta=(\lambda_H/\lambda_n)_0=$~constant determines the physical scale of the perturbations at the start of the dust era. When $\alpha\neq1,\sqrt{2/3}$, the system of (\ref{d2Delta}) and (\ref{dSigma}) solves analytically giving
\begin{equation}\label{gensol}
\Delta_{(n)}= C_1\,t^{s_1}+ C_2\,t^{s_2}+ C_3\left[\frac{\beta^2}{6(\alpha^2-1)} +\frac{1}{\alpha^2-{2/3}}\left({t_0\over t}\right)^{2/3}\right]\,,
\end{equation}
where $s_{1,2}=-[1\mp\sqrt{25-24\alpha^2}]/6$ exactly as before (see solution (\ref{sol1})). Consequently, the introduction of the tension stresses has added two extra modes (one constant and one decaying) to the linear evolution of magnetised density perturbations. Then, depending on the scale of the perturbation, we may consider the cases:
\begin{itemize}
\item $\lambda_n\gg\lambda_J$: In this case $\alpha\ll1$ and the above solution reduces to\footnote{Although we use the same symbols for the integration constants in all our solutions, these generally differ.}
\begin{equation}\label{s2.1}
\Delta= C_1\,t^{2/3}+ C_2\,t^{-1}+ C_3+ C_4\,t^{-2/3}\,.
\end{equation}
Hence, on scales much larger than the magnetic Jeans length, the incorporation of the field's tension has not changed the standard picture. The density perturbations keep growing as $\Delta\propto t^{2/3}$, like their magnetic-free counterparts (compare to solution (\ref{1a})).
\item $\lambda_n\ll\lambda_J$: Here $\alpha\gg1$ and expression (\ref{gensol}) becomes
\begin{equation}\label{s2.2}
\Delta_{(n)}= t^{-1/6}\left(C_1\,t^{\imath\alpha\sqrt{2/3}}+ C_2\,t^{-\imath\alpha\sqrt{2/3}}\right)+ C_3+ C_4\,t^{-2/3}\,.
\end{equation}
As in solution (\ref{1b}) before, on small scales the magnetic pressure still forces the perturbations to oscillate with an amplitude that drops as $t^{-{1/6}}$. This time, however, the oscillations do not decay to zero but to a finite constant value that depends on the initial conditions.
\item $\lambda_n=\lambda_J$: This special case corresponds to $\alpha=1$, when we can no longer use solution (\ref{gensol}). Instead, setting $\alpha=1$ into Eq.~(\ref{d2Delta}), the system of (\ref{d2Delta}), (\ref{dSigma}) gives
\begin{equation}\label{s2.3}
\Delta= C_1\ln t+ C_2+ C_3\,t^{-1/3}+ C_4\,t^{-2/3}\,.
\end{equation}
Therefore, at the magnetic Jeans length, where the field's pressure cancels out the gravitational pull of the matter, the magnetic tension becomes the sole player, takes over and leads to a weak (logarithmic) growth of the perturbations. Recall that, in the absence of tension stresses, perturbations with wavelength equal to the Jeans length remain constant (see solution (\ref{1c}) before). The growth seen in solution (\ref{s2.3}) demonstrates the opposing action between the field's pressure and tension on the linear evolution of density perturbations, which lies at the core of this investigation.
\item $\lambda_n=\sqrt{3/2}\,\lambda_J$: This is our second special case, corresponding to $\alpha=\sqrt{2/3}$ and to $\beta^2(t/t_0)^{2/3}\gg1$, in which case the system (\ref{d2Delta}) and (\ref{dSigma}) accepts the solution
\begin{equation}\label{s2.4}
\Delta= C_1\,t^{1/3}+ C_2+ C_3\,t^{-2/3}\,.
\end{equation}
Consequently on scales that are only slightly larger than the magnetic Jean's length the perturbations grow as $t^{1/3}$, instead of following the $\Delta\propto t^{2/3}$-law associated with much larger wavelengths (see solutions (\ref{1a}) and (\ref{s2.1})). This implies that the growth-rate of density perturbations increases gradually as we move on to scales progressively larger than $\lambda_J$ and the overall magnetic effect weakens (which is to be expected).\footnote{A closer look into the study of~\cite{VT} reveals that solutions (\ref{s2.3}) and (\ref{s2.4}) reside in the Newtonian equations as well, although not as distinct special cases, which is probably the reason they were not identified there.}
\end{itemize}
\subsection{Including the magneto-curvature effects}\label{ssIM-CEs}
In order to incorporate the magneto-curvature stresses seen in Eqs.~(\ref{D''}) and (\ref{D harmonic}) into our solutions, we need to involve the evolution formula of the rescaled 3-Ricci scalar (see expression (\ref{lcK})). Then, taking the time derivative of (\ref{D''}) and using (\ref{lcK}) and (\ref{S}), we arrive at the differential equation
\begin{equation}\label{D'''}
\dddot{\Delta}= - 6H\ddot{\Delta}- {7\over6}\,\bar{\rho}\dot{\Delta}+ {1\over2}\,H\bar{\rho}\Delta+ {2\over3}\,Hc_{\rm a}^2{\rm D}^2\Delta+ {2\over3}\,c_{\rm a}^2{\rm D}^2\dot{\Delta}- {2H\over\bar{\rho}}\,{\rm D}^2 \left(\Sigma_B^2-\Omega_B^2\right)\,.
\end{equation}
Harmonically decomposed the above reads
\begin{eqnarray}\label{D''' hd}
\nonumber \dddot{\Delta}_{(n)}&=& -6H\ddot{\Delta}_{(n)}- {7\over6}\,\bar{\rho} \left(1+\frac{3}{7}\,\alpha^2\right)\dot{\Delta}_{(n)}+ \frac{1}{2}\,H\bar{\rho}\left(1-\alpha^2\right)\Delta_{(n)} \nonumber\\ &&+{2\over3}\,H\beta^2\left({t\over t_0}\right)^{2/3} \left(\Sigma_B^2-\Omega_B^2\right)_{(n)}\,,
\end{eqnarray}
with $\alpha=\lambda_J/\lambda_n=$~constant and $\beta=(\lambda_H/\lambda_n)_0$. After equipartition, the above differential equation takes the form
\begin{eqnarray}\label{D''' eq}
\nonumber \frac{{\rm d}^3\Delta_{(n)}}{{\rm d}t^3}&=& -\frac{4}{t}\,\frac{{\rm d}^2\Delta_{(n)}}{{\rm d}t^2}- \frac{14}{9t^2}\,\left(1+\frac{3}{7}\,\alpha^2\right)\frac{{\rm d}\Delta_{(n)}}{{\rm d}t}+ \frac{4}{9t^3}\left(1-\alpha^2\right)\Delta_{(n)} \\&& +\frac{4}{9t}\,\beta^2\left({t\over t_0}\right)^{2/3} \left(\Sigma_B^2-\Omega_B^2\right)_{(n)}\,.
\end{eqnarray}
Finally, when $\alpha\neq1$, the system of (\ref{dSigma}) and (\ref{D''' eq}) solves to give
\begin{equation}\label{gen sol3}
\Delta_{(n)}= C_1\,t^{s_1}+ C_2\,t^{s_2}+ {1\over\alpha^2-1}\left(C_3+C_4\,t^{-2/3}\right)\,.
\end{equation}
with $s_{1,2}=-[1\mp\sqrt{25-24\alpha^2}]/6$. When $\alpha=1$, on the other hand we obtain
\begin{equation}
\Delta= C_1\,\ln t+ C_2- C_3\,t^{-1/3}+ C_4\,t^{-2/3}\,.
\end{equation}
For all practical purposes, the above results are identical to solutions (\ref{gensol}) and (\ref{s2.3}), implying that the inclusion of the magneto-curvature effects does not alter the linear evolution of the density perturbations. This is not surprising, since the spatial flatness of the FRW background ensures that the magneto-curvature stresses are too weak to make a noticeable difference.
\section{Discussion}\label{sD}
With the exception of the Cosmic Microwave Background (CMB), magnetic fields have been observed nearly everywhere in the cosmos. The idea of primordial magnetism has also been gaining ground because it could in principle explain all the large-scale $B$-fields seen in the universe today. If present, cosmological magnetic fields could have played a role during structure formation, since they can in principle generate and affect the evolution of all types of perturbations, namely scalar, vector and tensor distortions (see \S~\ref{ssTIs}). When it comes to scalar (density) perturbations, however, half of the magnetic effects are excluded, since all the available cosmological studies (with the exception of~\cite{VT} -- to the best of our knowledge) account only for field's pressure and bypass the magnetic tension.\footnote{In astrophysics the implications of the magnetic tension have been investigated in a number of studies looking at the physics of star formation, accretion discs and compact stars (e.g.~see~\cite{NCVR}-\cite{MM} and references therein).} Moreover, technically speaking, it is more straightforward to obtain analytic solutions before rather than after equipartition. The main difficulty comes from the Alfv\'en speed, which is constant throughout the radiation era but acquires a time-dependence after equilibrium. As result, the available dust-epoch solutions were obtained after imposing certain simplifying assumptions~\cite{TB1}-\cite{BMT}. In the present work we re-examine the magnetic implications for the evolution of baryonic density perturbations and try to address both of the aforementioned issues. Our study uses full general relativity, incorporates the effects of the field's tension and focuses on the post-recombination universe. The aim was to refine and extend previous relativistic studies, as well as provide a direct comparison with the existing Newtonian treatments of the issue. Above all, however, we wanted to investigate and reveal the as yet unknown role of the magnetic tension.
At the centre of our analysis is the wave-like equation monitoring the linear evolution of magnetised density perturbations. In contrast to previous approaches, this formula carries the effects of the magnetic tension, in addition to those of the field's pressure. After equipartition, the latter is the sole source of support against the gravitational pull of the matter. This leads to a purely magnetic Jeans length, which means that the magnetic pressure could in principle determine the first gravitationally bound formations. The tension stresses, on the other hand, are triggered by the elasticity of the field lines and by their natural tendency to react against any agent that distorts them from equilibrium. Among these are the magneto-curvature stresses, which result from the purely geometrical coupling between the $B$-field and the spatial geometry of the host spacetime. We have incorporated all the aforementioned effects into our analytic solutions in three successive steps of increasing inclusiveness. At first, we only considered the effects of the field's pressure, in which case our results were in full agreement with those of the previous Newtonian study. We then also accounted for the role of the magnetic tension and finally, to complete the picture, we incorporated the magneto-curvature stresses as well. Our results showed that the field's pressure and tension act against each other. The magnetic pressure, in particular, inhibits the growth of the perturbations, while the tension tends to enhance it. These effects were also found to be scale-dependent, with the pressure dominating well inside the (purely magnetic) Jeans length and with the tension taking over near the Jeans threshold. On much larger wavelengths, on the other hand, neither of these agents had a measurable effect and the perturbations evolved unaffected by the field's presence
More specifically, well inside the magnetic Jeans length and in the absence of any tension input, we found that the field's pressure forces the perturbations to oscillate with an amplitude that decreases as $\Delta\propto t^{-1/6}$ and decays (asymptotically) to zero. When the magnetic tension was included the oscillations still decayed (at the same rate), though now to a finite value instead of zero. Near the Jeans length the support of the field's pressure and the gravitational pull of the matter cancel each other out, thus leaving the magnetic tension as the sole player. This resulted into a slow logarithmic growth for the density perturbations, which revealed the (as yet unknown) opposing action of the aforementioned two magnetic agents on the linear evolution of density gradients. We expect an analogous effect near the Jeans length during the radiation era as well. Qualitatively speaking, the role of the magnetic tension demonstrated how versatile and unconventional the $B$-fields can be. Quantitatively, the tension effects were relatively weak because their contribution decays quickly (faster than that of the field's pressure) with the universal expansion. Nevertheless, it is conceivable that there can be physical situations where the field's tension could play a more prominent role. This should probably happen in the nonlinear phase of structure formation on scales considerably smaller than the magnetic Jeans length and more likely during the (typically) anisotropic collapse of a magnetised protogalactic cloud. Given the complexity of the nonlinear regime, however, one should have to employ numerical methods to complement the analytical work. Beyond the Jeans length, the overall magnetic effect was found to gradually fade away and the standard (non-magnetised) linear growth-rate of density perturbations was eventually re-established. Finally, the magneto-curvature stresses (which also result from the field's tension) were found to be too weak to leave a measurable imprint. This was largely expected, however, given the (assumed) spatial flatness of our FRW background.\footnote{In order to study the coupling between magnetism and spacetime geometry in detail and to investigate its potential implications in depth, one needs to allow for FRW backgrounds with nonzero spatial curvature.} What is particularly interesting about these stresses, is that their effect reverses depending on the sign of the spatial curvature (i.e.~on whether it is positive or negative -- see Eq.~(\ref{D''}) in \S~\ref{ssW-LE}). Therefore, if these magneto-geometrical effects were to be detected, they should also provide information about the universe's spatial geometry.\\
\textbf{Acknowledgments} JDB was supported by the the Science and Technology Facilities Council (STFC) of the United Kingdom.
|
{
"timestamp": "2018-05-25T02:11:13",
"yymm": "1803",
"arxiv_id": "1803.02747",
"language": "en",
"url": "https://arxiv.org/abs/1803.02747"
}
|
\section{Lubin-Tate $(\varphi,\Gamma)$-modules }
In the first two subsections we generalize some constructions and results from the theory of cyclotomic $(\varphi,\Gamma)$-modules over $k$ (i.e. with underlying Lubin Tate group ${\mathbb G}_m$) to the more general context of $(\varphi,\Gamma)$-modules over $k$ with arbitrary underlying Lubin Tate group. Namely, we define an exact functor from admissible torsion $k[[t]]$-modules with commuting semilinear actions by $\Gamma={\mathcal O}_F^{\times}$ and $\varphi$ to \'{e}tale $(\varphi,\Gamma)$-modules over $k$. The former category is closely related to that of $\psi$-stable lattices in \'{e}tale $(\varphi,\Gamma)$-modules ${\bf D}$, and we are lead to transpose some of Colmez's constructions \cite{col} involving the $\psi$-stable lattices ${\bf D}^{\natural}$ and ${\bf D}^{\sharp}$ to our context. The difference is that we no longer assume that the $\psi$-operator on
$k((t))$ satisfies $\psi(1)=1$, but this necessitates only minor modifications.
We then identify a category of admissible torsion $k[[t]]$-modules with $\Gamma$- and $\varphi$-action on which the above functor is fully faithful.
\label{muluta}
\subsection{$(\varphi,\Gamma)$-modules and torsion $k[[t]]$-modules}
Fix a Lubin-Tate formal power series $\Phi(t)$ for $F$ with respect to $\pi$. Put $\Gamma={\mathcal O}_F^{\times}$. The formula $\gamma\cdot
t=[\gamma]_{\Phi}(t)$ with $\gamma\in\Gamma$ defines an action of $\Gamma$ on
$k[[t]]$ and on $k((t))$. Consider the $k$-algebra$${\mathfrak O}=k[[t]][\varphi,\Gamma]$$ with
commutation rules given by$$\gamma\cdot
\varphi=\varphi\cdot\gamma,\quad\quad \gamma\cdot
t=[\gamma]_{\Phi}(t)\cdot\gamma,\quad\quad\varphi\cdot
t=\Phi(t)\cdot \varphi$$for $\gamma\in\Gamma$. Of course,
$\Phi(t)=[\pi]_{\Phi}(t)$ is congruent to $t^q$ in $k[[t]]$.\\
{\bf Definition:} A $\psi$-operator on $k[[t]]$ is an additive map $\psi:k[[t]]\to
k[[t]]$ such that
$\psi(\gamma\cdot t)=\gamma\cdot (\psi(t))$
for all $\gamma\in\Gamma$ and such
that the following holds true\footnote{We
do not require $\psi(1)=1$.}: If we view $\varphi$ as acting on
$k[[t]]$, then\begin{gather}\psi(\varphi(a)x)=a\psi(x)\quad\quad\mbox{ for }a,x\in k[[t]].\label{psichar}\end{gather}
\begin{lem}\label{erleicht} There is a surjective $\psi$-operator on
$k[[t]]$ which extends to
a surjective operator $\psi=\psi_{k((t))}$ on $k((t))$ satisfying formula (\ref{psichar}) analogously.
If $\Phi(t)=\pi t+t^q$ and $F\ne{\mathbb Q}_p$ we may choose $\psi_{k((t))}$ on $k((t))$ such that for $m\in{\mathbb Z}$ and
$0\le i\le q-1$ we have\begin{gather}\psi_{k((t))}(t^{mq+i})=\left\{\begin{array}{l@{\quad:\quad}l}0
& 0\le i\le q-2\\t^m& i=q-1\end{array}\right.\label{psineqp}.\end{gather}
If $\Phi(t)=\pi t+t^q$ and $F={\mathbb Q}_p$ we may choose $\psi_{k((t))}$ on $k((t))$ such that for $m\in{\mathbb Z}$ and
$0\le i\le q-1$ we have\begin{gather}\psi_{k((t))}(t^{mq+i})=\left\{\begin{array}{l@{\quad:\quad}l}\frac{q}{\pi}t^m& i=0\\ensuremath{\overrightarrow{0}}
& 1\le i\le q-2\\t^m& i=q-1\end{array}\right.\label{psieqqp}.\end{gather}
\end{lem}
{\sc Proof:} A construction of $\psi$ is explained in \cite{schven} section
3: First, the given formula defines $\varphi$ as an injective
endomorphism of ${\mathcal O}_F[[t]]$. Next, the map $$\varphi({\mathcal O}_F[[t]])^q\longrightarrow {\mathcal O}_F[[t]],\quad (a_0,\ldots,a_{q-1})\mapsto\sum_{i=0}^{q-1}a_it^{i}$$is surjective: this can be checked modulo $\pi$, hence follows from $\Phi(t)\equiv t^q$ modulo $\pi$. The map is also injective, as follows from Proposition 1.7.3 in \cite{peterlec}. It follows that
$\psi=\frac{1}{\pi}\varphi^{-1}{\rm tr}_{{\mathcal O}_F[[t]]/\varphi({\mathcal
O}_F[[t]])}$ defines an operator on ${\mathcal O}_F[[t]]$ satisfying
$\psi(\varphi(a)x)=a\psi(x)$. It induces an operator $\psi$ on $k[[t]]$,
extending to $\psi=\psi_{k((t))}$ on $k((t))$ according to formula
(\ref{psichar}). To see the commutation with the
$\Gamma$-action we proceed similarly in as \cite{schven} Remark 3.2 iv. Let ${\mathcal Z}$ denote the set of $\pi$-torsion points (in the maximal ideal of ${\mathcal O}_F$) for $\Phi$. Let $F_1$ denote the extension of $F$ generated by the elements of ${\mathcal Z}$. For $z\in{\mathcal Z}$ we have the ${\mathcal O}_F$-algebra morphism $\sigma_z:{\mathcal O}_F[[t]]\to {\mathcal O}_{F_1}[[t]]$ with $t\mapsto z+_{\Phi}t$ (where $z+_{\Phi}t$ indicates addition with respect to the formal group law $\Phi$). It follows from \cite{schven} formula (10) that ${\rm tr}_{{\mathcal O}_F[[t]]/\varphi({\mathcal
O}_F[[t]])}=\sum_{z\in{\mathcal Z}}\sigma_z$. For $\gamma\in\Gamma$ and $a=a(t)\in {\mathcal O}_F[[t]]$ we thus compute$$\varphi(\gamma\cdot (\psi(a(t))))=\varphi(\psi(a([\gamma]_{\Phi}(t))))=\psi(a([\gamma]_{\Phi}([\pi]_{\Phi}(t))))$$$$=\psi(a([\pi]_{\Phi}([\gamma]_{\Phi}(t))))=\varphi(\psi(a))([\gamma]_{\Phi}(t))=\frac{1}{\pi}\sum_{z\in{\mathcal Z}}(\sigma_z(a))([\gamma]_{\Phi}(t))$$$$=\frac{1}{\pi}\sum_{z\in{\mathcal Z}}\sigma_{[\gamma^{-1}]_{\Phi}(z)}(a([\gamma]_{\Phi}(t)))=\frac{1}{\pi}\sum_{z\in{\mathcal Z}}\sigma_{z}(a([\gamma]_{\Phi}(t)))=\varphi(\psi(\gamma\cdot a(t)))$$hence $\gamma\cdot (\psi(a))=\psi(\gamma\cdot a)$ as $\varphi$ is injective.
To see surjectivity of $\psi$ we may assume $\Phi(t)=\pi t+t^q$. (All Lubin Tate formal groups with respect to $\pi$ are isomorphic.) We then compute ${\rm tr}_{{\mathcal O}_F[[t]]/\varphi({\mathcal
O}_F[[t]])}(t^{mq+i})$ by looking at the matrix of $t^{mq+i}$ with respect to the basis $1,t,\ldots,t^{q-1}$. Namely, for $0\le i, j\le q-1$ and $m\in{\mathbb Z}$
we
have\begin{gather}t^{mq+i}t^j\equiv\left\{\begin{array}{l@{\quad:\quad}l}(t^q+\pi
t)^mt^{i+j}-m\pi(t^q+\pi
t)^{m-1}t^{i+j+1}& 0\le i+j\le q-2\\(t^q+\pi
t)^mt^{q-1}-m\pi(t^q+\pi
t)^{m}& i+j= q-1\\(t^q+\pi
t)^{m+1}t^{i+j-q}-(m+1)\pi(t^q+\pi
t)^m t^{i+j-q+1}& q\le i+j\le 2q-2\end{array}\right.\end{gather}modulo
$(\pi^2)$. For $1\le i\le q-2$ none of the $t^{mq+i}t^j$ contributes to $\frac{1}{\pi}\varphi^{-1}{\rm tr}_{{\mathcal O}_F[[t]]/\varphi({\mathcal
O}_F[[t]])}$. For $i=0$ the respective first summands in the first and
second line (in the above case distinction) together contribute $q$ many identical
summands. Thus, even when divided by $\pi$ their sum disappears modulo $\pi$
if $F\ne{\mathbb Q}_p$, whereas if $F={\mathbb Q}_p$ their sum is as
stated. For $i=q-1$ the second summand in the second line contributes once,
and the second summand in the third line contributes $(q-1)$ times. When comparing
the respective coefficients $m$ resp. $m+1$ we see that the outcome is as stated.\hfill$\Box$\\
In the following, we fix a surjective $\psi$-operator $\psi$ on $k[[t]]$ and
extend it to $\psi=\psi_{k((t))}$ on $k((t))$ as in Lemma \ref{erleicht}.\\
{\bf Definition:} An \'{e}tale $(\varphi,\Gamma)$-module over
$k((t))$ is an ${\mathfrak O}\otimes_{k[[t]]}k((t))$-module ${\bf D}$
which is finitely dimensional over $k((t))$ such that the
$k((t))$-linearized structure map is bijective:$${\rm id}\otimes\varphi:k((t))\otimes_{\varphi,k((t))} {\bf
D}\stackrel{\cong}{\longrightarrow}{\bf
D}.$$
For a $k[[t]]$-module $\Delta$ we write $\Delta^*={\rm
Hom}_k(\Delta,k)$ (algebraic dual). A $k[[t]]$-module $\Delta$ is called admissible if
$$\Delta[t]=\{x\in\Delta\,;\,tx=0\}$$
is a finite dimensional $k$-vector space.
\begin{pro}\label{nopsi} Let $\Delta$ be an ${\mathfrak O}$-module which is finitely
generated over $k[[t]][\varphi]$, admissible over
$k[[t]]$ and torsion over $k[[t]]$, and suppose that $t$ acts surjectively on $\Delta$. Then
$\Delta^*\otimes_{k[[t]]}k((t))$ is
in a natural way an \'{e}tale $(\varphi,\Gamma)$-module over
$k((t))$ and we have$${\rm dim}_k\Delta[t]={\rm dim}_{k((t))}(\Delta^*\otimes_{k[[t]]}k((t))).$$The contravariant functor\begin{gather}\Delta\mapsto
\Delta^*\otimes_{k[[t]]}k((t))\label{wienschluss}\end{gather}is exact.
\end{pro}
{\sc Proof:}\footnote{For $F={\mathbb Q}_p$ and $\Phi(t)=(1+t)^p-1$ this is
a construction of Colmez and Emerton, as recalled in \cite{breuil} Lemma
2.6.} We endow $\Delta^*$ with
a $k[[t]][\Gamma]$-action by putting $$(a\cdot
\ell)(\delta)=\ell(a\delta)\quad\quad \mbox{
and }\quad\quad (\gamma\cdot \ell)(\delta)=\ell(\gamma^{-1}\delta)$$for $a\in
k[[t]]$, $\ell\in \Delta^*$, $\delta\in\Delta$ and
$\gamma\in\Gamma$. Let $C$ be the cokernel of
$k[[t]]\otimes_{\varphi,k[[t]]}\Delta\stackrel{{\rm
id}\otimes \varphi}{\longrightarrow}\Delta$. As $\Delta$ is finitely generated over $k[[t]][\varphi]$ we see that $C$ is
finite dimensional over $k$ and is killed by some power of $t$, hence
$C^*\otimes_{k[[t]]}k((t))=0$. It follows that the $k((t))$-linear
map\begin{gather}\Delta^*\otimes_{k[[t]]}k((t))\stackrel{({\rm
id}\otimes \varphi)^*\otimes k((t))
}{\longrightarrow}(k[[t]]\otimes_{\varphi,k[[t]]}\Delta)^*\otimes_{k[[t]]}k((t))\label{emcobrza1}\end{gather}is
injective. We will show that it is also surjective. Let $\ell\in {\Delta}^*$ be non-zero. Choose $x\in{\Delta}$ with
$\ell(x)\ne0$. As $t$ acts surjectively on ${\Delta}$ we find for each $n\ge0$ some $y\in{\Delta}$
with $t^ny=x$ , hence $(t^n\ell)(y)=\ell(t^ny)=\ell(x)\ne0$, hence
$t^n\ell\ne0$. Therefore ${\Delta}^*$ is torsion free as a
$k[[t]]$-module. Choose a $k$-vector space complement of ${\Delta}[t]$ in
${\Delta}$ and let ${\Delta}_0^*$ be the sub vector space of ${\Delta}^*$
consisting of linear forms vanishing on it. It is easy to see that
${\Delta}_0^*$ generates ${\Delta}^*$ as a $k[[t]]$-module, and ${\rm dim}_k({\Delta}^*/t{\Delta}^*)={\rm dim}_k{\Delta}_0^*$. Moreover, ${\rm dim}_k{\Delta}_0^*={\rm dim}_k{\Delta}[t]$ is finite by assumption. Together we see that ${\Delta}^*$ is a free $k[[t]]$-module of rank ${\rm dim}_k{\Delta}[t]$. The same applies to $(k[[t]]\otimes_{\varphi,k[[t]]}{\Delta})^*$. It follows that the source and the target of the $k((t))$-linear map (\ref{emcobrza1}) have the same finite $k((t))$-vector
space dimension (namely ${\rm dim}_k{\Delta}[t]$), hence it is bijective. Next, we use the $\psi$-operator on $k[[t]]$ to define
the $k[[t]]$-linear map\begin{gather}
k[[t]]\otimes_{\varphi,k[[t]]}({\Delta}^*)\longrightarrow(k[[t]]\otimes_{\varphi,k[[t]]}{\Delta})^*\label{anneh}\end{gather}$$a\otimes\ell\quad\mapsto\quad[b\otimes
x\mapsto \ell(\psi(ab)x)].$$Let $a\in k[[t]]$ and $\ell\in {\Delta}^*$ both be non-zero. We find some
$b\in k[[t]]$ with $\psi(ab)\ne0$. Indeed, given some $\tilde{a}\in k[[t]]$ with $\psi(\tilde{a})\ne0$, we find non-zero $b,c\in k[[t]]$ with $ab=\tilde{a}\varphi(c)$, hence $\psi(ab)=\psi(\tilde{a}\varphi(c))=c\psi(\tilde{a})\ne0$. As $t$ acts surjectively on ${\Delta}$, so does $\psi(ab)$. Thus the map (\ref{anneh}) does not vanish on $a\otimes\ell$. Since $k[[t]]$ is free over
$\varphi(k[[t]])$ this proves that the map (\ref{anneh}) is
injective. As above we see that after base extension to $k((t))$ the $k((t))$-vector
space dimensions of its source and its target coincide. Thus, its base extension to $k((t))$ is an
isomorphism. Composing the latter with the
inverse of (\ref{emcobrza1}) gives a $k((t))$-linear
isomorphism$$k((t))\otimes_{\varphi,k((t))}(\Delta^*\otimes_{k[[t]]}k((t)))=(k[[t]]\otimes_{\varphi,k[[t]]}(\Delta^*))\otimes_{k[[t]]}k((t))\longrightarrow\Delta^*\otimes_{k[[t]]}k((t))$$which yields the desired $\varphi$-operator on $\Delta^*\otimes_{k[[t]]}k((t))$. The exactness of $\Delta\mapsto
\Delta^*\otimes_{k[[t]]}k((t))$ is clear.\hfill$\Box$\\
\begin{satz}\label{sosego} (Schneider) There is an equivalence between the category of \'{e}tale $(\varphi,\Gamma)$-modules over
$k((t))$ and the category of continuous representations of ${\rm Gal}(\overline{F}/F)$ on finite dimensional $k$-vector spaces.
\end{satz}
{\sc Proof:} A detailed proof can be found in \cite{peterlec}.\hfill$\Box$\\
\begin{lem} \label{abstrtor} Let $N$ be a $k$-vector space, and suppose that we are given a $k$-linear automorphism $\tau$ of $N$, a basis
${\mathcal N}$ of $N$ and monomials $g_{\nu}(t)\in k[t]$ of degree $0\le k_{\nu}\le q-1$ for ${\nu}\in {\mathcal
N}$. View $N$ as a $k[[t]]$-module with $t\cdot N=0$ and let $\Delta$ denote
the quotient of $k[[t]][\varphi]\otimes_{k[[t]]}N$ by the $k[[t]][\varphi]$-submodule $\nabla$
generated by the elements $$1\otimes
{\nu}+g_{\nu}(t)\varphi\otimes\tau({\nu})$$with ${\nu}\in {\mathcal
N}$. We then have:
(a) $k[[t]][\varphi]\otimes_{k[[t]]}N$ is
a torsion $k[[t]]$-module.
(b) The map $N\to \Delta[t]$ sending $n\in N$ to the class of $1\otimes n$ is
an isomorphism. In particular, $\Delta$ is admissible if $N$ is a finite dimensional $k$-vector space.
\end{lem}
{\sc Proof:} (a) As $\varphi\cdot t=t^q\cdot \varphi$ in $k[[t]][\varphi]$ we may write any element in $k[[t]][\varphi]\otimes_{k[[t]]}N$ as a
finite sum of elements of the form $a\varphi^n\otimes x$ with $a\in k[[t]]$,
$n\ge 0$ and $x\in N$. It is therefore enough to show \begin{gather}a\varphi^n\otimes x=0\quad\quad\mbox{ for each }a\in t^{q^n}k[[t]]\label{tdmneu}\end{gather}where $n\ge 0$ and $x\in N$. We may write
$a=a_0 t^{q^n}$ with $a_0\in k[[t]]$ and compute$$a\varphi^n\otimes x=a_0 t^{q^n}\varphi^n\otimes
x=a_0\varphi^nt\otimes x=0.$$
(b) It follows from formula (\ref{tdmneu}) that we may
write$$k[[t]][\varphi]\otimes_{k[[t]]}N\cong \bigoplus_{\nu\in{\mathcal
N}}\bigoplus_{i\ge 0}\bigoplus_{0\le\theta\le
q^{i}-1}k.t^{\theta}\varphi^i\otimes \tau(\nu).$$Consider the three $k$-sub
vector spaces \begin{gather}1\otimes N=\bigoplus_{\nu\in{\mathcal N}}k\otimes
\tau(\nu)=\bigoplus_{\nu\in{\mathcal N}}k\otimes
\nu,\notag\\ensuremath{\mathbb{C}}=\bigoplus_{\nu\in{\mathcal N}}\bigoplus_{i>0}\bigoplus_{0\le\theta<q^{i-1}k_{\nu}}k.t^{\theta}\varphi^i\otimes \tau(\nu),\label{bohrnerv}\\\nabla=\bigoplus_{\nu\in{\mathcal
N}}\bigoplus_{i>0}\bigoplus_{\epsilon\ge 0}k.t^{\epsilon}\varphi^{i-1}(1\otimes
{\nu}+g_{\nu}(t)\varphi\otimes\tau({\nu})).\label{bohrnerv0}\end{gather}Using the formula $\varphi\cdot
t=t^q\cdot \varphi$ we see$$t^{\epsilon}\varphi^{i-1}(1\otimes
{\nu}+g_{\nu}(t)\varphi^i\otimes\tau({\nu}))\in
k^{\times}.t^{\epsilon+q^{i-1}k_{\nu}}\varphi^i\otimes
\tau({\nu})+k[[t]]\varphi^{i-1}\otimes\nu.$$and that the sum in
(\ref{bohrnerv0}) only runs over the $0\le\epsilon<(q-1)q^{i-1}k_{\nu}-1$, or equivalently, the corresponding
$\theta=\epsilon+q^{i-1}k_{\nu}$ run over the $q^{i-1}k_{\nu}\le \theta\le
q^{i}-1$. Thus we
find\begin{gather}k[[t]][\varphi]\otimes_{k[[t]]}N\quad\cong\quad 1\otimes
N\quad\bigoplus\quad \nabla\quad\bigoplus\quad
C.\label{bohrnerv1}\end{gather}Consider the composed
map$$C\quad\longrightarrow\quad
k[[t]][\varphi]\otimes_{k[[t]]}N\quad\stackrel{t\cdot }{\longrightarrow}\quad
k[[t]][\varphi]\otimes_{k[[t]]}N\quad\longrightarrow\quad 1\otimes
N\quad\bigoplus\quad C$$where the first arrow is the inclusion, the last arrow
is the projection. This map is bijective, the critical point being the
computation$$t\cdot (k.t^{q^{i-1}k_{\nu}-1}\varphi^i\otimes
\tau(\nu))=k.t^{q^{i-1}k_{\nu}}\varphi^i\otimes
\tau(\nu)=k.\varphi^{i-1}t^{k_{\nu}}\varphi\otimes \tau(\nu)\equiv
k.\varphi^{i-1}\otimes \nu$$ modulo $\nabla$ (for $i>0$). It follows that indeed the image of $1\otimes N$ in $\Delta$ is the kernel of $t$ acting on $\Delta$.\hfill$\Box$\\
\subsection{$\psi$-stable lattices in $(\varphi,\Gamma)$-modules}
\begin{lem} An \'{e}tale $(\varphi,\Gamma)$-module ${\bf D}$ over
$k((t))$ naturally carries an additive operator $\psi$
satisfying $$\psi(a\varphi(x))=\psi(a)x\quad\quad\mbox{and}\quad\quad\psi(\varphi(a)x)=a\psi(x)$$for
all $a\in
k((t))$ and all $x\in {\bf
D}$, and commuting with the action of $\Gamma$.
\end{lem}
{\sc Proof:} We define the composed map$$\psi:{\bf D}\longrightarrow
k[[t]]\otimes_{\varphi,k[[t]]}{\bf D}\longrightarrow{\bf D}$$where the first
arrow is the inverse of the structure isomorphism ${\rm id}\otimes\varphi$,
and where the second arrow is given by $a\otimes x\mapsto \psi(a)x$. By
construction, it
satisfies $\psi(a\varphi(x))=\psi(a)x$. To see $\psi(\varphi(a)x)=a\psi(x)$
observe that by assumption we may write $x=\sum_ia_i\varphi(d_i)$ with $d_i\in
{\bf D}$ and $a_i\in k((t))$. We then
compute$$\psi(\varphi(a)x)=\sum_i\psi(\varphi(a)a_i\varphi(d_i))=\sum_i\psi(\varphi(a)a_i)d_i=a\sum_i\psi(a_i)d_i=a\sum_i\psi(a_i\varphi(d_i))=a\psi(x).$$Finally, let
$\gamma\in\Gamma$. As $\gamma$ and $\varphi$ commute on $k[[t]]$, and as
$\Gamma$ acts semilinearly on ${\bf D}$, the additive map
$k[[t]]\otimes_{\varphi,k[[t]]}{\bf D}\to k[[t]]\otimes_{\varphi,k[[t]]}{\bf
D}$, $a\otimes d\mapsto \gamma(a)\otimes\gamma(b)$ is the map corresponding
to $\gamma$ on $D$ under the isomorphism ${\rm id}\otimes\varphi$, and under
$a\otimes x\mapsto \psi(a)x$ it commutes with $\gamma$ on $D$ since $\gamma$
and $\psi$ commute on $k((t))$.\hfill$\Box$\\
In the following, by a lattice in a $k((t))$-vector space ${\bf D}$ we mean a free $k[[t]]$-sub
module containing a $k((t))$-basis of ${\bf D}$.
\begin{lem}\label{hitze3} Let $D$ be a lattice in an \'{e}tale $(\varphi,\Gamma)$-module ${\bf D}$.
(a) $\psi(D)$ is a lattice.
(b) If $\varphi(D)\subset D$ then $D\subset\psi(D)$.
(c) If $D\subset k[[t]]\cdot \varphi(D)$ then $\psi(D)\subset D$.
(d) If $\psi(D)\subset D$ then $\psi(t^{-1}D)\subset t^{-1}D$, and for each $x\in {\bf D}$ there is some $n(x)\in{\mathbb N}$ such that for all $n\ge n(x)$ we have $\psi^n(x)\in t^{-1}D$.
\end{lem}
{\sc Proof:} (a) Use $\psi(\varphi(a)x)=a\psi(x)$ for $a\in
k((t))$ and $x\in {\bf
D}$ to see that $\psi(D)$ is a $k[[t]]$-module.
(b) Choose $a\in k[[t]]$ with $\psi(a)=1$. For $d\in D$ we have $d=\psi(a\varphi(d))$ which belongs to $\psi(D)$ since $\varphi(D)\subset D$.
(c) Let $d\in D$. By assumption there are $e_i\in D$ and $a_i\in k[[t]]$ with $d=\sum_ia_i\varphi(e_i)$, hence $\psi(d)=\sum_i\psi(a_i)e_i\in D$.
(d) For $i\ge0$ we
have \begin{gather}\psi(\varphi^i(t^{-1})D)\subset\varphi^{i-1}(t^{-1})\psi(D)\subset\varphi^{i-1}(t^{-1})D\label{colmepeda}\end{gather}where
the second inclusion uses the assumption. Formula (\ref{colmepeda}) for $i=1$
shows $\psi(t^{-1}D)\subset\psi(\varphi(t^{-1})M)\subset t^{-1}D$. Moreover,
if $n(x)\in{\mathbb N}$ is such that $x\in\varphi^n(t^{-1})D$ for $n\ge n(x)$,
then iterated application of formula (\ref{colmepeda}) shows $\psi^n(x)\in
\psi^n(\varphi^n(t^{-1})D)\subset
\psi^{n-1}(\varphi^{n-1}(t^{-1})D)\subset\ldots\subset t^{-1}D$ for $n\ge n(x)$.\hfill$\Box$\\
\begin{lem} (a) There are lattices $D_0$, $D_1$ in ${\bf D}$ with $$\varphi(D_0)\subset t D_0\subset D_0\subset D_1\subset k[[t]]\cdot \varphi(D_1).$$(b) For $D_0$, $D_1$ as in (a) and for $n\ge0$ we have $\psi^n(D_0)\subset\psi^{n+1}(D_0)\subset D_1$.
\end{lem}
{\sc Proof:} (a) See the proof of Lemma 2.2.10 in \cite{peterlec} (which follows \cite{col} Lemme II 2.3).
(b) Choose $a\in k[[t]]$ with $\psi(a)=1$. For $x\in D_0$ we have $\psi^n(x)=\psi^{n+1}(a\varphi(x))\in\psi^{n+1}(D_0)$ since $\varphi(D_0)\subset tD_0$ implies $\varphi(x)\in D_0$ and hence $a\varphi(x)\in D_0$. This shows $\psi^n(D_0)\subset\psi^{n+1}(D_0)$. As $D_0\subset D_1\subset k[[t]]\cdot\varphi(N_1)$, an induction using Lemma \ref{hitze3} (c) shows $\psi^{n+1}(D_0)\subset D_1$.\hfill$\Box$\\
\begin{pro}\label{goldabreise} There exists a unique lattice ${\bf D}^{\sharp}$ in ${\bf D}$ with $\psi({\bf D}^{\sharp})={\bf D}^{\sharp}$ and such that for each $x\in {\bf D}$ there is some $n\in{\mathbb N}$ with $\psi^n(x)\in {\bf D}^{\sharp}$.
For any lattice $D$ in ${\bf D}$ we have $\psi^n(D)\subset {\bf D}^{\sharp}$ for all $n>>0$.
For any lattice $D$ in ${\bf D}$ with $\psi(D)=D$ we have $t{\bf D}^{\sharp}\subset D\subset {\bf D}^{\sharp}$.
\end{pro}
{\sc Proof:} Using the previous Lemmata, the proof is the same as the one given in \cite{col} Proposition II.4.2. \hfill$\Box$\\
\begin{pro}\label{wienabsage} (a) For any lattice $D$ in ${\bf D}$ contained
in ${\bf D}^{\sharp}$ and stable under
$\psi$ we have $\psi(D)=D$.
(b) The intersection ${\bf D}^{\natural}$ of all lattices in ${\bf D}$
contained in ${\bf D}^{\sharp}$ and stable under
$\psi$ is itself a lattice, and it satisfies $\psi({\bf D}^{\natural})={\bf
D}^{\natural}$.
\end{pro}
{\sc Proof:} (cf. \cite{col} Proposition II.5.11 and Corollaire II.5.12)
(a) Since ${\bf D}^{\sharp}$ as well as $D$ and $\psi(D)$ are
lattices in ${\bf D}^{\sharp}$, both ${\bf D}^{\sharp}/D$ and ${\bf
D}^{\sharp}/\psi(D)$ are finite dimensional $k$-vector spaces. Therefore
$\psi({\bf D}^{\sharp})={\bf D}^{\sharp}$ and $\psi(D)\subset D$ immediately imply $\psi(D)=D$.
(b) For any $D$ as in (a) we have $t{\bf D}^{\sharp}\subset D$ by what we saw
in (a) together with proposition
\ref{goldabreise}. This shows $t{\bf D}^{\sharp}\subset{\bf D}^{\natural}$, hence
${\bf D}^{\natural}$ is indeed a lattice, and $\psi({\bf D}^{\natural})={\bf
D}^{\natural}$ follows by applying (a) once more.\hfill$\Box$\\
\begin{lem}\label{staga} ${\bf D}^{\natural}$ and ${\bf D}^{\sharp}$ are stable under the action of $\Gamma$.
\end{lem}
{\sc Proof:} If $D$ is a lattice in ${\bf D}$, then so is $\gamma\cdot D$ for
any $\gamma\in\Gamma$. If in addition $\psi(D)\subset D$, resp. $\psi(D)= D$,
then also $\psi(\gamma\cdot D)\subset \gamma\cdot D$, resp. $\psi(\gamma\cdot
D)= \gamma\cdot D$. From these observations we immediately get
$\gamma\cdot{\bf D}^{\natural}={\bf D}^{\natural}$ and $\gamma\cdot{\bf D}^{\sharp}={\bf D}^{\sharp}$.
\hfill$\Box$\\
\begin{lem}\label{irrnopsi} If $\Delta$ (in the situation of Proposition
\ref{nopsi}) is a simple ${\mathfrak O}$-module, then
$\Delta^*\otimes_{k[[t]]}k((t))$ is a simple $(\varphi,\Gamma)$-module.
\end{lem}
{\sc Proof:} By construction, $\psi$ on $\Delta^*\otimes_{k[[t]]}k((t))$, when
restricted to $\Delta^*$, is the dual of $\varphi$ on
$\Delta$. Therefore the simplicity of $\Delta$ as an ${\mathfrak O}$-module means that $\Delta^*$
admits no non-trivial $k[[t]]$-sub module stable under $\Gamma$ and $\psi$. If ${\bf D}$ is a non-zero $(\varphi,\Gamma)$-submodule of
$\Delta^*\otimes_{k[[t]]}k((t))$ then also ${\bf D}^{\natural}$ is non-zero
and stable under $\Gamma$ and $\psi$, cf. Proposition \ref{wienabsage} and
Lemma \ref{staga}. As ${\bf D}^{\natural}\subset
(\Delta^*\otimes_{k[[t]]}k((t)))^{\natural}\subset \Delta^*$ we get ${\bf
D}^{\natural}=\Delta^*$, as desired.\hfill$\Box$\\
\begin{lem}\label{goldhoch} Let $f:{\bf D}_1\to {\bf D}_2$ be a morphism of \'{e}tale $(\varphi,\Gamma)$-modules over
$k((t))$.
(a) $f({\bf D}_1^{\sharp})\subset {\bf D}_2^{\sharp}$ and $f({\bf D}_1^{\natural})\subset {\bf D}_2^{\natural}$.
(b) If $f:{\bf D}_1\to {\bf D}_2$ is injective (resp. surjective), then so is $f:{\bf D}_1^{\sharp}\to {\bf D}_2^{\sharp}$.
(c) If $f:{\bf D}_1\to {\bf D}_2$ is injective (resp. surjective), then so is $f:{\bf D}_1^{\natural}\to {\bf D}_2^{\natural}$.
\end{lem}
{\sc Proof:} (a) $f({\bf D}_1^{\sharp})$ is a free $k[[t]]$-submodule of ${\bf D}_2$ on
which $\psi$ acts surjectively. Thus $f({\bf D}_1^{\sharp})+{\bf
D}_2^{\sharp}$ is a lattice satisfying the defining condition for ${\bf
D}_2^{\sharp}$ given in \ref{goldabreise}, hence $f({\bf D}_1^{\sharp})+{\bf
D}_2^{\sharp}={\bf
D}_2^{\sharp}$, hence $f({\bf D}_1^{\sharp})\subset {\bf
D}_2^{\sharp}$. Next, let $D=\{x\in {\bf D}_1^{\natural}\,;\,f(x)\in {\bf
D}_2^{\natural}\}$. It is a lattice in ${\bf D}_1$ since ${\bf
D}_1^{\natural}$ is a lattice, $f({\bf D}_1^{\natural})\subset f({\bf
D}_1^{\sharp})\subset {\bf D}_2^{\sharp}$ and ${\bf D}_2^{\sharp}/{\bf
D}_2^{\natural}$ is a finite dimensional $k$-vector space. It is also stable
under $\psi$, hence contains ${\bf D}_1^{\natural}$, hence $f({\bf D}_1^{\natural})\subset {\bf D}_2^{\natural}$.
(b) and (c) If $f:{\bf D}_1\to {\bf D}_2$ is injective then obviously so are
$f:{\bf D}_1^{\sharp}\to {\bf D}_2^{\sharp}$ and $f:{\bf D}_1^{\natural}\to
{\bf D}_2^{\natural}$. If $f:{\bf D}_1\to {\bf D}_2$ is surjectice then
$f({\bf D}_1^{\natural})$ is a lattice in ${\bf D}_2$ stable under $\psi$,
hence contains ${\bf D}_2^{\natural}$. To see $f({\bf D}_1^{\sharp})={\bf
D}_2^{\sharp}$ we proceed as in \cite{col} Proposition II.4.6 (iii). Namely, choose a lattice $D'$ in ${\bf D}_1$ with $f(D')={\bf
D}_2^{\sharp}$. Put $D=\sum_{n\ge0}\psi^n(D')$. By construction we have $\psi(D)=D$
as well as $f(D)={\bf
D}_2^{\sharp}$ (since $\psi ({\bf
D}_2^{\sharp})={\bf
D}_2^{\sharp}$). Proposition \ref{goldabreise} shows that $D$ is
again a lattice. Let $x\in {\bf
D}_2^{\sharp}$. For any $n\ge0$ choose $x_n\in {\bf
D}_2^{\sharp}$ and $\tilde{x}_n\in D$ with $\psi^n(x_n)=x$ and
$f(\tilde{x}_n)=x_n$. Put $u_n=\psi^n(\tilde{x}_n)$. For all $n>>0$ we have
$\psi^n(D)\subset {\bf D}_1^{\sharp}$ by proposition \ref{goldabreise}, hence
$u_n\in {\bf D}_1^{\sharp}$ for all $n>>0$. As ${\bf D}_1^{\sharp}$ is compact, the
sequence $(u_n)_n$ has an accummulation point $u\in {\bf D}_1^{\sharp}$. By
construction, $f(u)=x$.\hfill$\Box$\\
\subsection{Partial full faithfulness of $\Delta\mapsto
\Delta^*\otimes_{k[[t]]}k((t))$}
\begin{lem}\label{goldhochzeit} Let $0\to {\bf D}_1\to {\bf D}_2\to{\bf D}_3\to0$ be an exact sequence of \'{e}tale $(\varphi,\Gamma)$-modules over
$k((t))$. For each $i$ let $D_i\subset {\bf D}_i$ be a lattice with $\psi(D_i)=D_i$, and suppose that the above sequence restricts to an exact sequence \begin{gather}0\longrightarrow {D}_1\longrightarrow {D}_2\longrightarrow{D}_3\longrightarrow0.\label{sommer}\end{gather}If $D_1={\bf D}_1^{\natural}={\bf D}_1^{\sharp}$ and $D_3={\bf D}_3^{\natural}={\bf D}_3^{\sharp}$, then also $D_2={\bf D}_2^{\natural}={\bf D}_2^{\sharp}$.
\end{lem}
{\sc Proof:} By Lemma \ref{goldhoch} the sequence $0\to {\bf
D}_1^{\natural}\to {\bf D}_2^{\natural}\to{\bf D}_3^{\natural}\to0$ is exact
on the left and on the right. Comparing it with the sequence (\ref{sommer})
via ${\bf D}_1^{\natural}=D_1$, ${\bf D}_2^{\natural}\subset D_2$ and ${\bf
D}_3^{\natural}=D_3$, we immediately get ${\bf D}_2^{\natural}=D_2$. Next,
by Lemma \ref{goldhoch} the sequence $0\to {\bf D}_1^{\sharp}\to {\bf
D}_2^{\sharp}\to{\bf D}_3^{\sharp}\to0$ is exact on the left and on the
right. We compare it with the sequence (\ref{sommer}) via $D_1={\bf
D}_1^{\sharp}$, $D_2\subset {\bf D}_2^{\sharp}$ and $D_3={\bf
D}^{\sharp}_3$. We claim $$\psi({\bf D}_1\cap {\bf D}_2^{\sharp})={\bf
D}_1\cap {\bf D}_2^{\sharp}.$$Of course, $\psi({\bf D}_1\cap {\bf D}_2^{\sharp})\subset{\bf
D}_1\cap {\bf D}_2^{\sharp}$ is clear. To see ${\bf
D}_1\cap {\bf D}_2^{\sharp}\subset\psi({\bf D}_1\cap {\bf D}_2^{\sharp})$
take $x\in {\bf D}_1\cap {\bf D}_2^{\sharp}$. Choose $y\in {\bf D}_2^{\sharp}$
with $\psi(y)=x$. Choose $y'\in D_2$ mapping to the same element in ${\bf
D}^{\sharp}_3=D_3$ as $y$. We then have $\psi(y')\in D_2\cap {\bf D}_1=D_1$,
hence $\psi(y-y')-x\in D_1$, hence there is some $z\in D_1$ with
$\psi(z)=\psi(y-y')-x$, hence $x=\psi(y-y'-z)\in \psi({\bf D}_1\cap {\bf
D}_2^{\sharp})$ since $y-y'\in {\bf D}_1\cap {\bf
D}_2^{\sharp}$ and $z\in {\bf D}_1\cap {\bf
D}_2^{\sharp}$.
The claim is proven. By the definition of ${\bf D}_1^{\sharp}$ it implies ${\bf D}_1\cap
{\bf D}_2^{\sharp}={\bf D}_1^{\sharp}$, hence ${\bf D}_1\cap
{\bf D}_2^{\sharp}=D_1$ since $D_1={\bf D}_1^{\sharp}$. Thus, $D_2={\bf D}_2^{\sharp}$.\hfill$\Box$\\
{\bf Definition:} We say that an ${\mathfrak O}$-module $\Delta$ is torsion standard
cyclic if it satisfies the following properties. It is torsion as a
$k[[t]]$-module, it is generated by ${\rm ker}(t|_{\Delta})=\Delta[t]$ as
a $k[[t]][\varphi]$-module, there is a basis $e_0,\ldots, e_d$ of
$\Delta[t]$ consisting of eigenvectors for the action of $\Gamma$, and
there are $0\le k_i\le q-1$ and $\rho_i\in k^{\times}$ for $0\le i\le d$ such
that $$t^{k_i}\varphi e_{i-1}=\rho_ie_i$$for all $0\le i\le d$, reading
$e_{-1}=e_d$. Finally, it is required that $k_i>0$ for at least one $i$, as well as $k_i<q-1$
for at least one $i$.
\begin{pro}\label{altfund} (a) $t$ acts surjectively on $\Delta$, and there is a canonical isomorphism of free $k[[t]]$-modules of rank
$d+1$\begin{gather}\Delta^*\cong k[[t]]\otimes_k(\Delta[t]^*).\label{saabvo50}\end{gather}
(b) If for any $1\le j\le d$ there is some $0\le i\le d$ with $k_i\ne k_{i+j}$, then $\Delta$ is irreducible as a $k[[t]][\varphi]$-module.
(c) For $0\le i\le d$ let $\eta_i:\Gamma\to k^{\times}$ be the character with
$\gamma\cdot e_i=\eta_i(\gamma)e_i$ for all $\gamma\in\Gamma$. Suppose that
for any $1\le j\le d$ which satisfies $k_i=k_{i+j}$ for all $0\le i\le d$
there is some $0\le i\le d$ with $\eta_{i}\ne\eta_{i+j}$. Then $\Delta$ is
irreducible as an ${\mathfrak O}$-module.
(d) At least after a finite extension of $k$ we have: $\Delta$ admits a filtration such that each associated
graded piece is an
irreducible torsion standard cyclic ${\mathfrak O}$-module. If $p$ does not divide $d+1$ then $\Delta$ is even the direct sum of
irreducible torsion standard cyclic ${\mathfrak O}$-modules.
\end{pro}
{\sc Proof:} (This is very similar to \cite{dfun} Proposition 6.2.) (a) For
$0\le j\le d$
consider $$w_j=k_j+qk_{j-1}+\ldots+q^jk_0+q^{j+1}k_d+\ldots+q^dk_{j+1}.$$Repeated
substitution of $\varphi\cdot t=t^q\cdot \varphi$ (recall $\Phi(t)=t^q$ modulo $\pi$) shows that $t^{w_j}\varphi^{d+1}e_j\in k^{\times}e_j$. As $k_i>0$ for at least one $i$ we have $w_j>0$, and hence $e_j\in t\Delta$. As $\Delta[t]$ is generated over $k$ by all $e_j$ it follows that $\Delta[t]\subset t\Delta$. As $\Delta$ is generated over $k[[t]][\varphi]$ by $\Delta[t]$, the equation $\varphi\cdot t=t^q\cdot \varphi$ therefore shows $\Delta\subset t\Delta$, i.e. $t$ acts surjectively on $\Delta$. As in the proof of Proposition \ref{nopsi} we deduce that $\Delta^*$ is a free $k[[t]]$-module of rank $d+1$. Consider the $k[[t]][\varphi]$-linear map \begin{gather}k[[t]][\varphi]\otimes_{k[[t]]}\Delta[t]/\nabla\longrightarrow\Delta\label{bohrlaerm}\end{gather}where $\nabla$ is the $k[[t]][\varphi]$-submodule of $k[[t]][\varphi]\otimes_{k[[t]]}\Delta[t]$ generated by the elements $t^{k_i}\varphi\otimes e_{i-1}=1\otimes \rho_ie_i$. By the assumption on $\Delta$ the map (\ref{bohrlaerm}) is surjective. But it is also injective, because Lemma \ref{abstrtor} tells us that it induces an isomorphisms between the respective kernels of $t$. We view the bijective map (\ref{bohrlaerm}) as an identification. The proof of Lemma \ref{abstrtor} yielded a canonical $k$-vector space
decomposition $\Delta=C\oplus \Delta[t]$ where the $k$-sub vector
space $C$ of $\Delta$ is generated by the image elements of the elements
$t^{\theta}\varphi^r\otimes e\in k[[t]][\varphi]\otimes_{k[[t]]}\Delta[t]$
which do not belong to $1\otimes \Delta[t]$ (for some $e\in\Delta[t]$, and
some $\theta, r\ge0$). We may thus identify $\Delta[t]^*={\rm Hom}_k(\Delta[t],k)$ with the subspace of
$\Delta^*={\rm Hom}_k(\Delta,k)$ consisting of all
$f\in \Delta^*$ with $f|_C=0$. This yields the
isomorphism (\ref{saabvo50}).
(b) Let $Z$ be a non zero $k[[t]][\varphi]$-sub module of $\Delta$. With $\Delta$ also $Z$ is a torsion $k[[t]]$-module, hence ${\rm ker}(t|_{Z})=Z[t]$ is non zero. For non zero elements $z=\sum_{0\le i\le d}x_ie_i$ of $Z[t]$ (with $x_i\in k$) put \begin{align}\eta(z)&={\rm max}\{k_i\,|\,0\le i\le d, x_i\ne0\},\notag\\\Lambda(z)&=t^{\eta(z)}\varphi z.\notag\end{align}Then $\Lambda(z)$ is again a non zero element of $Z[t]$. The hypothesis (for any $1\le j\le d$ there is some $0\le i\le d$ with $k_i\ne k_{i+j}$) shows that for sufficiently large $n\ge0$ we have $\Lambda^n(z)\in k^{\times}e_i$ for some $0\le i\le d$. For such $n$ we then even have $\Lambda^{n+j}(z)\in k^{\times}e_{i+j}$ for all $j\ge0$. It follows that $Z$ contains all $e_i$, hence $Z=H$.
(c) Let $0\ne
Z\subset \Delta$ be a nonzero
${\mathfrak O}$-submodule. Choose a non zero $z\in Z[t]$ such that, writing $z=\sum_{0\le i\le
m}x_{i}e_i$ with $x_i\in k$, the number $\nu(z)=|\{i\,|\,x_{i}\ne0\}|$ is minimal (for all non zero $z\in Z[t]$). If $\nu(z)=1$ then we obtain $Z=H$ as in the proof of (b). If $\nu(z)>1$ we use the function $\Lambda$ already employed in the proof of (b). For all $n\ge0$ we have $\nu(\Lambda^n(z))\le \nu(z)$, hence $\nu(\Lambda^n(z))= \nu(z)$ by the choice of $z$. Thus $x_{i}\ne0$ and $x_{i+j}\ne0$ for some $i,j$, with $j$ violating the hypothesis in (b). By the hypothesis in (c), replacing $i$ by $i+n$ and $z$ by $\Lambda^n(z)$ we may assume that
$\eta_i\ne\eta_{i+j}$. Pick $\gamma\in\Gamma$ with
$\eta_i(\gamma)\ne\eta_{i+j}(\gamma)$, and pick $a\in k^{\times}$ with $a
e_i=\gamma\cdot e_i$. Then $az-\gamma\cdot z$ is a non zero element in $Z[t]$
with $\nu(az-\gamma\cdot z)<\nu(z)$: a contradiction.
(d) Arguing by induction on $d$ we may assume by (c) that there is some $1\le j\le d$ which satisfies
$k_i=k_{i+j}$ and $\eta_{i}=\eta_{i+j}$ for all $0\le i\le d$. It necessarily
is a divisor of $d+1$. Passing to a
finite extension of $k$ if necessary we may assume that there is a $(d+1)$-st
root of $\prod_{i=0}^d\rho_i$ in $k$. Thus, rescaling the $e_i$ if necessary we may
assume $\rho_i=\rho_j$ for all $i, j$. Consider the $k$-sub vector space $V$ of $\Delta[t]$ spanned by the vectors $\epsilon_i=e_{ij}$ for $0\le i< \frac{d+1}{j}$. Then $$(\prod_{i=1}^j\rho_i^{-1})t^{k_j}\varphi\cdots t^{k_1}\varphi$$induces the automorphism $f$ of $V$ with $f(\epsilon_i)=\epsilon_{i+1}$ (where we understand $\epsilon_{\frac{d+1}{j}}=\epsilon_0$). Choose (after passing to a finite extension of $k$ if necessary) an $f$-stable filtration $0=V_0\subset V_1\subset\ldots\subset V_{\frac{d+1}{j}}=V$ such that each $V_i/V_{i-1}$ is one dimensional. Then define for $0\le s\le\frac{d+1}{j}$ the ${\mathfrak O}$-sub module $\Delta_s={\mathfrak O} V_{0}+\cdots+{\mathfrak O} V_{s}$ of $\Delta$. It induces on $\Delta[t]$ the filtration$$\Delta_s[t]= \Delta_{s-1}[t]+V_{s}+t^{k_1}\varphi V_s+\ldots+ t^{k_{j-1}}\varphi\cdots t^{k_1}\varphi V_s.$$By construction, each $\Delta_{i+1}/\Delta_{i}$ is torsion standard cyclic, and the induction hypothesis applies. If $p$ does not divide $\frac{d+1}{j}$ then there is even an $f$-stable direct sum decomposition $V=\oplus_sV_{[s]}$ with one dimensional $V_{[s]}$'s. Then $\Delta=\oplus_s\Delta_{[s]}$ with $\Delta_{[s]}={\mathfrak O}V_{[s]}$ is a direct sum decomposition of $\Delta$, and each $\Delta_{[s]}$ is torsion standard cyclic, and the induction hypothesis applies.\hfill$\Box$\\
\begin{lem}\label{papa82} Let ${\bf D}=\Delta^*\otimes_{k[[t]]}k((t))$ be the associated \'{e}tale $(\varphi,\Gamma)$-module over
$k((t))$, cf. Proposition \ref{nopsi}. We have ${\bf D}^{\natural}=\Delta^*={\bf D}^{\sharp}$.
\end{lem}
{\sc Proof:} As we noticed in the proof of Lemma \ref{altfund}, $\Delta^*$ is a free $k[[t]]$-module, hence the natural map $\Delta^*\to{\bf
D}=\Delta^*\otimes_{k[[t]]}k((t))$ is injective; we view it as an inclusion.
The $\varphi$-operator on $\Delta$ induces the $\psi$-operator on ${\bf D}$, in such a way that $\psi(\Delta^*)=\Delta^*$ since $\varphi$ acts injectively on $\Delta$. Thus, we know ${\bf D}^{\natural}\subset \Delta^*\subset{\bf D}^{\sharp}$.
From Proposition \ref{goldabreise} we get $t\Delta^*\subset{\bf
D}^{\natural}$, hence $t(\Delta^*/{\bf
D}^{\natural})=0$, hence $\Delta^*/{\bf
D}^{\natural}$ is dual to a subspace $W$ of $\Delta[t]$ stable under $\varphi$. To prove ${\bf D}^{\natural}=\Delta^*$ it is therefore enough to prove that $\Delta[t]$ does not contain a non-zero subspace $W$ stable under $\varphi$. Assume that such a $W$ does exist. A non-zero element $\beta\in W$ may be written as $\beta=\sum_{i=0}^{d}\alpha_i e_i$ with $\alpha_i\in k$. Let $k={\rm max}\{k_{i+1}\,|\,\alpha_i\ne0\}$. Since by assumption $k_i>0$ for at least one $i$, replacing $\beta$ by $\varphi^r\beta$ for some $r\in{\mathbb N}$ if necessary, we may assume $k>0$. But then $t^k\varphi\beta$ is a non-zero linear combination of the $e_i$, whereas we also have $t\varphi\beta=0$ since $\varphi\beta\in W\subset \Delta[t]$: a contradiction.
From Proposition \ref{goldabreise} we get $t{\bf
D}^{\sharp}\subset\Delta^*$. Similarly as above we therefore see that to prove
$\Delta^*={\bf D}^{\sharp}$ it is enough to prove that we cannot write
${\Delta}={\widetilde{\Delta}}/{W}$ for an ${\mathfrak O}$-module
$\widetilde{\Delta}$ and a non-zero $\varphi$-stable submodule $W$ of
${\widetilde{\Delta}}[t]$ such that $t$ acts surjectively on
$\widetilde{\Delta}$. Assume that such ${\widetilde{\Delta}}$ and $W$ do
exist. We then find some $\beta=\sum_{i=0}^{d}\alpha_i\widetilde{e}_i$ with
$\alpha_i\in k$, where $\widetilde{e}_i\in \widetilde{\Delta}$ lifts $e_i$,
such that $t\beta$ is a non-zero element in $W$. But then we may even assume $\beta=\widetilde{e}_i$ for some fixed $i$. As $t^q\varphi\beta=\varphi t\beta\ne0$ we successively find $t^{q-1}\varphi\beta\ne0$, $t^{q-1}\varphi t^{q-1}\varphi\beta\ne0$, $t^{q-1}\varphi t^{q-1}\varphi t^{q-1}\varphi\beta\ne0$ etc.. But this means $q-1=k_i$ for each $i$, contradicting the hypothesis.\hfill$\Box$\\
{\bf Definition:} Let ${\rm Mod}^{\clubsuit}({\mathfrak O})$ denote
the category of ${\mathfrak O}$-modules $\Delta$ which are finitely
generated over $k[[t]][\varphi]$, admissible and torsion over $k[[t]]$, and
which admit a filtration such that each associated graded piece is a torsion standard
cyclic ${\mathfrak O}$-module.
\begin{pro}\label{wienaus} The restriction of the functor (\ref{wienschluss}) to the category
${\rm Mod}^{\clubsuit}({\mathfrak O})$ is exact and fully faithful.
\end{pro}
{\sc Proof:} We already know that the functor is exact. Next, we claim ${\bf
D}^{\natural}=\Delta^*={\bf D}^{\sharp}$ for ${\bf
D}=\Delta^*\otimes_{k[[t]]}k((t))$ if $\Delta\in {\rm
Mod}^{\clubsuit}({\mathfrak O})$. Indeed, for torsion standard
cyclic $\Delta$ this is shown in Lemma \ref{papa82}, for general $\Delta\in {\rm
Mod}^{\clubsuit}({\mathfrak O})$ it then follows from Lemma
\ref{goldhochzeit}. Thus, invoking Pontrjagin duality (cf. e.g. Proposition 5.4 in \cite{schven}), the reverse functor (on the essential image of the
functor under discussion) is given by ${\bf D}\mapsto ({\bf
D}^{\natural})^*$, where $({\bf
D}^{\natural})^*$ denotes the topological dual of ${\bf
D}^{\natural}$.\hfill$\Box$\\
\subsection{Standard cyclic \'{e}tale $(\varphi,\Gamma)$-modules}
\label{standardcycet}
\begin{pro}\label{fudonsup} Assume
$\Phi(t)=\pi t+t^q$ and that $\psi_{k((t))}$ is as in Lemma \ref{erleicht}. Let $\Delta$ be a torsion standard
cyclic ${\mathfrak O}$-module, with $d$, $e_i$, $k_i$, $\rho_i$, $\eta_i$ as in the
definition resp. as in Proposition \ref{altfund}. The \'{e}tale $(\varphi,\Gamma)$-module $\Delta^*\otimes_{k[[t]]}k((t))$ over
$k((t))$ admits a $k((t))$-basis $f_0,\ldots, f_d$ such that for all $0\le j\le d$ we have\begin{gather}\varphi(f_{j-1})=\rho_{j-1}^{-1}t^{1+k_j-q}f_j\label{tobfrag}\end{gather} (reading
$f_{-1}=f_d$), and moreover \begin{gather}\gamma\cdot
f_j-\eta_j^{-1}(\gamma) f_j\in tk[[t]]f_j\quad\quad\mbox{ for all }\gamma\in\Gamma.\label{monueber}\end{gather}
\end{pro}
{\sc Proof:} We first assume $F\ne{\mathbb Q}_p$ and use formula (\ref{psineqp}). Put $N=\oplus_{i=0}^dk.e_i$. As explained in the proof of Proposition \ref{altfund}, we have a bijective map (\ref{bohrlaerm}) which we view as an identification. In particular, Lemma \ref{abstrtor} and its proof apply. In the context of that proof we identify $e_i$ with the class of $1\otimes e_i$ in
$\Delta$. By formula (\ref{bohrnerv1}) we have a $k$-linear isomorphism
$(1\otimes N)\oplus C
\cong \Delta$ with $C$ as in formula (\ref{bohrnerv}). For $0\le j\le d$ we
may therefore define $f_j\in \Delta^*$ by asking $f_j(C)=0$ and $f_j(e_i)=\delta_{ij}$ for
$0\le i\le d$. Proposition \ref{altfund} tells us that $f_0,\ldots, f_d$ is a
$k[[t]]$-basis of $\Delta^*$. For $\theta,r\ge0$ and any $i,j$ we have $f_j(t^{\theta}\varphi^r\otimes e_i)\ne0$ if and only if $r\equiv j-i$ modulo $(d+1){\mathbb Z}$ and $\theta=k_j+qk_{j-1}+\ldots+q^{r-1}k_{j-r+1}$. As before, $\psi\in{\rm End}_k(\Delta^*)$ is defined by $(\psi(f))(x)=f(\varphi(x))$ for $x\in\Delta$, $f\in\Delta^*$. We claim\begin{gather}\psi(t^{m+k_j+1}f_j)=\rho_{j-1}\psi_{k((t))}(t^{m})tf_{j-1}\label{neurech}\end{gather}for
all $j$, all $m\ge-k_j-1$. Indeed, for
$0\le i\le d$ and $\theta, r\ge0$ we have$$(\psi(t^{m+k_j+1}f_j))(t^{\theta}\varphi^r\otimes
e_i)=f_j(t^{m+k_j+1}\varphi t^{\theta}\varphi^r\otimes e_i).$$If
$m+1\notin{\mathbb Z}q$ then this shows $(\psi(t^{m+k_j+1}f_j))(t^{\theta}\varphi^r\otimes
e_i)=0$ by what we pointed out above. But $m+1\notin{\mathbb Z}q$ also implies
$\psi_{k((t))}(t^{m})=0$. In the
case $m+1=qn$ (some $n\in{\mathbb Z}$) we compute
$$(\psi(t^{m+k_j+1}f_j))(t^{\theta}\varphi^r\otimes
e_i)=f_j(t^{k_j+qn}\varphi t^{\theta}\varphi^r\otimes
e_i)=f_j(t^{k_j}\varphi t^{n+\theta}\varphi^r\otimes
e_i)$$$$=\rho_{j-1}f_{j-1}(t^{n+\theta}\varphi^r\otimes
e_i)=(\rho_{j-1}\psi_{k((t))}(t^{m})tf_{j-1})(t^{\theta}\varphi^r\otimes
e_i)$$where we used
$\psi_{k((t))}(t^{m})=t^{n-1}$. We have proven formula
(\ref{neurech}).
On the other hand, by tracing the construction in Proposition \ref{nopsi} we see
that $\varphi(tf_{j-1})$ is characterized by satisfying
\begin{gather}\psi(t^{m} \varphi(tf_{j-1}))=\psi_{k((t))}(t^{m})tf_{j-1}\label{sekrgeschwaetz}\end{gather} for all
$m$. Comparing formulae (\ref{neurech}) and (\ref{sekrgeschwaetz}) we find
$\varphi(tf_{j-1})=\rho_{j-1}^{-1}t^{k_j+1}f_j$ which is equivalent with formula (\ref{tobfrag}). Next, for $\gamma\in\Gamma$ we compute$$(\gamma\cdot
f_j)(e_i)=f_j(\gamma^{-1}\cdot
e_i)=f_j(\eta_i(\gamma^{-1})e_i)=(\eta_i(\gamma^{-1})f_j)(e_i)=(\eta_j(\gamma^{-1})f_j)(e_i).$$Here the last equation is trivial if $i=j$, whereas if $i\ne j$ both sides vanish. This shows $(\gamma\cdot
f_j-\eta_j(\gamma^{-1})f_j)|_N=0$, and hence $\gamma\cdot
f_j-\eta_j(\gamma^{-1})f_j\in t\Delta^*=tk[[t]]\{f_0,\ldots, f_d\}$. On the other hand, by what we pointed out above, $(\gamma\cdot f_j)(t^{\theta}\varphi^r\otimes e_i)=f_j([\gamma]_{\Phi}(t)^{\theta}\varphi^r\otimes e_i)$ vanishes if $r+i-j\notin (d+1){\mathbb Z}$, and this shows $\gamma\cdot f_j\in k[[t]]f_j$. We trivially have $\eta_j(\gamma^{-1})f_j\in k[[t]]f_j$, and hence altogether $\gamma\cdot
f_j-\eta_j(\gamma^{-1})f_j\in tk[[t]]\{f_0,\ldots, f_d\}\cap k[[t]]f_j=tk[[t]]f_j$, formula (\ref{monueber}).
Now we assume $F={\mathbb Q}_p$ and use formula (\ref{psieqqp}). Let us suppose for simplicity that $\pi=q$. For $0\le j\le d$ we
may define $f_j\in \Delta^*$ as follows. For $\theta, r\ge0$ (and any $i, j$) we require $f_j(t^{\theta}\varphi^r\otimes e_i)\ne0$ if and only if $r\equiv j-i$ modulo $(d+1){\mathbb Z}$ and there are $a_1,\ldots,a_{r-1}\in\{0,1\}$ such that $$\theta=k_j+qk_{j-1}+\ldots+q^{r-1}k_{j-r+1}+\sum_{i=1}^{r-1}a_iq^{i-1}(1-q);$$if this is the case we put $$f_j(t^{\theta}\varphi^r\otimes e_i)=\rho_{j-1}\rho_{j-2}\cdots\rho_{j-r}.$$(As usual, the subindices of the $\rho_?$ are read modulo $(d+1){\mathbb Z}$.) Again $f_0,\ldots, f_d$ is a
$k[[t]]$-basis of $\Delta^*$. Again we claim formula (\ref{neurech}). As before we see that both sides vanish if $m\notin{\mathbb Z}q-1\cup {\mathbb Z}q$, and coincide if $m\in{\mathbb Z}q-1$. But the same computation also shows their coincidence if $m=qn$ for some $n\in{\mathbb N}$, as follows:$$(\psi(t^{m+k_j+1}f_j))(t^{\theta}\varphi^r\otimes
e_i)=f_j(t^{k_j+1}\varphi t^{n+\theta}\varphi^r\otimes
e_i)$$$$=\rho_{j-1}f_{j-1}(t^{n+\theta+1}\varphi^r\otimes
e_i)=(\rho_{j-1}\psi_{k((t))}(t^{m})tf_{j-1})(t^{\theta}\varphi^r\otimes
e_i)$$where we used
$\psi_{k((t))}(t^{m})=t^{n}$. With formula (\ref{neurech}) being established, the remaining arguments are exactly as before.\hfill$\Box$\\
{\bf Definition:} We say that an \'{e}tale $(\varphi,\Gamma)$-module ${\bf D}$
of dimension $d+1$ over
$k((t))$ is standard cyclic if it admits a $k((t))$-basis $f_0,\ldots, f_d$ such that
there are $\sigma_j\in k^{\times}$, characters $\alpha_j:\Gamma\to k^{\times}$ and $m_j\in \{1-q,\ldots,-1,0\}$ for $0\le j\le d$ satisfying the following conditions:$$(m_0,\ldots, m_d)\notin\{(0,\ldots,0), (1-q,\ldots,1-q)\},$$$$\varphi(f_{j-1})=\sigma_j t^{m_j}f_j\quad\quad\mbox{ for all }j\mbox{ (reading }f_{-1}=f_d),$$$$\gamma\cdot f_j-\alpha_j(\gamma) f_j\in tk[[t]]\{f_0,\ldots,f_d\}\quad\quad\mbox{ for all }\gamma\in\Gamma.$$
\begin{lem}\label{ossa} (a) The constant $\prod_{j=0}^d\sigma_j\in k^{\times}$
as well as, up to cyclic permutation, the ordered tuple
$((\alpha_0,m_0),\ldots,(\alpha_d, m_d))$, are uniquely determined by the
isomorphism class of the $(\varphi,\Gamma)$-module ${\bf D}$.
(b) $\alpha_1,\ldots,\alpha_d$ are uniquely determined by $\alpha_0$ (and $m_0,\ldots,m_d$).
\end{lem}
{\sc Proof:} (a) In the following, for elements of ${\rm GL}_{d+1}(k((t)))$ we read the (two) respective indices of their entries always modulo $(d+1){\mathbb Z}$.
The effect of $\varphi$ on the basis $f_0,\ldots, f_d$ is described by $T=(T_{ij})_{0\le i,j\le d}\in{\rm GL}_{d+1}(k((t)))$ with $T_{i, i+1}=\sigma_it^{m_i}$ for $0\le i\le d$, but $T_{i, j}=0$ for $j\ne i+1$.
Let $\sigma'_j\in k^{\times}$ and $((\alpha'_0,m'_0),\ldots,(\alpha'_d, m'_d))$ be another datum as above, let ${\bf D}'$ be an \'{e}tale $(\varphi,\Gamma)$-module admitting a $k((t))$-basis $f'_0,\ldots, f'_d$ with $\varphi(f'_{j-1})=\sigma'_j t^{m'_j}f'_j$ and $\gamma\cdot f'_j-\alpha'_j(\gamma) f'_j\in tk[[t]]\{f'_0,\ldots,f'_d\}$ for $\gamma\in\Gamma$. Define $T'=(T'_{ij})_{0\le i,j\le d}\in{\rm GL}_{d+1}(k((t)))$ similarly as above.
Suppose that there is an isomorphism of $(\varphi,\Gamma)$-modules ${\bf
D}'\cong {\bf D}$. With respect to the bases $f_{\bullet}$ and
$f'_{\bullet}$ it is described by some $A(t)=(a_{i,j}(t))_{0\le i,j\le d}\in
{\rm GL}_{d+1}(k((t)))$. In view of $\varphi\cdot t= \Phi(t)\cdot \varphi$,
the compatibility of the isomorphism with the respective $\varphi$-actions
comes down to the matrix equation $$T\cdot A(t)=A(\Phi(t))\cdot T'.$$For the individual entries this is equivalent
with$$a_{i,j}(t)=\sigma'_j\sigma_i^{-1}t^{m'_j-m_i}a_{i-1,j-1}(\Phi(t))$$for
all $i,j$. Iteration of this equation yields$$a_{i,j}(t)=(\prod_{\ell=0}^d
\sigma'_{j-\ell}\sigma_{i-\ell}^{-1}(\Phi^{\ell}(t))^{m'_{j-\ell}-m_{i-\ell}})a_{i,j}(\Phi^{d+1}(t))$$for
all $i,j$. (Here $\Phi^{\ell}(t)$ resp. $\Phi^{d+1}(t)$ means
$\Phi(\Phi(\ldots \Phi(t)\ldots))$.) From this we deduce that for fixed $i, j$ either
$a_{i,j}$ is a non zero constant and $\prod_{\ell=0}^d
\sigma'_{j-\ell}\sigma_{i-\ell}^{-1}=1$ and $m'_{j-\ell}=m_{i-\ell}$ for all
$\ell$, or $a_{i,j}=0$. But since $A(t)$ is invertible we do find $i,j$ with
$a_{i,j}\ne 0$. It already follows that
$\prod_{j=0}^d\sigma_j=\prod_{j=0}^d\sigma'_j$ and that $(m'_0,\ldots,m'_d)$
coincides with $(m_0,\ldots, m_d)$ up to cyclic permutation. But since in
addition we just saw that $A$ is a constant matrix, with $a_{i,j}=0$ if and
only if $a_{i-1,j-1}=0$, we see that the same index permutation takes
$\alpha'_j$ to $\alpha_j$.
(b) This follows from the fact that, in view of the defining formulae, ${\bf D}$ is generated by
$f_0$ as a $\varphi$-module over $k((t))$.\hfill$\Box$\\
\begin{lem}\label{irrphiga} An absolutely simple \'{e}tale $(\varphi,\Gamma)$-module ${\bf D}$
of dimension $d+1$ over
$k((t))$ becomes standard cyclic over some finite field extension of $k$.
\end{lem}
{\sc Proof:} This statement has its analog in the well known classification of absolutely irreducible $(d+1)$-dimensional ${\rm Gal}(\overline{F}/F)$-representations over $k$, hence it follows from Theorem \ref{sosego}.\hfill$\Box$\\
{\bf Definition:} A $(d+1)$-dimensional standard cyclic ${\rm Gal}(\overline{F}/F)$-representation is a ${\rm Gal}(\overline{F}/F)$-representation over $k$ which corresponds, under the equivalence of categories in Theorem \ref{sosego}, to an \'{e}tale $(\varphi,\Gamma)$-module ${\bf D}$
of dimension $d+1$ over
$k((t))$ which is standard cyclic.
\section{Hecke algebras and supersingular modules}
\subsection{The pro-$p$-Iwahori Hecke algebra ${\mathcal H}$}
\label{defhsu}
We introduce the pro-$p$ Iwahori Hecke algebra ${\mathcal H}$ of ${\rm GL}_{d+1}(F)$ with coefficients in $k$ in a
slightly unorthodox way, which however is well suited for our later constructions.
Let $\overline{T}$ be a free ${\mathbb Z}/(q-1)$-module of rank $d+1$. Then
also ${\rm Hom}(\Gamma,\overline{T})$ (with $\Gamma={\mathcal O}_F^{\times}$)
is free of rank $d+1$ over ${\mathbb Z}/(q-1)$. We write the group law of
$\overline{T}$ multiplicatively, but the one of ${\rm
Hom}(\Gamma,\overline{T})$ we write additively. Let
$e^*,\alpha_1^{\vee},\ldots,\alpha_d^{\vee}$ be a ${\mathbb Z}/(q-1)$-basis of
${\rm Hom}(\Gamma,\overline{T})$. Put
$\alpha_{0}^{\vee}=-\sum_{i=1}^d\alpha_{i}^{\vee}$. We let the symmetric group
${\mathfrak S}_{d+1}$ act on ${\rm Hom}(\Gamma,\overline{T})$ as follows. We
think of ${\mathfrak S}_{d+1}$ as the permutation group of $\{0,1,\ldots,d\}$,
generated by the transposition $s=(01)\in {\mathfrak S}_{d+1}$ and the cycle
$\omega\in {\mathfrak S}_{d+1}$ with $\omega(i)=i+1$ for all $0\le i\le
d-1$. We then put$${\omega}\cdot e^* = e^*+\alpha_{0}^{\vee},\quad\quad{\omega}\cdot
\alpha_{0}^{\vee}=\alpha_{d}^{\vee}\quad\quad\mbox{ and }\quad\quad{\omega}
\cdot \alpha_{i}^{\vee}=\alpha_{i-1}^{\vee}\quad\mbox{ for }1\le i\le d.$$If $d=1$ we put$$s\cdot
e^* = e^*-\alpha_1^{\vee},\quad s\cdot
\alpha_i^{\vee}=-\alpha_i^{\vee}\quad\mbox{ for }i=0,1,$$but if $d\ge2$ we put$$s\cdot
e^* = e^*-\alpha_1^{\vee},\quad s\cdot
\alpha_0^{\vee}=\alpha_0^{\vee}+\alpha_1^{\vee},\quad s\cdot
\alpha_1^{\vee}=-\alpha_1^{\vee},\quad s\cdot
\alpha_2^{\vee}=\alpha_1^{\vee}+\alpha_2^{\vee},$$$$s\cdot
\alpha_i^{\vee}=\alpha_i^{\vee}\quad\mbox{ for }3\le i\le
d.$$ One easily checks that there is a unique action of ${\mathfrak S}_{d+1}$ on $\overline{T}$ such that for $\gamma\in\Gamma$ and $f\in {\rm
Hom}(\Gamma,\overline{T})$ we have $$\omega\cdot (f(\gamma))=(\omega\cdot f)(\gamma)\quad\quad\mbox{ and }\quad\quad s\cdot (f(\gamma))=(s\cdot f)(\gamma).$$
Define $\alpha_1^{\vee}({\mathbb
F}_q^{\times})$ to be the image of the composition ${\mathbb
F}_q^{\times}\to\Gamma\stackrel{\alpha_1^{\vee}}{\to} \overline{T}$ where the first map is the Teichm\"uller homomorphism.\\
{\bf Definition:} (a) The $k$-algebra ${\mathcal H}$ is generated by elements
$T_{\omega}^{\pm}$, $T_s$ and $T_t$ for $t\in \overline{T}$, subject
to the following relations (with $t,t'\in\overline{T}$):
\begin{gather}T_sT_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}=T_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}T_s\quad\quad\mbox{
if
}d>1,\label{braid1}\\T_sT_{\omega}^{-m}T_sT_{\omega}^m=T_{\omega}^{-m}T_sT_{\omega}^{m}T_s\quad\quad\mbox{
for all }1<m<d,\label{braid2}\\T_s^2=T_s\tau_s=\tau_sT_s\quad\quad\mbox{
with }\quad\tau_s=\sum_{t\in\alpha_1^{\vee}({\mathbb
F}_q^{\times})}T_{t},\label{sorge}\\T_{\omega}T_{\omega}^{-1}=1=T_{\omega}^{-1}T_{\omega},\label{hsv16},\\T_{\omega}^{d+1}T_s=T_sT_{\omega}^{d+1},\label{sanafabe}\\ T_{t}T_{t'}=T_{t'
t},\quad\quad T_{1_{\overline{T}}}=1,\label{hsv1}\\T_{t}T_{\omega}=T_{\omega}T_{{\omega}\cdot t},\label{hsvtabfue}\\T_{t}T_s=T_sT_{s\cdot t}.\label{hsvtabfue1}\end{gather}Notice that $T_{\omega}^{d+1}$ is central in ${\mathcal H}$.
(b) ${\mathcal H}_{\rm aff}$ is the $k$-subalgebra of ${\mathcal
H}$ generated by all $T_t$ for $t\in\overline{T}$, by
$T_{\omega}^{d+1}$, $T_{\omega}^{-d-1}$ and by
all $T_{\omega}^{m}T_sT_{\omega}^{-m}$ for $m\in{\mathbb Z}$.
(c) ${\mathcal H}^{\flat}$ is the quotient of ${\mathcal H}$ by
the two sided ideal spanned by all elements $T_t-1$ with $t\in\overline{T}$.\\
{\bf Caution:} ${\mathcal H}_{\rm aff}$ differs from the
similarly denoted algebra in \cite{vigneras}. (The difference is that here we
include $(T_{\omega}^{d+1})^{\mathbb Z}$.)\\
{\bf Remark:} Let $\overline{T}$ denote the subgroup of $G={\rm GL}_{d+1}(F)$ consisting of diagonal matrices with entries in the image of the Teichm\"uller homomorphism ${\mathbb F}_q^{\times}\to{\mathcal O}_F^{\times}$. For $\gamma\in\Gamma$ let $\overline{\gamma}$ be its image in ${\mathbb F}_q^{\times}$. In $\overline{T}$ define the elements $e^*(\gamma)={\rm diag}(\overline{\gamma},1_d)$ and $\alpha_i^{\vee}(\gamma)={\rm diag}(1_{i-1},\overline{\gamma},\overline{\gamma}^{-1},1_{d-i})$ for $1\le i\le d$. Define the elements $\omega=(\omega_{ij})_{0\le i,j\le d}$ and $s=(s_{ij})_{0\le i,j\le d}$ of $G$ by $\omega_{d0}=\pi$ and $\omega_{i,i+1}=1$ (for $0\le i\le d-1$) and $\omega_{ij}=0$ for all other pairs $(i,j)$, resp. by $s_{10}=s_{01}=s_{ii}=1$ for $i\ge 2$, and $s_{ij}=0$ for all other pairs $(i,j)$.
Let $I_0$ denote the pro-$p$-Iwahori subgroup of $G$ for which
$g=(g_{ij})_{0\le i,j\le d}\in G$ belongs to $I_0$ if and only if all the
following conditions are satisfied: $g_{ij}\in\pi{\mathcal O}_F$ for $i>j$,
and $g_{ij}\in{\mathcal O}_F$ for $i<j$, and $g_{ii}\in 1+\pi{\mathcal
O}_F$. The corresponding pro-$p$-Iwahori Hecke algebra $k[I_0\backslash G/I_0]$ is then
naturally isomorphic with ${\mathcal H}$, in such a way that the double coset
$I_0gI_0$ for $g\in\overline{T}\cup\{s,\omega\}$ corresponds to the element
$T_g\in {\mathcal H}$. If $I$ denotes the Iwahori subgroup of $G$ containing $I_0$, then ${\mathcal H}^{\flat}$
becomes isomorphic with the Iwahori Hecke algebra $k[I\backslash G/I]$. For all this, see \cite{vigneras}.\\
{\bf Definition:} A character $\chi:{\mathcal H}_{\rm aff}\to k$ is
called {\it supersingular} if the following two conditions are both satisfied:
(a) There is an $m\in{\mathbb Z}$ with $\chi(T_{\omega}^m T_sT_{\omega}^{-m})=0$.
(b) There is an $m\in{\mathbb Z}$ with either $\chi(T_{\omega}^m T_sT_{\omega}^{-m})=-1$ or
$\chi(T_{\omega}^m \tau_sT_{\omega}^{-m})=0$.\footnote{We have $\chi(T_{\omega}^m \tau_sT_{\omega}^{-m})=0$ if and only if $\chi(T_{\omega}^m T_tT_{\omega}^{-m})\ne 1$ for some $t\in\alpha_1^{\vee}({\mathbb
F}_q^{\times})$, if and only if $\chi(\alpha_{m+1}^{\vee}(\gamma))\ne 1$ for some $\gamma\in\Gamma$.}\\
{\bf Definition:} (a) An ${\mathcal
H}$-module $M$ is called {\it standard supersingular} if it is isomorphic with ${\mathcal
H}\otimes_{{\mathcal H}_{\rm aff},\chi}k.e$, where ${\mathcal H}_{\rm aff}$ acts on the one dimensional $k$-vector space $k.e$ through a supersingular character $\chi$.
Equivalently, $M$ is standard supersingular if and only if $M=\bigoplus_{0\le m\le d}T_{\omega}^m(M_1)$ with an ${\mathcal H}_{\rm
aff}$-module $M_1$ of $k$-dimension $1$ on which ${\mathcal H}_{\rm aff}$
acts through a supersingular character.\footnote{Then ${\mathcal H}_{\rm aff}$
acts on each $T_{\omega}^m(M_1)$ through a supersingular character.}
(b) An irreducible ${\mathcal
H}$-module is called {\it supersingular} if it is a subquotient of a standard supersingular ${\mathcal
H}$-module. An ${\mathcal H}$-module $M$ is called {\it supersingular} if it is the
inductive limit of finite dimensional ${\mathcal
H}$-modules and if each of its irreducible subquotients is supersingular.\\
{\bf Remark:} The above definition of supersingularity is equivalent with
the one given by Vign\'{e}ras. This follows from the discussion in section 6 of \cite{vigjuss}.\footnote{For example, that every supersingular ${\mathcal
H}{{{}}}$-module as defined in \cite{vigjuss} is the union of its finite dimensional submodules follows from the fact that, if $\xi\subset Z({\mathcal
H}{{{}}})$ is the ideal in the center which determines supersingularity, then ${\mathcal
H}{{{}}}/\xi$ is finite dimensional.}
\subsection{The coverings ${\mathcal H}^{\sharp\sharp}$ and ${\mathcal H}^{\sharp}$ of ${\mathcal H}$}
\label{monachstras}
{\bf Definition:} (a) Let ${\mathcal H}^{\sharp}$ denote the $k$-algebra generated by
elements $T_{\omega}^{\pm}$, $T_s$ and $T_{t}$ for $t\in
\overline{T}$, subject to
${\bullet}$ the relations
(\ref{sorge}), (\ref{hsv16}), (\ref{hsv1}), (\ref{hsvtabfue}),
${\bullet}$ the relations (\ref{hsvtabfue1}) for
$t=\alpha^{\vee}_{i}(\gamma)$ (all $0\le i\le d$, $\gamma\in\Gamma$),
${\bullet}$ the relation \begin{gather}
T_{\omega}^{d+1}T_s^2=T_s^2T_{\omega}^{d+1},\label{sanafabehsv17}\end{gather}
${\bullet}$ the relations \begin{gather}T_{t}T^2_s=T^2_sT_{t}\quad\quad\mbox{ for all }t\in\overline{T},\label{hsvtabfue11}\end{gather}
${\bullet}$ the relations \begin{gather}T^2_sT_{\omega}T^2_sT_{\omega}^{-1}T^2_sT_{\omega}=T_{\omega}T^2_sT_{\omega}^{-1}T^2_sT_{\omega}T^2_s\quad\quad\mbox{
if
}d>1,\label{braid11}\\T^2_sT_{\omega}^{-m}T^2_sT_{\omega}^m=T_{\omega}^{-m}T^2_sT_{\omega}^{m}T^2_s\quad\quad\mbox{
for all }1<m<d.\label{braid22}\end{gather}
(b) Let ${\mathcal H}^{\sharp\sharp}$ denote the $k$-algebra generated by
the elements $T_{\omega}^{\pm}$, $T_s$ and $T_{t}$ for $t\in
\overline{T}$, subject to
${\bullet}$ the relations
(\ref{sorge}), (\ref{hsv16}), (\ref{hsv1}), (\ref{hsvtabfue}),
${\bullet}$ the relations (\ref{hsvtabfue1}) for
$t=\alpha^{\vee}_{i}(\gamma)$ (all $0\le i\le d$, $\gamma\in\Gamma$),
${\bullet}$ the relations (\ref{hsvtabfue11}).
\begin{lem}\label{2vorstubei} In ${\mathcal H}$ we have the relations (\ref{sanafabehsv17}), (\ref{hsvtabfue11}), (\ref{braid11}) and (\ref{braid22}).
\end{lem}
{\sc Proof:} It is immediate that the relations (\ref{hsvtabfue1}), resp. (\ref{sanafabe}), imply the relations
(\ref{hsvtabfue11}), resp. (\ref{sanafabehsv17}). For $1<m<d$ and $t\in \alpha_1^{\vee}({\mathbb
F}_q)$ we have $s\omega^m\cdot t=\omega^m\cdot t$, hence $T_s\sum_{t\in\alpha_1^{\vee}({\mathbb
F}_q)}T_{\omega^m\cdot t}=\sum_{t\in\alpha_1^{\vee}({\mathbb
F}_q)}T_{\omega^m\cdot t}T_s$. The same applies with $-m$ instead of $m$, hence$$T_sT_{\omega}^{-m}\tau_sT_{\omega}^{m}=T_{\omega}^{-m}\tau_sT_{\omega}^{m}T_s\quad\quad\mbox{
and }\quad\quad
T_sT_{\omega}^{m}\tau_sT_{\omega}^{-m}=T_{\omega}^{m}\tau_sT_{\omega}^{-m}T_s.$$This,
together with $T_s^2=\tau_sT_s=T_s\tau_s$ (formula (\ref{sorge})), justifies (i) and (iii)
in$$T_s^2T_{\omega}^{-m}T_s^2T_{\omega}^{m}\stackrel{(i)}{=}\tau_s(T_{\omega}^{-m}\tau_sT_{\omega}^{m})T_sT_{\omega}^{-m}T_sT_{\omega}^{m}\stackrel{(ii)}{=}\tau_s(T_{\omega}^{-m}\tau_sT_{\omega}^{m})T_{\omega}^{-m}T_sT_{\omega}^{m}T_s\stackrel{(iii)}{=}T_{\omega}^{-m}T_s^2T_{\omega}^{m}T_s^2,$$whereas
(ii) is justified by (\ref{braid2}). We have shown (\ref{braid22}). Finally, to see (\ref{braid11}) comes down, using (\ref{sorge}), (\ref{hsvtabfue}) and (\ref{hsvtabfue1}), to comparing$$T_{\omega}T_s^2T_{\omega}^{-1}T_s^2T_{\omega}T_s^2=(\sum_{t_1,t_2,t_3\in\alpha_1^{\vee}({\mathbb
F}_q)}T_{\omega^{-1}\cdot
t_1}T_{\omega^{-1} s\omega\cdot t_2}T_{\omega^{-1} s
\omega s
\omega^{-1}\cdot t_3})T_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}T_s,$$$$T_s^2T_{\omega}^{-1}T_s^2T_{\omega}T_s^2T_{\omega}=(\sum_{t_1,t_2,t_3\in\alpha_1^{\vee}({\mathbb
F}_q)}T_{
t_1}T_{s \omega^{-1}\cdot t_2}T_{s\omega^{-1} s\omega\cdot t_3})T_sT_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}.$$That these are
equal follows from (\ref{braid1}) and equality of the bracketed
terms; for the latter observe $\omega s\omega^{-1} s \omega\cdot t=t$ for any $t\in\alpha_1^{\vee}({\mathbb
F}_q^{\times})$.\hfill$\Box$\\
In view of Lemma \ref{2vorstubei} we have natural surjections of $k$-algebras $${\mathcal H}^{\sharp\sharp}\longrightarrow {\mathcal H}^{\sharp}\longrightarrow {\mathcal H}\longrightarrow{\mathcal H}^{\flat}.$$
{\bf Remark:} ${\mathcal H}^{\sharp\sharp}$ (and in particular ${\mathcal H}^{\sharp}$ and ${\mathcal H}$) is generated as a $k$-algebra by $T_{\omega}^{\pm}$, $T_s$ and the $T_{e^*(\gamma)}$ for $\gamma\in\Gamma$.\\
\begin{lem}\label{involu} There are unique $k$-algebra involutions $\iota$ of ${\mathcal H}$, ${\mathcal H}^{\sharp}$ and ${\mathcal H}^{\sharp\sharp}$ with
$$\iota(T_{\omega})= T_{\omega},\quad\quad \iota(T_s)=\tau_s-T_s, \quad\quad \iota(T_t)=T_t\quad \mbox{ for }t\in\overline{T}.$$
\end{lem}
{\sc Proof:} This is a slightly tedious but straightforward computation. (For
${\mathcal H}$ see \cite{vigneras} Corollary 2.)\hfill$\Box$\\
{\bf Remark:} Besides $\iota$ consider the $k$-algebra involution $\beta$ of
${\mathcal H}$, ${\mathcal H}^{\sharp}$ and ${\mathcal H}^{\sharp\sharp}$
given on generators by$$\beta(T_{\omega})= T_{\omega}^{-1},\quad\quad
\beta(T_s)=T_s, \quad\quad \beta(T_t)=T_{s\cdot t}\quad \mbox{ for
}t\in\overline{T}.$$Moreover, for any automorphism ${\mathfrak o}$ of $\Gamma$
there is an associated automorphism $\alpha_{\mathfrak o}$ of ${\mathcal H}$,
${\mathcal H}^{\sharp}$ and ${\mathcal H}^{\sharp\sharp}$ given on generators by$$\alpha_{\mathfrak o}(T_{\omega})= T_{\omega},\quad\quad \alpha_{\mathfrak
o}(T_s)=T_s, \quad\quad \alpha_{\mathfrak o}(T_{\partial(\gamma)})=
T_{\partial({\mathfrak o}(\gamma))}\quad \mbox{ for }\gamma\in\Gamma,
\partial\in{\rm Hom}(\Gamma,\overline{T}).$$Do $\iota$, $\beta$ and the $\alpha_{\mathfrak o}$ generate the automorphism group of ${\mathcal H}$ (resp. of ${\mathcal H}^{\sharp}$, resp. of ${\mathcal H}^{\sharp\sharp}$) modulo inner automorphisms ?\\
\begin{lem}\label{quadrat} Let $M$ be an ${\mathcal H}^{\sharp\sharp}$-module. We have a direct sum decomposition$$M=M^{T_s=-{\rm
id}}\bigoplus M^{T_s^2=0}.$$
\end{lem}
{\sc Proof:} One computes $\tau_s^2=(q-1)\tau_s=-\tau_s$ and this shows $T_s=-{\rm id}$ on ${\rm im}(T^2_s)$ as well as $T_s^2=0$ on ${\rm im}(T^2_s-{\rm id})$.\hfill$\Box$\\
Let ${[0,q-2]}^{\Phi}$ be the set of tuples ${\epsilon}=(\epsilon_{i})_{0\le
i\le d}$ with $\epsilon_i\in\{0,\ldots, q-2\}$ and $\sum_{0\le i\le
d}{\epsilon}_{i}\equiv 0$ modulo $(q-1)$. We often read the indices as
elements of ${\mathbb Z}/(d+1)$, thus $\epsilon_i=\epsilon_j$ for $i,
j\in{\mathbb Z}$ whenever $i-j\in(d+1){\mathbb Z}$. We let the symmetric group
${\mathfrak S}_{d+1}$ (generated by $s$, $\omega$ as before) act on ${[0,q-2]}^{\Phi}$ as follows:$$({\omega}\cdot\epsilon)_0= \epsilon_{d}\quad\quad\mbox{ and
}\quad\quad({\omega}\cdot\epsilon)_i= \epsilon_{i-1} \quad \mbox{ for }1\le i\le d.$$If $d=1$ we put$$(s\cdot\epsilon)_i=-\epsilon_i\quad\mbox{ for
}i=0,1,$$but if $d\ge2$ we put$$(s\cdot\epsilon)_1=-\epsilon_1,\quad
(s\cdot\epsilon)_0=\epsilon_0+\epsilon_1,\quad
(s\cdot\epsilon)_2=\epsilon_1+\epsilon_2,\quad (s\cdot\epsilon)_i=\epsilon_i\quad\mbox{ for
}3\le i\le d.\footnote{Here and below we understand $-\epsilon_i$ to mean the representative in $[0,q-2]$ of the class of $-\epsilon_i$ in ${\mathbb Z}/(q-1)$, and similarly for $\epsilon_0+\epsilon_1$ and $\epsilon_1+\epsilon_2$.}$$
Throughout we assume that all
eigenvalues of the $T_{t}$ for $t\in{\overline{T}}$ acting on an ${\mathcal H}^{\sharp\sharp}$-module belong to $k$.
Let $M$ be an ${\mathcal H}^{\sharp\sharp}$-module. For $a\in[0,q-2]$ and ${\epsilon}=(\epsilon_{i})_{0\le i\le d}
\in{[0,q-2]}^{\Phi}$ and $j\in\{0,1\}$ put \begin{align}M^{{\epsilon}}&=\{x\in
M\,\,|\, T^{-1}_{\alpha_i^{\vee}(\gamma)}(x)=\gamma^{{\epsilon}_{i}}x\mbox{ for all }\gamma\in \Gamma,\mbox{ all }0\le i\le d\},\notag\\M^{\epsilon}_{\underline{a}}&=\{x\in M^{\epsilon}\,|\,T_{e^*(\gamma)}(x)=\gamma^ax\mbox{ for all }\gamma\in\Gamma\},\notag\\M^{\epsilon}_{\underline{a}}[j]&=\{x\in M^{\epsilon}_{\underline{a}}\,|\,T_{s}^2(x)=jx\}.\notag\end{align}
The $T_t$ for $t\in\overline{T}$ are of order divisible by $q-1$, hence are diagonalizable on the $k$-vector space $M$. Since they commute among each other and with $T_s^2$, we may simultaneously diagonalize all these operators (cf. Lemma \ref{quadrat} for $T_s^2$), hence\begin{gather}M=\bigoplus_{\epsilon, a, j}M^{\epsilon}_{\underline{a}}[j].\label{18okt}\end{gather}
\begin{lem}\label{samserswo} For any $\epsilon\in [0,q-2]^{\Phi}$ and $a\in
[0,q-2]$ we have$$T_{\omega}(M^{\epsilon}_{\underline{a}})=M^{\omega\cdot\epsilon}_{\underline{a-\epsilon_0}}\quad\quad\mbox{ and }\quad\quad T_{s}(M^{\epsilon})\subset
M^{s\cdot\epsilon}.$$If $M$ is even an ${\mathcal
H}$-module then\begin{gather} T_s(M^{\epsilon}_{\underline{a}})\subset M^{s\cdot\epsilon}_{\underline{\epsilon_1+a}}.\label{vorvigilallsaints}\end{gather}
\end{lem}
{\sc Proof:} $T_{\omega}(M^{\epsilon})=M^{\omega\cdot\epsilon}$ and $T_{s}(M^{\epsilon})\subset
M^{s\cdot\epsilon}$ follows from formula (\ref{hsvtabfue}) resp. from formula (\ref{hsvtabfue1}) for the $t=\alpha^{\vee}_{i}(\gamma)$. For
the following computation recall that $\omega\cdot e^{*}=e^{*}+\alpha_0^{\vee}$:
For $\gamma\in\Gamma$ and $x\in M^{\epsilon}_{\underline{a}}$ we
have$$T_{e^*(\gamma)}T_{\omega}(x)=T_{\omega}T_{(\omega\cdot e^*)(\gamma)}(x)=T_{\omega}T_{e^*(\gamma)}T_{\alpha_0^{\vee}(\gamma)}(x)=\gamma^{a-\epsilon_0}T_{\omega}(x).$$This shows $T_{\omega}(M^{\epsilon}_{\underline{a}})=M^{\omega\cdot\epsilon}_{\underline{a-\epsilon_0}}$. For formula (\ref{vorvigilallsaints}) recall that $s\cdot e^{*}=e^{*}-\alpha_1^{\vee}$ and employ formula (\ref{hsvtabfue1}). \hfill$\Box$\\
Any $x\in M$ can be uniquely written as $$x=\sum_{a\in[0,q-2]}x_{\underline{a}}\quad \mbox{ with }\quad x_{\underline{a}}\in\sum_{\epsilon\in [0,q-2]^{\Phi}} M^{\epsilon}_{\underline{a}}.$$Given $a\in{\mathbb Z}$ and $x\in M$, we write $x_{\underline{a}}=x_{\underline{\widetilde{a}}}$ where $\widetilde{a}\in[0,q-2]$ is determined by $a-\widetilde{a}\in(q-1){\mathbb Z}$.\\
{\bf Definition:} (a) An ${\mathcal
H}^{\sharp}$-module $M$ is called {\it standard supersingular} if the ${\mathcal
H}^{\sharp}$-action factors through ${\mathcal H}$, making it a standard supersingular ${\mathcal
H}$-module.
(b) An irreducible ${\mathcal
H}^{\sharp}$-module is called {\it supersingular} if it is a subquotient of a standard supersingular ${\mathcal
H}^{\sharp}$-module. An ${\mathcal H}^{\sharp}$-module $M$ is called {\it supersingular} if it is the
inductive limit of finite dimensional ${\mathcal
H}^{\sharp}$-modules and if each of its irreducible subquotients is supersingular.
(c) An ${\mathcal
H}^{\sharp\sharp}$-module $M$ is called {\it supersingular} if it satisfies
the condition analogous to (b).
(d) A supersingular ${\mathcal
H}^{\sharp}$-module is called {\it $\sharp$-supersingular} if for all
$e\in M^{\epsilon}_{\underline{a}}[0]$ with $\epsilon_1>0$ we have
$$(T_se)_{\underline{c+\epsilon_1+a}}=0\quad\quad \mbox{ for all }\quad q-1-\epsilon_1\le c\le q-2.$$
\begin{lem}\label{vialsa} An ${\mathcal
H}$-module is supersingular if and only if it is $\sharp$-supersingular when viewed as an ${\mathcal
H}^{\sharp}$-module.
\end{lem}
{\sc Proof:} This follows from formula (\ref{vorvigilallsaints}). \hfill$\Box$\\
\section{Reconstruction of supersingular ${\mathcal H}^{\sharp}$-modules}
Given an ${\mathcal
H}^{\sharp}$-module $M$ together with a submodule $M_0$ such that $M/M_0$ is supersingular, we address the problem of reconstructing the ${\mathcal
H}^{\sharp}$-module $M$ from the ${\mathcal
H}^{\sharp}$-modules $M_0$ and $M/M_0$ together with an additional set of data (intended to be sparse). Our proposed solution (Proposition \ref{schutzengel}) critically relies on the braid relations (\ref{braid11}), (\ref{braid22}).
\begin{lem}\label{brixen} Let $B_0,\ldots,B_n$ be linear operators on a
$k$-vector space $M$ such that\begin{gather}B_j^2=B_j\quad\mbox{ for all }
0\le j\le n,\notag\\B_jB_{j'}B_j=B_{j'}B_jB_{j'}\quad\mbox{ for all }0\le
j',j\le n, \notag\\B_jB_{j'}=B_{j'}B_j\quad\mbox{ for all }0\le j'< j\le
n\mbox{ with }j-j'\ge 2. \notag\end{gather}Put
$\beta=B_n\cdots B_{1}B_0$ and let $x\in M$ with $\beta^mx=x$ for some $m\ge1$. Then we have $B_jx=x$ for
each $0\le j\le n$.
\end{lem}
{\sc Proof:} We first claim\begin{gather}\beta
B_{j+1}=B_{j}\beta\quad\quad \mbox{ for
all }0\le j< n.\label{vorpiz}\end{gather}Indeed, \begin{align}\beta
B_{j+1}&=B_n\cdots B_{j+2}B_{j+1} B_{j}B_{j-1}\cdots B_{1}B_0 B_{j+1}\notag\\{}&=B_n\cdots B_{j+2}B_{j+1} B_{j} B_{j+1}B_{j-1} \cdots
B_{1}B_0\notag\\{}&=B_n\cdots B_{j+2} B_{j} B_{j+1}B_{j} B_{j-1} \cdots
B_{1}B_0\notag\\{}&=B_{j}\beta.\notag\end{align}Choose $\nu\ge1$ with
$m\nu\ge n$. For $0\le j\le n$ we then
compute$$x\stackrel{(i)}{=}\beta^{m\nu}x=\beta^{n-j}\beta^{m\nu-n+j}x\stackrel{(ii)}{=}\beta^{n-j}B_n\beta^{m\nu-n+j}
x\stackrel{(iii)}{=}B_j\beta^{n-j}\beta^{m\nu-n+ju} x{=}B_j\beta^{m\nu} x\stackrel{(iv)}{=}B_j x,$$where $(i)$
and $(iv)$ follow
from the hypothesis $\beta^m x=x$, where $(ii)$ follows
from $B_n\beta=\beta$ and where $(iii)$ follows from repeated application of
formula (\ref{vorpiz}).\hfill$\Box$\\
\begin{pro}\label{novordschutzengel} Let $M$ be an ${\mathcal
H}^{\sharp}$-module, let $M_0\subset M$ be an ${\mathcal
H}^{\sharp}$-submodule such that $M/M_0$ is supersingular. Let
$\overline{x}\in (M/M_0)^{\epsilon}$ (some $\epsilon\in [0,q-2]^{\Phi}$) be such that
$\overline{x}\{i\}=T_{\omega}^{i+1}\overline{x}$ is an eigenvector under
$T_s$, for each $i\in{\mathbb Z}$. For liftings $x\in M$ of $\overline{x}$ put $x\{i\}=T_{\omega}^{i+1}x$.
(a) If the ${\mathcal
H}^{\sharp}$-action on $M$ factors through ${\mathcal
H}$ then we may choose $x\in M^{\epsilon}$ such that for each $i$ with
$T_s(\overline{x}\{i\})= 0$ and $(\omega^{i+1}\cdot\epsilon)_1=0$ (all $\gamma\in\Gamma$) we have $T_s(x\{i\})=0$.
(b) If the ${\mathcal
H}^{\sharp}$-action on $M$ factors through ${\mathcal
H}$ then we may choose $x\in M^{\epsilon}$ such that for each $i$ with
$T_s(\overline{x}\{i\})= -\overline{x}\{i\}$ we have $T_s(x\{i\})=-x\{i\}$.
(c) We may choose $x\in M^{\epsilon}$ such that for each $i$ with
$T_s^2(\overline{x}\{i\})= 0$ we have $T_s^2(x\{i\})=0$.
(d) We may choose $x\in M^{\epsilon}$ such that for each $i$ with
$T_s^2(\overline{x}\{i\})=\overline{x}\{i\}$ we have $T_s^2(x\{i\})=x\{i\}$.
\end{pro}
{\sc Proof:} (a) Let $i_1<\ldots <i_r$ be the increasing enumeration of the
set of all $0\le i\le d$ with $T_sT_{\omega}^{i+1}(\overline{x})=0$ and $(\omega^{i+1}\cdot\epsilon)_1=0$ for all $\gamma\in\Gamma$. Since $M/M_0$ is supersingular this property is not fulfilled by all $i$; thus, after a cyclic index shift, we may assume $i_r<d$.
Start with an arbitrary lift $x\in M^{\epsilon}$ of $\overline{x}$.
We claim that for any $j$ with $0\le j\le r$, after modifying $x$ if
necessary, we can achieve $T_s(x\{i_s\})=0$ for all $s$ with $1\le
s\le j$. For $j=r$ this is the desired statement.
Induction on $j$. For $j=0$ there is nothing to do. Now fix $1\le j\le r$ and assume that $x$
satisfies the condition for $j-1$. For $-1\le i\le d$ and
$0\le m<j$ define inductively $$x\{i\}_0=x\{i\}=T_{\omega}^{i+1}x,$$$$x\{i\}_{m+1}=T_{\omega}^{i-i_{j-m}}T_s(x\{i_{j-m}\}_m).$$ We establish several subclaims.
(1) $x\{i\}_m\in M^{\omega^{i+1}\cdot \epsilon}$.
Induction on $m$. For $m=0$ there is nothing to do. Next, if the claim is true for an arbitrary $m$, then also for $x\{i_{j-m}\}_{m+1}=T_s(x\{i_{j-m}\}_m)$ as we have $T_s(M^{\omega^{i_{j-m}}\cdot\epsilon})=M^{\omega^{i_{j-m}}\cdot\epsilon}$ (since $(\omega^{i_{j-m}}\cdot\epsilon)_1=0$). Applying powers of $T_{\omega}$ to $x\{i_{j-m}\}_{m+1}$ we get the statement for all $x\{i\}_{m+1}$.
(2) $T_s(x\{i_s\}_m)=0$ for all $0\le s\le j$ and all $0\le m<j-s$.
Induction on $m$. For $m=0$ this is true by induction hypothesis (on $j$). Now
let $0<m <j-s$ and assume that we know the claim for $m-1$ instead of $m$. In
particular we then know $T_s(x\{i_s\}_{m-1})=0$. We deduce\begin{align}T_s(x\{i_s\}_m)&=T_sT_{\omega}^{i_s-i_{j-m+1}}T_sT_{\omega}^{i_{j-m+1}-i_s}T_{\omega}^{i_s-i_{j-m+1}}(x\{i_{j-m+1}\}_{m-1})\notag\\&=T_sT_{\omega}^{i_s-i_{j-m+1}}T_sT_{\omega}^{i_{j-m+1}-i_s}(x\{i_s\}_{m-1})\notag\\&=T_{\omega}^{i_s-i_{j-m+1}}T_sT_{\omega}^{i_{j-m+1}-i_s}T_s(x\{i_s\}_{m-1})\notag\\&=0\notag\end{align}where
we use the braid relation (\ref{braid2}) (which applies since
$|i_s-i_{j-m+1}|>1$ and $i_r<d$). The induction on $m$ is complete.
(3) $T_s(x\{i_s\}_m)=0$ for all $0\le s\le j$ and all $j-s+1<m\le j$.
Induction on $m+s-j$. The induction begins with $m+s-j=2$. By (2) we know
$T_s(x\{i_{j-m+1}\}_{m-2})=0$. Thus, if
$i_{j-m+1}+1<i_{j-m+2}$, the same argument as in (2) shows
$T_s(x\{i_{j-m+1}\}_{m-1})=0$ and hence $x\{i\}_m=0$ for all $i$, and
there is nothing more to do. If however $i_{j-m+1}+1=i_{j-m+2}$ we
compute\begin{align}T_s(x\{i_{j-m+2}\}_m)&=T_sT_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}(x\{i_{j-m+1}\}_{m-2})\notag\\&=T_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}T_s(x\{i_{j-m+1}\}_{m-2})\notag\\&=0\notag\end{align}where
we use the braid relation (\ref{braid1}). This settles the case $m+s-j=2$. For
$m+s-j>2$ we now argue exactly as in (2) again: $T_s(x\{i_s\}_m)=0$ implies
$T_s(x\{i_s\}_{m+1})=0$. The induction is complete.
(4) $T_s(x\{i_{j-m}\}_{m}+x\{i_{j-m}\}_{m+1})=0$ for all $0\le
m<j$.
Indeed, by (1) and our defining assumption on the $i_{j-m}$ we know that $x\{i_{j-m}\}_m$ is fixed under
$T_{\alpha_1^{\vee}(\Gamma)}$ and hence is killed by $T_s^2+T_s$, as follows
from the quadratic relation (\ref{sorge}). As
$x\{i_{j-m}\}_{m+1}=T_s(x\{i_{j-m}\}_m)$ this gives the claim.
(5) $$\tilde{x}=\sum_{0\le m\le j}x\{-1\}_m$$lifts $\overline{x}$.
Indeed, we have $T_s(x\{i_j\})\in M_0$ by our defining assumption on $i_j$. It follows that $x\{-1\}_{m}\in M_0$ for all $m\ge1$, hence $x-\tilde{x}\in M_0$.
(6) From (1) we deduce $\tilde{x}\{i\}\in M^{\omega^{i+1}\cdot\epsilon}$. Writing$$\tilde{x}\{i_s\}=(\sum_{0\le m<
j-s}x\{i_s\}_m)+(x\{i_s\}_{j-s}+x\{i_s\}_{j-s+1})+(\sum_{j-s+1<m\le j}x\{i_s\}_m)$$we see that (2), (3) and (4) imply $T_s(\tilde{x}\{i_s\})=0$ for all $s$ with $1\le s\le j$.
The induction on $j$ is complete: we may substitute $\tilde{x}$ for the old $x$.
(b) Composing the given ${\mathcal H}$-module
structure on $M$ with the involution $\iota$ of Lemma \ref{involu} we get a new ${\mathcal H}$-module structure on
$M$. Applying statement (a) to this new ${\mathcal H}$-module and then
translating back via $\iota$, we get statement (b).
(c) Statement (c) is proved in the same way as statement (a), with the
following minor modifications: Each occurence of $T_s$ must be replaced by
$T_s^2$, and in the definition of $x\{i\}_{m+1}$ the alternating sign
$(-1)^{m+1}$ must be included,
i.e.\begin{gather}x\{i\}_{m+1}=(-1)^{m+1}T_{\omega}^{i-i_{j-m}}T_s^2(x\{i_{j-m}\}_m)\label{nachbei}\end{gather}In
particular, we then have $x\{i_{j-m}\}_{m+1}=-T^2_s(x\{i_{j-m}\}_m)$. In (2)
and (3), the
appeal to the braid relations (\ref{braid1}), (\ref{braid2}) must be replaced
by an appeal to the braid relations (\ref{braid11}), (\ref{braid22}). In (4), the
appeal to $T_s^2+T_s=0$ on vectors fixed under
$T_{\alpha_1^{\vee}(\Gamma)}$ must be replaced by an appeal to $T_s^4-T_s^2=0$ (it
is here where the alternating sign in the defining formula (\ref{nachbei}) is
needed).
(d) Composing the given ${\mathcal H}^{\sharp}$-module
structure on $M$ with the involution $\iota$ of Lemma \ref{involu} we get a new ${\mathcal H}^{\sharp}$-module structure on
$M$. Applying statement (c) to this new ${\mathcal H}^{\sharp}$-module and then
translating back via $\iota$, we get statement (d).\hfill$\Box$\\
\begin{pro}\label{schutzengel} Let $M$ be an ${\mathcal
H}^{\sharp}$-module, let $M_0\subset M$ be an ${\mathcal
H}^{\sharp}$-sub module such that $M/M_0$ is supersingular. The action of
${\mathcal H}^{\sharp}$ on $M$ is uniquely determined by the following combined data:
(a) the action of ${\mathcal H}^{\sharp}$ on $M_0$ and on $M/M_0$,
(b) the action of $T_{e^*(\Gamma)}$ and of $T_sT_{\omega}$ on $M$,
(c) the restriction of $T_{\omega}$ to $(T_sT_{\omega})^{-1}(M_0)$, i.e. the map $$\{x\in M\,|\,T_sT_{\omega}(x)\in M_0\}\stackrel{T_{\omega}}{\longrightarrow} M,$$
(d) the subspace $\sum_{\epsilon\in
[0,q-2]^{\Phi}\atop\epsilon_1=0}M^{\epsilon}$ of $M$.
\end{pro}
{\sc Proof:} The $k$-algebra ${\mathcal
H}^{\sharp}$ is generated by $T_{e^*(\Gamma)}$, by $T_s$ and by
$T^{\pm}_{\omega}$. Therefore we only need to see that the action of $T_s$ and
$T_{\omega}$ on $M$ can be reconstructed from the given data (a), (b), (c), (d). Exhausting $M/M_0$ step by step we may assume that $M/M_0$ is an
irreducible supersingular ${\mathcal
H}^{\sharp}$-module.
We first show that $T_s$ is uniquely determined. For this we make constant use
of Lemma \ref{quadrat} (and the decomposition (\ref{18okt})). As $T_s|_{M_0}$ is given to us, it is enough to show that for any non-zero $\overline{x}$ in $M/M_0$ with
either $T_s(\overline{x})=-\overline{x}$ or $T_s(\overline{x})=0$ we find
some lifting $x\in M$ such that $T_s(x)$ can be reconstructed. Consider
first the case $T_s(\overline{x})=-\overline{x}$. By the quadratic relation (\ref{sorge}) (cf. Lemma \ref{quadrat}) we then have $\overline{x}\in\sum_{\epsilon\in
[0,q-2]^{\Phi}\atop\epsilon_1=0}(M/M_0)^{\epsilon}$, and using the datum
(d) as well as our knowledge of the subspace $T_sM$ (since $T_sM=T_sT_{\omega}M$ this is given to us
in view of datum (b)), we lift $\overline{x}$ to some $x\in T_sM\cap \sum_{\epsilon\in
[0,q-2]^{\Phi}\atop\epsilon_1=0}M^{\epsilon}$ (use the decomposition (\ref{18okt})). For such $x$ we have
$T_s(x)=-x$. Now consider the case $T_s(\overline{x})=0$. An arbitrary lifting
$x\in M$ of $\overline{x}$ then satisfies $T_s(x)\in M_0$, and $T_s(x)$ is determined by the
given data as
$T_s(x)=(T_sT_{\omega})T_{\omega}^{-1}(x)$ (notice that the datum (c) is
equivalent with the datum $T_s^{-1}(M_0)\stackrel{T_{\omega}^{-1}}{\rightarrow} M$).
To show that $T_{\omega}$ is uniquely determined, suppose that besides $T_{\omega}\in{\rm Aut}_k(M)$ there is another candidate
$\tilde{T}_{\omega}\in{\rm Aut}_k(M)$ extending the data (a), (b), (c), (d) to another ${\mathcal H}^{\sharp}$-action on $M$.
We find and choose some non-zero $\overline{x}\in M/M_0$ such that
$T_{\omega}^j(\overline{x})$ is an eigenvector under $T_s$, for each
$j\in{\mathbb Z}$. For any
$x\in M$ lifting $\overline{x}$ we
have\begin{gather}T_{\omega}=\tilde{T}_{\omega}\quad\mbox{ on
}M_0+k.T_{\omega}^{j-1}(x)\quad\mbox{ if
}T_sT_{\omega}^j(\overline{x})=0\label{nachorkan}\end{gather}as both
$\tilde{T}_{\omega}$ and ${T}_{\omega}$ respect the datum (c).
Let $i_0<\ldots<i_n$ be the increasing enumeration of the set $$\{0\le i\le
d\,|\,T_s^2{T}_{\omega}^i\overline{x}={T}_{\omega}^i\overline{x}\}.$$As
$M/M_0$ is a subquotient of a standard supersingular ${\mathcal H}$-module,
this set is not the full set $\{0\le i\le d\}$. Applying a suitable power of $T_{\omega}$ and reindexing we may assume
that $0$ does not belong to this set, i.e. that $i_0>0$.
Choose a lifting $x\in M$ of $\overline{x}$ such that for each $i\in\{i_0,\ldots,i_n\}+{\mathbb Z}(d+1)$ we have
$T_s^2{T}_{\omega}^ix={T}_{\omega}^ix$. This is possible by Proposition
\ref{novordschutzengel}. Put $z_{0}=x$. For $i\ge 1$
put\begin{gather}z_i=\left\{\begin{array}{l@{\quad:\quad}l}\tilde{T}_{\omega}z_{i-1}
& i\notin\{i_0,\ldots,i_n\}+{\mathbb Z}(d+1) \notag\\T_s^{2}\tilde{T}_{\omega}z_{i-1} &
i\in\{i_0,\ldots,i_n\}+{\mathbb Z}(d+1)\end{array}\right.\notag.\end{gather}We
claim \begin{gather}z_i={T}^{i}_{\omega}x\label{nachstur}\end{gather}for each
$i\ge 0$. Induction on $i$. The case $i=0$ is trivial. For $i\ge1$ with
$i\notin\{i_0,\ldots,i_n\}+{\mathbb Z}(d+1)$ we
compute$$z_{i}=\tilde{T}_{\omega}z_{i-1}\stackrel{(i)}{=}{T}_{\omega}z_{i-1}\stackrel{(ii)}{=}{T}^{i}_{\omega}x$$where
in $(i)$ we use statement (\ref{nachorkan}) and in (ii) we use the induction
hypothesis. For $i\ge1$ with $i\in\{i_0,\ldots,i_n\}+{\mathbb Z}(d+1)$ we
compute$$z_{i}=T^2_s\tilde{T}_{\omega}z_{i-1}\stackrel{(i)}{=}T^2_s{T}_{\omega}z_{i-1}\stackrel{(ii)}{=}{T}^{i}_{\omega}x$$where
in $(i)$ we use the assumption $T_sT_{\omega}=T_s\tilde{T}_{\omega}$, and in
(ii) we use the induction hypothesis ${T}_{\omega}z_{i-1}{=}{T}^{i}_{\omega}x$
and the assumption on $x$. The induction is
complete. Put $$B_{i_j}=\tilde{T}_{\omega}^{-i_j}T_s^2\tilde{T}_{\omega}^{i_j}.$$The relation (\ref{sanafabehsv17}) implies $B_{i_j}=\tilde{T}_{\omega}^{-i_j+(d+1)\nu}T_s^2\tilde{T}_{\omega}^{i_j-(d+1)\nu}$
for each $\nu\in{\mathbb Z}$. Thus$$(B_{i_n}\cdots
B_{i_1}B_{i_0})^mx\stackrel{(i)}{=}\tilde{T}_{\omega}^{-m(d+1)}z_{m(d+1)}\stackrel{(ii)}{=}\tilde{T}_{\omega}^{-m(d+1)}{T}_{\omega}^{m(d+1)}x$$for $m\ge 0$, where
$(i)$ follows from the definition of $z_{m(d+1)}$, whereas $(ii)$ follows from formula
(\ref{nachstur}). Choosing $m$ large enough we may assume
${T}_{\omega}^{m(d+1)}x=x$ and $\tilde{T}_{\omega}^{m(d+1)}x=x$ (as
${T}_{\omega}$ and $\tilde{T}_{\omega}$ are automorphisms of a finite vector space); then$$(B_{i_n}\cdots
B_{i_1}B_{i_0})^mx{=}x.$$The braid relations (\ref{braid11}), (\ref{braid22}) show that the
$B_{i_j}$ satisfy the hypotheses of Lemma \ref{brixen} (in particular, the commutation
$B_{i_0}B_{i_n}=B_{i_n}B_{i_0}$ follows from $i_0>0$). This
Lemma now tells us $B_{i_j}\cdots
B_{i_1}B_{i_0}x=x$ for each $0\le j\le n$. But by the
definition of the $z_i$ this
means \begin{gather}z_i=\tilde{T}^{i}_{\omega}x\label{nachstur1}\end{gather}for
each $0\le i\le d+1$. When compared with formula (\ref{nachstur}) this yields
${T}_{\omega}=\tilde{T}_{\omega}$ since $M$ is generated as a $k$-vector
space by $M_0$ together with the ${T}^{i}_{\omega}x$ (or: the
$\tilde{T}^{i}_{\omega}x$) for $0\le i\le d$.\hfill$\Box$\\
{\bf Remarks:} The above proof of Proposition \ref{schutzengel} shows the following:
(i) The subspace in (d) could be replaced by the subspace $\{x\in M\,|\,T_s^2(x)=x\}$.
(ii) If the ${\mathcal
H}^{\sharp}$-action factors through an ${\mathcal
H}$-action, then the datum (d) can be entirely left out ($T_{\omega}$ can
then be reconstructed without a priori knowledge of $T_s$).
\section{The functor}
Here we define a functor $M\mapsto \Delta(M)$ from supersingular ${\mathcal H}^{\sharp\sharp}$-modules to torsion $k[[t]]$-modules with $\varphi$- and $\Gamma$-actions, as outlined in the introduction. Its entire content is encapsulated in the explicit formula for the elements $h(e)$ introduced below.
We fix the Lubin-Tate formal power series $\Phi(t)=\pi t+t^q$ for $F$.
Let $M$ be an ${\mathcal H}^{\sharp\sharp}$-module. View $M$ as a $k[[t]]$-module with $t=0$
on $M$. Let $\Gamma$ act on $M$ by $$\gamma\cdot x=T^{-1}_{e^*(\gamma)}(x)$$
for $\gamma\in\Gamma$, making $M$ a $k[[t]][\Gamma]$-module. We have an isomorphism of $k[[t]][\varphi]$-modules$${\mathfrak O}\otimes_{k[[t]][\Gamma]}M\cong k[[t]][\varphi]\otimes_{k[[t]]}M$$and hence an action of ${\mathfrak O}$ on $k[[t]][\varphi]\otimes_{k[[t]]}M$.
For $e\in M^{\epsilon}_{\underline{a}}[j]$ (any $\epsilon\in{[0,q-2]}^{\Phi}$, any $a\in[0,q-2]$, any $j\in\{0,1\}$) define the element\begin{gather}h(e)=\left\{\begin{array}{l@{\quad:\quad}l}t^{\epsilon_1}\varphi\otimes T^{-1}_{{\omega}}(e)+1\otimes
e +\sum_{c=0}^{q-2}t^{c}\varphi\otimes
T^{-1}_{{\omega}}((T_se)_{\underline{c+\epsilon_1+a}}) & j=0 \notag\\t^{q-1}\varphi\otimes T^{-1}_{{\omega}}(e)+1\otimes
e & j=1\end{array}\right.\notag\end{gather}of $k[[t]][\varphi]\otimes_{k[[t]]}M$. Define $\nabla(M)$ to be the $k[[t]][\varphi]$-sub module of
$k[[t]][\varphi]\otimes_{k[[t]]}M$ generated by the elements $h(e)$ for all
$e\in M^{\epsilon}_{\underline{a}}[j]$ (all $\epsilon$, $a$,
$j$). Define $$\Delta(M)=\frac{k[[t]][\varphi]\otimes_{k[[t]]}M}{\nabla(M)}.$$
{\bf Remark:} If $M$ is even an ${\mathcal H}$-module, then in view of formula (\ref{vorvigilallsaints}) the definition of
$h(e)$ simplifies to become\begin{gather}h(e)=\left\{\begin{array}{l@{\quad:\quad}l}t^{\epsilon_1}\varphi\otimes T^{-1}_{{\omega}}(e)+1\otimes
e +\varphi\otimes
T^{-1}_{{\omega}}(T_se) & j=0 \notag\\t^{q-1}\varphi\otimes T^{-1}_{{\omega}}(e)+1\otimes
e & j=1\end{array}\right.\notag.\end{gather}In this case it is not
necessary to split up $M$ into eigenspaces under the action of
$T_{e^*(\Gamma)}$, and the {\it notation} of many of the subsequent
computations simplifies (no underlined subscripts are needed). However, they hardly simplify in mathematical complexity, not even if we restrict to ${\mathcal
H}^{\flat}$-modules only (in which case always $\epsilon_1=0$ and $T_{e^*(\gamma)}=1$).\\
\begin{lem}\label{strasheim} Let $e\in M^{\epsilon}_{\underline{a}}[j]$. The integer\begin{gather}k_e=\left\{\begin{array}{l@{\quad:\quad}l}\epsilon_{1}& j=0 \notag\\ q-1
& j=1 \notag \end{array}\right.\notag\end{gather}satisfies $k_e\equiv\epsilon_1$ modulo $(q-1)$.
\end{lem}
{\sc Proof:} $j=1$ means $T_s^2(e)=e$, hence the claim follows from the relation (\ref{sorge}).\hfill$\Box$\\
\begin{lem}\label{4stubai} For $e\in M^{\epsilon}_{\underline{a}}[j]$ we have $\gamma\cdot h(e)=h(T^{-1}_{e^*(\gamma)}(e))$ for all $\gamma\in \Gamma$. In particular, $\nabla(M)$ is stable under the action of $\Gamma$, hence is an ${\mathfrak O}$-sub module of $k[[t]][\varphi]\otimes_{k[[t]]}M$. Hence $\Delta(M)$ is even an ${\mathfrak O}$-module.
\end{lem}
{\sc Proof:} First notice that $T^{-1}_{e^*(\gamma)}(e)\in M^{\epsilon}_{\underline{a}}[j]$. In particular, $h(T^{-1}_{e^*(\gamma)}(e))$ is well defined. For $\gamma\in\Gamma$ we find\begin{gather}\gamma\cdot (1\otimes
e)=1\otimes \gamma\cdot e=1\otimes T^{-1}_{e^*(\gamma)}(e).\label{7vorstubai1}\end{gather}Next, we compute\begin{align}\gamma\cdot( t^{k_e}\varphi\otimes T^{-1}_{{\omega}}(e))&\stackrel{(i)}{=}\gamma^{k_e}t^{k_e}\varphi\otimes \gamma\cdot T^{-1}_{{\omega}}(e)\notag\\{}&\stackrel{(ii)}{=}t^{k_e}\varphi\otimes T^{-1}_{{\omega}}T^{-1}_{e^*(\gamma)}(e).\label{7vorstubai2}\end{align}In $(i)$ we used $\gamma\cdot
t=[\gamma]_{\Phi}(t)\cdot \gamma$ and $[\gamma]_{\Phi}(t)\equiv\gamma t$
modulo $t^qk[[t]]$ (Lemma \ref{dominikanerjub}) and the fact that, since
$\pi=0$ in $k$, we have $t^q\varphi\otimes M=\Phi(t)\varphi\otimes M=\varphi
t\otimes M=0$. To see (ii) observe$$\gamma\cdot
T^{-1}_{{\omega}}(e)=T^{-1}_{e^*(\gamma)}
T^{-1}_{{\omega}}(e)=T^{-1}_{{\omega}}T^{-1}_{({\omega}^{-1}\cdot
e^*)(\gamma)}(e)$$$$=T^{-1}_{{\omega}}T^{-1}_{(e^*-\alpha_1^{\vee})(\gamma)}(e)=T^{-1}_{{\omega}}T_{\alpha_1^{\vee}}T^{-1}_{e^*(\gamma)}(e)=\gamma^{-k_e}T^{-1}_{{\omega}}T^{-1}_{e^*(\gamma)}(e)$$where
in the last step we use Lemma \ref{strasheim}. Combining formulae (\ref{7vorstubai1}) and (\ref{7vorstubai2}) we are done in the case $j=1$. In the case $j=0$ we in addition need the formula\begin{gather}\gamma\cdot\sum_{c=0}^{q-2}t^{c}\varphi\otimes
T^{-1}_{{\omega}}((T_se)_{\underline{c+\epsilon_1+a}})=\sum_{c=0}^{q-2}t^{c}\varphi\otimes
T^{-1}_{{\omega}}((T_sT_{e^*(\gamma)}e)_{\underline{c+\epsilon_1+a}}).\label{7vorstubai4}\end{gather}Let us prove this (for $e\in M^{\epsilon}_{\underline{a}}[0]$). For $f\in{\mathbb Z}$ and $\gamma\in\Gamma$ we compute\begin{gather}T_{(\omega^{-1}\cdot
e^*)(\gamma)}((T_se)_{\underline{f}})\stackrel{(i)}{=}T_{e^*(\gamma)}T_{\alpha_1^{\vee}(\gamma^{-1})}((T_se)_{\underline{f}})\notag\\\stackrel{(ii)}{=}\gamma^{f-\epsilon_1}(T_se)_{\underline{f}}=\gamma^{f-\epsilon_1-a}(T_s(\gamma^a e))_{\underline{f}}=\gamma^{f-\epsilon_1-a}(T_sT_{e^*(\gamma)}e)_{\underline{f}}.\label{nachvor}\end{gather}In $(i)$ recall that $\omega^{-1}\cdot e^*=e^*-\alpha_1^{\vee}$, in $(ii)$ notice that $(T_se)_{\underline{f}}\in M^{s\cdot\epsilon}$ and $(s\cdot\epsilon)_1=-\epsilon_1$. For $c\in[0,q-2]$ we deduce\begin{align}\gamma\cdot (t^c\varphi\otimes T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}}))&=\gamma^c t^c\varphi\otimes\gamma\cdot (T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}}))\notag\\{}&=\gamma^ct^c\varphi\otimes T^{-1}_{e^*(\gamma)}T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}})\notag\\{}&=\gamma^ct^c\varphi\otimes T_{\omega}^{-1}T^{-1}_{(\omega^{-1}\cdot e^*)(\gamma)}((T_se)_{\underline{c+\epsilon_1+a}})\notag\\{}&=t^c\varphi\otimes T_{\omega}^{-1}((T_sT_{e^*(\gamma)}e)_{\underline{c+\epsilon_1+a}})\notag\end{align}where in the last equality we inserted formula (\ref{nachvor}).\hfill$\Box$\\
\begin{pro}\label{jishnusharp} Suppose that $M$ is supersingular.
(a) $\Delta(M)$ is a torsion $k[[t]]$-module, generated by $M$ as a
$k[[t]][\varphi]$-module, and $\varphi$ acts injectively on it. $\Delta(M)^*={\rm Hom}_k(\Delta(M),k)$ is a free $k[[t]]$-module of rank ${\rm
dim}_k(M)$. The map $M\to \Delta(M)$ which sends $m\in M$ to the class of
$1\otimes m$ induces a bijection \begin{gather}M\cong \Delta(M)[t].
\label{sofeendsharp}\end{gather}
(b) The assignment $M\mapsto \Delta(M)$ is an exact
functor from the category of supersingular ${\mathcal H}^{\sharp\sharp}$-modules to the category of ${\mathfrak O}$-modules.
(c) If $M$ is finite dimensional then $\Delta(M)$ belongs to ${\rm
Mod}^{\clubsuit}({\mathfrak O})$.
\end{pro}
{\sc Proof:} (a) Supersingularity implies that there is a separated
and exhausting descending filtration $(F^{\mu}M)_{{\mu}\in{\mathbb Z}}$ of $M$ by ${\mathcal
H}^{\sharp\sharp}$-submodules $F^{\mu}M$ such that \begin{gather}{T}_s
(F^{\mu-1}M\cap {\rm ker}(T_s^2))\subset F^{\mu}M\label{6vorstubaisharp}\end{gather}for each $\mu\in{\mathbb Z}$. Put$${\bf
F}^{\mu}=k[[t]][\varphi]\otimes_{k[[t]]}F^{{\mu}}M$$and denote by ${\bf
F}_{\nabla}^{\mu}$ the $k[[t]][\varphi]$-submodule of ${\bf F}^{\mu}$
generated by all $h(e)$ with $e\in F^{\mu}M\cap M^{\epsilon}_{\underline{a}}[j]$ for some $\epsilon$, $a$, $j$. We claim\begin{gather}{\bf F}_{\nabla}^{\mu}=\nabla(M)\cap{\bf F}^{\mu}.\label{savovasharp}\end{gather}Arguing by induction, we may assume
that this is known with ${\mu}-1$ instead of ${\mu}$. Let ${\mathcal E}$ be a family of elements $e\in F^{{\mu}-1}M\cap M^{\epsilon_e}_{\underline{a_e}}[j_e]$ (for suitable
$\epsilon_e\in [0,q-2]^{\Phi}$ and $a_e\in [0,q-2]$ and $j_e\in\{0,1\}$ depending on $e$), inducing a $k$-basis
of $F^{{\mu}-1}M/F^{{\mu}}M$. We consider an
expression\begin{gather}\sum_{j_1,j_2\in {\mathbb Z}_{\ge0},e\in{\mathcal E}}c_{j_1,j_2,e}t^{j_2}\varphi^{j_1}h(e)\label{safa3sharp}\end{gather}with $c_{j_1,j_2,e}\in k$. Assuming that the expression (\ref{safa3sharp})
belongs to ${\bf F}^{\mu}$ we need to see that it even belongs to
${\bf F}_{\nabla}^{\mu}$.
Suppose that this is false. We may then define$$j_1={\rm min}\{j\ge 0\,|\,c_{j,j_2,e}t^{j_2}\varphi^{j}h(e)\notin {\bf F}_{\nabla}^{\mu}\mbox{ for some }j_2\ge0,\mbox{ some } e\in{\mathcal E}\}.$$
{\it Claim: We find some $j_2$ and some $e$ with $c_{j_1,j_2,e}t^{j_2}\varphi^{j_1}h(e)\in {\bf
F}^{\mu}-{\bf F}_{\nabla}^{\mu}$.}
For $e\in {\mathcal E}$ the expression\begin{gather}1\otimes e+t^{k_e}\varphi\otimes T_{\omega}^{-1}(e)\label{freivorvansharp}\end{gather}is congruent to $h(e)$ modulo ${\bf
F}^{{\mu}}$, in view of $e\in F^{\mu-1}M$ and formula (\ref{6vorstubaisharp}). Therefore, modulo ${\bf F}^{{\mu}}$ the expression
(\ref{safa3sharp})
reads$$\sum_{j_1,j_2,e}c_{j_1,j_2,e}t^{j_2}\varphi^{j_1}\otimes
e+c_{j_1,j_2,e}t^{j_2}\varphi^{j_1}t^{k_e}\varphi\otimes
T_{\omega}^{-1}(e).$$Notice that $\varphi^{j_1}t^{k_e}\varphi\in
k[[t]]\varphi^{j_1+1}$. The claim now follows in view of\begin{gather}\frac{{\bf F}^{{\mu}-1}}{{\bf
F}^{{\mu}}}=\bigoplus_{j\ge0}k[[t]]\varphi^j\otimes_{k[[t]]}\frac{F^{{\mu}-1}M}{F^{\mu}M}.\label{savovalasharp}\end{gather}
The claim proven, we may argue by induction on the number
of summands in the expression (\ref{safa3sharp}) which do not belong to ${\bf
F}_{\nabla}^{\mu}$. We may thus assume from the start that the expression
(\ref{safa3sharp}) consists of a single summand $t^{j_2}\varphi^{j_1}h(e)$, and
that moreover
$e\notin {F}^{\mu}M$ for this $e$. The aim is then to deduce $t^{j_2}\varphi^{j_1}h(e)\in {\bf F}_{\nabla}^{\mu}$, which contradicts our
above assumption.
Let us write $\epsilon=\epsilon_e$ and $a=a_e$. The vanishing of
$t^{j_2}\varphi^{j_1}h(e)$ modulo ${\bf F}^{\mu}$ means, by the
decomposition (\ref{savovalasharp}) again,
that$$t^{j_2}\varphi^{j_1}\otimes
e\stackrel{(i)}{=}0\stackrel{(ii)}{=}t^{j_2}\varphi^{j_1}t^{k_e}\varphi\otimes
T_{\omega}^{-1}(e)$$(i.e. absolute vanishing, not just modulo
${\bf F}^{\mu}$). If $T_s^2(e)=e$ then this shows $t^{j_2}\varphi^{j_1} h(e)=0$. Now suppose $T_s^2(e)=0$ (and hence $k_e<q-1$). The definition of $h(e)$ together with the vanishings $(i)$ and $(ii)$ shows$$t^{j_2}\varphi^{j_1}h(e)=t^{j_2}\varphi^{j_1}\sum_{c=0}^{q-2}t^c\varphi \otimes
T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}}).$$Since the vanishing $(ii)$ also forces
$t^{j_2}\varphi^{j_1}t^{k_e}\varphi\in k[[t]]\varphi^{j_1+1}t$, there is some $i$ and
some $j'_2\ge0$ with $$t^{j_2}\varphi^{j_1}=t^{j'_2}\varphi^{j_1}t^i\quad \mbox{ and }\quad i\ge q-k_e.$$If
$k_e=0$ (and hence $i\ge q$) then again the conclusion is
$t^{j_2}\varphi^{j_1} h(e)=0$. It remains to discuss the case where
$0<k_e<q-1$. In this case, $(T_se)_{\underline{c+\epsilon_1+a}}\in
M^{s\cdot\epsilon}$ and $(s\cdot\epsilon)_1=-\epsilon_1$ implies $q-1-k_e=k_{(T_se)_{\underline{c+\epsilon_1+a}}}$
for each $c$. We thus see\begin{align}t^{q-k_e+c}\varphi \otimes
T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}})&=t^{1+c}(t^{k_{(T_se)_{\underline{c+\epsilon_1+a}}}}\varphi\otimes
T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}})+1\otimes
(T_se)_{\underline{c+\epsilon_1+a}})\notag\\{}&=t^{1+c}
h((T_se)_{\underline{c+\epsilon_1+a}})-\sum_{c'=0}^{q-2}t^{1+c+c'}\varphi\otimes
T_{\omega}^{-1}((T_s((T_se)_{\underline{c+\epsilon_1+a}}))_{\underline{c'+c+a}})\notag\end{align}by
the definition of $h((T_se)_{\underline{c+\epsilon_1+a}})$, again since
$(T_se)_{\underline{c+\epsilon_1+a}}\in M^{s\cdot\epsilon}$ and
$(s\cdot\epsilon)_1=-\epsilon_1$. For $0\le f\le q-2$ we have$$\sum_{0\le
c,c'\le q-2\atop c+c'=f}(T_s((T_se)_{\underline{c+\epsilon_1+a}}))_{\underline{f+a}}=\sum_{0\le
c\le q-2}(T_s((T_se)_{\underline{c+\epsilon_1+a}}))_{\underline{f+a}}=0$$as follows
from $T^2_s(e)=0$. This shows$$\sum_{c,c'=0}^{q-2}t^{1+c+c'}\varphi\otimes T_{\omega}^{-1}((T_s((T_se)_{\underline{c+\epsilon_1+a}}))_{\underline{c'+c+a}})=0.$$Since $e$ belongs to $F^{\mu-1}M$, formula (\ref{6vorstubaisharp}) shows $h((T_s e)_{\underline{c+\epsilon_1+a}})\in{\bf F}_{\nabla}^{\mu}$. Together we obtain $t^{q-k_e+c}\varphi \otimes
T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}})\in {\bf F}_{\nabla}^{\mu}$, hence $t^{i+c}\varphi\in \otimes
T_{\omega}^{-1}((T_se)_{\underline{c+\epsilon_1+a}})\in {\bf F}_{\nabla}^{\mu}$ for $0\le c\le q-2$. This gives$$t^{j_2}\varphi^{j_1}h(e)= \sum_{c=0}^{q-2}t^{j'_2}\varphi^{j_1}t^{i+c}\varphi \otimes T_{\omega}^{-1}((T_s e)_{\underline{c+\epsilon_1+a}})\in {\bf F}_{\nabla}^{\mu},$$as desired.
Formula (\ref{savovasharp}) is proven. We deduce that$$\frac{\nabla(M)\cap {\bf F}^{\mu-1}}{\nabla(M)\cap
{\bf F}^{{\mu}}}$$is
generated as a $k[[t]][\varphi]$-module by the classes
modulo ${\bf F}^{{\mu}}$
of the elements (\ref{freivorvansharp}). In view of the decomposition (\ref{18okt}) we may apply Lemma \ref{abstrtor} to
the subquotients of our filtration, proving the analogs of our claims for
these subquotients. They then
follow for the full space itself.
(b) It is clear that $M\mapsto \Delta(M)$ is a (covariant) right exact functor. To
see left exactness, let $M_1\to M_2$ be injective. Since the kernel of
$\Delta(M_1)\to \Delta(M_2)$ is a torsion $k[[t]]$-module it has, if non zero,
a non zero vector killed by $t$. By formula (\ref{sofeendsharp}) it must belong to (the image of)
$M_1$, contradicting injectivity of $M_1\to M_2$.
(c) If $M$ is a standard supersingular ${\mathcal
H}$-module then $\Delta(M)$ is a torsion standard cyclic ${\mathfrak
O}$-module (cf. also the discussion in subsection \ref{vialhe}). Therefore Proposition \ref{altfund} shows that for a subquotient $M$ of a standard supersingular ${\mathcal
H}$-module, $\Delta(M)$ admits a filtration such that each
associated
graded piece is a torsion standard cyclic ${\mathfrak
O}$-module. As the functor $\Delta$ is exact it therefore takes finite dimensional supersingular ${\mathcal
H}^{\sharp\sharp}$-modules to objects in ${\rm
Mod}^{\clubsuit}({\mathfrak O})$.\hfill$\Box$\\
{\bf Remark:} In Proposition \ref{jishnusharp}, the hypothesis that $M$ be supersingular may evidently be replaced by the following weaker hypothesis: $M$ is an inductive limit of ${\mathcal
H}^{\sharp\sharp}$-submodules which admit filtrations of finite
length such that on
each associated graded piece we have ${\rm ker}(T_s)={\rm ker}(T_s^2)$.
\section{Standard objects and full faithfulness}
\subsection{The bijection between standard supersingular Hecke modules and standard cyclic Galois representations}
\label{vialhe}
Let $M$ be a standard supersingular ${\mathcal
H}$-module, arising from the supersingular character $\chi:{\mathcal
H}_{\rm aff}\to k$. There is some $e_0\in M$ such that, putting $e_j=T^{-j}_{{\omega}}e_0$, we have $M=\bigoplus_{j=0}^dk.e_j$ and ${\mathcal
H}_{\rm aff}$ acts on $k.e_0$ by $\chi$. Denote by $\eta_j:\Gamma\to
k^{\times}$ the character through which
$T_{{e^*}(.)}^{-1}$ acts on $k.e_j$,
i.e. $T^{-1}_{{e^*}(\gamma)}(e_j)=\eta_j(\gamma)e_j$ for $\gamma\in\Gamma$.
\begin{lem} \label{irr} (a) There are $0\le k_{e_j}\le q-1$ for $0\le j\le d$, not all of them $=0$ and not all of them $=q-1$, such that\begin{gather}t^{k_{e_j}}\varphi \otimes T^{-1}_{{\omega}}(e_{j})=-1\otimes e_j\label{gabriel}\end{gather}for all $0\le j\le d$.
(b) If for any $1\le j\le d$ there is some $0\le i\le d$ with $k_{e_i}\ne k_{e_{i+j}}$, then $\Delta(M)$ is irreducible as a $k[[t]][\varphi]$-module.
(c) Suppose that for any $1\le j\le d$ which satisfies $k_{e_i}=
k_{e_{i+j}}$ for all $0\le i\le m$ there is some $0\le i\le d$ with $\eta_{i}\ne\eta_{i+j}$. Then $\Delta(M)$ is irreducible as an ${\mathfrak O}$-module.\end{lem}
{\sc Proof:} For $M$ as above, $\nabla(M)$ is generated by elements of the form
$h(e)=t^{k_e}\varphi\otimes T_{\omega}^{-1}(e)+1\otimes e$. They give rise to formula (\ref{gabriel}), hence statement (a). For statements (b) and (c) apply Proposition \ref{altfund}; in (c) notice that $\gamma\cdot(1\otimes e_j)=\eta_j(\gamma)\otimes e_j$ for $\gamma\in\Gamma$.\hfill$\Box$\\
\begin{lem}\label{mafr} (a) Conjugating $\chi$ by powers of
$T_{\omega}$ means cyclically permuting the ordered tuple
$((\eta_0,k_{e_0}),\ldots,(\eta_d,k_{e_d}))$ associated with $\chi$ as
above. Knowing the conjugacy class of $\chi$ (under powers of
$T_{\omega}$) is equivalent with knowing the tuple
$((\eta_0,k_{e_0}),\ldots,(\eta_d,k_{e_d}))$ up to cyclic permutations,
together with $\chi(T^{d+1}_{\omega})$.
(b) (Vign\'{e}ras) Two standard supersingular ${\mathcal
H}$-modules are isomorphic if and only if the element
$T^{d+1}_{\omega}\in {\mathcal
H}$ acts on them by the same constant in $k^{\times}$ and if they arise from two supersingular characters ${\mathcal
H}_{\rm aff}\to k$ which are conjugate under some power of $T_{\omega}$.
(c) (Vign\'{e}ras) A standard supersingular ${\mathcal
H}$-module $M$ arising from $\chi$ is simple if and only if the orbit of $\chi$ under conjugation by powers of $T_{\omega}$ has cardinality $d+1$.
\end{lem}
{\sc Proof:} Statement (a) is clear. For (b) and (c) see \cite{vigneras} Proposition 3 and Theorem 5.\hfill$\Box$\\
\begin{satz}\label{molola1} The assignment $M\mapsto\Delta(M)^*\otimes_{k[[t]]}k((t))$ induces bijections between sets of isomorphim classes of
(a) standard supersingular ${\mathcal
H}$-modules on the one hand, and standard cyclic \'{e}tale $(\varphi,\Gamma)$-modules of dimension $d+1$ on the other hand, as well as
(b) simple supersingular ${\mathcal
H}$-modules of $k$-dimension $d+1$ on the one hand, and simple \'{e}tale $(\varphi,\Gamma)$-modules of dimension $d+1$ on the other hand.\end{satz}
{\sc Proof:} By Lemma \ref{irr}(a) and Proposition \ref{fudonsup}, $\Delta(M)^*\otimes_{k[[t]]}k((t))$ (for a standard supersingular ${\mathcal
H}$-module $M$) is a standard cyclic \'{e}tale
$(\varphi,\Gamma)$-module of dimension $d+1$. Together with Lemma \ref{ossa}
and Lemma \ref{mafr} this shows, more precisely, that the assignment
$M\mapsto\Delta(M)^*\otimes_{k[[t]]}k((t))$ is injective on isomorphism
classes of standard supersingular ${\mathcal
H}$-modules. Here observe that we may rewrite
the equation (\ref{gabriel}) as \begin{align}t^{k_{e_j}}\varphi \otimes
e_{j+1}&=-1\otimes e_j\quad\quad\mbox{ for }0\le j\le
d-1\notag\\t^{k_{e_d}}\varphi \otimes \chi(T^{-d-1}_{\omega})e_{0}&=-1\otimes e_d\notag\end{align}where we used
$T_{{\omega}}^{-1}(e_d)=T_{{\omega}}^{-d-1}(e_0)=\chi(T^{-d-1}_{\omega})e_0$. Thus $(-1)^{d+1}\chi(T^{-d-1}_{\omega})\in k^{\times}$ is the constant referred to in Lemma \ref{ossa}. The
assignment $M\mapsto\Delta(M)^*\otimes_{k[[t]]}k((t))$ is also surjective: Indeed, for any given
set of parameter data of a standard cyclic \'{e}tale $(\varphi,\Gamma)$-module
of dimension $d+1$ as described in Lemma \ref{ossa} we find a suitable $\chi$
giving rise to these parameter data. Statement
(a) is proven. To prove statement (b) we use Lemma \ref{irr} and Lemma \ref{mafr} to see that if
$M$ is simple then the ${\mathfrak O}$-module $\Delta(M)$ is simple. By Lemma
\ref{irrnopsi} this means that $\Delta(M)^*\otimes_{k[[t]]}k((t))$ is
simple. To conclude one may now simply check that this simplicity argument in
fact can be reversed, using Lemma \ref{irrphiga}. (Of course, via Corollary \ref{molola}, this comes down to the well known description of the parameter sets for the isomorphim classes of irreducible $(d+1)$-dimensional ${\rm Gal}(\overline{F}/F)$-representations over $k$.) Alternatively, given the injectivity of the assignment $M\mapsto \Delta(M)^*\otimes_{k[[t]]}k((t))$, one might also just invoke the numerical version of Corollary \ref{molola} below, i.e. \cite{vigneras} Theorem 5.\hfill$\Box$\\
\begin{kor}\label{molola} The assignment $M\mapsto\Delta(M)^*\otimes_{k[[t]]}k((t))$, composed with the functor of Theorem \ref{sosego}, induces a bijection between the set of isomorphim classes of supersingular ${\mathcal
H}$-modules of $k$-dimension $d+1$, and the set of isomorphim classes of $(d+1)$-dimensional standard cyclic ${\rm Gal}(\overline{F}/F)$-representation.
\end{kor}
{\sc Proof:} Theorem \ref{molola1}. \hfill$\Box$\\
{\bf Remark:} (a) The "numerical Langlands correspondence" (for simple resp. irreducible objects) implied by
Corollary \ref{molola} was proven in \cite{vigneras} Theorem 5.
(b) There is an alternative and arguably more natural definition of
supersingularity for ${\mathcal
H}$-modules. Its agreement with the one given in subsection \ref{defhsu},
and hence the "numerical Langlands correspondence" with respect to
this alternative definition of supersingularity, was proven in \cite{ol}.
\subsection{Reconstruction of an initial segment of $M$ from $\Delta(M)$}
\label{allnov}
Let ${[0,q-1]}^{\Phi}$ be the set of tuples ${\mu}=(\mu_{i})_{0\le
i\le d}$ with $\mu_i\in\{0,\ldots, q-1\}$ and $\sum_{0\le i\le
d}{\mu}_{i}\equiv 0$ modulo $(q-1)$. We often read the indices as
elements of ${\mathbb Z}/(d+1)$, thus $\mu_i=\mu_j$ for $i,
j\in{\mathbb Z}$ whenever $i-j\in(d+1){\mathbb Z}$.
Let $\Delta$ be an ${\mathfrak O}$-module. For
$\mu\in[0,q-1]^{\Phi}$ let ${\mathcal F}{\Delta}[t]^{\mu}$ be the $k$-sub vector space of
$\Delta[t]=\{x\in \Delta\,|\,tx=0\}$ generated by all $x\in {\Delta}[t]$
satisfying $t^{\mu_{i}}\varphi\ldots t^{\mu_1}\varphi t^{\mu_0}\varphi x\in {\Delta}[t]$ for all $0\le i\le d$, as well as $t^{\mu_{d}}\varphi\ldots t^{\mu_1}\varphi t^{\mu_0}\varphi x\in k^{\times}x$.
Put ${\mathcal F}{\Delta}[t]=\sum_{\mu\in [0,q-1]^{\Phi}}{\mathcal F}{\Delta}[t]^{\mu}$ (sum in ${\Delta}[t]$).
\begin{lem}\label{auchwi} ${\mathcal F}{\Delta}[t]=\bigoplus_{\mu\in [0,q-1]^{\Phi}}{\mathcal F}{\Delta}[t]^{\mu}$, i.e. the sum is direct.
\end{lem}
{\sc Proof:} Consider the lexicographic enumeration $\mu(1),
\mu(2),\mu(3),\ldots $ of $[0,q-1]^{\Phi}$ such that
for each pair $r'>r$ there is some $0\le i_0\le d$ with $\mu_i(r)\ge \mu_i(r')$
for all $i<i_0$, and $\mu_{i_0}(r)> \mu_{i_0}(r')$. Let
$\sum_{r\ge1}x_{r}=0$ with $x_{r}\in
{\mathcal F}{\Delta}[t]^{\mu(r)}$. We prove $x_{r}=0$ for all $r$ by
induction on $r$. So, fix $r$ and assume $x_{r'}=0$ for all $r'<r$, hence $\sum_{r'\ge
r}x_{r'}=\sum_{r\ge1}x_{r}-\sum_{r'<r}x_{r}=0$. For $r'>r$ we have $t^{\mu_d(r)}\varphi\cdots
t^{\mu_0(r)}\varphi(x_{r'})=0$. Therefore \begin{align}0=t^{\mu_d(r)}\varphi\cdots
t^{\mu_0(r)}\varphi (\sum_{r'\ge r}x_{r'})&=\sum_{r'\ge r}t^{\mu_d(r)}\varphi\cdots
t^{\mu_0(r)}\varphi x_{r'}\notag\\{}&=t^{\mu_d(r)}\varphi\cdots t^{\mu_0(r)}\varphi x_{r}\in k^{\times}x_{r}\notag\end{align}and hence $x_r=0$.\hfill$\Box$\\
We define $k$-linear endomorphisms $T_{{\omega}}$, $T_s$ and $T_{e^*(\gamma)}$ (for $\gamma\in\Gamma$) of
${\mathcal F}{\Delta}[t]$ as follows. In view of Lemma \ref{auchwi} it is enough to
define their values on $x\in{\mathcal F}{\Delta}[t]^{\mu}$; we
put\begin{gather}T_{{\omega}}(x)=-t^{\mu_0}\varphi x,\quad\quad\quad\quad T_{e^*(\gamma)}(x)=\gamma^{-1}\cdot x,\notag\\T_s(x)=\left\{\begin{array}{l@{\quad:\quad}l}-x & \mu_d=q-1\notag\\ensuremath{\overrightarrow{0}}& \mu_d<q-1\end{array}\right.\notag.\end{gather}Here $\gamma^{-1}\cdot x$ is understood with respect to the $\Gamma$-action induced by the
${\mathfrak O}$-module structure on $\Delta(M)$.\\
{\bf Definition:} For an ${\mathcal
H}^{\sharp\sharp}$-module $M$ and $\mu\in[0,q-1]^{\Phi}$ let ${\mathcal F}M^{\mu}$
denote the $k$-sub vector space of $M$ consisting of $x\in M$ satisfying the following conditions for all $0\le i\le
d$:\begin{gather}T^{-1}_{\alpha_1^{\vee}(\gamma)}(T_{{\omega}}^i(x))=\gamma^{\mu_{i-1}}T_{{\omega}}^i(x)\quad\mbox{ for all }\gamma\in\Gamma,\label{nachsy}\\ T_s(T_{{\omega}}^i(x))=\left\{\begin{array}{l@{\quad:\quad}l}-T_{{\omega}}^i(x) & \mu_{i-1}=q-1\\ensuremath{\overrightarrow{0}}& \mu_{i-1}<q-1\end{array}.\right.\label{dochnachsy}\end{gather}Let ${\mathcal F}M$ denote the subspace of $M$ generated by the ${\mathcal F}M^{\mu}$ for all $\mu\in[0,q-1]^{\Phi}$.
\begin{lem}\label{ennoad} (a) For $\mu\in[0,q-1]^{\Phi}$ let $\epsilon_{\mu}\in{[0,q-2]}^{\Phi}$ be the unique element with\begin{gather}(\epsilon_{\mu})_{-i}\equiv \mu_{i}\,\, {\rm mod
}(q-1).\label{allsavi}\end{gather}for all $i$. Then we have ${\mathcal F}M^{\mu}\subset M^{\epsilon_{\mu}}$.
(b) ${\mathcal F}M$ is an ${\mathcal
H}^{\sharp\sharp}$-submodule of $M$. It contains each ${\mathcal
H}^{\sharp\sharp}$-sub module of $M$ which is a subquotient of a standard supersingular ${\mathcal
H}^{\sharp\sharp}$-module.
(c) Viewing the isomorphism ${\Delta(M)}[t]\cong M$ (Proposition \ref{jishnusharp}) as an identity, we have ${\mathcal F}M^{\mu}\subset
{\mathcal F}{\Delta(M)}[t]^{\mu}$ for each $\mu\in[0,q-1]^{\Phi}$, and in particular\begin{gather}{\mathcal F}M\subset {\mathcal F}{\Delta(M)}[t]\label{fiedro}.\end{gather} The operators $T_{{\omega}}$, $T_s$ and $T_{e^*(\gamma)}$ acting on ${\mathcal F}{\Delta(M)}[t]$ as defined above restrict to the operators $T_{{\omega}}, T_s, T_{e^*(\gamma)}\in{\mathcal H}^{\sharp\sharp}$ acting on ${\mathcal F}M$.\end{lem}
{\sc Proof:} This is a matter of inserting the definitions. Putting
$\mu'_i=\mu_{i+1}$ we first observe
$T_{\omega}({\mathcal F}M^{\mu})={\mathcal F}M^{\mu'}$, as well as
$T_{\omega}(x)\in M^{\epsilon_{\mu'}}$ for $x\in
M^{\epsilon_{\mu}}$. Moreover, for such $x$ we have
$k_{T_{\omega}(x)}\equiv (\epsilon_{\mu'})_1\equiv \mu'_{-1}=\mu_0$ modulo
$(q-1)$, and $k_{T_{\omega}(x)}=q-1$ if and only if
$T_sT_{\omega}(x)=-T_{\omega}(x)$, if and only if $\mu'_d=\mu_0=q-1$. Therefore the formula $T_{{\omega}}(x)=-t^{\mu_0}\varphi x$ becomes the formula for $h(e)$
with $e=T_{{\omega}}(x)$.\hfill$\Box$\\
{\bf Remark:} Some more effort yields that the inclusion
(\ref{fiedro}) is in fact an equality.
\subsection{Reconstruction of $\sharp$-supersingular ${\mathcal
H}^{\sharp}$-modules $M$ from $\Delta(M)$}
\begin{lem}\label{4vorstubai} Let $M$ be an irreducible supersingular ${\mathcal
H}$-module. Let $\mu\in[0,q-1]^{\Phi}$, let $x\in M$ and $u_{i,c}\in
M^{\omega^{-1} s\omega^{i+1}\cdot\epsilon_{\mu}}$ for $i\ge0$ and
$0\le c\le q-2$ (with $\epsilon_{\mu}$ given by formula (\ref{allsavi})). Assume $u_{i,c}=0$ if
(i) $\mu_i=0$, or
(ii) $\mu_i=q-1$ and $c>0$, or
(iii) $\mu_{i}<q-1$ and $c\ge q-1-\mu_{i}$.
Assume that, if we put $x\{-1\}=x$, then$$x\{i\}=t^{\mu_i}\varphi(x\{i-1\})-\sum_{c=0}^{q-2}t^c\varphi
u_{i,c}$$belongs to $M\cong \Delta(M)[t]$ for each $i\ge0$. Finally, assume that
$x\{D\}=x$ for some $D>0$ with $D+1\in{\mathbb Z}(d+1)$. Then there is some ${x}'\in M$ with $x-x'\in
M^{\epsilon_{\mu}}$ and such that$$x'\{i\}=t^{\mu_i}\varphi(\ldots
(t^{\mu_1}\varphi(t^{\mu_0}\varphi x'))\ldots )$$belongs to $M$ for each $i$,
and $x'\{D\}=x'$. Moreover, if $x$ is an eigenvector
for $T_{e^*(\Gamma)}$, then $x'$ can be chosen to be an eigenvector
for $T_{e^*(\Gamma)}$, with the same eigenvalues.
\end{lem}
{\sc Proof:} It is easy to see that all the irreducible subquotients of a standard supersingular ${\mathcal
H}$-module are isomorphic. In particular, an irreducible supersingular ${\mathcal
H}$-module is isomorphic with a submodule of a standard supersingular ${\mathcal
H}$-module. Therefore we may assume that $M$ itself is a (not necessarily
irreducible) standard supersingular ${\mathcal
H}$-module. We then have a direct sum decomposition
$M=\oplus_{j=0}^dM^{[j]}$ with ${\rm dim}_k(M^{[j]})=1$ and integers $0\le
k_j\le q-1$ such that \begin{gather}T_{\omega}(M^{[j+1]})=t^{k_j}\varphi
(M^{[j+1]})=M^{[j]}\label{3vorstubai}\end{gather}(always reading $j$ modulo
$(d+1)$). More precisely, we have $M^{[j]}\subset M^{\epsilon_j}$ for certain $\epsilon_j\in[0,q-2]^{\Phi}$, and $k_j\equiv (\omega\cdot\epsilon_{j+1})_1$ modulo $(q-1)$. It follows that\begin{gather}k[t]\varphi M=\bigoplus_{j=0}^dk[t]\varphi M^{[j]}.\label{3nachstubai}\end{gather}For $m\in
M$ write $m=\sum_jm^{[j]}$ with $m^{[j]}\in M^{[j]}$. By formulae
(\ref{3vorstubai}), (\ref{3nachstubai}), the defining formula for
$x\{i\}$ splits up into the formulae\begin{gather}x\{i\}^{[j]}=t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})-\sum_{c=0}^{q-2}t^c\varphi(u_{i,c}^{[j+1]})\label{3vorstubai1}\end{gather}for
all $j$. We use them to show\begin{gather}t^c\varphi(u_{i,c}^{[j+1]})= 0\quad \mbox{
if }c-\mu_i\notin(q-1){\mathbb Z}.\label{wienneu}\end{gather}If
$t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})=0$ and $x\{i\}^{[j]}=0$ or if
$t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})\ne 0$ then formula (\ref{wienneu}) follows from formulae
(\ref{3vorstubai}), (\ref{3nachstubai}) and (\ref{3vorstubai1}). If
$\mu_i\in\{0,q-1\}$ then formula (\ref{wienneu}) follows from our
assumptions on the $u_{i,c}$. Finally, we claim that the case where $t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})=0$ and
$x\{i\}^{[j]}\ne0$ and $\mu_i\notin\{0,q-1\}$ cannot occur. Indeed, the first two
conditions imply, by formula (\ref{3vorstubai1}), that $\sum_{c=0}^{q-2}t^c\varphi(u_{i,c}^{[j+1]})$ must be a
non-zero element in $M$. But we know $T_{\omega}(u_{i,c}^{[j+1]})\in M^{s\omega^{i+1}\cdot\epsilon_{\mu}}$
(which is implied by the assumption $u_{i,c}\in
M^{\omega^{-1} s\omega^{i+1}\cdot\epsilon_{\mu}}$) and hence $T_{\omega}(u_{i,c}^{[j+1]})=-t^{q-1-\mu_i}\varphi(u_{i,c}^{[j+1]})$ since$$q-1-\mu_i=q-1-\epsilon_{-i}=(s\omega^{i+1}\cdot\epsilon_{\mu})_1\quad\mbox{ if
}\mu_i\notin\{0,q-1\}.$$Therefore formulae
(\ref{3vorstubai}), (\ref{3nachstubai}) and (\ref{3vorstubai1}) show
that the only
candidate $c$ with $t^c\varphi(u_{i,c}^{[j+1]})\ne 0$ is then
$c=q-1-\mu_i$, but this violates the assumption $u_{i,c}=0$ for $c\ge
q-1-\mu_{i}$ in the present case. Formula (\ref{wienneu}) is proven. Arguing once more with formulae
(\ref{3vorstubai}), (\ref{3nachstubai}) and (\ref{3vorstubai1}) shows
\begin{gather}[t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})=0\quad\mbox{ or
}\quad\varphi(u_{i,0}^{[j+1]})=0]\quad\quad\quad\quad\mbox{ if
}\mu_i=q-1.\label{formelcrux}\end{gather}In the following, by $u_{i,q-1}$ we
mean $u_{i,0}$. If $t^{\mu_i}\varphi(u_{i,\mu_i}^{[j+1]})\ne 0$ we may write
$$t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})-t^{\mu_i}\varphi(u_{i,\mu_i}^{[j+1]})=\rho_{i,j}t^{\mu_i}\varphi(u_{i,j}^{[j+1]})$$
for some $\rho_{i,j}\in k$, since $t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})$ and
$t^{\mu_i}\varphi(u_{i,\mu_i}^{[j+1]})$ belong to the same one-dimensional
$k$-vector space. The upshot of formulae (\ref{wienneu})
and (\ref{formelcrux}) is then that formula (\ref{3vorstubai1}) simplifies to
become either $$x\{i\}^{[j]}=t^{\mu_i}\varphi(x\{i-1\}^{[j+1]})$$or$$x\{i\}^{[j]}=\rho_{i,j}t^{\mu_i}\varphi(u_{i,j}^{[j+1]})$$
for some $\rho_{i,j}\in k$. If (for fixed $j$) the first event occurs for all $i$ we obtain$$x^{[j]}=x\{D\}^{[j]}=t^{\mu_D}\varphi(\ldots
(t^{\mu_1}\varphi(t^{\mu_0}\varphi (x^{[j]})))\ldots )$$in which case we put
$n(j)=0$. Otherwise we pick the maximal $i\le D$ for which the second event occurs
to obtain$$x^{[j]}=x\{D\}^{[j]}=\rho_{j}t^{\mu_D}\varphi(\ldots
(t^{\mu_{n(j)}}\varphi(t^{\mu_{n(j)-1}}\varphi(u_{n(j)-1,\mu_{n(j)-1}}^{[j+1-n(j)]})))\ldots)$$
with $t^{\mu_{n(j)-1}}\varphi u_{n(j)-1,\mu_{n(j)-1}}^{[j+1-n(j)]}\ne0$, for some $1\le
n(j)\le D+1$ and some $\rho_j\in k$.
We study this second case $n(j)>0$ further. By construction,$$
w_j\{-1\}=t^{\mu_{n(j)-1}}\varphi(u_{n(j)-1,\mu_{n(j)-1}}^{[j+1-n(j)]})$$is
non-zero and belongs
to $M$. On the other hand, $u_{n(j)-1,\mu_{n(j)-1}}\in M^{\omega^{-1} s
\omega^{n(j)}\cdot\epsilon_{\mu}}$ implies $T_{\omega}(u^{[j+1-n(j)]}_{n(j)-1,\mu_{n(j)-1}})\in M^{s
\omega^{n(j)}\cdot\epsilon_{\mu}}$ and hence $$t^{(s
\omega^{n(j)}\cdot\epsilon_{\mu})_1}\varphi(u_{n(j)-1,\mu_{n(j)-1}}^{[j+1-n(j)]})=-T_{\omega}(u^{[j+1-n(j)]}_{n(j)-1,\mu_{n(j)-1}})\in M^{s
\omega^{n(j)}\cdot\epsilon_{\mu}}.$$Together this means $\mu_{n(j)-1}\equiv(s
\omega^{n(j)}\cdot\epsilon_{\mu})_1$ modulo $(q-1)$ and $w_j\{-1\}\in
M^{s
\omega^{n(j)}\cdot\epsilon_{\mu}}$. But we also have $\mu_{n(j)-1}\equiv(\omega^{n(j)}\cdot\epsilon_{\mu})_1$. Combining we see
$\mu_{n(j)-1}\equiv-\mu_{n(j)-1}$ modulo $(q-1)$. Hence, we
either have
$\mu_{n(j)-1}=0$ or $\mu_{n(j)-1}=\frac{q-1}{2}$ or $\mu_{n(j)-1}=q-1$. In
view of the
assumed vanishings of the $u_{i,c}$'s (and of
$u_{n(j)-1,\mu_{n(j)-1}}^{[j+1-n(j)]}\ne0$) this leaves $\mu_{n(j)-1}=q-1$
as the only possibility. It follows that$$s
\omega^{n(j)}\cdot\epsilon_{\mu}=\omega^{n(j)}\cdot\epsilon_{\mu}$$and hence $w_j\{-1\}\in M^{
\omega^{n(j)}\cdot\epsilon_{\mu}}$. Next, again by construction we know that$$ w_j\{s\}=t^{\mu_{n(j)+s}}\varphi (
w_j\{s-1\})$$belongs to $M$, for $0\le
s\le D-n(j)$. By what we learned about $w_j\{-1\}$ this implies $w_j\{s\}=(-1)^{s+1}T_{\omega}^{s+1}w_j\{-1\}\in M^{ \omega^{n(j)+s+1}\cdot\epsilon_{\mu}}$ by an induction on $s$ (and we also see $\mu_{n(j)+s}\in\{k_0,\ldots, k_d\}$ with the $k_{\ell}$ from formula (\ref{3vorstubai})). For $s=D-n(j)$ we obtain $x^{[j]}=x\{D\}^{[j]}\in
M^{\epsilon_{\mu}}$.
We now put $x'=\sum_{n(j)=0}x^{[j]}$.\hfill$\Box$\\
\begin{lem}\label{fabei} Let $M$ be an irreducible supersingular ${\mathcal
H}$-module. Let $\mu\in[0,q-1]^{\Phi}$ and $x\in M$ such
that$$x\{i\}=t^{\mu_i}\varphi(\ldots (t^{\mu_1}\varphi(t^{\mu_0}\varphi x
))\ldots)$$belongs to $M\cong \Delta(M)[t]$ for each $i\ge0$, and such
that $x\{D\}=x$ for some $D>0$ with $D+1\in{\mathbb Z}(d+1)$. Then $x\in
M^{\epsilon_{\mu}}$ and $x\{i\}=(-T_{\omega})^ix$ for each $i$.
\end{lem}
{\sc Proof:} This follows from the formulae (\ref{3vorstubai}) and
(\ref{3nachstubai}) in the proof of Lemma \ref{4vorstubai}. The argument is very similar to the one given in the proof of Lemma \ref{auchwi}.\hfill$\Box$\\
\begin{satz}\label{shmichaelmas} Let $M$ be a $\sharp$-supersingular ${\mathcal
H}^{\sharp}$-module. Via the isomorphism $M\cong \Delta(M)[t]$, the action of ${\mathcal H}^{\sharp}$ on $M$ can be recovered from the action of ${\mathfrak O}$ on $\Delta(M)$.
\end{satz}
{\sc Proof:} Define inductively the filtration $(F^iM)_{i\ge0}$ of $M$ by ${\mathcal
H}^{\sharp}$-sub modules as follows: $F^0M=0$, and $F^{i+1}M$ is the preimage of
${\mathcal F}(M/F^{i}M)$ under the projection $M\to M/F^{i}M$. The ${\mathcal
H}^{\sharp}$-action on the graded pieces can be recovered in view of Lemma
\ref{ennoad}. Exhausting $M$ step by step it is therefore enough to consider
the following setting: The action of ${\mathcal H}^{\sharp}$ has already been recovered
on an ${\mathcal H}^{\sharp}$-sub module $M_0$ of $M$ and on the quotient $M/M_0$, and
the latter is irreducible.
We reconstruct the action of $T_{e^*(\Gamma)}$ on $M$ by means
of $$T_{e^*(\gamma)}(x)=\gamma^{-1}\cdot x\quad \mbox{ for }\gamma\in
\Gamma$$as is tautological from our definitions. Next we are going to reconstruct the
decomposition\begin{gather}M=\bigoplus_{\epsilon\in[0,q-2]^{\Phi},\atop a\in
[0,q-2]}M^{\epsilon}_{\underline{a}}.\label{shletzthuerd}\end{gather}Let
$D>0$ be such that $D+1\in{\mathbb Z}(d+1)$ and $f^{D+1}={\rm id}$ for each
$k$-vector space automorphism $f$ of $M$ (as $M$ is finite, such a $D$ does exist). For
$\epsilon\in[0,q-2]^{\Phi}$ and $a\in [0,q-2]$ define $M^{[{\epsilon}]}_{\underline{a}}$ to be the
$k$-sub space of $M$ generated by all $x\in M$ with $\gamma\cdot x=\gamma^ax$
(all $\gamma\in\Gamma$) and
satisfying the following condition: There is some $\mu\in [0,q-1]^{\Phi}$
(depending on $x$) with $\epsilon_{\mu}=\epsilon$, and there are
$u_{i,c}\in M_0^{\omega^{-1}s\omega^{i+1}\cdot\epsilon}$ for
$i\ge0$ and $0\le c\le q-2$ with the following properties: Firstly,
$u_{i,c}=0$ if
(i) $\mu_i=0$, or
(ii) $\mu_i=q-1$ and $c>0$, or
(iii) $\mu_{i}<q-1$ and $c\ge q-1-\mu_{i}$.
Secondly, putting $x\{-1\}=x$ and\begin{gather}x\{i\}=t^{\mu_i}\varphi(x\{i-1\})-\sum_{c}t^c\varphi
u_{i,c},\label{shsabzwgo}\end{gather}we have $x\{i\}\in M\cong \Delta(M)[t]$ for any
$i$, as well as $x\{D\}=x$.
It will be enough to prove $M^{\epsilon}_{\underline{a}}=M^{[{\epsilon}]}_{\underline{a}}$. We first show\begin{gather}M^{\epsilon}_{\underline{a}}\subset
M^{[{\epsilon}]}_{\underline{a}}.\label{shcla1}\end{gather}
We have $(M/M_0)^{\epsilon}=\sum_{\mu\in
[0,q-1]^{\Phi}\atop\epsilon_{\mu}=\epsilon}{\mathcal F}(M/M_0)^{\mu}$ (cf. Lemma \ref{ennoad}), and this is respected by the action of $T_{e^*(\Gamma)}$. Thus,
we start with $\overline{x}\in {\mathcal F}(M/M_0)^{\mu}\cap (M/M_0)^{\epsilon}_{\underline{a}}$ for some $\mu$ with
$\epsilon_{\mu}=\epsilon$. By
Proposition \ref{novordschutzengel} we may lift it to some $x\in M^{\epsilon}$ such that for each $i$ with
$T_sT_{\omega}^{i+1}\overline{x}=0$ we have
$T_s^2T_{\omega}^{i+1}x=0$. As $T_{\omega}$ maps simultaneous eigenspaces for the $T_{t}$ (with $t\in\overline{T}$) again to such simultaneous eigenspaces, and as $T_s^2$ commutes with the $T_t$, we may assume $x\in M^{\epsilon}_{\underline{a}}$. Putting
$$x\{i\}=(-T_{\omega})^{i+1}x$$for $-1\le i\le D$, repeated application of Lemma \ref{samserswo} shows $x\{i\}\in
M^{\omega^{i+1}\cdot\epsilon}_{\underline{a_{\epsilon,i}}}$ with$$a_{\epsilon,-1}=a,\quad
a_{\epsilon,0}=a-\epsilon_0\quad\mbox{ and }\quad
a_{\epsilon,i}=a-\epsilon_0-\epsilon_{d-i+1}-\ldots-\epsilon_d$$for $i\le d$,
and then $a_{\epsilon,i}=a_{\epsilon,i'}$ for $i-i'\in{\mathbb Z}(d+1)$. If $0\le\mu_i<q-1$ put
$$u_{i,c}=T_{\omega}^{-1}((T_s(x\{i\}))_{\underline{c+\mu_i+a_{\epsilon,i}}}).$$As $\overline{x}\in {\mathcal F}(M/M_0)^{\mu}$ and $\mu_i<q-1$ we have $u_{i,c}\in M_0$, and as $x\{i\}\in
M^{\omega^{i+1}\cdot\epsilon}$ we have $u_{i,c}\in
M^{\omega^{-1}s\omega^{i+1}\cdot\epsilon}$, together $u_{i,c}\in
M_0^{\omega^{-1}s\omega^{i+1}\cdot\epsilon}$. From $\mu_i<q-1$ we furthermore
deduce $k_{x\{i\}}=(\omega^{i+1}\cdot\epsilon)_{1}= \mu_i$, and since
$T_s^2x\{i\}=0$ and $(T_s(x\{i\}))_{\underline{c+\mu_i+a_{\epsilon,i}}}=0$ for
$q-1-\epsilon_{-i}\le c\le q-2$ (by $\sharp$-supersingularity) we then see\begin{gather}t^{\mu_i}\varphi(x\{i-1\})-x\{i\}-\sum_{c}t^c\varphi
u_{i,c}=h(-x\{i\})=0.\label{shgruendo}\end{gather}
If $\mu_i=q-1$ we have $T_s^2(T_s^2x\{i\})=T_s^2x\{i\}$ and hence $k_{T_s^2x\{i\}}=q-1$ (independently of the value of $\mu_i$ we have $(\omega^{i+1}\cdot\epsilon)_{1}\equiv \mu_i$ modulo
$(q-1)$), hence$$t^{q-1}\varphi T_{\omega}^{-1}(T_s^2x\{i\})+T_s^2x\{i\}=h(T_s^2x\{i\})=0.$$Similarly we see $k_{(x\{i\}-T_s^2x\{i\})}=0$ and hence$$\varphi
T_{\omega}^{-1}(x\{i\} -T_s^2x\{i\})+x\{i\}-T_s^2x\{i\}=h(x\{i\}-T_s^2x\{i\})=0.$$Together this gives formula
(\ref{shgruendo}) again, this time with
$u_{i,0}=-T_{\omega}^{-1}(x\{i\}-T_s^2x\{i\})$ and $u_{i,c}=0$ for $c>0$. Moreover, $u_{i,0}$ belongs to
$M_0$ as $\overline{x}\in {\mathcal F}(M/M_0)^{\mu}$ and $\mu_i=q-1$; but it
also belongs to $M^{\omega^{-1}s\omega^{i+1}\cdot\epsilon}$
since $\mu_i=q-1$ implies
$\omega^{-1}s\omega^{i+1}\cdot\epsilon=\omega^{i}\cdot\epsilon$. By
construction, $x\{d\}=(-T_{\omega})^{d+1}(x)$, hence $x\{D\}=(-T_{\omega})^{D+1}x=x$.
It follows that $x\in M^{[{\epsilon}]}_{\underline{a}}$, and thus we have reduced our problem to showing $(M_0)^{\epsilon}_{\underline{a}}\subset
M^{[{\epsilon}]}_{\underline{a}}$. But for this we may appeal to an induction on ${\rm
dim}_k(M)$ (which we may assume to be finite).
We have shown formula (\ref{shcla1}). Now we show\begin{gather}M^{[{\epsilon}]}_{\underline{a}}\subset
M^{\epsilon}_{\underline{a}}.\label{shcla2}\end{gather}Let $x\in M^{[{\epsilon}]}_{\underline{a}}$,
$\mu\in [0,q-1]^{\Phi}$ (with $\epsilon_{\mu}=\epsilon$) and $u_{i,c}$ be as in the definition of
$M^{[{\epsilon}]}_{\underline{a}}$. Define $x\{i\}$ for $-1\le i\le D$ as in that definition. By
Lemma \ref{fabei} and the proof of the inclusion (\ref{shcla1}) we find
$\tilde{x}\in M^{\epsilon}_{\underline{a}}$ and $\tilde{u}_{i,c}\in M_0^{\omega^{-1}s\omega^{i+1}\cdot\epsilon}$ for $0\le
i\le D$ such that, after replacing $x$ by $x-\tilde{x}$ and $u_{i,c}$ by
$u_{i,c}-\tilde{u}_{i,c}$, we may assume $x\in M_0$.
{\it Claim: If $x\in M_0$ and if $M_0$ is irreducible, then there is some
${x}'\in (M_0)_{\underline{a}}$ with $x-{x}'\in
(M_0)_{\underline{a}}^{\epsilon}$ and such that\begin{gather}x'\{i\}=t^{\mu_i}\varphi(\ldots
(t^{\mu_1}\varphi(t^{\mu_0}\varphi x'))\ldots )\notag\end{gather}belongs to $M_0$ for all $i$, and $x'\{D\}=x'$.}
This follows from Lemma \ref{4vorstubai}.
If $M_0$ is not irreducible, choose an ${\mathcal
H}$-sub module $M_{00}$ in $M_0$ such that $M_0/M_{00}$ is irreducible. By the above claim and again invoking the proof of the inclusion (\ref{shcla1}), after
modifying $x$ by another element of $M^{\epsilon}_{\underline{a}}$ (now even
of $(M_0)^{\epsilon}_{\underline{a}}$) and suitably modifying the $u_{i,c}$, we may assume $u_{i,c}\in
M_{00}$. Thus,
it is now enough to solve the problem for the new $x\in (M_0)_{\underline{a}}$ (and the new $u_{i,c}\in
M_{00}$). We continue in this way. Since we may assume that ${\dim}_k(M)$
is finite, an induction on the dimension of $M$ allows us to conclude.
We have reconstructed the decomposition (\ref{shletzthuerd}) of $M$.
Now we reconstruct $T_sT_{\omega}$ acting on $M$. As we already know the decomposition (\ref{shletzthuerd}), it is enough to reconstruct
$T_sT_{\omega}(e)$ for $e\in M^{\epsilon'}_{\underline{a'}}$, all $\epsilon'$, $a'$. Given
such $e$, let $\overline{e}$ be its class in $M/M_0$. By Lemma \ref{samserswo}
there are then $\epsilon$, $a$ such that $T_{\omega}\overline{e}\in (M/M_0)^{\epsilon}_{\underline{a}}$.
First assume $\epsilon_1=0$. We then reconstruct $T_sT_{\omega}(e)$ as
$T_sT_{\omega}(e)=t^{q-1}\varphi(e)$. Indeed, to see this we may assume (by Lemma \ref{quadrat}) that
$T_{\omega}(e)$ is an eigenvector for $T_s^2$. If
$T_s^2T_{\omega}(e)=T_{\omega}(e)$ and hence $T_sT_{\omega}(e)=-T_{\omega}(e)$, the claim follows from the definition of
$h(T_{\omega}(e))$. If $T_s^2T_{\omega}(e)=0$ then in fact $T_sT_{\omega}(e)=0$
(since also $\epsilon_1=0$), and the definition of
$h(T_{\omega}(e))$ shows $t^{q-1}\varphi(e)=0$.
Now assume $\epsilon_1>0$. This implies $T_s^2T_{\omega}(e)=0$ and
$k_{T_{\omega}(e)}=\epsilon_1$, and by $\sharp$-supersingularity we
get$$t^{k_{T_{\omega}(e)}+1}\varphi e=-\sum_{0\le c<q-1-k_{T_{\omega}(e)}}t^{c+1}\varphi
T_{\omega}^{-1}((T_sT_{\omega}e)_{\underline{c+\epsilon_1+a}}).$$Here
$(T_sT_{\omega}e)_{\underline{c+\epsilon_1+a}}\in M_0^{s\cdot\epsilon}$ and
$q-1-k_{T_{\omega}(e)}=(s\cdot\epsilon)_1$. The
map $$\bigoplus_{0\le c<q-1-k_{T_{\omega}(e)}}M_0^{s\cdot\epsilon}\longrightarrow M_0,\quad (y_c)_c\mapsto\sum_{0\le c<q-1-k_{T_{\omega}(e)}}t^{c+1}\varphi
T_{\omega}^{-1}(y_c)$$is injective. This is first seen in the case where $M_0$
is irreducible; it then follows by an obvious devissage argument. We therefore
see that
the $(T_sT_{\omega}e)_{\underline{c+\epsilon_1+a}}$ for $0\le c<q-1-k_{T_{\omega}(e)}$ can be read off from $t^{k_{T_{\omega}(e)}+1}\varphi e$,
hence also $T_sT_{\omega}e$ can be read off from $t^{k_{T_{\omega}(e)}+1}\varphi e$ (by $\sharp$-supersingularity).
The restriction of $T_{\omega}$ to $\{x\in M\,|\,T_sT_{\omega}(x)\in M_0\}$ is
reconstructed as follows. Given
$\overline{x}\in (M/M_0)^{\omega^{-1}\cdot\epsilon}_{\underline{a-\epsilon_1}}$ (for some
$\epsilon$, some $a$) with $T_sT_{\omega}\overline{x}=0$, we use the decomposition (\ref{18okt}) to lift
$\overline{x}$ to some $x\in M^{\omega^{-1}\cdot\epsilon}_{\underline{a-\epsilon_1}}$. Since $(\omega^{-1}\cdot\epsilon)_0=\epsilon_1$, Lemma
\ref{samserswo} says $T_{\omega}{x}\in M^{\epsilon}_{\underline{a}}$. It then follows from the
definitions that$$T_{\omega}x=-t^{\epsilon_1}\varphi
x-\sum_{c\ge0}t^c\varphi T_{\omega}^{-1}((T_sT_{\omega}x)_{\underline{c+\epsilon_1+a}}).$$We have now collected all the data required in Proposition \ref{schutzengel} for reconstructing $M$ as an ${\mathcal H}^{\sharp}$-module.\hfill$\Box$\\
\subsection{Full faithfulness on $\sharp$-supersingular ${\mathcal
H}^{\sharp}$-modules}
\label{fullfafu}
Let ${\rm Rep}({\rm Gal}(\overline{F}/F))$
denote the category of representations of ${\rm
Gal}(\overline{F}/F)$ on $k$-vector spaces which are projective limits of finite
dimensional continuous ${\rm Gal}(\overline{F}/F)$-representations.
Let ${\rm Mod}_{ss}({\mathcal H}^{\sharp})$ denote the category of
$\sharp$-supersingular ${\mathcal H}^{\sharp}$-modules. Let ${\rm
Mod}_{ss}({\mathcal H})$, resp. ${\rm Mod}_{ss}({\mathcal
H}^{\sharp\sharp})$, denote the category of supersingular ${\mathcal
H}$-modules, resp. of supersingular ${\mathcal H}^{\sharp\sharp}$-modules.
Let $M\in{\rm Mod}_{ss}({\mathcal
H}^{\sharp\sharp})$ with ${\rm dim}_k(M)<\infty$. By Proposition \ref{jishnusharp} the ${\mathfrak O}$-module $\Delta(M)$ satisfies the hypotheses of Proposition \ref{nopsi}, hence $\Delta(M)^*\otimes_{k[[t]]}k((t))$ carries the structure of an \'{e}tale $(\varphi,\Gamma)$-module. Let $V(M)$ be the object in ${\rm Rep}({\rm Gal}(\overline{F}/F))$ assigned to $M$ by Theorem \ref{sosego}.
Exhausting an object in ${\rm Mod}_{ss}({\mathcal
H}^{\sharp\sharp})$ by its finite dimensional sub objects we see that this
construction extends to all of ${\rm Mod}_{ss}({\mathcal
H}^{\sharp\sharp})$.
\begin{satz}\label{7vorallhe} \begin{gather}{\rm Mod}_{ss}({\mathcal
H}^{\sharp\sharp})\longrightarrow {\rm Rep}({\rm
Gal}(\overline{F}/F)),\quad\quad M\mapsto
V(M)\label{torschluss}\end{gather}is an exact contravariant functor, with
${\rm dim}_k(M)={\rm dim}_k(V(M))$ for any $M$.\begin{gather}{\rm Mod}_{ss}({\mathcal H}^{\sharp})\longrightarrow {\rm Rep}({\rm Gal}(\overline{F}/F)),\quad\quad M\mapsto V(M),\notag\\{\rm Mod}_{ss}({\mathcal H})\longrightarrow {\rm Rep}({\rm Gal}(\overline{F}/F)),\quad\quad M\mapsto V(M)\label{torschluss1}\end{gather}are exact and fully faithful contravariant functors.
\end{satz}
{\sc Proof:} Exactness follows from exactness of $M\mapsto \Delta(M)$ (Proposition \ref{jishnusharp}), exactness of $\Delta\mapsto \Delta^*\otimes_{k[[t]]}k((t))$ (Proposition \ref{nopsi}) and exactness of the equivalence functor in Theorem \ref{sosego}. The same chain of arguments shows ${\rm dim}_k(M)={\rm dim}_k(V(M))$.
To prove faithfulness on ${\rm Mod}_{ss}({\mathcal H}^{\sharp})$, suppose that we are given finite dimensional objects $M$, $M'$ in ${\rm
Mod}_{ss}({\mathcal H}^{\sharp})$ and a morphism $\mu:V(M')\to V(M)$ in ${\rm Rep}({\rm Gal}(\overline{F}/F))$. By Theorem \ref{sosego}, the latter corresponds to a unique morphism of \'{e}tale $(\varphi,\Gamma)$-modules $$\mu:\Delta(M')^*\otimes_{k[[t]]}k((t)) \to\Delta(M)^*\otimes_{k[[t]]}k((t)).$$By Proposition \ref{wienaus} (which applies since Propostion \ref{jishnusharp} tells us $\Delta(M), \Delta(M')\in {\rm Mod}^{\clubsuit}({\mathfrak O})$) it is induced by a unique morphism of ${\mathfrak O}$-modules $\mu:\Delta(M)\to\Delta(M')$. Clearly $\mu$ takes ${\Delta(M)}[t]$ to ${\Delta(M')}[t]$, i.e. it takes
$M$ to $M'$. Applying Theorem \ref{shmichaelmas} to both $M$ and $M'$ we see that $\mu:M\to M'$ is ${\mathcal
H}^{\sharp}$-equivariant. If $M, M'\in{\rm Mod}_{ss}({\mathcal H}^{\sharp})$ are not necessarily finite dimensional, the same conclusion is obtained by exhausting $M$, $M'$ by its finite dimensional submodules. We deduce the stated full faithfulness on ${\rm Mod}_{ss}({\mathcal H}^{\sharp})$. It implies fully faithfulness on ${\rm Mod}_{ss}({\mathcal H})$ (cf. Lemma \ref{vialsa}).\hfill$\Box$\\
{\bf Example:} The analogs of Proposition
\ref{schutzengel} and Theorem \ref{7vorallhe} (b) fail for supersingular ${\mathcal
H}^{\sharp\sharp}$-modules. To see this, take $d=2$, and endow the $6$-dimensional
$k$-vector space $M$ with basis $e_0, e_1, e_2, f_0, f_1, f_2$ with the
structure of an ${\mathcal
H}^{\sharp\sharp}$-module as follows. $T_{t}$ for each $t\in
\overline{T}$ acts trivially. Put $T_s(f_0)=T_s(e_1)=T_s(e_2)=0$ and
$T_s(e_0)=-e_0$, $T_s(f_1)=-f_1$, $T_s(f_2)=-f_2$. Fix $\alpha\in k$ and put
$T_{\omega}(e_0)=e_1$, $T_{\omega}(e_1)=e_2$, $T_{\omega}(e_2)=e_0$,
$T_{\omega}(f_0)=f_1+\alpha e_1$, $T_{\omega}(f_1)=f_2-\alpha e_2$,
$T_{\omega}(f_2)=f_0$. This is even an ${\mathcal
H}^{\sharp}$-module if and only if $\alpha=0$, if and only if it is not
indecomposable (as an ${\mathcal
H}^{\sharp\sharp}$-module). The corresponding ${\mathfrak O}$-module
$\Delta(M)$ is
defined by the relations $\varphi e_0=-e_1$, $\varphi e_1=-e_2$,
$t^{q-1}\varphi e_2=-e_0$, $\varphi f_2=-f_0$, $t^{q-1}\varphi(f_0-\alpha
e_0)-f_1$, $t^{q-1}\varphi(f_1+\alpha
e_1)-f_2$. But this ${\mathfrak O}$-module is in fact independent of
$\alpha$, since $t^{q-1}\varphi e_1=t^{q-1}\varphi e_0=0$.\\
\subsection{The essential image}
\label{essim}
{\bf Definition:} Let ${\rm Hom}(\Gamma,k^{\times})^{\Phi}$ denote the group of $(d+1)$-tuples $\alpha=(\alpha_0,\ldots,\alpha_d)$ of characters $\alpha_j:\Gamma\to k^{\times}$. Let ${\mathfrak S}_{d+1}$ act on ${\rm Hom}(\Gamma,k^{\times})^{\Phi}$ by the formulae$$(\omega \cdot\alpha)_0=\alpha_d\quad\mbox{ and }\quad(\omega \cdot\alpha)_i=\alpha_{i-1}\mbox{ for }1\le i\le d,$$$$(s \cdot\alpha)_0=\alpha_1,\quad (s \cdot\alpha)_1=\alpha_0 \quad\mbox{ and }\quad(s \cdot\alpha)_i=\alpha_{i}\mbox{ for }2\le i\le d.$$Recall the action of ${\mathfrak S}_{d+1}$ on $[0,q-2]^{\Phi}$. Combining both (diagonally), we obtain an action of ${\mathfrak S}_{d+1}$ on ${\rm Hom}(\Gamma,k^{\times})^{\Phi}\times [0,q-2]^{\Phi}$.
In Lemma \ref{ossa} we attached to each standard cyclic \'{e}tale $(\varphi,\Gamma)$-module ${\bf D}$ of dimension $d+1$ an ordered tuple
$((\alpha_0,m_0),\ldots,(\alpha_d, m_d))$ (with integers $m_j\in[1-q,0]$ and characters $\alpha_j:\Gamma\to k^{\times}$), unique up to a cyclic permutation. Sending each $m_j$ to the representative in $[0,q-2]$ of its class in ${\mathbb Z}/(q-1)$, the tuple $(m_0,\ldots,m_d)$ gives rise to an element in $[0,q-2]^{\Phi}$. On the other hand, the tuple $(\alpha_0,\ldots,\alpha_d)$ constitutes an element in ${\rm Hom}(\Gamma,k^{\times})^{\Phi}$. Taken together we thus attach to ${\bf D}$ an element in ${\rm Hom}(\Gamma,k^{\times})^{\Phi}\times [0,q-2]^{\Phi}$, unique up to cyclic permutation. Equivalently, we attach to ${\bf D}$ an orbit in ${\rm Hom}(\Gamma,k^{\times})^{\Phi}\times [0,q-2]^{\Phi}$ under the action of the subgroup of ${\mathfrak S}_{d+1}$ generated by $\omega$.
Now let ${\bf D}'_1$, ${\bf D}'_2$ be irreducible \'{e}tale $(\varphi,\Gamma)$-modules over
$k((t))$. We say that ${\bf D}'_1$, ${\bf D}'_2$ are strongly ${\mathfrak S}_{d+1}$-linked if they are subquotients of $(d+1)$-dimensional standard cyclic \'{e}tale $(\varphi,\Gamma)$-modules ${\bf D}_1$, ${\bf D}_2$ respectively, and if ${\bf D}_1$, ${\bf D}_2$ give rise to the same ${\mathfrak S}_{d+1}$-orbit in ${\rm Hom}(\Gamma,k^{\times})^{\Phi}\times [0,q-2]^{\Phi}$. We say that
${\bf D}'_1$, ${\bf D}'_2$ are ${\mathfrak S}_{d+1}$-linked if they are subquotients of $(d+1)$-dimensional standard cyclic \'{e}tale $(\varphi,\Gamma)$-modules ${\bf D}_1$, ${\bf D}_2$ respectively, and if ${\bf D}_1$, ${\bf D}_2$ give rise to the same ${\mathfrak S}_{d+1}$-orbit in $[0,q-2]^{\Phi}$ (or equivalently, if the assigned tuples (up to cyclic permutation) in $[0,q-2]^{\Phi}$ coincide as {\it unordered} tuples (with multiplicities)).\\
{\bf Remark:} (a) Let ${\bf D}$ denote the \'{e}tale $(\varphi,\Gamma)$-module over
$k((t))$ corresponding to $V(M)$, for a finite dimensional supersingular ${\mathcal
H}^{\sharp\sharp}$-module $M$. Our constructions show:
{\bf (i)} Put $M=({\bf
D}^{\natural})^*[t]$ and consider the natural map of $k[[t]][\varphi]$-modules$$\kappa_{{\bf D}}:k[[t]][\varphi]\otimes_{k[[t]]}M\longrightarrow ({\bf
D}^{\natural})^*.$$Then ${\rm ker}(\kappa_{{\bf
D}})\cap (k\otimes M+k[[t]]\varphi\otimes M)$ generates ${\rm ker}(\kappa_{{\bf D}})$ as a
$k[[t]][\varphi]$-module.
{\bf (ii)} Each irreducible subquotient of ${\bf D}$ is a subquotient of a
$(d+1)$-dimensional standard cyclic \'{e}tale $(\varphi,\Gamma)$-module; more precisely:
{\bf (ii)(1)} If ${\bf D}$ (or equivalently, $M$) is indecomposable, then any two irreducible subquotients of ${\bf D}$ are ${\mathfrak S}_{d+1}$-linked.
{\bf (ii)(2)} If $M$ is even a supersingular ${\mathcal
H}$-module, and if ${\bf D}$ (or equivalently, $M$) is indecomposable,
then any two irreducible subquotients of ${\bf D}$ are strongly ${\mathfrak
S}_{d+1}$-linked.
{\bf (ii)(3)} If $M$ is even a supersingular ${\mathcal H}^{\flat}$-module, then each irreducible subquotient of ${\bf D}$ is a subquotient of a $(d+1)$-dimensional standard cyclic \'{e}tale $(\varphi,\Gamma)$-module with parameters $m_j\in\{1-q,0\}$ and $\alpha_j=1$ for all $j$.
{\bf (iii)} For any $(\varphi,\Gamma)$-sub module ${\bf D}_0$ of ${\bf D}$ the
$\psi$-operator on ${\bf D}_0\cap {\bf D}^{\natural}$ is surjective.
(b) Does property {\bf (i)} mean (at
least if property {\bf (iii)} is assumed) that ${\bf D}$ is the reduction of a crystalline $p$-adic ${\rm
Gal}(\overline{F}/F)$-representation with Hodge-Tate weights in $[-1,0]$ ?
(c) Property {\bf (iii)} means that the functor ${\bf D}_0\mapsto {\bf
D}_0^{\natural}$ is exact on the category of subquotients ${\bf D}_0$ of ${\bf D}$.
(d) It should not be too hard hard to show that properties {\bf (i)}, {\bf
(ii)(1)} and {\bf (iii)} together in fact {\it characterize} the essential image of the
functor (\ref{torschluss}).
(e) On the other hand, properties {\bf (i)}, {\bf
(ii)(2)} and {\bf (iii)} together do {\it not} characterize the essential image of the
functor (\ref{torschluss1}). To see this for $d=1$
consider the following \'{e}tale
$(\varphi,\Gamma)$-module ${\bf D}$ (which satisfies {\bf (i)}, {\bf (ii)(2)}, {\bf
(iii)}). It is given by a $k$-basis $e_0$, $e_1$, $f_0$, $f_1$, $g_0$,
$g_1$ of $({\bf
D}^{\natural})^*[t]$ and the following relations:$$\varphi e_1=e_0,\quad \varphi
f_1=f_0,\quad \varphi g_1=g_0,\quad
t^{q-1}\varphi e_0=e_1,\quad t^{q-1}\varphi f_0=f_1+e_1,\quad t^{q-1}\varphi
g_0=g_1+f_0.$$Another object not in the essential image is defined by the set of relations$$\varphi e_1=e_0,\quad \varphi
f_1=f_0,\quad \varphi g_1=g_0,\quad
t^{q-1}\varphi e_0=e_1,\quad t^{q-1}\varphi f_0=f_1+e_0,\quad t^{q-1}\varphi
g_0=g_1+f_1.$$
\section{From $G$-representations to ${\mathcal H}$-modules}
\subsection{Supersingular cohomology}
\label{susicoho}
Put $G={\rm GL}_{d+1}(F)$, let $I_0$ be a pro-$p$-Iwahori subgroup in $G$, and
fix an isomorphism between ${\mathcal H}$ and the pro-$p$-Iwahori Hecke
algebra $k[I_0\backslash G/I_0]$ corresponding to $I_0\subset G$. For a smooth
$G$-representation $Y$ (over $k$) the subspace $Y^{I_0}$ of $I_0$-invariants
then receives a natural action by ${\mathcal H}$. Let us denote by
$H^0_{ss}(I_0,Y)$ the maximal supersingular ${\mathcal H}$-submodule of
$Y^{I_0}$. It is clear that this defines a left exact functor$${\rm
Mod}(G)\longrightarrow {\rm Mod}_{ss}({\mathcal H}),\quad\quad Y\mapsto
H^0_{ss}(I_0,Y)$$where ${\rm Mod}(G)$ denotes the category of smooth
$G$-representations. Let $D^+(G)$ denote the derived category of complexes of
smooth $G$-representations vanishing in negative degrees, let
$D_{ss}^+({\mathcal H})$ denote the derived category of complexes of
supersingular ${\mathcal H}$-modules vanishing in negative degrees. The above
functor gives rise to a right derived
functor\begin{gather}R_{ss}(I_0,.):D^+(G)\longrightarrow D_{ss}^+({\mathcal
H}).\label{allesee}\end{gather}Let $D^+({\rm Gal}(\overline{F}/F)))$ denote
the derived category of complexes in ${\rm Rep}({\rm Gal}(\overline{F}/F))$
vanishing in negative degrees. Since the functor $V$ is exact, it induces a
functor$$V:D_{ss}^+({\mathcal H})\longrightarrow D^+({\rm
Gal}(\overline{F}/F))).$$We may compose them with $R_{ss}(I_0,.)$ to obtain a functor$$V\circ R_{ss}(I_0,.):D^+(G)\longrightarrow D^+({\rm Gal}(\overline{F}/F))).$$
{\bf Remark:} Of course, we expect the functor $V\circ R_{ss}(I_0,.)$ to be meaningful only when restricted to (complexes of) supersingular $G$-representations. The reason is the following theorem of Ollivier and Vign\'{e}ras \cite{ollvig17}: A smooth admissible irreducible $G$-representation $Y$ over an algebraic closure $\overline{k}$ of $k$ is supersingular if and only if $Y^{I_0}$ is a supersingular ${\mathcal H}\otimes_k{\overline{k}}$-module, if and only if $Y^{I_0}$ admits a supersingular subquotient.
It is known that, beyond the case where $G={\rm GL}_2({\mathbb Q}_p)$, a smooth admissible irreducible supersingular $G$-representation $Y$ over $k$ is not uniquely determined by the ${\mathcal H}$-module $Y^{I_0}$. Is it perhaps uniquely determined by the derived object $R_{ss}(I_0,Y)\in {\rm Mod}_{ss}({\mathcal H})$ ? It would then also be uniquely determined by the derived object ${V}(R_{ss}(I_0,Y)) \in D^+({\rm Gal}(\overline{F}/F)))$.
\subsection{An exact functor from $G$-representations to ${\mathcal H}$-modules}
We fix a $(d+1)$-st root of unity $\xi\in k^{\times}$ with
$\sum_{j=0}^{d}\xi^j=0$.
For an ${\mathcal H}$-module $M$ and $j\in{\mathbb Z}$ let $M^{{\xi}^j}$ be the ${\mathcal H}$-module which coincides with $M$ as a module over the $k$-sub algebra $k[T_s, T_{t}]_{t\in\overline{T}}$, but with $T_{\omega}|_{M^{{\xi}^j}}={\xi}^j T_{\omega}|_{M}$.
Let $\delta:M_0\to M_1$ be a morphism of ${\mathcal
H}$-modules. For $(x_0,x_1)\in M_0\oplus M_1$ put\begin{align}T_{\omega}((x_0,x_1))&=(T_{\omega}(x_0),T_{\omega}(\delta(x_0))+\xi T_{\omega}(x_1)),\notag\\T_s((x_0,x_1))&=(T_s(x_0),T_s(x_1)),\notag\\T_{t}((x_0,x_1))&=(T_{t}(x_0),T_{t}(x_1))\quad\mbox{ for }t\in\overline{T}.\notag\end{align}
\begin{lem}\label{14vorstubei} These formulae define an ${\mathcal
H}$-module structure on $M_0\oplus M_1$; we denote this new
${\mathcal H}$-module by $M_0\oplus^{\delta} M_1$. We have an exact
sequence of ${\mathcal H}$-modules \begin{gather}0\longrightarrow M_1^{\xi}\longrightarrow
M_0\oplus^{\delta} M_1\longrightarrow
M_0\to0.\label{lotharaugust}\end{gather}The morphism $\delta:M_0\to M_1$ can be recovered from the exact sequence (\ref{lotharaugust}).
If there is some $\lambda\in k^{\times}$ with $T_{\omega}^{d+1}=\lambda$ on $M_0$ and on $M_1$, then also $T_{\omega}^{d+1}=\lambda$ on $M_0\oplus^{\delta} M_1$,
\end{lem}
{\sc Proof:} By induction on $i$ one
shows $$T_{\omega}^i((x_0,x_1))=(T_{\omega}^i(x_0),\xi^iT_{\omega}^i(x_1)
+\sum_{j=0}^{i-1}\xi^jT_{\omega}^i(\delta(x_0)))$$for $i>0$, and hence
$T_{\omega}^{d+1}((x_0,x_1))=(T_{\omega}^{d+1}(x_0),T_{\omega}^{d+1}(x_1))$. From
here, all the required relations are straighforwardly verified, showing that
indeed we have defined an ${\mathcal H}$-module.
Obviously, from the exact sequence (\ref{lotharaugust}) both $M_0$ and $M_1$
can be recovered. That also $\delta$ can be recovered follows from the
following more general consideration. Suppose that we are given $\delta:M_0\to M_1$ and
$\epsilon:N_0\to N_1$ and a morphism of ${\mathcal H}$-modules
$f:M_0\oplus^{\delta} M_1\to N_0\oplus^{\epsilon} N_1$ with
$f(M_1^{\xi})\subset N_1^{\xi}$. Then there are ${\mathcal H}$-module
homomorphisms $f_0:M_0\to N_0$, $f_1:M_1^{\xi}\to N_1^{\xi}$ and
$\tilde{f}:M_0\to N_1^{\xi}$ with
$f((x_0,x_1))=(f_0(x_0),f_1(x_1)+\tilde{f}(x_0))$. For $x_0\in M_0$ we
compute $$f(T_{\omega}(x_0,0))=f(T_{\omega}(x_0),T_{\omega}(\delta(x_0)))=(T_{\omega}(f_0(x_0)),T_{\omega}(f_1(\delta(x_0)))+\xi
T_{\omega}(\tilde{f}(x_0))),$$$$T_{\omega}(f(x_0,0))=T_{\omega}(f_0(x_0),\tilde{f}(x_0))=(T_{\omega}(f_0(x_0)),T_{\omega}(\epsilon(f_0(x_0)))+\xi T_{\omega}(\tilde{f}(x_0))).$$As
$f(T_{\omega}(x_0,0))=T_{\omega}(f(x_0,0))$ we deduce
$T_{\omega}(\epsilon(f_0(x_0)))=T_{\omega}(f_1(\delta(x_0)))$, and since
$T_{\omega}$ is an isomorphism even $\epsilon(f_0(x_0))=f_1(\delta(x_0))$.\hfill$\Box$\\
Let $$(M_{\bullet},\delta_{\bullet})=[\ldots
\stackrel{\delta_{-2}}{\longrightarrow} M_{-1} \stackrel{\delta_{-1}}{\longrightarrow} M_0\stackrel{\delta_0}{\longrightarrow} M_1\stackrel{\delta_1}{\longrightarrow}M_2\stackrel{\delta_2}{\longrightarrow} \ldots]$$be a complex of ${\mathcal H}$-modules.
\begin{lem}\label{claudeles} (a) There is a unique ${\mathcal
H}$-module $\oplus^{\delta_{\bullet}}_{j\in\mathbb Z}M_j$ with the following properties:
${\bullet}$ As a $k$-vector space, $\oplus^{\delta_{\bullet}}_{j\in\mathbb Z}M_j=\oplus_{j\in\mathbb Z}M_j$.
${\bullet}$ For any $j$ we have $\tau(M_j)\subset M_j+M_{j+1}$ for each $\tau\in{\mathcal H}$; in particular, the sub space $M_{\ge j}=\oplus_{j'\ge j}M_{j'}$ is an ${\mathcal H}$-sub module.
${\bullet}$ The ${\mathcal H}$-module $M_{\ge j}/M_{\ge j+2}$ is isomorphic with $M_j^{\xi^j}\oplus^{\delta_j} M_{j+1}^{\xi^{j}}$ as defined in Lemma \ref{14vorstubei}.
(b) If there is some $\lambda\in k^{\times}$ with $T_{\omega}^{d+1}=\lambda$
on each $M_j$, then $T_{\omega}^{d+1}=\lambda$ on
$\oplus^{\delta_{\bullet}}_{j\in\mathbb Z}M_j$.
(c) The assignment $(M_{\bullet},\delta_{\bullet})\mapsto
(\oplus^{\delta_{\bullet}}_{j\in\mathbb Z}M_j,(M_{\ge j})_{j\in{\mathbb
Z}})$ is an exact and faithful functor
from the category of complexes of ${\mathcal H}$-modules to the
category of filtered ${\mathcal H}$-modules. The isomorphism class of
the complex
$(M_{\bullet},\delta_{\bullet})$ can be recovered from the isomorphism class
of the filtered ${\mathcal H}$-module $(\oplus^{\delta_{\bullet}}_{j\in\mathbb Z}M_j,(M_{\ge j})_{j\in{\mathbb
Z}})$.
\end{lem}
{\sc Proof:} This is clear from Lemma \ref{14vorstubei}.\hfill$\Box$\\
{\bf Definition:} (a) For a smooth $G$-representation $Y$ over $k$ and $i\ge0$ let us denote by $H_{ss}^i(I_0,Y)$ the $i$-th cohomology group of $R_{ss}(I_0,Y)$, cf. formula (\ref{allesee}).
(b) We say that a smooth $G$-representation $Y$ over $k$ is {\it exact} if for each $i\ge0$ the functor $Y'\mapsto H_{ss}^i(I_0,Y')$ is exact on the category of $G$-subquotients $Y'$ of $Y$.
(c) An exhaustive and
separated decreasing filtration $({Y}^{j})_{j\in{\mathbb Z}}$ of a smooth
$G$-representation $Y$ over $k$ is {\it exact} if ${Y}^{j}/{Y}^{j+1}$ is exact
for each $j$.
(d) Let $f:Y\to W$ be a morphism of smooth $G$-representations, let
$({Y}^{i})_{i\in{\mathbb Z}}$, $({W}^{j})_{j\in{\mathbb Z}}$ be exact
filtrations on $Y$, $W$. We say that $f$ is strict (with respect to these
filtrations) if there is some $s\in{\mathbb Z}$ such that for any
$i\in{\mathbb Z}$ we have \begin{gather}f(Y^i)=f(Y)\cap W^{i+s}\quad\mbox{ and }\quad f^{-1}(W^{i+s})=Y^i.\label{strictcon}\end{gather}
\begin{lem} The composition of strict morphisms is strict.
\end{lem}
{\sc Proof:} This is straightforward to verify.\hfill$\Box$\\
{\bf Example:} A semisimple smooth $G$-representation is exact. \\
Let ${\mathfrak R}_G$ denote the category whose objects are smooth
$G$-representation with an exact filtration, where two filtrations are
identified if they coincide up to shift; morphisms in ${\mathfrak R}_G$ are
supposed to be strict. We denote objects $(Y,({Y}^{i})_{i\in{\mathbb Z}})$ in ${\mathfrak R}_G$
simply by $Y^{\bullet}$.
Let ${\mathfrak E}({\mathcal H})$ denote the category of $E_1$-spectral sequences in the category of ${\mathcal H}$-modules.
For $Y^{\bullet}\in {\mathfrak R}_G$ we have the spectral
sequence$$E(Y^{\bullet})=[E_1^{m,n}(Y^{\bullet})=H^{m+n}_{ss}(I_0,{Y}^{m}/{Y}^{m+1})\Rightarrow
H_{ss}^{m+n}(I_0,Y)].$$Given a morphism $f:Y^{\bullet}\to W^{\bullet}$
in ${\mathfrak R}_G$, let $s\in{\mathbb Z}$ be such that formula (\ref{strictcon}) is
satisfied. Then $f$ induces morphisms
$H^{m}_{ss}(I_0,{Y}^{i}/{Y}^{i+1})\to H^{m}_{ss}(I_0,{W}^{i+s}/{W}^{i+s+1})$
for any $m$ and $i$, and these induce a morphism of spectral
sequences $E(Y^{\bullet})\to E(W^{\bullet})$. We thus obtain a functor $${\mathfrak R}_G\to {\mathfrak E}({\mathcal H}),\quad\quad Y^{\bullet}\mapsto E(Y^{\bullet}).$$
For $r\ge1$ let ${\mathcal Y}_r$ be the set of equivalence classes of pairs of
integers $(m,n)$, where $(m,n)$ is declared to be equivalent with $(m',n')$ if
and only if there is some $j\in{\mathbb Z}$ with $(m,n)=(m'+jr,n'-j(r-1))$. For
$y\in {\mathcal Y}_r$ let $E_r^y(Y^{\bullet})$ be the complex of ${\mathcal H}$-modules whose terms are the $E_r^{m,n}(Y^{\bullet})$ with $(m,n)\in y$, and whose
differentials $d_r:E_r^{m,n}(Y^{\bullet})\to E_r^{m+r,n-r+1}(Y^{\bullet})$ are given by the spectral sequence. We apply the functor of
Lemma \ref{claudeles} to $E_r^y(Y^{\bullet})$ to obtain a (filtered) supersingular ${\mathcal H}$-module ${\bf E}_r^y(Y^{\bullet})$.
For a morphism $f:Y^{\bullet}\to W^{\bullet}$
in ${\mathfrak R}_G$ we have induced ${\mathcal H}$-linear maps $f_r:\oplus_{y\in {\mathcal Y}_r}{\bf
E}_r^y(Y^{\bullet})\to \oplus_{y\in {\mathcal Y}_r}{\bf E}_r^y(
W^{\bullet})$. Notice however that, in general, for a given $y\in{\mathcal
Y}_r$ there is no $y'\in {\mathcal Y}_r$ such that $f_r({\bf
E}_r^y(Y^{\bullet}))\subset{\bf
E}_r^{y'}(W^{\bullet})$, even if $r=1$.
\begin{lem}\label{13forstubai} Let $0\to Y^{\bullet}\to W^{\bullet}\to
X^{\bullet}\to 0$ be an exact sequence in
${\mathfrak R}_G$. We then have an exact sequence of supersingular ${\mathcal H}$-modules$$0\longrightarrow\bigoplus_{y\in {\mathcal Y}_1}{\bf E}_1^y(
Y^{\bullet})\longrightarrow\bigoplus_{y\in {\mathcal Y}_1}{\bf E}_1^y(
W^{\bullet})\longrightarrow\bigoplus_{y\in {\mathcal Y}_1}{\bf E}_1^y(
X^{\bullet})\longrightarrow0.$$
\end{lem}
{\sc Proof:} This follow from the constructions.\hfill$\Box$\\
{\bf Remark:} The analog of Lemma \ref{13forstubai} is false for the maps $f_r$ for $r>1$.\\
{\bf Remark:} For a smooth $G$-representation $Y$ endowed with an exact filtration, we may apply the functor $V$ (resp. $\tilde{V}$) of subsection \ref{fullfafu} to the supersingular ${\mathcal H}$-module ${\bf E}_r^y(
Y^{\bullet})$ (any $r$). In this way, we assign a ${\rm Gal}(\overline{F}/F)$-representation to $Y$. We propose this construction as a non-derived alternative to that of subsection \ref{susicoho}. Of course, again it will be meaningful only on supersingular $G$-representations.
By Lemma \ref{13forstubai}, for $r=1$ it provides an {\it exact} functor from smooth $G$-representations (endowed with an exact filtration) to ${\rm Gal}(\overline{{\mathbb Q}_p}/{\mathbb Q}_p)$-representations. We expect that for $G={\rm GL}_2({\mathbb Q}_p)$ it essentially recovers the restriction of Colmez's functor to all \footnote{i.e. not only to those generated by their $I_0$-invariants} supersingular $G$-representations.
|
{
"timestamp": "2018-03-08T02:07:28",
"yymm": "1803",
"arxiv_id": "1803.02616",
"language": "en",
"url": "https://arxiv.org/abs/1803.02616"
}
|
\section{Introduction}
Automatic Machine Learning (AutoML\xspace) aims to find the best performing learning algorithms with minimal human intervention.
Many AutoML\xspace methods exist, including random search \citep{bergstra2012random}, performance modelling \citep{bergstra2011algorithms,bergstra2013making}, Bayesian optimization \citep{snoek2012practical}, genetic algorithms \citep{real2017large,miikkulainen2017evolving} and RL \citep{baker2017designing,ZophLe2017}.
We focus on neural AutoML\xspace, that uses deep RL to optimize architectures.
These methods have shown promising results.
For example, Neural Architecture Search has discovered novel networks that rival the best human-designed architectures on challenging image classification tasks~\citep{zhong2018practical,ZophVSL17}.
However, neural AutoML\xspace is expensive because it requires training many networks.
This may require vast computations resources;
\citet{ZophLe2017} report 800 concurrent GPUs to train on Cifar-10.
Further, training needs to be repeated for every new task.
Some methods have been proposed to address this cost, such as using a progressive search space~\cite{liu2017progressive}, or by sharing weights among generated networks~\cite{pham2018efficient,liu2018darts}.
We propose a complementary solution, applicable when one has multiple ML tasks to solve.
Humans can tune networks based on knowledge gained from prior tasks.
We aim to leverage the same information using transfer learning.
We exploit the fact that deep RL-based AutoML\xspace algorithms learn an explicit parameterization of the distribution over performant models.
We present Transfer Neural AutoML\xspace, a method to accelerate network design on new tasks based on priors learned on previous tasks.
To do this we design a network that performs neural AutoML\xspace on multiple tasks simultaneously.
Our method for multitask neural AutoML\xspace learns both hyperparameter choices common to multiple tasks and specific choices for individual tasks.
We then transfer this controller to new tasks and leverage the learned priors over performant models.
We reduce the time to converge in both text and image domains by over an order of magnitude in most tasks.
In our experiments we save $10$s of CPU hours for every task that we transfer to.
\section{Methods}
\subsection{Neural Architecture Search}
Transfer Neural AutoML\xspace is based on Neural Architecture Search (NAS)~\citep{ZophLe2017}.
NAS uses deep RL to generate models that maximize performance on a given task.
The framework consists of two components: a controller model and child models.
The controller is an RNN that generates a sequence of discrete actions.
Each action specifies a design choice; for example, if the child models are CNNs, these choices could include the filter heights, widths, and strides.
The controller is an autoregressive model, like a language model: the action taken at each time step is fed into the RNN as input for the next time step.
The recurrent state of the RNN maintains a history of the design choices taken so far.
The use of an RNN allows dependencies between the design choices to be learned.
The sequence of design choices define a child model that is trained and evaluated on the ML task at hand.
The performance of the child network on the validation set is used as a reward to update the controller via a policy gradient algorithm.
\subsection{Multitask Training}
We propose Multitask Neural AutoML\xspace, that searches for model on multiple tasks simultaneously.
It requires defining a generic search space that is shared across tasks.
Many deep learning models require the same common design decisions, such as choice of network depth, learning rate, and number of training iterations.
By defining a generic search space that contains common architecture and hyperparameter choices, the controller can generate a wide range of models applicable to many common problems.
Multitask training allows the controller to learn a broadly applicable prior over the search space by observing shared behaviour across tasks. The proposed multitask controller has two key features: learned task representations, and advantage normalization.
\paragraph{Learned task representations}
The multitask AutoML\xspace controller characterizes the tasks by learning a unique embedding vector for each task.
This task-embedding allows to condition model generation on the task ID.
The task-embeddings are analogous to word-embeddings commonly used for NLP, where each word is associated to a trainable vector~\citep{mikolov2013distributed}.
Figure~\ref{fig:taskebedding} (left)
shows the architecture of the multitask controller at each time step.
The task embedding is fed into the RNN at every time step.
In standard single-task training of NAS, only the embedding of the previous action is fed into the RNN.
In multitask training, the task embedding is concatenated to the action embedding.
We also add a skip connection across the RNN cell to ease the learning of action marginal distributions.
The task embeddings are the only task-specific parameters.
One embedding is assigned to each task;
these are randomly initialized and trained jointly with the controller.
At each iteration of multitask training, a task is sampled at random.
This task's embedding is fed to the controller, which generates a sequence of actions conditioned on this embedding.
The child model defined by these actions is trained and evaluated on the task, and the reward is used to update the task-agnostic parameters and the corresponding task embedding.
\begin{figure}[t]
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{cc}
\includegraphics[width=0.4\linewidth,trim={0 0 0 0}]{figures/M-AML-network.pdf} &
\includegraphics[width=0.6\linewidth,trim={0 0 0 0},clip]{figures/heatmap_multitask.pdf}
\end{tabular}}
\caption{\emph{Left:}A single time step of the recurrent multitask AutoML\xspace controller, in which a single action is taken.
The task embedding is concatenated with the embedding of the action sampled at the previous timestep and passed into the controller RNN.
All parameters, other than the task embeddings, are shared across tasks.
\emph{Right:} Cosine similarity between the task embeddings learned by the multitask neural AutoML\xspace model.
\label{fig:taskebedding}}
\end{figure}
\paragraph{Task-specific advantage normalization}
We train the controller using policy gradient.
Each task defines a different performance metric which we use as reward.
The reward affects the amplitude of the gradients applied to update the controller's policy, $\pi$ .
To maintain a balanced gradient updates across tasks, we ensure that the distribution of each task's rewards are scaled to have same mean and variance.
The mean of each task's reward distribution is centered on zero by subtracting the expected reward for the given task.
The centered reward, or advantage, $A_{\tau}(m)$, of a model, $m$, applied to a task, $\tau$, is defined as the difference between the reward obtained by the model, $R_{\tau}(m)$, and the expected reward for the given task, $b_{\tau}=\mathbb{E}_{m \sim \pi}[R_{\tau}(m)]$: $$A_{\tau}(m) = R_{\tau}(m) - b_{\tau}$$
Subtracting such a baseline is a standard technique in policy gradient algorithms used to reduce the variance of the parameter updates~\citep{greensmith2004variance}.
The variance of each task’s reward distribution is normalized by dividing the advantage by the standard deviation of the reward:
$A'_{\tau}(m) = (R_{\tau}(m) - b_{\tau})\sigma_\tau^{-1}$.
Where $\sigma_{\tau} = \sqrt{\mathbb{E}_{m \sim \pi}[(R_{\tau}(m)-b_{\tau})^2]}$.
We refer to $A'$ as the normalized advantage.
The gradient update to the parameters of the policy $\theta$ is the product of the advantage and expected derivative of the log probability of sampling an action: $A'_\tau(m)\mathbb{E}_\pi[ \nabla_\theta \log \pi_\theta(m)]$.
Thus, normalizing the advantage may also be seen as adapting the learning rate for each task.
In practice, we compute $b_{\tau}$ and $\sigma_{\tau}$ using exponential moving averages over the sequence of rewards:
$b_\tau^t = (1-\alpha)b_\tau^{t-1} + \alpha R_\tau(m)$,
$\sigma^{2,t}_\tau = (1-\alpha)\sigma_\tau^{2,t-1} + \alpha (R_\tau(m)-b_{\tau}^t)^2$,
where $t$ indexes the trial, and $\alpha=0.01$ is the decay factor.
\subsection{Transfer Learning}
The multitask controller is pretrained on a set of tasks and learns a prior over generic architectural and parameter choices, along with task-specific decisions encoded in the task embeddings.
Given a new task, we can perform transfer of the controller by: 1) reloading the parameters of the pretrained multitask controller, 2) adding a new randomly initialized task embedding for the new task. Then, architecture search is resumed, and the controller's parameters are updated jointly with the new task embedding.
By learning an embedding for the new task, the controller learns a representation that biases towards actions that performed well on similar tasks.
\section{Related Work}
A variety of optimization methods have been proposed to search over architectures, hyperparameters, and learning algorithms.
These include random search \citep{bergstra2012random},
parameter modeling \citep{bergstra2013making},
meta-learned hyperparameter initialization \citep{feurer2015initializing},
deep-learning based tree searches over a predefined model-specification language \citep{negrinho2017deeparchitect},
and learning of gradient descent optimizers \citep{wichrowska2017learned,bello2017neural}.
An emerging body of neuro-evolution research has adapted genetic algorithms for these complex optimization problems \citep{conti2017improving}, including to set the parameters of existing deep networks \citep{such2017deep},
evolve image classifiers \citep{real2017large},
and evolve generic deep neural networks \citep{miikkulainen2017evolving}.
Our work relates closest to NAS \citep{ZophLe2017}.
NAS was applied to construct CNNs for the CIFAR-10 image classification and RNNs for the Penn Treebank language modelling.
Subsequent work reduces the computational cost for more challenging tasks \citep{ZophVSL17}.
To engineer an architecture for ImageNet classification, \citet{ZophVSL17} train the NAS controller on the simpler CIFAR-10 task and then transfer the child architecture to ImageNet by stacking it.
However, they did not transfer the controller model itself, relying instead on the intuition that additional depth is necessary for the more challenging task.
Other works apply RL to automate architecture generation and also reduce the computation cost.
MetaQNN sequentially chooses CNN layers using Q-learning \citep{baker2016designing}. MetaQNN uses an aggressive exploration to reduce search time, though it can cause the resulting architectures to underperform.
\citet{cai2017reinforcement} transform existing architectures incrementally to avoid generating entire networks from scratch.
\citet{liu2017progressive} reduce search time by progressively increasing architecture complexity, and \cite{pham2018efficient} propose child-model weight sharing to reduce child training time.
Transfer learning has achieved excellent results as an initialization method for deep networks, including for models trained using RL \citep{yosinski2014transferable,sharif2014cnn,zhan2015online}.
Recent meta-learning research has broadened this concept to learn generalizable representations across classes of tasks \citep{finn2017model, mishra2017simple}.
Simultaneous multitask training can facilitate learning between tasks with a common structure, though retaining knowledge effectively across tasks is still an active area of research \cite{kirkpatrick2017overcoming,teh2017distral}.
There is also prior research on transfer of optimizers for Neural AutoML\xspace;
Sequential Model-based Optimizers have been transferred across tasks to improve hyperparameter tuning~\citep{bardenet2013collaborative,yogatama2014efficient},
we propose a parallel solution for neural methods.
\section{Experiments}
\paragraph{Child models}
Constructing the search space needs human input, so we choose wide parameter ranges to minimize injected domain expertise.
Our search space for child models contains two-tower feedforward neural networks (FFNN), similar to the wide and deep models in \citet{cheng2017}. One tower is a deep FFNN, containing an input embedding module, fully connected layers and a softmax classification layer. This tower is regularized with an L2 loss. The other is a wide-shallow layer that directly connects the one-hot token encodings to the softmax classification layer with a linear projection. This tower is regularized with a sparse L1 loss. The wide layer allows the model to learn task-specific biases for each token directly.
The deep FFNN's embedding modules are pretrained\footnote{The pretrained modules are distributed via TensorFlow Hub: \url{https://www.tensorflow.org/hub}\ .}
This results in child models with higher quality and faster convergence.
The single search space for all tasks is defined by the following sequence of choices:
1) Pretrained embedding module.
2) Whether to fine-tune the embedding module.
3) Number of hidden layers (HL).
4) HL size.
5) HL activation function.
6) HL normalization scheme to use.
7) HL dropout rate.
8) Deep column learning rate.
9) Deep column regularization weight.
10) Wide layer learning rate.
11) Wide layer regularization weight.
12) Training steps.
The Appendix contains the exact specification. The search space is much larger than the number of possible trials, containing $1.1$B configurations.
All models are trained using Proximal Adagrad with batch size 100.
Notice that this search space aims to optimize jointly the architecture and hyperparameters.
While standard NAS search spaces are defined strictly over architectural parameters.
\paragraph{Controller models}
The controller is a 2-layer LSTM with 50 units.
The action and task embeddings have size 25.
The controller and embedding weights are initialized uniformly at random, yielding an approximate uniform initial distribution over actions. The learning rate is set to $10^{-4}$ and it receives gradient updates after every child completes.
We tried four variants of policy gradient to train the controller: REINFORCE \citep{williams1992simple}, TRPO \citep{schulman2015trust}, UREX \citep{Nachum2017} and PPO \citep{schulman2017proximal}. In preliminary experiments on four NLP tasks, we found REINFORCE and TRPO to perform best and selected REINFORCE for the following experiments.
We evaluate three controllers.
First, Transfer Neural AutoML, our neural controller that transfers from multitask pre-training.
Second, Single-task AutoML, which is trained from scratch on each task.
Finally, a baseline, Random Search (RS), that selects action uniformly at random.
\paragraph{Metrics}
To measure the ability of the different AutoML\xspace controllers to find good models, we compute the average accuracy of the topN (accuracy-topN) child models generated during the search.
We select the best topN models according to accuracy on the validation set.
We then report the validation and test performance of these models.
We assess convergence rates with two metrics: 1) accuracy-topN achieved with a fixed budget of trials, 2) the number of trials required to attain a certain reward.
The latter can only be used with validation accuracy-topN since test accuracy-topN does not necessarily increase monotonically with the number of trials.
\subsection{Natural Language Processing}
\paragraph{Data}
We evaluate using 21 text classification tasks with varied statistics.
The dataset sizes range from $500$ to $420$k datapoints.
The number of classes range from $2$ to $157$,
and the mean length of the texts, in characters, range from $19$ to $20$k.
The Appendix contains full statistics and references.
Each child model is trained on the training set.
The accuracy on the validation set is used as reward for the controller.
The topN child models, selected on the validation set, are evaluated on the test set.
Datasets without a pre-defined train/validation/test split, are split randomly 80/10/10.
The multitask controller is pretrained on 8 randomly sampled tasks: Airline, Complaints, Economic News, News Aggregator, Political Message, Primary Emotion, Sentiment SST, US Economy.
We then transfer from this controller to each of the remaining 13 tasks.
\begin{table}[t]
\caption{
Performance of Random Search (RS), single-task Neural AutoML (NAML) and Transfer Neural AutoML (T-NAML). Bolding indicates the best controller, or within $\pm$2 s.e.m..
\emph{Left}: Number of trials needed to attain a validation accuracy-top10 equal to the best achieved by Random Search with 5000 trials (250/2500 for Brown and 20 Newsgroups, respectively).
\emph{Right}: Test accuracy-top10 given at a fixed budget $B$ of 500 trials ($B=250$ for Brown). Error bars show $\pm$2 s.e.m. computed across the top 10 models. Similar s.e.m. values are observed for all methods.
}
\label{tab:performance}
\begin{tabular}{p{0.5\textwidth}p{0.5\textwidth}}
\setlength\tabcolsep{2pt}
\begin{tabular}{lrrr}
\bf Dataset & \bf RS & \bf NAML & \bf T-NAML
\\ \hline \\
20 Newsgroups & 2470 & 1870 & \bf 435 \\
Brown Corpus & 245 & 235 & \bf 10 \\
SMS Spam & 4815 & 3390 & \bf 70 \\
Corp Messaging & 3850 & 1510 & \bf 80 \\
Disasters & 4970 & 2730 & \bf 25 \\
Emotion & 4995 & 1645 & \bf 195 \\
Global Warming & 4985 & 1935 & \bf 90 \\
Prog Opinion & 4200 & 3620 & \bf 60 \\
Customer Reviews & 4895 & 925 & \bf 15 \\
MPQA Opinion & 4965 & 1510 & \bf 15 \\
Sentiment Cine & 4520 & 3225 & \bf 535 \\
Sentiment IMDB & 4760 & \bf 630 & 690 \\
Subj Movie & 4745 & 1600 & \bf 105 \\
\hline
\end{tabular} &
\setlength\tabcolsep{2pt}
\begin{tabular}{lrrr}
\bf Dataset & \bf RS & \bf NAML & \bf T-NAML
\\ \hline \\
20 Newsgroups & 87.5 & 87.4 & \bf 88.1$\pm$0.4 \\
Brown Corpus & 37.0 & 38.2 & \bf 53.4$\pm$3.3 \\
SMS Spam & 97.9 & 97.8 & \bf 98.1$\pm$0.1 \\
Corp Messaging & \bf 90.0 & \bf 90.2 & \bf 90.2$\pm$0.3 \\
Disasters & 81.7 & 81.5 & \bf 82.1$\pm$0.3 \\
Emotion & 33.9 & 33.7 & \bf 35.3$\pm$0.3 \\
Global Warming & 82.4 & \bf 82.8 & \bf 82.9$\pm$0.3 \\
Prog Opinion & 68.9 & 66.3 & \bf 70.3$\pm$0.9 \\
Customer Reviews & 77.8 & 79.0 & \bf 81.4$\pm$0.5 \\
MPQA Opinion & 87.9 & 87.9 & \bf 88.6$\pm$0.3 \\
Sentiment Cine & 73.2 & \bf 76.3 & 75.4$\pm$0.4 \\
Sentiment IMDB & 85.8 & 87.3 & \bf 88.1$\pm$0.1 \\
Subj Movie & 92.6 & 93.2 & \bf 93.4$\pm$0.2 \\
\hline
\end{tabular}
\end{tabular}
\end{table}
\paragraph{Results}
To assess the controllers' ability to optimize the reward (validation set accuracy) we compute the speed-up versus the baseline, RS.
We first compute accuracy-top10 on the validation set for RS given a fixed budget of $B$ trials.
We use $B=5000$, except for the Brown Corpus and 20 Newsgroups where we can only use a $B=500,3500$, respectively, because these datasets were slower to train.
We then report the number of trials required by AutoML\xspace and T-AutoML\xspace to achieve the same validation accuracy-top10 as RS with $B$ trials.
Table~\ref{tab:performance} (left) shows the results.
Note that RS may exhibit fewer than $B=5000$ trials if it converged earlier.
These results shows that T-AutoML\xspace is effective at optimizing validation accuracy, offering a large reduction in time to attain a fixed reward.
In 12 of the 13 datasets T-AutoML\xspace achieves the desired reward fastest,
and in 9 cases achieves an order of magnitude speed-up.
\begin{figure*}[t]
\resizebox{\textwidth}{!}{
\huge{
\begin{tabular}{cccc}
20 Newsgroups & Brown Corpus & Corp Messaging & Customer Reviews \\
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_20_newsgroups.pdf} &
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_brown.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_crowdflower_corporate_messaging.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_customer_reviews.pdf}\\
Disasters & Emotion & Global Warming & MPQA Opinion \\
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_crowdflower_disasters.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_crowdflower_emotion.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_crowdflower_global_warming.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_mpqa.pdf}\\
Prog Opinion & Sentiment Cine & Sentiment IMDB & SMS Spam \\
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_crowdflower_progressive_opinion.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_sentiment_corpus_cine.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_sentiment_imdb.pdf}&
\includegraphics[trim={0 0 0 1cm},clip]{result_figs/test_top_10_transfer_cloudml_sms_spam_collection.pdf}
\end{tabular}
}}
\caption{Learning curves for Random Search (RS), single-task Neural AutoML (NAML), and Transfer (T-NAML).
\emph{x-axis}: Number of trials (child model evaluations).
\emph{y-axis}: Average test set accuracy of the 10 models with best validation accuracy (test accuracy-top10) found up to each trial. \label{fig:transfer-test-acc}}
\end{figure*}
Next, we assess the quality of the models on the test set.
Table~\ref{tab:performance} (right) shows test accuracy-top10 with a budget of 500 trials (250 for Brown Corpus).
Within this budget, T-AutoML\xspace performs best on all but one dataset.
T-AutoML\xspace outperforms single-task AutoML\xspace on 10 out of the 13 datasets, ties on one, and loses on two.
On the datasets where T-AutoML\xspace does not produce the best final model at 500 trials, it often produces better models at earlier iterations.
Figure~\ref{fig:transfer-test-acc} shows the full learning curves of test set accuracy-top10 versus number of trials.
Figure~\ref{fig:transfer-test-acc} shows that in most cases the controller with transfer starts with a much better prior over good models.
On some datasets the quality is improved with further training e.g. Emotion, Corp Messaging, but in others the initial configurations learned from the multitask model are not improved.
For reference, we put the learning curves for the initial multitask training phase in the Appendix.
We also ran RS and single-task AutoML\xspace on these datasets.
Slightly disappointingly, multitask training did not in itself yield substantial improvements over single-task; it attains a higher accuracy on two datasets, and in similar on the other six.
We aim to to attain good performance with fewest possible trials.
We do not seek to beat state-of-the-art all datasets because first,
although our search space is large,
it does not contain all performant model components (e.g. convolutions).
Second, we use embedding modules pretrained on large datasets
which makes the results incomparable to those that only uses in-domain training data.
However, to confirm that Neural AutoML\xspace generates good models we compare to some previous published results where available.
Overall we find that Transfer AutoML\xspace with the search space described above yields models competitive with the state-of-the-art.
For example,
\citet{almeida2013towards} use classical ML classifiers (Logistic Regression, SVMs, etc.) on SMS Spam and report best accuracy of 97.59\%. Transfer AutoML\xspace gets accuracy-top10 of 98.1\%.
\citet{LeM14} report 92.58\% accuracy on Sentiment IMDB with more complex architectures, Transfer AutoML's is a little behind, accuracy top-10 is 88.1\%.
\citet{li2016weighted} report 86.8\% accuracy using an ensemble of weighted neural BOWs on MPQA. Transfer AutoML\xspace achieve accuracy-top10 of 88.6\%.
\citet{li2016weighted} also evaluate their ensemble of weighted neural BOW models on Customer Reviews, and achieve 82.5\% best accuracy, though the best accuracy of any single model is 81.1\%. Comparably, T-AutoML\xspace gets an accuracy-top10 of 81.4\%.
\citet{barnes2017} compare many algorithms and report best accuracy on Sentiment-SST of 83.1\% using LSTMs. Multitask AutoML\xspace gets an accuracy-Top10 of 83.4\%.
The best performance achieved with a more complex architecture that is not in our search space is: 87.8\% \citep{LeM14}.
\citet{maas-EtAl:2011:ACL-HLT2011} report 88.1\% on Movie Subj, Transfer AutoML\xspace gets accuracy-top10 of 93.4\%.
\paragraph{Computational Cost and Savings}
The median cost to perform a single trial across all 21 datasets in our experiments is $T=268s$.
If we run $B$ trials with a speedup factor of $S$, we save $BT(1-S^{-1}) / 3600$CPU-h per task to attain a fixed reward (validation accuracy-top10).
Estimating the speedup factors from Table~\ref{tab:performance} (left) for transfer over single-task, we attain a median computational saving of $30$CPU-h per task when performing $B=500$ trials.
The mean is $89$CPU-h, but this is heavily influenced by the slow Brown Corpus.
The time to train the multitask controller is $15$h on 100 CPUs.
If we do not need the $M$ models for the tasks used to train the multitask controller, then we must run $>(1-1/S)^{-1}M$ new tasks to amortize this cost.
For the median speedup in our experiments $S=22$ that is $>1.05M$ new tasks.
\begin{SCfigure}
\resizebox{0.5\linewidth}{!}{
\begin{tabular}{cc}
\includegraphics[width=0.7\linewidth,trim={0cm 0cm 0cm 0cm},clip]{result_figs/cifar10_test.pdf}
\end{tabular}
}
\caption{Comparison on an image classification task, Cifar-10. Mean test accuracy of the top 10 models chosen on the validation set.}
\label{fig:image}
\end{SCfigure}
\subsection{Image classification}
To validate the generality of our approach we evaluate on image classification task: Cifar-10.
We compare the same three controllers: RS, AutoML\xspace trained from scratch, and Transfer AutoML\xspace pretrained on MNIST and Flowers\footnote{goo.gl/tpzfR1}.
Figure~\ref{fig:image} shows the mean accuracy-top-10 on the test set.
The transferred controller attains an accuracy-top-10 of $96.5\%$, similar to the other methods, but converges much faster as in the NLP tasks.
The best models embed images with a finetuned Inception v3 network, pretrained on ImageNet. Relu activations are preferred over Swish~\citep{ramachandran2017swish} and the dropout rate of converges to 0.3.
\subsection{Analysis}
\paragraph{Meta overfitting}
The controller is trained on the tasks' validation sets.
Overfitting of AutoML\xspace to the validation set is not often addressed.
This type of overfitting may seem unlikely because each trial is expensive,
and many trials may be required to overfit.
However, we observe it in some cases.
Figure~\ref{fig:overfitting} (left, center) shows the accuracy-top10 on the validation and test sets on the Prog Opinion dataset.
Transfer Neural AutoML\xspace attains good solutions in the first few trials,
but afterwards its validation performance grows while test performance does not.
The generalization gap between the validation and test accuracy increases over time.
This is the most extreme case we observed, but some other datasets exhibit some generalization gap also (see Appendix for all validation curves).
This effect is largest on Prog Opinion because the validation set is tiny, with only 116 examples.
Overfitting arises from bias due to selecting the best models on the validation set.
Child evaluation contains randomness due to the stochastic training procedure.
Therefore, over time we see an improved validation score, even after convergence, due to lucky evaluations.
However, those apparent improvements are not reflected on the test set.
Transfer AutoML\xspace exhibits more overfitting than single-task because it converges earlier.
We confirmed this effect; if we `cheat' and select models by their test-set performance, we observe the same artificial improvement on the test score as on the validation score.
Other than entropy regularization, we do not combat overfitting extensively.
Here, we simply emphasize that because our Transfer Neural AutoML\xspace model
observes many trials in total, meta-overfitting becomes a bigger issue.
We leave combatting this effect to future research.
\begin{figure}[t]
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{ccc}
\Huge Prog Opinion: val &
\Huge Prog Opinion: test &
\Huge Sentiment Cine module distribution \\
\includegraphics{result_figs/val_top_10_transfer_cloudml_crowdflower_progressive_opinion_lab.pdf}&
\includegraphics{result_figs/test_top_10_transfer_cloudml_crowdflower_progressive_opinion_lab.pdf}&
\includegraphics{figures/p_module.pdf}
\end{tabular}}
\caption{\emph{Left, Center}: Learning curves on the validation (left) and test sets (center) for the Prog Opinion dataset.
\emph{Right}: Evolution of the choice of pretrained embedding module for transfer to the Spanish Corpus-Cine task.
y-axis indicates the probability of sampling each table. This probability is estimated from the samples using a sliding window of width 100.
\label{fig:overfitting}}
\end{figure}
\paragraph{Distant transfer: across languages}
The more distant the tasks, the harder it is to perform transfer learning.
The Sentiment Cine task is an outlier because it is the only Spanish task.
Figure~\ref{fig:transfer-test-acc} and Table~\ref{tab:performance} show poorer performance of transfer on this task.
The most language-sensitive parameters are the pretrained word embeddings.
The controller selects from eight pretrained embeddings (see Appendix), six of which are English, and two Spanish.
In the first 1500 iterations, the transferred controller chooses English embeddings, limiting the performance.
However, after further training, the controller switches to Spanish tables at around 2000th trial,
Figure~\ref{fig:overfitting} (right).
At trial 2000, T-AutoML\xspace attains a test accuracy-top10 of 79.8\%, approximately equal to that or random search with 79.4\%, and greater than single-task with 78.1\%.
This indicates that although transfer works best on similar tasks, the controller is still able to adapt to outliers given sufficient training time.
\paragraph{Task representations and learned models}
We inspect the learned task similarities via the embeddings.
Figure~\ref{fig:taskebedding} (right) shows the cosine similarity between the task embeddings learned during multitask training.
The model assigns most tasks to two clusters.
It is hard to guess \emph{a priori} which tasks require similar models; the dataset sizes, number of classes and text lengths differ greatly.
However, the controller assigns the same model to tasks within the same cluster.
At convergence, the cluster \{Complaints, New Agg, Airline, Primary Emotion\} is assigned (with high probability) a 1-layer networks with 256 units, Swish activation function, wide-layer learning rate 0.01, and dropout rate 0.2.
The cluster \{Economic News, Political Emotion, Sentiment SST\} is assigned 2-layer networks with 64 units, Relu activation, wide-layer learning rate 0.003, and dropout rate 0.3.
Other choices follow similar distributions for each cluster.
For example, the same 128D word embeddings, trained using a Neural Language Model are chosen.
The controller also always chooses to fine-tune these embeddings.
The controller may remove either the deep or wide tower by setting the regularization very high, but in all cases it chooses to keep both active.
\paragraph{Ablation}
We consider two ablations of T-NAML.
First, we remove the task embeddings.
For this, we train a task-agnostic multitask controller without task embeddings, then transfer this controller as for T-NAML.
Second, we transfer a single architecture rather than the controller parameters.
For this, we train the task-agnostic multitask controller to convergence, and select the final child model.
We then re-train this single architecture on each new task.
Omitting task embeddings performs well on some tasks, but poorly on those that require a model different to the mode.
Overall, according to accuracy-top10 at 500 trials, T-NAML outperforms the version without task embeddings on 8 tasks, loses 4, and draws on 1.
The mean performance drop when ablating task embeddings is 1.8\%.
Using just a single model performs very poorly on many tasks, T-NAML wins
8 cases, loses 2, and draws 3, with a mean performance increase of 4.8\%.
\section{Conclusion}
Neural AutoML\xspace, whilst becoming popular, comes with a high computational cost.
To address this we propose transfer learning of the controller and show
a large reductions in convergence time across many datasets.
Extensions to this work include:
Broadening the search space to contain more models classes.
Attempting transfer across modalities; some priors over hyperparameter combinations learned on NLP tasks may be useful for images or other domains.
Making the controller more robust to evaluation noise, and addressing the potential to meta overfit on small datasets.
\subsubsection*{Acknowledgments}
We are very grateful to Quentin de Laroussilhe, Andrey Khorlin, Quoc Le, Sylvain Gelly, the Tensorflow Hub team and the Google Brain team Zurich for developing software frameworks and many useful discussions.
{\small
\bibliographystyle{unsrtnat}
{\small
|
{
"timestamp": "2019-01-29T02:27:44",
"yymm": "1803",
"arxiv_id": "1803.02780",
"language": "en",
"url": "https://arxiv.org/abs/1803.02780"
}
|
\section{Introduction}
Simulation is used extensively to facilitate decision-making processes related to complex systems. The popularity stems from its flexibility, allowing users to incorporate arbitrarily fine details of the system and estimate virtually any performance measure of interest. However, simulation models are often computationally expensive to execute, severely restricting the usefulness of simulation in settings such as real-time decision making and system optimization. In order to alleviate this computational inefficiency, metamodeling has been developed actively in the simulation community \citep{BartonMeckesheimer06}. The basic idea is that the user only executes the simulation model at some carefully selected design points. A metamodel, which runs much faster than the simulation model in general, is then built to approximate the true response surface -- the performance measure of the simulation model -- as a function of the design variables, by interpolating the simulation outputs properly. The responses at other locations are predicted by the metamodel without running additional simulation, thereby reducing the computational cost substantially.
Stochastic kriging (SK), proposed by \cite{AnkenmanNelsonStaum10}, is a particularly popular metamodel, thanks to its analytical tractability, ease of use, and capability of providing good global fit. It has been used successfully for quantifying input uncertainty in stochastic simulation \citep{BartonNelsonXie14,XieNelsonBarton14} and for optimizing expensive functions with noisy observations \citep{sun2014}. SK represents the response surface as a Gaussian process, which is fully characterized by its covariance function, and leverages the spatial correlations between the responses to provide prediction. However, one often encounters two numerical issues when implementing SK in practice, both of which are related to matrix inversion. Indeed, the inverse of the covariance matrix of the simulation outputs is essential for computing various quantities in SK, including the optimal predictor, the mean squared error of prediction, and the likelihood function.
An immediate issue regarding the inversion of a $n\times n$ matrix is that it typically requires $\mathcal O(n^3)$ computational time, which is prohibitive for large $n$, where $n$ is the number of the design points. For instance, it is reported in \cite{huang2006} that a major limitation of SK-based methods for simulation optimization is the high computational cost of fitting the SK metamodel, which, as the number of samples increases, eventually becomes even more expensive than running the original simulation model.
A second issue is that the covariance matrix involved in SK may become ill-conditioned (i.e., nearly singular), in which case the inversion is numerically unstable, resulting in inaccurate parameter estimation or prediction. This often occurs when $n$ is large, because then there are fairly likely two design points that are spatially close to each other, and thus the two corresponding columns in the covariance matrix are ``close to'' being linearly dependent.
These two numerical issues preclude the use of SK at a large scale, especially for problems with a high-dimensional design space. In geostatistics literature, inverting large covariance matrices that arise from Gaussian processes is a well-known numerical challenge and is sometimes referred to as ``the big $n$ problem'' informally. Typical solutions to this problem are based on approximations, that is, use another matrix that is easier to invert to approximate the covariance matrix; see \S \ref{sec:lit} for more details. This paper presents a new perspective. \emph{Instead of seeking good approximations for covariance matrices induced by an arbitrary covariance function, we will construct a specific class of covariance functions that induce computationally tractable covariance matrices.} In particular, the computational tractability stems from the following two properties of the covariance matrices induced by this class of covariance functions: (i) they can be inverted \emph{analytically}, and (ii) the inverse matrices are \emph{sparse}. Our novel approach will effectively reduce the computational complexity of SK to $\mathcal O(n^2)$, without resorting to approximation schemes. In situations where the simulation errors are negligible, our approach obviates the need of numerical inversion and further reduces the complexity to $\mathcal O(n)$.
We refer to this specific class of covariance functions as \emph{Markovian covariance functions} (MCFs), because the Gaussian processes equipped with them exhibit certain Markovian structure. Albeit seemingly restrictive, MCFs actually represent a broad class of covariance functions and can be constructed in a flexible, convenient fashion.
\subsection{Main Contributions}
First and foremost, we identify a simple but general functional form with which the covariance function of a 1-dimensional Gaussian process yields \emph{tridiagonal} precision matrices (i.e., the inverse of the covariance matrices), which are obviously sparse. In addition, the nonzero entries of the precision matrices can be expressed in terms of the covariance function in closed-form. To the best of our knowledge, there is no prior result establishing this kind of explicit connection between the form of covariance functions and sparsity in the corresponding precision matrices.
Second, we link MCFs to Sturm-Liouville (S-L) differential equations. Specifically, we show that the Green's function of an S-L equation has exactly the same form as MCFs. Not only does this connection provide a convenient tool to construct MCFs, but also implies that the number of MCFs having an analytical expression is potentially enormous, since any second-order linear ordinary differential equation can be recast in the form of an S-L equation.
Third, we extend MCFs to multidimensional design spaces in a ``composite'' manner, namely, defining the multidimensional covariance to be the product of 1-dimensional covariances along each dimension. This way of construction allows use of tensor algebra to preserve the sparsity in the precision matrices, provided that the design points form a regular lattice.
Last but not least, we demonstrate through extensive numerical experiments that MCFs can significantly outperform those that are commonly used such as the squared exponential covariance function in terms of accuracy in prediction of response surfaces. The improved accuracy can be attributed to two reasons: (i) the numerical stability of matrix inversion is enhanced greatly; (ii) the reduced computational complexity allows us to use more data.
\subsection{Related Literature}\label{sec:lit}
A great variety of techniques have been proposed to address the big $n$ problem in both geostatistics and machine learning literature, where Gaussian processes are widely used. Most of them focus on developing approximations of the covariance matrix that are computationally cheaper. Representative approximation schemes include reduced-rank approximation and sparse approximation. The former approximates the covariance matrix by a matrix having a much lower rank. The latter can be achieved by a method called covariance tapering. It forces the covariance to zero if the two design points involved are sufficiently far away from each other. The covariance matrix is then approximated by a sparse matrix. Both reduced-rank matrices and sparse matrices entail fast inversion algorithms. From a modeling perspective, these two approximation schemes emphasize long-scale and short-scale dependences respectively, but meanwhile fail to capture the other end of the spectrum \citep{SangHuang12}. We refer to \citet[Chapter 12]{BanerjeeCarlinGelfand14} and \citet[Chapter 8]{RasmussenWilliams06} for reviews with a focus on geostatistics and machine learning, respectively. Moreover, approximation schemes usually result in spurious quantification of the uncertainty about the prediction; see, e.g., \cite{ShahriariSwerskyWangAdamsdeFreitas16} and references therein.
Another popular approach to the big $n$ problem is to use Gaussian Markov random fields (GMRFs), which discard the concept of covariance function and model the precision matrix, i.e., the inverse of the covariance matrix, directly; see \citet{RueHeld05} for a thorough exposition on the subject and \cite{SalemiSongNelsonStaum17} for its application in large-scale simulation optimization. To construct a GMRF one first stipulates a graph, with nodes denoting locations of interest in the design space. The edges in the graph characterize the ``neighborhood'' of each node, and define a Markovian structure. In particular, given all its neighbors, each node is conditionally independent of its non-neighbors. A crucial property of GMRF is that entry $(i,j)$ of the precision matrix is nonzero if, and only if, node $i$ and node $j$ are neighbors. Hence, the precision matrix is sparse if each node has a small neighborhood. The sparsity is then taken advantage of to reduce the inversion-related computational time.
Despite its computational efficiency, GMRFs have clear disadvantages. First and foremost, they do not model association directly, and thus one cannot specify desired correlation behavior. Indeed, the relationship between entries in the precision matrix and the covariance matrix is very complex. This is because the joint distribution of the responses at two locations depends on the joint distribution of the responses at all the other locations. Second, GMRFs are built on graphs, and the discrete nature forbids predicting responses at locations that are not included in the graph, which is problematic for continuous design spaces.
The methodology developed in the present paper is closely related to GMRFs. Our work can be viewed as one way to extend GMRFs from discrete domains to continuous domains. But it is by no means a trivial extension, because we establish an explicit relationship between the form of a covariance function and the sparsity in the corresponding precision matrices. This allows us to combine the best of two worlds -- modeling association directly while preserving the computational tractability of GMRFs.
The rest of the paper is organized as follows. In \S\ref{sec:model}, we introduce the SK metamodel and motivate our approach to the big $n$ problem. In \S\ref{sec:MCF}, we introduce MCFs and characterize their essential structure, which effectively bridges the gap between Gaussian processes and GMRFs. In \S \ref{sec:SL}, we link MCFs with S-L differential equations. In \S\ref{sec:MLE}, we discuss maximum likelihood estimation of the unknown parameters, with an emphasis on the numerical stability as a result of the use of MCFs. We conduct extensive numerical experiments in \S\ref{sec:numerical} to demonstrate the scalability of SK in the presence of MCFs, and conclude in \S\ref{sec:conclusions}. The Appendices collect some technical proofs.
\section{Stochastic Kriging and the Big $n$ Problem}\label{sec:model}
Let $\bm x\in \mathscr{X}\subseteq\mathbb R^D$ denote the design variable of a computationally expensive simulation model, with $\mathscr{X}$ being the design space. Let $\mathsf Z(\bm x)$ denote the unknown response surface of that model. Suppose that the simulation model is run at design point $\bm x_i$ with $r_i$ independent replications, producing outputs $z_j(\bm x_i)$, $j=1,\ldots,r_i$, $i=1,\ldots,n$. Metamodeling is concerned with fitting $\mathsf Z(\bm x)$ based on the simulation outputs. The SK metamodel casts $\mathsf Z(\bm x)$ into a \textit{realization} of a Gaussian process,
\begin{equation}
\label{eq:uni_nriging}
\mathsf Z(\bm x) = \pmb \beta^\intercal \bm f(\bm x)+ \mathsf M(\bm x),
\end{equation}
where $\bm f(\bm x)$ is a vector of known functions (e.g., polynomial basis functions) and $\pmb \beta$ is a vector of unknown parameters of compatible dimension, and $\mathsf M$ is a mean zero Gaussian process that is randomly sampled from a space of functions mapping $\mathbb R^D \mapsto\mathbb R $. A particular feature of the SK metamodel \eqref{eq:uni_nriging} is the spatial correlation, i.e., $\mathsf M(\bm x)$ and $\mathsf M(\bm y)$ tend to be similar (resp., different) if $\bm x$ and $\bm y$ are close to (resp., distant from) each other in space. Let $k(\bm x,\bm y)\coloneqq\Cov(\mathsf M(\bm x), \mathsf M(\bm y))$ denote the covariance function of $\mathsf M$. It is crucial to specify $k(\bm x,\bm y)$ properly in order that SK provide a good fit globally over the design space $\mathscr{X}$.
The simulation outputs become
\begin{equation*}\label{eq:SK}
z_j(\bm x_i) = \mathsf Z(\bm x) + \varepsilon_j(\bm x_i)=\pmb \beta^\intercal \bm f(\bm x_i) + \mathsf M(\bm x_i) + \varepsilon_j(\bm x_i),
\end{equation*}
where $\varepsilon_1(\cdot),\varepsilon_2(\cdot),\ldots$ are normally distributed simulation errors. Define $\bar z(\bm x_i)\coloneqq r_i^{-1}\sum_{j=1}^{r_i}z_j(\bm x_i)$, $\bar \varepsilon(\bm x_i)\coloneqq r_i^{-1}\sum_{j=1}^{r_i}\varepsilon_j(\bm x_i)$, and $\bar \bm z \coloneqq (\bar z(\bm x_1),\cdots,\bar z(\bm x_n))^\intercal$. Let $\pmb \Sigma_\mathsf M$ denote the $n\times n$ covariance matrix of $(\mathsf M(\bm x_1),\ldots,\mathsf M(\bm x_n))$, i.e., entry $(i,j)$ of $\pmb \Sigma_\mathsf M$ is $\Cov(\mathsf M(\bm x_i),\mathsf M(\bm x_j))=k(\bm x_i,\bm x_j)$. Likewise, let $\pmb \Sigma_\varepsilon$ denote the covariance matrix of $(\bar \varepsilon(\bm x_1),\ldots,\bar \varepsilon(\bm x_n))$.
We assume that the simulation errors are mutually independent and are independent of $\mathsf M$. This assumption effectively rules out the use of common random numbers (CRN) because it will break the sparsity that our methodology critically hinges on. Nevertheless, this does not impose much practical restriction, since it is shown in \cite{ChenAnkenmanNelson12} that the use of CRN generally is detrimental to the prediction accuracy of SK.
Let $\bm x_0$ denote an arbitrary point in $\mathscr{X}$. SK is concerned with predicting $\mathsf Z(\bm x_0)$ based on $\bar \bm z$. The SK predictor that minimizes the mean squared error (MSE) of prediction is
\begin{equation}\label{eq:BLUP}
\widehat\mathsf Z(\bm x_0) = \pmb \beta^\intercal \bm f(\bm x_0) + \pmb\gamma^\intercal (\bm x_0) [\pmb \Sigma_\mathsf M + \pmb \Sigma_\varepsilon]^{-1}(\bar\bm z-\bm F\pmb \beta),
\end{equation}
with optimal MSE
\begin{equation}\label{eq:MSE}
\MSE^*(\bm x_0) = k(\bm x_0,\bm x_0) - \pmb\gamma^\intercal(\bm x_0)[\pmb \Sigma_\mathsf M + \pmb \Sigma_\varepsilon]^{-1}\pmb\gamma(\bm x_0),
\end{equation}
where $\pmb\gamma\coloneqq (k(\bm x_0, \bm x_1),\ldots,k(\bm x_0, \bm x_n))^\intercal$ and $\bm F\coloneqq (\bm f(\bm x_1),\ldots,\bm f(\bm x_n))^\intercal$, provided that $\pmb \beta$, $\pmb\gamma(\bm x_0)$, $\pmb \Sigma_\mathsf M$, and $\pmb \Sigma_\varepsilon$ are known. Clearly, they need to be estimated from the simulation outputs in practice. A typical method for estimating the unknown parameters is the maximum likelihood estimation (MLE), which maximizes the following log-likelihood function
\begin{equation}\label{eq:loglikelihood}
l(\pmb \beta,\pmb \theta) = -\frac{n}{2}\ln(2\pi) - \frac{1}{2}\ln|\pmb \Sigma_\mathsf M + \pmb \Sigma_\varepsilon| -\frac{1}{2}(\bar\bm z-\bm F\pmb \beta)^\intercal [\pmb \Sigma_\mathsf M + \pmb \Sigma_\varepsilon]^{-1} (\bar\bm z-\bm F\pmb \beta),
\end{equation}
where $|\cdot|$ denotes the determinant of a matrix and $\pmb \theta$ denotes the unknown parameter involved for specifying the covariance function $k$; see \S\ref{sec:MLE} for more discussion.
Obviously, computing \eqref{eq:BLUP}, \eqref{eq:MSE}, and \eqref{eq:loglikelihood} all requires inverting $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$, which comes with two numerical challenges and is referred to as the big $n$ problem in geostatistics literature \citep{BanerjeeCarlinGelfand14}. First, although $\pmb \Sigma_\varepsilon$ is diagonal due to the independence assumption, $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$ is a dense matrix in general and inverting it typically takes $\mathcal O(n^3)$ computational time, which becomes prohibitive for large $n$ (e.g., $n>10^3$). Second, this matrix often becomes ill-conditioned, and thus inverting it is prone to numerical instability. This may happen either if there are two design points spatially close to each other (so that the two corresponding columns of $\pmb \Sigma_\mathsf M$ are almost linearly dependent), or during the process of searching the parameter space for an estimate of $\pmb \theta$ for maximizing \eqref{eq:loglikelihood}. Moreover, both of the issues will be amplified by the dimensionality of the design space.
Existing solutions to the big $n$ problem heavily rely on approximation schemes, striving to approximate $\pmb \Sigma_\mathsf M$ by another matrix that can be inverted much faster. However, the reduction in computational time comes at the cost of inaccurate prediction of the responses and even invalid characterization their variances; see, e.g., \cite{Quinonero-CandelaRasmussen05}, \cite{SangHuang12}, and references therein.
By contrast, we propose in this paper a novel approach to the big $n$ problem. Instead of allowing any arbitrary covariance function and then seeking approximations of the associated covariance matrices, we will devise judiciously a specific but broad class of covariance functions having the following two properties: (i) $\pmb \Sigma_\mathsf M$ can be inverted analytically, and (ii) $ \pmb \Sigma_\mathsf M^{-1}$ is sparse.
These two properties make the computation of $[\pmb \Sigma_\mathsf M + \pmb \Sigma_\varepsilon]^{-1}$ substantially easier. To see this, notice that by the Woodbury matrix identity \cite[\S 0.7.4]{HornJohnson12},
\begin{equation}\label{eq:woodbury}
[\pmb \Sigma_\mathsf M + \pmb \Sigma_\varepsilon]^{-1} = \pmb \Sigma_\mathsf M^{-1} - \pmb \Sigma_\mathsf M^{-1}[ \pmb \Sigma_\mathsf M^{-1} + \pmb \Sigma_\varepsilon^{-1}]^{-1} \pmb \Sigma_\mathsf M^{-1}.
\end{equation}
Since $\pmb \Sigma_\mathsf M^{-1}$ has a known analytical expression and $\pmb \Sigma_\mathsf M^{-1} + \pmb \Sigma_\varepsilon^{-1}$ is sparse, $[\pmb \Sigma_\mathsf M^{-1} + \pmb \Sigma_\varepsilon^{-1}]^{-1}$ can be computed in $\mathcal O(n^2)$ time by leveraging a particular sparse structure that will become clear in \S\ref{sec:MCF}. The matrix multiplications in \eqref{eq:woodbury} require $\mathcal O(n^2)$ time also due to the sparsity of $\pmb \Sigma_\mathsf M$, as opposed to $\mathcal O(n^3)$ for multiplications of dense matrices. Therefore, computing \eqref{eq:woodbury} requires $\mathcal O(n^2)$ time, reducing one order of magnitude without resorting to any matrix approximation at all. Further, if the simulation errors are negligible, i.e., $\pmb \Sigma_\varepsilon\approx\bm 0$, then $[\pmb \Sigma_\mathsf M + \pmb \Sigma_\varepsilon]^{-1}\approx\pmb \Sigma_\mathsf M^{-1}$, which can be inverted analytically, then numerical inversion would become unnecessary. This implies that the computation of the SK predictor \eqref{eq:BLUP}, which is reduced to multiplications of vectors and sparse matrices, can be completed in $\mathcal O(n)$ time. The same goes for the computation of the optimal MSE \eqref{eq:MSE}.
Two central questions follow immediately. What structure needs to be imposed on the covariance function $k(\bm x,\bm y)$ so that the covariance matrix $\pmb \Sigma_\mathsf M$ has the two desirable properties? How broad is this specific class of covariance functions? This paper provides comprehensive answers.
\section{Markovian Covariance Functions} \label{sec:MCF}
In order to motivate the structure that we impose on the covariance function, we first introduce Gaussian Markov random fields (GMRFs) briefly and refer to \cite{RueHeld05} for a comprehensive treatment on the subject. Consider a graph consisting of $n$ nodes, each of which is labeled with $\bm x_i$ and has a random value $\mathsf M(\bm x_i)$, $i=1,\ldots,n$. Let $\bm X$ denote all the nodes and $\mathscr{N}(\bm x_i)$ denote the neighbors of $\bm x_i$, for each $i=1,\ldots,n$. Suppose that the joint distribution of $(\mathsf M(\bm x_1),\ldots,\mathsf M(\bm x_n))$ is multivariate normal. Then, $(\mathsf M(\bm x_1),\ldots,\mathsf M(\bm x_n))$ is called a GMRF if it has the Markovian structure (i.e., conditional independence structure) as follows. Given $\{\mathsf M(\bm x):\bm x\in\mathscr{N}(\bm x_i)\}$, the values of the neighbors of node $\bm x_i$, $\mathsf M(\bm x_i)$ is conditionally independent of the values of its non-neighbors, $\{\mathsf M(\bm x):\bm x\in E \setminus \mathscr{N}(\bm x_i)\}$. A critical property of GMRFs is that entry $(i,j)$ of the precision matrix $\pmb \Sigma_\mathsf M^{-1}$ is nonzero if, and only if, $\bm x_i$ and $\bm x_j$ are neighbors. Hence, $\pmb \Sigma_\mathsf M^{-1}$ is sparse if each node has a small neighborhood in the graph.
The fundamental cause for the sparsity of $\pmb \Sigma_\mathsf M^{-1}$ in GMRFs is obviously the Markovian structure. This inspires us to consider Gaussian processes that are Markovian. In particular, we consider three 1-dimensional examples -- Brownian motion, Brownian bridge, and the Ornstein-Uhlenbeck (O-U) process -- and calculate their associated precision matrices, respectively.
\begin{example}[Brownian Motion]\label{example:BM}
The covariance function of the standard 1-dimensional Brownian motion is $k_{\mathrm{BM}}(s,t)=\min(x,y)$, $x,y\geq 0$. Suppose that the design points $\{x_1,\ldots,x_n\}$ are equally spaced, i.e., $x_i=ih$ for some $h>0$. Then, it can be shown that $\pmb \Sigma_\mathsf M^{-1}$ is a tridiagonal matrix:
\[\pmb \Sigma_\mathsf M^{-1} = \frac{1}{h}
\begin{pmatrix}
2 & -1 & & & \\
-1 & 2 & -1 & & \\
\cdots & & \cdots & & \cdots\\
& & -1 & 2 & -1 \\
& & & -1 & 1
\end{pmatrix}.
\]
\end{example}
\begin{example}[Brownian Bridge]\label{example:BB}
The covariance function of the Brownian bridge defined on $[0,1]$ is $k_{\mathrm{BB}}(x,y) = \min(x,y)-xy$, $x,y\in[0,1]$. Suppose that the design points are $x_i=\frac{i}{n+1}$, $i=1,\ldots,n$. Then, it can be shown that $\pmb \Sigma_\mathsf M^{-1}$ is a tridiagonal matrix:
\[\pmb \Sigma_\mathsf M^{-1} = (n+1)
\begin{pmatrix}
2 & -1 & & & \\
-1 & 2 & -1 & & \\
\cdots & & \cdots & & \cdots\\
& & -1 & 2 & -1 \\
& & & -1 & 2
\end{pmatrix}.
\]
\end{example}
\begin{example}[O-U Process] \label{example:OU}
The O-U process is defined via the stochastic differential equation
\[\mathop{}\!\mathrm{d} X(t) = (\mu-\theta X(t))\,\mathop{}\!\mathrm{d} t + \sigma\,\mathop{}\!\mathrm{d} B(t), \quad t\geq 0\]
where $\mu$, $\theta>0$, and $\sigma>0$ are parameters, and $B(t)$ is the standard 1-dimensional Brownian motion. Then, the covariance function under the stationary distribution is $k_{\mathrm{OU}}(x,y) =\frac{\sigma^2}{2\theta} e^{-\theta|x-y|}$, $x,y\geq 0$. Using the same design points as Example \ref{example:BM}, it can be shown that $\pmb \Sigma_\mathsf M^{-1}$ is a tridiagonal matrix:\footnote{The discovery of the precision matrices associated with the O-U process being tridiagonal was initially made through several numerical trials. Together with Examples \ref{example:BM} and \ref{example:BB}, the tridiagonal pattern was already enough to motivate us to consider the functional form \eqref{eq:kernel}. The analytical expression of the precision matrix in Example \ref{example:OU} was calculated as a corollary of Theorem \ref{theo:tridiag} after we proved it.}
\[\pmb \Sigma_\mathsf M^{-1} = \frac{\theta }{\sigma^2 \sinh(\theta h)}
\begin{pmatrix}
\exp(\theta h) & -1 & \phantom{\exp(\theta h)} & & \\
-1 & 2\cosh(\theta h) & -1 & & \\
\cdots & & \cdots & & \cdots \\
& & -1 & 2\cosh(\theta h) & -1 \\
& & & -1 & \exp(\theta h)
\end{pmatrix}.
\]
\end{example}
Now that all the three examples have tridiagonal precision matrices, we naturally try to find the common feature in their covariance functions.
\subsection{Symmetric Tridiagonal Structure}
The key observation here is that the covariance functions in Examples \ref{example:BM}--\ref{example:OU} share the same form:
\begin{equation}\label{eq:kernel}
k(x,y)=p(x)q(y)\ind_{\{x\leq y\}} + p(y)q(x)\ind_{\{x>y\}},
\end{equation}
for some functions $p$ and $q$, where $\ind_{\{\cdot\}}$ is the indicator function. Specifically,
\begin{align*}
k_{\mathrm{BM}}(x,y) = & \min(x,y) = x \ind_{\{x\leq y\}} + y \ind_{\{x>y\}},\\
k_{\mathrm{BB}}(x,y) = & \min(x,y)-xy = x(1-y) \ind_{\{x\leq y\}} + y(1-x) \ind_{\{x>y\}}, \\
k_{\mathrm{OU}}(x,y) = & \frac{\sigma^2}{2\theta} e^{-\theta|x-y|} = \frac{\sigma^2}{2\theta}\left[e^{\theta x}e^{-\theta y}\ind_{\{x\leq y\}} + e^{\theta y}e^{-\theta x}\ind_{\{x> y\}}\right].
\end{align*}
Therefore, we conjecture that for Gaussian processes with a 1-dimensional domain, a covariance function of form \eqref{eq:kernel} would yield tridiagonal precision matrices. This turns out to be true in general under mild conditions and the design points do not need to be equally spaced. We present the result below as Theorem \ref{theo:tridiag}. The proof is done by induction on $n$ and is based on explicit calculations. We will use the Laplace expansion for the determinant of a square matrix. This is a classic result in linear algebra; see \citet[\S0.3.1]{HornJohnson12}.
\begin{lemma}[Laplace Expansion]\label{lemma:laplace}
Let $\bm K=(k_{i,j})$ be a $n\times n$ matrix and $M_{i,j}$ be its $(i,j)$ minor, i.e., the determinant of the submatrix formed by deleting the $i^{\mathrm{th}}$ row and $j^{\mathrm{th}}$ column of $\bm K$. Then,
\[|\bm K| = \sum_{\ell=1}^n (-1)^{i+\ell} k_{i,\ell} M_{i,\ell} = \sum_{\ell=1}^n (-1)^{\ell+j} k_{\ell,j} M_{\ell,j}.\]
\end{lemma}
To facilitate the presentation, we define several notations. Let $\mathcal X = \{x_1,\ldots,x_n\}$ denote a set of distinct points in $\mathbb R$, with $x_1<\cdots<x_n$. Fixing a function $k(x,y)$ of the form \eqref{eq:kernel}, let $\bm K=\bm K(\mathcal X,\mathcal X)$ be the $n\times n$ matrix whose entry $(i,j)$ is $k(x_i,x_j)$. For two subsets $\mathcal X',\mathcal X''\subseteq \mathcal X$, we use $\bm K(\mathcal X',\mathcal X'')$ to denote the submatrix of $\bm K$ formed by keeping the rows and columns that correspond to $\mathcal X'$ and $\mathcal X''$, respectively. Finally, let $p_i=p(x_i)$ and $q_i=q(x_i)$, $i=1,\ldots,n$.
\begin{theorem}\label{theo:tridiag}
Let $n\geq 3$. If $\bm K$ is nonsingular, then $\bm K^{-1}$ is a symmetric tridiagonal matrix.
\end{theorem}
\begin{proof} Since $k(x,y)=k(y,x)$, the symmetry of $\bm K$ is straightforward, and thus $\bm K^{-1}$ is symmetric.
To prove that $\bm K^{-1}$ is tridiagonal, i.e., $(\bm K^{-1})_{i,j}=0$ if $|j-i|\geq 2$, we use the relationship between the inverse and the minors of a square matrix \citep[\S0.8.2]{HornJohnson12},
\begin{equation}\label{eq:adjugate}
(\bm K^{-1})_{i,j} = \frac{1}{|\bm K|} (-1)^{i+j} M_{j,i},
\end{equation}
where $M_{j,i}$ is the $(j,i)$ minor of $\bm K$. Hence, it suffices to show that $M_{i,j}=0$ if $|j-i|\geq 2$, or equivalently,
\begin{equation}\label{eq:minor}
|\bm K(\mathcal X\setminus\{x_i\}, \mathcal X\setminus\{x_j\})|=0, \quad \mbox{if }j-i\geq 2,
\end{equation}
because of the symmetry of $\bm K$. We prove \eqref{eq:minor} by induction on $n$. For $n=3$,
\begin{equation*}
\bm K(\mathcal X,\mathcal X)=\begin{pmatrix}
p_1q_1 & p_1q_2 & p_1q_3\\
p_1q_2 & p_2q_2 & p_2q_3\\
p_1q_3 & p_2q_3 & p_3q_3\\
\end{pmatrix}.
\end{equation*}
Then,
\begin{equation*}
M_{1,3}=\begin{vmatrix}
p_1q_2 & p_2q_2 \\
p_1q_3 & p_2q_3 \\
\end{vmatrix}=0.
\end{equation*}
Now we suppose that \eqref{eq:minor} holds for any $n\leq N-1$. Then, for $n=N$ and $j\geq i+2$,
\begin{align}
M_{i,j} =& |\bm K(\mathcal X\setminus\{x_i\}, \mathcal X\setminus\{x_j\})| \nonumber\\
=& \sum_{\ell<j} (-1)^{(j-1)+\ell} k(x_j,x_\ell) |\bm K(\mathcal X\setminus\{x_i,x_j\}, \mathcal X\setminus\{x_j,x_\ell\})| \label{eq:minor_expan_1} \\
+ &\sum_{\ell>j} (-1)^{(j-1)+(\ell-1)} k(x_j,x_\ell) |\bm K(\mathcal X\setminus\{x_i,x_j\}, \mathcal X\setminus\{x_j,x_\ell\})| \nonumber
\end{align}
where the second equality follows from the Laplace expansion along the row of the submatrix $\bm K(\mathcal X\setminus\{x_i\}, \mathcal X\setminus\{x_j\})$ that corresponds to $x_j$. Here, $(j-1)$ and $(\ell-1)$ in the exponents reflect the necessary changes in the indices of the rows and columns of submatrix $\bm K(\mathcal X\setminus\{x_i\}, \mathcal X\setminus\{x_j\})$.
Let $\mathcal X'=\mathcal X\setminus\{x_j\}$. Then, the submatrix that appears in the Laplace expansion in \eqref{eq:minor_expan_1} can be rewritten as $\bm K(\mathcal X\setminus\{x_i,x_j\}, \mathcal X\setminus\{x_j,x_\ell\})=\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})$. Hence, its determinant is the $(i,\ell)$ minor of $\bm K(\mathcal X', \mathcal X')$ if $\ell<j$, or the $(i, \ell-1)$ minor if $\ell>j$. It follows that $\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})=0$ if $|\ell-i|\geq 2$ and $\ell<j$, or if $|\ell-1-i|\geq 2$ and $\ell>j$, by the induction assumption. Therefore, \eqref{eq:minor_expan_1} can be simplified to
\begin{equation}\label{eq:minor3}
M_{i,j} = \sum_{\ell=i-1}^{i+1} (-1)^{(j-1)+\ell} k(x_j,x_\ell) |\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})|,
\end{equation}
since $j\geq i+2$. Clearly, it suffices to show
\begin{equation}\label{eq:minor2}
\sum_{\ell=i-1}^{i+1} (-1)^{\ell} k(x_j,x_\ell) |\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})| = 0,
\end{equation}
in order to prove \eqref{eq:minor}. To that end, we further apply the Laplace expansion.
We now assume that $i\geq 4$. The cases $i=1,2,3$ can be proved in a similar fashion. For $\ell=i-1$, $\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})$ is
\setcounter{MaxMatrixCols}{20}
\begin{equation}\label{eq:bigmatrix}
\kbordermatrix{
& x_1 & x_2 & \cdots & x_{i-2} & x_{i} & x_{i+1} & \cdots & x_{j-1} & x_{j+1}&\cdots & x_N \\
x_1 & p_1q_1 & p_1q_2 & \cdots & p_1q_{i-2} & p_1q_{i} & p_1q_{i+1} & \cdots & p_1q_{j-1} & p_1q_{j+1}&\cdots & p_1q_N \\
x_2 & p_1q_2 & p_2q_2 & \cdots & p_2q_{i-2} & p_2q_{i} & p_2q_{i+1} & \cdots & p_2q_{j-1} & p_2q_{j+1} &\cdots & p_2q_N \\
\vdots& & & \vdots & & & & \vdots & & & \vdots &\\
x_{i-1} &p_1q_{i-1} & p_2q_{i-1} & \cdots & p_{i-2}q_{i-1} & p_{i-1}q_{i} & p_{i-1}q_{i+1} & \cdots & p_{i-1}q_{j-1} & p_{i-1}q_{j+1} &\cdots & p_{i-1}q_N \\
x_{i+1} &p_1q_{i+1} & p_2q_{i+1} & \cdots & p_{i-2}q_{i+1} & p_{i}q_{i+1} & p_{i+1}q_{i+1} & \cdots & p_{i+1}q_{j-1} & p_{i+1}q_{j+1} &\cdots & p_{i+1}q_N \\
\vdots & & & \vdots & & & & \vdots & & & \vdots &\\
x_{j-1} &p_1q_{j-1} & p_2q_{j-1} & \cdots & p_{i-2}q_{j-1} & p_{i}q_{j-1} & p_{i+1}q_{j-1} & \cdots & p_{j-1}q_{j-1} & p_{j-1}q_{j+1} &\cdots & p_{j-1}q_N \\
x_{j+1} &p_1q_{j+1} & p_2q_{j+1} & \cdots & p_{i-2}q_{j+1} & p_{i}q_{j+1} & p_{i+1}q_{j+1} & \cdots & p_{j-1}q_{j+1} & p_{j+1}q_{j+1} &\cdots & p_{j+1}q_N \\
\vdots & & & \vdots & & & & \vdots & & & \vdots &\\
x_N & p_1q_N & p_2q_N & \cdots & p_{i-2}q_N & p_{i}q_N & p_{i+1}q_N & \cdots & p_{j-1}q_N & p_{j+1}q_N &\cdots & p_Nq_N
}.
\end{equation}
With $i\geq 4$, the transpose of the submatrix of $\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_{i-1}\})$ formed by deleting the first row and keeping the first two columns in \eqref{eq:bigmatrix} is
\[
\bm K(\mathcal X'\setminus\{x_1,x_i\}, \{x_1,x_2\})^\intercal =
\kbordermatrix{
& x_2 & \cdots & x_{i-1} & x_{i+1} & \cdots & x_{j-1} & x_{j+1} &\cdots & x_N \\
x_1 & p_1q_2 & \cdots & p_1q_{i-1} & p_1q_{i+1} & \cdots & p_1q_{j-1} & p_1q_{j+1} & \cdots & p_1q_N\\
x_2 & p_2q_2 & \cdots & p_2q_{i-1} & p_2q_{i+1} & \cdots & p_2q_{j-1} & p_2q_{j+1} & \cdots & p_2q_N
},
\]
whose rows are linear dependent, obviously. Hence, if we apply the Laplace expansion to \eqref{eq:bigmatrix} along the first row, then only the first two terms in the expansion are nonzero. This is because the minors in the other terms all involve two linearly dependent columns, thereby being zero. Hence,
\begin{equation}\label{eq:expansion2}
\begin{aligned}
& |\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_{i-1}\})| \\ =& p_1q_1 |\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_1\})| - p_1q_2|\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_2\})|
\end{aligned}
\end{equation}
We next consider two cases, $p_2\neq 0$ and $p_2=0$, separately.
\textbf{Case 1 ($p_2\neq 0$).} Notice that $\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_1\})$ and $\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_2\})$ differ by only their first columns, and that the first column of the latter is a multiple of that of the former. In particular,
\begin{equation}\label{eq:multiple}
|\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_2\})| = \frac{p_1}{p_2} |\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_1\})|,
\end{equation}
and thus \eqref{eq:expansion2} becomes, for $\ell=i-1$,
\begin{equation}\label{eq:expansion3}
\begin{aligned}
|\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})| = \left(p_1q_1 - \frac{p_1^2q_2}{p_2} \right) |\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_\ell,x_1\})|.
\end{aligned}
\end{equation}
One can check easily that \eqref{eq:expansion3} holds for $\ell=i,i+1$ as well. Then, the left-hand-side of \eqref{eq:minor2} becomes
\begin{align}
& \sum_{\ell=i-1}^{i+1} (-1)^{\ell} k(x_j,x_\ell) |\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})| \nonumber \\
=&
\left(p_1q_1 - \frac{p_1^2q_2}{p_2} \right) \sum_{\ell=i-1}^{i+1}(-1)^\ell k(x_j,x_\ell)
|\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_\ell,x_1\})|. \label{eq:sum}
\end{align}
Let $\mathcal X''=\mathcal X'\setminus\{x_1\}$. Then, for the summation in \eqref{eq:sum},
\[\sum_{\ell=i-1}^{i+1}(-1)^\ell k(x_j,x_\ell)
|\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_\ell,x_1\})| = \sum_{\ell=i-1}^{i+1}(-1)^\ell k(x_j,x_\ell)
|\bm K(\mathcal X''\setminus\{x_i\},\mathcal X''\setminus\{x_\ell\})|,\]
which equals the $(i,j)$ minor of $\bm K(\mathcal X'', \mathcal X'')$ multiplied by $(-1)^{j-1}$, following the argument leading to \eqref{eq:minor3}. But the $(i,j)$ minor of $\bm K(\mathcal X'', \mathcal X'')$ is 0 by the induction assumption, since $j\geq i+2$. Therefore, \eqref{eq:sum} equals 0, which proves \eqref{eq:minor2}.
\textbf{Case 2 ($p_2= 0$).} It is easy to see that $\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_1\})$ is singular, since its first column is all zeros. Hence, \eqref{eq:expansion2} becomes
\[
|\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_{i-1}\})| = - p_1q_2|\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_2\})|.
\]
Since the first row of $\bm K(\mathcal X'\setminus\{x_i,x_1\},\mathcal X'\setminus\{x_{i-1},x_2\})$ is now $(p_1q_2,0,\ldots,0)$, we apply the Laplace expansion to this row to obtain
\begin{equation}\label{eq:expansion4}
|\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})| = - p_1^2q_2^2|\bm K(\mathcal X'\setminus\{x_i,x_1,x_2\},\mathcal X'\setminus\{x_\ell,x_2,x_1\})|,
\end{equation}
for $\ell=i-1$. Likewise, we can show that \eqref{eq:expansion4} holds for $\ell=i,i+1$ as well. Then, the left-hand-side of \eqref{eq:minor2} becomes, letting $\mathcal X'''=\mathcal X\setminus\{x_1,x_2\}$,
\[\sum_{\ell=i-1}^{i+1} (-1)^{\ell} k(x_j,x_\ell) |\bm K(\mathcal X'\setminus\{x_i\}, \mathcal X'\setminus\{x_\ell\})|
= -p_1^2q_2^2\sum_{\ell=i-1}^{i+1} (-1)^{\ell} k(x_j,x_\ell) |\bm K(\mathcal X'''\setminus\{x_i\}, \mathcal X'''\setminus\{x_\ell\})|.
\]
Then, we can prove \eqref{eq:minor2} using the same argument as the last paragraph of Case 1.
\end{proof}
Provided that $\bm K$ is nonsingular, not only can we show that $\bm K^{-1}$ is symmetric and tridiagonal, but also we can calculate the nonzero entries of $\bm K^{-1}$ analytically. The fact that $\bm K^{-1}$ is analytically invertible makes $\bm K$ highly computationally tractable. Before presenting the analytical expressions of the nonzero entries of $\bm K^{-1}$, we first calculate the determinant of $\bm K$.
\begin{proposition}\label{prop:determinant}
For $n\geq 2$,
\begin{equation}\label{eq:G_det}
|\bm K(\mathcal X,\mathcal X)|=p_1q_n\prod_{i=2}^n(p_iq_{i-1}-p_{i-1}q_i).
\end{equation}
\end{proposition}
\begin{proof}
We prove \eqref{eq:G_det} by induction on $n$. The base case $n=2$ is straightforward:
\[|\bm K(\mathcal X,\mathcal X)| =
\begin{vmatrix}
p_1q_1 & p_1q_2 \\
p_1q_2 & p_2q_2
\end{vmatrix}
= p_1q_1p_2q_2- p_1^2q_2^2 = p_1q_2(p_2q_1-p_1q_2).
\]
Now we suppose that \eqref{eq:G_det} holds for any $n\leq N-1$. Then, for $n=N$, applying the Laplace expansion to the first row of $\bm K(\mathcal X,\mathcal X)$,
\begin{align}
|\bm K(\mathcal X,\mathcal X)|&=\sum_{\ell=1}^N(-1)^{1+\ell} p_1q_\ell |\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_\ell\})| \nonumber \\
&= p_1q_1|\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_1\})|-p_1q_2|\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_2\})|, \label{eq:expansion5}
\end{align}
where the second equality follows from \eqref{eq:minor}. From the induction assumption,
\begin{equation}\label{eq:G_det_induction}
|\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_1\})|=p_2q_N\prod_{i=3}^N(p_iq_{i-1}-p_{i-1}q_i).
\end{equation}
Notice that
\begin{equation*}
\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_1\})=
\kbordermatrix{
& x_2 & x_3 & \cdots & x_N \\
x_2 & p_2q_2 & p_2q_3 & \dots & p_2q_N\\
x_3 & p_2q_3 & p_3q_3 & \dots & p_3q_N\\
\vdots& & & \vdots & \\
x_N & p_2q_N & p_3q_N & \dots & p_Nq_N\\
},
\end{equation*}
and
\begin{equation*}
\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_2\})=
\kbordermatrix{
& x_1 & x_3 & \cdots & x_N \\
x_2 & p_1q_2 & p_2q_3 & \dots & p_2q_N\\
x_3 & p_1q_3 & p_3q_3 & \dots & p_3q_N\\
\vdots& & & \vdots & \\
x_N & p_1q_N & p_3q_N & \dots & p_Nq_N\\
}.
\end{equation*}
Clearly, the above two matrices differ by only their first columns, and the first column of one matrix is a multiple of the other. Hence, if $p_2\neq 0$, then $|\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_2\})| = \frac{p_1}{p_2}|\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_1\})|$. Thus, by \eqref{eq:expansion5} and \eqref{eq:G_det_induction},
\begin{align*}
|\bm K(\mathcal X,\mathcal X)|=&\left(p_1q_1-\frac{p_1^2q_2}{p_2}\right) |\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_1\})| \\
=& \left(p_1q_1-\frac{p_1^2q_2}{p_2}\right) p_2q_N\prod_{i=3}^N(p_iq_{i-1}-p_{i-1}q_i)
= p_1q_N \prod_{i=2}^N(p_iq_{i-1}-p_{i-1}q_i).
\end{align*}
On the other hand, if $p_2=0$, then by \eqref{eq:expansion5} and \eqref{eq:G_det_induction},
\begin{align*}
|\bm K(\mathcal X,\mathcal X)|= & -p_1q_2|\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_2\})|\\
=& -p_1q_2 \cdot p_1q_2|\bm K(\mathcal X\setminus\{x_1,x_2\},\mathcal X\setminus\{x_2,x_1\})|\\
=& -p_1^2q_2^2 p_3q_N\prod_{i=4}^N(p_iq_{i-1}-p_{i-1}q_i),
\end{align*}
where the second equality follows from the Laplace expansion along the first row of $\bm K(\mathcal X\setminus\{x_1\},\mathcal X\setminus\{x_2\})$, whereas the last equality from the induction assumption. At last, notice that with $p_2=0$,
\begin{align*}
p_1q_N\prod_{i=2}^N(p_iq_{i-1}-p_{i-1}q_i) =& p_1q_N (p_2q_1-p_1q_2)(p_3q_2-p_2q_3)\prod_{i=4}^N(p_iq_{i-1}-p_{i-1}q_i)\\
=& -p_1^2q_2^2 p_3q_N\prod_{i=4}^N(p_iq_{i-1}-p_{i-1}q_i).
\end{align*}
Therefore, (\ref{eq:G_det}) holds for $n=N$. \hfill$\Box$
\end{proof}
By using the Laplace expansion and mathematical induction in a similar fashion, we can also prove the following result but defer the proof to Appendix \ref{app:A}.
\begin{proposition}\label{prop:minor}
For $n\geq 2$ and $2\leq i\leq n$,
\[|\bm K(\mathcal X\setminus\{x_{i-1}\}, \mathcal X\setminus\{x_i\})| =
p_1q_n\prod_{j=2,j\neq i}^n (p_j q_{j-1}-p_{j-1}q_j).
\]
\end{proposition}
With Propositions \ref{prop:determinant} and \ref{prop:minor}, the nonzero entries of $\bm K^{-1}$ can be readily calculated.
\begin{theorem}\label{theo:inverse}
For $n\geq 3$, if $\bm K$ is nonsingular, then the nonzero entries of $\bm K^{-1}$ are given as follows,
\[(\bm K^{-1})_{i,i} =
\left\{
\begin{array}{ll}
\displaystyle\frac{p_2}{p_1(p_2q_1-p_1q_2)},& \quad \mbox{if }i=1,\\[2.5ex]
\displaystyle\frac{p_{i+1}q_{i-1}-p_{i-1}q_{i+1}}{(p_iq_{i-1}-p_{i-1}q_i)(p_{i+1}q_i-p_iq_{i+1})},& \quad \mbox{if }2\leq i\leq n-1,\\[2.5ex]
\displaystyle\frac{q_{n-1}}{q_n(p_nq_{n-1}-p_{n-1}q_n)},& \quad \mbox{if }i=n,
\end{array}
\right.
\]
and
\[(\bm K^{-1})_{i-1,i} = (\bm K^{-1})_{i,i-1} = \frac{-1}{p_iq_{i-1}-p_{i-1}q_i}, \quad i=2,\ldots,n. \]
\end{theorem}
\begin{proof}
It follows from the identity \eqref{eq:adjugate} that
\[
(\bm K^{-1})_{i,i} = \frac{1}{|\bm K|}|\bm K(\mathcal X\setminus\{x_i\},\mathcal X\setminus\{x_i\})|\quad\mbox{and}\quad (\bm K^{-1})_{i-1,i} = \frac{-1}{|\bm K|}|\bm K(\mathcal X\setminus\{x_{i-1}\},\mathcal X\setminus\{x_i\})|.
\]
The results can then be shown by a straightforward calculation using Propositions \ref{prop:determinant} and \ref{prop:minor}. \hfill$\Box$ \end{proof}
\begin{remark}
There are two significant implications of Theorems \ref{theo:tridiag} and \ref{theo:inverse}. First, $\bm K^{-1}$ can be computed in $\mathcal O(n)$ time, since it is tridiagonal, having only $3n-2$ nonzero entries. Second, the numerical stability regarding the computation of $\bm K^{-1}$ is improved substantially, since its nonzero entries have simple analytical expressions and numerical algorithms for matrix inversion are no longer needed.
\end{remark}
\subsection{Positive Definiteness}
Theorem \ref{theo:tridiag} characterizes the essential structure of the covariance function of Gaussian processes with a 1-dimensional domain that yields sparse precision matrices. However, in order that a function of the form \eqref{eq:kernel} is a covariance function, the matrix $\bm K(\mathcal X,\mathcal X)$ must be positive semidefinite for any $\mathcal X=\{x_1,\ldots,x_n\}$. We further require $\bm K(\mathcal X,\mathcal X)$ to be positive definite so that it is invertible. The following conditions on $p$ and $q$ that constitute the function \eqref{eq:kernel} turn out to be both sufficient and necessary for the positive definiteness of $\bm K(\mathcal X,\mathcal X)$, provided that $p$ and $q$ are continuous.
\begin{assumption}\label{assump:MCF}
Let $(L,U)$ be an open interval in $\mathbb R$, where $L$ and $U$ are allowed to be $-\infty$ and $\infty$, respectively. For all $x,y\in(L,U)$,
\begin{enumerate}[label=(\roman*)]
\item
$p(x)q(y)-p(y)q(x)<0$ if $x<y$, and
\item
$p(x)q(y)>0$.
\end{enumerate}
\end{assumption}
\begin{remark}
It is straightforward to check that the covariance functions in Examples \ref{example:BM}--\ref{example:OU} all satisfy Assumption \ref{assump:MCF}.
\end{remark}
\begin{theorem}\label{theo:PD}
Suppose that $p:(L,U)\mapsto\mathbb R$ and $q:(L,U)\mapsto\mathbb R$ are both continuous. Then, $\bm K(\mathcal X,\mathcal X)$ is positive definite for any $\mathcal X\subset (L,U)$ with $|\mathcal X|=n\geq 2$ if and only if $p$ and $q$ satisfy Assumption \ref{assump:MCF}.
\end{theorem}
\begin{proof}
We first prove the ``if'' part. Fix an arbitrary $\mathcal X=\{x_1,\ldots,x_n\}$ with $x_1<\cdots<x_n$. The symmetry of $\bm K(\mathcal X,\mathcal X)$ is obvious. Then, the first leading principal minor of of $\bm K(\mathcal X,\mathcal X)$ is $p_1q_1=p(x_1)q(x_1)>0$. Moreover, for any $\ell=2,\ldots,n$, it follows from Proposition \ref{prop:determinant} that the $\ell^{\mathrm{th}}$ leading principal minor of $\bm K(\mathcal X,\mathcal X)$ is
\begin{align*}
|\bm K(\{x_1,\ldots,x_\ell\},\{x_1,\ldots,x_\ell\})| = & p_1q_\ell\prod_{i=2}^\ell (p_iq_{i-1}-p_{i-1}q_i) \\
=& p(x_1)q(x_\ell) \prod_{i=2}^\ell [p(x_i)q(x_{i-1})-p(x_{i-1})q(x_i)] > 0.
\end{align*}
Hence, $\bm K(\mathcal X,\mathcal X)$ is positive definite by Sylvester's criterion.
Now, we suppose that $\bm K(\mathcal X,\mathcal X)$ is positive definite for any $\mathcal X$, and prove the ``only if'' part by contradiction. Specifically, we show that if condition (i) or (ii) is false, then we can construct a matrix $\bm K(\mathcal X,\mathcal X)$ that violates Sylvester's criterion.
Assume that condition (i) is false, i.e., there exists $r<t$ for which $p(r)q(t)-p(t)q(r)\geq 0$. If $p(r)q(t)-p(t)q(r)= 0$, or if $p(r)q(t)-p(t)q(r)> 0$ and $p(r)q(t)\geq 0$, then
\[|\bm K(\{r,t\},\{r,t\})|=p(r)q(t)[p(t)q(r)-p(r)q(t)] \leq 0.\]
If $p(r)q(t)-p(t)q(r)> 0$ and $p(r)q(t)>0$, then we show that $h(s)\coloneqq p(r)q(s)-p(s)q(r)>0$ for any $s\in(r,t)$. To see this, notice that $h(r)=0$ and $h(t)>0$. It then follows from the continuity of $h(s)$ that $h(s)>0$, since $h(s)$ would has a zero $s_0\in(r,t)$ otherwise, which would imply that $|\bm K(\{r,s_0)\},\{r,s_0)\}|=0$. Likewise, we can show that $p(s)q(t)-p(t)q(s)>0$ for any $s\in(r,t)$. Hence,
\[|\bm K(\{r,s,t\},\{r,s,t\})|=p(r)q(t)[p(s)q(r)-p(r)q(s)][p(t)q(s)-p(s)q(t)]< 0.\]
Thus, we conclude that condition (i) must be true.
Assume that condition (ii) is false, i.e., there exist $r$ and $s$ such that $p(r)q(s)\leq 0$. If $r=s$, then for any $t>s$, the first leading principal minor of $\bm K(\{r,t\},\{r,t\})$ is $p(r)q(r)\leq 0$. If $r\neq s$, assuming $r<s$ without loss of generality, then $p(s)q(r)-p(r)q(s)>0$ since we have shown condition (i) must be true, and thus
\[|\bm K(\{r,s\},\{r,s\})| = p(r)q(s)[p(s)q(r)-p(r)q(s)] \leq 0,\]
which completes the proof.
\hfill$\Box$ \end{proof}
Through Theorems \ref{theo:tridiag}--\ref{theo:PD}, we have effectively characterized a class of computationally tractable covariance functions for Gaussian processes with a 1-dimensional domain. We call covariance functions of the form \eqref{eq:kernel} that satisfy Assumption \ref{assump:MCF} \emph{(1-dimensional) Markovian covariance functions} (MCFs).
\begin{remark}
MCFs establish an explicit connection between Gaussian processes and GMRFs. Let $\mathsf M(x)$ be a Gaussian process equipped with an MCF. Then, for any $\mathcal X=\{x_1,\ldots,x_n\}$, $\{\mathsf M(x):x\in\mathcal X\}$ forms a GMRF. Assuming that $x_1<\cdots<x_n$, the neighborhood structure of this GMRF is defined as follows: $x_i$ and $x_j$ are neighbors if and only if $|i-j|=1$, which is implied by the tridiagonal structure of the precision matrix $\pmb \Sigma_\mathsf M^{-1}$.
\end{remark}
\begin{corollary}\label{cor:change_var}
Let $\mathcal T:(L,U)\mapsto \mathbb R$ be a strictly increasing function and $\mathcal T^{-1}$ denotes is inverse. If $k(x,y)$ is an MCF for $x,y\in(L,U)$, then $k(\mathcal T(x),\mathcal T(y))$ is an MCF for $x,y\in(\mathcal T^{-1}(L), \mathcal T^{-1}(U))$.
\end{corollary}
\begin{proof}
Suppose that $k(x,y)=p(x)q(y)\ind_{\{x\leq y\}}+p(y)q(x)\ind_{\{x>y\}}$ with $p(x)$ and $q(x)$ satisfying Assumption \ref{assump:MCF}. Then,
\begin{align*}
k(h(x), h(y)) = & p(h(x))q(h(y))\ind_{\{h(x)\leq h(y)\}}+ p(h(y))q(h(x))\ind_{\{h(x)>h(y)\}}\\
= &\tilde p(x)\tilde q(y)\ind_{\{x\leq y\}}+\tilde p(y)\tilde q(x)\ind_{\{x>y\}}
\end{align*}
where $\tilde p(x)=p(h(x))$ and $\tilde q(x)=q(h(x))$. Here, the second equality follows from the strict increasing monotonicity of $h$. Moreover, it is easy to see that $\tilde p(x)$ and $\tilde q(x)$ satisfy Assumption \ref{assump:MCF}. \hfill$\Box$ \end{proof}
We will provide in \S\ref{sec:SL} a convenient approach to constructing MCFs based on ordinary differential equations (ODEs), provided that the ODE involved is analytically tractable. Corollary \ref{cor:change_var} provides an additional tool to construct new MCFs by modifying known ones.
\subsection{Multidimensional Extension}\label{sec:multidim}
So far, we have been focusing on Gaussian processes with a 1-dimensional domain. Unfortunately, there is no multidimensional analog to the S-L theory that we can take advantage of. We circumvent this difficulty by defining a $D$-dimensional MCF in the following ``composite'' manner: $k(\bm x,\bm y) = \prod_{i=1}^D k_i(x^{(i)},y^{(i)})$, where $\bm x=(x^{(1)},\ldots,x^{(D)})$, $\bm y=(y^{(1)},\ldots,y^{(D)})$, and $k_i(\cdot,\cdot)$ is a 1-dimensional MCF defined along the $i^{\mathrm{th}}$ dimension, $i=1,\ldots,D$. We remark that these 1-dimensional MCFs do not need to be the same and can be chosen to capture different correlation behaviors in each dimension.
The composite structure preserves the sparsity of the precision matrix, but it comes at the cost of restriction in selecting the design points $\mathcal X=\{\bm x_1,\ldots,\bm x_n\} $. In particular, we assume that $\mathcal X$ forms a regular lattice, that is, it can be expressed as a Cartesian product. But the coordinates along each dimensional do not need to be equally spaced.
\begin{assumption}\label{assump:lattice}
$\mathcal X= \bigtimes_{i=1}^D \{x^{(i)}_1,x^{(i)}_2,\ldots,x^{(i)}_{n_i}\}$ and $n=\prod_{i=1}^Dn_i$, where $n_i$ is the number of points along the $i^{\mathrm{th}}$ dimension and $x^{(i)}_1<x^{(i)}_2<\ldots<x^{(i)}_{n_i}$, $i=1,\ldots,D$.
\end{assumption}
It follows that the covariance matrix associated with $k(\cdot,\cdot)$, the $D$-dimensional MCF, can be written as $\bm K = \bigotimes_{i=1}^D \bm K_i$, where $\bm K_i$ is the covariance matrix corresponding to $k_i(\cdot,\cdot)$ and $\{x^{(i)}_1,\ldots,x^{(i)}_{n_i}\}$, and $\bigotimes$ denotes the tensor product of matrices. We refer to \citet[Chapter 13]{Laub05} for introduction of basic properties of tensor product. Then, the precision matrix can also be written as a tensor product: $\bm K^{-1} =\bigotimes_{i=1}^D \bm K_i^{-1} $. Hence, $\bm K^{-1}$ is also a sparse matrix since each $\bm K_i^{-1}$ is a tridiagonal matrix. The reduction in computational complexity suggested by \eqref{eq:woodbury} remains valid.
\section{Green's Function}\label{sec:SL}
The conditions in Assumption \ref{assump:MCF} can be trivially met by choosing a positive, strictly increasing function $p(x)$ and setting $q(x)\equiv 1$. The covariance function of a Brownian motion in Example \ref{example:BM} is indeed the case. However, this would mean that $k(x,y)=p(\min(x,y))$ is independent of $x$ for any $x>y$, which is not a reasonable feature in general. Despite the formal simplicity of the conditions in Assumption \ref{assump:MCF}, it is not immediately clear how to construct a wide spectrum of nontrivial functions $p(x)$ and $q(x)$ in a convenient way. We develop in this section a flexible, principled approach to constructing 1-dimensional MCFs. The key is to recognize that the function form \eqref{eq:kernel} resembles the Green's function of a Sturm-Liouville (S-L) differential equation. Since all second-order linear ODEs can be recast in the form of an S-L equation, the number of Green's functions that can be calculated analytically is potentially large; see \citet[Chapter 2.1]{ODEHandbook}.
The relation between Green's functions and covariances was also identified in \cite{DolphWoodbury52}. There are three critical differences between their work and ours. First, they work on higher-order Markov processes \cite[Appendix B]{RasmussenWilliams06} whereas we focus on the Markovian processes in the conventional sense, which is of order one. Second, this kind of generality instead restricts their analysis to the setting where the boundary condition of the S-L equation involved is imposed at infinity; further, their result which is similar to ours (Theorem \ref{theo:Green_MCF}) holds only for the case that the S-L equation has constant coefficients, which corresponds to the stationary O-U process. By contrast, in our analysis the boundary condition can be defined either on a finite interval or at infinity, and the coefficients of the S-L equation can be variable. Third, as a result of the last difference, the covariance functions constructed in their work are stationary, whereas our approach permits nonstationary covariance functions. In particular, we will construct an MCF that is nonstationary and even more computationally tractable than $k_{\mathrm{OU}}$, which is a stationary MCF; see the discussion in \S\ref{sec:MLE}. However, we do not discuss the nonstationarity from a modeling perspective in the present paper but refer interested readers to \cite{Sampson10}.
\subsection{Sturm-Liouville Equation}
Consider the following S-L equation defined on a finite interval $[L, U]$,
\begin{equation}\label{eq:SL}
\mathscr{L}f(x)\coloneqq \frac{1}{w(x)}\left[\frac{\mathop{}\!\mathrm{d} }{\mathop{}\!\mathrm{d} x}\left(-u(x)\frac{\mathop{}\!\mathrm{d} f(x)}{\mathop{}\!\mathrm{d} x}\right) + v(x)f(x)\right] = 0,
\end{equation}
with the boundary condition (BC)
\begin{equation}\label{eq:boundarycondition}
\left\{
\begin{aligned}
&\alpha_Lf(L)+\beta_Lf'(L)=0, \\
&\alpha_Uf(U)+\beta_Uf'(U)=0,
\end{aligned}
\right.
\end{equation}
where for some functions $\{u(x), v(x), w(x)\}$ and some constants $\{\alpha_L,\beta_L,\alpha_U,\beta_U\}$. We will consider three common BCs as follows.
\begin{itemize}
\item
Dirichlet BC: $\alpha_L=\alpha_U=1$ and $\beta_L=\beta_U=0$, i.e., $f(L)=f(U)=0$;
\item
Cauchy BC: $\alpha_L=\beta_U=1$ and $\alpha_U=\beta_L=0$, i.e., $f(L)=f'(U)=0$;
\item
Neumann BC: $\beta_L=\beta_U=1$ and $\alpha_L=\alpha_U=0$, i.e., $f'(L)=f'(U)=0$.
\end{itemize}
The Green's function $g(x,y)$ of the above S-L equation is the solution to $\mathscr{L}g(x,y)=\delta(x-y)$ with the same BC, where $\delta(\cdot)$ is the Dirac delta function. It is a classical result in S-L theory that the Green's function has the following form
\begin{equation}\label{eq:Green_MCF}
g(x,y) = Cf_1(x)f_2(y)\ind_{\{x\leq y\}} + Cf_1(y)f_2(x)\ind_{\{x>y\}},
\end{equation}
where $f_1$ and $f_2$ satisfy
\begin{equation}\label{eq:Green_components}
\left\{
\begin{array}{l}
\mathscr{L}f_1(x)=0,\mbox{ }x\in[L,U] \\
\alpha_Lf(L)+\beta_Lf'(L)=0
\end{array}
\right.
\quad
\mbox{ and }
\quad
\left\{
\begin{array}{l}
\mathscr{L}f_2(x)=0,\mbox{ }x\in[L,U] \\
\alpha_Uf(U)+\beta_Uf'(U)=0
\end{array}
\right.
.
\end{equation}
Here, the constant $C$ is determined in such a way that
\[\lim_{\epsilon\downarrow0} \bigg[\frac{\mathop{}\!\mathrm{d} g(x,y)}{\mathop{}\!\mathrm{d} x}\Big|_{x=y+\epsilon} - \frac{\mathop{}\!\mathrm{d} g(x,y)}{\mathop{}\!\mathrm{d} x}\Big|_{x=y-\epsilon}\bigg] = \frac{-1}{u(y)};\]
see \citet[Chapter 5.4]{Teschl12}. Consequently, the Green's function $g(x,y)$ has exactly the form \eqref{eq:kernel}.
Clearly, not every S-L equation has a Green's function that satisfies Assumption \ref{assump:MCF}. Proper conditions need to be imposed on the functions $\{u(x), v(x), w(x)\}$ in the S-L equation \eqref{eq:SL} as well as on the BC \eqref{eq:boundarycondition}, in order that the Green's function be positive definite.
\subsection{A General Result}
We show now that the Green's functions associated with a wide class of S-L equations are indeed MCFs. We assume that the S-L equation \eqref{eq:SL} is \emph{regular}, i.e., $u(x)$ is continuously differentiable, $v(x)$ and $w(x)$ are continuous, and $u(x)>0$ and $w(x)>0$ for $x\in[L,U]$; see \citet[Chapter 5.3]{Teschl12}. This is because the Green's function of a regular S-L equation enjoys an eigen-decomposition, which implies that the Green's function is positive semidefinite if the eigenvalues of the differential operator $\mathscr{L}$ are all positive.
\begin{theorem}\label{theo:Green_MCF}
Suppose that the S-L equation \eqref{eq:SL} is regular with $v(x)>0$ for $x\in[L,U]$ and the Dirichlet BC. Then, its Green's function is an MCF.
\end{theorem}
\begin{proof}
Fix a set of distinct points $\mathcal X=\{x_1,\ldots,x_n\}\subset (L,U)$. Let $\bm G(\mathcal X,\mathcal X)$ denote the matrix whose entry $(i,j)$ is $g(x_i,x_j)$. Given the fact that the Green's function has the form \eqref{eq:Green_MCF}, by Theorems \ref{theo:tridiag} and \ref{theo:PD} it suffices to show that $\bm G(\mathcal X,\mathcal X)$ is positive definite.
Consider the eigenvalue problem associated with the S-L equation \eqref{eq:SL} (i.e., the so-called S-L problem): $\mathscr{L}\phi(x) = \lambda \phi(x)$, with $\phi(x)$ satisfying the BC \eqref{eq:boundarycondition}. It is well known in ODE theory that if the S-L equation is regular and satisfies the BC \eqref{eq:boundarycondition}, then the S-L problem has a countable number of eigenvalues $\{\lambda_\ell:\ell= 1,2,\ldots\}$, and the normalized eigenfunctions $\{\phi_\ell(x):\ell=1,2,\ldots\}$ can be chosen real-valued and form an orthonormal basis in the space of functions
\[\mathsf{L}^2([L,U], w(x), \mathop{}\!\mathrm{d} x)\coloneqq \left\{h:[L,U]\mapsto\mathbb R\Big|\int_L^U h^2(x)w(x)\mathop{}\!\mathrm{d} x<\infty \right\},\]
endowed with the inner product $\langle h_1, h_2 \rangle\coloneqq \int_L^U h_1(x)h_2(x)w(x)\mathop{}\!\mathrm{d} x$. In particular, $\langle \phi_i,\phi_j\rangle$ equals 1 if $i=j$ and 0 otherwise. Moreover, the eigenvalues are all positive if $v(x)$ is positive on $[L, U]$ and the BC \eqref{eq:boundarycondition} is of the Dirichlet type. We refer to \citet[\S0.2.5]{ODEHandbook} for a discussion on the S-L problem and its properties.
Then, the Green's function can be expressed as the following eigen-decomposition
\[g(x,y) = \sum_{\ell=1}^\infty \lambda_\ell^{-1}\phi_\ell(x)\phi_\ell(y),\]
since $\lambda_\ell >0$ for each $\ell=1,2,\ldots$; see \citet[Chapter 10.1]{ArfkenWeberHarris12} for a proof. Notice that
\[\int_L^U\int_L^Uh(x)h(y)g(x,y)w(x)w(y)\mathop{}\!\mathrm{d} x\mathop{}\!\mathrm{d} y = \sum_{\ell=1}^\infty \lambda_\ell^{-1}\langle h,\phi_\ell\rangle^2 \geq 0,\]
for any $h\in \mathsf{L}^2([L,U], w(x), \mathop{}\!\mathrm{d} x)$. Hence, $g(x,y)$ is positive semidefinite, which implies that $\bm G(\mathcal X,\mathcal X)$ is positive semidefinite; see, e.g., \citet[Chapter 4.1]{RasmussenWilliams06}.
What remains is to prove $|\bm G(\mathcal X,\mathcal X)|\neq 0$. It follows from Sturm's comparison theorem \citep[Theorem 5.20]{Teschl12} that if $v(x)>0$, then any function that satisfies $\mathscr{L}f(x)=0$ has at most one zero in $[L, U]$. In particular, consider the functions $f_1$ and $f_2$ that constitute the Green's function in the expression \eqref{eq:Green_MCF}. Due to the Dirichlet BC, we know from \eqref{eq:Green_components} that $f_1(L)=f_2(U)=0$. Therefore, $f_1$ and $f_2$ have no other zeros in $(L,U)$, and thus
\begin{equation}\label{eq:PD_condition_1}
f_1(x)f_2(y)\neq 0,\quad \mbox{ for all $x,y\in(L,U)$}.
\end{equation}
Next, we show by contradiction that
\begin{equation}\label{eq:PD_condition_2}
f_1(x)f_2(y)-f_1(y)f_2(x)\neq 0,\quad \mbox{ for all $x,y\in(L,U)$ if $x>y$}.
\end{equation}
Assume that \eqref{eq:PD_condition_2} is false, i.e., there exist $s>t$ in $(L,U)$ such that $f_1(s)f_2(t)=f_1(t)f_2(s)$, or equivalently, $f_1(s)/f_2(s)=f_1(t)/f_2(t)$, since we have shown that $f_2(x)\neq 0$ for all $x\in(L,U)$.
Notice that for any $c\neq 0$, if we replace $f_1(x)$ by $cf_1(x)$ and adjust the constant $C$ to $C/c$ in the expression \eqref{eq:Green_components}, then we retain the functional form of an MCF. Hence, we can assume, without loss of generality, that $f_1$ is properly scaled so that $f_1(s)/f_2(s)=f_1(t)/f_2(t)=1$. This implies that $f_1(s)-f_2(s) = f_1(t)-f_2(t)=0$, i.e., $f_1(x)-f_2(x)$ has two zeros in $(L, U)$. However, since $f_1(x)-f_2(x)$ is a solution to $\mathscr{L}f(x)=0$, this contradicts the implication of Sturm's comparison theorem, namely, any solution to $\mathscr{L}f(x)=0$ has at most one zero in $[L, U]$ if $v(x)>0$ for $x\in[L,U]$.
At last, it follows from \eqref{eq:PD_condition_1}, \eqref{eq:PD_condition_2}, and Proposition \ref{prop:determinant} that $|\bm G(\mathcal X,\mathcal X)|\neq 0$. \hfill$\Box$
\end{proof}
\begin{remark}
It can be seen from the proof of Theorem \ref{theo:Green_MCF} that for a regular S-L equation, it suffices to assume $v(x)>0$ in order that its Green's function be a covariance function on the finite interval $[L,U]$. But the covariance matrix may be singular for BCs that are not of the Dirichlet type. Nevertheless, this does not mean that the Green's function cannot be a positive definite covariance function when $v(x)$ is not a positive function, or when other types of BCs are imposed. In general, if the Green's function of an S-L equation can be solved analytically in the form of \eqref{eq:Green_MCF}, then we can check whether it is an MCF by simply verifying verify Assumption \ref{assump:MCF}.
\end{remark}
\subsection{Some Examples}
We now use the Green's-function approach to construct several MCFs which turn out to have excellent performance when applied in SK for predicting response surfaces in the numerical experiments in \S\ref{sec:numerical}.
We assume that the domain of the S-L equation is $[L,U]=[0,1]$; otherwise, we use the change-of-variable technique to make it so. Consider the following ODE with constant coefficients
\begin{equation}\label{eq:Ising}
- f''(x)+\nu f(x)=0,
\end{equation}
by setting $u(x)\equiv 1$, $v(x)\equiv \nu$, and $w(x)\equiv 1$ in \eqref{eq:SL}. The Green's function has a different form, depending on the sign of $\nu$ and the BC. Theorem \ref{theo:Green_MCF} stipulates that the Green's function is an MCF if $\nu>0$ and the Dirichlet BC is imposed. For the other cases, we can easily verify that Assumption \ref{assump:MCF} is indeed satisfied if $\nu$ is above a (negative) threshold. Since it is a routine exercise to solve \eqref{eq:Ising} for the Green's function with a BC of the Dirichlet, Cauchy, or Neumann type, we omit the details and only present the results.
\begin{theorem}\label{theo:MCF_examples}
The Green's function of equation (\ref{eq:Ising}) is $g(x,y)=\eta^2 [p(x)q(y)\ind_{\{x\leq y\}} + p(y)q(x)\ind_{\{x>y\}}]$, where $\eta^2$, $p(x)$, and $q(x)$ are given in Table \ref{tab:Green}. Moreover, $g(x,y)$ is an MCF if any of the following three conditions is satisfied: (i) the Dirichlet BC is imposed and $\nu>-\pi^2$; (ii) the Cauchy BC is imposed and $\nu>-\frac{\pi^2}{4}$; (iii) the Neumann condition is imposed and $\nu>0$.
\end{theorem}
\begin{table}[t]
\small
\begin{center}
\caption{The Green's Function of Equation \eqref{eq:Ising}.} \label{tab:Green}
\begin{tabular}{ccccc}
\toprule
Boundary & $\nu$ & $\eta^2$ & $p(x)$ & $q(x)$ \\
\midrule
Dirichlet & $\displaystyle \nu\in(-\pi^2,0)$ & $\displaystyle \frac{1}{\gamma\sin(\gamma)}$ & $\sin(\gamma x)$ & $\sin(\gamma(1-x))$ \\
\arrayrulecolor{white}
\midrule
Dirichlet & $\nu=0$ & $1$ & $x$ & $1-x$ \\
\midrule
Dirichlet & $\displaystyle \nu>0$ & $\displaystyle \frac{1}{\gamma\sinh(\gamma)}$ & $\sinh(\gamma x)$ & $\sinh(\gamma(1-x))$ \\
\arrayrulecolor{black}
\midrule
Cauchy & $\displaystyle \nu\in(-\frac{\pi^2}{4},0)$ & $\displaystyle \frac{1}{\gamma\cos(\gamma)}$ & $\sin(\gamma x)$ & $\cos(\gamma(1-x))$ \\
\arrayrulecolor{white}
\midrule
Cauchy & $\nu=0$ & $1$ & $x$ & $1$ \\
\midrule
Cauchy & $\displaystyle \nu>0$ & $\displaystyle \frac{1}{\gamma\cosh(\gamma)}$ & $\sinh(\gamma x)$ & $\cosh(\gamma(1-x))$ \\
\arrayrulecolor{black}
\midrule
Neumann & $\displaystyle \nu>0$ & $\displaystyle \frac{1}{\gamma\sinh(\gamma)}$ & $\cosh(\gamma x)$ & $\cosh(\gamma(1-x))$ \\
\bottomrule
\end{tabular}
\end{center}
\small{\textit{Note.} {$\gamma=\sqrt{|\nu|}$.}}
\end{table}
It turns out that if the set of points $\mathcal X=\{x_1,\ldots,x_n\}$ are equally spaced, the precision matrix associated with the MCFs in Theorem \ref{theo:MCF_examples} has an even simpler structure than being symmetric tridiagonal. The proof relies on direct calculations suggested by Theorem \ref{theo:inverse} and is deferred to Appendix \ref{app:B}.
\begin{corollary}\label{cor:Dirichlet}
Let $g(x,y)=\eta^2[p(x)q(y)\ind_{\{x\leq y\}} + p(y)q(x)\ind_{\{x>y\}}]$, where $\eta^2>0$ is a free parameter, and $p(x)$ and $q(x)$ are the functions in Table \ref{tab:Green}. Suppose that $\mathcal X=\{x_1,\ldots,x_n\}\subset (0,1)$, where $x_i=x_1+(i-1)h$ with $h=\frac{x_n-x_1}{n-1}$, $i=1,\ldots,n$. Then, $\bm G^{-1}(\mathcal X,\mathcal X)$ is a symmetric, tridiagonal matrix:
\begin{equation}\label{eq:simple_precision}
\bm G^{-1}(\mathcal X,\mathcal X) =
\eta^{-2}a
\begin{pmatrix}
b & -1 & & & \\
-1 & c & -1 & & \\
\cdots & & \cdots & & \cdots\\
& & -1 & c & -1 \\
& & & -1 & d
\end{pmatrix},
\end{equation}
where the parameters $(a,b,c,d)$ are given in Table \ref{tab:inverse}.
\end{corollary}
\begin{table}[t]
\small
\begin{center}
\caption{Parameters in the Inverse Matrix \eqref{eq:simple_precision}.} \label{tab:inverse}
\begin{tabular}{cccccc}
\toprule
Boundary & $\nu$ & $a$ & $b$ & $c$ & $d$\\
\midrule
Dirichlet & $\displaystyle \nu\in(-\pi^2,0)$ & $\displaystyle \frac{1}{\sin(\gamma) \sin(\gamma h)}$ & $\displaystyle \frac{\sin(\gamma (x_1+h))}{\sin(\gamma x_1)}$ & $2\cos(\gamma h)$ & $\displaystyle \frac{\sin(\gamma(1-x_n+h))}{\sin(\gamma(1-x_n))}$ \\
\arrayrulecolor{white}
\midrule
Dirichlet & $\nu=0$ & $\displaystyle \frac{1}{h}$ & $\displaystyle 1+\frac{h}{x_1}$ & $2$ & $\displaystyle 1+\frac{h}{1-x_n}$ \\
\midrule
Dirichlet & $\displaystyle \nu>0$ & $\displaystyle \frac{1}{\sinh(\gamma) \sinh(\gamma h)}$ & $\displaystyle \frac{\sinh(\gamma (x_1+h))}{\sinh(\gamma x_1)}$ & $2\cosh(\gamma h)$ & $\displaystyle \frac{\sinh(\gamma(1-x_n+h))}{\sinh(\gamma(1-x_n))}$ \\
\arrayrulecolor{black}
\midrule
Cauchy & $\displaystyle \nu\in(-\frac{\pi^2}{4},0)$ & $\displaystyle \frac{1}{\sin(\gamma) \sin(\gamma h)}$ & $\displaystyle \frac{\sin(\gamma (x_1+h))}{\sin(\gamma x_1)}$ & $2\cos(\gamma h)$ & $\displaystyle \frac{\cos(\gamma(1-x_n+h))}{\cos(\gamma(1-x_n))}$ \\
\arrayrulecolor{white}
\midrule
Cauchy & $\nu=0$ & $\displaystyle \frac{1}{h}$ & $\displaystyle 1+\frac{h}{x_1}$ & $2$ & $\displaystyle 1$ \\
\midrule
Cauchy & $\displaystyle \nu>0$ & $\displaystyle \frac{1}{\sinh(\gamma) \sinh(\gamma h)}$ & $\displaystyle \frac{\sinh(\gamma (x_1+h))}{\sinh(\gamma x_1)}$ & $2\cosh(\gamma h)$ & $\displaystyle \frac{\cosh(\gamma(1-x_n+h))}{\cosh(\gamma(1-x_n))}$ \\
\arrayrulecolor{black}
\midrule
Neumann & $\displaystyle \nu>0$ & $\displaystyle \frac{1}{\sinh(\gamma) \sinh(\gamma h)}$ & $\displaystyle \frac{\cosh(\gamma (x_1+h))}{\cosh(\gamma x_1)}$ & $2\cosh(\gamma h)$ & $\displaystyle \frac{\cosh(\gamma(1-x_n+h))}{\cosh(\gamma(1-x_n))}$ \\
\bottomrule
\end{tabular}
\end{center}
\small{\textit{Note.} {$\gamma=\sqrt{|\nu|}$.}}
\end{table}
Corollary \ref{cor:Dirichlet} has two important implications from the computational perspective. First, by choosing a set of equally spaced design points, the precision matrix associated with the MCFs in Theorem \ref{theo:MCF_examples} can be computed in $O(1)$ time since its nonzero entries can be expressed in terms of only 4 quantities, regardless of the size of the matrix. This is a further reduction in complexity compared to computing the precision matrix of a general MCF, which amounts to $O(n)$.
Second, the expression \eqref{eq:simple_precision} allows reparameterization of the MCFs in Theorem \ref{theo:MCF_examples}. Instead of estimating the parameters of an MCF, we can express the likelihood function in terms of the parameters in the precision matrix. Under mild conditions, the resulting MLE can be solved without any matrix inversion, thereby improving substantially the computational efficiency and numerical stability. We discuss this matter in details in \S\ref{sec:MLE}.
\begin{table}[t]
\small
\begin{center}
\caption{Computational Complexity.} \label{tab:complex}
\begin{tabular}{cccc}
\toprule
Covariance Function & $\pmb \Sigma_\mathsf M^{-1}$ & $[\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon]^{-1}$ & SK Predictor + MSE \\
\midrule
General & $\mathcal O(n^3)$ & $\mathcal O(n^3)$ & $\mathcal O(n^3)$\\
MCF & $\mathcal O(n)$ & $\mathcal O(n^2)$ & $\mathcal O(n^2)$, or $\mathcal O(n)$ if $\pmb \Sigma_\varepsilon=\bm 0$ \\
CF in Table \ref{tab:Green} under Condition & $\mathcal O(1)$ & $\mathcal O(n^2)$ & $\mathcal O(n^2)$, or $\mathcal O(n)$ if $\pmb \Sigma_\varepsilon=\bm 0$ \\
\bottomrule
\end{tabular}
\end{center}
\small{\textit{Note.} {Condition: design points are equally spaced. }}
\end{table}
In order to highlight the computational enhancement of MCFs relative to general covariance functions, we summarize the complexity for computing various quantities using different covariance functions in Table \ref{tab:complex}. First, for computing $\pmb \Sigma_\mathsf M^{-1}$, MCFs reduce the complexity from $\mathcal O(n^3)$ to $\mathcal O(n)$ because of the sparsity of the inverse matrix and the analytical expression of its nonzero entries; the Green's function in Table \ref{tab:Green} further reduce the complexity to $\mathcal O(1)$ by taking advantage of the experiment design. Second, it can be seen that the existence of the simulation errors increases the computational complexity dramatically and offsets largely the benefit of MCFs. Third, once $[\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon]^{-1}$ is computed, the bulk of the computation of the SK predictor \eqref{eq:BLUP} and its MSE \eqref{eq:MSE} is to multiply the inverse matrix by a vector, which takes $\mathcal O(n^2)$ in general but is reduced to $\mathcal O(n)$ by the sparsity induced by MCFs.
\begin{remark}
The fact that entry $(i-1,i)$ of $\bm G^{-1}$ is independent of $i$ deserves an interpretation. Using the notations in Theorem \ref{theo:inverse}, this means that $p_iq_{i-1}-p_{i-1}q_i$ is a constant, which turns out to be related to the so-called Wronskian determinant $W(x)$ associated with the S-L equation. In particular, for two linearly independent solutions $p(x)$ and $q(x)$ to equation \eqref{eq:SL}, the Wronskian is defined as
\[W(x)=
\begin{vmatrix}
p(x) & q(x) \\
p'(x) & q'(x)
\end{vmatrix}
=p(x)q'(x)-p'(x)q(x).
\]
On the other hand, if we fix $x_{i-1}$, then
\[\frac{p_iq_{i-1}-p_{i-1}q_i}{h} = \frac{p_i(q_{i-1}-q_i)-q_i(p_{i-1}-p_i)}{h} \to -p(x_{i-1})q'(x_{i-1})+q(x_{i-1})p'(x_{i-1})=-W(x_{i-1}),\]
as $h\downarrow 0$. Hence, $p_iq_{i-1}-p_{i-1}q_i$ can be viewed as a ``discretized'' Wronskian. It is known in the theory of S-L equations that $u(x)W(x)$ is a constant for $x\in[L,U]$. Since $\mu(x)\equiv \mu$ in equation \eqref{eq:Ising}, $W(x)$ is a constant. Nevertheless, we must emphasize that in general, a constant Wronskian does not imply that $p_iq_{i-1}-p_{i-1}q_i$ is independent of $i$.
\end{remark}
\subsection{Illustration}
A particularly important application of SK, besides response surface prediction, is to facilitate the exploration-exploitation trade-off during the random search for solving simulation optimization problems \citep{sun2014}. To that end, the uncertainty about the prediction, which is a result of the interplay between the extrinsic uncertainty imposed by SK to the unknown response surface and the intrinsic uncertainty from the simulation errors, should be characterized meaningfully.
Given the fact that the squared exponential covariance function $k_{\mathrm{SE}}(x,y) = \eta^2 e^{-\theta(x-y)^2}$ is a standard choice in SK literature, we now compare MCFs with $k_{\mathrm{SE}}$ in terms of the performance in uncertainty quantification in stochastic simulation. Specifically, we consider two distinct MCFs: (i) the exponential covariance function $k_{\mathrm{Exp}}(x,y) = \eta^2 e^{-\theta|x-y|}$, which is the essentially same as the covariance function of the OU process in Example \ref{example:OU}; (ii) the Green's function associated with the Dirichlet BC in Theorem \ref{theo:MCF_examples}
\begin{equation}\label{eq:kernel_Diri}
k_{\mathrm{Dir}}(x,y) \coloneqq \left\{
\begin{array}{ll}
\eta^2\left[\sin(\gamma x)\sin(\gamma(1-y))\ind_{\{x\leq y\}}+\sin(\gamma y)\sin(\gamma(1-x))\ind_{\{x> y\}}\right],&\mbox{if } \nu <0, \\[1ex]
\eta^2\left[x(1-y)\ind_{\{x\leq y\}}+y(1-x)\ind_{\{x> y\}}\right],&\mbox{if }\nu =0, \\[1ex]
\eta^2\left[\sinh(\gamma x)\sinh(\gamma(1-y))\ind_{\{x\leq y\}}+\sinh(\gamma y)\sinh(\gamma(1-x))\ind_{\{x> y\}}\right], &\mbox{if } \nu>0,
\end{array}
\right.
\end{equation}
for $x,y\in(0,1)$, where $\gamma=\sqrt{|\nu|}$.
We assume that a 1-dimensional continuous surface is observed with errors having variance $\sigma^2$. Given the observations, we first fit the SK metamodel equipped with each of the three covariance functions using MLE which is detailed in \S\ref{sec:MLE}, and then predict the surface using the SK predictor \eqref{eq:BLUP} with the parameter estimates. We also compute the standard deviation (S.D.) of the prediction, i.e., the square root of the prediction MSE \eqref{eq:MSE}, in order to measure the uncertainty about the predicted surface. We consider both $\sigma=0$ and $\sigma=0.1$. The results are shown in Figure \ref{fig:Uncertainty}.
\begin{figure}[t]
\begin{center}
\caption{Uncertainty Quantification of the SK Prediction.} \label{fig:Uncertainty}
$\begin{array}{cc}
\includegraphics[width=0.45\textwidth]{figures/Uncertainty/SE_new.pdf} &
\includegraphics[width=0.45\textwidth]{figures/Uncertainty/SE_noise_new.pdf} \\
\includegraphics[width=0.45\textwidth]{figures/Uncertainty/OU_new.pdf} &
\includegraphics[width=0.45\textwidth]{figures/Uncertainty/OU_noise_new.pdf} \\
\includegraphics[width=0.45\textwidth]{figures/Uncertainty/Dir_new.pdf} &
\includegraphics[width=0.45\textwidth]{figures/Uncertainty/Dir_noise_new.pdf}
\end{array}$
\end{center}
\small{\textit{Note.} True surface (solid line), data ($+$), prediction (dashed line), $\pm$ standard deviation (shaded area).}
\end{figure}
Overall, all the three covariance functions can deliver meaningful uncertainty quantification of the unknown surface. For each covariance function, the 1-S.D. confidence band can mostly cover the true surface, and it is inflated by the observation noise. Moreover, the confidence band is wider for regions with fewer observations (e.g., the interval $[-6,0]$) than regions with more (e.g., $[0,6]$), and it is particularly wide for extrapolation (e.g., $|x|\geq 8$). A main difference between $k_{\mathrm{SE}}$ and the two MCFs that is revealed in Figure \ref{fig:Uncertainty} is that both the predicted surface and the confidence band are smoother for the former. But the lack of smoothness in the predicted surface does not appear to cause significant issues as far as the prediction accuracy is concerned, which will be shown in the extensive numerical experiments in \S\ref{sec:numerical}.
\section{Parameter Estimation}\label{sec:MLE}
Let $\pmb \theta$ denote the parameters used to specify the covariance function and $\bm K(\pmb \theta)$ denote the covariance matrix. We now discuss the estimation of $\pmb \theta$ and $\pmb \beta$, the parameters that determine the trend of the response surface. We develop a highly efficient and numerically stable MLE scheme for a specific class of MCFs. We assume in this section that $\pmb \Sigma_\varepsilon$, the variances of the simulation outputs, is known. This is a standard treatment regarding SK in simulation literature. In practice, $\pmb \Sigma_\varepsilon$ is replaced by the sample variances $\widehat\pmb \Sigma_\varepsilon$.
\subsection{Numerically Stable MLE}\label{sec:stable_MLE}
Recall the log-likelihood function \eqref{eq:loglikelihood},
\begin{equation}\label{eq:log-likelihood2}
l(\pmb \beta,\pmb \theta) = -\frac{n}{2}\ln(2\pi) - \frac{1}{2}\ln|\bm K(\pmb \theta) + \pmb \Sigma_\varepsilon| -\frac{1}{2}(\bar\bm z-\bm F\pmb \beta)^\intercal [\bm K(\pmb \theta) + \pmb \Sigma_\varepsilon]^{-1} (\bar\bm z-\bm F\pmb \beta).
\end{equation}
The first order optimality conditions are derived using standard results of matrix calculus in \cite{AnkenmanNelsonStaum10},
\begin{equation}\label{eq:MLE_conditions1}
\left\{
\begin{aligned}
\bm 0 = & \frac{\partial l(\pmb \beta,\pmb \theta)}{\partial \pmb \beta} = \bm F^\intercal \bm V^{-1}(\pmb \theta)(\bar \bm z-\bm F\pmb \beta), \\
\bm 0 = & \frac{\partial l(\pmb \beta,\pmb \theta)}{\partial \pmb \theta} = -\frac{1}{2}\mathrm{trace}\left[\bm V^{-1}(\pmb \theta)\frac{\partial \bm V(\pmb \theta)}{\partial \pmb \theta}\right]+\frac{1}{2}(\bar\bm z-\bm F\pmb \beta)^\intercal \left[\bm V^{-1}(\pmb \theta)\frac{\partial \bm V(\pmb \theta)}{\partial \pmb \theta}\bm V^{-1}(\pmb \theta)\right] (\bar\bm z-\bm F\pmb \beta),
\end{aligned}
\right.
\end{equation}
where $\bm V(\pmb \theta) = \bm K(\pmb \theta) + \pmb \Sigma_\varepsilon$. The Newton-Raphson algorithm or the Fisher scoring algorithm can be used to solve the above set of equations.
It is a well-known issue \cite[Chapter 5.4]{fang2006} that $\bm K(\pmb \theta)$ often becomes nearly singular when searching over the parameter space of $(\pmb \beta,\pmb \theta)$, causing serious numerical instability when numerically inverting $\bm V(\pmb \theta)$. Admittedly, the presence of $\pmb \Sigma_\varepsilon$ somewhat mitigates the issue, since it is a diagonal matrix whose diagonal entries are all positive. But unless all the diagonal entries of $\pmb \Sigma_\varepsilon$ are sufficiently large, which is not very likely when the number of design points is large, the numerical instability persists.
Nevertheless, if $\bm K(\pmb \theta)$ is constructed from an MCF, then $\bm K^{-1}(\pmb \theta)$ is a sparse matrix having closed-form entries, thanks to Theorem \ref{theo:inverse} and Assumption \ref{assump:lattice}. Instead of using numerical methods such as Gaussian elimination to invert $\bm K(\pmb \theta)$, we apply the Woodbury matrix identity \eqref{eq:woodbury},
\[\bm V^{-1}(\pmb \theta) = \bm K^{-1}(\pmb \theta) + \bm K^{-1}(\pmb \theta)[ \bm K^{-1}(\pmb \theta) + \pmb \Sigma_\varepsilon^{-1}]^{-1} \bm K^{-1}(\pmb \theta).
\]
Hence, numerical inversion is only needed for computing $[ \bm K^{-1}(\pmb \theta) + \pmb \Sigma_\varepsilon^{-1}]^{-1}$. Notice that the diagonal entries of $\pmb \Sigma_\varepsilon^{-1}$ are $r_i/\Var[\varepsilon(\bm x_i)]$, $i=1,\ldots,n$, which can be made sufficiently far away from 0 by increasing $r_i$, the number of simulation replications at $\bm x_i$. Therefore, $\bm K^{-1}(\pmb \theta) + \pmb \Sigma_\varepsilon^{-1}$ is not ill-conditioned in general. The numerical stability of MLE can be significantly improved.
\subsection{Further Enhancement}\label{sec:enhanced_MLE}
If the covariance function $k_{\mathrm{Dir}}(x,y)$ is adopted, we can further improve the computational efficiency and numerical stability of MLE. For notational simplicity, we focus on the 1-dimensional case but the result can be extended to the $D$-dimensional case without essential difficulty.
Suppose that the design points are $\mathcal X=\{x_i=ih:i=1,\ldots,n\}$ with $h=1/(n+1)$. By Corollary \ref{cor:Dirichlet}, the precision matrix associated with $k_{\mathrm{Dir}}(x,y)$ and $\mathcal X$ is
\begin{equation}\label{eq:Toeplitz}
\bm K^{-1}=\phi
\begin{pmatrix}
c & -1 & & & \\
-1 & c & -1 & & \\
\cdots & & \cdots & & \cdots\\
& & -1 & c & -1 \\
& & & -1 & c
\end{pmatrix},
\end{equation}
where $\phi=\eta^{-2}a$ and
\begin{equation}\label{eq:bijection}
\left\{
\begin{array}{lll}
a = \sin^{-1}(\gamma) \sin^{-1}(\gamma h), & c = 2\cos(\gamma h), & \mbox{ if } \nu<0, \\[1ex]
a = h^{-1}, & c = 2, & \mbox{ if } \nu=0, \\[1ex]
a = \sinh^{-1}(\gamma) \sinh^{-1}(\gamma h), & c = 2\cosh(\gamma h), & \mbox{ if } \nu>0.
\end{array}
\right.
\end{equation}
Namely, all the diagonal entries of $\bm K^{-1}$ are made equal by the specific values of $x_1$ and $x_n$ in $\mathcal X$, and thus $\bm K^{-1}$ becomes a Toeplitz matrix. (However, this property does not hold for the Green's functions that correspond to the Cauchy or Neumann BC in Theorem \ref{theo:MCF_examples}.)
A symmetric diagonal Toeplitz matrix enjoys a closed-form eigen-decomposition. Let $\{\lambda_i:i=1,\ldots,n\}$ be the eigenvalues of any matrix of the form \eqref{eq:Toeplitz} and $v_i^\intercal = (v_{i,1},\ldots,v_{i,n})$ be the eigenvector associated with $\lambda_i$, $i=1,\ldots,n$. Then,
\begin{equation}\label{eq:eigenparis}
\lambda_i = \phi\left[c + 2\cos\left(i\pi(n+1)^{-1}\right)\right] \quad\mbox{and}\quad v_{i,j} = \sin\left(ij\pi(n+1)^{-1}\right);
\end{equation}
see \cite{NoschesePasquiniReichel13}.
Notice that the mapping $(\eta^2, \nu)\mapsto(\phi,c)$ is bijective. Hence, we can reparameterize the MCF \eqref{eq:kernel_Diri} with $(\phi,c)$. Notice also that the eigenvector $v_i$ is independent of $(\phi,c)$, $i=1,\ldots,n$. Let $\bm P$ be the matrix whose $i^{\mathrm{th}}$ row is $v_i^\intercal$. Then, $\bm P^{-1}=\bm P^\intercal$, since $\bm K^{-1}$ is positive definite. Let $\pmb \Lambda(\phi,c)$ be the diagonal matrix whose $i^{\mathrm{th}}$ diagonal entry is $\lambda_i=\lambda_i(\phi,c)$. Then, $\bm K^{-1}(\phi,c)=\bm P^\intercal \pmb \Lambda(\phi,c)\bm P$.
We now assume that $\pmb \Sigma_\varepsilon$ has equal diagonal entries, i.e., $\Var[\varepsilon(x_i)]/r_i= \delta$, $i=1,\ldots,n$. This appears a reasonable assumption if (i) the simulation outputs have equal variances, i.e., $\Var[\varepsilon(x)]$ is a constant for all $x$, and the simulation budget is equally allocated, i.e., $r_1=\cdots=r_n$; or (ii) $r_i$ is chosen to be roughly proportional to $\Var[\varepsilon(x_i)]$. Under this assumption, $\bm K(\phi,c)+\pmb \Sigma_\varepsilon = \bm P^\intercal [\pmb \Lambda^{-1}(\phi,c) + \delta \bm I]\bm P$, where $\bm I$ denotes the identity matrix. Hence, $|\bm K(\phi,c)+\pmb \Sigma_\varepsilon| = \prod_{i=1}^n (\lambda_i^{-1}(\phi,c)+\delta) $ and
\[[\bm K(\phi,c)+\pmb \Sigma_\varepsilon]^{-1} = \bm P^\intercal \mathrm{Diag}\left(\frac{1}{\lambda_1^{-1}(\phi,c)+\delta},\ldots,\frac{1}{\lambda_n^{-1}(\phi,c)+\delta}\right)\bm P \coloneqq \bm P^\intercal \bm D(\phi,c)\bm P,\]
where $\bm D(\phi,c)$ is diagonal whose the $i^{\mathrm{th}}$ diagonal entry is $d_i(\phi,c)=1/(\lambda_i^{-1}(\phi,c)+\delta)$. It follows that the log-likelihood function \eqref{eq:log-likelihood2} can be rewritten as
\[l(\pmb \beta,\phi,c) = -\frac{n}{2}\ln(2\pi) + \frac{1}{2}\sum_{i=1}^n\ln(d_i(\phi,c))-\frac{1}{2}(\bar\bm z-\bm F\pmb \beta)^\intercal\bm P^\intercal\bm D(\phi,c)\bm P(\bar\bm z-\bm F\pmb \beta).
\]
The first order optimality conditions for maximizing $l(\pmb \beta,\phi,c)$ are
\begin{equation}\label{eq:MLE_conditions2}
\left\{
\begin{aligned}
\bm 0 = & \frac{\partial l(\pmb \beta,\phi,c)}{\partial \pmb \beta} = \bm F^\intercal\bm P^\intercal \bm D(\phi,c)\bm P(\bar \bm z-\bm F\pmb \beta),\\
0 = & \frac{\partial l(\pmb \beta,\phi,c)}{\partial \theta} = \frac{1}{2}\sum_{i=1}^nd_i^{-1}(\phi,c)\frac{\partial d_i(\phi,c)}{\partial \theta} -\frac{1}{2}(\bar\bm z-\bm F\pmb \beta)^\intercal\bm P^\intercal\frac{\partial \bm D(\phi,c)}{\partial \theta} \bm P(\bar\bm z-\bm F\pmb \beta),\quad \theta=\phi, c.
\end{aligned}
\right.
\end{equation}
Notice that $\frac{\partial \bm D(\phi,c)}{\partial \theta}$ , $\theta=\phi,c$, is diagonal and can be calculated easily given \eqref{eq:eigenparis}. In particular, the conditions in \eqref{eq:MLE_conditions2} do not involve any matrix inversion, thereby representing a further enhancement of computational efficiency relative to the optimality conditions of MLE for general MCFs. At last, with the maximum likelihood estimates of $(\phi, c)$, we can use \eqref{eq:bijection} to compute the estimates of $(\eta^2, a)$.
We summarize the differences in the use of MLE between MCFs and general covariance functions in Table \ref{tab:MLE}. It needs to be emphasized, however, that the two parametric families of MCFs in Table \ref{tab:Green} other than $k_{\mathrm{Dir}}$ do not yield the kind of numerical enhancement discussed in this section. This is because the inverse matrix induced by them does not have the Toeplitz structure by Corollary \ref{cor:Dirichlet}.
\begin{table}[t]
\small
\begin{center}
\caption{Comparison on MLE.}\label{tab:MLE}
\begin{tabular}{cccc}
\toprule
Covariance Function & Inversion Needed? & Optimality Conditions & Stability Enhanced?\\
\midrule
General & Yes & Eq. \eqref{eq:MLE_conditions1} & No \\
MCF & Yes & Eq. \eqref{eq:MLE_conditions1} & Yes \\
$k_{\mathrm{Dir}}$ under Conditions & No & Eq. \eqref{eq:MLE_conditions2} & Yes\\
\bottomrule
\end{tabular}
\end{center}
\small{\textit{Note.} {Conditions: (i) design points are equally spaced; (ii) $\pmb \Sigma_\varepsilon=\sigma^2\bm I$.}}
\end{table}
\begin{remark}
For a $D$-dimensional MCF $k(\bm x,\bm y)=\prod_{i=1}^Dk_i(x^{(i)},y^{(i)})$, where $k_i(\cdot,\cdot)$ is of the form \eqref{eq:kernel_Diri}, the optimality conditions of MLE can be derived in a similar manner. The key is to use the fact that the eigenvalues (resp., eigenvectors) of the tensor product $\bigotimes_{i=1}^D\bm K_i$ can be expressed as the tensor product of the eigenvalues (resp., eigenvectors) of each $\bm K_i$; see \citet[Theorem 13.12]{Laub05}.
\end{remark}
\begin{remark}
By applying Corollary \ref{cor:change_var}, we can relax the requirement on the form of the MCF from \eqref{eq:kernel_Diri} to $k_{\mathrm{Dir}}(\mathcal T(x),\mathcal T(y))$ for some strictly increasing function $\mathcal T$. However, we need to change the design points accordingly to $\{\mathcal T^{-1}(ih):i=1,\ldots,n\}$, where $\mathcal T^{-1}$ is the inverse function of $\mathcal T$.
\end{remark}
\section{Numerical Experiments}\label{sec:numerical}
The big $n$ problem of SK has two aspects -- computational inefficiency and numerical instability. We have shown rigorously that with use of MCFs, the computational time related to matrix inversion, which is the core of the computation of both MLE and the SK predictor \eqref{eq:BLUP}, can be reduced from $\mathcal O(n^3)$ to $\mathcal O(n^2)$; see Table \ref{tab:complex}. The numerical stability issue, on the other hand, is detrimental to the prediction accuracy of SK in a more subtle way. For instance, it may cause numerical optimization of the MLE to fail, returning erroneous estimates of the parameters and further producing unreasonable predictions. In this section, we demonstrate via extensive numerical experiments, with emphasis on the stability aspect, that MCFs represent an elegant solution to the big $n$ problem of SK.
We compare the following three covariance functions.
\begin{itemize}
\item
Squared exponential: $k_{\mathrm{SE}}(\bm x,\bm y)=\eta^2\exp\left(-\sum_{i=1}^D\theta_i(x_i-y_i)^2\right)$;
\item
Exponential: $k_{\mathrm{Exp}}(\bm x,\bm y)=\eta^2\exp\left(-\sum_{i=1}^D\theta_i|x_i-y_i|\right)$;
\item
Multidimensional extension of $k_{\mathrm{Dir}}(x,y)$ with distinct parameters in each dimension.
\end{itemize}
As discussed in \S\ref{sec:stable_MLE}, $k_{\mathrm{Exp}}$ can benefit from the tractability of MCFs, making its MLE significantly more stable than $k_{\mathrm{SE}}$. Further, $k_{\mathrm{Dir}}$ enjoys the ``inverse-free'' MLE scheme in \S\ref{sec:enhanced_MLE}, and thus has the highest computational tractability among the three competing alternatives. The computing environment of the following numerical experiments is a desktop PC with an Intel(R) Core(TM) i7-4790 3.60GHz processor and 16 GB of RAM, running Windows 7 Enterprise. The codes are written in Matlab R2015a. In the sequel, we assume that the SK metamodel \eqref{eq:uni_nriging} has a constant trend, i.e., $\mathsf Z(\bm x) = \beta+ \mathsf M(\bm x)$.
\subsection{Two-Dimensional Response Surfaces} \label{sec:2d_surface}
Consider three distinct 2-dimensional response surfaces which are defined and illustrated in Table \ref{tab:art} and Figure \ref{fig:art}, respectively.
\begin{table}[t]
\small
\begin{center}
\caption{Two-Dimensional Response Surfaces.} \label{tab:art}
\begin{tabular}{lll}
\toprule
Function Name & Expression & Domain \\
\midrule
Three-Hump Camel & $\mathsf Z(x,y)=2x^2-1.05x^4+\frac{x^6}{6}+xy+y^2$ & $x,y\in [-2,2]$ \\
Matyas & $\mathsf Z(x,y)=0.26(x^2+y^2)-0.48xy$ & $x,y\in[-10, 10]$ \\
Bohachevsky & $\mathsf Z(x,y)=x^2+2y^2-0.3\cos(3\pi x)-0.4\cos(4\pi y)+0.7$ & $x,y\in[-100,100]$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\begin{figure}[t]
\caption{Response Surfaces of the Functions in Table \ref{tab:art}.} \label{fig:art}
\begin{center}
\includegraphics[width=0.32\textwidth]{figures/Surfaces/Three-Hump.pdf}
\includegraphics[width=0.32\textwidth]{figures/Surfaces/Matyas.pdf}
\includegraphics[width=0.32\textwidth]{figures/Surfaces/Bohachevsky.pdf}
\end{center}
\end{figure}
For each surface, we choose $n=m^2$ design points and let them form an equally spaced lattice within the design space, for some integer $m\geq 3$. For instance, for the three-hump camel function whose domain is $[-2,2]^2$, we set the design points to be $\{(x_i,y_j)|x_i=\frac{4i}{m+1}-2, \frac{4j}{m+1}-2, i,j=1,\ldots,m\}$. We set the number of prediction points to be $K=100^2$ and place them equally spaced in the same way. For simplicity, we assume that the sampling variance is $\sigma^2$ for each design point, implying that the covariance matrix of the sampling errors is $\pmb \Sigma_\varepsilon=\sigma^2 \bm I$, where $\bm I$ denotes the $n\times n$ identity matrix. Given a covariance function (i.e., $k_{\mathrm{Dir}}$, $k_{\mathrm{Exp}}$, or $k_{\mathrm{SE}}$), we first estimate the unknown parameters with MLE as discussed in \S\ref{sec:MLE}, and then compute the SK predictor $\hat\mathsf Z(\bm x_i)$ for each prediction point $\bm x_i$, $i=1,\ldots,K$ by plugging the parameter estimates into \eqref{eq:SK}. In order to assess the prediction accuracy, we compute the standardized root mean squared error (SRMSE) as follows
\begin{equation*}\label{eq:SRMSE}
\mathrm{SRMSE} = \frac{\sqrt{\sum_{i=1}^K \left[\mathsf Z(\bm x_i) - \hat\mathsf Z(\bm x_i)\right]^2 }}{\sqrt{\sum_{i=1}^K \left[\mathsf Z(\bm x_i)-K^{-1}\sum_{h=1}^K\mathsf Z(\bm x_h)\right]^2}}.
\end{equation*}
since the three surfaces are of substantially different scales and the standardization facilitates the comparison. We repeat the experiment for both noiseless ($\sigma=0$) and noisy ($\sigma>0$) data, for each of the three surfaces, each of the three covariance functions and $m=3,4,\ldots, 12$. The results are presented in Figure \ref{fig:Matyas}.
\begin{figure}[t]
\begin{center}
\caption{Accuracy for Predicting the Surfaces in Figure \ref{fig:art}.} \label{fig:Matyas}
$
\begin{array}{cc}
\includegraphics[width=0.45\textwidth]{figures/TestFunc/3HumpCamelSigma0_SRMSE.pdf} &
\includegraphics[width=0.45\textwidth]{figures/TestFunc/3HumpCamelSigma1_SRMSE.pdf} \\
\includegraphics[width=0.45\textwidth]{figures/TestFunc/MatyasSigma0_SRMSE.pdf} &
\includegraphics[width=0.45\textwidth]{figures/TestFunc/MatyasSigma10_SRMSE.pdf} \\
\includegraphics[width=0.45\textwidth]{figures/TestFunc/BohachevskySigma0_SRMSE.pdf} &
\includegraphics[width=0.45\textwidth]{figures/TestFunc/BohachevskySigma100_SRMSE.pdf}
\end{array}
$
\end{center}
\end{figure}
Clearly, in the absence of simulation errors, i.e., $\sigma=0$, $k_\mathrm{SE}$ has yields highest prediction accuracy especially when $n$ is small, while $k_{\mathrm{Exp}}$ and $k_{\mathrm{Dir}}$ have almost identical performance. However, when $n$ is large, $k_{\mathrm{SE}}$ will encounter the serious numerical instability issue, as reflected by the sudden ``blow-up'' in SRMSE. This is because for large $n$, e.g., $n>50$, $\pmb \Sigma_\mathsf M$ becomes highly ill-conditioned during the numerical procedure of solving MLE, in which case the numerical error associated with computing $\pmb \Sigma_\mathsf M^{-1}$ is overwhelming, and both the parameter estimates and the prediction are unreliable.
On the other hand, in the presence of simulation errors, the numerical instability issue is mitigated greatly and we do not observe the ``blow-up'' behavior in SRMSE in our experiments even for large $n$. This is because the matrix that needs to be inversed now in order to compute the MLE and the SK predictor is $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$, which is far away from being singular despite the ill-condition of $\pmb \Sigma_\mathsf M$. Nevertheless, the simulation errors degrade the prediction accuracy of SK in general, and $k_{\mathrm{SE}}$ appear to suffer the most. Specifically, the SRMSE associated with $k_{\mathrm{SE}}$ is significantly higher than the other two. The performances of $k_{\mathrm{Exp}}$ and $k_{\mathrm{Dir}}$ are comparable with the former noticeably better.
In order to reveal clearly the possible numerical instability associated matrix inversion, we compute the \emph{condition number} (associated with the $\mathsf{L}^2$ vector norm) of $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$, which measures how roundoff errors during computation impact the entries of the computed inverse matrix; see \citet[Chapter 5.8]{HornJohnson12} for exposition on the subject. The positive definiteness of $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$ implies that its condition number is the ratio between its largest eigenvalue to its smallest eigenvalue. The larger the condition number is, the more ill-conditioned the matrix is. In Figure \ref{fig:CondNum}, we plot the condition number of $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$ with plug-in parameter estimates from MLE for fitting the samples from the three-hump camel function. The plots for the Matyas function and the Bohachevsky function are highly similar, thereby omitted.
\begin{figure}[t]
\begin{center}
\caption{Condition Number of $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$.}\label{fig:CondNum}
\includegraphics[width=0.45\textwidth]{figures/TestFunc/3HumpCamelSigma0_CondNum.pdf}
\includegraphics[width=0.45\textwidth]{figures/TestFunc/3HumpCamelSigma1_CondNum.pdf}
\end{center}
\small{\textit{Note.} Three-hump camel function; $\pmb \Sigma_\varepsilon=\bm 0$ if $\sigma=0$.}
\end{figure}
Figure \ref{fig:CondNum} shows that the condition number of $\pmb \Sigma_\mathsf M$ for $k_{\mathrm{SE}}$ basically increases exponentially fast in $n$. For example, it is larger than $10^{10}$ for $n= 12^2$, which means that $\pmb \Sigma_{\mathsf M}$ is severely ill-conditioned and explains the erroneous prediction results revealed in Figure \ref{fig:Matyas}. By contrast, the condition number of $\pmb \Sigma_\mathsf M$ grows dramatically slower for the other two covariance functions. However, in the presence of simulation errors, the condition number of $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$ is reduced substantially, especially for $k_{\mathrm{SE}}$. Indeed, it has been well documented in geostatistics literature that the condition number of the covariance matrix associated with $k_{\mathrm{SE}}$ is particularly large. A typical treatment is to add artificially the so-called ``nugget effect'' which plays essentially the same role as $\pmb \Sigma_\varepsilon$ mathematically; see, e.g., \cite{AbabouBagtzoglouWood94} and references therein.
\subsection{Scalability Demonstration}
We now demonstrate the scalability of SK when equipped with MCFs. In the experiments that follow, we do not incorporate $k_{\mathrm{SE}}$ in the comparison, because with it SK scales poorly as $n$ increases and almost certainly ends up with numerical failure as shown in \S\ref{sec:2d_surface}. We consider two response surfaces. One is the Griewank function
\[\mathsf Z(\bm x) = \sum_{i=1}^4 \left(\frac{x^{(i)}}{20}\right)^2 - 10\prod_{i=1}^D\cos\left(\frac{x^{(i)}}{\sqrt{i}}\right) + 10, \quad \bm x\in[-5,5]^4,\]
with $D=4$; see Figure \ref{fig:Griewank} (left panel) for its 2-dimensional projections.
\begin{figure}[t]
\begin{center}
\caption{Two-Dimensional Projections of the Griewank Function and the Expected Cycle Time.} \label{fig:Griewank}
\includegraphics[width=0.32\textwidth]{figures/Surfaces/Griewank.pdf}
\includegraphics[width=0.32\textwidth]{figures/Surfaces/JackNet_slice1.pdf}
\includegraphics[width=0.32\textwidth]{figures/Surfaces/JackNet_slice2.pdf}
\end{center}
\end{figure}
The experiment is set up in the same way as \S\ref{sec:2d_surface}. We choose $n=m^4$ design points and $K=100^4$ prediction points, and make both sets of points equally spaced within the design space. The sampling variance at each design point is set to be $\sigma^2=1$. In addition to the prediction accuracy based on SRMSE, we compare $k_{\mathrm{Dir}}$ and $k_{\mathrm{Exp}}$ in terms of the computational efficiency as well. Notice that the implementation of SK comprises primarily two steps -- parameter estimation and computation of the predictor. Inversion of $\pmb \Sigma_\mathsf M+\pmb \Sigma_\varepsilon$, which is the scalability bottleneck, is performed repeatedly in the former step. By contrast, given the estimated parameters, the matrix inversion is a one-time operation and thus can be stored to compute the predictor \eqref{eq:BLUP} at different design points, since the matrix is independent of the design point.
As discussed in \S\ref{sec:MLE}, $k_{\mathrm{Dir}}$ enjoys a more efficient MLE scheme than general MCFs such as $k_{\mathrm{Exp}}$. We therefore compare their computational efficiency by measuring the CPU time used to solve the MLE. Specifically, we use the Matlab function \texttt{fsolve} to solve numerically the first-order optimality conditions \eqref{eq:MLE_conditions1} and \eqref{eq:MLE_conditions2} for $k_{\mathrm{Exp}}$ and $k_{\mathrm{Dir}}$, respectively. We set the initial point randomly, repeat the experiment 100 times, and compute the average CPU time. The results are presented in Figure \ref{fig:scalability} (upper panel).
A second surface arises from a queueing context and is adopted from \cite{YangLiuNelsonAnkenmanTongarlak11}. Consider a $N$-station Jackson network in which both the interarrival times and the service times are exponentially distributed. The arrivals consist of $D$ different types of products and the fraction of product $i$ is $\alpha_i$, $i=1,\ldots,D$. Suppose that station $j$ has a service rate $\mu_j$, regardless of the product type, $j=1,\ldots,N$. The station having the largest utilization among all is called the bottleneck station. Let $\rho$ denote the utilization of the bottleneck station. The design variable is $(\alpha_1,\ldots,\alpha_D, \rho)$, for $\alpha_i\in[0,1]$ with $\alpha_1+\cdots+\alpha_D=1$ and $\rho\in[0.5,0.9]$. The response surface of interest is the expected cycle time (CT) of, say, product 1. It is shown in \cite{YangLiuNelsonAnkenmanTongarlak11} that
\begin{equation}\label{eq:CT}
\E[\mathrm{CT}_1]= \sum_{j=1}^N \frac{\delta_{1j}}{\mu_j\left[1- \rho\left(\frac{\sum_{i=1}^D\alpha_i\delta_{ij}/\mu_j}{\max_h \sum_{i=1}^D\alpha_i\delta_{ih}/\mu_h}\right)\right]},
\end{equation}
where $\delta_{i,j}$ is the expected number of visits to station $j$ by product $i$. The parameters $\mu_j$ and $\delta_{i,j}$ are generated randomly and given as follows:
\[\mu =
\begin{pmatrix}
1.25 \\
1.85 \\
1.97 \\
1.45
\end{pmatrix},
\quad
\delta =
\begin{pmatrix}
1.553 & 1.012 & 0.926 & 0.242 \\
0.127 & 1.066 & 1.115 & 0.536 \\
1.182 & 1.597 & 1.486 & 1.850 \\
1.800 & 1.310 & 1.029 & 1.179
\end{pmatrix}.
\]
Notice that the design space is not a hyperrectangle. To accommodate the requirement that the design points form a regular lattice, we conduct the following change of variables. Define
$x^{(1)}=\sqrt{\alpha_1}$, $x^{(i)}=\sqrt{\alpha_{i}/(1-\sum_{h=1}^{i-1}\alpha_h)}$, $i=2,\ldots,D-2$, $x^{(D)}=\rho$. Then, $x^{(i)}\in[0,1]$ for $i=1,\ldots,D-1$ because $\alpha_1+\cdots+\alpha_D=1$. Let $\bm x=(x^{(1)},\ldots,x^{(D)})\in[0,1]^{D-1}\times[0.5,0.9]$ and $\mathsf Z(x)$ be the expression of \eqref{eq:CT} after the change of variables; see Figure \ref{fig:Griewank} (middle and right panels) for its 2-dimensional projections. A critical difference between this surface and the others is that it is not differentiable everywhere. This is because the bottleneck station varies as the product-mix vector $(\alpha_1,\ldots,\alpha_D)$ changes.
We assume $D=N=4$. The experiment setup is the same as that for the Griewank function, except that we choose $n=5m^3$ design points as follows: $x^{(i)}\in\{\frac{1}{m+1},\ldots,\frac{m}{m+1}\}$, $i=1,2,3$, and $x^{(D)}\in\{0.5,0.6,\ldots,0.9\}$, for $m=5,6,\ldots,15$. The results are presented in Figure \ref{fig:scalability} (lower panel).
\begin{figure}[t]
\begin{center}
\caption{Efficiency for Solving MLE and Prediction Accuracy of SK with MCFs.} \label{fig:scalability}
$
\begin{array}{cc}
\includegraphics[width=0.45\textwidth]{figures/Griewank/Griewank_CPUtime.pdf} &
\includegraphics[width=0.45\textwidth]{figures/Griewank/Griewank_SRMSE.pdf} \\
\includegraphics[width=0.45\textwidth]{figures/JackNet/JackNet_CPUtime.pdf} &
\includegraphics[width=0.45\textwidth]{figures/JackNet/JackNet_SRMSE.pdf}
\end{array}
$
\end{center}
\end{figure}
We see from Figure \ref{fig:scalability} that SK can scale up dramatically with the use of MCFs. It can easily handle large-scale problems in a computationally efficient and numerically stable fashion. For example, even with $10^4\times 10^4$ covariance matrices, the MLE can be solved within a minute on an average desktop computer and does not encounter any numerical instability issue. This is a consequence of the analytical invertibility and the sparsity structure induced by MCFs.
Moreover, between the two MCFs tested here, $k_{\mathrm{Dir}}$ outperforms $k_{\mathrm{Exp}}$ substantially in terms of computational efficiency, due to the enhanced MLE scheme in \S\ref{sec:enhanced_MLE}. However, we stress that such enhancement comes at the cost of flexibility in the allocation of simulation budget across design points, because $\pmb \Sigma_\varepsilon$ needs to be in the form of $\sigma^2\bm I$. In terms of prediction accuracy, $k_{\mathrm{Dir}}$ is also noticeably better than $k_{\mathrm{Exp}}$.
\section{Concluding Remarks} \label{sec:conclusions}
The present paper addresses the poor scalability of the popular SK metamodel using a novel approach. By imposing a Markovian structure on the Gaussian random field, we identify the form of the covariance function that leads to analytically invertible covariance matrices with sparsity in the inverse. We further develop a connection between such MCFs and the Green's functions of S-L equations, which effectively provides a flexible, principled approach to constructing MCFs. With the use of MCFs, the computational complexity related to matrix inversion is reduced from $\mathcal O(n^3)$ to $\mathcal O(n^2)$ in general without any matrix approximations, to $\mathcal O(n)$ in the absence of simulation errors, and even to $\mathcal O(1)$ for some specific MCFs with carefully chosen design points.
Extensive numerical experiments demonstrate that for small-scale problems, MCFs have comparable performance as the squared exponential covariance function, a standard choice for SK, in terms of the prediction accuracy; however, the true advantage of MCFs resides in large-scale problems, which can be handled in a timely and stable manner without suffering from the numerical instability issue that SK normally exhibits under general covariance functions.
Several follow-up problems should be investigated to realize the full potential of the methodology. For example, the condition number of the covariance matrix is examined numerically in the present paper. The observation that MCFs yield a small condition number ought to be addressed theoretically to further strengthen the foundation of the methodology. For another example, using gradient information to enhance the prediction accuracy of SK is a technique that receives much attention; see \cite{ChenAnkenmanNelson13} and \cite{QuFu14}. However, in the presence of the gradient, the size of the covariance matrix that needs to be inverted becomes $(D+1)n\times (D+1)n$, since there is a distinct derivative surface for the partial derivative along each dimension in addition to the response surface itself. Hence, the big $n$ problem is even more severe in this context and our methodology can potentially be of great help.
\section*{Acknowledgment}
The first author is supported by the Hong Kong PhD Fellow Scheme (Ref. No. PF14-13781). The second author is supported by the Hong Kong Research Grant Council (Project No. 16211417).
|
{
"timestamp": "2018-03-08T02:06:03",
"yymm": "1803",
"arxiv_id": "1803.02575",
"language": "en",
"url": "https://arxiv.org/abs/1803.02575"
}
|
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\begin{document}
\twocolumn[
\aistatstitle{Gaussian Process Latent Variable Alignment Learning}
\aistatsauthor{ Ieva Kazlauskaite \And Carl Henrik Ek \And Neill D. F. Campbell}
\aistatsaddress{ University of Bath, UK \\ Electronic Arts \\ \small{\texttt{i.kazlauskaite@bath.ac.uk}} \And University of Bristol, UK \And University of Bath, UK \\ Royal Society \\\small{\texttt{n.campbell@bath.ac.uk}}}
]
\begin{abstract}
We present a model that can automatically learn alignments between high-dimensional data in an unsupervised manner. Our proposed method casts alignment learning in a framework where both alignment and data are modelled simultaneously. Further, we automatically infer groupings of different types of sequences within the same dataset. We derive a probabilistic model built on non-parametric priors that allows for flexible warps while at the same time providing means to specify interpretable constraints. We demonstrate the efficacy of our approach with superior quantitative performance to the state-of-the-art approaches and provide examples to illustrate the versatility of our model in automatic inference of sequence groupings, absent from previous approaches, as well as easy specification of high level priors for different modalities of data.
\end{abstract}
\input{includes/intro.tex}
\input{includes/background.tex}
\input{includes/methodology.tex}
\input{includes/experiments.tex}
\input{includes/conclusions.tex}
\subsubsection*{Acknowledgments}
This work has been supported by EPSRC CDE (EP/L016540/1) and CAMERA (EP/M023281/1) grants as well as the Royal Society. IK would like to thank the Frostbite Physics team at EA.
\subsection*{Datasets with quantifiable comparisons}
\subsection*{Motion capture dataset}
In this experiment we use the full set of joint motions to align a set of sports actions (see \S 4 for further information on the motion capture dataset). In Fig.~\ref{fig:gen_model} we provide an illustration of the power of using a generative model for alignment. New locations in the manifold encode novel motion sequences that are supported by the data. By allowing the model to align the data, it greatly improves the generative power as the model is capable of producing a wider range of plausible motions.
\begin{figure}[h]
\centering
\includegraphics[width=0.49\textwidth]{supp/gen_model.pdf}
\caption{An advantage of our approach is that it not only aligns the data but is also a generative probabilistic model. Here we show novel sequences generated at new locations in the manifold. The black dots indicate the embedded locations of the training sequences. We note that, while we have only shown still images, each manifold location describes an entire time series. A video showing this is included with the supplementary material.} \label{fig:gen_model}
\end{figure}
Fig.~\ref{fig:toys_aligns} and Fig.~\ref{fig:toys_warps} give an example of the alignments and the warps produced by our method on the quantifiable dataset, see \S 4 in the paper. The detailed results of our experiments on this dataset are provided in Table~\ref{table:main_results}.
\subsection*{iPhone motion data}
This dataset contains aerobic actions recorded using the Inertial Measurement Unit (IMU) on a smartphone~\cite{McCall:2012}, which contain high frequency variations. Unlike previous methods~\cite{Tucker:2013}, which require the data to be smoothed first, our framework allows us to take into account the prior belief about the dataset in a principled way. By replacing the smooth RBF kernels for modeling the data with a Mat\'{e}rn $1/2$ kernel and taking into account the periodic nature of the actions by also including an additive periodic kernel, we are able to model the data without the need for preprocessing. Furthermore, by removing the smoothing prior from the warping functions, we allow the warps to be more flexible improving the alignment accuracy.
The alignment results for the iPhone motion data are shown in Fig.~\ref{fig:gait}. The IMU includes a 3D accelerometer, a gyroscope, and a magnetometer, and records samples at 60 Hz. As in~\cite{Tucker:2013}, for our experiment we take the accelerometer data in the x-direction for the jumping actions from subject $3$, and, in particular, we look at $5$ sequences each of which contains $400$ frames. A Mat\'{e}rn $1/2$ kernel and a periodic kernel are used to fit the sequences as they contain high frequency variations, and we remove the smoothness constraint from the model of the warping functions to allow them to be more flexible.
\subsection*{Shift task}
A common task in functional data alignment is that of estimating uniform translations of the time axis. One particular problem described by Marron \emph{et al.} is that of aligning nuclear magnetic resonance (NMR) spectrum corresponding to different chemical components (e.g. ethanol) for a set of wines~\cite{Marron:2015}. It is known that pH differences in wines induce a shift in values of the components and impedes their identification~\cite{Larsen:2006}. As shown by Marron \emph{et al.} the alignment may be achieved using uniform shifts and minimizing the loss that requires sequences to be proportional to each other. Such operation is included in our model allowing us to perform the task of NMR spectrum alignment, and we are able to demonstrate a separation in the phase between the red wines and the white and ros\'{e} wines, see Fig.~\ref{fig:wines}.
\begin{figure*}[h]
\begin{minipage}{0.49\textwidth}
\centering
\begin{subfigure}[h]{\textwidth}
\includegraphics[width=\textwidth]{supp/align_23_col_2_s.pdf}
\end{subfigure}
\caption{Original inputs and aligned sequences estimated by DTW, DDTW, IMW, CTW, GTW, SRVF, our approach and its three variants.} \label{fig:toys_aligns}
\end{minipage}%
\hspace{4pt}
\begin{minipage}{0.49\textwidth}
\centering
\begin{subfigure}[h]{\textwidth}
\includegraphics[width=\textwidth]{supp/warps_23_col_2_s.pdf}
\end{subfigure}
\caption{True warps and warps estimated by DTW, DDTW, IMW, CTW, GTW, SRVF, our approach and its three variants.} \label{fig:toys_warps}
\end{minipage}
\end{figure*}
\begin{table*}[h]
\centering
\input{includes/results_table.tex}%
\caption{Datasets used for our evaluation where $J$ and $T$ refer to the number of sequences and dimensionality. Results are presented as MSE of warpings. The summary of the results presented in this table is given in Fig. $3$ in the paper. }
\label{table:main_results}
\end{table*}
\begin{figure*}[h]
\centering
\begin{subfigure}[h]{\textwidth}
\includegraphics[width=\textwidth]{supp/f_j_gait}
\end{subfigure} \\ \vspace{5pt}
\begin{subfigure}[h]{\textwidth}
\includegraphics[width=\textwidth]{supp/aligned_gait}
\end{subfigure}
\caption{The top row shows the observed data and the fitted Gaussian Processes. The bottom row shows the corresponding warps (left) and the aligned functions.}\label{fig:gait}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[h]{0.49\textwidth}
\includegraphics[height=0.2\textheight]{supp/wines_alignment_fixed_s.pdf}
\end{subfigure}
\begin{subfigure}[h]{0.49\textwidth}
\includegraphics[height=0.2\textheight]{supp/wines_warpings_fixed_s.pdf}
\end{subfigure}
\caption{Alignment of NMR spectrum data~\cite{Marron:2015}. The zoom of the warping functions show the separation of the white/ros\'{e} wines (shown in blue) and red wines (shown in red).}\label{fig:wines}
\end{figure*}
\section{Methodology}
\label{sec:methodology}
\label{sec:our_model}
\input{includes/overview_figure.tex}
Alignment learning is the task of recovering a set of monotonic warping functions that have been used to create samples of a latent sequence.
Fig.~\ref{fig:overview} provides an overview of our approach. We are provided with a number of noisy time warped observations of a set of unobserved latent sequences and our task is to infer both this set of sequences and the time warps that give rise to the observations.
Let us assume that we have $J$ noisy sequence observations $\{{Y}_{j}\}$ (Fig.~\ref{fig:overview_obs}) where each observed sequence comprises $N$ time samples, $\mathbf{Y} = (y_{(j,n)}) \in \mathbb{R}^{J \times N}$. We consider each sequence to be generated as a sample from a latent function $f_j(x)$ (Fig.~\ref{fig:overview_fit}) under a monotonic warping $g_j(x)$ as $y_{(j,n)} = f_j( \, g_j(x_n) ) + \varepsilon_{jn}$ where the samples have been corrupted by additive Gaussian noise $\varepsilon_{jn} \sim \mathcal{N}(0, \beta_j^{-1})$. Due to the close association of sequences and temporal data, we use the word time to refer to the input domain of the sequence, however our method is general and applicable to any ordered index set.
The aligned sequences, which are unobserved, are given by the corresponding functions without the time warp $s_{(j,n)} = f_j( x_n ) + \varepsilon_{jn}, \, \mathbf{S} = (s_{(j,n)}) \in \mathbb{R}^{J \times N}$ (as illustrated in Fig.~\ref{fig:overview_true} and Fig.~\ref{fig:overview_aligned}).
This means that we can encode our warping function as the transformation from a \emph{known} sampling of an \emph{unknown} aligned sequence to the \emph{unknown} sampling of the \emph{known} observations for each sequence. However, as described in the introduction and illustrated in Fig.~\ref{fig:overview_true}, we wish to design a model that is not restricted to the case where all the observations arise from a \emph{single} latent function (for example, in Fig.~\ref{fig:overview_true} there are two unknown true sequences).
To account for the possible existence of multiple latent functions, we consider a generative model for the aligned sequences themselves. By specifying that the generative process be as simple as possible, we encourage the clustering of these sequences, which allows to automatically find the smallest number of latent functions explaining the data. We encode this as the aligned sequences being generated via a smooth mapping $h(\cdot)$ from a low dimensional space $Z \in \mathbb{R}^{Q}$ as
\begin{equation}
S_j = h(Z_j) + \hat{\epsilon} \quad \text{s.t.} \quad S_j = f_j(x) + \epsilon_j \: \forall \; j \; \label{eqn:such_that}
\end{equation} where $\hat{\epsilon} \sim \NormalDistrib{0, \gamma^{-1} I}$ and $\epsilon_j \sim \NormalDistrib{0, \beta_j^{-1} I}$.
This low dimensional manifold is visualised, for our toy example, in Fig.~\ref{fig:overview_manifold} where $Z$ is a 2D space and the locations of the aligned sequences $S_j$ are shown as coloured points matching the corresponding aligned sequences in Fig.~\ref{fig:overview_all_aligned}. We see that the two different sequences are clustered appropriately by their location in the manifold and the sequences are correctly aligned. We use a probabilistic model of the aligned sequences, which allows us to quantify uncertainty of the low-dimensional manifold representations (the heatmap in Fig.~\ref{fig:overview_manifold}).
\subsection{Probabilistic Model}
In this section we specify the two components of our model that correspond to the constraint introduced in Eq.~\eqref{eqn:such_that}.
The first part corresponds to fitting the data that explains the observed sequences and specifies the latent functions, while the second part enforces a simple, low-dimensional structure of the aligned sequences. Given noisy observed data, we do not impose this constraint exactly, but rather define both model components as probabilistic models and interpret this constraint as one of the aligned sequences having high likelihood under both model components simultaneously. If the aligned sequences $\mathbf{S}$ were known, this interpretation would correspond exactly to fitting a model to the observed data by maximising the data likelihood.
Since $\mathbf{S}$ are unobserved, we refer to them as \emph{pseudo-observations}; similarly to~\cite{Titsias:2009}, we augment the probability space with a set of pseudo-observations which are constrained by the two components of our model. We then fit the model by maximising the joint marginal likelihood of observations $\mathbf{Y}$ and pseudo-observations $\mathbf{S}$, while optimising not only w.r.t.~the model parameters, but also w.r.t.~the pseudo-observations $\mathbf{S}$.
\paragraph{Model over time} We have $Y_j \in \mathbb{R}^{N}$ as the observed sequences, and let $X \in \mathbb{R}^{N}$ denote an observed uniform sampling of time. We introduce a random variable $G_j \in \mathbb{R}^{N}$ to encode the time warp function sampled at $X$ such that $G_j \sim g_j(X)$. The random variables for the functions $f_j(\cdot)$ are more involved since the functions are evaluated at different locations. Let $F_j^{\mathrm{G}} \sim f_j(G_j)$ denote the output of the function sampled at the time warped locations $G_j$ and let $F_j^{\mathrm{X}} \sim f_j(X)$ denote the function evaluated at the uniform sample locations $X$. The observations $Y_j$ are the noise-corrupted versions of $F^G$, and similarly, we call the noise-corrupted version of $F^X$ pseudo-observations, since they are not observed and should be inferred.
We now define the priors over the generating and warping functions $f_j(\cdot)$ and $g_j(\cdot)$. Specifying a parametric mapping is challenging and it severely limits the possible functions we can recover. In this paper, we make use of flexible non-parametric Gaussian Process (GP) priors which allows us to provide significant structure to the learning problem without reducing the possible solution space. The two random variables connected with $f_j(\cdot)$ may then be jointly specified under a GP prior where the covariance, with hyperparameters $\theta$, is evaluated at $G_j$ and $X$ for $F_j^{\mathrm{G}}$ and $F_j^{\mathrm{X}}$ respectively as
\begin{gather}
p\left(\begin{bmatrix} F_j^{\mathrm{X}} \\ F_j^{\mathrm{G}}\end{bmatrix} \middle| \, G_j, X_j, \theta_j \right) \sim
\mathcal{N}\left(\mathbf{0},
\begin{bmatrix} k_{\theta_j}(X, X) & k_{\theta_j}(X, G_j) \\ k_{\theta_j}(G_j, X) & k_{\theta_j}(G_j, G_j) \end{bmatrix}
\right)
\end{gather}
\paragraph{Warping functions} We encode our preference for smooth warping functions $g_j(\cdot)$ by making $p(G_j \mid X)$ a GP prior with a smooth kernel function. We can ensure monotonicity by an appropriate parametrisation of the $G_j$ using an auxiliary input. Without loss of generality, these are constrained to be monotonic in the range $[-1,1]$ using a set of auxiliary variables $U_j \in \mathbb{R}^{N}$ such that
\begin{align}
[G_j]_n &:= 2 \,\sum_{k=1}^n\!\left[ \,\mathrm{softmax}\!\left( U_j \right) \right]_{k} - 1\ .
\label{eq:warps}
\end{align}
Importantly, all warping functions are continuous and generative which means we are able to resample the data. Therefore, we write that $p(G_j \mid X, \omega) \sim \mathcal{N}(0, k_{\omega_j}(X, X))$ with hyperparameters $\omega_j$. An alternative to our parameterisation is GPs with monotonicity information~\cite{Riihimaki:2010}, however, this approach does not guarantee that the posterior predictive is monotonic.
\paragraph{Model over sequences} We would like a constraint that aligns similar sequences to each other while keeping dissimilar sequences apart without us specifying which sequences belong together. We consider using dimensionality reduction as a means of preserving similarities in the prediction space as well as imposing the preference for dissimilar data points to be placed far apart in the latent space. In particular, we propose to use a GP-LVM that places independent GPs over the data features and optimises the locations of the low-dimensional latent points that correspond to each sequence.
To this end, we let the random variable $Z_j \in \mathbb{R}^{Q}$ be the embedded manifold location of the sequence. The random variable $H_j$ denotes the output of the mapping function evaluated at $Z_j$ such that $H_j \sim h(Z_j)$. To ease notation, we use bold symbols to denote the concatenation across $J$ such that, for example, $\mathbf{Z} = [Z_1, \dots, Z_J]$. We encode the preference for a smooth mapping by placing a GP prior over the mapping $h(\cdot)$ so that we have $p(\mathbf{H} \mid \mathbf{Z}, \psi) \sim \mathcal{N}(0, k_{\psi}(\mathbf{Z}, \mathbf{Z}))$ where $\psi$ are the hyperparameters of the covariance kernel. The pseudo-observations are modelled by the GPLVM by adding independent Gaussian noise to $\mathbf{H}$. Next we consider the joint distribution of the model to derive an objective which simultaneously ensures that (i) the observed data $\mathbf{Y}$ is fitted well by the corresponding GPs $f$ at the warped locations, (ii) the pseudo observations are fitted well by the corresponding GPs $f$ at the fixed sampling locations, (iii) the pseudo observations are such that they exhibit a simple structure that is captured by the latent variable model.
\paragraph{Joint distribution} The joint distribution (ignoring the hyperparameters and noise terms for clarity) decomposes as
\begin{multline}
p(\mathbf{S}, \mathbf{Y},\mathbf{F}^{\mathrm{X}}, \mathbf{F}^{\mathrm{G}}, \mathbf{G}, \mathbf{H}, \mathbf{Z} | \mathbf{X}) = p(\mathbf{Y} | \mathbf{F}^{\mathrm{G}} ) \, p(\mathbf{S} | \mathbf{H}, \mathbf{F}^{\mathrm{X}}) \\ p(\mathbf{H} | \mathbf{Z}) \, p(\mathbf{F}^{\mathrm{X}}, \mathbf{F}^{\mathrm{G}} | \mathbf{G}, \mathbf{X}) \, p( \mathbf{G} | \mathbf{X}) \, p(\mathbf{Z}) \ . \label{eqn:joint}
\end{multline}
The terms $p(\mathbf{H} | \mathbf{Z})$, $p(\mathbf{F}^{\mathrm{X}}, \mathbf{F}^{\mathrm{G}} | \mathbf{G}, \mathbf{X})$ and $p( \mathbf{G} | \mathbf{X})$ are the GP priors defined previously and $p(\mathbf{Z}) \sim \mathcal{N}(\mathbf{0}, I)$ is the latent prior . We note that $p(\mathbf{F}^{\mathrm{X}}, \mathbf{F}^{\mathrm{G}} | \mathbf{G}, X)$ and $p( \mathbf{G} | X)$ factorise fully over $J$.
\paragraph{Likelihood terms} The likelihood of the observations under i.i.d. Gaussian noise with precision $\beta_j$ is $p(\mathbf{Y} | \mathbf{F}^{\mathrm{G}} ) = \prod_j p(Y_j | F_j^{\mathrm{G}} ) = \prod_j \mathcal{N}(Y_j | F_j^{\mathrm{G}}, \beta_j^{-1}I)$. The likelihood of the pseudo-observations is more involved since it encodes the relationship of Eq.~\eqref{eqn:such_that}. We define the likelihood $p(\mathbf{S} \mid \mathbf{H}, \mathbf{F}^{\mathrm{X}})$ as an equal mixture:
\begin{gather}
p(\mathbf{S} \mid \mathbf{H}, \mathbf{F}^{\mathrm{X}}) =
\frac{1}{2} \left( \prod_n \mathcal{N}(S_n |\mathbf{H}_n, \gamma^{-1} I_J) + \prod_j \mathcal{N}(S_j | \mathbf{F}_j^X, \beta_j^{-1} I_N) \right) \label{eq:likelihood}
\end{gather} where $S_j$ refers to the rows and $S_n$ refers to the columns of $\mathbf{S}$.
This explicitly encourages the two components (the one over $h$ and the one over $f$) to coincide so that the pseudo-observations are explained by both components of the model simultaneously. In order to find the maximum likelihood solution, we use the fact that $\log(\rfrac{1}{2} \: a + \rfrac{1}{2} \:b) \geq \rfrac{1}{2}\log(a) + \rfrac{1}{2}\log(b)$, and we maximise the lower bound on the log-likelihood to find the parameters of the two models, the latent variables, and the pseudo-observations.
\paragraph{Approximations} The integrals over $\mathbf{H}$ and $\mathbf{F}^X$ are regular GP marginalisations, which can be computed in closed-form. The integral corresponding to $\mathbf{F}^{\mathrm{G}}$ includes a composition of GPs, ($f \circ g$), which does not have a closed-form solution. Following \cite{Lawrence:2007hierarchical}, we approximate this integral using a point estimate, and since $\mathbf{G}$ is directly optimised, it allows us to use the monotonic parametrisation of Eq.~\eqref{eq:warps} without the need to integrate over the corresponding parameters.
\paragraph{Learning}We place priors over the hyperparameters $\{\gamma, \psi, \theta_{j}, \beta_{j}\}$ as log-Normal distributions with zero mean and unit variance. We also place an additional prior on the raw sample points $U_j$ to encourage smooth warps, and improve training as,
\begin{equation}
\log \, \p{\{U_j\}} = \displaystyle\sum_{j=1}^{J} \, \log \,\NormalDistrib{U_j\mid \mathbf{0}, I_{N}}.
\end{equation} We optimise the following marginal log-likelihood (excluding the terms corresponding to the priors on the hyperparameters): $\log p(\mathbf{S}, \mathbf{Y}, \mathbf{Z} \mid X) = \log p(\mathbf{S}, \mathbf{Y} \mid X) + \log p(\mathbf{S} \mid \mathbf{Z}) + \log p(\mathbf{Z})$ w.r.t. the pseudo observations $\mathbf{S}$, the latent variables $\mathbf{Z}$ and the hyperparameters of the model to obtain the MAP estimates.
\paragraph{Implementation}We implement our model using the TensorFlow~\cite{Tensorflow} framework and minimise the negative marginal log-likelihood objective using the Adam optimizer~\cite{AdamOpt}. By default, we used standard squared exponential covariance functions for all the Gaussian process priors.
In some of the experiments, different covariance functions were used when the data or warping functions were less smooth (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot the Mat\'{e}rn covariance).
The complexity of our method is limited by the inversion of the covariance matrices and therefore scales with $\mathcal{O}(J N^3 + J^3)$.
However, there are standard sparse approaches available to scale to longer sequences. We also implemented the sparse variational method of Titsias~\cite{Titsias:2009} which reduces the complexity to $\mathcal{O}\!\left(J\,NM^2 + J^3\right)$, where $M$ is a specified number of inducing points for the sparse approximation. This method performed well for $M$ an order of magnitude smaller than the full $N$. We note that the use of a sparse approximation fits naturally with the rest of our model as it increases the smoothness of the observations, which may simplify the alignment task.
\subsection{Comparison of Variants of our Model}
\label{sec:param}
Our proposed model is fully non-parametric and models both the warping and the generating functions at the same time. Methods that rely on the standard $\mathbb{L}^2$ metric in the input space are ill-posed and thus require a regularisation term. This leads to an optimisation problem that suffers from poor local minima and relies on the use of a coarse-to-fine approach. In order to highlight the limitations of using the standard $\mathbb{L}^2$ metric in the input space, we describe a model that performs a parametric re-sampling of the data which corresponds to removing our model of the warpings but retaining a model of the data. In effect we take a traditional pairwise minimisation approach but include a probabilistic model of the data which has the effect of regularising the optimisation problem.
\paragraph{Parametric warps}We use a parametric re-sampling function $\tilde{g}^{(j)}(\cdot)$ similar to \cite{Zhou:2016} consisting of $K$ monotonically increasing basis functions. For each input sequence $Y_j$, we learn a set of weights $\mathbf{w}^{(j)}\in\mathbb{P}^{K}$. By enforcing that the weights lie on the surface of the $k^{\text{th}}$ order probability simplex $\mathbb{P}$ the resulting function is guaranteed to be monotonic. The task is now to find the set of weights $\{\mathbf{w}_k^{j}\}_{k=1}^K$ such that resampling the data according to the warping functions results in the aligned sequences. As we do not have access to $\mathbf{S}$, we use the same latent variable model as previously and refer to this model as \emph{GP-LVM+basis}. The model can be learned using gradient descent. The parametric model described above, as well as some previous approaches, rely on hand-picked basis functions to define the warps. This results in poor accuracy when the set of basis functions is small and in high computational complexity when the set is large.
\paragraph{Energy alignment}We demonstrate the efficacy of using the alignment GP-LVM to perform simultaneous clustering and alignment by replacing it with an energy minimisation objective that is similar to the previous literature, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot~\cite{Kurtek:2011}. The latent variable model part of the objective is replaced with an energy minimisation term between each of the $S_j$ and the mean of all the sequences $\{S_j\}_{j=1}^J$. In \S~\ref{sec:experiments} we show the results of this method with the GP warping functions (\emph{energy+GP}) and with the basis function warpings as described above (\emph{energy+basis}).
\section{Introduction}
\label{intro}
Learning from sequential data is challenging as data might be sampled at different and uneven rates, sequences might be collected out of phase, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot.
Consider the following scenarios: humans performing a task may take more or less time to complete parts of it, climate patterns are often cyclic though particular events take place at slightly different times in the year, the mental ability of children varies depending on their age, neuronal spike waveforms contain temporal jitter, replicated scientific experiments often vary in timing.
However, most sample statistics, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot~mean and variance, are designed to capture variation in amplitude rather than phase/timing. This leads to increased sample variance, blurred fundamental data structures and an inflated number of principal components needed to describe the data. Therefore, the data needs to be aligned in order for dependencies such as these to be recovered. This is a non-trivial task that is often performed as a pre-processing stage to modelling.
Traditionally, the notion of sequence similarity comes from a measure of pairwise similarity integrated across the sequences. This local measure often leads to highly non-convex optimisations problems making alignments challenging to learn. In this paper we take a different approach where we encapsulate alignment and modelling within a single framework. By simultaneously modelling the sequences and the alignment we can capture global structure thereby circumventing the difficulties associated with an objective function based on pairwise similarity.
Further difficulties arise when the the dataset contains observations from several distinct functions. Consider, for example, a set of motion capture experiments that include tasks such as running, jumping and sitting down. Data for each of these three types of sequences can be aligned to themselves but a global alignment between them may not exist. In traditional approaches, the observed data must be grouped into the distinct sequence types before alignment. To overcome this limitation, our approach also produces a generative model over the sequences themselves. This means we can simultaneously infer both alignments and their grouping.
Methods for learning alignments can broadly be classified into two categories. The first learns a function to warp the input dimension while the second directly learns the transformed sequences. There are several benefits to learning a warping function as it allows us to resample the data and, by constraining the class of functions, we can also incorporate global constraints on the alignment. However, specifying a parametric function is challenging and often results in difficult optimisation tasks.
Directly learning transformed sequences avoids having to specify a parametrisation. However, this comes at the cost of removing all but the most rudimentary global constraints on the warping function since the optimal alignment is completely specified by the pairwise similarity.
In contrast, we propose a novel approach that learns the warping function using a probabilistic model. Underpinning our methodology is the use of Gaussian process priors that allows us to approach this learning in a Bayesian framework achieving principled regularisation without reducing the solution space.
Our proposed model overcomes a number of problems with the existing literature and confers three main contributions:
\begin{enumerate}[topsep=-3pt,itemsep=-3pt,partopsep=0pt]
\item We model the observed data directly with a generative process, rather than interpolating between observations, that allows us to reject noise in a principled manner.
\item The generative model of the aligned data allows a fully unsupervised approach that performs simultaneous clustering and alignment.
\item We use continuous, non-parametric processes to model explicitly the warping functions throughout; this allows the specification of sensible priors rather than unintuitive or heuristic choices of parametrisations.%
\end{enumerate}
\section{Experiments} \label{sec:experiments}
We now discuss the experimental evaluation of our proposed model. We use standard squared exponential covariance functions for all the GP priors, unless stated otherwise. We show comparisons to current state-of-the-art approaches from data mining and functional data analysis communities using publicly available reference implementations\footnote{See~\cite{CTW:implement} for the implementation of DTW, DDTW, IMW, CTW, GTW, and~\cite{SRVF:implement} for the implementation of SRVF.}. The accuracy is primarily measured in terms of the warping error, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot~the mean squared error (MSE) between the known true warps and the estimated warps. The alignment error, the MSE between the pairs of aligned sequences, is easily misinterpreted since it is a local measurement. In particular, it does not capture the degenerate cases where the local maxima and minima in the input sequences are shifted to non-corresponding extrema; this is particularly true in datasets with periodic components. Other examples of degenerate behaviour are multiple dimensions collapsing to a single point and warps that rely on translating and rescaling every input in each dimension that leads to over-fitting (an example of this is IMW alignment~\cite{Zhou:2012}). All of these result in high alignment accuracy but produce poor quality results.
\paragraph{Datasets with quantifiable comparisons} \label{sec:toys}
For this experiment, we use the dataset proposed by Zhou and De la Torre~\cite{Zhou:2012}. It consists of sequences that are generated by temporally transforming latent 2D shapes under known warping transformations that allow quantitative evaluation of the estimated warps. To better assess the quality of the alignments, we run $25$ tests with randomly selected size of the dataset, dimensionality and temporal transformations. Our approach outperforms other methods on these datasets, see Fig.~\ref{fig:toys} and the supplementary material, and produces accurate alignments irrespective of the size of the dataset, dimensionality and structure of the sequences.
\begin{figure}[t!]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=0.92\textwidth]{warps_test21_n.png}
\caption{Comparison to state-of-the-art: average error on $25$ datasets proposed by Zhou \emph{et al}\onedot~\cite{Zhou:2012}. } \label{fig:toys}
\end{minipage}
\end{figure}
\begin{table}
\begin{minipage}{0.49\textwidth}
\setlength{\tabcolsep}{4pt}
\centering
\small
\scalebox{0.9}{
\begin{sc}
\begin{tabular}{lcccr}
\hline \\[-1pt]
MSE (SD) & srvf & gp-lvm+basis & \textbf{Ours} \\[1pt]
\hline \\[-1pt]
Alignment & 6.4 ($\pm$1.7) & 8.4 ($\pm$2.7) & \textbf{5.9} ($\pm$1.1) \\[2pt]
Warping & 30.0 ($\pm$10.4) & \textbf{9.7} ($\pm$4.9) & \textbf{9.7} ($\pm$5.7) \\[1pt]
\hline \\[-12pt]
\end{tabular}
\end{sc}
\caption{Quantitative comparison of alignments and warps for the best competing method on dataset with multiple true sequences (alignment and grouping task).}\label{table:quantitative}
\end{minipage}
\end{table}
The variant of our method that uses parametric warps (\emph{gplvm+basis}) performs competitively on these datasets, motivating the use of a Gaussian process objective for alignment. Furthermore, we see that our non-parametric approach to modelling the time warps improves the flexibility of the model; out of the two models that rely on energy minimisation as the alignment objective, \emph{energy+basis} and \emph{energy+gplvm}, the latter one demonstrates lower warping error and significantly lower standard deviation on this dataset. This result supports the premise that even though the non-parametric representation allows for any smooth monotonic warp, the probabilistic framework places sufficient structure to make the problem well posed and avoid over-fitting. An example of warps and alignments for this experiment are available in the supplementary material.
\paragraph{Dataset for clustering}
In our second experiment, we consider a dataset that contains multiple clusters of sequences. This task requires the sequences to be aligned within each cluster. None of PDTW, PCTW, GTW nor the energy minimisation methods are able to perform this task as they have no knowledge of the underlying structure of the dataset. The SRVF algorithm performs clustering by first aligning the data in terms of amplitude and phase, then performing fPCA based on the estimated summary statistics, and finally modelling the original data using joint Gaussian or non-parametric models on the fPCA representations. We compare the performance of the SRVF algorithm with our approach as well as the variant of our approach with fixed basis functions.
We consider a dataset that contains three distinct groups of functions that were generated by temporally transforming three random 2D curves as described previously. All three approaches rely on the structure of the data alone to recognise the existence of the clusters and Fig.~\ref{fig:clusters_align} shows that all three methods are able to align the data within clusters.
\begin{figure}[t!]
\centering
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth, height=7.cm]{cl_results_1pick_test_11_new_s.pdf}%
\end{subfigure}
\begin{subfigure}[h]{0.48\textwidth}
\includegraphics[width=\textwidth, height=7.cm]{cl_results_2pick_test_11_new_s.pdf}%
\end{subfigure}
\caption{Alignment of $15$ sequences that belong to $3$ different clusters (top 4 graphs) and the corresponding warping functions.} \label{fig:clusters_align}
\end{figure}
\begin{figure*}[t!]
\centering
\begin{minipage}{\textwidth}
\centering
\begin{subfigure}[h]{0.31\textwidth}
\includegraphics[width=\textwidth, height=3.3cm]{combined_mocap/results_no_align_data_2.pdf}
\end{subfigure}
\begin{subfigure}[h]{0.34\textwidth}
\includegraphics[width=\textwidth, height=3.3cm]{combined_mocap/results_7_1_aligned.pdf}
\end{subfigure}
\begin{subfigure}[h]{0.34\textwidth}
\includegraphics[width=\textwidth, height=3.3cm]{combined_mocap/results_7_1_warps.pdf}
\end{subfigure}\\
\caption{GP-LVM alignment demonstrates the preference for a simplified explanation when the model is given the ability to align the data.} \label{fig:exp2}
\end{minipage}%
\hspace{5pt}
\begin{minipage}{\textwidth}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[height=3.2cm]{combined_mocap/results_no_align_manifold_2_new.pdf}
\caption{Without alignment.}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[height=3.2cm]{combined_mocap/results_7_1_manifold_new.pdf}
\caption{With alignment.}
\end{subfigure}
\caption{2D manifolds produced without and with alignment in the GP-LVM. Using the alignments emphasizes the existence of multiple clusters of data and aligns data points within each cluster.}
\label{fig:manifolds}
\end{minipage}
\end{figure*}
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{0.39\textwidth}\centering
\includegraphics[height=2.8cm]{images/heartbeats_no_align3_crop.png}
\caption{The clustering of the unaligned observed sequences does not reveal the two types of heartbeats.} \end{subfigure}\hfill%
\begin{subfigure}[t]{0.59\textwidth}\centering
\includegraphics[height=2.8cm]{images/heartbeats_align2.png}
\caption{Accounting for the alignment of sequences allows us to discover automatically the two different types of heartbeats.} \end{subfigure}
\caption{Alignment of heartbeats data~\cite{Bentley:2011}.}\label{fig:heartbeats}
\end{figure*}
The performance of the methods is contrasted by calculating the MSE among all pairs of sequences within each group (alignment error) and the MSE between the true warping functions and the warping functions calculated using each of the methods (warping error). For this comparison we repeat the test $25$ times with randomly selected initial curves, number of dimensions and number of sequences per group. The quantitative comparison in Table~\ref{table:quantitative} shows that our method consistently achieves the lowest alignment errors (i.e. with lowest standard deviation (SD) on the set of datasets).
Our method, as well as the parametric variant of it, also achieves low warping errors in comparison to SRVF which implies that they are able to reconstruct the original temporal transformations more accurately than SRVF. This behaviour is apparent in Fig.~\ref{fig:clusters_align} where the warping functions produced by our method, and the parametric version of it, resemble the true warps while SRVF estimates noticeably different warping functions; this results in unpredictable distortions in the aligned dataset. These results reflects the differences between the SRVF method and our approach; while SRVF is cast as an optimisation problem over a constrained domain, the domain of our probabilistic formulation is much larger but, importantly, structured from the assumptions encoded in the prior. This provides a better regularisation ultimately leading to the improvement in the recovered warpings.
\paragraph{Motion capture data}We evaluate the performance of our model on a set of motion capture data from the CMU database~\cite{CMU:mocap}, where each input sequence corresponds to a short clip of motion and the data is represented as quaternion locations of the joints of the subject performing the motion. We use the motion of subject no $64$ from the CMU dataset that correspond to golf related motions such as a swing, a putt, and placing and picking up of a ball. We consider five instances of three different motions that need to be temporally aligned within the three groups. Fig.~\ref{fig:exp2} illustrates how our model favours the simplified, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot~aligned, inputs. The corresponding manifolds produced using a traditional GP-LVM (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot~without alignment) and a manifold produced using our approach are shown in Fig.~\ref{fig:manifolds}. Our model produces a fine alignment of the input sequences within each of the groups, and consequently the resulting two-dimensional manifold offers a good separation of the three groups. We note that the manifold produced using GP-LVM without alignment contains more isolated areas, which means the model is less capable of generalising between the warps. Therefore, our implicitly aligned model is able to generate smoother transitions in the manifold, producing high quality predicted outputs of novel alignments.
\paragraph{Heartbeats data} This dataset contains heartbeat sounds, and it is known that a normal heart sound has a clear "lub dub, lub dub” pattern which varies temporally depending on the age, health, and state of the subject~\cite{Bentley:2011}. Our approach automatically aligns and clusters the heart sounds recorded by a digital stethoscope. Instead of using a pre-processing step with a low-pass filter to account for the noise in the high frequencies, we use a Mat\'{e}rn $3/2$ kernel that takes into count the rapid variations in the recordings while also limiting the effect of the uninformative high frequency noise. Fig.~\ref{fig:heartbeats} illustrates how simultaneous fitting and alignment allows us to correctly discover and cluster the two types of heartbeats.
\section{Conclusion and Future Work}
\label{sec:conclusions}
We have presented a probabilistic model that is able to implicitly align inputs that contain temporal variations. Our approach models the observed data directly producing a generative model of the functions rather than interpolating between observations. In addition, using a GP-LVM for alignment builds an unsupervised generative model that has the benefit of simultaneous clustering and aligning the input sequences. Furthermore, we proposed a continuous, non-parametric explicit model of the time warping functions that removes issues such as quantisation artefacts and the need for ad-hoc pre-processing. We demonstrated that the proposed approaches perform competitively on alignment tasks, and outperform the existing methods on the task of simultaneous alignment and clustering. In the future we will consider the use of Bayesian GP-LVM for automatic model selection and will test the framework on additional datasets, including multi-modal data.
\section{Background}
\label{sec:background}
\label{sec:related_work}
\paragraph{Pairwise similarity}There has been a significant amount of work in learning alignments from data. Most approaches are based on the assumption of the existence of a pairwise similarity measure between the instances of each sequence.
The classical approach to minimise the distance between two sequences is called Dynamic Time Warping (DTW), and is based on a computing an affinity matrix
of the two sequences to be aligned~\cite{Berndt:1994}. The solution corresponds to the path through this matrix that leads to the minimal combined pairwise cost. The optimal solution is found by backtracking through the affinity matrix and can be estimated using Dynamic Programming~\cite{Muller:2007}. DTW finds the optimal alignment based on a pairwise distance between each element in two sequences. Such formulation imposes a number of limitations. DTW returns an alignment but not a parametrised warping, and it is not trivial to encode a preference towards different warps as this would be a global characteristic while DTW is a local algorithm.
\paragraph{Multiple sequences}In its original form DTW aligns two sequences only but several extensions allow it to process multiple sequences at once, most notably Procrustes dynamic time warping (PDTW), Procrustes derivative dynamic time warping (PDDTW), and Iterative Motion Warping (IMW)~\cite{Keogh:2001, Drydmard:2016, Hsu:2005}. All of these methods are applied directly in the observation space which is a limitation when the data contains a significant amount of noise. The main algorithms that address this limitation are Canonical Time Warping (CTW) and Generalized Time Warping (GTW)~\cite{Zhou:2009, Zhou:2012}. Both of these approaches perform feature extraction and find a subspace that maximises the linear correlation of data samples. Similarly to our approach, GTW is parametrised using monotonic warping functions. However, in all these methods the spatial alignment and time warping are coupled. Another extension, called Generalized Canonical Time Warping (GCTW) combines CCA with DTW to simultaneously align multiple sequences of multi-modal data~\cite{Zhou:2016}. GCTW relies on additional heuristic energy terms and on coarse-to-fine optimisation to get the energy method to converge to a good local minimum.
\paragraph{Feature extraction}More recently, deep neural networks were employed to perform temporal alignments~\cite{Trigeorgis:2016}, ~\cite{Trigeorgis:2017}. The proposed method, called Deep Canonical Time Warping (DCTW), performs non-linear feature extraction and it performs competitively on larger audio-visual datasets. A different method proposed by ~\cite{Listgarten:2004} uses continuous hidden Markov models, where the latent trace is an underlying representation of the set of observable sequences. \cite{Haxby:2011} introduced hyperalignment that finds isometric transformations of trajectories in voxel space that result in an accurate match of the time-series data. An extension to this model was proposed by~\cite{Lorbert:2012} who address the issues of scalability and feature extension through the use of the kernel trick. The authors note that classification accuracy relies on intelligent feature selection.
\paragraph{Manifold alignment}Similar to our approach,~\cite{Cui:2014} propose an unsupervised manifold alignment method. It is based on finding alignment by enforcing several constraints such as geometry structure and feature matching, geometry preservation and integer constraints. The approach shows promising results but is very computationally expensive. Another non-linear feature extraction method~\cite{Vu:2012} named Manifold Time Warping relies on constructing a k-nearest neighbour graph and then performing DTW to align a pair of sequences.
\paragraph{Implicit transformation}Another approach to alignment is to use an implicit transformation of the sequences. In \cite{Cuturi:2007,Cuturi:2011} the authors propose a kernel function that is capable of mapping sequences of different length to an implicit feature space. Another similar approach is \cite{Baisero:2015tm} which describes a range of different kernels on sequences, this method is flexible and allows for learning implicit feature space mappings for sequences of not only different lengths but also different dimensionality. These methods work well experimentally but as the alignment is implicit we cannot re-align sequences or construct novel ones.
\paragraph{Shape analysis}A different line of work, often referred to as elastic registration or shape analysis is considered in the functional data analysis literature. In \cite{Garreau:2014} the authors propose an extension to DTW by replacing the Euclidean distance with a Mahalanobis distance. By having a parametrisable distance function the authors are able to learn the metric function from a set of paired observations. \cite{Kurtek:2011} study the group-theoretic approach to warps by using the group of warping functions to describe the equivalence relation between signals. In particular, the authors use the Fisher-Rao Riemannian metric and the resulting geodesic distance to align signals with random warps, scalings and translations. Square root velocity function (SRVF) facilitates the use of Fisher-Rao distance between functions by estimating the $\mathbb{L}^2$ norm between their SRVFs~\cite{Srivastava:2011, Kurtek:2012}. \cite{Tucker:2013} proposed a generative model that combines elastic shape analysis of curves and functional principal component analysis (fPCA). Another recent extension called Elastic functional coding relies on trajectory embeddings on Riemannian manifolds and results in manifold functional variant of PCA~\cite{Anirudh:2015}.
\paragraph{Gaussian processes}\label{sec:gaussian_processes}%
Our model makes use of Gaussian processes as priors over warpings, sequences and their groupings. A Gaussian process (GP)~\cite{Rasmussen:2005} is a random process specified by a mean $m(x)$ and a covariance function $k_{\theta}(x,x')$. The covariance function is parametrised by a set of hyper-parameters $\theta$ while the mean is often considered as constant zero. The index set of the two functions is infinite which allows GPs to be interpreted as non-parametric priors over the space of functions. Even though the process is infinite, an instantiation of the process is finite and reduces to a Gaussian distribution. In a regression setting we observe a set of noisy samples $\mathcal{D} = \{x_{n}, y_n\}_{n=1}^N$ of a latent function $f(\cdot)$ such that $y_{n} = f(x_{n}) + \varepsilon_{n}$. By placing a GP prior over the latent function $f\sim \mathcal{GP}(m(x),k(x,x'))$ the instantiations of the function at the training data $\{f_n=f(x_n)\}_{n=1}^N$ are Gaussian as $F\sim\mathcal{N}(m(X),k(X,X))$ where $F$ and $X$ are concatenations of the function instantiations and the observed input locations respectively. By choosing a Gaussian noise model the functions can be marginalised out in closed form due to the self-conjugate property of the Gaussian distribution. The Gaussian process latent variable model (GP-LVM)~\cite{Lawrence:2005vk} is a model that uses GP priors to learn latent variables. The model assumes that each dimension of the observed data $Y$ have been generated from a latent variable $X$ through some latent function $f$. By placing a GP prior over $f$ and marginalising out this mapping, the latent representation of the data can be recovered. The model is very flexible and has been implemented across a wide range of different applications, for example~\cite{Campbell:2014, Grochow:2004, Urtasun:2005}.
\paragraph{Warped GPs} In~\cite{Snelson:2004} and~\cite{Lazaro:2012} the authors construct a GP with a warped input space to account for differences in observations (e.g.~inputs may vary over many orders of magnitude), and show that a warped GP finds the standard preprocessing transforms, such as the logarithm, automatically. In comparison, our approach leads to a warped output space of the GP-LVM, and uses the additional knowledge of possible misalignments in the high-dimensional space to regularise the problem of building a low-dimensional latent space. Concurrent with our work,~\cite{Duncker:2018} use GPs for modelling sequences of neural population spike-trains and the corresponding temporal warps. The proposed approach is an extension to GP factor analysis~\cite{Yu:2009} and uses a linear combination of shared and private latent processes to encourage alignment of sequences for different trails. Unlike our work,~\cite{Duncker:2018} do not recover a clustering of the sequences and thus require the groups of sequences for alignment to be known a-priori.
|
{
"timestamp": "2019-03-04T02:14:31",
"yymm": "1803",
"arxiv_id": "1803.02603",
"language": "en",
"url": "https://arxiv.org/abs/1803.02603"
}
|
\section{Introduction}
A common parameter used to extract mass composition information is \ensuremath{X_\mathrm{max}}\xspace, the atmospheric depth in \ensuremath{\mathrm{g/cm}^2}\xspace from the top of the atmosphere where the longitudinal development of an air shower reaches the maximum number of particles or the maximum of the energy deposited in the atmosphere. Different cosmic ray primaries propagate through the atmosphere differently, resulting in different observed distributions of \ensuremath{X_\mathrm{max}}\xspace~\cite{1977ICRC....8..353G}. Due to statistical variability in the interaction between cosmic rays of a specific primary mass and the atmosphere, a cosmic ray's primary mass cannot be determined on an event by event basis by examining \ensuremath{X_\mathrm{max}}\xspace. Instead we study the \ensuremath{X_\mathrm{max}}\xspace distribution of cosmic rays of similar energy to infer the mass composition distribution of the events. Differences in the mode, width and tail of the \ensuremath{X_\mathrm{max}}\xspace distribution provide information on the mass composition distribution of the events and on the hadronic interaction properties~\cite{Kampert:2012mx,Collaboration:2012wt}.
\begin{figure}[!htb]
\centering
\vspace{0.6cm}
\includegraphics[trim={0 1.3cm 0 0}, width=0.4\textwidth]{conex_proof_p_EPOS_QGSJET.pdf}
\caption{An \ensuremath{X_\mathrm{max}}\xspace distribution of 750 {Epos-LHC}\xspace simulated proton events (red), and separately 750 {QGSJetII-04}\xspace simulated protons events (blue), of energy \energy{18}.}
\label{fig:conex_proof_p_EPOS_QGSJET}
\end{figure}
\fig{fig:conex_proof_p_EPOS_QGSJET} shows the \ensuremath{X_\mathrm{max}}\xspace distribution resulting from the {CONEX v4r37}\xspace simulation of 750 proton events according to the {Epos-LHC}\xspace model, and separately 750 proton events according to the {QGSJetII-04}\xspace model, of energy \energy{18}. The figure illustrates the differences in the \ensuremath{X_\mathrm{max}}\xspace distribution predicted by different hadronic interaction models. Most noticeable is the difference in the modes of the distributions, but there are also marginal differences in the width and tails of the distributions. These differences between the hadronic interaction models change with energy to some degree. Although the dissimilarity between these predicted distributions may appear minor, applying a parameterisation based on these different predictions to data can have a considerable impact on the mass composition inferred. Consequently, typical mass composition studies of \ensuremath{X_\mathrm{max}}\xspace are strongly dependent on the hadronic interaction model assumed.
The algorithm {CONEX v4r37}\xspace~\cite{Bergmann:2006yz,Pierog:2004re}, along with the hadronic interaction packages {Epos-LHC}\xspace \cite{Pierog:2013ria}, {QGSJetII-04}\xspace \cite{Ostapchenko:2010vb} and {Sibyll2.3}\xspace \cite{sib}, were used to simulate air showers to obtain \ensuremath{X_\mathrm{max}}\xspace distributions according to each of these models. We have developed a parameterisation for des\-cri\-bing these expected \ensuremath{X_\mathrm{max}}\xspace distributions for cosmic rays of some energy and mass. Our parameterisation of the \ensuremath{X_\mathrm{max}}\xspace distributions can then be used to fit observed \ensuremath{X_\mathrm{max}}\xspace distributions, to extract primary mass information (composition fractions) from each energy bin. By in\-clu\-ding some of the coefficients of our \ensuremath{X_\mathrm{max}}\xspace parameterisation in the fit, mass composition results are obtained which are somewhat independent of the hadronic interaction model assumed.
Assuming the {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace hadronic models, the Auger \ensuremath{X_\mathrm{max}}\xspace distributions can be well reproduced assuming a composition of at least four components consisting of proton, Helium, Nitrogen and Iron~\cite{Aab:2014kda,Aab:2014aea,AugerCombinedFit}. Therefore, in this work we have used mock data sets to evaluate the performance of our method for retrieving the true relative amounts of p, He, N, Fe (composition fractions). The results of applying this method to interpret the published Auger \ensuremath{X_\mathrm{max}}\xspace distributions in~\cite{Aab:2014kda} in terms of the mass composition of cosmic rays are presented.
\section{\label{sec:param} Parameterisation of \ensuremath{\bf{X_\mathrm{max}\;}} distributions}
An \ensuremath{X_\mathrm{max}}\xspace distribution of some primary energy and mass can be modelled as the convolution of a Gaussian with an exponential \cite{Peixoto:2013tu}. Three shape parameters $(t_{0},\sigma,\lambda)$ define the \ensuremath{X_\mathrm{max}}\xspace distribution:
\begin{equation} \label{eq:Xmaxbasic}
\frac{dN}{d\text{X}_{\text{max}}}(t)\ = \frac{1}{2\lambda} \exp\left({\frac{t_{0} -t}{\lambda} + \frac{\sigma^2}{2\lambda^2}}\right)Erfc\left(\frac{t_{0}-t+\frac{\sigma^2}{\lambda}}{\sigma\sqrt{2}}\right)
\end{equation}
where $t_{0}$ defines the mode of the Gaussian component, $\sigma$ defines the width of the Gaussian component and $\lambda$ defines the exponential tail of the \ensuremath{X_\mathrm{max}}\xspace distribution, and $t$ is the \ensuremath{X_\mathrm{max}}\xspace bin. The mode and spread of the distribution defined in Equation~\eqref{eq:Xmaxbasic} is sensitive to $t_{0}$ and $\sigma$ respectively.
We fit Equation~\eqref{eq:Xmaxbasic} to the \ensuremath{X_\mathrm{max}}\xspace distributions from {CONEX v4r37}\xspace simulations of cosmic rays of a particular primary energy, mass (either proton, Helium, Nitrogen or Iron primaries) and hadronic interaction model, obtaining the values of $t_{0}$, $\sigma$ and $\lambda$ for that distribution (see Appendix \ref{AppA}). The fit results as a function of energy are displayed in \figsThree{fig:conex_epos}{fig:conex_qgs}{fig:conex_sib}. The solid lines are fits to the shape parameters ( $t_{0}$, $\sigma$ and $\lambda$ ) as a function of energy. The functions fitted are defined as follows:
\begin{equation} \begin{split} \label{eq:Xmaxbasicshape}
t_{0}(E) &= \ensuremath{t_{0_\mathrm{norm}}}\xspace + B\cdot \log_{10}\left(\frac{\log_{10}E}{\log_{10}E_0}\right),
\\
\sigma(E) &= \ensuremath{\sigma_{\mathrm{norm}}}\xspace + C\cdot \log_{10}\left(\frac{E}{E_0}\right),
\\
\lambda(E) &= \ensuremath{\lambda_{\mathrm{norm}}}\xspace - K + K \cdot \left(\frac{\log_{10}E}{\log_{10}E_0}\right)^{\frac{L}{\ln10}} \; ,
\end{split}\end{equation}
where E is the energy in eV and $E_0 = \energy{18.24} $, the energy at which we choose to normalise the equations. This energy corresponds to the energy at which Auger has measured $\lambda$ for a proton dominated composition~\cite{Collaboration:2012wt}. This means that $ \ensuremath{\lambda_{\mathrm{norm}}}\xspace $ for proton can be directly compared with $\Lambda_\eta$, the exponential tail measured by Auger, which is shown in Equation~\eqref{eq:AugerLambda}. We even considered adopting $\Lambda_\eta$ as the value for \ensuremath{\lambda_{\mathrm{norm}}}\xspace, but this could potentially break self consistency in the models.
\begin{equation} \label{eq:AugerLambda}
\Lambda_\eta = [55.8 \pm 2.3(stat) \pm 1.6(sys)]\; \ensuremath{\mathrm{g/cm}^2}\xspace
\end{equation}
The coefficients in Equation~\eqref{eq:Xmaxbasicshape} are specified in Appendix \ref{AppB} for each mass component and hadronic model.
\begin{figure}
\includegraphics[width=0.48\textwidth]{conexprofilesall_e.pdf}
\caption{Fits to the shape parameter as a function of energy according to the {Epos-LHC}\xspace model.}
\label{fig:conex_epos}
\end{figure}
\begin{figure}
\includegraphics[width=0.48\textwidth]{conexprofilesall_q.pdf}
\caption{Fits to the shape parameter as a function of energy according to the {QGSJetII-04}\xspace model.}
\label{fig:conex_qgs}
\end{figure}
\begin{figure}
\includegraphics[width=0.48\textwidth]{conexprofilesall_s_reduced.pdf}
\caption{Fits to the shape parameter as a function of energy according to the {Sibyll2.3}\xspace model.}
\label{fig:conex_sib}
\end{figure}
The functions of Equation~\eqref{eq:Xmaxbasicshape} consist of two parts, the first part defining the value of a shape parameter at the normalisation energy, and the second part defining the change in the shape parameter as a function of energy. For example, for protons \ensuremath{t_{0_\mathrm{norm}}}\xspace would be the value of $t_0$ for protons at \energy{18.24}, and similarly \ensuremath{\sigma_{\mathrm{norm}}}\xspace would be the value of $\sigma$ at \energy{18.24}.
\subsection{\label{sec.reso_acc} Accounting for the detector resolution and acceptance}
The expected \ensuremath{X_\mathrm{max}}\xspace distributions are affected by the detector resolution and the detector acceptance. The Pierre Auger \ensuremath{X_\mathrm{max}}\xspace publication~\cite{Aab:2014kda} provides parametrisations for the average detector \ensuremath{X_\mathrm{max}}\xspace resolution as a function of energy ($Res(E)$) and the detector acceptance as a function of \ensuremath{X_\mathrm{max}}\xspace for each energy bin, $Acc(E,t)$, where $t$ is the \ensuremath{X_\mathrm{max}}\xspace bin as in Equation~\eqref{eq:Xmaxbasic}.
The detector \ensuremath{X_\mathrm{max}}\xspace resolution is accounted for by adding it in quadrature with the corresponding $\sigma(E)$, to provide the total expected value of $\sigma(E)_{tot}$ for some primary:
\begin{equation} \label{eq:AcoountReso}
\sigma(E)_{tot} = \sqrt{\sigma(E)^2 + Res(E)^2}
\end{equation}
We can combine Equations~\eqref{eq:Xmaxbasic}, \eqref{eq:Xmaxbasicshape}, \eqref{eq:AcoountReso} and the detector acceptance $Acc(E,t)$ to obtain the expected \ensuremath{X_\mathrm{max}}\xspace distribution for cosmic rays of a mixture of primary masses in a particular energy bin according to a hadronic interaction model:
\begin{equation}
\begin{split}\label{eq:Xmaxfinal}
\frac{dN}{d\text{X}_{\text{max}}}(E,t)\bigg|_{\text{total}} &= \\
N(E)Acc(E,t)&\sum_{i=p,He,N,Fe}f_i(E)\:\frac{dN}{d\text{X}_{\text{max}}}(E,t)\bigg|_i
\end{split}
\end{equation}
where $f_p(E)$, $f_{He}(E)$, $f_{N}(E)$ and $f_{Fe}(E)$ are the fractions of proton, Helium, Nitrogen and Iron events respectively, and $N(E)$ is the total number of events. The fractions $f_p$, $f_{He}$, $f_{N}$ and $f_{Fe}$ are all correlated. Furthermore, the range of allowed values is not always $[0,1]$. This range changes depending on the values of the other fractions. For example, if $f_p$ were $0.9$, the allowed range for any of the other fractions would be $[0,0.1]$. In order to avoid changing the fraction limits in an iterative way, we have expressed the fractions $f_p$, $f_{He}$, $f_{N}$ and $f_{Fe}$ in terms of $\eta_1$, $\eta_2$ and $\eta_3$ as follows:
\begin{align}
\label{eq:massparameters}
f_p(E) &= \eta_1 \nonumber \\
f_{He}(E) &= (1-\eta_1)\eta_2 \nonumber \\
f_N(E) &= (1-\eta_1)(1-\eta_2)\eta_3 \nonumber \\
f_{Fe}(E) &= 1 - f_p(E) - f_{He}(E) - f_{N}(E)
\end{align}
Therefore, each energy bin has a set of $\eta_1$, $\eta_2$ and $\eta_3$ which defines the mass fractions of that energy bin. The allowed range for $\eta_1$, $\eta_2$ and $\eta_3$ is always $[0,1]$, consequently the mass fractions are constrained to values between 0 and 1 whilst the sum of the mass fractions equals 1. So, in practice we fit $\eta_1$, $\eta_2$ and $\eta_3$ to determine the corresponding fractions ($f_p$, $f_{He}$, $f_{N}$, $f_{Fe}$).
\fig{fig:predicted_Xmax_RMS} displays the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace and \ensuremath{\sigma(X_\text{max})}\xspace predictions of the three parameterisations for each primary. The predicted \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace separation of each adjacent mass component (eg. proton vs. helium, helium vs. nitrogen) within a parameterisation is approximately \SI{30}{\ensuremath{\mathrm{g/cm}^2}\xspace} to \SI{40}{\ensuremath{\mathrm{g/cm}^2}\xspace}. The predicted \ensuremath{\sigma(X_\text{max})}\xspace of the primaries is much larger for the {QGSJetII-04}\xspace and {Sibyll2.3}\xspace parameterisations than the {Epos-LHC}\xspace parameterisation.
\begin{figure*}
\centering \includegraphics[width=0.98\textwidth]{predicted_Xmax_RMS.pdf}
\caption{The \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace and \ensuremath{\sigma(X_\text{max})}\xspace predictions of the {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisations for proton (black), helium (red), nitrogen (green) and iron (blue).}
\label{fig:predicted_Xmax_RMS}
\end{figure*}
\subsection{\label{sec:validation}Validation of the parameterisation}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{realdata_comp_0_all.pdf}%
\caption{Fitting only the mass fractions of our parameterisations to FD \ensuremath{X_\mathrm{max}}\xspace data measured by the Pierre Auger Observatory. The error bars represent the statistical error of the fits. Included is the mass composition results for each hadronic model from the Pierre Auger Observatory analysis (labelled `Auger fits'). \cite{Aab:2014aea}.}
\label{fig:realdata_comp_0_all}
\end{figure}
\fig{fig:realdata_comp_0_all} displays the mass composition results of fitting the mass fractions using our {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisations and the \ensuremath{X_\mathrm{max}}\xspace data measured by the Pierre Auger Observatory fluorescence detector (FD) \cite{Aab:2014kda}. The fits took into account the detector resolution and acceptance. The mass composition obtained using our \ensuremath{X_\mathrm{max}}\xspace parameterisations are consistent with the Auger analysis of the 2014 FD \ensuremath{X_\mathrm{max}}\xspace data set \cite{Aab:2014aea}, where \ensuremath{X_\mathrm{max}}\xspace distribution templates from hadronic interaction models were compared to the data. The compatibility of our results with the 2014 Auger analysis validates the accuracy of our \ensuremath{X_\mathrm{max}}\xspace parameterisations.
\section{Method}\label{sec.method}
The parameters of Equation~\eqref{eq:Xmaxfinal} are fitted to energy binned \ensuremath{X_\mathrm{max}}\xspace distributions. The coefficients of Equation~\eqref{eq:Xmaxbasicshape} shown in Appendix \ref{AppB} were obtained with a global fit which included all energy bins.
When fitting (the \ensuremath{X_\mathrm{max}}\xspace distribution data) for the mass fraction parameters using our {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace parameterisation with the coefficients fixed (as in \fig{fig:realdata_comp_0_all}), the resulting mass composition reflects the characteristics of the corresponding hadronic model. Therefore, the estimated composition depends on which hadronic model is used. Additionally, the mass composition fitted to each energy bin is independent of the mass composition fitted to other energy bins. However, by including some of the coefficients shown in Appendix \ref{AppB} in the fit, in addition to the mass composition fractions, the mass composition obtained has a reduced dependence on the hadronic interaction model assumed. In this alternative case the mass composition fitted at each energy bin has some dependence with the fits at other energy bins. This is because the fitted coefficients (from the \ensuremath{X_\mathrm{max}}\xspace parameterisation) are fitted using all energy bins, while in the first case these coefficients were fixed.
\begin{figure}
\includegraphics[width=0.48\textwidth]{model_diff_comparison_eq.pdf}
\caption{{Epos-LHC}\xspace shape parameter value minus {QGSJetII-04}\xspace shape parameter value for some mass and energy.}
\label{fig:model_diff_comparison_eq}
\end{figure}
\begin{figure}
\includegraphics[width=0.48\textwidth]{model_diff_comparison_es.pdf}%
\caption{{Epos-LHC}\xspace shape parameter value minus {Sibyll2.3}\xspace shape parameter value for some mass and energy.}
\label{fig:model_diff_comparison_es}
\end{figure}
\begin{figure}
\includegraphics[width=0.48\textwidth]{model_diff_comparison_sq.pdf}%
\caption{{Sibyll2.3}\xspace shape parameter value minus {QGSJetII-04}\xspace shape parameter value for some mass and energy.}
\label{fig:model_diff_comparison_sq}
\end{figure}
In principle, if we were able to use the Auger \ensuremath{X_\mathrm{max}}\xspace data to perform a global fit of the mass composition and all of the coefficients from Equation~\eqref{eq:Xmaxbasicshape}, the resulting composition would be independent of the hadronic models, depending only on the assumed functional forms of the equations. However, the degeneracy between the fitted mass fractions and the coefficients makes it impossible to unambiguously constrain all of these parameters (i.e. the solution would be degenerate). Therefore, we need to identify which coefficients are most relevant for interpreting the mass composition, and evaluate whether we can unambiguously fit these coefficients and the mass composition. One way to identify which coefficients to include in a global fit is to compare the values of $t_{0}$, $\sigma$ and $\lambda$ between different models. This comparison will identify the parameters that are well or poorly constrained by our current knowledge of the high energy hadronic interaction physics.
\figsThree{fig:model_diff_comparison_eq}{fig:model_diff_comparison_es}{fig:model_diff_comparison_sq} illustrates the $t_{0}$, $\sigma$ and $\lambda$ difference between the {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace parameterisations at some energy and mass. The differences as a function of energy are relatively small. For example, the slope of $\Delta t_{0}$ as a function of energy is less than $\sim5\;\ensuremath{\mathrm{g/cm}^2}\xspace$ /energy-decade, which is small compared with an elongation rate of $60\;\ensuremath{\mathrm{g/cm}^2}\xspace$ /energy-decade. We have also verified that the separation between different primaries in the $t_{0}$, $\sigma$ and $\lambda$ space is similar for the three tested models. The main differences between our {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisations are the normalisation of $t_0$ and $\sigma$. The difference in the normalization of $\lambda$ is not negligible, but it has little impact on the mass composition interpretation. Therefore, when including \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in the global fit, we should obtain a similar interpretation of the mass composition with either the {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace \ensuremath{X_\mathrm{max}}\xspace distribution parameterisation. We choose to fit \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in the following way:
\begin{itemize}
\item \ensuremath{t_{0_\mathrm{norm}}}\xspace is fitted such that the absolute values of \ensuremath{t_{0_\mathrm{norm}}}\xspace for each primary change by the same amount. Therefore, the difference in \ensuremath{t_{0_\mathrm{norm}}}\xspace between primaries is conserved.
\item \ensuremath{\sigma_{\mathrm{norm}}}\xspace is fitted such that the ratio of $\sigma$ between primaries remains similar to the initial ratio over the energy range (differences in $C$ between primaries prevents the exact conservation of the initial ratio). Therefore, if \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons changes by $\Delta$, \ensuremath{\sigma_{\mathrm{norm}}}\xspace for other primaries will change by $\Delta$ multiplied by the initial average ratio of $\sigma$ between that primary and proton.
\end{itemize}
Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in this way assumes the hadronic models are correctly predicting the separation in $t_0$ between different species, and the ratio of $\sigma$ between different species, over the fitted energy range.
In Equation~\eqref{eq:Xmaxbasicshape}, the values of the shape parameters for Helium, Nitrogen and Iron can be expressed in terms of the corresponding values for protons, therefore fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in the way described above can be implemented by simply fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons.
In order to avoid unphysical fit results, we constrain the possible fitted values for \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. These constraints are significantly wider than the separation between the {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisation predictions for these coefficients. The predicted value of \ensuremath{t_{0_\mathrm{norm}}}\xspace for protons according to {Epos-LHC}\xspace is $\sim \SI{703}{\ensuremath{\mathrm{g/cm}^2}\xspace}$, according to {QGSJetII-04}\xspace is $\sim \SI{688}{\ensuremath{\mathrm{g/cm}^2}\xspace}$, and according to {Sibyll2.3}\xspace is $\sim \SI{714}{\ensuremath{\mathrm{g/cm}^2}\xspace}$. The minimum and maximum limits of \ensuremath{t_{0_\mathrm{norm}}}\xspace for protons are set to \depth{670} and \depth{765} respectively. The predicted value of \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons according to {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace is $\sim$ \depth{22}, $\sim$ \depth{25} and $\sim$ \depth{28} respectively. The minimum and maximum limits of \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons are set to \depth{5} and \depth{55} respectively.
With a suitable shift in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, many primary mixtures which produce a fairly smooth total distribution can be fitted well with a single dominant distribution, instead of a sum of distributions. On the other hand, a distribution dominated by a single primary can be well fitted by a balanced mixture of distributions when \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace are shifted appropriately. It is common that \ensuremath{X_\mathrm{max}}\xspace distributions can be fitted with a value of \ensuremath{t_{0_\mathrm{norm}}}\xspace for protons much larger than the true \ensuremath{t_{0_\mathrm{norm}}}\xspace of the distributions, which results in the primary mass of the events being overestimated (i.e. biased towards heavier masses). Therefore, it is important that appropriate shape coefficient limits are chosen.
We have evaluated the performance of fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in addition to the mass fractions using simulated \ensuremath{X_\mathrm{max}}\xspace distributions of a known composition (see details in Sec. \ref{sec.performance}). Provided there is enough dispersion of masses in the data, it is possible to fit with good accuracy, \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the corresponding abundance (fractions) of p, He, N and Fe. An important achievement from including \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in the fit is that the mass composition interpretation becomes consistent whether using the predicted {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace parameterisation.
The requirement of a large dispersion of masses is evaluated over the entire energy range. For example, a data set consisting of a pure proton composition at higher energies can be fitted, provided that at lower energies we have populations consisting of other primaries. If the statistics or mass dispersion were not large enough, there would be some degeneracy in the fit between the mass fractions and \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. A greater change in the mass composition with energy improves the accuracy of the fit.
Apart from the dispersion of masses in the data, the performance of the fit depends on the intrinsic values for $\sigma$ of the data. This is nature's width for the \ensuremath{X_\mathrm{max}}\xspace distribution of the different primaries. The separation of the distribution modes between primaries remains unchanged in the fit, therefore primary \ensuremath{X_\mathrm{max}}\xspace distributions of larger width will increase the \ensuremath{X_\mathrm{max}}\xspace distribution overlap of adjacent primaries, resulting in the fit of \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass composition becoming more uncertain.
We have also evaluated the performance of fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, $B$, and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in addition to the mass fractions, where $B$ defined in Equation~\eqref{eq:Xmaxbasicshape} describes the change in $t_0$ with energy. As the predicted mass composition is particularly sensitive to the predicted values of $t_0$, $B$ is a powerful coefficient which can significantly affect the fitted mass composition. We fit $B$ such that for each primary the value of $B$ changes by the same amount from the initial predicted value, thus the initial predicted differences among primaries in the rate of change of $t_0$ with energy are conserved (identical to how \ensuremath{t_{0_\mathrm{norm}}}\xspace is fitted). Our \ensuremath{X_\mathrm{max}}\xspace parameterisations have similar values for $B$, therefore we do not expect fits of $B$ to yield results significantly different from the initial prediction of $B$ when we are fitting {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace simulated \ensuremath{X_\mathrm{max}}\xspace data. However, if the values of $B$ predicted by our parameterisations are significantly incorrect for the data being fitted, considerable systematics would be introduced to the reconstructed mass composition if $B$ remains fixed.
Data sets that can be fitted with \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace may not be accurately fitted when $B$ is included in the fit, as fitting extra coefficients increases the degeneracy between the fitted variables. Fitting these three coefficients accurately requires a greater spread of primaries and/or statistics than fitting just \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. The predicted value of $B$ for protons according to {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace is $\sim$ \depth{2533}, $\sim$ \depth{2445} and $\sim$ \depth{2666} respectively. With \ensuremath{t_{0_\mathrm{norm}}}\xspace normalised at \energy{18.24}, a change in $B$ of \depth{350} corresponds to a change in $t_0$ at \energy{19.5} of $\sim$\depth{10}. The fitting range limits of $B$ for protons is \depth{1000} to \depth{4000}.
We have also considered constraining $t_0$ at \energy{14}, where the hadronic models are more reliable, and fitting $B$ and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. Fitting $B$ in this way can also provide a consistent mass fraction result between the {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace parameterisation fits of simulated \ensuremath{X_\mathrm{max}}\xspace data, as the $t_0$ prediction of the fitted energy range adjusts in a way that is similar to the \ensuremath{t_{0_\mathrm{norm}}}\xspace fit, with the added advantage that unlike the \ensuremath{t_{0_\mathrm{norm}}}\xspace fit, the resulting fitted parameterisation of $t_0$ is consistent with the hadronic model predictions at lower energies. We have found that over the energy range of interest (\energy{17.8} to \energy{20}), fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace results in a more accurate mass composition reconstruction compared to fitting $B$ and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. This is because there is less degeneracy between the fitted mass fractions and shape parameters when fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. Additionally, a $t_0$ parameterisation constrained at \energy{18.24} describes the energy range of interest better than a $t_0$ parameterisation extrapolated from \energy{14}. If a wider energy range was being fitted, then a \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fit would be less accurate, because the $t_0$ and $\sigma$ parameterisations of different models do not adequately align over a wider energy range by only adjusting their normalisations. It is also important to recognise that this fit of $B$ is restricted, as we are fixing how $t_0$ changes with energy, and only fitting the rate of change of the $\log_{10}\left(\frac{\log_{10}E}{\log_{10}E_0}\right)$ factor. To properly fit the slope of $t_0$ with energy would require the fit of a third $t_0$ parameter (for example, fitting $B$ and $x$ in $B\cdot\log_{10}\left(\frac{\log_{10}E}{\log_{10}E_0}\right)^x$, where $x$ currently equals 1).
We have evaluated the effect of different \ensuremath{X_\mathrm{max}}\xspace bin sizes and energy bin sizes on the performance of the fit. When fitting only the mass fractions, \depth{1} \ensuremath{X_\mathrm{max}}\xspace binning gives marginally more accurate results than \depth{20} \ensuremath{X_\mathrm{max}}\xspace binning (\depth{20} is the \ensuremath{X_\mathrm{max}}\xspace bin size of the Auger \ensuremath{X_\mathrm{max}}\xspace distributions published in~\cite{Aab:2014kda}). The absolute improvement in the fitted mass fractions is no greater than $3\%$ in an energy bin. However, when fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in addition to the mass fractions, using a small \ensuremath{X_\mathrm{max}}\xspace binning is more important, otherwise the chosen center of the \ensuremath{X_\mathrm{max}}\xspace bins may significantly affect the fitted results, especially if the statistics are not large. The predicted separation between different primaries in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace can be very small. For example, our {Epos-LHC}\xspace parameterisation predicts the difference in \ensuremath{t_{0_\mathrm{norm}}}\xspace between proton and helium is only $\sim \depth{6}$.
Therefore, a \depth{20} \ensuremath{X_\mathrm{max}}\xspace binning (as published in \cite{Aab:2014kda}) can be too coarse, and can shift the apparent \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace of the distribution, which affects the fit of \ensuremath{t_{0_\mathrm{norm}}}\xspace.
Due to similar reasons, the energy bin size is also important. Energy binning that is too large can result in data from the same primary mass, but on opposite extremes of the energy bin, being evaluated as data from different primaries. This is because the separation between the predicted \ensuremath{X_\mathrm{max}}\xspace distributions of different primaries is small compared to the shift in these \ensuremath{X_\mathrm{max}}\xspace distributions with energy. We find that an energy binning of $0.1$ in $\log_{10}(E/\text{eV})$ is reasonable.
\section{Performance}\label{sec.performance}
Using {CONEX v4r37}\xspace, 100 \ensuremath{X_\mathrm{max}}\xspace data sets were generated according to both the {Epos-LHC}\xspace and {QGSJetII-04}\xspace hadronic interaction models for a number of different mass compositions. The data consists of 17 energy bins, of which there are 13 energy bins of a width of $0.1$ in $\log_{10}(E/\text{eV})$ between \energy{17} and \energy{18.3}, and 4 fixed energy bins at \energy{18.5}, \energy{18.7}, \energy{19} and \energy{19.5}. Each energy bin contains approximately 750 events. The binning of the simulated \ensuremath{X_\mathrm{max}}\xspace distributions is \depth{1}.
We have fitted only the mass fractions (all coefficients from the \ensuremath{X_\mathrm{max}}\xspace parameterisation were kept fixed) to data of a single primary generated with the same hadronic interaction model the parameterisation fitted is based on. Figs.~\ref{fig:EPOSLHC_pureP} to \ref{fig:EPOSLHC_pureFe} summarises the results (of these 100 fits) for the {Epos-LHC}\xspace hadronic model and Figs.~\ref{fig:QGSJET_pureP} to \ref{fig:QGSJET_pureFe} for the {QGSJetII-04}\xspace model. The markers represent the medians of the fitted mass fractions, and the error bars represent the standard deviation. The results show that our \ensuremath{X_\mathrm{max}}\xspace parameterisations are an accurate description of the expected \ensuremath{X_\mathrm{max}}\xspace distribution of a primary according to the {Epos-LHC}\xspace or {QGSJetII-04}\xspace hadronic interaction models. Both our {Epos-LHC}\xspace and {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisation fits can accurately determine the mass composition of data from the same hadronic model.
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100proton_EPOS_30_fitmass.pdf}
\caption{ Fitting only the mass fractions to mock data sets of \ensuremath{X_\mathrm{max}}\xspace distributions. The data sets have been generated using the {Epos-LHC}\xspace model and assuming a {\bf{proton}} primary composition over the whole energy range. The composition fits were performed using our \ensuremath{X_\mathrm{max}}\xspace parameterisations for the {Epos-LHC}\xspace model predictions. `Rec. mass' refers to the mass fractions fitted to the data.}
\label{fig:EPOSLHC_pureP}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100He_EPOS_30_fitmass.pdf}
\caption{Same as Fig.\,~\ref{fig:EPOSLHC_pureP}, but assuming a {\bf{Helium}} primary composition over the whole energy range.}
\label{fig:EPOSLHC_pureHe}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100N_EPOS_30_fitmass.pdf}
\caption{Same as Fig.\,~\ref{fig:EPOSLHC_pureP} but assuming a {\bf{Nitrogen}} primary composition over the whole energy range.}
\label{fig:EPOSLHC_pureN}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100Fe_EPOS_30_fitmass.pdf}
\caption{Same as Fig.\,~\ref{fig:EPOSLHC_pureP}, but assuming an {\bf{Iron}} primary composition over the whole energy range.}
\label{fig:EPOSLHC_pureFe}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100proton_QGSJET_31_fitmass.pdf}
\caption{ Fitting only the mass fractions to mock data sets of \ensuremath{X_\mathrm{max}}\xspace distributions. The data sets have been generated using the {QGSJetII-04}\xspace model and assuming a {\bf{proton}} primary composition over the whole energy range. The composition fits were performed using our \ensuremath{X_\mathrm{max}}\xspace parameterisations for the {QGSJetII-04}\xspace model predictions.}
\label{fig:QGSJET_pureP}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100He_QGSJET_31_fitmass.pdf}
\caption{Same as Fig.\,~\ref{fig:QGSJET_pureP}, but assuming a {\bf{Helium}} primary composition over the whole energy range.}
\label{fig:QGSJET_pureHe}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100N_QGSJET_31_fitmass.pdf}
\caption{Same as Fig.\,~\ref{fig:QGSJET_pureP} but assuming a {\bf{Nitrogen}} primary composition over the whole energy range.}
\label{fig:QGSJET_pureN}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width = 0.48\textwidth]{100Fe_QGSJET_31_fitmass.pdf}
\caption{Same as Fig.\,~\ref{fig:QGSJET_pureP}, but assuming an {\bf{Iron}} primary composition over the whole energy range.}
\label{fig:QGSJET_pureFe}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=0.48\textwidth]{50proton50He_50He50N_EPOS_30_fitmass.pdf}%
\caption{Fitting only the mass fractions of our {Epos-LHC}\xspace parameterisation to {Epos-LHC}\xspace \ensuremath{X_\mathrm{max}}\xspace data.}
\label{fig:50proton50He_50He50N_EPOS_30_fitmass}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=0.48\textwidth]{50proton50He_50He50N_QGSJET_31_fitmass.pdf}%
\caption{Fitting only the mass fractions of our {QGSJetII-04}\xspace parameterisation to {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace data.}
\label{fig:50proton50He_50He50N_QGSJET_31_fitmass}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=0.48\textwidth]{50proton50He_50He50N_QGSJET_31_fitt0sigma.pdf}%
\caption{Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions of our {QGSJetII-04}\xspace parameterisation to {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace data.}
\label{fig:50proton50He_50He50N_QGSJET_31_fitt0sigma}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=0.48\textwidth]{50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma.pdf}
\caption{Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions of our {QGSJetII-04}\xspace parameterisation to {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace data. Helium has been replaced by Iron in the last energy bin to increase the mass dispersion.}
\label{fig:50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=0.48\textwidth]{50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma.pdf}
\caption{Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions of our {Epos-LHC}\xspace parameterisation to {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace data. Helium has been replaced by Iron in the last energy bin to increase the mass dispersion.}
\label{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma}
\end{figure}
\begin{figure}[htbp!]
\includegraphics[width=0.48\textwidth]{50proton50He_50He50N_50N50Fe_QGSJET_30_fitmass.pdf}
\caption{Fitting only the mass fractions (i.e. \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace are kept fixed) of our {Epos-LHC}\xspace parameterisation to {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace data. Compare this Fig. with \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma} where \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace were included in the fit.}
\label{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitmass}
\end{figure}
\fig{fig:50proton50He_50He50N_EPOS_30_fitmass} to \fig{fig:50proton50He_50He50N_QGSJET_31_fitt0sigma} summarises the results of fits to 100 \ensuremath{X_\mathrm{max}}\xspace data sets with a true mass composition consisting of $50\%$ proton and helium in the first 8 energy bins, and $50\%$ helium and nitrogen in the remaining 9 energy bins. When fitting only the mass fractions (i.e. keeping fixed the coefficients of the \ensuremath{X_\mathrm{max}}\xspace distribution parameterisation) of our parameterisations to {CONEX v4r37}\xspace \ensuremath{X_\mathrm{max}}\xspace data based on the same model, the fits are able to reconstruct the mass composition to within an absolute offset in the median of $10\%$ from the true mass (as seen in \figs{fig:50proton50He_50He50N_EPOS_30_fitmass}{fig:50proton50He_50He50N_QGSJET_31_fitmass}).
\fig{fig:50proton50He_50He50N_QGSJET_31_fitt0sigma} shows the results of fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, in addition to the mass fractions, of the {QGSJetII-04}\xspace parameterisation to {QGSJetII-04}\xspace data. These {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace distributions do not provide sufficient constraints on our fitted parameterisation, resulting in a mass composition reconstruction that does not resemble the true mass composition. In order to successfully fit \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace to data of a similar distribution, a wider range of primary masses over the energy range of the data is required (wider than the one in the given example). For example, in \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma} we have increased the range of primary masses by replacing helium with iron in the last energy bin. The resulting fit of the mass fractions (with \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace also fitted) have an absolute offset in the median of less than $\sim 15\%$ from the true values, which is comparable to a fit of only the mass fractions to data of a similar composition.
\begin{figure}
\includegraphics[width=0.48\textwidth]{histcorr_50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma.pdf}
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma}.}
\label{fig:hist_50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma}
\end{figure}
\begin{figure}
\includegraphics[width=0.48\textwidth]{histcorr_50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma.pdf}
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma}.}
\label{fig:hist_50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma}
\end{figure}
\subsection{Fitting data originating from a different model.}
Compare \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma} with \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitmass}, which shows the composition fits when using the {Epos-LHC}\xspace parameterisation to fit {QGSJetII-04}\xspace data, with \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fitted in the former, and \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fixed in the latter. Fitting these two coefficients is enough to result in a reconstructed mass much closer to the true mass, despite the fitted data originating from a different model. By fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, there is no longer a significant iron component where there should only be $50\%$ helium and nitrogen, and in the $50\%$ proton and helium range there is no longer a fitted nitrogen component larger than the helium fraction.
\figs{fig:hist_50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma}{fig:hist_50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma} show the difference between the fitted values and initial values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace (and their correlation) when fitted to the data with iron added in the last energy bin. \fig{fig:hist_50proton50He_50He50N_50N50Fe_QGSJET_31_fitt0sigma} displays the results of fitting {QGSJetII-04}\xspace data with our {QGSJetII-04}\xspace parameterisation, and as expected the difference between the reconstructed and initial values of our coefficients is minimal. \fig{fig:hist_50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma} displays the results of fitting the same {QGSJetII-04}\xspace data with our {Epos-LHC}\xspace parameterisation (the reconstructed mass is shown in \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma}), and we see that \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace are shifted towards the {QGSJetII-04}\xspace values for these coefficients. The initial {Epos-LHC}\xspace proton \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace values are $\sim \depth{703}$ and $\sim \depth{22}$ respectively, while the initial {QGSJetII-04}\xspace proton \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace values (and therefore the approximate values of the {QGSJetII-04}\xspace MC data) are $\sim \depth{688}$ and $\sim \depth{25}$ respectively.
Notice that in \fig{fig:50proton50He_50He50N_EPOS_30_fitmass} to \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma} the bins containing a helium and nitrogen mix are reconstructed better than the bins containing a proton and helium mix. Proton and helium distributions are harder to reconstruct due to their wider spread and their larger overlap. A wider spread means that for a given number of events, less events will populate individual \ensuremath{X_\mathrm{max}}\xspace bins. Therefore, proton and helium fits have larger statistical uncertainties. Additionally, the \ensuremath{X_\mathrm{max}}\xspace parameterisations for lighter masses do not describe the {CONEX v4r37}\xspace \thinspace {Epos-LHC}\xspace and {QGSJetII-04}\xspace simulated data as accurately. \fig{fig:EPOS_QGS_diff_mean_sigma_all} in Appendix \ref{AppA} illustrates that as the primary mass of the distribution increases, the \ensuremath{X_\mathrm{max}}\xspace parameterisations reproduce the true \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace and \ensuremath{\sigma(X_\text{max})}\xspace of the distributions with better accuracy. Appendix \ref{AppA} shows that for proton and helium data especially, the fits of Equation~\eqref{eq:Xmaxbasic} to MC data of either hadronic model tend to overestimate the number of events at the mode of the distribution. When fitting mixes of protons and helium, our fits tend to have a reconstruction bias towards protons.
As the absolute separation between $\sigma$ for different primaries is similar in the {Epos-LHC}\xspace and {QGSJetII-04}\xspace parameterisations (like $t_0$), marginally better results would be obtained in \fig{fig:50proton50He_50He50N_50N50Fe_QGSJET_30_fitt0sigma} if instead of fitting \ensuremath{\sigma_{\mathrm{norm}}}\xspace such that the initial ratios of $\sigma$ among primaries are conserved, \ensuremath{\sigma_{\mathrm{norm}}}\xspace was fitted such that the initial separation between \ensuremath{\sigma_{\mathrm{norm}}}\xspace among primaries was conserved (like \ensuremath{t_{0_\mathrm{norm}}}\xspace). However, conserving the initial ratios of $\sigma$ is the more physical approach, because if \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons changes by \depth{10}, we would not expect that \ensuremath{\sigma_{\mathrm{norm}}}\xspace for iron would also change by \depth{10}. Additionally, nature does not necessarily conform to the {Epos-LHC}\xspace or {QGSJetII-04}\xspace predictions of the absolute separation of \ensuremath{\sigma_{\mathrm{norm}}}\xspace among primaries.
\section{\ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace parameter space scan of the Auger FD \ensuremath{X_\mathrm{max}}\xspace data}
\label{tnorm and sigmanorm parameter space scan of the Auger FD Xmax data}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.48\textwidth]{FDdata_t0sigma_scan_extended.pdf}
\caption{The \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace parameter space scan over the Auger FD \ensuremath{X_\mathrm{max}}\xspace data. For each model parameterisation, at specific values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, the mass fractions are fitted to the data, and the first $5\sigma$ contours of the minimised Poisson log likelihood are shown. The scanned shape coefficient values for proton are shown. The coefficient values of the heavier nuclei change (relative to protons) in the way the shape coefficient would be fitted, outlined in Section~\ref{sec.method}.}
\label{fig:FDdata_t0sigma_scan}
\end{figure}
\fig{fig:FDdata_t0sigma_scan} shows the minimised Poisson log likelihood space of the mass fraction fit of a parameterisation to Auger FD \ensuremath{X_\mathrm{max}}\xspace data, where \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace have been fixed to some particular value (indicated by the x and y axes). The z-axis shows the difference between the minimised probability for some value of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, and the absolute minimised probability obtained from the \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace values which best fitted the data for a particular parameterisation. A difference of 1 in the minimised Poisson log likelihood corresponds to $1\sigma$. The absolute minima of the {Epos-LHC}\xspace and {QGSJetII-04}\xspace fits to the Auger FD data correspond to a similar value of \ensuremath{t_{0_\mathrm{norm}}}\xspace for protons, whereas the absolute minimum of the {Sibyll2.3}\xspace fit is located at a significantly larger value of \ensuremath{t_{0_\mathrm{norm}}}\xspace for protons. Between the three fitted parameterisations, when estimating the heavier nuclei \ensuremath{t_{0_\mathrm{norm}}}\xspace values there is more similarity. This is because the separation between the proton $t_0$ prediction and heavier nuclei is larger in the {Sibyll2.3}\xspace parameterisation than {Epos-LHC}\xspace or {QGSJetII-04}\xspace (see \figsThree{fig:model_diff_comparison_eq}{fig:model_diff_comparison_es}{fig:model_diff_comparison_sq}). This is also true for $\sigma$.
These scans show that the fits of the Auger FD \ensuremath{X_\mathrm{max}}\xspace data performed in Section~\ref{sec.res} did not become stuck in a local minimum. The scans can also reveal secondary solutions which are not as deep as the deepest minimum.
\section{Evaluating the fit performance for a mass composition consistent with the Auger results}
\label{sec.augerperformance}
The performance of fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions of our parameterisations to the Auger FD \ensuremath{X_\mathrm{max}}\xspace data is evaluated by fitting mock \ensuremath{X_\mathrm{max}}\xspace data sets that resemble the Auger FD \ensuremath{X_\mathrm{max}}\xspace distributions. This was achieved by fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions of a particular parameterisation to the Auger FD \ensuremath{X_\mathrm{max}}\xspace data, and then using this fitted parameterisation to generate the mock data sets. Appendix~\ref{AppB} displays the \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace values fitted to the Auger data, values which correspond to the absolute minima found from the scans in Section~\ref{tnorm and sigmanorm parameter space scan of the Auger FD Xmax data}. These mock data sets have a true mass composition which is defined by the parameterisation used to generate them, therefore we can evaluate the ability of our \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and mass fraction fit to accurately reconstruct the true mass fractions. The binning of the mock Auger \ensuremath{X_\mathrm{max}}\xspace distributions is \depth{20}.
The measured FD \ensuremath{X_\mathrm{max}}\xspace distributions are broadened by the \ensuremath{X_\mathrm{max}}\xspace resolution of the detector, and are affected by the detector acceptance, therefore the mock \ensuremath{X_\mathrm{max}}\xspace data generated from the fitted parameterisation are convolved with the same detector effects. The \ensuremath{X_\mathrm{max}}\xspace resolution and acceptance of the Auger data is taken into account when fitting this mock Auger \ensuremath{X_\mathrm{max}}\xspace data. Our mock \ensuremath{X_\mathrm{max}}\xspace distributions and the \ensuremath{X_\mathrm{max}}\xspace distributions measured by Auger are treated with exactly the same approach.
\subsection{Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions}
\figsThree{fig:EPOS_30_fitt0sigma_mod}{fig:EPOS_31_fitt0sigma_mod}{fig:EPOS_32_fitt0sigma_mod} display the mass composition results from fitting the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of either the {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace parameterisations respectively, to 100 data sets generated from the parameterisation which resulted when the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of the {\bf{Epos-LHC}\xspace} parameterisation were fitted to Auger FD \ensuremath{X_\mathrm{max}}\xspace data (as will be shown in Section~\ref{sec.res}). The true mass composition of the mock data is therefore the mass composition which resulted from the {Epos-LHC}\xspace fit to the Auger FD \ensuremath{X_\mathrm{max}}\xspace data. \figsThree{fig:hist_EPOS_30_fitt0sigma}{fig:hist_EPOS_31_fitt0sigma}{fig:hist_EPOS_32_fitt0sigma} display the fitted proton values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace relative to the original values of the model applied, compared to the change required to match the true proton values of the mock data. The red lines indicate the mock data input values and the blue histograms are the reconstructed values. The correlations between the reconstructed \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace are also shown in \figsThree{fig:hist_EPOS_30_fitt0sigma}{fig:hist_EPOS_31_fitt0sigma}{fig:hist_EPOS_32_fitt0sigma}. There are no reconstruction systematics when using the {Epos-LHC}\xspace parameterisation to fit {Epos-LHC}\xspace generated data (\fig{fig:hist_EPOS_30_fitt0sigma}), but there are some systematics when using the {QGSJetII-04}\xspace or {Sibyll2.3}\xspace parameterisations to fit {Epos-LHC}\xspace generated data (\figs{fig:hist_EPOS_31_fitt0sigma}{fig:hist_EPOS_32_fitt0sigma}). These systematics in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace translate into relative small systematics of the reconstructed mass fractions (as seen in \figs{fig:EPOS_31_fitt0sigma_mod}{fig:EPOS_32_fitt0sigma_mod}).
\figs{fig:EPOS_31_fitt0sigma_mod}{fig:hist_EPOS_31_fitt0sigma} show that despite the differences between the {Epos-LHC}\xspace and {QGSJetII-04}\xspace parameterisations (which are not limited to different \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace predictions), by allowing \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of the {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisation to be fitted to mock data based on the {Epos-LHC}\xspace parameterisation, the true mass fractions are reconstructed with an overall accuracy comparable to the {Epos-LHC}\xspace fits of {Epos-LHC}\xspace data. The absolute offsets in the median mass fractions from the true mass are less than $10\%$ in most energy bins. This demonstrates that fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace significantly reduces the differences between the {Epos-LHC}\xspace and {QGSJetII-04}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisations. As we are fitting the {QGSJetII-04}\xspace parameterisation to mock data based on the {Epos-LHC}\xspace parameterisation, we do not expect the average fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace to be centred on the red lines even if no systematic offset was present in the mass fractions reconstruction. This is because the separation of these coefficients between masses differs between the {Epos-LHC}\xspace and {QGSJetII-04}\xspace parameterisations, thus if the fitted {QGSJetII-04}\xspace value of \ensuremath{t_{0_\mathrm{norm}}}\xspace for protons was equal to the {Epos-LHC}\xspace value of \ensuremath{t_{0_\mathrm{norm}}}\xspace for protons, the accordingly adjusted \ensuremath{t_{0_\mathrm{norm}}}\xspace values of other masses would differ between these parameterisations.
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance30fit1100_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{{Epos-LHC}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {Epos-LHC}\xspace parameterisation fit of Auger data. }
\label{fig:EPOS_30_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance31fit1100_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{{QGSJetII-04}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {Epos-LHC}\xspace parameterisation fit of Auger data.}
\label{fig:EPOS_31_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance32fit1100_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{{Sibyll2.3}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {Epos-LHC}\xspace parameterisation fit of Auger data.}
\label{fig:EPOS_32_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance30fit1100_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:EPOS_30_fitt0sigma_mod}.}
\label{fig:hist_EPOS_30_fitt0sigma}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance31fit1100_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:EPOS_31_fitt0sigma_mod}.}
\label{fig:hist_EPOS_31_fitt0sigma}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance32fit1100_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:EPOS_32_fitt0sigma_mod}.}
\label{fig:hist_EPOS_32_fitt0sigma}
\end{figure}
The mass composition reconstruction accuracy of the {Epos-LHC}\xspace fit to {Epos-LHC}\xspace based data changes less with energy than the accuracy of the {QGSJetII-04}\xspace fit to the {Epos-LHC}\xspace data. This is because the {Epos-LHC}\xspace $t_0$ parameterisation fit to the {Epos-LHC}\xspace based data is offset by a constant value at all energies from the true $t_0$ of the mock data, whereas the difference between the fitted {QGSJetII-04}\xspace $t_0$ parameterisation and the true $t_0$ of the mock data (based on {Epos-LHC}\xspace) changes with energy.
\fig{fig:EPOS_32_fitt0sigma_mod} shows the {Sibyll2.3}\xspace fit to the {Epos-LHC}\xspace data results in a reconstructed mass that is very representative of the true mass, but this mass reconstruction is not as accurate as the {Epos-LHC}\xspace and {QGSJetII-04}\xspace fits to this data. This is because a \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace shift of the {Sibyll2.3}\xspace parameterisation does not align the {Sibyll2.3}\xspace $t_0$ and $\sigma$ parameterisations with the {Epos-LHC}\xspace (or {QGSJetII-04}\xspace) descriptions as adequately as the {Epos-LHC}\xspace or {QGSJetII-04}\xspace descriptions can be aligned with each other (compare \figsThree{fig:model_diff_comparison_eq}{fig:model_diff_comparison_es}{fig:model_diff_comparison_sq}). Larger differences in the $\lambda$ {Sibyll2.3}\xspace parameterisation relative to the other parameterisations further hinders an accurate mass reconstruction of data based on these other parameterisations.
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance30fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm.pdf}%
\caption{{Epos-LHC}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {QGSJetII-04}\xspace parameterisation fit of Auger data. }
\label{fig:QGS_30_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm.pdf}%
\caption{{QGSJetII-04}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {QGSJetII-04}\xspace parameterisation fit of Auger data.}
\label{fig:QGS_31_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance32fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm.pdf}%
\caption{{Sibyll2.3}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {QGSJetII-04}\xspace parameterisation fit of Auger data.}
\label{fig:QGS_32_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance30fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:QGS_30_fitt0sigma_mod}.}
\label{fig:hist_QGS_30_fitt0sigma}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:QGS_31_fitt0sigma_mod}.}
\label{fig:hist_QGS_31_fitt0sigma}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance32fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:QGS_32_fitt0sigma_mod}.}
\label{fig:hist_QGS_32_fitt0sigma}
\end{figure}
Similar to the earlier figures presented, \figsThree{fig:QGS_30_fitt0sigma_mod}{fig:QGS_31_fitt0sigma_mod}{fig:QGS_32_fitt0sigma_mod} display the mass composition results from fitting the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of either the {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace parameterisations respectively, to 100 data sets generated from the parameterisation which resulted when the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of the \textbf{{QGSJetII-04}\xspace} parameterisation were fitted to Auger FD \ensuremath{X_\mathrm{max}}\xspace data. The true mass composition of the mock data is the mass composition from this {QGSJetII-04}\xspace fit to the Auger FD \ensuremath{X_\mathrm{max}}\xspace data. The {QGSJetII-04}\xspace based mock \ensuremath{X_\mathrm{max}}\xspace distributions will be slightly different to the {Epos-LHC}\xspace based mock distributions, because the \ensuremath{X_\mathrm{max}}\xspace parameterisations do not perfectly fit the Auger data, and the respective parameterisations consist of differences which can not be compensated for by an appropriate \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace shift. \figsThree{fig:hist_QGS_30_fitt0sigma}{fig:hist_QGS_31_fitt0sigma}{fig:hist_QGS_32_fitt0sigma} display the fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for the {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace fits respectively to the {QGSJetII-04}\xspace based data.
The fits to {QGSJetII-04}\xspace based mock data produce similar results to the fits of {Epos-LHC}\xspace based mock data. The mass fraction, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fit of the {Epos-LHC}\xspace parameterisation to {QGSJetII-04}\xspace based mock data reconstructs the mass composition above \energy{18.2} with an accuracy almost as good as the {QGSJetII-04}\xspace parameterisation fit to the same data. For both the {Epos-LHC}\xspace and {QGSJetII-04}\xspace fits, the absolute offsets in the median mass fractions from the true mass are less than $10\%$ in most energy bins. As noted before, due to the differences between the {Epos-LHC}\xspace and {QGSJetII-04}\xspace $t_0$ descriptions as a function of energy, the mass reconstruction accuracy of the {Epos-LHC}\xspace fit varies more with energy than the {QGSJetII-04}\xspace fit. Again the {Sibyll2.3}\xspace fit, in this case to {QGSJetII-04}\xspace based data, does not reconstruct the mass composition as accurately as the {Epos-LHC}\xspace or {QGSJetII-04}\xspace fits.
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance30fit1100_twentyrealdata_extendlimitmodel32old_fitt0normsigmanorm.pdf}%
\caption{{Epos-LHC}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {Sibyll2.3}\xspace parameterisation fit of Auger data.}
\label{fig:SIB_30_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance31fit1100_twentyrealdata_extendlimitmodel32old_fitt0normsigmanorm.pdf}%
\caption{{QGSJetII-04}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {Sibyll2.3}\xspace parameterisation fit of Auger data.}
\label{fig:SIB_31_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance32fit1100_twentyrealdata_extendlimitmodel32old_fitt0normsigmanorm.pdf}%
\caption{{Sibyll2.3}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {Sibyll2.3}\xspace parameterisation fit of Auger data.}
\label{fig:SIB_32_fitt0sigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance30fit1100_twentyrealdata_extendlimitmodel32old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:SIB_30_fitt0sigma_mod}.}
\label{fig:hist_SIB_30_fitt0sigma}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance31fit1100_twentyrealdata_extendlimitmodel32old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:SIB_31_fitt0sigma_mod}.}
\label{fig:hist_SIB_31_fitt0sigma}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance32fit1100_twentyrealdata_extendlimitmodel32old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:SIB_32_fitt0sigma_mod}.}
\label{fig:hist_SIB_32_fitt0sigma}
\end{figure}
\figsThree{fig:SIB_30_fitt0sigma_mod}{fig:SIB_31_fitt0sigma_mod}{fig:SIB_32_fitt0sigma_mod} display the mass composition results from fitting the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of either the {Epos-LHC}\xspace, {QGSJetII-04}\xspace or {Sibyll2.3}\xspace parameterisations respectively, to 100 data sets generated from the parameterisation which resulted when the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of the \textbf{{Sibyll2.3}\xspace} parameterisation were fitted to Auger FD \ensuremath{X_\mathrm{max}}\xspace data. \figsThree{fig:hist_SIB_30_fitt0sigma}{fig:hist_SIB_31_fitt0sigma}{fig:hist_SIB_32_fitt0sigma} display the respective \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace from these fits. The {Epos-LHC}\xspace and {QGSJetII-04}\xspace fits to the {Sibyll2.3}\xspace based data do not reconstruct the true mass composition as accurately as the {Sibyll2.3}\xspace fit, but they do accurately represent the general transition of the mass composition. The {Sibyll2.3}\xspace fit to {Sibyll2.3}\xspace based data (see \figs{fig:SIB_32_fitt0sigma_mod}{fig:hist_SIB_32_fitt0sigma}) results in absolute offsets in the median mass fractions from the true mass of less than $10\%$.
The data fitted in this section sufficiently constrains the fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, regardless of the parameterisation fitted. If different populations of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace were present in a histogram plot, it would indicate the data is unable to adequately constrain the fit, due to the degeneracy between the fitted shape coefficients and the mass fractions.
Data consisting of predominantly iron, such as the data sets fitted in this section, are easier to fit than data consisting of predominately protons and helium.
The ability of a \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fit of these parameterisations to reconstruct the general mass composition trend of data based on any of these three parameterisations, indicates that the normalisations of $t_0$ and $\sigma$ are the most relevant differences between these parameterisations in regards to reconstructing the mass composition. The results of the \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and mass fraction fits of the Auger FD \ensuremath{X_\mathrm{max}}\xspace data ~\cite{Aab:2014kda} are presented in Section~\ref{sec.res}.
\subsection{Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, $B$, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions}
The coefficient $B$ (which defines the energy dependence of $t_0$) can also be fitted with \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace provided the data consists of an adequate dispersion of masses and statistics. This three-coefficient fit will generally be less precise than the two-coefficient fit of only \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. Fitting additional coefficients increases the degeneracy between the fitted variables, unless there is significant mass diversity and statistics. Our {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace predictions of $B$ are fairly similar among primaries, therefore we do not expect to see a significant improvement in the systematics of the reconstructed mass composition when adding $B$ to our parameterisation fits of data based on any of these three models. However, it is possible that nature has a different energy dependence for $t_0$ (different from the three models), so by including $B$ in the fit we reduce considerably the model dependence of the mass composition interpretation of the \ensuremath{X_\mathrm{max}}\xspace distributions.
\figs{fig:EPOS_30_fitt0Asigma_mod}{fig:hist_EPOS_30_fitt0Asigma} display the reconstructed mass composition and fitted coefficient values from fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, $B$ and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of our {Epos-LHC}\xspace parameterisations to data generated from the {Epos-LHC}\xspace \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fit of the FD \ensuremath{X_\mathrm{max}}\xspace data set. Comparing this result to \fig{fig:EPOS_30_fitt0sigma_mod}, the systematic offsets in the median reconstructed mass composition from the true mass for the three-coefficient fit are similar to the two-coefficient fit. \fig{fig:hist_EPOS_30_fitt0Asigma} shows that the three fitted shape coefficients are accurately fitted and are well constrained.
However, as mentioned previously, data consisting of predominantly iron are easier to fit than data consisting of predominately proton and helium. The \ensuremath{t_{0_\mathrm{norm}}}\xspace, $B$, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and mass fraction fit of the latter data can result in a reconstructed mass composition which is considerably less accurate than a fit where $B$ is fixed to the true value of the data. This is because the degeneracy between the fitted parameters can result in the fitted shape coefficients shifting away from the true values.
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{performance30fit1101_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{{Epos-LHC}\xspace fit of \ensuremath{X_\mathrm{max}}\xspace data generated from the {Epos-LHC}\xspace parameterisation fit of Auger data.}
\label{fig:EPOS_30_fitt0Asigma_mod}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_performance30fit1101_twentyrealdata_extendlimitmodel30old_fitt0normsigmanorm.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace, $B$ and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:EPOS_30_fitt0Asigma_mod}.}
\label{fig:hist_EPOS_30_fitt0Asigma}
\end{figure}
\subsection{Effect of \ensuremath{X_\mathrm{max}}\xspace systematic uncertainties when fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace}
Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace can compensate for systematic offsets in \ensuremath{X_\mathrm{max}}\xspace, while fitting \ensuremath{\sigma_{\mathrm{norm}}}\xspace can compensate for systematic errors in the estimation of the detector resolution of \ensuremath{X_\mathrm{max}}\xspace. \figs{fig:performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10}{fig:histcorr_performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10} shows the results of fitting the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of our {QGSJetII-04}\xspace parameterisation to 100 data sets generated from the parameterisation which resulted when the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of the {QGSJetII-04}\xspace parameterisation were fitted to Auger FD \ensuremath{X_\mathrm{max}}\xspace data. Across the whole energy range, the mock data was shifted by a systematic offset of \depth{-10}, and also smeared by a Gaussian distributed random variable of $\sigma = \depth{10}$ (this additional smearing is not accounted for in the resolution of the applied \ensuremath{X_\mathrm{max}}\xspace parameterisation), to test if the fit of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace can compensate for these systematics. The red lines in \fig{fig:histcorr_performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10} indicate the true \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace values of the data (relative to the initial {QGSJetII-04}\xspace parameterisation being fitted) before the \ensuremath{X_\mathrm{max}}\xspace systematics were applied.
The mean shift in the fitted \ensuremath{t_{0_\mathrm{norm}}}\xspace values from the original \ensuremath{t_{0_\mathrm{norm}}}\xspace values of the data is $\sim$\;\depth{-10} (\fig{fig:histcorr_performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10}), to compensate mainly for the \depth{-10} \ensuremath{X_\mathrm{max}}\xspace systematic offset applied to the data. As $t_0$ changes by the same amount for each primary when \ensuremath{t_{0_\mathrm{norm}}}\xspace is fitted, and the \ensuremath{X_\mathrm{max}}\xspace systematic was applied consistently to all data, the \ensuremath{t_{0_\mathrm{norm}}}\xspace fit is capable of completely accounting for the \ensuremath{X_\mathrm{max}}\xspace systematic offset. However, \ensuremath{\sigma_{\mathrm{norm}}}\xspace for each primary is changed by different absolute amounts when fitting this coefficient, but all of the data is smeared (all masses are consistently smeared), consequently the correct \ensuremath{\sigma_{\mathrm{norm}}}\xspace cannot be fitted for each primary, which may also effect the fit of \ensuremath{t_{0_\mathrm{norm}}}\xspace. The shift in \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the original \ensuremath{\sigma_{\mathrm{norm}}}\xspace is only $\sim$\;\depth{+2}. Despite the fit of \ensuremath{\sigma_{\mathrm{norm}}}\xspace being unable to thoroughly account for the \depth{10} systematic in the resolution, the absolute offsets in the median reconstructed mass fractions from the true mass are less than $10\%$ in most energy bins, due to a combined shift of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace in the appropriate directions.
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10.pdf}%
\caption{Fits of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace to \ensuremath{X_\mathrm{max}}\xspace data consisting of a $\SI{-10}{\ensuremath{\mathrm{g/cm}^2}\xspace}$ systematic offset in \ensuremath{X_\mathrm{max}}\xspace. The \ensuremath{X_\mathrm{max}}\xspace data was also smeared by a Gaussian distributed random variable of $\sigma = \SI{10}{\ensuremath{\mathrm{g/cm}^2}\xspace}$, which was unaccounted for in the initial \ensuremath{X_\mathrm{max}}\xspace parameterisation fitted.}
\label{fig:performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10}
\end{figure}
\begin{figure}[htb!]
\includegraphics[width=0.48\textwidth]{histcorr_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10.pdf}%
\caption{Change in \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace for protons from the fits in \fig{fig:performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10}.}
\label{fig:histcorr_performance31fit1100_twentyrealdata_extendlimitmodel31old_fitt0normsigmanorm_Xmaxoffset10_Xmaxres10}
\end{figure}
The accuracy of the reconstructed mass fractions from the fit of this shifted and smeared data is similar to the same fit of the un-shifted and un-smeared data in \fig{fig:QGS_31_fitt0sigma_mod}. Reasonable detector resolution systematics and systematic offsets in \ensuremath{X_\mathrm{max}}\xspace will not significantly effect the accuracy of the reconstructed mass composition.
If the data was not smeared by a Guassian random variable, and only shifted by a constant \ensuremath{X_\mathrm{max}}\xspace offset, the \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fit of this shifted data would result in a change in the fitted \ensuremath{t_{0_\mathrm{norm}}}\xspace (compared to the \ensuremath{t_{0_\mathrm{norm}}}\xspace fitted to the un-shifted data) which is very close to the value of the \ensuremath{X_\mathrm{max}}\xspace offset. Shifting the \ensuremath{X_\mathrm{max}}\xspace data by a constant value has essentially the same effect on the fit as shifting the parameterisation by a constant value, with a very minuscule difference arising if the detector acceptance of \ensuremath{X_\mathrm{max}}\xspace is not shifted by the same offset to account for the applied \ensuremath{X_\mathrm{max}}\xspace offset (this is not an issue when fitting the measured Auger data).
\section{\label{sec.res}Results}
We have applied our {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace \ensuremath{X_\mathrm{max}}\xspace parameterisations separately to \ensuremath{X_\mathrm{max}}\xspace data measured by the Pierre Auger Observatory fluorescence detector (FD) \cite{Aab:2014kda}.
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{realdatamerge_4_extended.pdf}%
\caption{ Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions of our parameterisations to FD \ensuremath{X_\mathrm{max}}\xspace data measured by the Pierre Auger Observatory. The fitted mass fractions and p-values for each fitted model are shown. The red solid squares show the p-values for {QGSJetII-04}\xspace when fitting only the mass fractions (\ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fixed).}
\label{fig:realdata_comp_2_all}
\end{figure*}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.48\textwidth]{realdatamerge_lnARMS_extended.pdf}%
\caption{First two moments of the \ensuremath{\ln A}\; distribution estimated from the fitted fractions of the \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and mass fraction fit of the FD \ensuremath{X_\mathrm{max}}\xspace distributions measured by the Pierre Auger Observatory.}
\label{fig:lnAmoments}
\end{figure}
\begin{figure}[!ht]
\centering
\hbox{\hspace{-0.5cm}
\includegraphics[width=0.48\textwidth]{realdatamerge_XmaxRMS_extended.pdf}}
\caption{ The black lines show the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace and \ensuremath{\sigma(X_\text{max})}\xspace initially predicted by the \ensuremath{X_\mathrm{max}}\xspace parameterisations for proton and iron. The red, blue and green lines show the new predictions for the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace and \ensuremath{\sigma(X_\text{max})}\xspace after fits of \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and the mass fractions to FD \ensuremath{X_\mathrm{max}}\xspace distributions measured by the Pierre Auger Observatory.}
\label{fig:realdata_moments_2_all}
\end{figure}
\fig{fig:realdata_comp_2_all} displays the results from fitting the mass fractions and the coefficients \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace of our {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace \ensuremath{X_\mathrm{max}}\xspace distribution parameterisations. The top three panels display the fitted mass fractions for each model, and the bottom panel shows the p-values for these fits. The fits of these parameterisations to the \ensuremath{X_\mathrm{max}}\xspace distributions are shown in Appendix~\ref{AppFDfits_t0sigma}.
The p-value is defined as the probability of obtaining a worse fit (larger likelihood ratio $\mathcal{L}$) than that obtained with the data. The resulting parameterisation and fractions from the fit of the \ensuremath{X_\mathrm{max}}\xspace distributions were used to generate sets of mock \ensuremath{X_\mathrm{max}}\xspace distributions to determine the p-values, and to calculate the mass composition statistical errors. Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace improves the goodness of the fit of the \ensuremath{X_\mathrm{max}}\xspace distributions (bottom panel \fig{fig:realdata_comp_2_all}). This is evident by comparing the {QGSJetII-04}\xspace p-values for the \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fit to the {QGSJetII-04}\xspace p-values for the fit of only the mass fractions
We find that the {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace parameterisation fits of the \ensuremath{X_\mathrm{max}}\xspace distributions give a consistent mass composition result. \fig{fig:lnAmoments} shows the corresponding moments of the \ensuremath{\ln A}\; distribution. The results suggest a composition consisting of predominantly iron. Below \energy{18.8}, the small proportions of proton, helium and nitrogen vary. Above \energy{18.8}, there is little proton or helium, and with increasing energy the nitrogen component gradually gives way to the growing iron component, which dominates at the highest energies. There does not appear to be a distinct feature near the ankle ($\sim \energy{18.2}$), where it is assumed cosmic rays transition from Galactic to extragalactic \cite{Linsley}. Considering the upper limits on the large scale anisotropy \cite{2012ApJS..203...34P} indicate protons below \energy{18.5} are most likely of extragalactic origin, the fitted proton fractions below the ankle are suitably small if cosmic rays below the ankle are Galactic. A significant modification of the hadronic models is required to accommodate a proton dominant composition above \energy{18} \cite{Berezinsky:1988wi}.
The first two moments of the Auger \ensuremath{X_\mathrm{max}}\xspace distributions from \cite{Aab:2014kda} and their predictions (for proton and Fe) as a function of energy are shown in \fig{fig:realdata_moments_2_all}. It shows that the \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fits reduce the difference between the predictions from the {Epos-LHC}\xspace and {QGSJetII-04}\xspace hadronic models. For $t_0$ and $\sigma$, the separation between the proton prediction and heavier nuclei is larger in the {Sibyll2.3}\xspace parameterisation than the {Epos-LHC}\xspace or {QGSJetII-04}\xspace parameterisations, consequently the {Sibyll2.3}\xspace proton predictions from the fit are in disagreement with the two other parameterisations. The values of the coefficients in Equation~\eqref{eq:Xmaxbasicshape} for proton, helium, nitrogen and iron primaries for the {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace models (assuming a normalisation energy of $E_0 = \energy{18.24}$) can be found in \tab{tab:comp} of Appendix~\ref{AppB}. The values fitted to the data for \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace are also shown in \tab{tab:comp}. The statistical errors in the estimated value of \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace for protons or iron over the energy range are the same as the statistical error in the fitted value of \ensuremath{t_{0_\mathrm{norm}}}\xspace, while for \ensuremath{\sigma(X_\text{max})}\xspace the statistical error is less than \depth{1} for protons and iron.
The fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace are much larger than the initial parameterisation predictions, consequently the predicted \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace from the fits are much larger than the initial predictions. The fitted \ensuremath{\sigma_{\mathrm{norm}}}\xspace values are also larger than the initial predictions, consequently the predicted \ensuremath{\sigma(X_\text{max})}\xspace from the fit is larger. After the fit of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, our {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace parameterisations still have different predictions for the \ensuremath{X_\mathrm{max}}\xspace distribution shape properties as a function of mass and energy, but despite this there is reasonable agreement on the reconstructed mass composition from these fits. An observed shift in the fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace from the initial parameterisation prediction could be due to the initial parameterisation inadequately describing nature, systematics in the measured \ensuremath{X_\mathrm{max}}\xspace values, or a combination of both factors. Degeneracy between the fitted parameters could also contribute to a shift in the fitted coefficients, however the performance analysis in Section~\ref{sec.augerperformance} indicates that the results presented here are unlikely to be affected by degeneracy.
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.9\textwidth]{realdatamerge_4_extended_t0.pdf}%
\caption{ Fitting \ensuremath{t_{0_\mathrm{norm}}}\xspace and the mass fractions of our parameterisations to FD \ensuremath{X_\mathrm{max}}\xspace data measured by the Pierre Auger Observatory. The fitted mass fractions and p-values for each fitted model are shown.}
\label{fig:realdata_comp_2_all_t0}
\end{figure*}
The mass composition results are sensitive to the assumed values of the \ensuremath{X_\mathrm{max}}\xspace distribution properties which are not affected by the fit of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace (such as the elongation rate and the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace separation between p and Fe). The results are also sensitive to the fitting range limits. As our knowledge of the hadronic physics occurring at the highest energies progresses, the coefficients which are fitted and the fitting range limits applied may change. For example, a reduced upper limit of \ensuremath{t_{0_\mathrm{norm}}}\xspace would result in the \ensuremath{t_{0_\mathrm{norm}}}\xspace, \ensuremath{\sigma_{\mathrm{norm}}}\xspace and mass fraction fit of the Auger data reconstructing a mass composition consisting of predominantly proton and helium. An increase in the statistics of the Auger \ensuremath{X_\mathrm{max}}\xspace data, and/or an increased energy range, can reveal additional information regarding the shape coefficients.
Using the fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace, the parameters of the equations in \cite{Abreu:2013env}, to convert the \ensuremath{X_\mathrm{max}}\xspace moments into \ensuremath{\ln A}\; moments, have been determined and are shown in Tables~\ref{tab:lnA_coeff_3} and \ref{tab:lnA_coeff_4} of Appendix~\ref{AppC}.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.48\textwidth]{realdatamerge_lnARMS_extended_t0.pdf}%
\caption{First two moments of the \ensuremath{\ln A}\; distribution estimated from the fitted fractions of the \ensuremath{t_{0_\mathrm{norm}}}\xspace and mass fraction fit of the FD \ensuremath{X_\mathrm{max}}\xspace distributions measured by the Pierre Auger Observatory.}
\label{fig:lnAmoments_t0}
\end{figure}
\begin{figure}[!htb]
\centering
\hbox{\hspace{-0.5cm}
\includegraphics[trim={0 0 0cm 0}, width=0.48\textwidth]{realdatamerge_XmaxRMS_extended_t0.pdf}}
\caption{The black lines show the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace and \ensuremath{\sigma(X_\text{max})}\xspace initially predicted by the \ensuremath{X_\mathrm{max}}\xspace parameterisations for proton and iron. The red, blue and green lines show the new predictions for the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace and \ensuremath{\sigma(X_\text{max})}\xspace after fits of the mass fractions and \ensuremath{t_{0_\mathrm{norm}}}\xspace (applying the standard {QGSJetII-04}\xspace $\sigma$ prediction) to FD \ensuremath{X_\mathrm{max}}\xspace distributions measured by the Pierre Auger Observatory.}
\label{fig:realdata_moments_2_all_t0}
\end{figure}
Given the large \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace values fitted to the Auger data when the mass fractions, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace are fitted, a second set of fits were performed where only \ensuremath{t_{0_\mathrm{norm}}}\xspace and the mass fractions were fitted to the Auger data, using the same \ensuremath{t_{0_\mathrm{norm}}}\xspace fitting range. These fits of the three parameterisations each used the standard {QGSJetII-04}\xspace $\sigma$ prediction. The resulting mass composition, \ensuremath{\ln A}\; and \ensuremath{X_\mathrm{max}}\xspace moments are shown in \figsThree{fig:realdata_comp_2_all_t0}{fig:lnAmoments_t0}{fig:realdata_moments_2_all_t0} respectively. The fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace are shown in \tab{tab:comp} of Appendix~\ref{AppB}, and using these values the parameters of the equations in \cite{Abreu:2013env} have been determined and are shown in Tables~\ref{tab:lnA_coeff_5} and \ref{tab:lnA_coeff_6} of Appendix~\ref{AppC}.
As the fitted values of \ensuremath{t_{0_\mathrm{norm}}}\xspace are not as large compared to the two-coefficient fit, the predicted \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace of the fits are not as large, but still quite large compared to the initial parameterisation predictions. The reconstructed mass composition from the fits of only \ensuremath{t_{0_\mathrm{norm}}}\xspace (\fig{fig:realdata_comp_2_all_t0}) consists of a larger abundance of nitrogen and protons, at the expense of iron and helium, compared to that of the \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fit (\fig{fig:realdata_comp_2_all}). The general transition of the mass composition for the three parameterisations is consistent between the one-coefficient and two-coefficient fits.
\clearpage
\section{Conclusions}
We have presented a novel method to estimate the mass composition (from \ensuremath{X_\mathrm{max}}\xspace distributions) which is less dependent on hadronic models. The method uses parameterisations of \ensuremath{X_\mathrm{max}}\xspace distributions according to different hadronic interaction models. Provided that the measured \ensuremath{X_\mathrm{max}}\xspace distributions consist of different primary masses and sufficient statistics over a large energy range (which seems to be the case for the Auger \ensuremath{X_\mathrm{max}}\xspace data), two shape coefficients, of the \ensuremath{X_\mathrm{max}}\xspace distribution parameterisation, can be fitted together with the mass fractions, reducing the model dependency in the mass composition interpretation (we have tested the {Epos-LHC}\xspace, {QGSJetII-04}\xspace and {Sibyll2.3}\xspace models). The main differences between the predicted \ensuremath{X_\mathrm{max}}\xspace distributions from different models are the normalisation values of the mode and spread for each primary. So, by fitting two coefficients (\ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace) which adjust the normalisation of the mode and spread for each primary in an appropriate manner, the resulting mass composition is consistent for the three hadronic models tested here. A third coefficient, ``$B$'', which adjust the energy dependence of the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace can be fitted, further reducing the systematic model uncertainty in the fitted mass composition. However, given the current statistics and limited energy range of the published Auger \ensuremath{X_\mathrm{max}}\xspace distributions and the possible distribution of masses, fitting this third parameter may introduce large systematic uncertainties in the composition.
The mass fraction, \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace fits reconstruct a mass composition trend with energy that is consistent between the three models. There is a dominant abundance of iron over the energy range, particularly at the highest energies where there is almost pure iron. By fitting only \ensuremath{t_{0_\mathrm{norm}}}\xspace and adopting the {QGSJetII-04}\xspace $\sigma$ prediction for the three models, the relative abundance of protons increases.
The results are sensitive to the other model parameters that we keep fixed, such as the elongation rate and the \ensuremath{\left\langle \Xmax \negthickspace \; \right\rangle}\xspace separation between p and Fe. It is important to note that systematics in the measured \ensuremath{X_\mathrm{max}}\xspace values are absorbed by the fits of \ensuremath{t_{0_\mathrm{norm}}}\xspace and \ensuremath{\sigma_{\mathrm{norm}}}\xspace. Thus, the composition fractions are not significantly affected by systematics in \ensuremath{X_\mathrm{max}}\xspace.
\input{Blaess_Composition.bbl}
\onecolumngrid
\newpage
|
{
"timestamp": "2018-03-08T02:04:14",
"yymm": "1803",
"arxiv_id": "1803.02520",
"language": "en",
"url": "https://arxiv.org/abs/1803.02520"
}
|
\section{Introduction}
\label{intro}
Over the last two decades, quantum entanglement \cite{horodecki2009} has emerged as a crucial resource in a plethora of quantum information processing tasks, including quantum teleportation \cite{horodecki2009,bennett1993,bouwmeester1997}, quantum dense coding \cite{bennett1992,mattle1996,sende2010}, quantum cryptography \cite{ekert1991,jennewein2000}, and measurement-based quantum computation \cite{raussendorf2001,raussendorf2003,briegel2009}. It has also been proven useful in areas other than quantum information science, such as in detecting quantum phase transitions in quantum many-body systems \cite{osterloh2002,osborne2002,amico2008,chiara2017}, in characterizing topologically ordered states \cite{kitaev2006,pollmann2010,chen2010,jiang2012}, in studying the AdS/CFT correspondence \cite{hubeny2015,pastawski2015,almheiri2015,jahn2017}, and even in areas other than physics, such as in describing the transport properties in photosynthetic complexes \cite{sarovar2010,zhu2012,lambert2013,chanda2014}. Impressive experimental advancement in creating entangled quantum states in the laboratory, by using current technology and substrates such as ions \cite{leibfried2003,leibfried2005,brown2016}, photons \cite{raimond2001,prevedel2009,barz2015}, superconducting qubits \cite{clarke2008,barends2014}, nuclear magnetic resonance molecules \cite{negrevergne2006}, and cold atoms in optical lattices \cite{mandel2003,bloch2005,bloch2008} has enabled the realisation of a wide range of entanglement-based quantum protocols.
Studying the properties of entanglement confined in a subsystem of a increasingly larger multipartite quantum systems remains a pressing task. Many studies aiming at investigating such entanglement have followed two popular approaches. In one, an appropriate entanglement measure is computed for the reduced state $\rho_{N-m}$ of a chosen subsystem $\Omega$ that contains $ N-m $ qubits, obtained by tracing out the $ m $ qubits in the rest of the multipartite system, $\overline{\Omega}$, from the $N$-qubit state $\rho$, such that $\rho_{N-m}=\text{Tr}_{\overline{\Omega}}\rho$ \cite{horodecki2009}. In the other approach, one attempts to obtain entangled post-measurement states over the region $\Omega$ by performing local projection measurements over $\overline{\Omega}$, so that the average entanglement of the states in the post-measurement ensemble over $\Omega$ is non-zero \cite{divincenzo1998,verstraete2004,popp2005,sadhukhan2017}. For instance, an $N$-qubit Greenberger-Horne-Zeilinger (GHZ) state \cite{greenberger1989} given by $\ket{\mbox{GHZ}}=\frac{1}{\sqrt{2}}\left(\ket{0}^{\otimes N}+\ket{1}^{\otimes N}\right)$ is a classic example where the second approach is particularly useful. Here, the reduced state of $N-m$ qubits for any $m\leq N-2$, given by $\rho_{N-m}^{\text{GHZ}}=\frac{1}{2}\left[(\ket{0}\bra{0})^{\otimes (N-m)}]+(\ket{1}\bra{1})^{\otimes (N-m)}\right]$ has zero entanglement. On the other hand, the post-measurement states of, say, two qubits, obtained by performing local projection measurements on any one qubit in, say, a three-qubit GHZ state in the basis of Pauli $X$ matrix, are maximally entangled Bell sates $\ket{\phi^{\pm}}=(\ket{00}+\ket{11})/\sqrt{2}$. This motivates one to define \textit{localizable entanglement} as the maximum average entanglement, as measured by an appropriate entanglement measure, localized over $\Omega$ by performing local projection measurements over $\overline{\Omega}$ \cite{verstraete2004,popp2005,sadhukhan2017}. Localizable entanglement has been proven to be indispensable in investigating the correlation length in quantum many-body systems \cite{verstraete2004,popp2005,verstraete2004a,jin2004}, in studying quantum phase transitions in cluster-Ising \cite{skrovseth2009,smacchia2011} and cluster-XY models \cite{montes2012}, in protocols like percolation of entanglement in quantum networks \cite{acin2007}, and in quantifying local entanglement in stabilizer states \cite{raussendorf2003,hein2006,fujii2015,van-den-nest2004}.
One major challenge with respect to localizable entanglement, even in qubit systems, is its computability, due to the maximization that needs to be performed over all possible local projection measurements on the $m$ measured parties in the $N$-partite system \cite{verstraete2004,popp2005,sadhukhan2017}. Since the number of independent real parameters over which the maximization is to be performed increases with increasing number of measured qubits in the multipartite state \cite{sadhukhan2017}, the computation of localizable entanglement becomes in general difficult even in states of a small number of qubits. Also, in experiments, performing all possible local projection measurements on a set of qubits and determining the post-measurement states by performing state tomography is resource-intensive and becomes certainly impractical for systems of a large number of qubits. Moreover, an additional complication arises from the fact that one needs to deal with experimental $N$-qubit states which due to noise necessarily deviate to some degree from ideal, often pure target states. In such cases, determination of the localizable entanglement becomes difficult also due to the limited number of computable measures of entanglement in multipartite mixed states \cite{sadhukhan2017}, if one is interested in localizable entanglement in sets involving more than two qubits.
In this situation, a promising approach towards understanding the behaviour of localizable entanglement under noise for large stabilizer states is to develop non-trivial as well as computable lower bounds of the actual quantity. This may provide useful information about the system and the dependence of localizable entanglement over different relevant parameters. For example, in the case of the dependence of the localizable entanglement on the noise strength, a non-zero value of the lower bound of the localizable entanglement at a specific value of the noise strength implies sustenance of the actual localizable entanglement for that noise strength. Note that a similar approach of determining computable lower bounds has been adopted in the case of concurrence and entanglement of formation \cite{bennett1996,hill1997,wootters1998,coffman2000,wootters2001}, where the optimization involved in the computation of the actual quantity is difficult to achieve \cite{mintert2004,mintert2004a,mintert2005,mintert2005a,huang2014}. However, in order to satisfy practical purposes, one requires the lower bound of localizable entanglement to be easily computable from limited knowledge of the quantum state, and without performing a full state tomography, for which the required measurement resources increase if the system size is large. It is therefore also imperative to develop bounds that can be computed by performing least number of local measurements.
There have been attempts to determine the entanglement content and to characterize the dynamics of entanglement in noisy stabilizer states. Methods have been developped in order to obtain lower as well as upper bounds of entanglement between two subparts in an arbitrarily large graph state under noise \cite{cavalcanti2009,aolita2010}. Also, the behaviour of long-range entanglement \cite{raussendorf2005}, relative entropy of entanglement \cite{hajdusek2010}, and macroscopic bound entanglement \cite{cavalcanti2010} in cluster states under thermal noise has been investigated. The problem of efficiently estimating relative entropy of entanglement in an experimentally created noisy graph state by stabilizer measurement has also been addressed \cite{wunderlich2010}. Since localizable entanglement is the natural choice for quantifying entanglement between two parties in a multiqubit graph state with or without noise, an in-depth analysis of localizable entanglement in general noisy large-scale graph states is now necessary.
In this paper, we show how computable lower bounds of localizable entanglement can be constructed. For concreteness, we focus on stabilizer states \cite{raussendorf2003,hein2006,fujii2015,van-den-nest2004} and, more specifically, within this class of states, on graph states \cite{hein2006, raussendorf2001,raussendorf2003,briegel2009}, since the characterization of graph states and their properties is well developed and a versatile language for the description of these systems exists. However, since any stabilizer state can be mapped on to a graph state by local unitary operation \cite{van-den-nest2004,hein2006}, our results are either directly translatable, or derivable in a similar way for arbitrary stabilizer states.
We adopt two different, yet related approaches to obtain computable lower bounds for localizable entanglement in the case of mixed quantum states. The first approach is based on entanglement witnesses \cite{terhal2002,guhne2002,bourennane2004,guhne2009,guhne2005,alba2010,amaro} that are local observables whose expectation value signals the presence of entanglement. We use a class of witnesses, called \textit{local} witnesses \cite{guhne2005,alba2010,amaro}, and we show how they can be used to estimate a lower bound of the value of localizable entanglement in subsystems of qubits. Lower bounds of the localizable entanglement can be computed from the expectation values of the witness operators evaluated in the noisy quantum state \cite{brandao2005,brandao2006,eisert2007,guehne2007,guehne2008}. We show that the entanglement measure estimated by the expectation values of these witness operators serve as a faithful lower bound to the actual localizable entanglement on chosen subsystems of specific size. In the second scheme that we explore, we obtain a lower bound of localizable entanglement by considering a specific measurement strategy, thereby restricting the full set of local projection measurement required to compute the localizable entanglement. More specifically, for noisy graph states, we show that a computable lower bound of localizable entanglement is obtained by performing local $Z$ measurements over all qubits in the graph except for the qubits in the region of interest. We establish a relation between these two seemingly unrelated approaches, and test the performance of the obtained lower bounds by benchmarking them for graph states undergoing uncorrelated Pauli noise.
The paper is organized as follows. In Sec. \ref{sec:def}, we introduce the notation we use and review key concepts of localizable entanglement and graph states, including graph-diagonal states, used throughout this paper. Section \ref{sec:lb} contains a discussion on the local witness-based and local measurement-based lower bounds of localizable entanglement and the interrelation between these bounds. In Sec. \ref{sec:perf}, we demonstrate and compare the performances of the lower bounds in the case of specific noise models, and determine an analytical formula for the measurement-based lower bound in terms of noise-strength and the system size of the analyzed states. Sec. \ref{sec:conclude} contains concluding remarks.
\section{Definitions}
\label{sec:def}
\subsection{Localizable and restricted localizable entanglement}
\label{subsec:le}
The localizable entanglement (LE) \cite{verstraete2004,popp2005,sadhukhan2017} over a number $ N_{\Omega}\geq2 $ of selected qubits forming the region $\Omega$ in a multi-qubit system is defined as the maximum average entanglement that can be accumulated over $\Omega$ by performing local measurements over the qubits in the set $\overline{\Omega}$, where $\Omega\cap\overline{\Omega}=\emptyset$, and $\Omega\cup\overline{\Omega}$ represents the multiqubit system. We denote the state of an $N$-qubit system by $\rho$, where the qubits constituting the system are labelled from $ 1 $ to $ N $ such that $\Omega=\{1,2,3,\cdots,N_{\Omega}\}$, and $\overline{\Omega}=\{N_{\Omega}+1,N_{\Omega}+2,\cdots,N\}$. We label the $m$ ($m=N-N_{\Omega}\leq N-2$) qubits in $\overline{\Omega}$ by $\{r_1,r_2,\cdots,r_m\}$, with $r_i\in\{N_{\Omega}+1,N_{\Omega}+2,\cdots,N\}$, and perform local measurements on them. We restrict ourselves to rank-$1$ projection measurements $\mathcal{M}\equiv\{\mathcal{M}_k;\;k=0,1,2,\cdots,2^{m}-1\}$, in the Hilbert space of $\overline{\Omega}$, which is of dimension $2^{m}$. The post-measurement ensemble $\{p^k,\rho^k_{\Omega}\}$ is represented by the $N$-qubit post-measurement state $\rho^k_{\Omega}$, given by
\begin{eqnarray}
\rho^k_{\Omega}=\frac{\mbox{Tr}_{\overline{\Omega}}[\mathcal{M}_k\rho{\mathcal{M}_k}^\dagger]}{\mbox{Tr}[\mathcal{M}_k\rho{\mathcal{M}_k}^\dagger]},
\label{eq:le-pmstate}
\end{eqnarray}
and the probability with which $\rho^k_{\Omega}$ is obtained, given by
\begin{eqnarray}
p^k=\mbox{Tr}[\mathcal{M}_k\rho{\mathcal{M}_k}^\dagger].
\end{eqnarray}
Here, $k$ denotes the measurement outcome, and $\sum_{k=0}^{2^{m}-1}p^k=1$. The LE over the $N-m$ qubits in the region $\Omega$ in the $N$-qubit system is given by
\begin{eqnarray}
E_{\Omega}(\rho)=\underset{\mathcal{M}}{\mbox{sup}}\sum_{k=0}^{2^{m}-1}p^k E(\rho_{\Omega}^k),
\label{eq:localizable_entanglement}
\end{eqnarray}
where $E$ is a pre-decided entanglement measure. The supremum in Eq.(\ref{eq:localizable_entanglement}) is taken over the complete set of rank-$1$ projection measurements over the qubits in $\overline{\Omega}$.
Rank-$1$ projection measurements on the qubits in $\overline{\Omega}$ can be parametrized as $\mathcal{M}\equiv\{\mathcal{M}_k=\bigotimes_{r_i\in\overline{\Omega}}\ket{k_{r_i}}\bra{k_{r_i}}\}$, where $k_{r_i}\in\{\mathbf{0},\mathbf{1}\}\;\forall r_i\in\overline{\Omega}$, and $\{\ket{k_{r_i}}\}$ are given by \cite{nielsen2010}
\begin{eqnarray}
\ket{\mathbf{0}}_{r_i}&=&\cos (\theta_{r_i}/2)\ket{0}+e^{\text{i}\phi_{r_i}}\sin(\theta_{r_i}/2)\ket{1},\nonumber \\
\ket{\mathbf{1}}_{r_i}&=&\sin (\theta_{r_i}/2)\ket{0}-e^{\text{i}\phi_{r_i}} \cos (\theta_{r_i}/2)\ket{1},
\end{eqnarray}
with $\{\ket{0},\ket{1}\}$ being the computational basis, and $\{(\theta_{r_i},\phi_{r_i}); i=1,2,\cdots,m\}$ are $2m$ real parameters, such that $0 \leq \theta_{r_i} \leq \pi$, $0 \leq \phi_{r_i} < 2\pi$. Here, one can interpret the outcome index $k$ as the multi-index $k_{r_1}k_{r_2}\cdots k_{r_m}$. Therefore, the optimization in Eq.(\ref{eq:localizable_entanglement}) reduces to an optimization over a space of $2m$ real parameters. In general, such optimizations are hard problems when $m$ is large, and can be analytically performed only for a handful of quantum states even in the case of qubit systems \cite{verstraete2004,popp2005,sadhukhan2017}.
Instead of computing the actual localizable entanglement, one may define a restricted localizable entanglement (RLE) (see \cite{chanda2015} for similar quantities defined in context of quantum information-theoretic measures, such as quantum discord \cite{ollivier2001,henderson2001}), where only single-qubit projection measurements corresponding to the basis of the Pauli operators are allowed. This implies that for each qubit in $\overline{\Omega}$, the possible values of $(\theta_{r_i},\phi_{r_i})$ are \textbf{(i)} $(\theta_{r_i}=0,\phi_{r_i}=0)$ corresponding to the basis $\{\ket{0}_{r_i},\ket{1}_{r_i}\}$ of $Z_{r_i}$, \textbf{(ii)} $(\theta_{r_i}=\pi/2,\phi_{r_i}=0)$ corresponding to the basis $\{\ket{\pm}_{r_i}\}$ of $X_{r_i}$, and \textbf{(iii)} $(\theta_{r_i}=\pi/2,\phi_{r_i}=\pi/2)$ corresponding to the basis $\{\ket{y_{\pm}}_{r_i}\}$ of $Y_{r_i}$, where $\{X,Y,Z\}$ denote the standard Pauli operators.
We denote the complete set of all possible Pauli measurement settings over the $m$ qubits in $\overline{\Omega}$ by $\mathcal{M}^{\mathcal{P}}\equiv\{\mathcal{M}^{\mathcal{P}}_l;l=0,1,2,\cdots,3^m-1\}$. Corresponding to a specific value of $l$, there can be $2^m$ measurement outcomes, denoted by the index $k$, corresponding to each of which the projection operator is given by
\begin{eqnarray}
\mathcal{M}^{\mathcal{P}}_{(l,k)}=\bigotimes_{r_i\in\overline{\Omega}}\frac{1}{2}\left[I+(-1)^{k_{r_i}}\sigma_{l_{r_i}}\right]
\label{eq:projform}
\end{eqnarray}
where $l_{r_i}\in\{0,1,2\}$ represents the direction of local projection with $\sigma_{0}=Z$, $\sigma_1=X$, and $\sigma_{2}=Y$ for a specific $r_i$, and $k_{r_i}=0$ $(k_{r_i}=1)$ corresponds to the outcome $+1(-1)$ of the projection measurement. Here, we interpret the index $l$ as the multi-index $l\equiv l_{r_1}l_{r_2}\cdots l_{r_m}$, where the value of $l$ is the base $3$ representation of the string $l_{r_1}l_{r_2}\cdots l_{r_m}$, and the outcome index $k$ as the multi-index $k\equiv k_{r_1}k_{r_2}\cdots k_{r_m}$, where the value of $k$ is the base $2$ representation of the string $k_{r_1}k_{r_2}\cdots k_{r_m}$. Using this notation and following Eq. (\ref{eq:localizable_entanglement}), the RLE is given by
\begin{eqnarray}
E_{\Omega}^\mathcal{P}(\rho)=\underset{\mathcal{M^{\mathcal{P}}}}{\mbox{sup}}\sum_{k=0}^{2^{m}-1}p^{(l,k)} E(\rho_{\Omega}^{(l,k)}),
\end{eqnarray}
where
\begin{eqnarray}
\rho_{\Omega}^{(l,k)}&=& \frac{\mbox{Tr}_{\overline{\Omega}}\left[\mathcal{M}^{\mathcal{P}}_{(l,k)}\rho{\mathcal{M}^{\mathcal{P}\dagger}_{(l,k)}}\right]}{\mbox{Tr}[\mathcal{M}^{\mathcal{P}}_{(l,k)}\rho{\mathcal{M}^{\mathcal{P}^\dagger}_{(l,k)}}]},
\label{eq:res_pmstate}
\end{eqnarray}
and
\begin{eqnarray}
p^{(l,k)}&=&\mbox{Tr}[\mathcal{M}^{\mathcal{P}}_{(l,k)}\rho{\mathcal{M}^{\mathcal{P}\dagger}_{(l,k)}}].
\label{eq:le-pmstate-pauli}
\end{eqnarray}
Clearly, $E_{\Omega}\geq E^\mathcal{P}_{\Omega}$, thereby providing a lower bound to the LE when the optimization is not achieved by Pauli measurements. However, there are important examples and large classes of quantum states, for which $E_{\Omega} = E^\mathcal{P}_{\Omega}$. These include \textbf{(i)} graph states \cite{hein2006}, \textbf{(ii)} $N$-qubit generalized GHZ and generalized W states \cite{sadhukhan2017}, \textbf{(iii)} Dicke states and superposition of Dicke states with different excitations and a fixed number of qubits \cite{sadhukhan2017}, \textbf{(iv)} ground states of paradigmatic quantum spin models like the one-dimensional anisotropic $XY$ model in a magnetic field and the $XXZ$ model \cite{verstraete2004,popp2005,venuti2005,sadhukhan2017}, and also \textbf{(v)} the ground states of quantum spin systems described by stabilizer Hamiltonians in the presence of external perturbations in the form of magnetic field or spin-spin interaction, such as the cluster-Ising model \cite{skrovseth2009}.
\subsection{Graph states and stabilizer formalism}
\label{subsec:stab}
A mathematical graph \cite{hein2006,diestel2000,west2001} $\mathcal{G}\left( \mathcal{V},\mathcal{E}\right)$ is composed of a set $ \mathcal{V} $ of $ N $ nodes, labelled by $1,2,\cdots,N-1,N$ and a set $ \mathcal{E} $ of edges or links $(i,j)$ ($i\neq j$) connecting the nodes $i$ and $j$, where $i,j\in\mathcal{V}$. A graph is represented by the adjacency matrix $ \Gamma $, given by
\begin{eqnarray}
\Gamma_{ij}=\begin{cases}
1, & \text{for } (i,j) \in \mathcal{E}, \\
0, & \text{for } (i,j) \notin\mathcal{E},
\end{cases}
\end{eqnarray}
which is an $ N\times N $ binary matrix. In this paper, we consider \emph{simple}, \emph{undirected}, and \emph{connected} graphs \cite{hein2006,diestel2000,west2001} only. A simple graph does not contain a loop, i.e., a link connecting a node to itself, and multiple edges between a pair of nodes. A graph $\mathcal{G}$ is connected if for each pair of sites $ \{i,j\}\in\mathcal{V} $, there exists a path $\mathcal{L}$, constituted of a set of links $\{(k,l)\}\in\mathcal{E}$ with $k,l\in\mathcal{V}$, which connects the nodes $i$ and $j$. Also, in an undirected graph, the links $(i,j)$ and $(j,i)$ are equivalent. We denote the neighbourhood of a node $i$ by $\mathcal{N}_{i}\subset\mathcal{V}$, which is the set of nodes $\{j\}$ in which each node is connected to $i$ by a link, i.e., $(i,j)\in\mathcal{E}$ $\forall$ $j\in\mathcal{N}_i$.
Let us now consider a region in the graph $\mathcal{G}$, denoted by $\Omega$, which is designated by only the nodes in $\Omega$. For the subgraph $\mathcal{G}_\Omega(\Omega,\mathcal{E}_\Omega)$ corresponding to a region $\Omega$, with $\Omega\subset\mathcal{V}$ and $\mathcal{E}_\Omega\subset\mathcal{E}$, all the above definitions remain valid, and $\mathcal{E}_{\Omega}$ contains only the links $\{(i,j)\}$ such that $i,j\in\Omega$. We denote the cardinality of $\Omega$ by $N_\Omega$ ($ N_\Omega \leq N $). In agreement with the notation used in Sec. \ref{subsec:le}, the rest of the graph is denoted by $\mathcal{G}_{\overline{\Omega}}(\overline{\Omega},\mathcal{E}_{\overline{\Omega}})$, where $\mathcal{E}_{\overline{\Omega}}$ has a definition similar to that of $\mathcal{E}_{\Omega}$ and the set of all nodes is $ \mathcal{V}=\Omega\cup\overline{\Omega} $. The set of links $ \{(i,r_{j})\} $ between a node $ i\in\Omega $ and a node $ r_{j}\in\overline{\Omega} $ is denoted by $ \mathcal{E}_{\gamma} $, so that the complete set of existing links is $ \mathcal{E}=\mathcal{E}_{\Omega}\cup\mathcal{E}_{\overline{\Omega}}\cup\mathcal{E}_{\gamma} $. The boundary $\partial\Omega\subset\overline{\Omega}$ of the region $ \Omega $ is composed by the nodes in $\overline{\Omega}$ that are linked with nodes in $ \Omega $ (see Fig. \ref{fig:graph}(a) for examples of $\mathcal{G}_{\Omega}$, $\mathcal{G}_{\overline{\Omega}}$, $\partial\Omega$, and $\mathcal{E}_{\gamma}$ in a simple graph). Without loss of generality, one can label the nodes such that $\Omega=\{1,2,3,\cdots,N_{\Omega}\}$, and $\overline{\Omega}=\{N_{\Omega}+1,N_{\Omega}+2,\cdots,N\}$, which leads to
\begin{eqnarray}
\Gamma=\left( \begin{array}{cc} \Gamma_{\Omega} & \gamma^{T} \\
\gamma & \Gamma_{\overline{\Omega}}\end{array}\right).
\end{eqnarray}
Here, $\Gamma_{\Omega}$ and $\Gamma_{\overline{\Omega}}$ are the adjacency matrices corresponding to $\mathcal{G}_{\Omega}$ and $\mathcal{G}_{\overline{\Omega}}$, respectively, while the $(N-N_{\Omega})\times N_{\Omega}$ matrix $\gamma$ represents the set of links connecting $\Omega$ and $\overline{\Omega}$. In order to keep parity between the notations in Secs. \ref{subsec:le} and \ref{subsec:stab}, we would like to determine the LE over the region $\Omega$ in $\mathcal{G}$, implying $N_{\Omega}=N-m$.
A graph state $\ket{\mathcal{G}}$ is a multiqubit stabilizer quantum state associated to an undirected graph $\mathcal{G}$, where a qubit is placed at every node in the graph. The state is defined by a set, $G\in\mathcal{P}^{N} $, of mutually commuting generators \cite{hein2006}, $g_i$, where $g_{i}\ket{\mathcal{G}}=\ket{\mathcal{G}}$ $\forall$ $i=1,2,\cdots,N$. Here, $\mathcal{P}^{N}$ denotes the Pauli group \cite{hein2006,nielsen2010}, and the form of the generators $\{g_i\}$, given by
\begin{eqnarray}
g_{i}=X_{i}\otimes\big[\bigotimes_{j\in \mathcal{V}}Z_{j}^{\Gamma_{ij}}\big],
\label{eq:generators}
\end{eqnarray}
is determined by the underlying graph structure (see Fig. \ref{fig:graph}(a) for an explicit example in a five-qubit graph). The generators $\{g_i\}$ share common eigenstates, and the state $\ket{\mathcal{G}}$ is the common eigenstate of $\{g_i\}$ with eigenvalue $+1$. The rest of the $2^N-1$ eigenstates of $\{g_i\}$ are local unitary equivalent to $\ket{\mathcal{G}}$, given by $\{\ket{\mathcal{G}^\nu}=Z_\nu\ket{\mathcal{G}}\}$, where $\nu=0,1,2,\cdots,2^N-1$, and $Z_\nu=\bigotimes_{j\in\mathcal{G}}Z^{\nu_{j}}$, where $\nu_j\in\{0,1\}$. The index $\nu$ is a multi-index $\nu\equiv\nu_1\nu_2\cdots\nu_N$, and can be interpreted as the decimal representation of the binary sequence $\nu_1\nu_2\cdots\nu_N$. In this representation, $\ket{\mathcal{G}}=\ket{\mathcal{G}^0}$. The set of eigenstates $\{\ket{\mathcal{G}^\nu}\}$ forms a complete orthonormal basis of the Hilbert space of the system, and any state that is diagonal in this basis, written as \cite{hein2006,cavalcanti2009,aolita2010,kay2010,kay2011,guhne2011a,guhne2011b}
\begin{eqnarray}
\rho_{\text{GD}}=\sum_{\nu=0}^{2^N-1}p_\nu\ket{\mathcal{G}^\nu}\bra{\mathcal{G}^\nu},
\label{eq:gdstate}
\end{eqnarray}
is a graph-diagonal (GD) state, where $\langle \mathcal{G}^\nu |\mathcal{G}^{\nu^\prime}\rangle=\delta_{\nu,\nu^\prime}$, $\delta_{\nu,\nu^\prime}$ being the Kronecker delta, and $\{p_\nu\}$ is any probability distribution. From now on, we shall use the words qubits and nodes interchangeably, and denote them with the same labels, since each node in $\mathcal{G}$ accounts for a specific qubit in $\ket{\mathcal{G}}$.
\begin{figure}
\includegraphics[scale=0.3]{fig_graph}
\caption{{(Color online.) \textbf{Graph state, stabilizers, and local complementation operation.} (a) A five-qubit graph $\mathcal{G}\left( \mathcal{V},\mathcal{E}\right)$, constituted of nodes $\mathcal{V}=\{1,2,3,4,5\}$ and links $\mathcal{E}=\{(1,2),(1,4),(2,3),(2,4),(3,4),(4,5)\}$ is depicted, and the corresponding stabilizer generators $\{g_1,g_2,g_3,g_4,g_5\}$, according to Eq. (\ref{eq:generators}) are explicitly shown. As an example, we consider the subgraph $\mathcal{G}_{\Omega}=(\Omega,\mathcal{E}_{\Omega})$ corresponding to the region $\Omega$ constituted of nodes $\mathcal{V}_{\Omega}=\{1,5\}$ and no links, i.e., $\mathcal{E}_{\Omega}=\emptyset$. On the other hand, $\mathcal{G}_{\overline{\Omega}}=(\overline{\Omega},\mathcal{E}_{\overline{\Omega}})$ is constituted of nodes $\overline{\Omega}=\{2,3,4\}$ and links $\mathcal{E}_{\overline{\Omega}}=\{(2,3),(2,4),(3,4)\}$. The boundary $\partial\Omega$, in this case, is given by $\partial\Omega=\{2,4\}$, and $\mathcal{E}_{\gamma}=\{(1,2),(1,4),(4,5)\}$. (b) A LC operation w.r.t. the node $2$ leads to the graph $\mathcal{G}^\prime$ with modified connectivity, and the corresponding transformation of the graph states, $\ket{\mathcal{G}}\rightarrow\ket{\mathcal{G}^\prime}$ is given by a local unitary transformation according to Eq. (\ref{eq:lcu_def}), as shown explicitly in the figure.}}
\label{fig:graph}
\end{figure}
There exist graph states that are connected to each other by local unitary operations, thereby having identical entanglement properties \cite{hein2006}. A specific set of such states are of particular interest, which correspond to the different graphs connected to each other by the local complementation (LC) operation \cite{hein2006,bouchet1991-93,van-den-nest2004}. The LC operation with respect to a qubit $i$, denoted by $\tau_i(.)$, on a graph $\mathcal{G}$ deletes all the links $\{(j,k)\}$ if $j,k\in\mathcal{N}_i$, and $(j,k)\in\mathcal{E}$, and creates all the links $\{(j,k)\}$ if $j,k\in\mathcal{N}_i$, and $(j,k)\notin\mathcal{E}$. The operation $\tau_i$ that transforms $\mathcal{G}$ into a new graph $\mathcal{G}'$ is equivalent to a set of local unitary operations, denoted by $U_C^i$, on the corresponding graph state so that $\ket{\mathcal{G}}\rightarrow U_C^i\ket{\mathcal{G}}=\ket{\mathcal{G}'}$, where
\begin{eqnarray}
U_C^i=u_i^x\otimes\left[\bigotimes_{j\in\mathcal{N}_i}u_j^z\right],
\label{eq:lcu_def}
\end{eqnarray}
with $u_i^x=\exp[(-\text{i}\pi/4)X_i]$ and $u_j^z=\exp[(\text{i}\pi/4)Z_j]$ being local Clifford operations (for an example, see Fig. \ref{fig:graph}(b)). For a fixed value of $N$, the set of all possible graphs connected by (sequences of) LC operations over different nodes in the graph is called an orbit \cite{hein2006}. There may exist more than one orbit for a specific value of $N$. The orbits are mutually disjoint sets, and the union of all the orbits corresponding to a fixed value of $N$ provides the complete set of all possible connected graphs.
\section{Lower bounds of localizable entanglement}
\label{sec:lb}
In this section, we establish a relation between the LE over a region $\Omega$ in a graph $\mathcal{G}$ with local entanglement witnesses, and provide a hierarchy of bounds of LE based on suitably chosen local measurements and the expectation values of local entanglement witnesses.
\subsection{Witness- and measurement-based lower bounds}
\label{subsec:wlb_mlb}
An entanglement witness \cite{terhal2002,guhne2002,bourennane2004,guhne2009,guhne2005,alba2010,amaro} $\mathcal{W}$ is an operator with non-negative expectation values in all separable states, implying that a negative expectation value ($\mathrm{Tr}\left(\rho\mathcal{W}\right)<0$) of the witness operator unambiguously signals the presence of genuine entanglement in $\rho$. A witness operator $\mathcal{W}^g$ that detects the genuine $N$-partite entanglement in a multiparty pure state $\ket{\psi}$ and a state $\rho$ that is close to $\ket{\psi}$ is called a global witness operator, and can be chosen to be of the form \cite{bourennane2004}
\begin{eqnarray}
\mathcal{W}^{g}=\alpha I-\ket{\psi}\bra{\psi}.
\label{eq:global_witness}
\end{eqnarray}
Here, $I$ is the identity operator in the Hilbert space of $\ket{\psi}$, and $\alpha$ is the largest Schmidt coefficient of $\ket{\psi}$, given by $\alpha=\max_{\{\ket{\phi}\in S_B\}}|\langle\phi |\psi\rangle|^2$, $S_B$ being the complete set of all biseparable states. If $\ket{\psi}$ is a graph state $\ket{\mathcal{G}}$, then it is genuinely multiparty entangled if the underlying graph is connected, and $\mathcal{W}^g$ with $\alpha=\frac{1}{2}$ provides the global entanglement witness operator that can detect entanglement of a noisy state $\rho$ close to the ideal state $\ket{\mathcal{G}}$. Here, $\rho$ may originate from the exposure of an already prepared state $\ket{\mathcal{G}}$ to noise (where we assume that the state $\ket{\mathcal{G}}$ has been prepared with a high fidelity with the actual target state), or in an experiment, where the target state is $\ket{\mathcal{G}}$, but one ends up with a mixed state $\rho$ due to noise in the experimental apparatus. Assuming that the effect of noise in both scenarios can be simulated by known physical noise models, we consider $\rho=\Lambda(\rho_{\mathcal{G}})$, where $\rho_{\mathcal{G}}=\ket{\mathcal{G}}\bra{\mathcal{G}}$, and the operation $\Lambda(\cdot)$ describes the transformation $\ket{\mathcal{G}}\rightarrow\rho$.
A local witness $\mathcal{W}_{\Omega}$ is an operator that detects the entanglement in a subset $\Omega$ of qubits constituting the state $\rho$. If the subgraph $\mathcal{G}_{\Omega}$ is connected, a local witness can be constructed from the generators $\{g_i\}$ as \cite{guhne2005,alba2010,amaro}
\begin{eqnarray}
\mathcal{W}_{\Omega}=\frac{1}{2} I-\prod_{i\in\Omega}\frac{I+g_{i}}{2},
\label{eq:local_witness}
\end{eqnarray}
with the property that the expectation value of $\mathcal{W}_{\Omega}$ in the state $\rho$ is the same as the expectation value of the witness operator $\mathcal{W}^g_{\Omega}$ in the reduced state $\rho_{\Omega}$, i.e.,
\begin{eqnarray}
\omega=\mathrm{Tr}\left(\rho\mathcal{W}_{\Omega}\right) =\mathrm{Tr}\left( \rho_{\Omega}\mathcal{W}^{g}_{\Omega}\right).
\label{eq:witness_property}
\end{eqnarray}
Here the witness operator $\mathcal{W}^g_{\Omega}$ is \textit{global} with reference to the region $\Omega$ in $\mathcal{G}$, so that \cite{guhne2005,alba2010,amaro}
\begin{eqnarray}
\mathcal{W}^{g}_{\Omega}=\frac{1}{2}\,I-\ket{\mathcal{G}_{\Omega}}\bra{\mathcal{G}_{\Omega}},
\label{eq:witness}
\end{eqnarray}
$\ket{\mathcal{G}_{\Omega}}$ being the graph state corresponding to the subgraph $\mathcal{G}_{\Omega}$. The reduced state $\rho_{\Omega}$ lives only in $\Omega$, and is given by
\begin{eqnarray}
\rho_{\Omega}=\mathrm{Tr}_{\overline{\Omega}}\left(U_{\gamma} \rho U_{\gamma}^{-1}\right),
\label{eq:rho_omega}
\end{eqnarray}
where the unitary operator $ U_{\gamma} $ disentangles $ \ket{\mathcal{G}_\Omega} $ from $\ket{\mathcal{G}_{\overline{\Omega}}}$, so that $ U_{\gamma}\ket{\mathcal{G}}=\ket{\mathcal{G}_{\Omega}}\otimes\ket{\mathcal{G}_{\overline{\Omega}}}$ \cite{hein2006}. The unitary operator $U_{\gamma}$, written as
\begin{eqnarray}
U_{\gamma}=\prod_{(i,r_j)\in\mathcal{E}_{\gamma}}U_{ir_j}^{CZ},
\label{eq:u_omega_b}
\end{eqnarray}
is constituted of controlled phase unitaries acting on the links $(i,r_j)\in\mathcal{E}_{\gamma}$ with $i\in\Omega$ and $r_j\in\overline{\Omega}$, given by $ U_{ir_j}^{CZ}=\frac{1}{2}[(I_{r_j}+Z_{r_j})+Z_{i}( I_{r_j}-Z_{r_j})]$. Note here that the operator $\mathcal{W}_{\Omega}$ (Eq. (\ref{eq:local_witness})) is constituted of generators $\{g_i\}$ with $i\in\Omega$. Under the transformation $U_\gamma g_i U_\gamma^{-1}$, the resulting generator no longer has support on $\overline{\Omega}$. Therefore, the unitary operator $U_{\gamma}$ transforms $\mathcal{W}_\Omega$ into $\mathcal{W}_\Omega^g$ as
\begin{eqnarray}
U_{\gamma}\mathcal{W}_{\Omega}U_{\gamma}^{-1}=\mathcal{W}_{\Omega}^{g}\otimes I_{\overline{\Omega}}.
\label{eq:witness_unitary}
\end{eqnarray}
Next, we notice that the unitary operator $U_{\gamma}$ is constituted of controlled phase unitaries $U_{ir_j}^{CZ}$ which involve operators $\frac{1}{2}(I_{r_j}\pm Z_{r_j})$ corresponding to the qubits $r_j\in\partial\Omega$ in $Z$. Therefore, writing the identity operator corresponding to the Hilbert space of a specific qubit $r_j\in\overline{\Omega}\mbox{\textbackslash}\partial\Omega$ as $I_{r_j}=[(I_{r_j}+Z_{r_j})+(I_{r_j}-Z_{r_j})]/2$, the form of the unitary operator can be expanded as
\begin{eqnarray}
U_{\gamma}&=&\sum_{k} \mathcal{Z}^k_{\Omega}\prod_{r_j\in{\overline{\Omega}}}\left(\frac{I+(-1)^{k_{r_j}}Z_{r_j}}{2}\right),
\label{eq:unitary_expanded}
\end{eqnarray}
where the correction unitaries $\{\mathcal{Z}^k_{\Omega}\}$ are given by
\begin{equation}\label{correction_witnesses}
\mathcal{Z}^{k}_{\Omega}=\prod_{i\in\Omega}Z_{i}^{\textbf{k}\cdot\gamma_{i}},
\end{equation}
where $ \gamma_{i} $ is the $i$-th column of $ \gamma $, $\mathbf{k}$ is a row matrix constituted of the individual measurement outcomes $ k_{r_j} $ corresponding to the qubits $r_{j}\in\overline{\Omega}$, and $\mathbf{u}\cdot\mathbf{v}$ indicates a matrix product calculated modulo $2$ for the matrices $\mathbf{u}$ and $\mathbf{v}$. Note here that $\mathcal{Z}^k_{\Omega}$ acts only on $ \Omega $, and it is determined entirely according to the links in $\mathcal{E}_{\gamma}$, and the values of $\{k_{r_j}\}$ for $r_{j}\in\partial\Omega$. Then,
\begin{eqnarray}
\rho_{\Omega}&=&\mbox{Tr}_{\overline{\Omega}}(U_{\gamma}\rho U_{\gamma})
= \sum_{k}p^{(0,k)} \mathcal{Z}_{\Omega}^{k} \rho_{\Omega}^{(0,k)} \mathcal{Z}_{\Omega}^{k},
\label{eq:relation_mixed}
\end{eqnarray}
where $\rho_{\Omega}^{(0,k)}$ and $p^{(0,k)}$ are for $l=0$ in Eqs. (\ref{eq:res_pmstate}) and (\ref{eq:le-pmstate-pauli}) respectively.
\subsubsection*{Hierarchy of lower bounds}
We are now in a position to establish a hierarchy between a set of quantities that are relevant in investigating the behaviour of localizable entanglement. It is clear from the definition of RLE that although the computational complexity of RLE is less than the same corresponding to a computation of the exact LE, one has in principle still to consider $3^{m}$ possible Pauli measurement settings, which grows exponentially with $m$. For large $m$, where this becomes impractical, one may compute the average entanglement that can be localized on $\Omega$, obtained by choosing a particular setting of Pauli measurement, say, $\mathcal{M}^{\mathcal{P}}_{l}$, in $\overline{\Omega}$, instead of considering the full set of $3^m$ elements of $\mathcal{M}^{\mathcal{P}}$. Here, we have adopted the notation used in Sec.~\ref{subsec:le}. The value of the average entanglement computed in this way depends completely on the choice of the value of $l$. In the scenarios where the choice is not an optimal setting, the average entanglement serves as a lower bound of the RLE, and by extension a lower bound of LE, i.e.,
\begin{eqnarray}
E_{\Omega}(\rho)\geq E^\mathcal{P}_{\Omega}(\rho)\geq E^{l}_{\Omega}(\rho).
\label{eq:ineq}
\end{eqnarray}
We call such a lower bound the \emph{measurement-based lower bound} (MLB) in the following. Unless otherwise stated, throughout this paper, we shall consider Pauli measurements only, and discard the superscript $\mathcal{P}$ from all the operators to keep them uncluttered. Note that a poor choice of the setting may result in vanishing average entanglement corresponding to a trivial lower bound of LE, which highlights the importance of an informed choice of measurement setting from within the full set of Pauli measurements.
In the case of $l=0$, the lower bound $E^0_{\Omega}$ corresponds to local $Z$ measurements on all qubits in $\overline{\Omega}$, and Eq.~(\ref{eq:ineq}) becomes
\begin{eqnarray}
E_{\Omega}(\rho)\geq E^\mathcal{P}_{\Omega}(\rho)\geq E^0_{\Omega}(\rho).
\label{eq:ineq_1}
\end{eqnarray}
A non-zero value of $E^0_{\Omega}$ is likely when $\Omega$ in $\mathcal{G}$ is connected because $\mathcal{M}_0$ is an optimal measurement setting in the absence of noise (i.e., for $\rho=\ket{\mathcal{G}}\bra{\mathcal{G}}$). The use of $E^0_{\Omega}$ as the MLB is justified in scenarios where the state $\rho$ is very close to the graph state $\ket{\mathcal{G}}$, i.e., when the noise acting on the state has very low strength, or when in an experiment the prepared state has very high fidelity with the target state $\ket{\mathcal{G}}$. In such situations, one expects the optimal measurement to not deviate much from the optimal one in the absence of noise. However, in subsequent sections, we shall demonstrate that there exist situations in which $E^0_{\Omega}$ serves as a good choice for MLB even when the noise strength is considerably high.
A clear connection between $E^0_{\Omega}$ and the local entanglement witnesses can now be drawn by using Eq. (\ref{eq:relation_mixed}). The local-unitary invariance of entanglement measures \cite{horodecki2009} implies $E(\mathcal{Z}_{\Omega}^{k}\rho_{\Omega}^{(0,k)}\mathcal{Z}_{\Omega}^{k})=E(\rho_{\Omega}^{(0,k)})$, which leads to
\begin{eqnarray}
E^0_{\Omega}(\rho)=\sum_{k}p^{(0,k)}E(\mathcal{Z}_{\Omega}^{k}\rho_{\Omega}^{(0,k)} \mathcal{Z}_{\Omega}^{k}),
\end{eqnarray}
for a specific choice of the entanglement measure $E$. Using the convexity property of entanglement measures \cite{horodecki2009,horodecki2001} results in $E^0_{\Omega}(\rho)\geq E(\rho_{\Omega})$, where $\rho_{\Omega}$ is given by Eq.~(\ref{eq:rho_omega}), and one can modify Eq.~(\ref{eq:ineq_1}) as
\begin{eqnarray}
E_{\Omega}(\rho)\geq E^\mathcal{P}_{\Omega}(\rho)\geq E^0_{\Omega}(\rho)\geq E(\rho_{\Omega}).
\label{eq:ineq_2}
\end{eqnarray}
The quantity $E(\rho_{\Omega})$ may still be difficult to compute in the general case if the region $\Omega$ is large and if $\rho_{\Omega}$ is a mixed state. However, the expectation value $\omega=\mbox{Tr}(\rho_{\Omega}\mathcal{W}_{\Omega}^g)$, which is obtained by measuring $ \mathcal{W}_{\Omega} $ on $ \rho $, can typically be determined, say, in an experiment, with a number of resources that depends only on the size of $ \Omega $, unlike obtaining $\rho_{\Omega}$ from $ \rho $ and the posterior full state tomography for it, which require an effort that depends on the total size of system. From the definition of witness operators, one expects $\omega$ corresponding to a good witness operator and a specific quantum state to be highly negative if the state is highly entangled. Motivated by this, one may use a minimal set of data, and solve an optimization problem which aims to answer the question as to what the minimum amount of entanglement, $E^{\mbox{\scriptsize min\normalsize}}(\rho_{\Omega})$, as measured by any bipartite or multipartite measure $E$, is among all states $\varrho$, subject to $\varrho$ that are consistent with the data of $\omega$. In other words, one aims to find the quantity given by \cite{eisert2007,guehne2007,guehne2008}
\begin{eqnarray}
E^{\mbox{\scriptsize min\normalsize}}(\rho_{\Omega})=&&\underset{\varrho}{\inf} E(\varrho),
\end{eqnarray}
subject to
\begin{eqnarray}
\omega=\text{Tr}\left(\varrho \mathcal{W}_{\Omega}^g\right)=\mbox{Tr}(\rho_{\Omega}\mathcal{W}_{\Omega}^g),
\end{eqnarray}
where $\varrho$ is in the Hilbert space of $\Omega$, $\varrho\geq 0$, and $\text{Tr}(\varrho)=1$. In the most general scenario, the expectation values of the local witness operators would provide a lower bound of $E^{\mbox{\scriptsize min\normalsize}}(\rho_{\Omega})$, given by $E^{\mathcal{W}}_{\Omega}(\omega)$, so that the inequality in (\ref{eq:ineq_2}) can be further appended as
\begin{eqnarray}
E_{\Omega}(\rho)\geq E^\mathcal{P}_{\Omega}(\rho)\geq E^0_{\Omega}(\rho)\geq E(\rho_{\Omega})\geq E^{\mathcal{W}}_{\Omega}(\omega),
\label{eq:ineq_3}
\end{eqnarray}
where we refer the quantity $E^{\mathcal{W}}_{\Omega}(\omega)$ as the \emph{witness-based lower bound} (WLB) of LE, which is a function of only the expectation value of a local witness $ \omega=\mathrm{Tr}\left( \rho\mathcal{W}_{\Omega} \right) $.
In the following Secs. \ref{subsec:lb_unitary} and \ref{subsec:lb_gdstate} we provide technically detailed discussions of modifications of the hierarchy of lower bounds given in (\ref{eq:ineq_3}) in particular situations, such as under local unitary transformations and for GD states. More specifically, we show that for GD states, $E^0_{\Omega}(\rho)= E(\rho_{\Omega})$, and we use logarithmic negativity \cite{lee2000,vidal2002,plenio2005} as a bipartite entanglement measure to show that for GD states and a region $\Omega$ constituted of two qubits only, $E(\rho_{\Omega})= E^{\mathcal{W}}_{\Omega}(\omega)$. Readers interested in the demonstration of the different lower bounds in the case of graph states under physical noise can skip these discussions, and move on to Sec. \ref{sec:perf}, where we demonstrate the behaviour of the lower bounds under local Pauli noise as functions of the noise strength.
\subsection{Lower bounds under local unitary transformation}
\label{subsec:lb_unitary}
\begin{figure*}
\includegraphics[width=0.7\textwidth]{square_graph.pdf}
\caption{(Colour online) \textbf{Creation of a link $(a,b)$ by successive application of LC operations.} (a) A square graph $\mathcal{G}_S$ with a region $\Omega$ of two disconnected qubits $a$ and $b$ denoted by black nodes, joined by a path $\mathcal{L}$, constituted of the qubits $\{a,1,2,b\}$ and the links $\{(a,1),(1,2),(2,b)\}$, denoted by thick black continuous lines. (b) LC operation on qubit ``$1$" (blue) leading to the graph $\tau_1(\mathcal{G}_S)$). The new links created by the operation are denoted by blue continuous lines. Note that the link $(a,2)$ has been created in this LC operation, which is crucial for the creation of the link $(a,b)$ in the next step. No links are deleted in the operation $\tau_1$. (c) LC operation on qubit ``$2$" (red) in the graph $\tau_1(\mathcal{G}_S)$ result in the modified graph $\mathcal{G}^\prime=\tau_2\circ\tau_1(\mathcal{G}_S)$, in which the link $(a,b)$ is present. The new links created by this operation are denote by red continuous lines. Note that four of the blue links created in the previous step along with four links from the original graph are deleted by this operation.}
\label{fig:square_graph}
\end{figure*}
An important requirement for the construction of the local witness operator $\mathcal{W}_{\Omega}$ is that the region $\Omega$ in the graph has to be connected. Also, in the case of low noise strength, the value of $E^0_{\Omega}$ can be expected to be non-zero iff $\Omega$ is connected in $\mathcal{G}$, since in the absence of noise, computing $E^0_{\Omega}$ yields zero if $\Omega$ is not connected. However, there may arise situations where the chosen region $\Omega$ in a graph $\mathcal{G}$ is not connected. In that scenario, one may arrive at a graph $\mathcal{G}^\prime$ by performing LC operations over a set of chosen qubits in the graph, so that the region $\Omega$ becomes connected in $\mathcal{G}^\prime$, and the hierarchies given in (\ref{eq:ineq_3}) hold good. For example, let us consider a region $\Omega$ of two disconnected qubits $a$ and $b$. The fact that the original graph $\mathcal{G}$ is connected ensures the existence of a path $\mathcal{L}$ constituted of links $\{(i,j)\}\in\mathcal{E}$ that connects $a$ and $b$. A series of LC operations on the selected qubits $\{i\}\subseteq\mathcal{L}$, where $i\neq a,b$, creates a link between the qubits $a$ and $b$, thereby resulting in a new graph $\mathcal{G}^\prime$ with modified connectivity, where the link $(a,b)$ is present. We illustrate this in Fig. \ref{fig:square_graph} with the example of a square graph. However, a series of LC operations over a graph is equivalent to a local Clifford unitary transformation of the graph state, as demonstrated in Sec. \ref{subsec:stab}. Therefore, in order to check whether Eq.~(\ref{eq:ineq_3}) is valid in the case of a graph where the selected region is not connected, one has to check whether the inequalities remain invariant under such local unitary transformation.
Remembering that the LC operation on a set of qubits in a graph is equivalent to the application of local Clifford unitaries on a set of qubits in the graph state \cite{van-den-nest2004,hein2006}, without loss of generality, one may write
\begin{eqnarray}
\ket{\mathcal{G}^\prime}=U_L\ket{\mathcal{G}},
\label{eq:local_unitary}
\end{eqnarray}
where $U_L=\otimes_{i=1}^N U_i$, $\{U_i\}$ being the set of local Clifford unitary operators acting on the qubits $i\in\mathcal{G}$. In the case of a quantum state $\rho$ originating from the graph state due to noise or some error in the experimental setup, without any loss in generality, $\rho^\prime=U_L\rho U_L^{-1}$, where $\rho^\prime$ is the quantum state resulting when $\ket{\mathcal{G}^\prime}$ has undergone the same transformation as $\ket{\mathcal{G}}$ up to the local unitary $ U_{L} $. Note that since $\rho$ and $\rho^\prime$ are connected by local unitary operators, and since LE is invariant under local unitary transformation of the quantum state, $E_{\Omega}(\rho)=E_{\Omega}(\rho^\prime)$ for any connected region $\Omega\in\mathcal{G}$. Moreover, we note that the Clifford unitary operators have the property
\begin{eqnarray}
\sigma_i =U_i^{-1}\sigma_i^\prime U_i,
\label{eq:clifford_property}
\end{eqnarray}
where both $\sigma_i$ and $\sigma_i^\prime$ are Pauli operators corresponding to the qubit $i$, up to the multiplicative factors $\{\pm 1,\pm\text{i}\}$, while $\sigma_i$ is not necessarily equal to $\sigma^\prime_i$. Since computing the RLE includes all possible Pauli measurement settings, this implies $E_{\Omega}^\mathcal{P}(\rho)=E_{\Omega}^\mathcal{P}(\rho^\prime)$.
Clearly, the optimal measurement bases for computing LE for $\rho$ and $\rho^\prime$ are not identical. However, the measurement basis corresponding to $\rho$ can be determined by using the knowledge of $U_L$, and an appropriate measurement basis for $\rho^\prime$. In this scenario, we expect $ \rho^\prime $ to be close to the graph state $\ket{\mathcal{G}^\prime}$ where the region $ \Omega $ is connected, so that the appropriate measurement basis for $ \rho^\prime $ should be $\mathcal{M}^0$, which involves only local $Z$ measurement over all qubits in $\overline{\Omega}$. But due to their local unitary connection, the localizable entanglement $ E^0_{\Omega}(\rho^\prime) $ equals $ E^l_{\Omega}(\rho) $, where the value of $l\equiv l_{r_1}l_{r_2}\cdots l_{r_{m}}$ is such that for all $r_i\in\overline{\Omega}$, $\sigma_{l_{r_i}}=U_{L}^{-1}Z_{r_i}U_{L}$, up to the multiplicative factors $\{\pm 1,\pm\text{i}\}$.
In connection with the local witness operator, one has to now consider
\begin{equation}
\mathcal{W}_{\Omega}^\prime=\frac{1}{2} I-\left( \prod_{i\in\Omega}\frac{I+U_L^{-1} g_{i}^\prime U_L}{2}\right),
\label{eq:wit_unitary}
\end{equation}
with $\{g^\prime_i\}$ being the generators of $\ket{\mathcal{G}^\prime}$ and $ \{U_L^{-1} g_{i}^\prime U_L\} $ are products of the generators $ \{g_{j}\} $ of $\ket{\mathcal{G}}$. Note that the state $\rho_{\Omega}^\prime$ corresponding to $\mathcal{G}^\prime$ is obtained from $ \rho^\prime $ according to Eqs. (\ref{eq:rho_omega}) and (\ref{eq:u_omega_b}), but using a different unitary operator $U_{\gamma^\prime}$, which is defined according to the connectivity of $\mathcal{G}^\prime$. In light of this, the hierarchies of lower bounds in Eq.~(\ref{eq:ineq_3}), in the case of $\mathcal{G}^\prime$, become
\begin{eqnarray}
E_{\Omega}(\rho)\geq E^\mathcal{P}_{\Omega}(\rho)\geq E^l_{\Omega}(\rho)\geq E(\rho_{\Omega}^\prime)\geq E^{\mathcal{W}}_{\Omega}(\omega^\prime),
\label{eq:ineq_4}
\end{eqnarray}
where $ \omega'=\mathrm{Tr}\left( \rho\mathcal{W}_{\Omega}^\prime\right) $ and $ \rho_{\Omega}^\prime=\mathrm{Tr}\left( U_{\gamma^\prime}\rho^\prime U_{\gamma^\prime}^{\dagger}\right) $, with $ U_{\gamma^\prime} $ being the disentangling unitary of Eq.~(\ref{eq:u_omega_b}) for $ \ket{\mathcal{G}^\prime} $.
In scenarios where $\Omega$ is not connected, in the absence of noise, an optimal measurement setting for computing the LE over the region $\Omega$ is the one that corresponds to a sequence of graph operations that results in a connected region $\Omega$. For example, in the case of a disconnected region $\Omega$ constituted of only two qubits, say, ``$a$", and ``$b$", one of the optimal measurement settings corresponds to (i) $X$ measurements on all the qubits that are situated on a path connecting qubits ``$a$" and ``$b$", and (ii) $Z$ measurements on rest of the qubits in the graph \cite{hein2006}. However, there may exist more than one such Pauli measurement setting. Note also that there may exist different sets of local unitary operations that connect $\ket{\mathcal{G}}$ to different graph states where $\Omega$ is connected. Both MLB and WLB described above can therefore be made tighter by considering all such possible cases, and then choosing the maximum of the values.
\subsection{Lower bounds in graph-diagonal states}
\label{subsec:lb_gdstate}
In this section, we focus on the hierarchies of lower bounds in the case of GD states. The motivation behind determining the structure of lower bounds for GD states stems from the fact that these states occur naturally when graph states are subjected to Pauli noise \cite{cavalcanti2009,aolita2010}, as is demonstrated in Sec. \ref{sec:perf}. Also, any quantum state can be transformed into a GD state by local operations, as demonstrated in \cite{kay2010,kay2011,guhne2011a}.
Let us first consider the measurement operation $\mathcal{M}_{0}=\{\mathcal{M}_{(0,k)}\}$ with $l=0$ for the $N$-qubit graph state, where the form of $\mathcal{M}_{(l,k)}$ is defined in Eq.~(\ref{eq:projform}) (see Sec.~\ref{subsec:le}). Unless otherwise stated, we keep the value of $l$ fixed at $l=0$ here and throughout the rest of the paper. To keep notation simple, we discard the subscript $l$ from now on, and denote the measurement operation by $\mathcal{M}_0\equiv\{\mathcal{M}_{k}\}$. Here, $\mathcal{M}_{k}=\bigotimes_{r_i\in\overline{\Omega}}\mathcal{M}_{k_{r_i}}$, with $k_{r_i}\in\{0,1\}$. Denoting the graph state as $\rho_{\mathcal{G}}=\ket{\mathcal{G}(\overline{\Omega},\Omega)}\bra{\mathcal{G}(\overline{\Omega},\Omega)}$, implying that $\ket{\mathcal{G}(\overline{\Omega},\Omega)}$ consists of the qubits in $\overline{\Omega}$ and $\Omega$, the effect of operating $\mathcal{M}_{k_{r_i}}$ on $\rho_{\mathcal{G}}$ for a specific $r_i\in\overline{\Omega}$ is given by \cite{hein2006}
\begin{eqnarray}
\hspace{-3mm}\rho_{(\mathcal{G}-r_i)}^{k_{r_{i}}}=\mbox{Tr}_{r_i}\left(\mathcal{M}_{k_{r_i}}\rho \mathcal{M}_{k_{r_i}}\right)
=\frac{1}{2} \mathcal{Z}^{k_{r_i}}\rho_{(\mathcal{G}-r_{i})} \mathcal{Z}^{k_{r_i}}
\label{eq:meas_rule_gen}
\end{eqnarray}
with
\begin{eqnarray}
\mathcal{Z}^{k_{r_i}}=\bigotimes_{j\in\mathcal{N}_{r_i}}Z_j^{k_{r_i}}.
\label{eq:correction}
\end{eqnarray}
Here, $\mathcal{N}_{r_i}$ represents the neighbourhood of the qubit $r_i$, and $\rho_{(\mathcal{G}-r_i)}=\ket{\mathcal{G}(\overline{\Omega}-r_i,\Omega)}\bra{\mathcal{G}(\overline{\Omega}-r_i,\Omega)}$ corresponds to the graph $\mathcal{G}(\overline{\Omega}-r_i,\Omega)$, obtained from $\mathcal{G}(\overline{\Omega},\Omega)$ by deleting the qubit $r_i$ and all the links that are connected to it. Performing local $Z$-measurement over all qubits in $\overline{\Omega}$, the normalized post-measurement state $\rho_{\mathcal{G}}^{k}$ corresponding to the measurement outcome $k$ can be written as $\rho_{\mathcal{G}}^{k}=\mathcal{Z}^{k} \left( \rho_{\mathcal{G}_{\Omega}} \otimes \mathcal{M}_k\right) \mathcal{Z}^{k}$, where $\rho_{\mathcal{G}_{\Omega}}=\ket{\mathcal{G}(\Omega)}\bra{\mathcal{G}(\Omega)}$ is the graph state corresponding to the subgraph $\mathcal{G}_{\Omega}$, and the corresponding probability is $p^k=2^{-m}$, which is independent of $k$. The correction is a local operator that can be factorized in a part acting on $ \Omega $ and a part acting on the rest of the qubits, i.e., $ \mathcal{Z}^{k}=\mathcal{Z}^{k}_{\Omega}\otimes\mathcal{Z}^{k}_{\overline{\Omega}} $. Here, $\mathcal{Z}_{\Omega}^{k}$ is the outcome-dependent correction applied to the qubits in $\Omega$ due to the local $Z$ measurements over the qubits in $\overline{\Omega}$ (see Eq.~(\ref{correction_witnesses})). Therefore, tracing out the qubits in $\overline{\Omega}$, the post-measurement state on $\Omega$ corresponding to outcome $k$ is
\begin{eqnarray}
\rho_{\mathcal{G}_\Omega}^{k}=\mathcal{Z}_{\Omega}^{k} \rho_{\mathcal{G}_{\Omega}} \mathcal{Z}_{\Omega}^{k}.
\end{eqnarray}
Similar to Eqs.~(\ref{eq:unitary_expanded}) and (\ref{correction_witnesses}), $ \mathcal{Z}_{\Omega}^{k} $ only depends on the links in $ \mathcal{E}_{\gamma} $.
In the case of GD states, the $N$-qubit post-measurement state, $\rho_{GD}^{k}=\mathcal{M}_k\rho_{GD} \mathcal{M}_k$, corresponding to a specific outcome $k$, can be written as
\begin{eqnarray}
{\rho_{GD}^{k}} &=&\sum_{\nu} p_\nu \mathcal{M}_{k} \ket{\mathcal{G}^\nu}\bra{\mathcal{G}^\nu} \mathcal{M}_{k}.
\label{eq:pm_gdstate}
\end{eqnarray}
Using Eq. (\ref{eq:meas_rule_gen}) in Eq. (\ref{eq:pm_gdstate}), one obtains the normalized post-measurement state corresponding to the outcome $k$ as
\begin{eqnarray}
{\rho_{\text{GD}}^{k}}=\sum_{\nu}p_\nu Z_{\nu} \mathcal{Z}^{k}\left( \rho_{\mathcal{G}_{\Omega}}^0\otimes \mathcal{M}_k\right) \mathcal{Z}^{k} Z_\nu,
\end{eqnarray}
where $\rho_{\mathcal{G}_{\Omega}}^0$ is given by $\rho_{\mathcal{G}_{\Omega}}^0=\rho_{\mathcal{G}_{\Omega}}=\ket{\mathcal{G}(\Omega)}\bra{\mathcal{G}(\Omega)}$. Without loss of generality, we write $Z_{\nu}$ as $Z_{\nu_{\Omega}}\otimes Z_{\nu_{\overline{\Omega}}}$, where the indices $\nu_{\Omega}$ ($\nu_{\Omega}=0,1,2,\cdots,2^{N-m}-1$) and $\nu_{\overline{\Omega}}$ ($\nu_{\overline{\Omega}}=0,1,2,\cdots,2^m-1$) are such that
\begin{eqnarray}
Z_{\nu_{\Omega}}&=&\bigotimes_{i\in\Omega}Z_{i}^{\nu_{\Omega}^i},\nonumber \\
Z_{\nu_{\overline{\Omega}}}&=&\bigotimes_{r_j\in\overline{\Omega}}Z_{r_{j}}^{\nu_{\overline{\Omega}}^{r_j}},
\end{eqnarray}
with $\nu_{\Omega}^i,\nu_{\overline{\Omega}}^{r_j}\in\{0,1\}$. Tracing out the qubits in $\overline{\Omega}$, the post-measurement state ${\rho_{\text{GD},\Omega}^{k}}$ corresponding to the region $\Omega$ can be written as
\begin{eqnarray}
\rho_{\text{GD},\Omega}^{k}=\mathcal{Z}_{\Omega}^{k}\rho_{\text{GD},\Omega}^{0} \mathcal{Z}_{\Omega}^{k},
\label{eq:k-dependent}
\end{eqnarray}
with
\begin{eqnarray}
\rho_{\text{GD},\Omega}^0&=&\sum_{\nu_{\Omega}}\tilde{p}_{\nu_{\Omega}} Z_{\nu_{\Omega}} \rho_{\mathcal{G}_{\Omega}}^0 Z_{\nu_{\Omega}},\nonumber\\
&=&\sum_{\nu_{\Omega}}\tilde{p}_{\nu_{\Omega}}\ket{\mathcal{G}^{\nu_{\Omega}}(\Omega)}\bra{\mathcal{G}^{\nu_{\Omega}}(\Omega)}
\end{eqnarray}
being the post-measurement state corresponding to $k=0$ (i.e., $\mathcal{Z}_{\Omega}^{k}=I_{\Omega}$), where $\tilde{p}_{\nu_{\Omega}^\prime}=\sum_{\nu}p_{\nu}\delta_{\nu_{\Omega},\nu_{\Omega}^\prime}$. Note here that the measurement outcome is reflected only through the correction $\mathcal{Z}_{\Omega}^{k}$. Therefore, the post-measurement states ${\rho_{\text{GD},\Omega}^{k}}$ corresponding to different measurement outcomes $k\neq 0$ are connected to $\rho_{\text{GD}, \Omega}^0$ by local unitary operators of the form $\mathcal{Z}_{\Omega}^{k}$.
Next, we determine the form of $U_{\gamma}\rho_{\text{GD}}U_{\gamma}^{-1}$, given by
\begin{eqnarray}
U_{\gamma}\rho_{\text{GD}}U_{\gamma}^{-1}=\sum_{\nu}p_{\nu}Z_\nu U_{\gamma}\rho_{\mathcal{G}}^0 U_{\gamma}^{-1}Z_\nu.
\end{eqnarray}
Since by the definition of $U_{\gamma}$, $U_{\gamma}\rho_{\mathcal{G}}^0 U_{\gamma}^{-1}=\rho_{\mathcal{G}_{\Omega}}^0\otimes \rho_{\mathcal{G}_{\overline{\Omega}}}^0$, $\rho_{\text{GD},\Omega}=\mbox{Tr}_{\overline{\Omega}}(U_{\gamma}\rho_{\text{GD}}U_{\gamma}^{-1})$ leads to
\begin{eqnarray}
\rho_{\text{GD},\Omega}=\sum_{\nu_{\Omega}}\tilde{p}_{\nu_{\Omega}} Z_{\nu_{\Omega}} \rho_{\mathcal{G}_{\Omega}}^0 Z_{\nu_{\Omega}}=\rho_{\text{GD},\Omega}^0,
\label{eq:gd_cz}
\end{eqnarray}
with the definitions of $\nu_{\Omega}$ as given above.
We now consider the hierarchy of bounds given in (\ref{eq:ineq_3}), and observe that $E^0_{\Omega}(\rho_{\text{GD}})=E(\rho_{\text{GD},\Omega}^0)$ due to Eq. (\ref{eq:k-dependent}) and the local unitary invariance of entanglement measures. Also, from Eq.~(\ref{eq:gd_cz}), $E(\rho_{\text{GD},\Omega})=E(\rho_{\text{GD},\Omega}^0)$. Combining these observations, the relation in (\ref{eq:ineq_3}) is modified as
\begin{eqnarray}
E_{\Omega}(\rho_{\text{GD}})\geq E^\mathcal{P}_{\Omega}(\rho_{\text{GD}})\geq E^0_{\Omega}(\rho_{\text{GD}})= E(\rho_{\text{GD},\Omega})\geq E^{\mathcal{W}}_{\Omega}(\omega_{\text{GD}}) \nonumber \\
\label{eq:ineq_5}
\end{eqnarray}
for GD states, where $ \omega_{\text{GD}}=\mathrm{Tr}\left(\rho_{\text{GD}}\mathcal{W}_{\Omega}\right) $.
\subsubsection*{Witness-based lower bound for regions of size two}
We now focus on the WLB in the case of GD states where the region $\Omega$ of interest has size two. For concreteness, we choose logarithmic negativity \cite{lee2000,vidal2002,plenio2005} as the measure of bipartite entanglement. For bipartite quantum states $\varrho_{AB}$ of two parties $A$ and $B$, logarithmic negativity is defined as
\begin{eqnarray}
L_g(\varrho_{AB})=\log_2(N_g(\varrho_{AB})+1),
\label{eq:logneg}
\end{eqnarray}
where $N_g(\varrho_{AB})$ is the negativity of $\varrho_{AB}$, based on the Peres-Horodecki separability criterion \cite{peres1996,horodecki1996}, given by
\begin{eqnarray}
N_{g}=\|\varrho_{AB}^{T_{A}}\|_1-1.
\label{eq:neg}
\end{eqnarray}
Here, $\varrho_{AB}^{T_{A}}$ is the partial transposition of the state $\varrho_{AB}$ with respect to $A$ performed in the computational basis, and $\|\varrho\|_1=\mbox{Tr}\sqrt{\varrho^\dagger \varrho}$ is the trace-norm of $\varrho$. The negativity of the state $\varrho_{AB}$ can then be computed as
\begin{eqnarray}
N_{g}=2\,\sum_{\lambda_i<0}|\lambda_i|,
\label{eq:neg_comp}
\end{eqnarray}
where $\{\lambda_i\}$ are the eigenvalues of $\varrho_{AB}^{T_A}$. In the case of witness operators $\mathcal{W}^g_{\Omega}$ given by Eq.~(\ref{eq:witness}), the lower bound $E^{\mathcal{W}}_{\Omega}(\omega)$ of $N_g$, corresponding to a region $\Omega$ of two or three qubits, is given by (see Appendix \ref{ap:wlb_opt})
\begin{eqnarray}
E^{\mathcal{W}}_{\Omega}(\omega) &=&
\begin{cases}
-2\omega, & \text{for } \omega < 0, \\
0, & \text{for } \omega \geq 0.
\end{cases}.
\label{eq:wit_func_form}
\end{eqnarray}
We demonstrate the following results for negativity, which can be straightforwardly extended in the case of logarithmic negativity.
Using the form of $\rho_{\text{GD},\Omega}$ in Eq.~(\ref{eq:gd_cz}) and the witness operator $\mathcal{W}^g_{\Omega}$ in Eq.~(\ref{eq:witness}), one can determine $\omega_{\text{GD}}=\mbox{Tr}(\rho_{\text{GD},\Omega}\mathcal{W}^g_{\Omega})=\frac{1}{2}-\tilde{p}_0$, implying $E^{\mathcal{W}}_{\Omega}(\omega_{\text{GD}})=2\tilde{p}_0-1$ when $\tilde{p}_0>\frac{1}{2}$ (i.e., $\omega_{\text{GD}}<0$), and $E^{\mathcal{W}}_{\Omega}(\omega_{\text{GD}})=0$ for $\tilde{p}_0\leq\frac{1}{2}$ (i.e., $\omega_{\text{GD}}\geq 0$).
Considering now the two qubits in $\Omega$ to be the two parties $A$ and $B$, $\rho_{\text{GD},\Omega}^{T_{A}}$ is also diagonal in the graph-state basis, similar to $\rho_{\text{GD},\Omega}$, with the eigenvalues of $\rho_{\text{GD},\Omega}^{T_{A}}$ given by
\begin{eqnarray}
\lambda_{0} &=& 1/2-\tilde{p}_{3},\, \lambda_{1}=1/2-\tilde{p}_{2},\nonumber \\
\lambda_{2} &=& 1/2-\tilde{p}_{1},\, \lambda_{3}=1/2-\tilde{p}_{0}.
\label{eq:eigen_ptran}
\end{eqnarray}
If $\tilde{p}_i\le \frac{1}{2}$ $\forall$ $i\in\{0,1,2,3\}$, $\lambda_i\geq0$, implying $N_g(\rho_{\text{GD},\Omega})=0$. On the other hand, if any of the weights $\{\tilde{p}_i\}$, say $\tilde{p}_j=\max\{\tilde{p}_i\}$ is $>\frac{1}{2}$, then $\tilde{p}_{i\neq j}<\frac{1}{2}$. If $j=0$, then $\lambda_3<0$, implying $N_g(\rho_{\text{GD},\Omega})=2\tilde{p}_0-1$.
Therefore, $N_{g}(\rho_{\text{GD},\Omega})=E^{\mathcal{W}}_{\Omega}(\omega_{\text{GD}})$ if $\tilde{p}_0=\max\{\tilde{p}_{i}\}$, $i=0,1,2,3$, implying that in case of negativity as the entanglement measure, and for $\Omega$ having size two, Eq.~(\ref{eq:ineq_5}) for GD states becomes
\begin{eqnarray}
E_{\Omega}(\rho_{\text{GD}})\geq E^\mathcal{P}_{\Omega}(\rho_{\text{GD}})\geq E^0_{\Omega}(\rho_{\text{GD}})&=& E(\rho_{\text{GD},\Omega})\nonumber \\ &=& E^{\mathcal{W}}_{\Omega}(\omega_{\text{GD}}).
\label{eq:ineq_6}
\end{eqnarray}
The corresponding logarithmic negativity of $\rho_{\text{GD},\Omega}$ is given by $L_g(\rho_{\text{GD},\Omega})=\log_2(2\tilde{p}_0)$, following Eq.~(\ref{eq:logneg}). In Sec.~\ref{sec:perf}, we consider local, spatially uncorrelated Pauli noise, giving rise to GD states in which $\tilde{p}_0>\frac{1}{2}$ is a common occurrence.
As a final comment, in a region $\Omega$ constituted of two qubits, the bipartite and the genuine multipartite entanglements coincide, but this is not the case if $\Omega$ contains more than two qubits. We shall demonstrate that the use of a bipartite entanglement measure for a region of two qubits results in a tighter WLB where $E_{\Omega}^{\mathcal{W}}$ matches with $E^0_{\Omega}(\rho^\prime)$, while such property is absent when $\Omega$ is bigger (see Fig.~\ref{fig:linear_23}(a)--(b) and the subsequent discussions). The procedure of obtaining a WLB for localizable entanglement over a region $\Omega$ having size bigger than two qubits remains the same as described in Secs. \ref{subsec:wlb_mlb}-\ref{subsec:lb_gdstate} and Appendix \ref{ap:wlb_opt}, the only difference being in the functional form of $E^{\mathcal{W}}_{\Omega}(\omega)$ (Eq. (\ref{eq:wit_func_form})), which depends explicitly on the chosen entanglement measure. For demonstration, in this paper, we have chosen logarithmic negativity as the measure of bipartite entanglement between the two qubits in $\Omega$ due to the computability of the measure. The main challenge in obtaining a proper WLB for a region $\Omega$ of size larger than two qubits remains in the scarcity of computable genuine multipartite measure of entanglement for mixed multiparty states. However, given such a computable multiparty entanglement measure exists, WLB corresponding to that measure for a region larger than two qubits can be computed by determining $E_{\Omega}^{\mathcal{W}}(\omega)$.
\section{Performance of the lower bounds}
\label{sec:perf}
\begin{figure*}
\includegraphics[width=\textwidth]{3-4-linear-graph.pdf}
\caption{(Colour online) \textbf{Localizable entanglement over regions of different size against noise-parameter for linear graphs.} (a) Variations of $E_{13}(\rho)$, $E_{13}^{\mathcal{P}}(\rho)$, $E^0_{13}(\rho^\prime)$ and $E^{\mathcal{W}}_{13}(\omega)$ as functions of the noise parameter $q$ for the region $\Omega\equiv \{1,3\}$ in the linear graph $\mathcal{G}_{L}=\{\mathcal{V}_L,\mathcal{E}_L\}$ composed of four qubits, where $\mathcal{V}_{L}=\{1,2,3,4\}$, and $\mathcal{E}_{L}=\{(1,2),(2,3),(3,4)\}$. We consider BF noise applied to all the qubits. (b) Variations of $E_{1|23}(\rho)$, $E_{1|23}^{\mathcal{P}}(\rho)$, $E^0_{1|23}(\rho)$ and $E^{\mathcal{W}}_{1|23}(\omega)$ as functions of $q$ for the region $\Omega\equiv \{1,2,3\}$ with the bipartition $1|23$ in the linear graph $\mathcal{G}_L$ under BF noise. (c) Variations of $E_{13}(\rho)$, $E_{13}^{\mathcal{P}}(\rho)$, $E^0_{13}(\rho^\prime)$ and $E^{\mathcal{W}}_{13}(\omega)$ as functions of $q$ for the region $\Omega\equiv \{1,3\}$ in the linear graph $\mathcal{G}_{L}$ under AD noise.}
\label{fig:linear_23}
\end{figure*}
In this section, we discuss the performance of the MLB and the WLB discussed in Sec. \ref{sec:lb}. For concreteness, to this end we consider graph states $\mathcal{G}$ under local uncorrelated Pauli noise and local amplitude-damping (AD) noise \cite{nielsen2010}, and discuss how the MLB and the WLB can be computed over a connected region $\Omega$ in the $N$-qubit system. We employ the Kraus operator representation \cite{nielsen2010,cavalcanti2009,aolita2010,holevo2012}, where the evolution of the graph state $\rho_{\mathcal{G}}$ under noise is given by $\rho_\mathcal{G}\rightarrow\rho=\Lambda(\rho_\mathcal{G})$, and where the operation $\Lambda(.)$ can be expressed by an operator-sum decomposition \cite{nielsen2010,holevo2012} given by
\begin{eqnarray}
\rho = \Lambda(\rho_\mathcal{G}) &=& \sum_{\alpha=0}^{4^N-1}K_\alpha\rho_\mathcal{G} K^\dagger_\alpha \nonumber \\
&=& \sum_{\alpha=0}^{4^N-1}q_\alpha J_\alpha\rho_\mathcal{G} J^\dagger_\alpha.
\label{eq:evolve}
\end{eqnarray}
Here, $\{K_\alpha=\sqrt{q_\alpha}J_\alpha\}$ are the Kraus operators satisfying the completeness condition $\sum_{\alpha}K_\alpha^\dagger K_\alpha=I$, with $I$ being the identity operator in the Hilbert space of the system. The map $\Lambda(.)$ in Eq.~(\ref{eq:evolve}) is a completely positive trace-preserving (CPTP) map, and $q$ is the driving parameter of the noise model, which introduces the notion of time, $t$, depending on the type of the physical process through which the system evolves.
For uncorrelated Pauli noise, the individual Kraus operators, $K_{\alpha}$ can be written as the product of identity, $I$, and the three Pauli operators, $X,Y$, and $Z$ acting on the individual qubits. The operators $\{J_{\alpha}\}$ in Eq. (\ref{eq:evolve}) now have the form
\begin{eqnarray}
J_\alpha=\bigotimes_{i=1}^N\sigma_{\alpha_i},
\label{eq:kraus_pauli}
\end{eqnarray}
and
\begin{eqnarray}
q_\alpha=\prod_{i=1}^Nq_{\alpha_i},
\label{eq:kraus_prob}
\end{eqnarray}
with $\alpha_i\in\{0,1,2,3\}$, $\sum_{\alpha_i=0}^3q_{\alpha_i}=1$, and $\sigma_{0}=I_i$, $\sigma_{1}=X_i$, $\sigma_{2}=Y_i$, and $\sigma_{3}=Z_i$. Note here that the index $\alpha$ on the left hand side can be interpreted as the multi-index $\alpha\equiv\alpha_1\alpha_2\cdots\alpha_N$, where $\alpha$ is represented in base $4$ by the string $\alpha_1\alpha_2\cdots\alpha_N$. Examples of Pauli noise include bit-flip (BF), bit-phase-flip (BPF), phase-flip (PF), and depolarizing (DP) channels, with the corresponding values of the probability $q_{\alpha_i}$ given for completeness as follows:
\begin{eqnarray}
\mbox{BF:} &\mbox{\hspace{0.2cm}}& q_{0}=1-\frac{q}{2},q_{1}=\frac{q}{2},q_{2}=0,q_{3}=0; \\
\mbox{BPF:} &\mbox{\hspace{0.2cm}}& q_{0}=1-\frac{q}{2},q_{1}=0,q_{2}=\frac{q}{2},q_{3}=0; \\
\mbox{PF:} &\mbox{\hspace{0.2cm}}& q_{0}=1-\frac{q}{2},q_{1}=0,q_{2}=0,q_{3}=\frac{q}{2}; \\
\mbox{DP:} &\mbox{\hspace{0.2cm}}& q_{0}=1-\frac{3q}{4},q_{1}=\frac{q}{4},q_{2}=\frac{q}{4},q_{3}=\frac{q}{4}.
\label{eq:channel_prob}
\end{eqnarray}
All of these channels induce a complete decoherence on the input quantum state at probability $q = 1$, without any energy exchange with environments, thereby representing non-dissipative noisy channels. Note here that an operation $\sigma_{\alpha_i}$, $\alpha_i=1,2$, on the qubit $i$ of a pure graph state is equivalent to a Pauli $Z$ operator on the qubit $i$ and its neighbourhood, as shown in the following equations:
\begin{eqnarray}
\sigma_{\alpha_i=1} &\leftrightarrow & \bigotimes_{j\in\mathcal{N}_i}Z_j,\nonumber\\
\sigma_{\alpha_i=2} &\leftrightarrow & Z_i\otimes \left[\bigotimes_{j\in\mathcal{N}_i}Z_j\right].
\end{eqnarray}
This implies that a graph state under local uncorrelated Pauli noise is a graph-diagonal state \cite{cavalcanti2009,aolita2010}. Hence the discussions in Sec. \ref{subsec:lb_gdstate} apply.
On the other hand, in the case of local AD noise, the single-qubit Kraus operators are given by
\begin{eqnarray}
K_0&=&\left(\begin{array}{cc}
1 & 0 \\
0 & \sqrt{1-q}\\
\end{array}\right),
K_1=\left(\begin{array}{cc}
0 & \sqrt{q} \\
0 & 0\\
\end{array}\right),
\end{eqnarray}
with $K_2$ and $K_3$ being null operators. Note that although the single-qubit Kraus operators in the case of AD channel can be expanded in terms of Pauli operators, the resulting state $\rho$ due to the application of AD noise to all the qubits in a graph state is not a GD state.
We now illustrate the behaviour of the different quantities in Eq.~(\ref{eq:ineq_3}) for the specific example of a linear graph $\mathcal{G}_{L}=\{\mathcal{V}_L,\mathcal{E}_L\}$ of size $N=4$, where $\mathcal{V}_L=\{1,2,3,4\}$, and $\mathcal{E}_L=\{(1,2),(2,3),(3,4)\}$. We consider two specific cases -- one with a region $\Omega$ of size $2$, constituted of qubits $1$ and $3$ that are not connected by a direct link (see Fig. \ref{fig:linear_23}(a)), and the other with a connected region $\Omega$ of three-qubits, constituted of the qubits $1$, $2$, and $3$. In the first case, one may consider a LC operation on the qubit $2$ to create the link $(1,3)$, so that $\Omega$ becomes connected in the new graph $\mathcal{G}^\prime=\tau_2(\mathcal{G}_L)$. We determine $E_{13}(\rho)$, $E^{\mathcal{P}}_{13}(\rho)$, $E^0_{13}(\rho^\prime)$, and $E^{\mathcal{W}}_{\Omega}(\omega)$ as per the discussions in Sec.~\ref{sec:lb}, when BF noise is applied to all the qubits. Note here that the transformation $\tau_2(.)$ corresponds to the local unitary operation $U_L=\exp (i\pi Z_1/4)\exp(-i\pi X_2/4)\exp(i\pi Z_3/4)$ on $\ket{\mathcal{G}_L}$ (see Sec. \ref{subsec:stab}). Therefore, computing $E^0_{13}(\rho^\prime)$ for the state $\rho^\prime$ is equivalent to computing $E^{l=6}_{13}(\rho)$ for the state $\rho$ by performing $Y$ measurement on the qubit $2$ and $Z$ measurement on the qubit $4$. Recall that the value $l=6$ is the decimal representation of the multi-index $l_{r_1}l_{r_2}$ in base $3$ ($l_{r_1}=2$ for $r_1\equiv 2$, implying $Y$ measurement, and $l_{r_2}=0$ for $r_2\equiv 4$, implying $Z$ measurement), following the notation for measurement bases as introduced in Sec. \ref{subsec:le}. Note also that this differs from the index convention for designating Pauli operators used in this section. In Fig. \ref{fig:linear_23}(a), we have plotted the variations of $E_{13}(\rho)$, $E^{\mathcal{P}}_{13}(\rho)$, $E^0_{13}(\rho^\prime)$, and $E^\mathcal{W}(\omega)$ as functions of $q$. We observe that irrespective of the structure of the graph, the LE over two and three-qubit regions in graph states under local uncorrelated Pauli noise is always optimized by local Pauli measurements, implying $E_{\Omega}(\rho)=E^{\mathcal{P}}_{\Omega}(\rho)$. Also, in accordance with the results obtained in Sec. \ref{subsec:lb_gdstate}, we find that $E^{0}_{13}(\rho^\prime)=E^{\mathcal{W}}(\omega)$ for all values of $q$. We point out here that the quantity $E^{l=3}_{13}(\rho)$, corresponding to an $X$ measurement on qubit $2$ ($l_{r_1}=1$) and a $Z$ measurement on qubit $4$ ($l_{r_2}=0$), is equal to $E_{13}(\rho)$, as $l=3$ provides the optimal measurement basis in the noiseless case. This is understandable from the fact that the measurement over qubit $2$ commutes with the BF noise applied to it, thereby neutralizing the effect of the noise. This will be discussed in more detail in Sec. \ref{subsec:mlb_arb}.
On the other hand, in the second example, the region of interest $\Omega\equiv\{1,2,3\}$ is already connected. Since we consider a bipartite measure, namely, logarithmic negativity as the measure of entanglement, we focus on the bipartition $1|23$ of the region $\Omega$. However, the results to be reported remain unchanged in the case of other two bipartitions, $2|13$ and $12|3$ also. The variations of $E_{1|23}(\rho)$, $E_{1|23}^{\mathcal{P}}(\rho)$, $E^0_{1|23}(\rho)$, and $E^\mathcal{W}_{1|23}(\omega)$ against the noise parameter $q$ are depicted in Fig. \ref{fig:linear_23}(b). Note here that in contrast to the former example, here $E^0_{1|23}(\rho)>E^\mathcal{W}(\omega)$ for all values of $q$ except at $q=0$, therefore ensuring the validity of the results obtained in Sec.~\ref{subsec:lb_gdstate}. Lastly, we consider the local AD noise as an example of non-Pauli noise, and determine the variations of $E_{13}(\rho)$, $E^{\mathcal{P}}_{13}(\rho)$, $E^0_{13}(\rho^\prime)$, and $E^{\mathcal{W}}_{\Omega}(\omega)$ as functions of $q$. The results are depicted in Fig. \ref{fig:linear_23}(c). The reconstruction of the graph and the corresponding change in the measurement directions are as the same as in Fig. \ref{fig:linear_23}(a).
\subsection{Measurement-based lower bound under Pauli noise for arbitrary graphs}
\label{subsec:mlb_arb}
From the results presented in Fig. \ref{fig:linear_23}(b), it is clear that there exists situations in which $E^{0}_{\Omega}$ may provide a tighter lower bound than $E^\mathcal{W}(\omega)$. However, in the case of noisy graph states of large size, the computation of the quantity $E_{\Omega}^0$ as a lower bound of $E_{\Omega}$ may turn out to be difficult. In this subsection, we shall describe how $E_{\Omega}^0$, in the case of uncorrelated local Pauli noise and a specific connected region $\Omega$, can be computed as a function of the noise parameter, $q$, by using only the knowledge of the connectivity of the underlying graph. For the purpose of demonstration, we consider a region consisting of two qubits $a$ and $b$ only, so that $\Omega\equiv \{a,b\}$. However, the methodology discussed here can be applied to regions of any size in arbitrary graphs.
Let us consider the general situation where $a$ and $b$ are not connected in $\mathcal{G}$. In such a case, one may obtain a graph $\mathcal{G}^\prime$ with the link $(a,b)$ by the prescriptions discussed in Sec.~\ref{subsec:lb_unitary}. Application of Eq.~(\ref{eq:local_unitary}) in Eq.~(\ref{eq:evolve}) leads to
\begin{eqnarray}
\rho &=&\sum_{\alpha=0}^{4^N-1}q_\alpha J_\alpha \rho_{\mathcal{G}} J^\dagger_\alpha = U_L^{-1}\rho^\prime U_L,
\label{eq:transform}
\end{eqnarray}
where
\begin{eqnarray}
\rho^\prime &=& \sum_{\alpha=0}^{4^N-1}q_\alpha J^\prime_{\alpha} \rho_{\mathcal{G}^\prime} J^\prime_{\alpha},
\label{eq:evolve2}
\end{eqnarray}
with $J_{\alpha}^\prime=U_L J_{\alpha}U_L^{-1}$, and $\rho_{\mathcal{G}^\prime}=\ket{\mathcal{G}^\prime}\bra{\mathcal{G}^{\prime}}$. The property of the Clifford operators (Eq.~(\ref{eq:clifford_property})) implies that the operators $J^\prime_{\alpha}=J_{\alpha^\prime}$ in Eq.~(\ref{eq:evolve2}), where $J_{\alpha^\prime}$ is now given by
\begin{eqnarray}
J_{\alpha^\prime}=\bigotimes_{i=1}^N\sigma_{\alpha_i^\prime}
\end{eqnarray}
with $\alpha_i^\prime=0,1,2,3$, and $\sigma_{\alpha_i^\prime}=U_i^{-1}\sigma_{\alpha_i}U_i$, where the index $\alpha^\prime$ can be interpreted as the multi-index $\alpha^\prime\equiv\alpha_1^\prime\alpha_2^\prime\cdots\alpha_{N}^\prime$, in the same way as $\alpha$. Note that $\rho^\prime$ is also a GD state.
For reasons that will become clear in the subsequent discussion, we write the modified operators, $\{J_{\alpha^\prime}\}$, and the probabilities $q_\alpha$ in Eq.~(\ref{eq:evolve2}) as
\begin{eqnarray}
J_{\alpha^\prime}&=&J_{\alpha_{ab}^\prime}\otimes\left[\bigotimes_{r_i\in\overline{\Omega}}\sigma_{\alpha_{r_i}^\prime}\right],\nonumber \\
q_\alpha &=& q_{\alpha_{ab}}q_{\overline{\Omega}}
\label{eq:kraus_pauli_2}
\end{eqnarray}
where $J_{\alpha_{ab}^\prime}=\sigma_{\alpha_{a}^\prime}\otimes\sigma_{\alpha_{b}^\prime}$, $q_{\alpha_{ab}}=q_{\alpha_a}q_{\alpha_b}$, and
\begin{eqnarray}
q_{\overline{\Omega}}=\prod_{r_i\in\overline{\Omega}}q_{\alpha_{r_i}}.
\end{eqnarray}
Here, $\alpha_{r_i},\alpha_a,\alpha_b \in\{0,1,2,3\}$, and $\sum_{\alpha_i=0}^3q_{\alpha_i}=1$. The index $\alpha^\prime$, $\alpha_{ab}$, and $\alpha_{ab}^\prime$ can be interpreted as the multi-indices $\alpha^\prime\equiv\alpha^\prime_a\alpha^\prime_b\alpha^\prime_{r_1}\cdots\alpha^\prime_{r_{N-2}}$, $\alpha_{ab}\equiv\alpha_a\alpha_b$, and $\alpha_{ab}^\prime\equiv\alpha_a^\prime \alpha_b^\prime$, in the same way as $\alpha$ in Eq.~(\ref{eq:kraus_pauli}).
Let us now consider the measurement operation $\mathcal{M}_0$, as a result of which the $N$-qubit post-measurement state, ${\rho^\prime}^k=\mathcal{M}_k\rho^\prime \mathcal{M}_k$, corresponding to a specific outcome $k$, can be written as
\begin{eqnarray}
{\rho^\prime}^k &=&\sum_{\alpha}q_\alpha J_{\alpha^\prime} \mathcal{M}_{k^\prime} \rho_{\mathcal{G}^\prime} \mathcal{M}_{k^\prime} J_{\alpha^\prime},
\label{eq:pm_state}
\end{eqnarray}
with $\mathcal{M}_{k^\prime}=\bigotimes_{r_i\in\overline{\Omega}}\mathcal{M}_{k^\prime_{r_i}}$, $k^\prime_{r_i}\in\{0,1\}$, where
\begin{eqnarray}
\mathcal{M}_{k_{r_i}^\prime}=\sigma_{\alpha_{r_i}^\prime}\mathcal{M}_{k_{r_i}}\sigma_{\alpha_{r_i}^\prime}.
\label{eq:form_projectors_2_single}
\end{eqnarray}
The interpretation of the index $k^\prime$ in terms of the indices $\{k_{r_{i}}\}$ corresponding to the outcomes of the measurements on the individual qubits is similar to the other indices, such as $\alpha$, $\alpha^\prime$, $l$, and $k$. Note that the transformation in Eq.~(\ref{eq:form_projectors_2_single}) does not change the basis of the measurement, but changes its outcome.
We proceed along the same line as in Sec.~\ref{subsec:lb_gdstate}, and write the graph state as $\rho_{\mathcal{G}^\prime}=\ket{\mathcal{G}^\prime(\overline{\Omega},a,b)}\bra{\mathcal{G}^\prime(\overline{\Omega},a,b)}$. Use of Eq.~(\ref{eq:meas_rule_gen}) in Eq.~(\ref{eq:pm_state}) over qubits in $\overline{\Omega}$, and then tracing out $\overline{\Omega}$ lead to the two-qubit post-measurement state corresponding to qubits $a$ and $b$, given by
\begin{eqnarray}
{\rho^\prime}^k_{ab}&=&\sum_{\alpha}q_{\overline{\Omega}}\left[q_{\alpha_{ab}}J_{\alpha_{ab}^\prime} \rho_{\mathcal{G}_{ab}}^{\beta}J_{\alpha_{ab}^\prime} \right],
\label{eq:pm_state_ab_2}
\end{eqnarray}
where $\rho_{\mathcal{G}_{ab}}^\beta=\mathcal{Z}^\beta_{ab} \rho_{\mathcal{G}_{ab}} \mathcal{Z}^\beta_{ab}$ and $\rho_{\mathcal{G}_{ab}}=\ket{\mathcal{G}_{ab}}\bra{\mathcal{G}_{ab}}$ is the two-qubit graph state. Here, the set $\{\mathcal{Z}^{\beta}_{ab}=Z^{\beta_a}_{a}\otimes Z_b^{\beta_b}\}$ is constituted of all possible outcome-dependent corrections on $\rho_{\mathcal{G}_{ab}}$ due to different values of $k^\prime$, where $\beta_{a},\beta_{b}\in\{0,1\}$, $Z_{a,b}^0=I_{a,b}$, $Z_{a,b}^1=Z_{a,b}$, and $\beta\equiv \beta_a\beta_b$ is a multi-index given by the decimal representation of the binary string $\beta_a\beta_b$.
Note here that $J_{\alpha_{ab}^\prime}$ and $q_{\alpha_{ab}}$ are independent of the measurement outcome, and depend respectively on the local unitary operator $U_L$ (Eq.~(\ref{eq:evolve2})), and the probability corresponding to the Kraus operators acting on the qubit-pair $(a,b)$ only. Therefore, for a specific graph $\mathcal{G}$, further simplification of the form of the state ${\rho^\prime}^k_{ab}$ is possible by grouping the terms with identical $\rho_{\mathcal{G}_{ab}}^{\beta}$ (i.e., $\rho_{\mathcal{G}_{ab}}^{\beta}$ with the same value of $\beta$) together. Let us introduce the noise local to the qubit pair $(a,b)$ as $\Lambda_{ab}$, where
\begin{eqnarray}
\Lambda_{ab}(\rho_{\mathcal{G}_{ab}}^\beta)&=&\sum_{\alpha_{ab}}q_{\alpha_{ab}}J_{\alpha_{ab}^\prime}\rho_{\mathcal{G}_{ab}}^\beta J_{\alpha_{ab}^\prime}.
\label{eq:loc_noise}
\end{eqnarray}
Using this notation, Eq.(\ref{eq:pm_state_ab_2}), for a specific graph $\mathcal{G}$, can be written as
\begin{eqnarray}
{\rho^\prime}^k_{ab}&=&\sum_{\beta}q_{\overline{\Omega}}^\beta \Lambda_{ab}(\rho_{\mathcal{G}_{ab}}^\beta)=\Lambda_{ab}\big(\tilde{\rho}_{ab}),
\label{eq:pm_state_ab_3}
\end{eqnarray}
where
\begin{eqnarray}
\tilde{\rho}_{ab}=\sum_{\beta}q_{\overline{\Omega}}^\beta\rho_{\mathcal{G}_{ab}}^\beta
\label{eq:rhok}
\end{eqnarray}
and for a fixed value of $\beta=\beta^\prime$, $q_{\overline{\Omega}}^{\beta^\prime}$ is the sum of the probabilities $q_{\overline{\Omega}}$ corresponding to all the values of $\alpha$, where $\beta=\beta^\prime$. Note that ${\rho^\prime}^k_{ab}=\tilde{\rho}_{ab}$ iff $\alpha_{a}^\prime=\alpha_b^\prime=0$, implying $\alpha_a=\alpha_b=0$, i.e., qubits $a$ and $b$ are free from noise. If local uncorrelated Pauli noise is present on qubits $a$ and $b$, then the entanglement of the qubit-pair $(a,b)$ decays, implying $E({\rho^\prime}^k_{ab})\leq E(\tilde{\rho}_{ab})$, $E$ being any entanglement measure. We further note that Eq. (\ref{eq:meas_rule_gen}) suggests that the corrections over the qubit pair $(a,b)$ are fully determined by the neighbourhood of the qubit pair, denoted by $\mathcal{N}_{ab}=\mathcal{N}_a\cup\mathcal{N}_b$, where $\mathcal{N}_{a}(\mathcal{N}_b)$ is the neighbourhood of qubit $a$ ($b$). Therefore, the probability corresponding to the Kraus operator acting on qubit $r_i\notin\mathcal{N}_{ab}$ does not affect the post-measurement state. Since the separability of Pauli maps indicates that $\sum_{\alpha_{{\overline{\Omega}}^\prime}}q_{{\overline{\Omega}}^\prime}=1$ for any ${\overline{\Omega}}^\prime\subset{\overline{\Omega}}$, where $\alpha_{{\overline{\Omega}}^\prime}$ is the multi-index involving the indices $\{\alpha_{r_i}\}$ such that $r_{i}\in{\overline{\Omega}}^\prime$, $q_{\overline{\Omega}}^{\beta}$ can be expressed as
\begin{eqnarray}
q_{\overline{\Omega}}^{\beta}&=&\sum_{\underset{\overline{\Omega}\in\mathcal{N}_{ab}}{\alpha_{\overline{\Omega}}}}q_{\overline{\Omega}|\beta}.
\label{eq:qr}
\end{eqnarray}
\begin{figure}
\includegraphics[scale=0.4]{ab_neigh.pdf}
\caption{(Colour online) \textbf{General structure of the neighbourhood of a connected two-qubit region in an arbitrary graph.} (a) The neighbourhood $\mathcal{N}_{ab}$ of the connected qubits $a$ and $b$ in the graph $\mathcal{G}^\prime=\tau_2\circ\tau_1(\mathcal{G_S})$ shown in Fig.~\ref{fig:square_graph}(b). The links that are connected directly to either of the qubits $a$ or $b$ are depicted by continuous lines, while the links $\{(i,j)\}$ with $i,j\in\mathcal{N}_{ab}$ are represented by broken lines. (b) General structure of $\mathcal{N}_{ab}$ in an arbitrary graph, where the red qubits are the connected qubits of interest, labelled by $a$ and $b$. The neighbourhood $\mathcal{N}_{ab}$ is constituted of three types of qubits : (1) the qubits that are connected to both $a$ and $b$ (the set $\tilde{\mathcal{N}}_{ab}$, denoted by yellow nodes), (2) the qubits that are connected to only $a$ (the set $\tilde{\mathcal{N}}_a$, denoted by blue nodes), and (3) those connected to only $b$ (the set $\tilde{\mathcal{N}}_b$, denoted by green nodes).}
\label{fig:neigh}
\end{figure}
\subsubsection*{MLB as a function of noise strength and system size}
The dependence of $q_{\overline{\Omega}}^\beta$ on the noise strength and the system size can be explicitly determined by considering a general form of the neighbourhood $\mathcal{N}_{ab}$ in an arbitrary graph $\mathcal{G}^\prime$, where the qubits $a$ and $b$ are connected. Let us consider, for example, the neighbourhood $\mathcal{N}_{ab}$ in the graph $\tau_2\circ\tau_1(\mathcal{G}_S)$ (Fig.\ref{fig:square_graph}(b)). In Fig.~\ref{fig:neigh}(a), we present $\mathcal{N}_{ab}$ corresponding to $\tau_2\circ\tau_1(\mathcal{G}_S)$, where the black (colour online) qubits are the qubits of interest, and $\mathcal{N}_{ab} $ is constituted of the gray (colour online) qubits. The broken links indicate the connectivity of the neighbourhood qubits that are irrelevant in the context of the corrections applied to the qubit pair $(a,b)$ due to local Pauli measurements over the qubits in $\mathcal{N}_{ab}$. On the other hand, the continuous links are the links that connect a qubit in $\mathcal{N}_{ab}$ with either $a$, or $b$, or both, which represent the three types of qubits constituting $\mathcal{N}_{ab}$. Evidently, the corrections on $(a,b)$ according to Eq.~(\ref{eq:meas_rule_gen}) are determined by the connectivity of the qubits in $\mathcal{N}_{ab}$ represented by the continuous links. These features remain unaltered even in the case of a pair of connected qubits in an arbitrary graph.
In Fig. \ref{fig:neigh}(b), we present the most general form of an isolated neighbourhood $\mathcal{N}_{ab}$ of a connected qubit-pair $(a,b)$ in an arbitrary $\mathcal{G}^\prime$. The qubits in $\mathcal{N}_{ab}$ are categorized into three classes according to their connectivity. \textbf{Class 1} consists of the qubits in $\mathcal{N}_{ab}$, denoted by $\tilde{\mathcal{N}}_a$ and represented by the blue (color online) nodes, that are connected to only qubit $a$. The qubits in $\mathcal{N}_{ab}$ that are denoted by $\tilde{\mathcal{N}}_b$ and are connected to only qubit $b$, form the \textbf{Class 2}, and are shown by he green (color online) nodes. And the rest of the qubits in $\mathcal{N}_{ab}$, denoted by $\tilde{\mathcal{N}}_{ab}$, that are connected to both of the qubits $a$ and $b$ is denoted by \textbf{Class 3}. Clearly, $\mathcal{N}_{ab}=\tilde{\mathcal{N}}_a\cup\tilde{\mathcal{N}}_b\cup\tilde{\mathcal{N}}_{ab}$, $\mathcal{N}_{a}=\tilde{\mathcal{N}}_a\cup\tilde{\mathcal{N}}_{ab}$, and $\mathcal{N}_{b}=\tilde{\mathcal{N}}_b\cup\tilde{\mathcal{N}}_{ab}$. From Eq.~(\ref{eq:form_projectors_2_single}), one can also categorize the noise on each qubit in $\mathcal{N}_{ab}$ into two categories. In the first category denoted by \textbf{Type 1}, $k_{r_i}^\prime\neq k_{r_i}$ with a finite probability when the transformation in Eq.~(\ref{eq:form_projectors_2_single}) is carried out (bit-flip and depolarizing channel for example), while $k_{r_i}^\prime$ always equals to $k_{r_i}$ when the noise is of \textbf{Type 2} (for example, phase-flip noise). We denote the set of qubits in $\mathcal{N}_{ab}$ experiencing \textbf{Type 1} (\textbf{Type 2}) noise by $\mathcal{N}_{ab}^1$ ($\mathcal{N}_{ab}^2$), where $\mathcal{N}_{ab}=\mathcal{N}_{ab}^1\cup\mathcal{N}_{ab}^2$, and $\mathcal{N}_{ab}^1\cap\mathcal{N}_{ab}^2=\emptyset$. Similar notations are adopted for qubits in $\tilde{\mathcal{N}}_{a}$, $\tilde{\mathcal{N}}_b$ and $\tilde{\mathcal{N}}_{ab}$ also.
\begin{figure*}
\includegraphics[width=\textwidth]{qc.pdf}
\caption{(Colour online) \textbf{Measurement-based lower bounds against noise strength for fixed neighbourhood size.} (a) Variation of $E^0_{ab}$ as a function of $q$ for $n_a=n_{ab}=n_b=n$ with $n=1$, when $\alpha_a^\prime\alpha_{b}^\prime=00$ (no noise on qubits $a$ and $b$, Eq.~(\ref{eq:lneg_analytic})), $\alpha_a^\prime\alpha_{b}^\prime=01$ (no noise on qubit $a$ and BF noise on qubit $b$), $\alpha_a^\prime\alpha_{b}^\prime=11$ (BF noise on both qubits $a$ and $b$), and $\alpha_a^\prime\alpha_{b}^\prime=13$ (BF noise on qubit $a$ and PF noise on qubit $b$). (b) Variation of $E^0_{ab}$ as a function of $q$ for $n_a=n_{ab}=n_b=n$ with $n=10$, when $\alpha_a^\prime\alpha_{b}^\prime=00$, $\alpha_a^\prime\alpha_{b}^\prime=01$, $\alpha_a^\prime\alpha_{b}^\prime=11$, and $\alpha_a^\prime\alpha_{b}^\prime=13$. (c) Variation of $q_c$ as a function of $n$ in the case of $\alpha_a^\prime\alpha_{b}^\prime=00$ (Eq.~(\ref{eq:pcrit})), $01$, $11$, and $13$.}
\label{fig:qc}
\end{figure*}
Let us first determine the form of ${\rho^\prime}^k_{ab}$ when only the set $\mathcal{N}_{ab}^1$ is populated, and $\mathcal{N}_{ab}^2=\emptyset$. Non-zero contribution in $q_{\overline{\Omega}}^\beta$ is provided by the qubits in $\mathcal{N}_{ab}^1$ due to the probabilistic change of the outcome from $k_{r_i}$ to $k_{r_i}^\prime$, along with the application of appropriate corrections $\mathcal{Z}^{\beta}_{ab}$ on $\rho_{\mathcal{G}_{ab}}$. Without loss of generality, let us denote the number of qubits in $\tilde{\mathcal{N}}_{a}^1$, $\tilde{\mathcal{N}}_{b}^1$, and $\tilde{\mathcal{N}}_{ab}^1$ by $n_a$, $n_b$, and $n_{ab}$, respectively.
Let us also assume that corresponding to a specific outcome $k$ in Eq.~(\ref{eq:pm_state}), $n^{0}_a$ of the outcomes $\{k_{r_i};r_i\in\overline{\Omega}\in\tilde{\mathcal{N}}_{a}^1\}$ are $0$, while $n_a^1$ are $1$, such that $n_a=n_a^0+n_a^1$. Similar definitions apply for $n_{b}^{0,1}$ and $n_{ab}^{0,1}$. Interpreting $q_{\overline{\Omega}}^\beta$ as the probability that the correction $\mathcal{Z}^{\beta}_{ab}$ is applied to $\rho_{\mathcal{G}_{ab}}$, its explicit form can be determined as (see Appendix \ref{ap:pmstate} for a detailed derivation)
\begin{eqnarray}
q_{\overline{\Omega}}^{0} &=& P_{a}^- P_{ab}^-P_{b}^- + P_{a}^+ P_{ab}^+ P_{b}^+,\nonumber\\
q_{\overline{\Omega}}^{1} &=& P_{a}^- P_{ab}^-P_{b}^+ + P_{a}^+ P_{ab}^+ P_{b}^-,\nonumber\\
q_{\overline{\Omega}}^{2} &=& P_{a}^+ P_{ab}^-P_{b}^- + P_{a}^- P_{ab}^+ P_{b}^+,\nonumber\\
q_{\overline{\Omega}}^{3} &=& P_{a}^- P_{ab}^+P_{b}^- + P_{a}^+ P_{ab}^- P_{b}^+,
\label{eq:mix_prob}
\end{eqnarray}
with
\begin{eqnarray}
P_a^\pm &=&\frac{1}{2}\left[1\pm (-1)^{n_{a}^1}(1-q)^{n_a}\right],\nonumber\\
P_b^\pm &=&\frac{1}{2}\left[1\pm (-1)^{n_{b}^1}(1-q)^{n_b}\right],\nonumber \\
P_{ab}^\pm &=&\frac{1}{2}\left[1\pm (-1)^{n_{ab}^1}(1-q)^{n_{ab}}\right],
\label{eq:expl}
\end{eqnarray}
where we have assumed the noise to be of BF, BPF, or DP type. Therefore, ${\rho^\prime}^k_{ab}$ (Eq.~(\ref{eq:pm_state_ab_3})), in its explicit form, can be determined as a function of the size of $\mathcal{N}_{ab}^1$ and $q$ by using Eqs.~(\ref{eq:mix_prob})-(\ref{eq:expl}) as ${\rho^\prime}^k_{ab}=\Lambda_{ab}(\tilde{\rho}_{ab})$ with
\begin{eqnarray}
\tilde{\rho}_{ab}&=&q_{\overline{\Omega}}^0\mathcal{Z}^0_{ab}\rho_{\mathcal{G}_{ab}}\mathcal{Z}^0_{ab}+q_{\overline{\Omega}}^1\mathcal{Z}^1_{ab}\rho_{\mathcal{G}_{ab}}\mathcal{Z}^1_{ab}\nonumber \\
&&+q_{\overline{\Omega}}^2\mathcal{Z}^2_{ab}\rho_{\mathcal{G}_{ab}}\mathcal{Z}^2_{ab}+q_{\overline{\Omega}}^3\mathcal{Z}^3_{ab}\rho_{\mathcal{G}_{ab}}\mathcal{Z}^3_{ab},
\label{eq:final_form}
\end{eqnarray}
where the form of $\Lambda_{ab}$ is given in Eq.~(\ref{eq:loc_noise}). In the general scenario where $\mathcal{N}_{ab}^2\neq\emptyset$, its only contribution to ${\rho^\prime}_{ab}^k$ is an extra correction belonging to the set $\{\mathcal{Z}^\beta_{ab}\}$ according to the connectivity of the qubits in $\mathcal{N}_{ab}$. However, $\mathcal{Z}^\beta_{ab}$ being a local unitary operator, the entanglement properties of ${\rho^{\prime}}^k_{ab}$ remain unchanged, and Eq.~(\ref{eq:expl}) represents the effective form of ${\rho^\prime}^k_{ab}$ as far as entanglement is concerned. Therefore, the dependence of the entanglement of ${\rho^{\prime}}^k_{ab}$ on the noise strength and the size of the system is solely determined by the qubits in $\mathcal{N}_{ab}^1$. Note here that the two-qubit post-measurement states corresponding to different values of $k$ are connected by local unitary operators (see Sec.~\ref{subsec:lb_gdstate}), implying that it is sufficient to consider ${\rho^\prime}^0_{ab}$, or any other value of $k$, since $E_{ab}^0(\rho^\prime)=E({\rho^\prime}^k_{ab})=E({\rho^\prime}^0_{ab})$ (see Eq.~(\ref{eq:ineq_5})).
To investigate the features of the MBL as a function of the noise strength and the system size, we choose logarithmic negativity as the measure of bipartite entanglement, $E$. From the expression of ${\rho^\prime}_{ab}^{k}$ (Eq. (\ref{eq:final_form})), it is clear that $L_g({\rho^\prime}_{ab}^{k})\leq L_g(\tilde{\rho}_{ab})$ (see Eqs. (\ref{eq:pm_state_ab_3}) -- (\ref{eq:rhok}) and subsequent discussions). For the purpose of demonstration, we consider the scenario where noise is absent on qubits $a$ and $b$, i.e., ${\rho^\prime}^k_{ab}=\tilde{\rho}_{ab}$.
One can compute the logarithmic negativity of the state $\tilde{\rho}_{ab}$ from Eq.~(\ref{eq:logneg}). The negativity of the state $\tilde{\rho}_{ab}$, for a fixed value of $q$ is given by Eq. (\ref{eq:neg_comp}), where $\{\lambda_i;i=0,1,2,3\}$ are the eigenvalues of $\tilde{\rho}_{ab}^{T_a}$. These eigenvalues can be explicitly computed in a similar fashion as in Eq.~(\ref{eq:eigen_ptran}) by identifying $\tilde{p}_i$ to be equivalent to $q^\beta_{\overline{\Omega}}$, where both $i,\beta=0,1,2,3$. As functions of $q$, $n_a$, $n_b$, and $n_{ab}$, $\{\lambda_i\}$ are given by
\begin{eqnarray}
\lambda_0 &=&\frac{1}{4}[1+\tilde{q}^{n_a+n_{ab}}-\tilde{q}^{n_a+n_b}+\tilde{q}^{n_{ab}+n_{b}}],\nonumber\\
\lambda_1 &=&\frac{1}{4}[1+\tilde{q}^{n_a+n_{ab}}+\tilde{q}^{n_a+n_b}-\tilde{q}^{n_{ab}+n_{b}}],\nonumber\\
\lambda_2 &=&\frac{1}{4}[1-\tilde{q}^{n_a+n_{ab}}+\tilde{q}^{n_a+n_b}+\tilde{q}^{n_{ab}+n_{b}}],\nonumber\\
\lambda_3 &=&\frac{1}{4}[1-\tilde{q}^{n_a+n_{ab}}-\tilde{q}^{n_a+n_b}-\tilde{q}^{n_{ab}+n_{b}}],
\end{eqnarray}
where $q+\tilde{q}=1$. For the purpose of illustration, let us now consider the situation where $n_{a}=n_{ab}=n_b=n$. In this case, the eigenvalues of $\tilde{\rho}_{ab}^{T_a}$ are $\lambda_0=\lambda_1=\lambda_2=\frac{1}{4}[1+\tilde{q}^{2n}]$, and $\lambda_{3}=\frac{1}{4}[1-3\tilde{q}^{2n}]$, of which the negative eigenvalue is $\lambda_{3}$ in the range $0\leq q <1-\left(\frac{1}{3}\right)^{\frac{1}{2n}}$. In this range, $E^0_{ab}$ as a function of $q$ and $n$ can be expressed as
\begin{eqnarray}
E^0_{ab}=\log_{2}\left[3(1-q)^{2n}+1\right]-1.
\label{eq:lneg_analytic}
\end{eqnarray}
For a specific value of $n$, $E^0_{ab}$ goes to zero at a critical value
\begin{eqnarray}
q_c=1-\left(\frac{1}{3}\right)^{\frac{1}{2n}}.
\label{eq:pcrit}
\end{eqnarray}
For $q>q_c$, $\lambda_3$ becomes positive, and the logarithmic negativity vanishes.
In Fig.~\ref{fig:qc}(a), we plot the variation of $E^0_{ab}$ as a function of the noise strength $q$ with $n=1$, for different types of noise present on the qubit pair $(a,b)$. We conveniently denote the different types of noise on $(a,b)$ by the multi-index $\alpha_{ab}^\prime\equiv\alpha_a^\prime\alpha_{b}^\prime$, where, for example, $\alpha_a^\prime\alpha_b^\prime=11$ implies bit-flip noise applied to both qubits $a$ and $b$. We find that the variation of $E^0_{ab}$ with $q$ in the case of $\{\alpha_a^\prime\alpha_b^\prime=01,02,03,10,20,30\}$ are quantitatively identical. Similar behaviour is observed in the case of $\{\alpha_a^\prime\alpha_b^\prime=11,12,21,23,32,33\}$ and $\{\alpha_a^\prime\alpha_b^\prime=13,22,31\}$. With an increase in the value of $n$, the value of $E_{ab}^0$ for a fixed value of $q$ decreases, and the effect of the noise on the region $\Omega\equiv \{a,b\}$ becomes less prominent. This is clearly shown by the coincidence of the variations of $E^0_{ab}$ against $q$, when the neighbourhood size is increased to $n=10$ (see Fig.~\ref{fig:qc}(b)). The variation of $E^0_{ab}$ with $q$ remains qualitatively unchanged if one considers different relations between $n_a$, $n_{ab}$, and $n_b$ instead of $n_a=n_{ab}=n_b=n$. However, identical dynamics is now shown by groups of noise channels, denoted by specific values of $\alpha_a^\prime\alpha_b^\prime$, which are different from that in the former case. In Fig.~\ref{fig:qc}(b), we plot the variation of $q_c$ as a function of increasing $n$ for different types of noise on the qubits $a$ and $b$, where the data for $\alpha_a^\prime\alpha_{b}^\prime=00$ corresponds to Eq.~(\ref{eq:pcrit}), and the data corresponding to the rest of the noise models are obtained numerically, by considering $E^0_{ab}=0$ for values below a numerical cut-off, concretely, if $E_{ab}^0<10^{-6}$. The qualitative behaviour of $q_c$ against the system size is found to remain invariant for different relations between $n_a$, $n_{ab}$, and $n_b$ instead of $n_a=n_{ab}=n_b=n$.
In the regime of low noise strengths, $q\rightarrow 0$, upon expanding the logarithm and keeping terms up to second order in $q$, Eq. ~(\ref{eq:lneg_analytic}) leads to
\begin{eqnarray}
E_{ab}^0&\approx & 1-\frac{3nq}{2\ln 2}+\frac{3n(n-2)q^2}{8\ln 2},\nonumber \\
&=&\mathcal{O}_0(n)+\mathcal{O}_1(n)+\mathcal{O}_2(n),
\label{eq:approx}
\end{eqnarray}
$\mathcal{O}_k(n)$ being the term involving $n$ in order $k$. The variation of $E^0_{ab}$ as a function of $n$ for fixed values of $q$ is depicted in Fig.~\ref{fig:en}, when the noise strength is small. To determine the leading order of $n$ that describes $E^0_{ab}$ for small values of $q$, we plot, in Fig.~\ref{fig:en}, $E^0_{ab}\approx \mathcal{O}_0(n)+\mathcal{O}_1(n)$ (up to first order in $n$, shown by broken line) and $E^0_{ab}\approx \mathcal{O}_0(n)+\mathcal{O}_1(n)+\mathcal{O}_2(n)$ (up to second order in $n$, shown by continuous line) as functions of $n$. It is clear from Fig.~\ref{fig:en} that for a fixed small value of $q$, $E^0_{ab}\approx \mathcal{O}_0(n)+\mathcal{O}_1(n)$ matches the actual variation of $E^0_{ab}$ satisfactorily when $n$ is very small $(\sim 10)$. When $n$ increases, the second order term in $n$ starts to become prominent, and $E^0_{ab}\approx \mathcal{O}_0(n)+\mathcal{O}_1(n)+\mathcal{O}_2(n)$ describes entanglement satisfactorily.
We would like to point out here that the prescription for computing the post-measurement density matrix to obtain a form equivalent to Eq. (\ref{eq:final_form}) remains unchanged for a region $\Omega$ having size larger than two qubits also. The major step in this calculation is the determination of the mixing probabilities according to the general structure of the neighborhood of $\Omega$ in a graph where $\Omega$ is connected, which can be achieved following procedure similar to that described in this Section and the Appendix \ref{ap:pmstate}. As mentioned earlier in Sec. \ref{subsec:lb_gdstate}, the main difficulty of estimating localizable multipartite entanglement over a region larger than two qubits in the presence of noise is the lack of computable measures of genuine multipartite entanglement for mixed states. In this paper, we have considered a computable bipartite measure of entanglement, namely, logarithmic negativity, which is equivalent to the genuine multiparty entanglement when $\Omega$ is constituted of two qubits only. However, given a computable multiparty entanglement measure for mixed states, the MLB to the localizable multipartite entanglement over a chosen region $\Omega$ constituted of any number of qubits can, in principle, be computed by following a procedure same as in the case of a two-qubit region.
\begin{figure}
\includegraphics[scale=0.4]{en.pdf}
\caption{(Colour online) \textbf{Measurement-based lower bound as a function of system size.} The variations of $E^0_{ab}$ (Eq.~(\ref{eq:lneg_analytic})) as functions of $n$ for different small values of $q$, with $n_a=n_{ab}=n_{b}=n$. The broken (continuous) lines correspond to the variations of $E^{0}_{ab}$ with $n$ when $E^0_{ab}=\mathcal{O}_0(n)+\mathcal{O}_1(n)$ ($E^0_{ab}=\mathcal{O}_0(n)+\mathcal{O}_1(n)+\mathcal{O}_2(n)$) (see Eq.~(\ref{eq:approx})).}
\label{fig:en}
\end{figure}
\begin{figure}
\includegraphics[scale=0.45]{linear.pdf}
\caption{(Colour online) \textbf{Schematic representation of local complementation operations on a linear graph under phase-flip noise.} (a) On the left, a linear graph $\mathcal{G}_L$ with two bulk qubits $a$ and $b$, separated by $n_{\mathcal{L}}=5$ qubits, is shown. The noise on each qubit is of PF type ($Z$-type), and is indicated by the labels. A series of LC operations on the qubits in $\mathcal{L}$, given by Eq.~(\ref{eq:lc_linear}), takes $\mathcal{G}_L$ to $\mathcal{G}^\prime$ (on the right), where the link $(a,b)$ exists. The operation also changes the noise on individual qubits according to Eqs.~(\ref{eq:transform})-(\ref{eq:evolve2}) and Appendix \ref{ap:pauli}, which is indicated by the different labels, where label $X$ and $Y$ indicate BF and BPF noise, respectively. (b) A similar transformation is described for a linear graph with $n_{\mathcal{L}}=6$.}
\label{fig:linear_graph}
\end{figure}
\subsubsection*{Linear graph}
We conclude the discussion on the MLB with the example of a linear graph $\mathcal{G}_L$, in which we intend to determine the MLB over two qubits $a$ and $b$, where the total number of qubits along the path connecting $a$ and $b$ is $n_{\mathcal{L}}$. Note here that the qubit pair $(a,b)$ can either be (i) the boundary qubits, so that in $\mathcal{G}_L$, both $\mathcal{N}_a$ and $\mathcal{N}_b$ have size $1$, or they can be (ii) bulk qubits (as in Fig.~\ref{fig:linear_graph}(a)-(b)), where both $\mathcal{N}_a$ and $\mathcal{N}_b$ have size $2$. For the purpose of demonstration, we consider the scenario where $a$ and $b$ are bulk qubits, $n_{\mathcal{L}}\geq 3$, and PF noise is applied to each of the qubits in $\mathcal{G}_L$. The transformation $\mathcal{G}_L\rightarrow\mathcal{G}^\prime$, where $\{a,b\}$ are connected in $\mathcal{G}^\prime$, is constituted of successive LC operations on the qubits in $\mathcal{L}$, starting from the qubit nearest to $a$ and ending at the qubit nearest to $b$ without skipping any qubit in the middle, so that
\begin{eqnarray}
\mathcal{G}^\prime=\tau_{n_{\mathcal{L}}}\circ\tau_{n_\mathcal{L}-1}\circ\cdots\circ\tau_2\circ\tau_1(\mathcal{G}_L).
\label{eq:lc_linear}
\end{eqnarray}
The structure of $\mathcal{G}^\prime$ is shown for $n_{\mathcal{L}}=5$ ($n_\mathcal{L}=6$) in Fig.~\ref{fig:linear_graph}(a) (\ref{fig:linear_graph}(b)). The Eq.~(\ref{eq:lc_linear}) can equivalently be represented as $\ket{\mathcal{G}^\prime}=U_L\ket{\mathcal{G}_L}$, with
\begin{eqnarray}
U_L=U_a\otimes\big(\bigotimes_{i\in\mathcal{L}}V_i\big)\otimes U_b,
\label{eq:lcu_linear}
\end{eqnarray}
where
\begin{eqnarray}
U_a&=&(u^z_a)^{n_{\mathcal{L}}},\, U_b=u^z_b,\, V_1=(u^z_1)^{n_{\mathcal{L}}-1}u^x_1,\, V_{n_{\mathcal{L}}}=u^{x}_{n_{\mathcal{L}}}u^z_{n_{\mathcal{L}}},\nonumber \\
V_j&=&(u^z_j)^{n_{\mathcal{L}}-j}u^x_ju^z_j; \,\,2\leq j\leq (n_{\mathcal{L}}-1),
\label{eq:lcu_linear_def}
\end{eqnarray}
with $u_i^x$ and $u_i^z$ defined in Sec.~\ref{subsec:stab}. Note here that in the case of $n_{\mathcal{L}}=1$, $U_a=u_a^z$, $U_b=u_b^z$, $V_1=u_1^x$, while for $n_{\mathcal{L}}=2$, $U_a=(u_a^z)^2$, $U_b=u_b^z$, $V_1=u_1^zu_1^x$, and $V_2=u_2^xu_2^z$. The transformation of the Pauli operators due to the unitary operators $\{U_a,U_b,V_j;\,j=1,\cdots,n_{\mathcal{L}}\}$ are given in Appendix \ref{ap:pauli}, which describes the change of the type of noise on individual qubits according to Eqs.~(\ref{eq:transform}) and (\ref{eq:evolve2}). The post-LC operation structures of the graphs, as demonstrated in the case of $n_{\mathcal{L}}=5,6,$ in Fig.~\ref{fig:linear_graph}, is such that for $n_{\mathcal{L}}$ odd, $n_a=0$, $n_{ab}=(n_{\mathcal{L}}+1)/2$, and $n_b=(n_{\mathcal{L}}-1)/2$, while for $n_{\mathcal{L}}$ even, $n_a=0$ and $n_{ab}=n_b=n_{\mathcal{L}}/2$. Therefore, $E^0_{ab}$ as a function of $q$ and $n_{\mathcal{L}}$ can be computed by following the methodology discussed in Sec.~\ref{subsec:mlb_arb}. Note here that the values of $n_a$, $n_b$, and $n_{ab}$ in terms of $n_{\mathcal{L}}$ depend on the structure of the graph $\mathcal{G}^\prime$ as well as the noise on the qubits in $\mathcal{N}_{ab}$ in $\mathcal{G}^\prime$. For instance, in the case of the BF noise on all the qubits, irrespective of the value of $n_{\mathcal{L}}$, $n_a=n_{ab}=n_b=1$. The invariance of $E^0_{ab}$ with $n_{\mathcal{L}}$ in the case of BF noise on all the qubits in $\mathcal{G}_{L}$ can be understood by noticing the fact that the optimal measurement basis in the absence of noise corresponds to $X$ measurements on qubits in $\mathcal{L}$, and $Z$ measurements on the rest of the qubits except $a$ and $b$, and the measurement on qubits in $\mathcal{L}$ commutes with the noise.
\section{Conclusions and Outlook}
\label{sec:conclude}
In this paper, we have considered two different approaches of determining computable lower bounds of localizable entanglement for large stabilizer states under noise. One of the approaches is based on local witnesses, whose expectation values can be used to obtain a lower bound of the localizable entanglement. The other approach restricts the allowed directions of the local projection measurements over the qubits outside the specific region of interest over which the localizable entanglement is to be computed. By establishing a relation between the disentangling operation that reduces the full quantum state to the quantum state corresponding to the specific regime, and local $Z$ measurements over qubits outside the region, we have been able to connect these two seemingly different approaches, and have proposed a hierarchy of lower bounds of localizable entanglement.
Using graph states for demonstration, we show that in the case of graph states exposed to noise, the measurement-based lower bound is greater or equal to the witness-based lower bound. The equality occurs in the case of graph diagonal states, when localizable entanglement over a region constituted of two qubits is to be determined. We have demonstrated how the hierarchy of lower bounds of localizable entanglement is modified due to local unitary transformation, and discussed the behaviour of the lower bounds under physical noise models, such as the local uncorrelated Pauli noise. We have demonstrated that for two-qubit regions, in the case of graph states under local Pauli noise, which form a subset of the complete set of graph-diagonal states, the witness-based lower bound coincides with the measurement-based lower bound. But in the case of three-qubit regions, the measurement-based lower bound is a tighter lower bound for localizable entanglement. We have also proposed an analytical approach to determine the measurement-based lower bound for quantum states of arbitrary size under Pauli noise, and discussed the behaviour of the measurement-based lower bound by performing $Z$-measurement over the qubits outside a two-qubit region as a function of noise strength and system size. The results discussed in this paper are either valid for, or can be translated to more general stabilizer states due to their connection with graph states by local unitary operation. The witness-based lower bounds of localizable entanglement proposed in this paper can be evaluated experimentally without performing a full state tomography, and by considering only one local witness-operator expectation value, which makes it a quantity feasible to be computed in experiments. Also, the measurement-based lower bound discussed in this paper does not require a full optimization with all possible local measurement bases over the qubits outside the region, but needs only local measurement in the computational basis, and can be determined by only knowing the structure of the graph and the type of noise applied to the qubits. Therefore, we expect the quantities and methods introduced in this work to be valuable for the investigation of localizable entanglement in experimental medium- and large-scale noisy stabilizer states.
\acknowledgements
We acknowledge support by U.S. A.R.O. through Grant No. W911NF-14-1-010. The research is also based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office Grant No. W911NF-16-1-0070. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the U.S. Army Research Office.
|
{
"timestamp": "2018-08-06T02:08:51",
"yymm": "1803",
"arxiv_id": "1803.02753",
"language": "en",
"url": "https://arxiv.org/abs/1803.02753"
}
|
\section{Introduction}
In a typical Japanese agricultural town, the length of water canals extends to dozens of kilometers while the dry season during which inspection and repair can be conducted lasts only $1-2$ months per year. However, the current inspection process involves technicians walking along the canals and manually measuring and marking the damaged areas in a log book. Each technician can inspect only $0.5$ km a day. Furthermore, it takes over one month to convert the recorded data into digital information for guiding repairs. Water canal inspection using UAVs (Figure \ref{fig:uav_inspection}) would significantly reduce inspection time and labor cost by automatically identifying defects and registering their GPS coordinates.
We will focus our efforts on a town in Niigata, an agricultural district along the northwest coast of Japan. In the area of interest shown in Figure \ref{fig:map_of_water_canals}, there are about $46.2$ km of water canals spread over tens of square kilometers of farmland. The scale of water canals is beyond the cruise and communication range of most commercially available UAVs, which can typically fly for $20-40$ minutes within $7$ km of the ground station. As a result, inspection of all water canals requires \textit{heterogeneous} vehicles, i.e. a combination of multiple UAVs and ground vehicles.
\begin{figure}[htp]
\centering
\includegraphics[width=1.56in]{wide_field.png}
\includegraphics[width=1.74in]{narrow_field.png}
\caption{UAV taking photos of water canals for damage assessment}
\label{fig:uav_inspection}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[width=3.3in]{map_clean.png}
\caption{Water canal map of a town in Niigata ($10$ km $\times 7$ km)}
\label{fig:map_of_water_canals}
\end{figure}
A viable inspection strategy first divides all water canals into sub-regions that a UAV (or a fleet of UAVs) can cover within one battery charge. Once the UAVs complete inspection of one sub-region, they return to ground vehicles (cars driven by human operators) and are transported to the next sub-region while their batteries are swapped.
There are several constraints that the UAVs and cars have to satisfy. First, as the cars are used as ground stations to monitor and recharge the UAVs, each UAV needs to have at least one car within its communication range at all time, and each UAV will return to a car when it runs out of battery. Second, as the cars travel on public roads, they have to abide by local traffic regulations and their speed is limited by real-time traffic conditions. Moreover, it is also necessary to quickly re-plan the paths should unexpected situations (such as UAV breakdown or change of traffic conditions) arise.
This paper presents an MIQP-based planning framework that generates optimal inspection plans for a given fleet of UAVs and cars, accounting for the aforementioned constraints. The plan includes recharging and transportation between different regions. The proposed framework can also quickly generate feasible new plans if a current plan is interrupted.
The structure of this paper is as follows: Section \ref{sec:problem_statement} describes the coverage planning problem we are trying to solve; Section \ref{ch:Algorithms} provides detailed descriptions of our proposed planning framework; simulation results are presented in Section \ref{ch:Result}; the conclusion and future work are stated in Section \ref{ch:Conclusion}.
\section{Related Works}
Cooperative control of a multi-agent system reduces operation time, introduces redundancy and robustness to the system to better handle adversarial situations such as malfunction or failure of one or more agents. Araki \textit{et al.} built a robot swarm that can function as both ground and aerial vehicles, and developed planning methods using mixed integer program (MIP) \cite{araki2017multi}. Schillinger \textit{et al.} studied optimal planning algorithms for multiple ground robots using linear temporal logic \cite{schillinger2017multi}. Kim \textit{et al.} generates dynamically-feasible trajectories for multiple UAVs to cooperatively transport large objects \cite{kim2017motion}.
Due to the expressiveness of integer variables, MIPs have been used to solve planning problems with complex constraints. Although current MIP solvers have worst-case exponential complexity \cite{bertsimas1997introduction}, MIPs can be solved fast enough for many nontrivial problems that are practically useful. Classical graph planning problems such as traveling salesman problem (TSP) \cite{hoffman2013traveling} and vehicle routing problem (VRP) \cite{laporte1992vehicle} can be written down and solved as mixed-integer linear programs (MILP). Avellar \textit{et al.} solves a minimum-time ground area coverage problem by converting it to a VRP \cite{avellar2015multi}. Dynamically-feasible, obstacle-free UAV trajectories can also be obtained by solving MIPs \cite{deits2015efficient,richards2002coordination}. Grotli \textit{et al.} studied UAV planning problems with fuel and communication constraints, but the communication ground stations are fixed \cite{grotli2012path}. Evers \textit{et al.} deals with the coverage problem and considers the uncertainty of fuel usage and weather conditions to provide a robust planning solution \cite{evers2014robust}. However, no refueling is planned, and only target nodes instead of edges are required to be visited. Lim \textit{et al.} covers both edges and nodes of a graph representing a power network using a fleet of UAVs and minimizes overall inspection time \cite{lim2016multi}.
To the best of our knowledge, although a wide variety of planning problems has been solved with MIPs, the planning problem of heterogeneous vehicles, which allows multiple recharging/refueling of some of the vehicles and has constraints involving multiple types of vehicles, has yet to be addressed.
\section{Problem statement \label{sec:problem_statement}}
\subsection{Graph representations of water canals and roads\label{sec:graph_generation}}
The first step in our planning framework is to abstract canal and road networks from a map into graphs (Figure \ref{fig:Graph_Generation}). GPS information of roads is extracted from the open-source map named OpenStreetMap \cite{OpenStreetMap} using OSMnx \cite{boeing2017osmnx}. NetworkX \cite{hagberg2008exploring} is used to assign nodes to end and intersection points on the road network and identify the adjacency matrix of the graph.
Water canals and roads are represented as two weighted graphs:
\begin{itemize}
\item $G_{canal}=(V_{canal},E_{canal})$, where $v \in V_{canal}$ is an intersection of or a point along the canals, and $e \in E_{canal}$ is a canal segment. The canal graphs are constructed in such a way that all edges in the graph have comparable weights, i.e. $weight(e_{canal}) \approx w_c$ for all $e_{canal} \in E_{canal}$, so that it takes a UAV the same amount of time to inspect all edges in $G_{canal}$. An important observation is that $G_{canal}$ is usually a tree;
\item $G_{road}=(V_{road},E_{road})$, where $v \in V_{road}$ is an intersection of roads, and $e \in E_{canal}$ a road segment.
\end{itemize}
\begin{figure}[h]
\centering
\subfigure[Graphs of water canals with $78$ nodes and $77$ edges] {\label{fig:graph_canal}\includegraphics[width=3.0in]{3b.png}}
\subfigure[Graphs of roads with $1391$ nodes and $1398$ edges] {\label{fig:graph_road}\includegraphics[width=3.0in]{road_graph.png}}
\subfigure[Graph of water canals and roads]{\label{fig:graph_canal_road}\includegraphics[width=3.0in]{33.png}}
\caption{Representation of water canals and roads}
\label{fig:Graph_Generation}
\end{figure}
\subsection{Heterogeneous planning problem \label{sec:coverage planning}}
Given the following information:
\begin{itemize}
\item graphs of water canals and roads;
\item number of available cars and UAVs;
\item UAV battery life and transmission range;
\end{itemize}
we want to find paths for UAVs and cars that inspect all water canals as quickly as possible, subject to the following constraints:
\begin{itemize}
\item cars drive on the road graph $G_{road}$;
\item UAVs fly on the canal graph $G_{canal}$, except when taking off from and returning to the cars;
\item every UAV needs to have at least one car within its transmission distance, as shown in Figure \ref{fig:nearby};
\item every UAV takes off with a fully-charged battery, and needs to return to a car when its battery is depleted.
\end{itemize}
\begin{figure}[h]
\centering
\includegraphics[width=3.3in]{n1.png}
\caption{Every UAV needs a car within its transmission distance.}
\label{fig:nearby}
\end{figure}
\section{Proposed computational methods}\label{ch:Algorithms}
\subsection{Overview}
Using the canal and road graphs defined in Section \ref{sec:graph_generation}, the inspection paths for UAVs and cars are found through three steps:
\subsubsection{graph partitioning} as the entire canal graph is usually much larger than what a UAV fleet can inspect with a single battery charge, the canal graph is partitioned into subgraphs that a given fleet can possibly inspect within the UAV's battery life ($20-40$ minutes). In addition, partitioning the canal graph significantly reduces the size and computation time of each of the path planning problems.
\subsubsection{heterogeneous vehicle path planning} for each subgraph generated from partitioning, the heterogeneous vehicle path planning algorithm generates paths for UAVs and cars that inspect all canals in the given subgraph, subject to the constraints given in Section \ref{sec:coverage planning}.
\subsubsection{car routing between subgraphs} while being recharged, the UAVs are also being transported from one subgraph to another by cars. The car routing algorithm gives the minimum-length path that visits and inspects all canal subgraphs.
\subsection{Graph partitioning}
To minimize the time for swapping batteries and traveling between partitioned subgraphs, the objective is that partitions have comparable sizes. Let $K$ be the total number of UAVs and $M$ the number of edges a UAV can inspect with one fully charged battery. The water canal graph $G_{canal}$ consists of $N_n$ vertices and $N_e$ edges. Therefore, the maximum number of edges in a subgraph is $KM$, and an initial guess for the number of subgraphs is $s_o=\lceil\frac{N_e}{KM}\rceil$.
The actual number of subgraphs is increased from $s_o$ until the following mixed-integer quadratic program (MIQP) becomes feasible. The MIQP of minimizing the traveling distance of UAVs has the following objective function and constraints:
\begin{equation}
\underset{x}{\mathrm{min.}}\sum_{s=1}^S(\sum_{i=1}^N x_{si})^2,
\end{equation}
subject to:
\begin{equation}
\forall e, \sum_{s=1}^S w_{se}=1,
\label{equ:coverage}
\end{equation}
\begin{equation}
\forall s, K\leq\sum_{e=1}^E w_{se}\leq KM,
\label{equ:size}
\end{equation}
\begin{equation}
\forall s, \sum_{e=1}^E w_{se}=\sum_{i=1}^N x_{si}-1, \mathrm{and}
\label{equ:connected}
\end{equation}
\begin{equation}
\forall s,x_{si}+x_{sj}-2w_{se}\geq0,
\label{equ:node_edge}
\end{equation}
where $x_{si}\in\{0,1\}$ represents whether Node $i$ is in Subgraph s, $w_{se}\in\{0,1\}$ represents whether Edge $e$ in Subgraph $s$. Constraint (\ref{equ:coverage}) requires that each edge belongs to one and only one subgraph. Constraint (\ref{equ:size}) sets $K$ and $KM$ as the lower and upper bounds of the number of edges in every subgraph. As a connected subgraph of a tree is also a tree, Constraint (\ref{equ:connected}) implies that each subgraph must be connected. Constraint (\ref{equ:node_edge}) shows that if Edge $e_{ij}$ is in Subregion s, both Node $i$ and Node $j$ are also in Subgraph $s$.
For example, given a fleet of 4 UAVs, in which every UAV can cover $3$ edges on a single battery charge ($K = 4$, $M = 3$), it is calculated that the graph of water canal should be partitioned into 8 subgraphs. The partitioning result is shown in Figure \ref{fig:Graph_Partition}.
\begin{figure}[h]
\centering
\includegraphics[width=3.3in]{6b.png}
\caption{Graph partitioning result, different colors represent different subgraphs.}
\label{fig:Graph_Partition}
\end{figure}
\vspace{2mm}
\subsection{Heterogeneous vehicles path planning for subgraphs \label{sec:heterogeneous_planning}}
Given fixed numbers of UAVs and cars, the problem of generating paths for UAVs and cars to inspect a canal subgraph can be formulated as an MIQP with the objective of minimizing the traveling distance of UAVs. In addition to the constraints in Section \ref{sec:coverage planning}, we make the following assumptions:
\begin{itemize}
\item UAVs and cars only travel to adjacent nodes over one time step;
\item All canal edges in a subgraph must be inspected by a UAV;
\item It takes a UAV one time unit to inspect an edge in the canal subgraph; and
\item The time taken to fly between the canals and the vehicles at the beginning and end of the inspection is negligible compared to the inspection time. This is reasonable because the UAVs need to fly very slowly in a zig-zag pattern in order to acquire clear images of the canal walls. In comparison, the UAVs can fly reasonably fast when traveling between the canals and the cars.
\end{itemize}
Let $N_{s}$ be the total number of edges in the canal subgraph indexed by $s$, the planning time horizon $T$ is increased from the initial guess $t_0=\lceil \frac{N_{s}}{KM}\rceil$ until the following MIQP becomes feasible. The MIQP of minimizing the traveling distance of UAVs has the following objective function:
\begin{equation}
\min_{x,y}\sum_{k=1}^K\sum_{t=1}^T\sum_{i=1}^{N_{canal}}(x_{k(t+1)i}-x_{kti})^2,
\end{equation}
and constraints:
\begin{equation}
\forall e\in E_s,\sum_{k=1}^K\sum_{t=1}^T\sum_{d=1}^2w_{ektd}=1,
\label{equ:coverage_constraint}
\end{equation}
$\forall k,t,e,$
\begin{equation}
x_{kti}+x_{k(t+1)j}-2w_{ekt1}\geq0,
\label{equ:node_edge_constraint1}
\end{equation}
\begin{equation}
x_{ktj}+x_{k(t+1)i}-2w_{ekt2}\geq0,
\label{equ:node_edge_constraint}
\end{equation}
$\forall k,k_{car},t $
\begin{equation}
\left[\begin{array}{c}y_{k_{car}t1} \\ \vdots \\ y_{k_{car}tN_{road}}\end{array}\right]\leq R\left[\begin{array}{c}x_{kt1} \\ \vdots \\ x_{ktN_{canal}}\end{array}\right]+(1-\omega_{ktk_{car}})\left[\begin{array}{c}1 \\ \vdots \\1\end{array}\right],
\label{equ:communication_constraint1}
\end{equation}
\begin{equation}
\forall t,k \sum_{k_{car}=1}^{K_{car}}\omega_{ktk_{car}}\geq1,
\label{equ:communication_constraint}
\end{equation}
\begin{equation}
\forall k,t,k_{car} \sum_{i=1}^{N_{canal}}x_{kti}=1,
\sum_{i=1}^{N_{road}}y_{k_{car}ti}=1,
\label{equ:position_constraint}
\end{equation}
\begin{equation}
\forall k,t, k_{car} \left[\begin{array}{c} x_{k(t+1)1}\\ \vdots \\ x_{k(t+1)N_{canal}}\end{array}\right]\leq A_{canal}\left[\begin{array}{c}x_{kt1}\\ \vdots \\ x_{ktN_{canal}}\end{array}\right]
\label{equ:constraint_connectivity_UAV},
\end{equation}
\begin{equation}
\left[\begin{array}{c} y_{k_{car}(t+1)1}\\ \vdots \\ y_{k_{car}(t+1)N_{road}}\end{array}\right]\leq A_{road}\left[\begin{array}{c}y_{k_{car}t1}\\ \vdots \\ y_{k_{car}tN_{road}}\end{array}\right],
\label{equ:constraint_connectivity_car}
\end{equation}
where $x_{kti}\in \{ 0,1 \}$ represents whether UAV $k$ at time $t$ is at Node $i$ of the canal subgraph; $y_{kti} \in \{ 0,1 \}$ represents whether Car $k$ at time $t$ is at Node $i$ of the road subgraph; $\omega_{ktk_{car}}\in\{0,1\}$ represents whether UAV $k$ is within the communication range of Car $k_{car}$ at time $t$. $K_{car}$ is the total number of cars in the fleet. $R \in \mathbb{R}^{N_{road}\times N_{canal}}$ is the transmission matrix containing only $0$s and $1$s. $R(i,j) =1 $ if the Euclidean distance between Node $v_i \in V_{road}$ and Node $v_j \in V_{canal}$ is less than the UAV's maximum transmission distance. $A_{canal}$ and $A_{road}$, containing only $0$s and $1$s, are the adjacency matrices of the water canal and road subgraphs, respectively. $A(i,j)=1$ if node $i$ and node $j$ are adjacent.
Constraint (\ref{equ:coverage_constraint}) means that all edges must be inspected once. $w_{ektd} \in \{ 0,1 \}$ represents whether Edge $e$ is visited by the UAV $k$ starting at time $t$ in the direction $d$. For an edge $e_{ij} \quad (i<j)$, $d=1$ means that when the edge is inspected by an UAV, the UAV first visits Node $i$ and then Node $j$. Similarly, $d=2$ means that the UAV visits first Node $j$ and then $i$. $w_{ekt1}=1$ implies that $x_{kti}=1$, $x_{k(t+1)j}=1$, and $w_{ekt2}=1$ implies that $x_{ktj}=1$ and $x_{k(t+1)i}=1$. This relationship between $w_{ektd}$ and $x_{kti}$ is summarized in Constraint (\ref{equ:node_edge_constraint1}) and (\ref{equ:node_edge_constraint}).
Constraints (\ref{equ:communication_constraint1}) and (\ref{equ:communication_constraint}) imply that for every UAV at all time, there must be at least one car within its transmission distance. Constraint (\ref{equ:position_constraint}) shows that at any time step, all cars and UAVs must appear at only one node. Constraints (\ref{equ:constraint_connectivity_UAV}) and (\ref{equ:constraint_connectivity_car}) require that UAVs and cars only travel to adjacent nodes between consecutive time steps.
The planning result for one subgraph is shown in Figure \ref{fig:plan_result}. The magenta graph is a subgraph of $G_{canal}$ and the grey graph a subgraph of $G_{road}$. The fleet has 4 UAVs and 2 cars. The UAV inspection paths are shown on top of $G_{canal}$ as dashed lines with different colors. The car paths are shown as dotted lines on top of $G_{road}$.
\begin{figure}[h]
\centering
\includegraphics[width=3.0in]{72.png}
\caption{Planning result for one subgraph}
\label{fig:plan_result}
\end{figure}
\subsection{Re-planning}
When executing a plan on a subgraph, re-planning is necessary in the event of UAV failure or traffic jam. To re-plan at $t>0$, an MIQP similar to the one in Section \ref{sec:heterogeneous_planning} is constructed, with the following modifications:
\begin{itemize}
\item Canal edges already inspected at time $t$ are removed from the canal subgraph.
\item Congested roads are removed from the road subgraph.
\item Battery life of UAVs is updated to the remaining battery life.
\item All UAVs and cars start at their positions in the original plan at time $t$.
\end{itemize}
As shown in Figure \ref{fig:Replanning_fail}, the proposed re-planning method successfully generates new paths for UAVs and cars when a UAV fails or the traffic changes.
\begin{figure}[h]
\captionsetup{justification=centering}
\centering
\subfigure[Original plan (4 UAVs and 2 cars)]{\label{fig:r1}\includegraphics[width=1.50in]{r1.png}}
\hspace{2mm}
\subfigure[Re-planning at $t=1$. 4 edges have been inspected and removed from the canal subgraph.]{\label{fig:r2}\includegraphics[width=1.55in]{r2.png}}
\subfigure[Re-planning at $t=1$ due to traffic condition changes (some edges in the road graph are removed.)]{\label{fig:r3}\includegraphics[width=1.55in]{r3.png}}
\hspace{2mm}
\subfigure[Re-planning at $t=1$ due to UAV failure (only 3 UAVs remain functioning)]{\label{fig:r4}
\includegraphics[width=1.55in]{r4.png}}
\caption{Re-planning results}
\label{fig:Replanning_fail}
\end{figure}
\vspace{2mm}
\subsection{Car routing between subgraphs}
After finding an optimal plan to inspect a single subgraph, we need to find the shortest car route that visits all subgraphs. It is also required that the car route starts from and ends at a fixed location called the office. To plan car routes between canal subgraphs, a new graph $G_{subgraphs}$ whose nodes correspond to the canal subgraphs is constructed. As the cars are free to travel from any canal subgraph to any other canal subgraph, $G_{subgraphs}$ is fully connected but asymmetric (because of one-way roads). After determining the weights of the edges of the new fully-connected graph, the car routing problem, which searches for the shortest cycle that visits all nodes in $G_{subgraphs}$, can be solved as an Asymmetric Traveling Salesman Problem (ATSP).
\subsubsection{Determining edge weights in $G_{subgraphs}$}
Let $S$ be the total number of subgraphs generated from canal graph partitioning. The total number of nodes in $G_{subgraphs}$ is $S+1$ because the office is also included as a node. Let $Q \in R^{(S+1)\times(S+1)}$ be the matrix of edge weights of $G_{subgraphs}$, in which $Q_{AB}\coloneqq Q(A,B) $ is the weight of the directed edge pointing from Node $A$ to Node $B$. We want a $Q$ that minimizes the total travel distances of all cars when they carry UAVs from one subgraph to another (i.e. Subgraph A to Subgraph B).
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.3in]{9.png}
\caption{Transform into ATSP problem}
\label{fig:atsp}
\end{center}
\end{figure}
In the inspection planning of subgraphs (Section \ref{sec:heterogeneous_planning}), the planner gives a starting and leaving node for every car in every subgraph. The UAVs take off at the starting nodes (shown as $\bigcirc$ in Figure \ref{fig:atsp}) and land at the leaving positions ($\times$ in Figure \ref{fig:atsp}). $Q_{AB}$ can be determined by finding the shortest of all paths connecting a leaving node ($\times$) of Subgraph A and a starting node ($\bigcirc$) in Subgraph B. This can be formulated as the following MIQP:
\begin{equation}
Q_{AB}=\min\sum_{i=1}^{K_{car}}\sum_{j=1}^{K_{car}}d_{ij}x_{ij},
\end{equation}
subject to:
\begin{equation}
\forall i, \sum_{j=1}^{K_{car}}x_{ij}=1,
\forall j, \sum_{i=1}^{K_{car}}x_{ij}=1,
\label{equ:complete}
\end{equation}
where $d_{ij}$ is the length of the shortest path between Node $i$ and Node $j$ in the road graph, and $x_{ij} \in \{0,1\}$ is whether car travels from Node $i$ in Subgraph $A$ to Node $j$ in Subgraph $B$. Constraint (\ref{equ:complete}) says that all ending nodes of Subgraph $A$ must be visited only once and that all starting nodes of Subgraph $B$ must be visited only once.
\subsubsection{ATSP}
After finding $Q$, the car routing problem is reduced to an ATSP for a complete directed graph. The objective is to find the shortest closed path that starts from the office and visits all subgraphs:
\begin{equation}
\min_x\sum_{t=0}^{S+1}\begin{bmatrix}x_{0t} & \dots & x_{(S+1)t} \end{bmatrix} Q \begin{bmatrix}x_{0(t+1)} \\ \vdots \\ x_{(S+1)(t+1)} \end{bmatrix},
\end{equation}
subject to:
\begin{equation}
x_{00}=1,x_{(S+1)(S+1)}=1, and
\label{equ:start_end_pos}
\end{equation}
\begin{equation}
t\in\{1,\dots,S\}, \sum_{s=1}^S x_{st}=1,
s\in\{1,\dots,S\}, \sum_{t=1}^S x_{st}=1,
\label{equ:sub_visit_cons}
\end{equation}
where $x_{st}\in\{0,1\}$ denotes whether subgraph s is visited at time step $t$. Constraint (\ref{equ:start_end_pos}) says that cars must begin and end the journey from/at the office. Constraint (\ref{equ:sub_visit_cons}) means at the same time, only one subgraph is visited, and each subgraph can be visited only once. Figure \ref{fig:Car_Routing_between_Subgraphs} shows the shortest paths for cars.
\begin{figure}[h]
\centering
\captionsetup{justification=centering}
\subfigure[Start and end nodes on graph]{\label{fig:original}\includegraphics[width=1.65in]{91.png}}
\subfigure[Start and end nodes]{\label{fig:start_end}\includegraphics[width=1.65in]{92.png}}
\subfigure[Car-route between 2 subgraphs]{\label{fig:short2path}\includegraphics[width=1.65in]{93.png}}
\subfigure[ATSP problem]{\label{fig:subgraphs}\includegraphics[width=1.65in]{95.png}}
\subfigure[ATSP result]{\label{fig:short_path}\includegraphics[width=1.65in]{11d.png}}
\subfigure[Car routing result]{\label{fig:car_result}\includegraphics[width=1.65in]{94.png}}
\caption{Car routing between subgraphs}
\label{fig:Car_Routing_between_Subgraphs}
\end{figure}
\section{Results and Discussion} \label{ch:Result}
The computation time of all methods in our inspection planning framework is benchmarked and summarized in Table \ref{table_bench}. The problem used for benchmarking considers the inspection of a water canal graph with $78$ nodes and $77$ edges. The inspection is conducted by a fleet consisting of $4$ UAVs and $2$ cars. Every UAV can cover 3 edges with a fully-charged battery. The MIPs are solved using GUROBI\cite{gurobi} on a laptop with Intel® Core™ i7-3520M CPU (dual core four threads).
\begin{table}[h]
\caption{Benchmarking results}
\label{table_bench}
\centering
\begin{tabular}{|c||c||c||c|}
\hline
Graph& Heterogeneous planning& Car &\\ partition & for subgraphs & routing & Total \\
\hline
0.02s& 412.22s & 5.22s& 417.26s \\
\hline
\end{tabular}
\end{table}
The planning result shown in Figure \ref{fig:overall_result} illustrates that our algorithm ensures the coverage of water canals and solves the problem caused by limited transmission distance and UAV battery life. Colored lines on graph of water canal represent the path of UAVs and colored lines on graph of road denote path of cars. Dotted lines between graph of water canal and graph of road are path of UAVs between cars and water canal. The entire inspection plan involving all cars and UAVs is also simulated in Simulink, as shown in Figure \ref{fig:3dSimulation}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.3in]{overall.png}
\caption{Planning result for 4 UAVs and 2 Cars}
\label{fig:overall_result}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.0in]{3d.png}
\caption{Simulation environment}
\label{fig:3dSimulation}
\end{center}
\end{figure}
A 3D reconstruction of a water canal subgraph using Octomap \cite{hornung2013octomap} (Figure \ref{fig:recons}) is generated by simulating a UAV with stereo-camera flying inside the CAD model of water canals (Figure \ref{fig:cad}) following a planned path. The local octomap for all the UAVs are obtained from the disparity depth-map of a virtual stereo camera sensor in Gazebo, a realistic physics simulator. These local octomap are stitched using the ground truth poses from the simulator to obtain a unified octomap of the model. Figure \ref{fig:reconstruction} proves that our method inspects both walls of water canals and is expected to work effectively in real world scenarios.
\begin{figure}[h]
\begin{center}
\subfigure[CAD model of a water canal subgraph]{\label{fig:cad}\includegraphics[width=3.3in]{cad.png}}
\subfigure[Octomap reconstruction from simulated on-board stereo-camera]{\label{fig:recons}\includegraphics[width=3.3in]{reconstruction.png}}
\caption{Water canal subgraph reconstruction}
\label{fig:reconstruction}
\end{center}
\end{figure}
It is shown in Figure \ref{fig:solving_time_} that the maximum planning time for canal subgraphs (algorithm in Section \ref{sec:heterogeneous_planning}) is 140, which can be further reduced by parallelization. Based on experiments, the time needed by cars to travel from one subgraph to another is about 10 minutes. If a new inspection plan is needed for the next subgraph due to different traffic conditions or the malfunctioning of some of the UAVs, the new plan will be ready before arriving at the next subgraph.
It has been assumed in Section \ref{sec:heterogeneous_planning} that the time needed by one UAV to inspect one edge in $G_{canal}$ is 1 unit time, which is 10 minutes based on experiments. If we further assume that the time taken by cars to travel from one subgraph to another is 10 minutes, then the total inspection time for UAV-car fleets of different sizes can be calculated, and is summarized in Table \ref{table_time}. There is a 75\% reduction in inspection time by increasing the fleet size from 1 UAV and 1 car to 4 UAVs and 2 cars.
Based on experimental results, the efficiencies of different inspection methods are summarized in Table \ref{table_inspection_comparison}. Manual inspection involves a technician walking along the canals and looking for defects. In manual navigation, an operator flies a UAV manually along the canals, and another inspector looks for defects from the UAV's video feed. In our approach (heterogeneous vehicle routing), the UAV flies autonomously, records video of the canal walls, which are then analyzed with computer vision algorithms. Compared with current manual inspection and manual UAV flying method, the proposed heterogeneous vehicle planning framework using $4$ UAVs and $2$ cars would reduce inspection time by $90\%$ and $60\%$, respectively. These results could be improved further by using more UAVs and cars.
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.3in]{141.png}
\caption{Planning time for all subgraphs}
\label{fig:solving_time_}
\end{center}
\end{figure}
\begin{table}[h]
\caption{Fleet size and inspection Time}
\label{table_time}
\centering
\begin{tabular}{|c||c||c||c|}
\hline
Fleet size &1 UAV 1 car &3 UAVs 3 cars& 4 UAVs 2 cars\\
\hline
Inspection time & & &\\
(minutes) &1320& 490 &330 \\
\hline
\end{tabular}
\end{table}
\begin{table}[h]
\caption{Comparisons of different inspection methods}
\label{table_inspection_comparison}
\centering
\begin{tabular}{|c||c||c||c|}
\hline
&Manual& Manual& Heterogeneous \\
&inspection & navigation& vehicle routing\\
\hline
Water canal& & &\\inspection time & & &\\($s/m^2$)
&4.11 & 1.35 & 0.42\\
\hline
Resolution &NA &0.2 mm-25 mm& 0.2 mm\\
\hline
\end{tabular}
\end{table}
\section{Conclusion}\label{ch:Conclusion}
Using UAV and car combinations to inspect large area of water canal reduces inspection time. A innovative algorithm is formulated to solve the new heterogeneous planning problem resulted from the UAV and car combinations. For the current map of water canal in an agricultural town in Japan, the planning algorithm runs online. The planning time is $140$ seconds for each subgraphs and re-planning time is less than $3$ seconds. Compared with manual inspection, using $4$ UAVs and $2$ cars can be expected to reduce inspection time by at most $90\%$.
In the future, collision avoidance between vehicles, aerial constraints (weather, flying zones, busy aerial traffic, aerial regulations, etc.), ground traffic regulations (one-way, non-parking zones, speed limits, etc.) and uncertainty in maps and measurements of UAV/car localization will be added to the system to increase safety during inspection flights.
\section*{ACKNOWLEDGMENT}
The authors would like to thank TOPRISE Co., LTD. for their sponsor for this work and
Mayuko Nemoto for help on problem formulation and running experiments.
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-03-08T02:10:08",
"yymm": "1803",
"arxiv_id": "1803.02723",
"language": "en",
"url": "https://arxiv.org/abs/1803.02723"
}
|
\section{Introduction}
Quantum hypothesis testing~\cite{QHT1} is a central area in quantum
information theory~\cite{Watrous,HolevoBOOK}, with many studies for both
discrete variable (DV)~\cite{NiCh} and continuous variable (CV)
systems~\cite{RMP}. A number of tools~\cite{QCB1,QCB2,QCB3,QHB1,QHB2} have
been developed for its basic formulation, known as quantum state
discrimination. In particular, since the seminal work of Helstrom in the
70s~\cite{QHT1}, we know how to bound the error probability affecting the
symmetric discrimination of two arbitrary quantum states. Remarkably, after
about 40 years, a similar bound is still missing for the discrimination of two
arbitrary quantum channels. There is a precise motivation for that: The main
problem in quantum channel discrimination
(QCD)~\cite{QCD1,QCD2,QCD3,QCD4,QCD6} is that the strategies involve an
optimization over the input states and the output measurements, and this
process may be adaptive in the most general case, so that feedback from the
output can be used to update the input.
Not only the ultimate performance of adaptive QCD\ is still
unknown due to the difficulty of handling feedback-assistance, but
it is also known that adaptiveness needs to be considered in QCD.
In fact, apart from the cases where two channels are
classical~\cite{HayaCLASS}, jointly programmable or
teleportation-covariant~\cite{PirCo,ReviewMETRO}, feedback may
greatly improve the discrimination. For instance, Ref.~\cite{Harrow
\ presented two channels which can be perfectly distinguished by
using feedback in just two adaptive uses, while they cannot be
perfectly discriminated by any number of uses of a block
(non-adaptive) protocol, where the channels are probed in an
identical and independent fashion. This suggests that the best
discrimination performance is not directly related to the diamond
distance~\cite{Diamond}, when computed over multiple copies of the
quantum channels.
In this work we finally fill this fundamental gap by deriving a
universal computable lower bound for the error probability
affecting the discrimination of two \textit{arbitrary} quantum
channels. To derive this bound we adopt a technique which reduces
an adaptive protocol over an arbitrary finite-dimensional quantum
channel into a block protocol over multiple copies of the
channel's Choi matrix. This is obtained by using port-based
teleportation (PBT)~\cite{PBT,PBT1,PBT2,Brau} for channel
simulation and suitably generalizing the technique of
teleportation stretching~\cite{PLOB,TQC,Uniform}. This\ reduction
is shown for adaptive protocols with any task (not just QCD). When
applied to QCD, it allows us to bound the ultimate error
probability by using the Choi matrices of the channels.
As a direct application, we bound the ultimate adaptive
performance of quantum
illumination~\cite{Qill0,Qill1,Qill2,Qill3,Qill4,Qill6,Qill7,Qill8}
and the ultimate adaptive resolution of any single-photon
diffraction-limited optical system, setting corresponding no-go
theorems for these applications. We then apply our result to
adaptive quantum metrology showing an ultimate bound which has an
asymptotic Heisenberg scaling. As an example, we also study the
adaptive discrimination of amplitude damping channels, which are
the most difficult channels to be simulated. Finally, other
implications are for the two-way assisted capacities of quantum
and private communications.
\section{Results}
\subsection{Adaptive protocols}
Let us formulate the most general adaptive protocol over an arbitrary quantum
channel $\mathcal{E}$ defined between Hilbert spaces of dimension $d$ (more
generally, this can be taken as the dimension of the input space). We first
provide a general description and then we specify the protocol to the task of
QCD. A general adaptive protocol involves an unconstrained number of quantum
systems which may be subject to completely arbitrary quantum operations (QOs).
More precisely, we may organize the quantum systems into an input register
$\mathbf{a}$ and an output register $\mathbf{b}$, which are prepared in an
initial state $\rho_{0}$ by applying a QO $\Lambda_{0}$ to some fundamental
state of $\mathbf{a}$ and $\mathbf{b}$. Then, a system $a_{1}$ is picked from
the register $\mathbf{a}$ and sent through the channel $\mathcal{E}$. The
corresponding output $b_{1}$ is merged with the output register $b_{1
\mathbf{b\rightarrow b}$. This is followed by another QO $\Lambda_{1}$ applied
to $\mathbf{a}$ and $\mathbf{b}$. Then, we send a second system $a_{2
\in\mathbf{a}$ through $\mathcal{E}$ with the output $b_{2}$ being
merged again $b_{2}\mathbf{b\rightarrow b}$ and so on. After $n$
uses, the registers will be in a state $\rho_{n}$ which depends on
$\mathcal{E}$ and the sequence of QOs
$\{\Lambda_{0},\Lambda_{1},\ldots,\Lambda_{n}\}$ defining the
adaptive protocol $\mathcal{P}_{n}$ with output state $\rho_{n}$
(see Fig.~\ref{figada}).
\begin{figure}[ptbh]
\begin{center}
\vspace{-2cm} \includegraphics[width=0.49\textwidth]{adaptive} \vspace{-2.4cm}
\end{center}
\caption{General structure of an adaptive quantum protocol, where
channel uses $\mathcal{E}$ are interleaved by QOs $\Lambda$'s. See
text for more details.}
\label{figada
\end{figure}
In a protocol of quantum communication, the registers belong to remote users
and, in absence of entanglement-assistance, the QOs are local operations (LOs)
assisted by two-way classical communication (CC), also known as adaptive
LOCCs. The output is generated in such a way to approximate some target
state~\cite{PLOB}. In a protocol of quantum channel estimation, the channel is
labelled by a continuous parameter $\mathcal{E}=\mathcal{E}_{\theta}$ and the
QOs include the use of entanglement across the registers. The output state
will encode the unknown parameter $\rho_{n}=\rho_{n}(\theta)$, which is
detected and the outcome processed into an optimal estimator~\cite{PirCo}.
Here, in a protocol of binary and symmetric QCD, the channel is labelled by a
binary digit, i.e., $\mathcal{E}=\mathcal{E}_{u}$ where $u\in\{0,1\}$ has
equal priors. The QOs are generally entangled and they generate an output
state encoding the information bit, i.e., $\rho_{n}=\rho_{n}(u)$.
The output state $\rho_{n}(u)$ of an adaptive discrimination protocol
$\mathcal{P}_{n}$\ is finally detected by an optimal positive-operator valued
measure (POVM). For binary discrimination, this is the Helstrom POVM, which
leads to the conditional error probability
\begin{equation}
p(\mathcal{E}_{0}\neq\mathcal{E}_{1}|\mathcal{P}_{n})=\frac{1-D\left[
\rho_{n}(0),\rho_{n}(1)\right] }{2}, \label{protPROB
\end{equation}
where $D(\rho,\sigma):=||\rho-\sigma||/2$ is the trace distance~\cite{NiCh}.
The optimization over all discrimination protocols $\mathcal{P}_{n}$ defines
the minimum error probability affecting the $n$-use adaptive discrimination of
$\mathcal{E}_{0}$ and $\mathcal{E}_{1}$, i.e., we may writ
\begin{equation}
p_{n}(\mathcal{E}_{0}\neq\mathcal{E}_{1}):=\inf_{\mathcal{P}_{n
}~p(\mathcal{E}_{0}\neq\mathcal{E}_{1}|\mathcal{P}_{n}). \label{protINF
\end{equation}
This is generally less than the $n$-copy diamond distance between the two
channels $\mathcal{E}_{0}^{\otimes n}$ and $\mathcal{E}_{1}^{\otimes n}
\begin{equation}
p_{n}(\mathcal{E}_{0}\neq\mathcal{E}_{1})\leq\frac{1-\frac{1}{2
||\mathcal{E}_{0}^{\otimes n}-\mathcal{E}_{1}^{\otimes n}||_{\diamond}}{2},
\label{eqDD
\end{equation}
where~\cite{Watrous}
\begin{equation}
||\mathcal{E}_{0}^{\otimes n}-\mathcal{E}_{1}^{\otimes n}||_{\diamond
:=\sup_{\rho_{ar}}||\mathcal{E}_{0}^{\otimes n}\otimes\mathcal{I}(\rho
_{ar})-\mathcal{E}_{1}^{\otimes n}\otimes\mathcal{I}(\rho_{ar})||,
\end{equation}
with $\mathcal{I}$ being an identity map acting on a reference
system $r$. The upper bound in Eq.~(\ref{eqDD}) is achieved by a
non-adaptive protocol, where an (optimal) input state $\rho_{ar}$
is prepared and its $a$-parts transmitted through
$\mathcal{E}_{u}^{\otimes n}$. Note that Eq.~(\ref{eqDD}) is very
difficult to compute, which is why we usually compute larger but
simpler single-letter upper bounds such as
\begin{equation}
p_{n}(\mathcal{E}_{0}\neq\mathcal{E}_{1})\leq\frac{F(\rho_{\mathcal{E}_{0
},\rho_{\mathcal{E}_{1}})^{n}}{2}, \label{fidSIM
\end{equation}
where $F$ is the fidelity between the Choi matrices,
$\rho_{\mathcal{E}_{0}}$ and $\rho_{\mathcal{E}_{1}}$, of the two
channels.
Our question is: Can we complete Eq.~(\ref{eqDD}) with a
corresponding lower bound? Up to today this has been only proven
for jointly-programmable channels, i.e., channels
$\mathcal{E}_{0}$ and
$\mathcal{E}_{1}$ admitting a simulation $\mathcal{E}_{u}(\rho)=\mathcal{S
(\rho\otimes\pi_{u})$ with a trace-preserving QO $\mathcal{S}$ and different
program states $\pi_{0}$ and $\pi_{1}$. In this case, we have $p_{n
\geq\lbrack1-D(\pi_{0}^{\otimes n},\pi_{1}^{\otimes n})]/2$~\cite{PirCo}. In
particular, this is true if the channels are jointly teleportation-covariant,
so that $\mathcal{S}$ becomes teleportation and the program state is a Choi
matrix $\rho_{\mathcal{E}_{u}}$. For these channels, Ref.~\cite{PirCo} found
that Eq.~(\ref{eqDD}) holds with an equality and we may write $||\mathcal{E
_{0}^{\otimes n}-\mathcal{E}_{1}^{\otimes n}||_{\diamond}=||\rho
_{\mathcal{E}_{0}}^{\otimes n}-\rho_{\mathcal{E}_{1}}^{\otimes n}||$. More
precisely, the question to ask is therefore the following: Can we establish a
\textit{universal} lower bound for $p_{n}(\mathcal{E}_{0}\neq\mathcal{E}_{1})$
which is valid for\textit{ arbitrary} channels? As we show here, this is
possible by resorting to a more general (multi-program) simulation of the
channels, i.e., of the type $\mathcal{S}(\rho\otimes\pi_{u}^{\otimes M})$.
\begin{figure*}[ptbh]
\begin{center}
\vspace{-4.6cm} \includegraphics[width=1.05\textwidth]{PBT} \vspace{-0.4cm}
\vspace{-5.2cm}
\end{center}
\caption{From port-based teleportation (PBT) to Choi-simulation of a quantum
channel (see also Ref.~\cite{PBT}). \textbf{(a)} Schematic representation of
the PBT protocol. Alice and Bob share an $M\times M$ qudit state which is
given by $M$ maximally-entangled states $\Phi_{\mathbf{AB}}^{\otimes M}$. To
teleport an input qubit state $\rho_{C}$, Alice applies a suitable POVM
$\{\Pi_{i}\}$ to the input qubit $C$ and her $\mathbf{A}$ qubits. The outcome
$i$ is communicated to Bob, who selects the $i$-th among his $\mathbf{B}$
qubits (tracing all the others). The performance does not depend on the
specific \textquotedblleft port\textquotedblright\ $i$ selected and the
average output state is given by $\Gamma_{M}(\rho_{C})$ where $\Gamma_{M}$ is
the PBT channel. The latter reduces to the identity channel in the limit of
many ports $M\rightarrow\infty$. \textbf{(b)} Suppose that Bob applies a
quantum channel $\mathcal{E}$ on his teleported output. This produces the
output state $\mathcal{E}^{M}(\rho_{C})$ of Eq.~(\ref{effective}). For large
$M$, one has $\mathcal{E}^{M}\rightarrow\mathcal{E}$ in diamond norm.
\textbf{(c)} Equivalently, Bob can apply $\mathcal{E}^{\otimes M}$ to all his
qubits $\mathbf{B}$ in advance to the CC from Alice. After selection of the
port, this will result in the same output as before. \textbf{(d)} Now note
that Alice's LO and Bob's port selection form a global LOCC $\mathcal{T}^{M}$
(trace-preserving by averaging over the outcomes). This is applied to a
tensor-product state $\rho_{\mathcal{E}}^{\otimes M}$ where $\rho
_{\mathcal{E}}$ is the Choi matrix of the original channel $\mathcal{E}$. Thus
the approximate channel $\mathcal{E}^{M}$ is simulated by applying
$\mathcal{T}^{M}$ to $\rho_{C}\otimes\rho_{\mathcal{E}}^{\otimes M}$ as in
Eq.~(\ref{LOCCsim}).
\label{fig1
\end{figure*}
\subsection{PBT and simulation of the identity}
Let us describe the protocol of PBT with qudits of arbitrary dimension
$d\geq2$. More technical details can be found in the original
proposals~\cite{PBT1,PBT2}. The parties exploit two ensembles of $M\geq2$
qudits, i.e., Alice has $\mathbf{A}:=\{A_{1},\ldots,A_{M}\}$ and Bob has
$\mathbf{B}:=\{B_{1},\ldots,B_{M}\}$ representing the output \textquotedblleft
ports\textquotedblright. The generic $i$th pair $(A_{i},B_{i})$ is prepared in
a maximally-entangled state, so that we have the global stat
\begin{equation}
\Phi_{\mathbf{AB}}^{\otimes M}=\bigotimes_{i=1}^{M}|\Phi\rangle_{i}\langle
\Phi|,~~|\Phi\rangle_{i}:=d^{-1/2}\sum_{k}\left\vert k\right\rangle _{A_{i
}\otimes\left\vert k\right\rangle _{B_{i}}. \label{nonOPT
\end{equation}
To teleport the state of a qudit $C$, Alice performs a joint measurement on
$C$ and her ensemble $\mathbf{A}$. This is a POVM $\{\Pi_{C\mathbf{A}
^{i}\}_{i=1}^{M}$ with $M$ possible outcomes (see
Refs.~\cite{PBT1,PBT2} for the details). In the standard protocol
considered here, this POVM is a square root measurement (known to
be optimal in the qubit case). Once Alice communicates the outcome
$i$ to Bob, he discards all the ports but the $i$th one, which
contains the teleported state (see Fig.~2a).
The measurement outcomes are equiprobable and independent of the input, and
the output state is invariant under permutation of the ports (this can be
understood by the fact that the scheme is invariant under permutation of the
Bell states and, therefore, of the ports). Averaging over the outcomes, we
define the teleported state $\rho_{B}^{M}=\Gamma_{M}(\rho_{C})$, where
$\Gamma_{M}$ is the corresponding PBT\ channel. Explicitly, this channel takes
the form
\begin{equation}
\Gamma_{M}(\rho_{C})=\sum_{i=1}^{M}\mathrm{Tr}_{\mathbf{A}\bar{B_{i}}C
[\Pi_{C\mathbf{A}}^{i}\left( \rho_{C}\otimes\Phi_{\mathbf{AB}}^{\otimes
M}\right) ],
\end{equation}
where $\text{Tr}_{\bar{B_{i}}}$ denotes the trace over all ports $\mathbf{B}$
but $B_{i}$.
As shown in Ref.~\cite{PBT1}, the standard protocol gives a
depolarizing channel~\cite{NiCh} whose probability $\xi_{M}$
decreases to zero for increasing number of ports $M$. Therefore,
in the limit of many ports $M\gg1$, the $M$-port PBT channel
$\Gamma_{M}$ tends to an identity channel $\mathcal{I}$, so that
Bob's output becomes a perfect replica of Alice's input. Here we
prove a stronger result in terms of channel uniform
convergence~\cite{TQC,Uniform}. In fact, for any $M$, we show that
the simulation error, expressed in terms of the diamond distance
between $\Gamma_{M}$ and $\mathcal{I}$, is one-to-one with the
entanglement fidelity of the PBT channel $\Gamma_{M}$. In turn,
this result allows us to write a simple upper bound for this
error. Moreover, we can fully characterize the simulation error
with an exact analytical expression for qubits (see Methods for
the proof, with further details being given in Supplementary
Section~I).
\begin{lemma}
\label{lemmaPBT}In arbitrary (finite) dimension $d$, the diamond distance
between the $M$-port PBT channel $\Gamma_{M}$ and the identity channel
$\mathcal{I}$ satisfie
\begin{equation}
\delta_{M}:=||\mathcal{I}-\Gamma_{M}||_{\diamond}=2[1-f_{e}(\Gamma_{M})],
\label{firstEQQ
\end{equation}
where $f_{e}(\Gamma_{M}):=\langle\Phi|[\mathcal{I}\otimes\Gamma_{M
(|\Phi\rangle\langle\Phi|)]|\Phi\rangle$ is the entanglement fidelity of
$\Gamma_{M}$. This gives the upper boun
\begin{equation}
\delta_{M}\leq2d(d-1)M^{-1}~. \label{kbound
\end{equation}
More precisely, we can write the exact result
\begin{equation}
\delta_{M}=\frac{2\left( d^{2}-1\right) }{d^{2}}\xi_{M}, \label{exactBB
\end{equation}
where $\xi_{M}$ is the depolarizing probability of the PBT\
channel $\Gamma_{M}$. For qubits ($d=2$), the \textquotedblleft
PBT\ number\textquotedblright\ $\xi_{M}$ has the closed analytical
expression
\begin{align}
\xi_{M} & =\frac{1}{3}\frac{M+2}{2^{M-1}}+\frac{1}{3}\sum_{s=s_{min
}^{(M-1)/2}\frac{s(s+1)}{2^{M-4}}\binom{M}{\frac{M-1}{2}-s}\times\nonumber\\
& \frac{\left( M+2\right) -\sqrt{\left( M+2\right)
^{2}-\left( 2s+1\right) ^{2}}}{\left( M+2\right) ^{2}-\left(
2s+1\right) ^{2}}, \label{PBTnumbers}
\end{align}
where $s_{min}=1/2$ for even $M$ and $0$ for odd $M$.
\end{lemma}
\subsection{General channel simulation via PBT}
Let us discuss how PBT can be used for channel simulation. This was first
shown in Ref.~\cite{PBT}\ where PBT was introduced as a possible design for a
programmable quantum gate array~\cite{Array}. As depicted in Fig.~\ref{fig1}b,
suppose that Bob applies an arbitrary channel $\mathcal{E}$ to the teleported
output, so that Alice's input $\rho_{C}$ is subject to the approximate
channel
\begin{equation}
\mathcal{E}^{M}(\rho_{C}):=\mathcal{E}\circ\Gamma_{M}(\rho_{C}).
\label{effective
\end{equation}
Note that the port selection commutes with $\mathcal{E}$, because the POVM
acts on a different Hilbert space~\cite{PBT}. Therefore, Bob can equivalently
apply $\mathcal{E}$ to each port before Alice's CC, i.e., apply $\mathcal{E
^{\otimes M}$ to his $\mathbf{B}$ qudits before selecting the output port, as
shown in Fig.~\ref{fig1}c. This leads to the following simulation for the
approximate channe
\begin{equation}
\mathcal{E}^{M}(\rho_{C})=\mathcal{T}^{M}(\rho_{C}\otimes\rho_{\mathcal{E
}^{\otimes M})~, \label{LOCCsim
\end{equation}
where $\mathcal{T}^{M}$ is a trace-preserving LOCC and $\rho_{\mathcal{E}}$ is
the channel's Choi matrix (see Fig.~\ref{fig1}d). By construction, the
simulation LOCC $\mathcal{T}^{M}$ is universal, i.e., it does not depend on
the channel $\mathcal{E}$. This means that, at fixed $M$, the channel
$\mathcal{E}^{M}$ is fully determined by the program state $\rho_{\mathcal{E
}$. One can bound the accuracy of the simulation. From Eq.~(\ref{effective})
and the monotonicity of the diamond norm, we get
\begin{equation}
||\mathcal{E}-\mathcal{E}^{M}||_{\diamond}\leq\delta_{M},
\label{simulationERROR
\end{equation}
where $\delta_{M}$ is the simulation error in Eq.~(\ref{kbound}), with the
dimension $d$ being the one of the input Hilbert space. It is worth to remark
that, while the simulation in Eq.~(\ref{LOCCsim}) relies on a number of copies
of the channel's Choi matrix, it can be applied to an arbitrary quantum
channel $\mathcal{E}$ without the condition of teleportation
covariance~\cite{PLOB}.
\begin{figure*}[ptb]
\begin{center}
\vspace{-4.6cm} \includegraphics[width=0.99\textwidth]{Fig2} \vspace{-5cm}
\end{center}
\caption{Port-based teleportation stretching of a generic adaptive protocol
over a quantum channel $\mathcal{E}$. This channel is fixed in quantum/private
communication, while it is unknown and parametrized in
estimation/discrimination problems. \textbf{(a)}~We show the last transmission
$a_{n}\rightarrow b_{n}$ through $\mathcal{E}$, which occurs between two
adaptive QOs $\Lambda_{n-1}$ and $\Lambda_{n}$. This last step produces the
output state $\rho_{n}$. \textbf{(b)}~In each transmission, we replace
$\mathcal{E}$ with its $M$-port simulation $\mathcal{E}^{M}$ so that the
output of the protocol becomes $\rho_{n}^{M}$ which approximates $\rho_{n}$
for large $M$. Note that, in the last transmission, the register state
$\rho_{\mathbf{ab}a_{n}}$ undergoes the transformation $\rho_{\mathbf{ab
b_{n}}=\mathcal{I}_{\mathbf{ab}}\otimes\mathcal{E}^{M}(\rho_{\mathbf{ab}a_{n
})$. \textbf{(c)}~Each propagation through $\mathcal{E}^{M}$ is replaced by
its PBT\ simulation. For the last transmission, this means that $\rho
_{\mathbf{ab}b_{n}}=\mathcal{I}_{\mathbf{ab}}\otimes\mathcal{T}^{M
(\rho_{\mathbf{ab}a_{n}}\otimes\rho_{\mathcal{E}}^{\otimes M})$ where
$\mathcal{T}^{M}$ is the LOCC of the PBT and $\rho_{\mathcal{E}}$ is the Choi
matrix of the original channel. \textbf{(d)}~All the adaptive QOs $\Lambda
_{i}$ and the simulation LOCCs $\mathcal{T}^{M}$ are collapsed into a single
(trace-preserving) QO $\bar{\Lambda}$. Correspondingly, $n$ instances of
$\rho_{\mathcal{E}}^{\otimes M}$ are collected. As a result, the approximate
output $\rho_{n}^{M}$ is given by $\bar{\Lambda}$ applied to the
tensor-product state $\rho_{\mathcal{E}}^{\otimes nM}$ as in Eq.~(\ref{sss}).
\label{stretch
\end{figure*}
\subsection{PBT stretching of an adaptive protocol}
Channel simulation is a preliminary tool for the following technique of
teleportation stretching, where an arbitrary adaptive protocol is reduced into
a simpler block version. There are two main steps. First of all, we need to
replace each channel $\mathcal{E}$ with its $M$-port approximation
$\mathcal{E}^{M}$\ while controlling the propagation of the simulation error
$\delta_{M}$ from the channel to the output state. This step is crucial also
in simulations via standard teleportation~\cite{TQC,ReviewMETRO} (see also
Refs.~\cite{networkPIRS,Multipoint,finiteStretching,nonPauli,HWchannels}).
Second, we need to \textquotedblleft stretch\textquotedblright\ the
protocol~\cite{PLOB} by replacing the various instances of the approximate
channel $\mathcal{E}^{M}$ with a collection of Choi matrices $\rho
_{\mathcal{E}}^{\otimes M}$ and then suitably re-organizing all the remaining
QOs. Here we describe the technique for a generic task, before specifying it
to QCD.
Given an adaptive protocol $\mathcal{P}_{n}$ over a channel $\mathcal{E}$ with
output $\rho_{n}$, consider the same protocol over the simulated channel
$\mathcal{E}^{M}$, so that we get the different output $\rho_{n}^{M}$. Using a
\textquotedblleft peeling\textquotedblright\ argument (see Methods), we bound
the output error in terms of the channel simulation error
\begin{equation}
||\rho_{n}-\rho_{n}^{M}||\leq n||\mathcal{E}-\mathcal{E}^{M}||_{\diamond}\leq
n\delta_{M}. \label{outputERROR
\end{equation}
Once understood that the output state can be closely approximated, let us
simplify the adaptive protocol over $\mathcal{E}^{M}$. Using the simulation in
Eq.~(\ref{LOCCsim}), we may replace each channel $\mathcal{E}^{M}$ with the
resource state $\rho_{\mathcal{E}}^{\otimes M}$, iterate the process for all
$n$ uses, and collapse all the simulation LOCCs and QOs as shown in
Fig.~\ref{stretch}. As a result, we may write the multi-copy Choi
decomposition\
\begin{equation}
\rho_{n}^{M}=\bar{\Lambda}(\rho_{\mathcal{E}}^{\otimes nM})~, \label{sss
\end{equation}
for a trace-preserving QO $\bar{\Lambda}$. Now, we can combine the two
ingredients of Eqs.~(\ref{outputERROR}) and~(\ref{sss}), into the following.
\begin{lemma}
[PBT stretching]\label{lemma}Consider an adaptive quantum protocol (with
arbitrary task) over an arbitrary $d$-dimensional quantum channel
$\mathcal{E}$ (which may be unknown and parametrized). After $n$ uses, the
output $\rho_{n}$ of the protocol can be decomposed as follows
\begin{equation}
||\rho_{n}-\bar{\Lambda}(\rho_{\mathcal{E}}^{\otimes nM})||\leq n\delta_{M},
\label{LemmaEQ
\end{equation}
where $\bar{\Lambda}$ is a trace-preserving QO, $\rho_{\mathcal{E}}$ is the
Choi matrix of $\mathcal{E}$, and $\delta_{M}$ is the $M$-port simulation
error in Eq.~(\ref{kbound}).
\end{lemma}
When we apply the lemma to protocols of quantum or private communication,
where the QOs $\Lambda_{i}$ are LOCCs, then we may write Eq.~(\ref{LemmaEQ})
with $\bar{\Lambda}$ being a LOCC. In protocols of channel estimation or
discrimination, where $\mathcal{E}$ is parametrized, we may write
Eq.~(\ref{LemmaEQ}) with $\rho_{\mathcal{E}}$ storing the parameter of the
channel. In particular, for QCD we have $\{\mathcal{E}_{u}\}_{u=0,1}$ and the
output $\rho_{n}(u)$ of the adaptive protocol $\mathcal{P}_{n}$ can be
decomposed as follows
\begin{equation}
||\rho_{n}(u)-\bar{\Lambda}(\rho_{\mathcal{E}_{u}}^{\otimes nM})||\leq
n\delta_{M}.
\end{equation}
\subsection{Ultimate bound for channel discrimination}
We are now ready to show the lower bound for minimum error probability
$p_{n}(\mathcal{E}_{0}\neq\mathcal{E}_{1})$ in Eq.~(\ref{eqDD}). Consider an
arbitrary protocol $\mathcal{P}_{n}$, for which we may write
Eq.~(\ref{protPROB}). Combining Lemma~2 with the triangle inequality leads t
\begin{align}
||\rho_{n}(0)-\rho_{n}(1)|| & \leq2n\delta_{M}+||\bar{\Lambda
(\rho_{\mathcal{E}_{0}}^{\otimes nM})-\bar{\Lambda}(\rho_{\mathcal{E}_{1
}^{\otimes nM})||\nonumber\\
& \leq2n\delta_{M}+||\rho_{\mathcal{E}_{0}}^{\otimes nM}-\rho_{\mathcal{E
_{1}}^{\otimes nM}||, \label{eqBOUND
\end{align}
where we also use the monotonicity of the trace distance under channels.
Because $\bar{\Lambda}$\ is lost, the bound does no longer depend on the
details of the protocol $\mathcal{P}_{n}$, which means that it applies to all
adaptive protocols. Thus, using Eq.~(\ref{eqBOUND}) in Eqs.~(\ref{protPROB})
and~(\ref{protINF}), we get the following.
\begin{theorem}
\label{mainTHEO}Consider the adaptive discrimination of two channels
$\{\mathcal{E}_{u}\}_{u=0,1}$ in dimension $d$. After $n$ probings, the
minimum error probability satisfies the boun
\begin{equation}
p_{n}(\mathcal{E}_{0}\neq\mathcal{E}_{1})\geq B:=\frac{1-n\delta_{M
-D(\rho_{\mathcal{E}_{0}}^{\otimes nM},\rho_{\mathcal{E}_{1}}^{\otimes nM
)}{2}, \label{surprise
\end{equation}
where $M$ may be chosen to maximize the right hand side.
\end{theorem}
\noindent Not only this is the first universal bound for adaptive QCD, but
also its analytical form is rather surprising. In fact, its tighest value is
given by an optimal (finite) number of ports $M$ for the underlying protocol
of PBT.
Let us bound the trace distance in Eq.~(\ref{surprise}) as
\begin{equation}
D^{2}\leq1-F^{2nM},~F:=\mathrm{Tr}\sqrt{\sqrt{\rho_{\mathcal{E}_{0}}
\rho_{\mathcal{E}_{1}}\sqrt{\rho_{\mathcal{E}_{0}}}}, \label{letus
\end{equation}
where $F$ is the fidelity between the Choi matrices of the
channels. This comes from the Fuchs-van de Graaf
relations~\cite{Fuchs} and the multiplicativity of the fidelity
over tensor products. Other bounds that can be written are
\begin{equation}
D\leq nM\left\Vert
\rho_{\mathcal{E}_{0}}-\rho_{\mathcal{E}_{1}}\right\Vert ,
\end{equation}
from the subadditivity of the trace distance, and
\begin{equation}
D\leq\sqrt{nM(\ln\sqrt{2})\min\{S(\rho_{\mathcal{E}_{0}}||\rho_{\mathcal{E}_{1}}),S(\rho_{\mathcal{E}_{1}}||\rho_{\mathcal{E}_{0}})\}},
\end{equation}
from the Pinsker inequality~\cite{Pin1,Pin2}, where $S(\rho
||\sigma)=\mathrm{Tr}[\rho(\log_{2}\rho-\log_{2}\sigma)]$ is the
relative entropy~\cite{NiCh}.
If we exploit Eqs.~(\ref{kbound}) and~(\ref{letus}) in Eq.~(\ref{surprise}),
we may write the following simplified boun
\begin{equation}
B\geq\frac{1}{2}-\frac{\sqrt{1-F^{2nM}}}{2}-\frac{d(d-1)n}{M}\,.
\end{equation}
In the previous formula there are terms with opposite monotonicity in $M$, so
that the maximum value of the bound $B$ is achieved at some intermediate value
of $M$. Setting $M=xd(d-1)n$ for some $x>2$, we get
\begin{equation}
B\geq\frac{1}{2}-\frac{1}{x}-\frac{1}{2}\sqrt{1-F^{2xd(d-1)n^{2}}}.
\end{equation}
One good choice is therefore $M=4d(d-1)n$, so that
\begin{equation}
B\geq(1-2\sqrt{1-F^{8d(d-1)n^{2}}})/4.
\end{equation}
In particular, consider two infinitesimally-close channels, so that
$F\simeq1-\epsilon$ where $\epsilon\simeq0$ is the infidelity. By expanding in
$\epsilon$ for any finite $n$, we may write
\begin{equation}
B\geq\frac{1}{4}-n\sqrt{2d(d-1)\epsilon}\simeq\frac{\exp(-4n\sqrt
{2d(d-1)\epsilon})}{4}. \label{boundB
\end{equation}
For instance, in the case of qubits this becomes $[\exp
(-8n\sqrt{\epsilon})]/4$, to be compared with the upper bound
$[\exp(-2n\epsilon)]/2$ computed from Eq.~(\ref{fidSIM}).
Discriminating between two close quantum channels is a problem in
many physical scenarios. For instance, this is typical in quantum
optical resolution~\cite{Tsang15,Lupo16,Tsang2} (discussed below),
quantum
illumination~\cite{Qill0,Qill1,Qill2,Qill3,Qill4,Qill5,Qill6,Qill7,dd3,Qill8}
(discussed below), ideal quantum
reading~\cite{Qread,QreadCAP,Arno12,ArnoIJQI,GaeENTROPY}, quantum
metrology~\cite{Sam1,Sam2,Paris,Giova,ReviewNEW} (discussed
below), and also tests of quantum field theories in non-inertial
frames~\cite{Doukas}, e.g., for detecting effects such as the
Unruh or the Hawking radiation.
\subsection{Limits of single-photon quantum optical resolution}
Consider a microscope-type problem where we aim at locating a
point in two possible positions, either $s/2$ or $-s/2$, where the
separation $s$ is very small. Assume we are limited to use probe
states with at most one photon and an output finite-aperture
optical system (this makes the optical process to be a
qubit-to-qutrit channel, so that the input dimension is $d=2$).
Apart from this, we are allowed to use an arbitrary large quantum
computer and arbitrary QOs to manipulate its registers. We may
apply Eq.~(\ref{boundB}) with $\epsilon\simeq\eta s^{2}/16$, where
$\eta$ is a diffraction-related loss parameter. In this way, we
find that the error probability affecting the discrimination of
the two positions is approximately bounded by $B\gtrsim
\frac{1}{4}\exp(-2ns\sqrt{\eta})$. This bound establishes a no-go
for perfect quantum optical resolution. See Supplementary
Section~II for more mathematical details on this specific
application.
\subsection{Limits of adaptive quantum illumination}
Consider the protocol of quantum illumination in the DV setting~\cite{Qill0}.
Here the problem is to discriminate the presence or not of a target with low
reflectivity $\eta\simeq0$ in a thermal background which has $b\ll1$ mean
thermal photons per optical mode. One assumes that $d$ modes are used in each
probing of the target and each of them contains at most one photon. This means
that the Hilbert space is $(d+1)$-dimensional with basis $\{\left\vert
0\right\rangle ,\left\vert 1\right\rangle ,\ldots,\left\vert d\right\rangle
\}$, where $\left\vert i\right\rangle :=\left\vert 0\cdots010\cdots
0\right\rangle $ has one photon in the $i$th mode. If the target is absent
($u=0$), the receiver detects thermal noise; if the target is present ($u=1$),
the receiver measures a mixture of signal and thermal noise.
In the most general (adaptive) version of the protocol, the
receiver belongs to a large quantum computer where the
$(d+1)$-dimensional signal qudits are picked from an input
register, sent to target, and their reflection stored in an output
register, with adaptive QOs performed between each probing. After
$n$ probings, the state of the registers $\rho_{n}(u)$ is
optimally detected. Assuming the typical regime of quantum
illumination~\cite{Qill0}, we find that the error probability
affecting target detection is approximately bounded by
$B\gtrsim\frac{1}{4}\exp(-4nd\sqrt{\eta})$. This bound establishes
a no-go for exponential improvement in quantum illumination.
Entanglement and adaptiveness can \textit{at most} improve the
error exponent with respect to separable probes, for which the
error probability is $\lesssim\frac{1}{2}\exp [-n\eta/(8d)]$. See
also Supplementary Section~III.
\subsection{Limits of adaptive quantum metrology}
Consider the adaptive estimation of a continuous parameter $\theta$ encoded in
a quantum channel $\mathcal{E}_{\theta}$. After $n$ probings, we have a
$\theta$-dependent output state $\rho_{n}(\theta)$ generated by an adaptive
quantum estimation protocol $\mathcal{P}_{n}$. This output state is then
measured by a POVM $\mathcal{M}$ providing an optimal unbiased estimator
$\tilde{\theta}$ of parameter $\theta$. The minimum error variance
Var$(\tilde{\theta}):=\langle(\tilde{\theta}-\theta)^{2}\rangle$ must satisfy
the quantum Cramer-Rao bound \textrm{Var}$(\tilde{\theta})\geq1/
\textrm{QFI}$_{\theta}(\mathcal{P}_{n})$, where \textrm{QFI}$_{\theta
}(\mathcal{P}_{n})$ is the quantum Fisher information~\cite{Sam1} associated
with $\mathcal{P}_{n}$. The ultimate precision of adaptive quantum metrology
is given by the optimization over all protocols
\begin{equation}
\overline{\text{\textrm{QFI}}}_{\theta}^{n}:=\sup_{\mathcal{P}_{n
}\text{\textrm{QFI}}_{\theta}(\mathcal{P}_{n}). \label{fisheroptm
\end{equation}
This quantity can be simplified by PBT\ stretching. In fact, for any input
state $\rho_{C}$, we may write the simulation $\mathcal{E}_{\theta}^{M
(\rho_{C})=\mathcal{T}^{M}(\rho_{C}\otimes\rho_{\mathcal{E}_{\theta}}^{\otimes
M})$ which is an immediate extension of Eq.~(\ref{LOCCsim}). In this way, the
output state can be decomposed following Lemma~2, i.e., we may write
$||\rho_{n}(\theta)-\bar{\Lambda}(\rho_{\mathcal{E}_{\theta}}^{\otimes
nM})||\leq n\delta_{M}$. Exploiting the latter inequality for large $n$, we
find that the ultimate bound of adaptive quantum metrology takes the form
\begin{equation}
\overline{\text{\textrm{QFI}}}_{\theta}^{n}\lesssim n^{2}\mathrm{QFI
(\rho_{\mathcal{E}_{\theta}}), \label{cubem
\end{equation}
where $\mathrm{QFI}(\rho_{\mathcal{E}_{\theta}})$ is computed on
the channel's Choi matrix. In particular, we see that PBT allows
us to write a simple bound in terms of the Choi matrix and implies
a general no-go theorem for super-Heisenberg scaling in quantum
metrology. See Supplementary Section~IV for a detailed proof of
Eq.~(\ref{cubem}).
\subsection{Tightening the main formula}
Let us note that the formula in Theorem~\ref{mainTHEO} is expressed in terms
of the universal error $\delta_{M}$ coming from the PBT simulation of the
identity channel (Lemma~\ref{lemmaPBT}). There are situations where the
diamond distance $\Delta_{M}:=||\mathcal{E}-\mathcal{E}^{M}||_{\diamond}$
between a quantum channel $\mathcal{E}$ and its $M$-port simulation
$\mathcal{E}^{M}$ is exactly computable. In these cases, we can certainly
formulate a tighter version of Eq.~(\ref{surprise}) where $\delta_{M}$ is
suitably replaced. In fact, from the peeling argument, we have $||\rho
_{n}-\rho_{n}^{M}||\leq n\Delta_{M}$, so that a tighter version of
Eq.~(\ref{LemmaEQ}) is simply $||\rho_{n}-\bar{\Lambda}(\rho_{\mathcal{E
}^{\otimes nM})||\leq n\Delta_{M}$. Then, for the two possible outputs
$\rho_{n}(0)$ and $\rho_{n}(1)$ of an adaptive discrimination protocol over
$\mathcal{E}_{0}$ and $\mathcal{E}_{1}$, we can replace Eq.~(\ref{eqBOUND})
with
\begin{equation}
||\rho_{n}(0)-\rho_{n}(1)||\leq2n\bar{\Delta}_{M}+||\rho_{\mathcal{E}_{0
}^{\otimes nM}-\rho_{\mathcal{E}_{1}}^{\otimes nM}||,
\end{equation}
where $\bar{\Delta}_{M}:=(||\mathcal{E}_{0}-\mathcal{E}_{0}^{M}||_{\diamond
}+||\mathcal{E}_{1}-\mathcal{E}_{1}^{M}||_{\diamond})/2$. It is now easy to
check that Eq.~(\ref{surprise}) becomes the followin
\begin{equation}
p_{n}(\mathcal{E}_{0}\neq\mathcal{E}_{1})\geq\frac{1-n\bar{\Delta}_{M
-D(\rho_{\mathcal{E}_{0}}^{\otimes nM},\rho_{\mathcal{E}_{1}}^{\otimes nM
)}{2}. \label{improvedB
\end{equation}
In the following section, we show that $\bar{\Delta}_{M}$, and therefore the
bound in Eq.~(\ref{improvedB}), can be computed for the discrimination of
amplitude damping channels.
\subsection{Discrimination of amplitude damping channels}
As an additional example of application of the bound, consider the
discrimination between amplitude damping channels. These channels are not
teleportation covariant, so that the results from Ref.~\cite{PirCo} do not
apply and no bound is known on the error probability for their adaptive
discrimination. Recall that an amplitude damping channel $\mathcal{E}_{p}$
transforms an input state $\rho$ as follow
\begin{equation}
\mathcal{E}_{p}(\rho)
{\textstyle\sum\nolimits_{i=0,1}}
K_{i}\rho K_{i}^{\dagger},
\end{equation}
with Kraus operator
\begin{equation}
K_{0}:=\left\vert 0\right\rangle \left\langle 0\right\vert +\sqrt
{1-p}\left\vert 1\right\rangle \left\langle 1\right\vert ,~K_{1}:=\sqrt
{p}\left\vert 0\right\rangle \left\langle 1\right\vert ,
\end{equation}
where $\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle \}$ is the
computational basis and $p$ is the damping probability or rate.
Given two amplitude damping channels, $\mathcal{E}_{p_{0}}$ and $\mathcal{E
_{p_{1}}$, first assume a discrimination protocol where these
channels are probed by $n$ maximally-entangled states and the
outputs are optimally measured. The optimal error probability for
this (non-adaptive) block protocol is given by
$p_{n}^{\text{block}}=[1-D(\rho_{\mathcal{E}_{p_{0}}}^{\otimes
n},\rho_{\mathcal{E}_{p_{1}}}^{\otimes n})]/2$ and satisfies
\begin{equation}
\frac{1-\sqrt{1-F(p_{0},p_{1})^{2n}}}{2}\leq p_{n}^{\text{block}}\leq
\frac{F(p_{0},p_{1})^{n}}{2}, \label{iidPROB
\end{equation}
where $F(p_{0},p_{1}):=F(\rho_{\mathcal{E}_{p_{0}}},\rho_{\mathcal{E}_{p_{1}
})$ is the fidelity between the Choi matrices. In particular, we explicitly
comput
\begin{equation}
F=\frac{1+\sqrt{(1-p_{0})(1-p_{1})}+\sqrt{p_{0}p_{1}}}{2}.
\end{equation}
It is clear that $p_{n}^{\text{block}}$ in Eq.~(\ref{iidPROB}) is an upper
bound to ultimate (adaptive) error probability $p_{n}(\mathcal{E}_{p_{0}
\neq\mathcal{E}_{p_{1}})$ for the discrimination of the two channels.
To lowerbound the ultimate probability we employ
Eq.~(\ref{improvedB}). In fact, for the $M$-port simulation
$\mathcal{E}_{p}^{M}$ of $\mathcal{E}_{p}$, we
comput
\begin{equation}
\Delta_{M}(p)=||\mathcal{E}_{p}-\mathcal{E}_{p}^{M}||_{\diamond} =\xi
_{M}\left( \frac{1-p}{2}+\sqrt{1-p}\right) ,\label{Deltadamp
\end{equation}
where $\xi_{M}$ are the PBT numbers defined in Eq.~(\ref{PBTnumbers}). For any
two amplitude damping channels, $\mathcal{E}_{p_{0}}$ and $\mathcal{E}_{p_{1
}$, we can then compute $\bar{\Delta}_{M}(p_{0},p_{1})$ and use
Eq.~(\ref{improvedB}) to bound $p_{n}(\mathcal{E}_{p_{0}}\neq\mathcal{E
_{p_{1}})$. More precisely, we can also exploit Eq.~(\ref{letus}) and write
the computable lower bound
\begin{equation}
p_{n}(\mathcal{E}_{p_{0}}\neq\mathcal{E}_{p_{1}})\geq\frac{1-n\bar{\Delta
_{M}(p_{0},p_{1})-\sqrt{1-F(p_{0},p_{1})^{2nM}}}{2}. \label{LBcompute
\end{equation}
In Fig.~\ref{Figcomp} we show an example of discrimination between two
amplitude damping channels. In particular, we show how large is the gap
between the upper bound $p_{n}^{\text{block}}$ of Eq.~(\ref{iidPROB}) and the
lower bound in Eq.~(\ref{LBcompute}) suitably optimized over the number of
ports $M$. It is an open question to find exactly $p_{n}(\mathcal{E}_{p_{0
}\neq\mathcal{E}_{p_{1}})$. At this stage, we do not know if this result may
achieved by tightening the upper bound or the lower bound.\begin{figure}[ptb]
\begin{center}
\vspace{+0.2cm} \includegraphics[width=0.45\textwidth]{ADchannels}
\vspace{-0.1cm}
\end{center}
\caption{Error probability in the discrimination of two amplitude
damping channels, one with damping rate $p\geq0.8$ and the other
with rate $p+1\%$. We assume $n=20$ probings of the unknown
channel. The upper dark region identifies the region where the
error probability $p_{n}^{\text{block}}$ of
Eq.~(\ref{iidPROB}) lies. The adaptive error probability $p_{n}(\mathcal{E
_{p_{0}}\neq\mathcal{E}_{p_{1}})$ lies below this dark region and
above the dotted points, which represent our lower bound of
Eq.~(\ref{LBcompute}) optimized over the number of ports $M$. For
comparison, we also plot the lower bound for specific $M$.}
\label{Figcomp
\end{figure}
\section{Discussion}
In this work we have established a general and fundamental lower
bound for the error probability affecting the adaptive
discrimination of two arbitrary quantum channels acting on a
finite-dimensional Hilbert space. This bound is conveniently
expressed in terms of the Choi matrices of the channels involved,
so that it is very easy to compute. It also applies to many
scenarios, including adaptive protocols for quantum-enhance
optical resolution and quantum illumination. In order to derive
our result, we have employed port-based teleportation as a tool
for channel simulation, and developed a methodology which
simplifies adaptive protocols performed over an arbitrary
finite-dimensional channel. This technique can be applied to many
other scenarios. For instance, in quantum metrology we are able to
prove that adaptive protocols of quantum channel estimation are
limited by a bound simply expressed in terms of the Choi matrix of
the channel and following the Heisenberg scaling in the number of
probings. Not only this shows that our bound is asymptotically
tight but also draws an unexpected connection between port-based
teleportation and quantum metrology. Further potential
applications are in quantum and private communications, which are
briefly discussed in our Supplementary Section~V.
\section{Methods}
\subsection{Simulation error in diamond norm (proof of Lemma~1)\label{SUP0}}
It is easy to check that the channel $\Gamma_{M}$ associated with the qudit
PBT protocol of Ref.~\cite{PBT} is covariant under unitary transformations,
i.e.,
\begin{equation}
\Gamma_{M}(U\rho U^{\dag})=U\Gamma_{M}(\rho)U^{\dag},
\end{equation}
for any input state $\rho$ and unitary operator $U$. As discussed
in Ref.~\cite{Majenz}, for a channel with such a symmetry, the
diamond distance with the identity map is saturated by a maximally
entangled state, i.e.,
\begin{equation}
\Vert\mathcal{I}-\Gamma_{M}\Vert_{\diamond}=\Vert|\Phi\rangle\langle
\Phi|-\mathcal{I}\otimes\Gamma_{M}\left( |\Phi\rangle\langle\Phi|\right)
\Vert\,, \label{CMajenz
\end{equation}
where $|\Phi\rangle=d^{-1/2}\sum_{k=1}^{d}|k\rangle|k\rangle$. Here we first
show that
\begin{equation}
\Vert|\Phi\rangle\langle\Phi|-\mathcal{I}\otimes\Gamma_{M}\left( |\Phi
\rangle\langle\Phi|\right) \Vert=2[1-f_{e}(\Gamma_{M})]~. \label{showCL
\end{equation}
In fact, note that the map $\Lambda_{M}=\mathcal{I}\otimes\Gamma_{M}$ is
covariant under twirling unitaries of the form $U\otimes U^{\ast}$, i.e.,
\begin{align}
& \Lambda_{M}\left[ (U\otimes U^{\ast})\rho(U\otimes U^{\ast})^{\dag
}\right] \nonumber\\
& =(U\otimes U^{\ast})\Lambda_{M}(\rho)(U\otimes U^{\ast})^{\dag},
\end{align}
for any input state $\rho$ and unitary operator $U$. This implies that the
state $\Lambda_{M}(|\Phi\rangle\langle\Phi|)$ is invariant under twirling
unitaries, i.e.,
\begin{equation}
(U\otimes U^{\ast})\Lambda_{M}(|\Phi\rangle\langle\Phi|)(U\otimes U^{\ast
})^{\dag}=\Lambda_{M}(|\Phi\rangle\langle\Phi|)~.
\end{equation}
This is therefore an isotropic state of the for
\begin{equation}
\Lambda_{M}(|\Phi\rangle\langle\Phi|)=(1-p)|\Phi\rangle\langle\Phi|+\frac
{p}{d^{2}}\mathbb{I},\label{eq:ADchoi1
\end{equation}
where $\mathbb{I}$ is the two-qudit identity operator.
We may rewrite this state as follow
\begin{equation}
\Lambda_{M}(|\Phi\rangle\langle\Phi|)=F|\Phi\rangle\langle\Phi|+(1-F)\rho
^{\perp},\label{deco1
\end{equation}
where $\rho^{\perp}$ is state with support in the orthogonal complement of
$\Phi$, and $F$ is the singlet fractio
\begin{equation}
F:=\langle\Phi|\Lambda_{M}(|\Phi\rangle\langle\Phi|)|\Phi\rangle=1-p+pd^{-2}.
\end{equation}
Thanks to the decomposition in Eq.~(\ref{deco1}) and using basic properties of
the trace norm~\cite{NiCh}, we may then write
\begin{align}
& \Vert|\Phi\rangle\langle\Phi|-\Lambda_{M}\left( |\Phi\rangle\langle
\Phi|\right) \Vert\nonumber\\
& =\Vert(1-F)|\Phi\rangle\langle\Phi|-(1-F)\rho^{\perp}\Vert\nonumber\\
& =(1-F)\Vert|\Phi\rangle\langle\Phi|\Vert+(1-F)\Vert\rho^{\perp
\Vert\nonumber\\
& =2(1-F)\nonumber\\
& =2[1-f_{e}(\Gamma_{M})],
\end{align}
where the last step exploits the fact that the singlet fraction $F$ is the
channel's entanglement fidelity $f_{e}(\Gamma_{M})$. This completes the proof
of Eq.~(\ref{showCL}).
Therefore, combining Eqs.~(\ref{CMajenz}) and (\ref{showCL}), we obtain
\begin{equation}
\Vert\mathcal{I}-\Gamma_{M}\Vert_{\diamond}=2[1-f_{e}(\Gamma_{M
)],\label{tttt
\end{equation}
which is Eq.~(\ref{firstEQQ}) of the main text. Then, we know that the
entanglement fidelity of $\Gamma_{M}$ is bounded as~\cite{PBT}
\begin{equation}
f_{e}(\Gamma_{M})\geq1-d(d-1)M^{-1}.\label{Hiro
\end{equation}
Therefore, using Eq.~(\ref{Hiro}) in Eq.~(\ref{tttt}), we derive the following
upper bound
\begin{equation}
\Vert\mathcal{I}-\Gamma_{M}\Vert_{\diamond}\leq2d(d-1)M^{-1},
\end{equation}
which is Eq.~(\ref{kbound}) of the main text.
Let us now prove Eq.~(\ref{exactBB}). It is known~\cite{PBT1} that
implementing the standard PBT\ protocol over the resource state of
Eq.~(\ref{nonOPT}) leads to a PBT channel $\Gamma_{M}$ which is a
qudit depolarizing channel. Its isotropic Choi matrix
$\rho_{\Gamma_{M}}$, given in Eq.~(\ref{eq:ADchoi1}), can be
written in the form
\begin{equation}
\rho_{\Gamma_{M}}=\left( 1-\frac{d^{2}-1}{d^{2}}\xi_{M}\right) |\Phi
\rangle^{0}\langle\Phi|+\sum_{i=1}^{d^{2}-1}\frac{\xi_{M}}{d^{2}}|\Phi
\rangle^{i}\langle\Phi|,
\end{equation}
where $\xi_{M}$ is the probability $p$ of depolarizing, $|\Phi\rangle
^{0}\langle\Phi|$ is the projector onto the initial maximally-entangled state
of two qudits (one system of which was sent through the channel), and
$|\Phi\rangle^{i}\langle\Phi|$ are the projectors onto the other $d^{2}-1$
maximally-entangled states of two qudits (generalized Bell states). Since the
Choi matrix of the identity channel is $\rho_{\mathcal{I}}=|\Phi\rangle
^{0}\langle\Phi|$, it is easy to compute
\begin{align}
\left\vert \rho_{\mathcal{I}}-\rho_{\Gamma_{M}}\right\vert & :=\sqrt{\left(
\rho_{\mathcal{I}}-\rho_{\Gamma_{M}}\right) \left( \rho_{\mathcal{I}
-\rho_{\Gamma_{M}}\right) ^{\dag}}\nonumber\\
& =\frac{d^{2}-1}{d^{2}}\xi_{M}|\Phi\rangle^{0}\langle\Phi|+\sum_{i=1
^{d^{2}-1}\frac{\xi_{M}}{d^{2}}|\Phi\rangle^{i}\langle\Phi|.
\end{align}
From the previous equation, we derive
\begin{equation}
\mathrm{Tr}_{2}\left\vert \rho_{\mathcal{I}}-\rho_{\Gamma_{M}}\right\vert
=\frac{2\left( d^{2}-1\right) }{d^{3}}\xi_{M}\sum_{j=0}^{d-1}|j\rangle
\langle j|,\label{null
\end{equation}
where we have used $\mathrm{Tr}_{2}|\Phi\rangle^{i}\langle\Phi|=d^{-1
\sum_{j=0}^{d-1}|j\rangle\langle j|$ in the qudit computational
basis $\{|j\rangle\}$ and we have summed over the $d^{2}$
generalized Bell states. It is clear that Eq.~(\ref{null}) is a
diagonal matrix with equal non-zero elements, i.e., it is a
scalar. As a result, we can apply Proposition~1 of
Ref.~\cite{nechita} over the Hermitian operator $\rho_{\mathcal{I}
-\rho_{\Gamma_{M}}$, and write
\begin{gather}
\Vert\mathcal{I}-\Gamma_{M}\Vert_{\diamond}=\Vert\rho_{\mathcal{I}}-\rho_{\Gamma_{M}}\Vert\nonumber\\
=\mathrm{Tr}\left\vert \rho_{\mathcal{I}}-\rho_{\Gamma_{M}}\right\vert
=\frac{2\left( d^{2}-1\right) }{d^{2}}\xi_{M}~.
\end{gather}
The final step of the proof is to compute the explicit expression
of $\xi_{M}$ for qubits, which is the formula given in
Eq.~(\ref{PBTnumbers}). Because this derivation is technically
involved, it is reported in Supplementary Section~I.
\subsection{Propagation of the simulation error\label{SUP3}}
For the sake of completeness, we provide the proof of the first
inequality in~Eq.~(\ref{outputERROR}) (this kind of proof already
appeared in Refs.~\cite{PLOB,TQC}). Consider the adaptive protocol
described in the main
text. For the $n$-use output state we may compactly writ
\begin{equation}
\rho_{n}=\Lambda_{n}\circ\mathcal{E}\circ\Lambda_{n-1}\circ\cdots
\circ\mathcal{E}\circ\Lambda_{1}\circ\mathcal{E}(\rho_{0}),
\end{equation}
where $\Lambda$'s are adaptive QOs and $\mathcal{E}$ is the channel applied to
the transmitted signal system. Then, $\rho_{0}$ is the preparation state of
the registers, obtained by applying the first\ QO$\ \Lambda_{0}$ to some
fundamental state. Similarly, for the $M$-port simulation of the protocol, we
may writ
\begin{equation}
\rho_{n}^{M}=\Lambda_{n}\circ\mathcal{E}^{M}\circ\Lambda_{n-1}\circ\cdots
\circ\mathcal{E}^{M}\circ\Lambda_{1}\circ\mathcal{E}^{M}(\rho_{0}),
\end{equation}
where $\mathcal{E}^{M}$ is in the place of $\mathcal{E}$.
Consider now two instances ($n=2$) of the adaptive protocol. We may bound the
trace distance between $\rho_{2}$ and $\rho_{2}^{M}$ using a \textquotedblleft
peeling\textquotedblright\ argument~\cite{PirCo,PLOB,TQC,Uniform,ReviewMETRO}
\begin{align}
\left\Vert \rho_{2}-\rho_{2}^{M}\right\Vert & =\left\Vert \Lambda_{2
\circ\mathcal{E}\circ\Lambda_{1}\circ\mathcal{E}(\rho_{0})\right. \nonumber\\
& -\Lambda_{2}\circ\mathcal{E}^{M}\circ\Lambda_{1}\circ\mathcal{E}^{M
(\rho_{0})||\nonumber\\
& \overset{{\tiny (1)}}{\leq}||\mathcal{E}\circ\Lambda_{1}\circ
\mathcal{E}(\rho_{0})-\mathcal{E}^{M}\circ\Lambda_{1}\circ\mathcal{E}^{M
(\rho_{0})||\nonumber\\
& \overset{{\tiny (2)}}{\leq}||\mathcal{E}\circ\Lambda_{1}\circ
\mathcal{E}(\rho_{0})-\mathcal{E}\circ\Lambda_{1}\circ\mathcal{E}^{M}(\rho
_{0})||\nonumber\\
& +||\mathcal{E}^{M}\circ\Lambda_{1}\circ\mathcal{E}(\rho_{0})-\mathcal{E
^{M}\circ\Lambda_{1}\circ\mathcal{E}^{M}(\rho_{0})||\nonumber\\
& \overset{{\tiny (3)}}{\leq}||\mathcal{E}(\rho_{0})-\mathcal{E}^{M}(\rho
_{0})||\nonumber\\
& +||\mathcal{E}[\Lambda_{1}\circ\mathcal{E}^{M}(\rho_{0})]-\mathcal{E
^{M}[\Lambda_{1}\circ\mathcal{E}^{M}(\rho_{0})]||\nonumber\\
& \overset{{\tiny (4)}}{\leq}2||\mathcal{E}-\mathcal{E}^{M}||_{\diamond}~.
\label{DiamondV
\end{align}
In $(1)$ we use the monotonicity of the trace distance under
completely-positive trace-preserving (CPTP) maps (i.e., quantum
channels); in $(2)$ we employ the triangle inequality; in $(3)$ we
use the monotonicity with respect to the the CPTP map
$\mathcal{E}\circ\Lambda_{1}$ whereas in $(4)$ we exploit the fact
that the diamond norm is an upper bound for the trace norm
computed on any input state. Generalizing the result of
Eq.~(\ref{DiamondV}) to arbitrary $n$, we achieve the first
inequality in Eq.~(\ref{outputERROR}). Note that the previous
reasoning also applies to a classically-parametrized channel
$\mathcal{E}_{u}$.
\subsection{PBT simulation of amplitude damping channels}
Here we show the result in Eq.~(\ref{Deltadamp}) for $\Delta_{M
(p)=||\mathcal{E}_{p}-\mathcal{E}_{p}^{M}||_{\diamond}$, which is the error
associated with the $M$-port simulation of an arbitrary amplitude damping
channel $\mathcal{E}_{p}$. From Ref.~\cite{PBT1}, we know that the PBT channel
$\Gamma^{M}$ is a depolarizing channel. In the qubit computational basis
$\{\left\vert i,j\right\rangle \}_{i,j=0,1}$, it has the following Choi
matrix
\begin{equation}
\rho_{\Gamma^{M}}
\begin{pmatrix}
\frac{1}{2}-\frac{\xi_{M}}{4} & 0 & 0 & \frac{1}{2}-\frac{\xi_{M}}{2}\\
0 & \frac{\xi_{M}}{4} & 0 & 0\\
0 & 0 & \frac{\xi_{M}}{4} & 0\\
\frac{1}{2}-\frac{\xi_{M}}{2} & 0 & 0 & \frac{1}{2}-\frac{\xi_{M}}{4
\end{pmatrix}
,
\end{equation}
where $\xi_{M}$ are the PBT\ numbers of Eq.~(\ref{PBTnumbers}). Note that
these take decreasing positive values, for instanc
\begin{align}
\xi_{2} & =\frac{6-\sqrt{3}}{6}\simeq0.71,\nonumber\\
\xi_{3} & =1/2,\nonumber\\
\xi_{4} & =\frac{13-2\sqrt{2}-2\sqrt{5}}{16},\nonumber\\
\xi_{5} & =\frac{35-4\sqrt{6}-4\sqrt{10}}{48},\nonumber\\
\xi_{6} & =\frac{70-15\sqrt{3}-5\sqrt{7}-3\sqrt{15}}{96}\simeq0.2.
\end{align}
By applying the Kraus operators $K_{0}$ and $K_{1}$ of $\mathcal{E}_{p}$
locally to $\rho_{\Gamma^{M}}$ we obtain the Choi matrix of the $M$-port
simulation $\mathcal{E}_{p}^{M}$, which is
\begin{equation}
\rho_{\mathcal{E}_{p}^{M}}
\begin{pmatrix}
x & 0 & 0 & y\\
0 & \left( 1-p\right) \xi_{M} & 0 & 0\\
0 & 0 & w & 0\\
y & 0 & 0 & z
\end{pmatrix}
,
\end{equation}
where $x:=\frac{1}{2}-\left( 1-p\right) \frac{\xi_{M}}{4}$, $y:=\sqrt
{1-p}\left( \frac{1}{2}-\frac{\xi_{M}}{2}\right) $, $z:=\left( \frac{1
{2}-\frac{\xi_{M}}{4}\right) \left( 1-p\right) $, and $w:=\left( \frac
{1}{2}-\frac{\xi_{M}}{4}\right) p+\frac{\xi_{M}}{4}$. This has to be compared
with the Choi matrix of $\mathcal{E}_{p}$, which i
\begin{equation}
\rho_{\mathcal{E}_{p}}
\begin{pmatrix}
\frac{1}{2} & 0 & 0 & \frac{\sqrt{1-p}}{2}\\
0 & 0 & 0 & 0\\
0 & 0 & \frac{p}{2} & 0\\
\frac{\sqrt{1-p}}{2} & 0 & 0 & \frac{1-p}{2
\end{pmatrix}
.
\end{equation}
Now, consider the Hermitian matrix $J=\rho_{\mathcal{E}_{p}^{M}
-\rho_{\mathcal{E}_{p}}$. If the matrix $\phi=\mathrm{Tr}_{2}\sqrt{J^{\dag
J}=\mathrm{Tr}_{2}\sqrt{JJ^{\dag}}$ is scalar (i.e., both of its
eigenvalues are equal), then the trace distance between the Choi
matrices $||J||$ is
equal to the diamond distance between the channels $\Delta_{M}(p)
~\cite[Proposition~1]{nechita}. After simple algebra we indeed
find
\begin{equation}
\phi=\frac{\xi_{M}}{8}\left[ 2(1-p)+a_{-}+a_{+}\right]
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
,
\end{equation}
where $a_{\pm}=\sqrt{1-p}\sqrt{5\pm4\sqrt{1-p}-p}$. Because $\phi$
is scalar, the condition above is met and the expression of
$\Delta_{M}(p)$ is twice the
(degenerate) eigenvalue of $\phi$, i.e.
\begin{equation}
\Delta_{M}(p)=\frac{\xi_{M}}{4}\left[ 2(1-p)+a_{-}+a_{+}\right] ,
\end{equation}
which simplifies to Eq.~(\ref{Deltadamp}).
\subsection*{Aknowledgements}
This work has been supported by the EPSRC via the `UK Quantum
Communications Hub' (EP/M013472/1) and by the European Commission
via `Continuous Variable Quantum Communications' (CiViQ, Project
ID: 820466).~The authors would like to thank Satoshi Ishizaka, Sam
Braunstein, Seth Lloyd, Gaetana Spedalieri, and Zhi-Wei Wang for
feedback.
|
{
"timestamp": "2019-06-06T02:14:13",
"yymm": "1803",
"arxiv_id": "1803.02834",
"language": "en",
"url": "https://arxiv.org/abs/1803.02834"
}
|
\section{Introduction}
\label{sec:intro}
\subsection{Presentation of the equation and preceding work}
We consider the homogeneous fractional Fokker-Planck Equation
\begin{equation*}\label{eq:FFP}\tag{FFP}
\partial_t f = \Lambda f := \I(f) + \divg\left(E f\right),
\end{equation*}
where $E$ is a given force field with polynomial growth at infinity and \begin{equation*}
\I = \lapfrac \text{ with } \alpha\in (0,2)
\end{equation*}
is the fractional Laplacian. The fractional Laplacian is a generalization of the Laplacian that can be seen as the opposite of a fractional iteration of the positive operator $-\Delta$. It can be defined for any nice function $f$ through its Fourier transform by
\begin{equation}\label{def:I}
\widehat{\I(f)} = -|2\pi\xi|^\alpha \widehat{f}.
\end{equation}
Alternatively, it is also defined up to a constant depending on $\alpha$ and $d$ for sufficiently smooth functions $f$ by the following integral expression (see e.g. \cite[Chapter 1, \S 1] {landkof_foundations_1972})
\begin{equation}\label{eq:I_u_intg1}
\I(f) = \vp\intd \frac{f(y)-f(x)}{|y-x|^{d+\alpha}} \d y,
\end{equation}
where $\vp$ indicates that it is a principal value when $\alpha\geq1$.
It can be seen as the infinitesimal generator of a Levy process. A probabilistic point of view about fractional diffusion can for example be found in \cite{jourdain_nonlinear_2008}. The integral representation can be seen in the perspective of the dynamic associated with this Levy process as it represents the fact that particles will jump from $x$ to $y$ proportionally to the difference of value of $f$, from the high to the low densities, and proportionally to the inverse of a power of the distance. It highlights the non-local behavior of this operator.
It is in our case in competition with the force field $E$. For $\alpha<1$, this force field will be stronger in small scales, resulting in possibly discontinuous solutions (see for example \cite{silvestre_regularity_2005}). We restrict ourselves to a force field with at most polynomial growth at infinity.
We mention that another reason for the recent interest about the factional Laplacian is the fact that it can also be seen as a simplified version of the Boltzmann linearized operator, see for example \cite{desvillettes_smoothness_2005}, \cite{mouhot_rate_2006}, \cite{mischler_stability_2009}, \cite{mischler_stability_2009-1}, \cite{tristani_boltzmann_2016}, \cite{canizo_exponential_2016}, \cite{canizo_rate_2017}, \cite{herau_short_2017}. It was for example used extensively in \cite{imbert_weak_2016} and in \cite{silvestre_new_2016} to retrieve Harnack's inequalities and regularity for the Boltzmann equation without cutoff.
\subsection{Main results}\label{ss:MainResults}
In all this paper, we will denote by $d\in\N^*$ the dimension of the space for the space variable, $\Omega\subset\R^d$ will be an open subset, $\mu$ a measure (or its identification to a Lebesgue measurable function when it is absolutely continuous with respect to the Lebesgue measure) and $m$ a nonnegative weight function which will often be of the form $\weight{x}^k$ for $k\in\R$ where $\weight{x} = \sqrt{1+|x|^2}$. We will often denote by $C$ constants whose exact value have no importance, or write for example $C_a$ when we want to emphasize that the constant depends on $a$, but also use the following notations
\begin{align*}
a\lesssim b &\ \overset{\text{def}}{\ssi}\ \exists C>0,\, a\leq Cb
\\
a\simeq b &\ \overset{\text{def}}{\ssi}\ a\lesssim b \text{ and } b\lesssim a.
\end{align*}
Notice that $f,g$ will usually denote functions of time and space while $u,v$ will usually only depend on the space variable $x$. Moreover, $q = p' := \frac{p}{p-1}$ will denote the Hölder conjugate of $p$ and $a\wedge b := \min(a,b)$.
We will mainly work in weighted Lebesgue spaces denoted by $L^p(m)$ for $p \in [1,\infty]$, associated to the norm
\begin{equation*}
\|u\|_{L^p(m)} \ := \ \|um\|_{L^p}.
\end{equation*}
We also recall the extension of Sobolev Spaces (see \cite{campanato_proprieta_1963}) to fractional order of derivation, which can be defined through the following semi-norms, generalization of the Hölder property to the Lebesgue spaces for $s\in (0,1)$
\begin{equation}\label{def:seminorm}
|u|_{W^{s,p}}^p \ := \ c_{s,d}\iintd \frac{|u(y)-u(x)|^p}{|y-x|^{d+ps}}\d y\d x.
\end{equation}
Those are Banach Spaces for the norm $\|u\|^p_{W^{s,p}} := |u|^p_{W^{s,p}} + \|u\|^p_{L^p}$. When $s\in(1,2)$, the norm becomes $\|u\|^p_{W^{s,p}} := \|\nabla u\|^p_{W^{s-1,p}} + \|u\|^p_{L^p}$. See for example \cite{triebel_theory_1992},\cite{triebel_theory_2010},\cite{mazya_sobolev_2011} or \cite{di_nezza_hitchhikers_2012} for a more complete study of these spaces.\vspace{\baselineskip}
We are interested here in a confining force field with polynomial growth taking the form
\begin{equation}\label{eq:E_ex}
E \ = \ \langle x\rangle^{\gamma-2} x \ = \ \nabla\left(\frac{\langle x\rangle^{\gamma}}{\gamma}\right),
\end{equation}
with $\gamma \in \R$. To simplify the notations, we will sometimes use $\beta := \gamma-2$. The case $E = x = \nabla V(x)$ with $V(x) = \frac{|x|^2}{2}$ is the most studied in the literature (see for example \cite{biler_generalized_2003}, \cite{gentil_levy-fokker-planck_2008}, \cite{gentil_logarithmic_2009}, \cite{tristani_fractional_2015}). In this case the steady state can be computed explicitly and the equation is equivalent up to a scaling to the fractional heat equation (see for example \cite{biler_asymptotics_2001}). Since our method do not use the explicit formula for $E$, we will always assume the following more general hypotheses for a given $\gamma\in\R$.
\paragraph{\textbf{Hypotheses on $E$:}}
\begin{align}\label{hyp_2bis:grad_E}
|\nabla E| &\ \lesssim\ \weight{x}^{\gamma-2}
\\\label{hyp_3:E_confining}
E\cdot x &\ \gtrsim\ \weight{x}^{\gamma-2}|x|^2.
\end{align}
Remark also that the kernel in the definition~\eqref{eq:I_u_intg1} of the fractional Laplacian, $\kappa_\alpha : z \mapsto \frac{c_{\alpha,d}}{|z|^{d+\alpha}}$, could be replaced by any symmetric kernel $\kappa_\alpha$ verifying
\begin{equation*}
\kappa_\alpha(z) \simeq \frac{1}{|z|^{d+\alpha}}.
\end{equation*}
Our first result is about existence and uniqueness of a solution.
\begin{thm}\label{th:existence}
Let $m:=\langle x\rangle^k$ with $ k\in(0,\alpha\wedge1)$. Then there exists $p_\gamma > 1$ such that for all $p\in[1,p_\gamma)$, if $f^\mathrm{in}\in L^p(m)$, there exists a unique solution
\begin{equation*}
f\in C^0(\R_+,L^p(m))
\end{equation*}
to the \eqref{eq:FFP} equation such that $f(0,\cdot) = f^\mathrm{in}$. Moreover, $\Lambda$ is the generator of a $C^0$-semigroup in $L^p(m)$.
\end{thm}
This result generalizes the results obtained by Wei and Tian in \cite{wei_well-posedness_2015}, where the existence was proved for divergence-bounded force fields. The a priori estimates on weighted spaces, from where come the relations between $E$ and $p$, have been already used in the case of the classical Fokker-Planck equation (for example by Gualdani and al in \cite{gualdani_factorization_2013}).
As it can be seen in the proof, to prove the existence of a solution, hypotheses \eqref{hyp_2bis:grad_E} and \eqref{hyp_3:E_confining} can be weakened to the existence of $k\in(0,\alpha\wedge 1)$ and $p>1$ such that
\begin{align}\nonumber
E &\ \in\ W^{1,r}_\mathrm{loc} \cap L^\infty_\mathrm{loc} &&\text{for a given } r>2
\\\nonumber
E\cdot x &\ \geq\ 0
\\\label{eq:varphi}
\varphi_{m,p} &\ := \ \frac{\divg(E)}{q} - E\cdot\frac{\nabla m}{m} \ \leq \ C.
\end{align}
In particular, it implies that we do not need to control $|\nabla E|$ but only $\divg(E)$. Moreover, when $\gamma \leq 2$, \eqref{hyp_3:E_confining} is unnecessary.
Remark that when \eqref{hyp_2bis:grad_E} and \eqref{hyp_3:E_confining} hold, then \eqref{eq:varphi} holds for $\gamma\leq2$ or $p$ smaller than a given $p_\gamma\in (1,+\infty)$ which is such that
\begin{equation}\label{eq:strict_confinement}
\forall p\in(1,p_\gamma),\,\varphi_{m,p} \ \leq \ b\mathds{1}_\Omega - a \langle x\rangle^{\gamma-2},
\end{equation}
for a given $(a,b)\in \R_+^*\times\R$ and a given bounded set $\Omega$. This relation is similar to the Foster-Lyapunov condition for Harris recurrence (see \cite{meyn_stability_1993}, \cite{bakry_rate_2008}, \cite{hairer_asymptotic_2011} and \cite{eberle_quantitative_2016}). When $E$ takes the form \eqref{eq:E_ex}, we can quantify explicitly the value of $p_\gamma = 1 + \frac{k}{d+\gamma-2-k}$.
\begin{thm}\label{th:regu}
Let $m:=\langle x\rangle^k$ with $ k\in(0,\alpha\wedge1)$ and $f\in L^1(m)$ be a solution to the \eqref{eq:FFP} equation. Then there exists $p_\gamma > 1$ such that $f$ is immediately in all $L^{p}(m)$ for $p<p_\gamma$ and, if $\gamma\leq 2$, $f\in L^\infty(m)$.
\end{thm}
There has been some recent interest in the regularity theory for integro-differential equations. In \cite{silvestre_holder_2010}, \cite{silvestre_differentiability_2012}, \cite{schwab_regularity_2016}, it is proved that under some regularity conditions on $E$ and if $f\in L^\infty$ is the solution to \eqref{eq:FFP}, then $f$ is actually Hölder continuous or even more differentiable. However, it is also proved in \cite{silvestre_loss_2013} that there can be some loss of regularity when $E$ is not regular enough. As proved in \cite{chamorro_fractional_2016} for divergence free drifts or in Proposition \ref{prop:regu}, we can still obtain fractional Besov or Sobolev regularity in these cases. Theorem \ref{th:regu} gives in particular the regularization from $L^1$ to $L^\infty$ in the case when $E\in C^1_b$, which then allows to use the theorems cited above.
\begin{thm}\label{th:unicite_equilibre}
Assume $\gamma>2-\alpha$ and $m=\langle x\rangle^k$ with $0\leq k<\alpha\wedge1$. Then there exists $p^*>1$ such that for any $p\in(1,p^*)$, there exists a unique $F\in L^p(m)\cap L^1_+$ of mass $1$ such that
\begin{equation*}
\Lambda F \ = \ 0.
\end{equation*}
\end{thm}
This result generalizes the results obtained by Mischler and Mouhot in \cite{mischler_exponential_2016} and Kavian and Mischler in \cite{kavian_fokker-planck_2015} where it is proved for the classical Laplacian and respectively $\gamma\geq 1$ and $\gamma \leq 1$. It is also close to the result obtained by Mischler and Tristani in \cite{mischler_uniform_2017} where the fractional Laplacian is replaced by integral operators with integrable kernel.
The last and main result is the following rate of convergence towards equilibrium.
\begin{thm}\label{th:cv}
Assume $\gamma>2-\alpha$ and let $m:=\langle x\rangle^k$ with $0\leq k < (\alpha\wedge1)$. Then, if $\gamma\geq 2$, there exists $a>0$ such that for any $p\in[1,p_\gamma)$,
\begin{equation*}
\|f-F\|_{L^p(m)} \ \lesssim \ e^{-at}\|f^\mathrm{in}-F\|_{L^p(m)}.
\end{equation*}
If $\gamma\in(2-\alpha,2)$, there exists $p^*>1$ such that for any $p\in(1,p^*)$ and any $\bar{k}<k$, the following rate holds
\begin{equation*}
\|f-F\|_{L^p(\bar{m})} \ \lesssim \ \weight{t}^{-\frac{k-\bar{k}}{2-\gamma}}\|f^\mathrm{in}-F\|_{L^p(m)},
\end{equation*}
where $\bar{m} = \weight{x}^{\bar{k}}$.
\end{thm}
This result generalizes the one obtained by Wang in \cite{wang_phi-entropy_2014} where, following the techniques of \cite{gentil_levy-fokker-planck_2008}, exponential convergence of the relative entropy is obtained for force fields $E\in C^1_b$ such that $\forall v\in\R^d, v\cdot\nabla E\cdot v\simeq|v|^2$ and the one obtained by Tristani in \cite{tristani_fractional_2015} where exponential convergence towards equilibrium is proved in $L^p(m)$ in the case $E(x)=x$. It is also the natural extension to the fractional case of the results obtained by Kavian and Mischler in \cite{kavian_fokker-planck_2015} and Mouhot and Mischler in \cite{mischler_exponential_2016}, which correspond respectively to the case $\gamma\in(0,1)$ and $\gamma\geq 1$ for the classical Laplacian. The reason of the lower bound on $\gamma > 2 - \alpha$ is due to the strong nonlocal behavior of the fractional Laplacian which seems to compensate the confining effect of the force field.\vspace{\baselineskip}
The paper is organized as follows. The second section proves some properties of the fractional Laplacian and of the operator $\Lambda$ which will be useful for the various results of the paper.
Section~\ref{sec:existence} proves the existence and uniqueness in the weighted $L^p(m)$ spaces for $p\in(1,2)$. We first create a solution for an approximated problem and then use a priori estimates and compactness properties to obtain a solution to the original problem.
Following the ideas of Nash in \cite{nash_continuity_1958}, section~\ref{sec:gain} of this article generalizes the regularization property of the semigroup associated to the \eqref{eq:FFP} equation as established in \cite{tristani_fractional_2015}. Moreover, a gain of integrability as well as a gain of positivity are also proved, which are useful to deal with convergence without any $L^\infty$ bound.
In section~\ref{sec:steady}, the existence of a stationary state is proved by using an adequate splitting of the operator. It follows the general idea of writing operators as a regularizing part and a dissipative part, as explained in \cite{gualdani_factorization_2013}. We then prove a weak and strong maximum principle and deduce the uniqueness of the equilibrium from the Krein-Rutman Theorem.
The fifth section deals with polynomial convergence when $E$ is not confining enough to create a spectral gap. It uses techniques inspired from \cite{bakry_rate_2008} by using both Foster-Lyapunov estimates introduced by Meyn and Tweedie in \cite{meyn_stability_1993} and a local Poincaré inequality. It proves the first part of Theorem \ref{th:cv}.
Last section is devoted to the proof of the exponential convergence when $E$ is strongly confining (i.e. $\gamma>2$) and follows a different approach as it replaces the use of the Poincaré inequality by the gain of positivity property, following the work of Hairer and Mattingly in \cite{hairer_yet_2011}. It proves the second part of Theorem \ref{th:cv}.
\paragraph{\bf Acknowledgments:} I wish to acknowledge the help provided by my supervisor, Mr. Stephane Mischler. He gave me very useful advice and I used a lot his course on evolution PDEs \cite{mischler_introduction_2015}. I would also like to thank Ms. Isabelle Tristani and the members of the CEREMADE for their advice.
\section{Main inequalities}
\subsection{Preliminary results about fractional Laplacian}
We first recall the standard notations that we will use on this paper. We will denote by $\B(E,F)$ the space of continuous linear mappings from $E$ to $F$, by $u_+ := \max(u,0)$ the positive part of $u$. Moreover, we will identify bounded measures on measurable sets of $\R^d$ with bounded radon measures $\mu \in \mathcal{M}(\Omega) := C_0(\Omega)'$ and write
\begin{align*}
\int f\mu & \ := \ \int u(x)\mu(\d x), & \mu(A) & \ := \ \int_A \mu,
\end{align*}
for any $\mu$-measurable function $u$ and $\mu$-measurable set $A$. We will write the mass of a measure $\langle u\rangle_{\R^d} := \intd u$. We also recall that $\mathcal{D}(\Omega) = C^\infty_c(\Omega)$ and $\mathcal{D}'(\Omega)$ is the space of distributions on $\Omega$. Moreover, we will not write $\Omega$ when $\Omega = \R^d$.
Notice that in order to simplify the computations, we will use the following definition for the power of a vector, $x^a := |x|^{a-1}x$ for any $a\in \R$, and we will use a short notation to simplify the writing of the integrals,
\begin{align*}
\ka &:= \kappa_\alpha(x_*-x) & u &:= u(x) & u_* &:= u(x_*),
\end{align*} where $x_*$ denote the first variable of integration. We can write for example
\begin{align*}
\iint F(u,u_*) = \iint F(u(x),u(x_*)) \d x_* \d x.
\end{align*}
With these notations and since $\alpha\in(0,2)$, for sufficiently smooth and decaying functions $u$, we can write the fractional Laplacian as a principal value
\begin{equation*}
\I(u) \ = \ \vp\left(\intd \ka(u_*-u)\right) \ = \ \lim\limits_{\eps\to0} c_{\alpha,d}\int_{|x-y|>\eps} \frac{u(y)-u(x)}{|y-x|^{d+\alpha}} \d y.
\end{equation*}
Remark that the principal value can be removed when $\alpha\in(0,1)$. An other useful expression is
\begin{align}
\I(u) \ &= \ \intd \frac{u(y)-u(x)-(y-x)\nabla u(x)}{|z|^{d+\alpha}} \d z,
\label{eq:I_u_intg2}
\end{align}
By duality, it can also be defined on more general spaces of tempered distributions with a growth smaller than $|x|^{\alpha}$ at infinity by the formula $\langle \I(u),\varphi\rangle_{\mathcal{D}',\mathcal{D}} := \langle u,\I(\varphi)\rangle_{\I(\mathcal{D})',\I(\mathcal{D})}$. In particular, we will mostly use the fractional Laplacian of weight functions of the form $m(x)=\weight{x}^k$ with $k<\alpha$.
Following the model of the Laplacian, we define for $p>1$
\begin{align}
\G(u,v) \ &:= \ \intd \frac{\ka}{2}\ (u_*-u)(v_*-v)
\label{def:prod_grad}\\
\Dp{p}(u) \ &:= \ \G(u,u^{p-1}) \ \geq \ 0.
\label{def:Gp}
\end{align}
The first quantity can be seen as a generalization of $\nabla u\cdot \nabla v$. It is known as the "Carré du Champs" operator in Probabilities. The second can be seen as a generalization of $\left|\nabla |u|^{p/2}\right|^2$.
The quantity \eqref{def:prod_grad} comes naturally when considering the fractional Laplacian of a product of (sufficiently smooth) functions, since the following formula holds
\begin{equation}\label{eq:product}
\I(uv) \ = \ u\I(v)+v\I(u)+2\G(u,v).
\end{equation}
Moreover, we have the following integration by parts formula
\begin{equation}\label{eq:ipp}
\intd u \I(v) \ = \ \intd \I(u) v \ = \ - \intd \G(u,v).
\end{equation}
So that in particular, by definition \eqref{def:Gp}
\begin{equation}\label{eq:I_dissip}
\intd \I(u)u^{p-1} \ = \ - \intd \Dp{p}(u) \ \leq \ 0.
\end{equation}
Remark that these relations also holds when replacing $\kappa_\alpha(x-x_*)$ by a general symmetric kernel $\kappa(x,x_*)$.
It will be useful to remark that the following quantities are equivalent.
\begin{prop}\label{prop:Gp}
Let $u$ be such that $\Dp{p}(u)$ is bounded for a given $p\in(1,\infty)$. Then
\begin{align}\label{eq:Gp_Dp}
\Dp{p}(u) \ &\simeq \ \frac{1}{p}\I(|u|^p)-u^{p-1}\I(u)
\\\label{eq:Gp_Dpp}
&\simeq \ \frac{1}{q}\I(|u|^p)-u\I(u^{p-1})
\\\label{eq:Gp_grad}
&\simeq \ \intd \ka |u_*^{p/2}-u^{p/2}|^2,
\end{align}
where we recall that $q=p'$ and $a\simeq b$ means here that $a/b$ is bounded by above and below by positive constants depending only on $p$.
\end{prop}
\begin{demo}[Proposition~\ref{prop:Gp}]
For the first line, we remark that
\begin{align*}
\Dp{p}(u) &= \iintd \ka (u_*-u)(u^{p-1}_* - u^{p-1})
\\
&= \iintd \ka\,d_1(u_*/u) |u|^p
\\
\frac{1}{p}\I(|u|^p)-u^{p-1}\I(u) &= \frac{1}{p}\iintd \ka (|u_*|^p-|u|^p - p u^{p-1}(u_* - u))
\\
&= \frac{1}{p}\iintd \ka\,d_2(u_*/u) |u|^p,
\end{align*}
where we recall that $u^p = |u|^{p-1}u$ and we defined for any $z\in\R$,
\begin{align*}
d_1(z) &= (z-1)(z^{p-1}-1) \geq 0
\\
d_2(z) &= |z|^p - 1 - p (z-1) \geq 0.
\end{align*}
Then we remark that $d_1/d_2$ is a bounded positive function since it is continuous on $\R\backslash\{1\}$, converges to $1$ when $|z|\to\infty$ and to $2/p$ when $z\to 1$. Therefore, $d_1\simeq d_2$ and it implies \eqref{eq:Gp_Dp}. The other inequalities are treated in the same way.
\end{demo}
Another useful result is the estimation of the growth of the fractional Laplacian of weight functions.
\begin{prop}[Fractional Derivation of weight functions]\label{prop_I_m2}
Let $k\in(0,\alpha\wedge1)$ and $m : x\mapsto\weight{x}^k$ defined for $x\in\R^d$. Then, the following inequality holds
\begin{equation}\label{eq:I_m2}
\left|\I(m)\right| \ \leq \ \frac{C}{\weight{x}^{\alpha-k}},
\end{equation}
where $C$ is of the form $\frac{C_k\,\omega_d}{(\alpha-k)(2-\alpha)}$. Moreover, when $\alpha<1$
\begin{align}\label{eq:grad_m}
\grad{\alpha}m \ &\leq \ \frac{C_{\alpha,k}}{\weight{x}^{\alpha-k}},
\\\label{eq:grad_m2}
\grad{\alpha}\left(m^{-1}\right) \ &\leq \ \frac{C_{\alpha,|k|}}{\weight{x}^{\alpha}},
\end{align}
where $C_{\alpha,k}$ is of the form $\frac{C_k\,\omega_d}{(\alpha-k)(1-\alpha)}$ and $\grad{\alpha}$ is defined by
\begin{equation}\label{def:grad}
\grad{\alpha}u \ := \ \intd \ka\,|u_*-u|.
\end{equation}
\end{prop}
\begin{demo}[Proposition~\ref{prop_I_m2}]
We first look at the case $\alpha\in(0,1)$ and then at the case $\alpha\in(0,2)$ which works only for $\I(m)$.
\step{1. Case $\alpha\in(0,1)$}
Let $x\in\R^d$ and $R>1$. We split $\grad{\alpha}$ into two parts
\begin{equation*}
\grad{\alpha}m \ \leq \ \int_{|x-y|>R} \frac{|m(x)-m(y)|}{|x-y|^{d+\alpha}}\d y + \int_{|x-y|\leq R} \frac{|m(x)-m(y)|}{|x-y|^{d+\alpha}}\d y \ =: \ \mathcal{I}_1 + \mathcal{I}_2.
\end{equation*}
For the first part, we remark that since $k\in(0,1)$ and $\forall y\in\R,|\nabla\weight{y}|\leq 1$, we obtain
\begin{align*}
|\weight{x}^k-\weight{y}^k| \ \leq \ |\weight{x}-\weight{y}|^k \ \leq \ |x-y|^k.
\end{align*}
It leads to
\begin{equation*}
\mathcal{I}_1 \ \leq \ \int_{|z|>R} \frac{\d z}{|z|^{d+\alpha-k}}
\ \leq \ \frac{\omega_d}{(\alpha-k)R^{\alpha-k}}.
\end{equation*}
$\bullet$ If $|x|\geq1$, we take $R := |x|/2$. Then $|x|^{-1}\leq \sqrt{2}\weight{x}^{-1}$, from what we deduce
\begin{equation*}
\mathcal{I}_1 \ \leq \ \frac{C\,\omega_d}{(\alpha-k)} \frac{1}{\weight{x}^{\alpha-k}}.
\end{equation*}
Let $y\in\R^d$ be such that $|x-y|<|x|/2$. For $w\in [x,y] \subset \R^d$, we have $|w| \geq |x| - |x-w| \geq |x|/2$. Thus, we obtain
\begin{align}
|m(x)-m(y)| \ & \leq \ |x-y|\sup_{[x,y]}|\nabla m| \label{eq:DL}
\\
& \leq \ |x-y|\sup_{w\in[x,y]}|k\weight{w}^{k-2}w|
\nonumber\\
& \leq \ 2^{1-k}k\weight{x}^{k-1}|x-y|,
\nonumber
\end{align}
where we used $|x|\leq\weight{x}$ and $\weight{x/2} \geq \weight{x}/2$. It implies the following upper bound
\begin{equation*}
\mathcal{I}_2 \ \leq \ C\,\weight{x}^{k-1} \int_{|z|\leq |x|/2} \frac{\d z}{|z|^{d+\alpha-1}} \ \leq \ \frac{C\,\omega_d}{1-\alpha} \weight{x}^{k-\alpha}.
\end{equation*}
$\bullet$ If $|x|\leq1$, we take $R := 1$ and we deduce
\begin{equation*}
\mathcal{I}_1 \ \leq \ \frac{\omega_d}{(\alpha-k)}.
\end{equation*}
Moreover, as $k\weight{x}^{k-1}\leq 1$, \eqref{eq:DL} gives us
\begin{equation*}
|m(x)-m(y)| \ \leq \ |x-y|.
\end{equation*}
Therefore
\begin{equation*}
\mathcal{I}_2 \ \leq \ \int_{|z|\leq 1} \frac{\d z}{|z|^{d+\alpha-1}} \ \leq \ \frac{\omega_d}{1-\alpha}.
\end{equation*}
$\bullet$ We end the proof of \eqref{eq:grad_m} by gathering the two parts together. Since $m\geq 1$, we get $\eqref{eq:grad_m2}$ by remarking that
\begin{equation*}
\grad{\alpha}(m^{-1}) \ = \ \intd \ka \left|\frac{m_*-m}{m_*m}\right| \ \leq \ \frac{\grad{\alpha}m}{m}.
\end{equation*}
\step{2. Proof of \eqref{eq:I_m2}}
We use the integral representation \eqref{eq:I_u_intg2} to change $\mathcal{I}_2$ by
\begin{equation*}
\mathcal{I}_2 = \int_{|x-y|\leq R} \frac{|m(x)-m(y)-(x-y)\cdot\nabla m(x)|}{|x-y|^{d+\alpha}}\d y.
\end{equation*}
Then \eqref{eq:DL} is replaced by a second order Taylor inequality, which gives
\begin{equation*}
|m(x)-m(y)-(x-y)\cdot\nabla m(x)| \ \leq \ C_k\weight{x}^{k-2}|x-y|^2.
\end{equation*}
The other parts of the proof are similar to the step $1$.
\end{demo}
\subsection{Inequalities for the generator of the semigroup}\label{subseq:apriori}
To get existence, uniqueness and additional gains of weight and regularity on the solutions to the \eqref{eq:FFP} equation, the main inequalities are given in the following
\begin{prop}\label{prop:estim}
Let $m=\weight{x}^k$ with $k\in(0,1)$ and $u\in L^p(m\weight{x}^{(\gamma-2)_+})$. If $k<\alpha<1$, the following holds
\begin{equation}\label{eq:estim_a_priori_1}
\intd \Lambda(u)u^{p-1}m^p + \intd \Dp{p}(um) \ \leq \ \intd |u|^pm^p\left(\frac{C_k}{\weight{x}^{\alpha-k}}+\varphi_{m,p}\right),
\end{equation}
where $\varphi_{m,p}$ is defined by \eqref{eq:varphi} and $\Dp{p}\geq 0$ is defined by \eqref{def:Gp}. If $kp< (\alpha\wedge1)$, we also have
\begin{equation}\label{eq:estim_a_priori_2}
\intd \Lambda(u)u^{p-1}m^p + C_p\intd \Dp{p}(um) \ \leq \ \intd |u|^pm^p\left(\frac{C_{k,p}}{\weight{x}^\alpha}+\varphi_{m,p}\right).
\end{equation}
\end{prop}
\paragraph{\textbf{Remarks:}} In particular, as already pointed out in introduction, $\varphi_{m,p}$ is always bounded above when $\gamma\leq 2$. When $\gamma>2$, there exists $p_\gamma>1$ such that $\varphi_{m,p}$ is bounded for any $p\in(1,p_\gamma)$. Moreover, in this case, there exists $(a,b)\in\R_+^*\times\R$ such that
\begin{equation*}
\varphi_{m,p} \leq b-a\weight{x}^{\gamma-2}.
\end{equation*}
Inequality~\eqref{eq:estim_a_priori_2} is more restrictive on $k$ since it needs $k<\alpha/p$, but it has the advantage to work for all $\alpha\in(0,2)$ and to give a second term with a smaller weight.
\begin{lem}\label{lem:Jmp_estimate}
Let $m=\weight{x}^k$ with $|k|<\alpha\leq 1$ and $u\in L^p(m\weight{x}^{(k-\alpha)/p})$. Then the following inequality holds true
\begin{eqnarray}\label{eq:Jmp_estimate}
\left|\intd (\I(mu)-m\I(u))(um)^{p-1}\right| & \leq & C_{k} \left\|u\right\|_{L^p(m\weight{x}^{(k-\alpha)/p})}^p.
\end{eqnarray}
\end{lem}
\begin{demo}[Lemma~\ref{lem:Jmp_estimate}]
Using the integral definition \eqref{eq:I_u_intg1} of $\I$, we have
\begin{align*}
\intd j_m(u)v \ & = \ \iintd \ka\, \left((u_*m_*-um)-u(m_*-m)\right)v
\\
& = \ \iintd \ka\,\frac{m_*-m}{m_*}(u_*m_*)v.
\end{align*}
Thus, by Hölder's inequality, we get
\begin{equation*}
\left|\intd j_m(u)v\right| \ \leq \ \left(\iintd \ka\, \frac{|m_*-m|}{m_*}|u_*m_*|^p\right)^\frac{1}{p} \left(\iintd \ka\, \frac{|m_*-m|}{m_*}|v|^q\right)^\frac{1}{q}.
\end{equation*}
By the fact that $\frac{|m_*-m|}{m_*} = \frac{|m_*^{-1}-m^{-1}|}{m^{-1}}$ and exchanging $x$ and $x_*$ in the first integral, we obtain
\begin{equation*}
\left|\intd j_m(u)v\right| \ \leq \ \left(\intd \frac{\grad{\alpha}m}{m}|um|^p\right)^\frac{1}{p} \left(\intd \frac{\grad{\alpha}(m^{-1})}{m^{-1}}|v|^q\right)^\frac{1}{q}.
\end{equation*}
where $\grad{\alpha}$ is defined by \eqref{def:grad}. In particular, if $m=\weight{x}^k$ with $|k|<\alpha$, we obtain from Proposition~\ref{prop_I_m2}
\begin{equation*}
\left|\intd j_m(u)v\right| \ \leq \ C_k \left\|u\right\|_{L^p(m\weight{x}^{-\alpha/p})} \left\|v\right\|_{L^q(\weight{x}^{(k-\alpha)/q})},
\end{equation*}
which implies \eqref{eq:Jmp_estimate} by taking $v=(um)^{p-1}$.
\end{demo}
\begin{demo}[Proposition~\ref{prop:estim}]
Let $\Phi = \frac{|\cdot|^p}{p}$ and $u\in C_c^\infty$. Then, by definition
\begin{equation*}
\intd\Lambda(u)\Phi'(u)m^p\ = \ \intd \I(u)\Phi'(u)m^p+\divg(Eu)\Phi'(u)m^p.
\end{equation*}
Let first focus on the term containing the force field $E$.
We expand the divergence of the product, use the fact that $\Phi'(u)\nabla u = \nabla \Phi(u)$ and integrate by parts the second term to find
\begin{equation*}
\intd \divg(E u)\Phi'(u)m^p \ = \ \intd \divg(E)(u\Phi'(u)-\Phi(u))m^p - \Phi(u) E\cdot\nabla m^p.
\end{equation*}
By definition of $\Phi$, we obtain
\begin{equation}\label{eq:estim_a_priori_E}
\intd \divg(E u)u^{p-1}m^p \ = \ \intd |u|^pm^p \varphi_{m,p},
\end{equation}
where $\varphi_{m,p}$ is given by \eqref{eq:varphi}. Let now look at the term containing $\I$. By using \eqref{eq:I_dissip}, we have
\begin{equation*}
\intd \I(u)u^{p-1}m^p \ = \ - \intd \Dp{p}(um) + \intd (\I(mu)-m\I(u))(um)^{p-1}.
\end{equation*}
By \eqref{eq:Jmp_estimate}, when $|k|<\alpha \leq 1$, we deduce the following inequality for $\I$
\begin{equation}\label{eq:estim_a_priori_I}
\intd \I(u)u^{p-1}m^p \ \leq \ - \intd \Dp{p}(um) + C_k \intd |u|^pm^p\weight{x}^{k-\alpha}.
\end{equation}
For $\alpha\in (0,2)$, when $kp\in(0,\alpha\wedge1)$, we recall that by relation~\eqref{eq:Gp_Dp},
\begin{equation*}
\Dp{p}(u) \ \simeq \ \frac{1}{p}\I(|u|^p)-\I(u)u^{p-1}.
\end{equation*}
Hence, using the fractional integration by parts formula \eqref{eq:ipp}, we get
\begin{equation*}
\intd \I(u)u^{p-1}m^p \ = \ -C_p\intd \Dp{p}(u)m^p + \frac{1}{p}\intd |u|^p\I(m^p).
\end{equation*}
By formula~\eqref{eq:I_m2}, it leads to
\begin{equation}\label{eq:estim_a_priori_I_2_tmp}
\intd \I(u)u^{p-1}m^p + C_p\intd \Dp{p}(u)m^p \ \leq \ C_k \intd |u|^pm^p\weight{x}^{-\alpha}.
\end{equation}
Now we remark that, by relation~\eqref{eq:Gp_grad}
\begin{align*}
\intd \Dp{p}(um) \ &\simeq \ \iintd |(um)_*^{p/2}-(um)^{p/2}|^2
\\
&\leq \ 2\iintd |u_*^{p/2}-u^{p/2}|^2m_*^p + |m_*^{p/2}-m^{p/2}|^2|u|^p
\\
&\lesssim \ \intd \Dp{p}(u)m^p + \Dp{p}(m)|u|^p.
\end{align*}
Moreover, since $\Dp{p}(m) \simeq \G(m^{p/2},m^{p/2})$, by the bound \eqref{eq:ipp}, we obtain
\begin{align*}
2\Dp{p}(m) \ \lesssim \I(m^p) + m^{p/2}\left|\I(m^{p/2})\right| \lesssim \frac{m^p}{\weight{x}^\alpha},
\end{align*}
where we used \eqref{eq:I_m2} since $kp<\alpha$. Therefore, inequality~\eqref{eq:estim_a_priori_I_2_tmp} becomes
\begin{equation}\label{eq:estim_a_priori_I_2}
\intd \I(u)u^{p-1}m^p + C_p\intd \Dp{p}(um) \ \leq \ C_{k,p} \intd |u|^pm^p\weight{x}^{-\alpha}.
\end{equation}
We conclude that \eqref{eq:estim_a_priori_1} and \eqref{eq:estim_a_priori_2} hold by combining the inequality for the part with $E$, equation~\eqref{eq:estim_a_priori_E} with the inequalities for the parts with $\I$, \eqref{eq:estim_a_priori_I} and \eqref{eq:estim_a_priori_I_2}.
All these manipulation can be justified by taking $u_n = \chi_n(\rho_n*u) \to u$ where $\chi_n\in C^\infty_c$ is a cutoff function and $\rho_n\in C^\infty_c$ an approximation of $\delta_0$.
The main technical point is to obtain an estimate on the following commutator
\begin{equation*}
r_n(u) \ := \ (E\cdot\nabla u)*\rho_n - E\cdot\nabla(u*\rho_n).
\end{equation*}
In the spirit of DiPerna-Lions commutator estimate (see \cite{diperna_ordinary_1989}) and Lemma~\ref{lem:Jmp_estimate}, we obtain
\begin{equation*}
r_n(u) \ \underset{n\to+\infty}{\longrightarrow} 0 \ \mathrm{\ in\ } L^p(m\weight{x}^{-(\gamma-2)_+/p}),
\end{equation*}
which ends the proof.
\end{demo}
\section{Well-posedness}\label{sec:existence}
This section is devoted to the proof of the part of Theorem~\ref{th:existence} concerning existence and uniqueness of a continuous semigroup. In order to prove the existence of a solution to the \eqref{eq:FFP} equation, we use a viscosity approximation of the equation and a truncation of $E$ and $\I$. We first prove the existence for the approximated problem in $L^2(M)$. We can identify the dual of $V:= H^1(M) = \{u\in L^2(M),\nabla u\in L^2(M)\}$ to $H^{-1}(M)$ by defining $\langle f,g\rangle_{V',V} = \langle fM,gM\rangle_{H^{-1},H^1} = \intd fgM^2$. Moreover, $L^2(M)$ is a Hilbert space for the scalar product $\langle f,g\rangle_{L^2(M)} = \intd fgM^2$. Remark that in the case $\alpha>1$, proving the existence is simpler as the divergence operator is bounded in $H^\alpha$, so that we do not need to use a viscosity approximation.
\begin{lem}[Viscosity Approximation]\label{lem_viscosity}
Let $M:=\weight{x}^k$ with $k\in\R$ and for $\eps\in(0,1)$ define $\kappa_\alpha^\eps(x) := \kappa_\alpha(x)\mathds{1}_{\{\eps<|x|<1/\eps\}}$ and $\I_\eps(u) := \intd \ka^\eps\left(u_*-u\right)$. Then, there exists a unique solution in
\begin{equation*}
C^0([0,T],L^2(M)) \ \cap \ L^2((0,T),H^1(M)) \ \cap \ H^1((0,T),H^{-1}(M)),
\end{equation*}
to the problem
\begin{equation}
\partial_tf\ =\ \Lambda_\eps f\ =\ \eps\Delta f + \I_\eps(f) + \divg(E_\eps f),
\end{equation}
with $f(0,\cdot)=f^{\mathrm{in}}\in L^2(M)$, $E_\eps\in L^\infty$ and \begin{equation*}
\left(\divg(E_\eps)-E_\eps\cdot\dfrac{\nabla M^2}{M^2}\right)_+\in L^\infty.
\end{equation*}
\end{lem}
\begin{demo}[Lemma~\ref{lem_viscosity}]
The result is an application of J.L.Lions Theorem (see for example \cite[Théorème X.9]{brezis_analyse_2005}). We thus prove that the hypotheses of this theorem hold.
\step{1. Continuity of $\Lambda_\eps$}
Let $(f,g)\in H^1(M)^2$. Then
{\small\begin{align*}
\langle\Lambda_\eps f,g\rangle_{V',V} & = \intd-\eps\nabla f \cdot\nabla(g M^2) + \I_\eps(f)g M^2 + \divg(E_\eps f)g M^2
\\
& = \intd\left(-\eps\nabla f\cdot\nabla g - \eps g\nabla f\cdot\dfrac{\nabla M^2}{M^2} + \I_\eps(f)g - E_\eps f\left(\nabla g+g\dfrac{\nabla M^2}{M^2}\right)\right) M^2.
\end{align*}}Since $\kappa_\alpha^\eps\in L^1$, we can write $I_\eps(f) = \kappa_\alpha^\eps * f - K_\eps f$ where $K_\eps = \|\kappa_\alpha^\eps\|_{L^1}$. Using Peetre's inequality which tells that
\begin{align*}
\weight{x+y} \ & \leq \ \sqrt{2}\weight{x}\weight{y},
\end{align*}
and the fact that $\kappa_\alpha^\eps$ is compactly supported, we get after a short computation
\begin{equation}\label{eq:bound_I_eps}
\left|\intd \I_\eps(f)gM^2\right|
\ \leq \ C_\eps K_\eps \|f\|_{L^2(M)}\|g\|_{L^2(M)}.
\end{equation}
Thus, using the Cauchy-Schwartz inequality, there exists $C_\eps>0$ such that
\begin{equation*}
|\langle\Lambda_\eps f,g\rangle_{V',V}| \ \leq \ \left(C_k(\eps+\|E_\eps\|_{L^\infty})+C_\eps K_\eps\right) \|f\|_{H^1(M)}\|g\|_{H^1(M)},
\end{equation*}
where we used $|\nabla M^2| \leq 2|k|M^2$. It proves that $\Lambda_\eps\in\B(V,V')$.
\step{2}
For $f\in H^1(M)$, using \eqref{eq:bound_I_eps} and the a priori estimate \eqref{eq:estim_a_priori_E}, we get
\begin{align*}
\langle\Lambda_\eps f,f\rangle_{V',V} & = \intd\left(\I_\eps(f)f -\eps|\nabla f|^2 + f^2 \left(\divg(E_\eps)-E_\eps\cdot\dfrac{\nabla M^2}{M^2}- \eps \dfrac{\Delta M^2}{2M^2}\right)\right) M^2
\\
&\leq -\eps\|f\|_{H^1(M)}^2 + C_{k,\eps,E_\eps,\kappa_\alpha^\eps} \|f\|_{L^2(M)}^2,
\end{align*}
where we used $|\nabla M^2| \leq 2|k|M^2$ and $|\Delta M^2| \leq 6|k|M^2$. Therefore, we can apply J.L.Lions Theorem.
\end{demo}
To get results in the good spaces, we will use the following injection that is a straightforward application of Hölder's inequality and the density of $C^\infty_c$ in $L^p$.
\begin{lem}\label{lem_inclusion}
Let $(p,q)\in[1,+\infty]^2$ and $(l,k)\in\R^2$ such that $p\leq q$ and $(l-k)>d\left(\frac1p-\frac1q\right)$.
Let $M=\weight{x}^l$ and $m=\weight{x}^k$, then
\begin{equation*}
L^q(M) \hookrightarrow L^p(m),
\end{equation*}
with dense and continuous embedding. In particular, if $l>k+\frac d2$ and $p\in[1,2]$, we have the following embedding $L^2(M) \hookrightarrow L^p(m)$.
\end{lem}
We now can prove the existence of a weak solution by letting $\eps\to 0$.
\begin{lem}\label{lem:existence}
Let $m=\weight{x}^k$ with $k\in(0,\alpha\wedge 1)$ and $p\in(1,p_\gamma)$ as defined by \eqref{eq:strict_confinement} (or $p>1$ if $\gamma\leq 2$). Then there exists a unique weak solution $f\in L^\infty_{\mathrm{loc}}(\R_+,L^p(m))$ to the \eqref{eq:FFP} equation.
\end{lem}
\begin{demo}[Lemma~\ref{lem:existence}]
We prove first existence of a solution in $L^p(m)$ by using the approximation in $L^2(M)$ and then we use it to prove existence in $L^1(m)$.
\step{1. Existence in $L^p(m)$ for $p>1$}
Assume that $f^\mathrm{in}\in L^p(m)$ for $p\in(1,2]$. Then, by Lemma~\ref{lem_inclusion}, there exists a family of functions $f_\eps^\mathrm{in}\in L^2(M)$ such that
\begin{equation*}
f_\eps^\mathrm{in} \overset{L^p(m)}{\underset{\eps\to 0}{\longrightarrow}} f^\mathrm{in}.
\end{equation*}
For a fixed $\eps>0$, let $\chi_\eps\in C^\infty_c$ be a radial function such that $\chi_\eps(x) = \tilde{\chi}_\eps(|x|)$ where $\tilde{\chi}_\eps$ is a decreasing function and $\mathds{1}_{\oball(0,1/\eps)} \leq \chi_\eps \leq \mathds{1}_{\oball(0,2/\eps)}$. Let $f_\eps\in C([0,T],L^2(M))$ be a solution of $\partial_t f_\eps = \Lambda_\eps f_\eps$ as given by Lemma~\ref{lem_viscosity}, with $E_\eps = E\chi_\eps$. For such a $E_\eps$, we have indeed $\divg(E_\eps)-E_\eps\cdot\dfrac{\nabla M^2}{M^2}$ bounded above because of the fact that $E\in L^\infty_\mathrm{loc}$ and $\divg(E)_+\in L^\infty_\mathrm{loc}$.
Let $\rho\in\mathcal{D}(\R^{d+1},\R_+)$ be such that $\int\rho = 1$ and $\mathrm{supp}(\rho)\subset(-1,0)\times\oball(0,1)$ so that $\rho_n(t,x):=n^{d+1}\rho(nt,n^dx)$ is an approximation of identity. The fractional Laplacian commutes with the convolution by smooth functions (which is an immediate property by using its Fourier definition \eqref{def:I}), thus the regularized function defined by $f_{\eps,n} := f_\eps*\rho_n\in C^\infty(\R_+\times\R^d)\cap L^2(M)$ verifies in the classical sense the equation
\[
\partial_t f_{\eps,n} = \Lambda_\eps f_{\eps,n} + r_n,
\]
where
\[
r_n = (E_\eps\cdot\nabla f_\eps)*\rho_n-E_\eps\cdot\nabla f_{\eps,n}.
\]
As proved in \cite[Lemma~II.1]{diperna_ordinary_1989}, since $E_\eps\in L^1((0,T),W^{1,r}_\mathrm{loc})$ for $r>1$ such that $\frac{1}{p}=\frac{1}{2}+\frac{1}{r}$ and $f_\eps\in L^\infty((0,T),L^2_\mathrm{loc})$, it holds
\[
r_n \underset{n\to\infty}{\longrightarrow} 0 \mathrm{\ in \ } L^1((0,T),L^p_{\mathrm{loc}}).
\]
Moreover the convergence also holds in $L^1((0,T),L^p(m))$ because $E_\eps$ is compactly supported. Using inequality \eqref{eq:estim_a_priori_1} or \eqref{eq:estim_a_priori_2} for $\I=\I_\eps$ and the fact that $\varphi_{m,p}$ is bounded from above, we obtain
\begin{equation*}
\partial_t\left(\intd \frac{|f_{\eps,n}|^p}{p}m^p\right) \ \leq \ \intd |f_{\eps,n}|^pm^p \left(C_k +\frac{\divg(E_\eps)}{q}- k\frac{E_\eps\cdot x}{\weight{x}^2}\right) + |f_{\eps,n}|^{p-1}|r_n|m^p.
\end{equation*}
For the part containing $E_\eps$, we have
\begin{equation*}
\frac{\divg(E_\eps)}{q} - k\frac{E_\eps\cdot x}{\weight{x}^2} \ = \ \left(\frac{\divg(E)}{q} - k\frac{E\cdot x}{\weight{x}^2}\right)\chi_\eps + \frac{E\cdot\nabla(\chi_\eps)}{q}.
\end{equation*}
By hypothesis, the first term is bounded above and the second term is negative since
\begin{equation}
E\cdot\nabla(\chi_\eps) \ = \ E\cdot\dfrac{x}{|x|}\tilde{\chi}'(|x|) \ \leq \ 0.
\end{equation}
Using Hölder's inequality to control the error term, we obtain
\begin{align*}
\partial_t\left(\intd \frac{|f_{\eps,n}|^p}{p}m^p\right) \ &\leq \ C\intd |f_{\eps,n}|^pm^p + \|r_n\|_{L^p(m)} \left(\intd |f_{\eps,n}|^pm^p\right)^{1/p'}
\\
&\leq \ \left(C + \|r_n\|_{L^p(m)}\right)\intd |f_{\eps,n}|^pm^p + \|r_n\|_{L^p(m)},
\end{align*}
where we used the fact that for $\forall x\geq 0,\ x^{1/p'}\leq 1+x$ since $p'\in[2,\infty)$. Grönwall's inequality gives
\begin{align*}
\|f_{\eps,n}\|_{L^p(m)}^p \ &\leq \ e^{CT+p\|r_n\|_{L^1((0,T),L^p(m))}} \left(\|f_{\eps,n}^\mathrm{in}\|_{L^p(m)}^p + p\|r_n\|_{L^1((0,T),L^p(m))}\right).
\end{align*}
Passing to the limit in $n$, as $f_{\eps,n}\to f_\eps$ in $L^p(m)$, the error term cancels, hence
\begin{equation*}
\|f_\eps\|_{L^p(m)}^p \ \leq \ e^{CT} \|f_\eps^\mathrm{in}\|_{L^p(m)}^p.
\end{equation*}
Thus, up to a subsequence, it converges in $\mathcal{D}'([0,T]\times\R^d)$ to $f\in L^\infty([0,T],L^p(m))$.
Let $\varphi\in\mathcal{D}([0,T]\times\R^d)$. Then $\ka^\eps|\varphi_*-\varphi| \leq \ka|\varphi_*-\varphi|$ which is integrable, thus $\I_\eps(\varphi)$ converges to $\I(\varphi)$ by the Lebesgue dominated convergence Theorem. Therefore, we have
\begin{equation*}
\langle f_\eps,\I_\eps(\varphi)\rangle_{\mathcal{D}',\mathcal{D}}
\ \underset{\eps\to 0}{\longrightarrow} \ \langle f,\I(\varphi)\rangle.
\end{equation*}
It implies that $\I_\eps(f_\eps)\underset{\eps\to0}{\longrightarrow} \I(f)$ in $\mathcal{D}'([0,T]\times\R^d)$. We can also easily check that $\divg(E_\eps f_\eps)\underset{\eps\to0}{\longrightarrow} \divg(Ef)$ and $(\partial_t-\eps\Delta)f_\eps \underset{\eps\to0}{\longrightarrow} \partial_tf$ in $\mathcal{D}'(\R_+\times\R^d)$.
Therefore, we obtain the existence of $f\in L^\infty([0,T],L^p(m))$ verifying the \eqref{eq:FFP} equation. Uniqueness follows directly by remarking that $f^\mathrm{in} = 0 \implies f=0$.
\step{2. Existence in $L^1(m)$}
Consider now the case where $f^\mathrm{in}\in L^1(m)$. As $k<\alpha$, by Lemma~\ref{lem_inclusion} we can find $k<l<\alpha$ and $p\in(1,2)$ such that with $M=\weight{x}^l$, we have $L^p(M)\hookrightarrow L^1(m)$. Let $f_n^\mathrm{in}\underset{n\to\infty}{\longrightarrow} f^\mathrm{in}$ in $L^1(m)$ and $f_n$ be the corresponding solution of the \eqref{eq:FFP} given by the existence in the $L^p$ case. Then, the same proof, but with the $L^1(m)$ estimates, gives
\begin{eqnarray}\label{eq:cauchy_seq_l1}
\|f_{n_1}-f_{n_2}\|_{L^1(m)} & \leq & e^{(C_0+C)T} \|f_{n_1}^\mathrm{in}-f_{n_2}^\mathrm{in}\|_{L^1(m)} \underset{n\to\infty}{\longrightarrow} 0.
\end{eqnarray}
Therefore, $f_n$ is a Cauchy sequence and we can again verify that it converges to a solution in $L^\infty((0,T),L^1(m))$ of the equation.
\end{demo}
\begin{lem}\label{lem:w_continuity}
Let $E\in L^\infty_{\mathrm{loc}}$, $m\in L^0(\R,\R_+^*)$ and $f\in L^\infty((0,T),L^p(m))$ for $p\in(1,+\infty)$ be a weak solution of the \eqref{eq:FFP} equation. Then we have the following continuity in time
\begin{equation*}
f\in C^0([0,T],w-L^p(m)),
\end{equation*}
where $w-L^p(m)$ indicates that we take the weak topology on $L^p(m)$.
\end{lem}
\begin{demo}[Lemma~\ref{lem:w_continuity}]
Let $\varphi\in\mathcal{D}(\R^d)$. As $f$ is solution of \eqref{eq:FFP} in $\mathcal{D}'((0,T)\times\R^d)$, taking $\psi\otimes\varphi \in \mathcal{D}((0,T)\times\R^d)$ as test function, we can write
\begin{equation*}
-\int_0^T\intd f(t,x)\partial_t\psi(t)\varphi(x)\d x\d t \ = \ \int_0^T\intd f(t,x)\psi(t)(\I(\varphi)-E\cdot\nabla\varphi)(x)\d x\d t,
\end{equation*}
or equivalently
\begin{equation*}
\partial_tu_\varphi = v_\varphi\mathrm{\ in\ }\mathcal{D}'(0,T),
\end{equation*}
with
\begin{align*}
u_\varphi &: t\mapsto\intd f(t,\cdot)\varphi &\mathrm{and}&& v_\varphi &: t\mapsto\intd f(t,\cdot)(\I(\varphi)-E\cdot\nabla\varphi).
\end{align*}
For $\varphi\in\mathcal{D}(\R^d)$ and $E\in L^\infty_{\mathrm{loc}}$, we have $\I(\varphi)-E\cdot\nabla\varphi\in L^\infty(\weight{x}^{d+\alpha})$. Thus, as by Lemma~\ref{lem_inclusion}, $f\in L^\infty((0,T),L^p(m)) \subset L^\infty((0,T),L^1(\weight{x}^{-(d+\alpha)}))$, we obtain that $u_\varphi\in L^\infty(0,T)$ and $v_\varphi\in L^\infty(0,T)$. Hence, $u_\varphi\in W^{1,\infty}(0,T)\subset C^0([0,T])$.
Let $p\neq 1$. We now show that the result is still true by replacing $\varphi$ by $g\in L^{p'}(m^{-1})$. First, we remark that $u_g$ is well defined in $L^\infty(0,T)$. Then, by the density of $\mathcal{D}(\R^d)$ in $L^{p'}$, there exists a sequence $(\tilde{\varphi}_n)_{n\in\N}\in\mathcal{D}(\R^d)^\N$ such that $\tilde{\varphi}_n \underset{n\to+\infty}{\longrightarrow}gm^{-1}$ in $L^{p'}$, or equivalently, there exists $\varphi_n := m\tilde{\varphi}_n\in\mathcal{D}(\R^d)$ such that $\varphi_n \underset{n\to+\infty}{\longrightarrow}g$ in $L^{p'}(m^{-1})$. We now look at the sequence of $u_{\varphi_n}$ and write
\begin{align*}
\left\|u_{\varphi_n}-u_g\right\|_{C^0([0,T])} \ & = \ \left\|\intd f(t,\cdot)(\varphi_n-g)\right\|_{L^\infty(0,T)}
\\
& \leq \ \left\|f\right\|_{L^\infty((0,T),L^p(m))}\left\|\varphi_n-g\right\|_{L^{p'}(m^{-1})} \underset{n\to+\infty}{\longrightarrow} 0.
\end{align*}
It proves that $u_g\in C^0(0,T)$.
\end{demo}
We can now combine the previous lemmas to give the proof of Theorem~\ref{th:existence}.
\begin{demo}[Theorem~\ref{th:existence}]
Since the time continuity in the weak topology $\sigma(X,X')$ implies the continuity in the strong $X$ topology (see e.g. \cite{engel_one-parameter_1999}), combining Lemmas~\ref{lem:existence} and Lemma~\ref{lem:w_continuity} gives the result in the case $p>1$.
If $p=1$, we prove the time continuity differently. Using again an $L^p(M)$ approximation sequence $f_n$, we obtain from equation \eqref{eq:cauchy_seq_l1} that it is a Cauchy sequence in $C^0([0,T],L^1(m))$, since
\begin{align*}
\|f_{n_1}-f_{n_2}\|_{C^0([0,T],L^1(m))} \ & = \ \sup\limits_{t\in(0,T)} \|f_{n_1}-f_{n_2}\|_{L^1(m)}
\\
& \leq \ e^{(C_0+C)T} \|f_{n_1}^\mathrm{in}-f_{n_2}^\mathrm{in}\|_{L^1(m)} \underset{n\to\infty}{\longrightarrow} 0,
\end{align*}
from what we conclude that $f\in C^0([0,T],L^1(m))$.
\end{demo}
\section{Additional properties for solutions to the equation}\label{sec:gain}
In this section, we prove that the semigroup associated to the \eqref{eq:FFP} equation actually gives gains of regularity, integrability, weight and positivity, which is useful to retrieve quantitative estimates about the regularity of solutions, to prove uniform in time estimates in weighted Lebesgues spaces and existence and uniqueness of the steady state, as well as quantitative rate of decay towards equilibrium.
\subsection{Gain of regularity and integrability}
\begin{prop}\label{prop:regu}
Let $f\in L^1(m)$ be a solution of the \eqref{eq:FFP} equation as given in Theorem~\ref{th:existence} for $m=\weight{x}^k$ with $k\in(0,\alpha\wedge 1)$. Then there exists $c>0$ such that the following inequality holds
\begin{equation}\label{eq:gain_regu}
\|f\|_{L^p(m)} \lesssim \left(c+\frac{d(p-1)}{\alpha t}\right)^{\frac{d}{q\alpha}} e^{t\lambda_1} \|f^\mathrm{in}\|_{L^1(m)},
\end{equation}
where $\lambda_1$ is the growth bound of $e^{t\Lambda}$ in $L^1(m)$, $q'=p\in[1,p_\gamma)$ and if $\alpha\geq 1$, $p<\alpha/k$.
Moreover, if $f^\mathrm{in}\in L^p(m)$, we obtain the following Sobolev regularity
\begin{align}\label{eq:regu_H}
(fm)^{p/2} \ &\in \ L^2((0,T),H^{\alpha/2}).
\end{align}
\end{prop}
\paragraph{\bf Remarks:} Formula \eqref{eq:gain_regu} can also be written in other words
\begin{eqnarray}
\|e^{t\Lambda}\|_{L^1(m)\to L^p(m)} & \lesssim & \left(c+t^{\frac{-d}{\alpha q}}\right) e^{t\lambda_1}.
\end{eqnarray}
In order to show regularizing properties of the \eqref{eq:FFP} equation, one possibility is to use a fractional variant of the Nash inequality in $L^p(m)$ spaces. In the case of $L^2$ spaces, it is proved for example in \cite[Lemma~5.2]{tristani_fractional_2015}.
\begin{lem}[Fractional Nash inequality in $L^p(m)$]\label{lem:Nash_frac_m_p}
Let $p\in[1,2]$ and $m=\weight{x}^k$ with $kp\in(0,\alpha\wedge1)$ or $0<k<\alpha<1$. Then for any $u\in L^p(m)$, we have
\begin{align}\label{eq:Nash_Lpm_W_alpha}
\intd \I(u)u^{p-1}m^p \ &\lesssim \ C_{k,p}\left\|u\right\|_{L^p(m)}^p - \left|(um)^\frac{p}{2}\right|_{H^{\alpha/2}}^2
\\\label{eq:Nash_Lpm_L1}
&\lesssim \ C_{k,p}\left\|u\right\|_{L^p(m)}^p - \left\|u\right\|_{L^p(m)}^{p+\frac{q\alpha}{d}} \left\|u\right\|_{L^1(m)}^{\frac{-q\alpha }{d}}.
\end{align}
\end{lem}
\begin{demo}[Lemma~\ref{lem:Nash_frac_m_p}]
By the definition of the Sobolev seminorm \eqref{def:seminorm} and the relation \eqref{eq:Gp_grad}, we remark that
\begin{equation*}
\intd \Dp{p}(v) \ \simeq \ |v^\frac{p}{2}|_{H^{\alpha/2}}.
\end{equation*}
Therefore, \eqref{eq:Nash_Lpm_W_alpha} is a consequence of inequalities \eqref{eq:estim_a_priori_I} or \eqref{eq:estim_a_priori_I_2}. By using the following Gagliardo-Nirenberg inequalities (see for example \cite{mazya_sobolev_2011})
\begin{equation*}
\left\|(um)^{p/2}\right\|_{L^2} \ \lesssim \ \left|(um)^{p/2}\right|_{H^{\frac{\alpha}{2}}}^{\theta} \left\|(um)^{p/2}\right\|_{L^{2/p}}^{1-\theta},
\end{equation*}
with $\theta \ = \ \frac{p}{p+q\alpha/d}$, which can also be written
\begin{equation*}
\left\|u\right\|_{L^p(m)}^{p/\theta} \ \lesssim \ \left|(um)^{p/2}\right|_{H^{\frac{\alpha}{2}}}^{2} \left\|u\right\|_{L^1(m)}^{p(1/\theta-1)},
\end{equation*}
we deduce \eqref{eq:Nash_Lpm_L1} from \eqref{eq:Nash_Lpm_W_alpha}.
\end{demo}
Nash type inequalities let appear the following family of ordinary differential inequalities that can be solved explicitly and lead to the growth in time given by the following application of Gronwall's inequality.
\begin{lem}\label{lem:ODE_ineq}
Let $(A,B,C,b)\in \R^4$ and $y\in L^1_+(0,T)$ verifying in the weak sense $\partial_t X \leq B X - A e^{-bCt} X^{1+C}$. Then, the following upper bound holds
\begin{equation*}
X \leq \frac{e^{-bt}}{A^{1/C}} \left((B-b)+\frac{1}{Ct}\right)^{1/C}.
\end{equation*}
\end{lem}
We can now combine Lemma~\ref{lem:ODE_ineq} with previous Nash type inequalities \eqref{eq:Nash_Lpm_W_alpha} and \eqref{eq:Nash_Lpm_L1} to prove Proposition~\ref{prop:regu}.
\begin{demo}[Proposition~\ref{prop:regu}]
Let $X = X(t) := \|f\|_{L^p(m)}^p$, $Y := \|f\|_{L^1(m)}^p$ and $\theta := \frac{\alpha}{d(p-1)}>0$. The second fractional Nash inequality~\eqref{eq:Nash_Lpm_L1} can be written
\begin{equation*}
\intd \I(f)f^{p-1}m^p \ \leq \ \bar{C}X - \tilde{C} Y^{-\theta}X^{1+\theta}.
\end{equation*}
Thus, using the inequality~\eqref{eq:estim_a_priori_E} for the $\divg(E\cdot)$ part of the operator $\Lambda$, we obtain
\begin{align*}
\partial_t X \ & = \ p\intd \I(f) f^{q-1} m^p + p\intd f^p m^p \varphi_{m,p}
\\
& \leq \ (p\bar{C}+C)X - p\tilde{C} Y^{-\theta}X^{1+\theta}.
\end{align*}
Using the fact that $Y\leq e^{q\lambda_1 t}Y(0)$ and Lemma~\ref{lem:ODE_ineq}, we obtain
\begin{equation*}
X(t) \ \leq \ e^{q\lambda_1 t} \left(\frac{1}{p\tilde{C}}\right)^{\frac{1}{\theta}} \left(c_p+\frac{1}{\theta t}\right)^{\frac{1}{\theta}} Y(0),
\end{equation*}
with $c_p = p\bar{C}+C-q\lambda_1$. It proves \eqref{eq:gain_regu}. Let now $Z := |(fm)^{p/2}|_{H^{\alpha/2}}$ and assume $X(0)$ is bounded. Then by Theorem~\ref{th:existence}, we know that $X\leq e^{tp\lambda_p}X(0)$ for a given $\lambda_p\in\R$. Using now the first fractional Nash inequality~\eqref{eq:Nash_Lpm_W_alpha}, we have
\begin{equation*}
\intd \I(f)f^{p-1}m^p \ \leq \ \bar{C}X - Z^q.
\end{equation*}
It gives us, by integrating the a priori estimates with respect to time
\begin{equation*}
\int_0^TZ^q \ \leq \ X(0) - X(T) + (p\bar{C}+C)\int_0^TX.
\end{equation*}
Therefore, we obtain
\begin{equation*}
\int_0^T\|(fm)^{p/2}\|_{H^{\alpha/2}}^q \ \leq \ X(0) \left( 1 + (p\bar{C}+C+1)p\lambda_pe^{Tp\lambda_p}\right),
\end{equation*}
which gives \eqref{eq:regu_H}.
\end{demo}
\subsection{$L^1(m)\to L^\infty(m)$ Regularization when $\gamma \leq 2$}
When $\gamma\leq 2$, we have a stronger regularization than Proposition~\ref{prop:regu} since the solutions are globally bounded in space. This property, which will hold also for the equilibrium, will be particularly useful to get the polynomial decay of Theorem~\ref{th:cv}.
\begin{prop}\label{prop:regu_L_infty}
Assume $\gamma \leq 2$. Let $f\in L^1(m)$ be a solution of the \eqref{eq:FFP} equation as given in Theorem~\ref{th:existence} with $m:=\weight{x}^k$ with $2k\in(0,(\alpha\wedge 1))$ or $0\leq k<\alpha\leq1$. Then the following inequality holds
\begin{equation}\label{eq:gain_regu_L_infty}
\|f\|_{L^\infty(m)} \ \lesssim \ \left(C+t^{\frac{-d}{\alpha}}\right) e^{\frac{t}{2}(\lambda_1^*+\lambda_1)} \|f^\mathrm{in}\|_{L^1(m)},
\end{equation}
where $\lambda_1$ is the growth bound of $e^{t\Lambda}$ in $L^1(m)$, $\lambda_1^*$ the growth bound of $e^{t\Lambda^*}$ and $C\in\R$.
\end{prop}
\begin{demo}[Proposition~\ref{prop:regu_L_infty}]
Since $\gamma \leq 2$, then Theorem~\ref{th:existence} and the inequalities \eqref{eq:estim_a_priori_1} or \eqref{eq:estim_a_priori_2} hold in $L^p(m)$ for all $p\in [1,2]$ and Proposition~\ref{prop:regu} holds for $p=2$. It implies
\begin{equation}\label{eq:regu_L1_L2}
\|e^{t\Lambda}\|_{L^1(m)\to L^2(m)} \ \lesssim \ t^{\frac{-d}{2\alpha}}e^{t\lambda_1}.
\end{equation}
Moreover, for $g$ solution of the dual equation $\partial_t g = \Lambda^* g := \I(g) - E\cdot\nabla g$, we have
\begin{align*}
\intd -(E\cdot \nabla g) g^{p-1}m^{-p} \ & = \ \frac{1}{p}\intd |g|^p\divg(Em^{-p})
\\
& = \ \intd |g|^pm^{-p}\left(\frac{\divg(E)}{p} - E\cdot \frac{\nabla m}{m}\right)
\\
& \leq \ \frac{\left\|\divg(E)\right\|_{L^\infty}}{p} \intd |g|^pm^{-p},
\end{align*}
by combining with formula~\eqref{eq:estim_a_priori_I} that still holds, we obtain the estimate
\begin{equation*}
\partial_t\left(\intd |g|^pm^{-p}\right) \ = \ - \intd \Dp{p}(gm^{-1}) + \intd |g|^pm^{-p} \left(C_k+\frac{\left\|\divg(E)\right\|_{L^\infty}}{p}\right).
\end{equation*}
Which is the equivalent of \eqref{eq:estim_a_priori_1} for the dual equation in $L^p(m^{-1})$. With the same proof, we get that Theorem~\ref{th:existence} and Proposition~\ref{prop:regu} also hold in $L^p(m^{-1})$ for $p\in [1,2]$, from what we deduce
\begin{equation}\label{eq:regu_L1_L2_*}
\|e^{t\Lambda^*}\|_{L^1(m^{-1})\to L^2(m^{-1})} \ \lesssim \ \left(c+t^{\frac{-d}{2\alpha }}\right)e^{t\lambda_1^*},
\end{equation}
where $\lambda_1^*$ is the growth bound of $e^{t\Lambda^*}$ in $L^1(m^{-1})$. Since the dual of $L^1(m^{-1})$ and $L^2(m^{-1})$ can be identified with $L^\infty(m)$ and $L^2(m)$, we deduce from \eqref{eq:regu_L1_L2_*} that
\begin{equation*}
\|e^{t\Lambda}\|_{L^2(m)\to L^\infty(m)} \ \lesssim \ \left(c+t^{\frac{-d}{2\alpha }}\right)e^{t\lambda_1^*}.
\end{equation*}
And combining with \eqref{eq:regu_L1_L2}, by writing $e^{t\Lambda} = e^{\frac{t}{2}\Lambda}e^{\frac{t}{2}\Lambda}$, we end up with
\begin{equation*}
\|e^{t\Lambda}\|_{L^1(m)\to L^\infty(m)} \ \lesssim \ \left(C+t^{\frac{-d}{\alpha}}\right) e^{\frac{t}{2}(\lambda_1^*+\lambda_1)},
\end{equation*}
which ends the proof.
\end{demo}
\subsection{Gain of positivity}
We prove in this section the gain and the propagation of strict positivity. It will be useful to prove the uniqueness of the steady state and also, as explained in Proposition \ref{prop:cv_harris}, to get asymptotic estimates when we are not able to prove that the steady state is bounded and use Poincaré inequality. The first proposition is the classical maximum principle.
\begin{prop}[Weak Parabolic Maximum Principle]\label{prop:wPMP} Assume that the conditions of Proposition~\ref{prop:estim} are satisfied and let $f\in L^p(\R_+,L^p(m\weight{x}^{(\gamma-2)_+/p}))$ be such that
\begin{itemize}
\item[$\circ$] $(\partial_t -\Lambda)f \geq 0$,
\item[$\circ$] $f(0,\cdot)=f^\mathrm{in}\geq0$.
\end{itemize}
Then $f\geq 0$.
\end{prop}
\begin{demo}[Proposition \ref{prop:wPMP}]
Let $g\in L^p(m\weight{x}^{(\gamma-2)_+/p})$, $g_- := (-g)_+$ its negative part and $\Phi(g) := g_+^p$. We remark that
\begin{equation*}
\intd \I(g)\Phi'(g)m^p \ \leq \ p\intd \I(g_+)g_+^{p-1}m^p,
\end{equation*}
because, as $(g_-)(g_+) = 0$, we have
\begin{align*}
-\intd \I(g_-)g_+^{p-1}m^p \ & = \ -\iintd \ka((g_-)_* - g_-)g_+^{p-1}m^p
\\
& = \ -\iintd \ka(g_-)_*(g_+^{p-1})m^p \ \leq \ 0.
\end{align*}
Thus, if $g$ is such that $\partial_t g \leq \Lambda g$, we get
\begin{equation*}
\partial_t\left(\intd |g_+|^pm^p\right) \ \leq\ p\intd (\Lambda g)g_+^{p-1}m^p \ \leq\ p\intd \Lambda(g_+)g_+^{p-1}m^p.
\end{equation*}
Using the a priori estimates \eqref{eq:estim_a_priori_1} or \eqref{eq:estim_a_priori_2}, we obtain
\begin{equation*}
\intd |g_+|^pm^p \ \leq \ e^{\lambda t}\intd |g_+^\mathrm{in}|^pm^p.
\end{equation*}
We conclude by taking $f=-g$ and remarking that $f^\mathrm{in}_- = 0 \implies f_- =0$.
\end{demo}
The second proposition claims that the solutions to the \eqref{eq:FFP} equations are actually bounded by below by a strictly positive function as soon as they have positive mass in a compact set. It implies in particular the strong maximum principle.
\begin{prop}\label{prop:positivity}
Let $f$ be a solution to the \eqref{eq:FFP} equation with initial condition $f^\mathrm{in}\in L^1_+\cap L^p(m)$. Then for any $a>d+\alpha+\gamma-2$ and $R>0$ sufficiently large, there exists an increasing function $\psi_R\in C^0\cap L^\infty(\R_+^*,\R_+^*)$ such that
\begin{equation*}
f(t,x) \ \geq \ \frac{\psi_R(t)}{\weight{x}^a}\int_{B_R} f^\mathrm{in},
\end{equation*}
where $B_R$ denotes the ball of size $R$.
\end{prop}
For a given $r>0$, we define $\chi := \mathds{1}_{B_r}$, $\chi^c := 1-\chi$, $\kappa^c := \kappa_\alpha \chi^c + \kappa_\alpha(r)\chi = \min(\kappa_\alpha,\kappa_\alpha(r))$ and $\kappa := \kappa_\alpha - \kappa^c \geq 0$. As $\kappa^c \in L^1$, we will denote by $K^c := \|\kappa^c\|_{L^1}$ and will decompose $\I$ into
\begin{align*}
\I_c(u) \ & := \ \intd \kappa^c_\*\,(u_*-u) \ =\ \kappa^c * u - K^c u
\\
\I_\chi(u) \ & := \ \intd \kappa_\*\,(u_*-u) \ =\ \int_{|x-y|<r} \kappa(x-y)(u(y)-u(x))\d y.
\end{align*}
Then we define the splitting
\begin{equation*}
\Lambda = A + B,
\end{equation*}
where
\begin{equation*}
A u = \kappa^c * u \text{ and } B = (I_\chi + \divg(E\ \cdot) - K^c).
\end{equation*}
Since the second operator still generates a positive semigroup, the strategy is to use the following Duhamel's formula (see e.g. \cite{arendt_one-parameter_1986})
\begin{equation*}
e^{t\Lambda} \ = \ e^{tB} + e^{t\Lambda}\star Ae^{tB},
\end{equation*}
where we defined the time convolution of two operators by
\begin{equation*}
U\star V : t \mapsto \int_0^t U(t-s)V(s)\d s,
\end{equation*}
and to prove that $A$ gives a gain of positivity while $e^{t\Lambda}$ propagates the lower bound. These properties are given in the following lemmas. We will need the following bound by below
\begin{lem}[Bound by below for $\I(m)$]\label{lem:I_m_below}
Let $m(x):=\weight{x}^k$ with $k<\alpha$.
Then
\begin{equation*}
\I(m) \ \geq \ C_k\weight{x}^{-(d+\alpha)}-\tilde{C}_km.
\end{equation*}
\end{lem}
\begin{demo}[Proposition~\ref{lem:I_m_below}]
We use the above splitting of the fractional Laplacian into $I = I_\chi + I_c$ for $\chi = \mathds{1}_{B_1}$.We first deal with $\I_\chi(m)$ and remark that
\begin{align}\label{def:lap_tronque}
\I_\chi(u) \ & := \ \intd \kappa_\*\left(u_*-u-(x_*-x)\cdot\nabla u\right).
\end{align}
By a second order Taylor approximation, for $z\in B_1$, we obtain
\begin{align*}
|m(x+z)-m(x)-z\cdot\nabla m(x)| \ &\leq \ \frac{|z|^2}{2} \left\|\nabla^2 m\right\|_{L^\infty(B_1(x))}.
\end{align*}
Thus, by the change of variable $z=x-x_*$ in \eqref{def:lap_tronque}, we can write
\begin{align*}
|\I_\chi(m)| \ & \leq \ \frac{1}{2}\left\|\nabla^2 m\right\|_{L^\infty(B_1(x))} \int_{|z|<1} \frac{\chi(z)\d z}{|z|^{d+\alpha-2}}
\\
& \leq \ \frac{\omega_d}{2(2-\alpha)} \left\|\chi\right\|_{L^\infty}\,\left\|\nabla^2 m\right\|_{L^\infty(B_1(x))}.
\end{align*}
In particular, since $m = \weight{x}^k$, we have
\begin{equation*}
\left\|\nabla^2 m\right\|_{L^\infty(B_1(x))} \ \leq \ \sup_{|z|<R}|k(|k|+3)\weight{x+z}^{k-2}|.
\end{equation*}
Peetre's inequality tells that for all $(x,z)\in\R^{2d}$, we have
\begin{align*}
\weight{x+z}^{k-2} \ & \leq \ \sqrt{2}^{|k-2|}\weight{x}^{k-2}\weight{z}^{|k-2|}.
\end{align*}
Since $\weight{z}\leq\weight{1}$, we obtain
\begin{equation}\label{eq:i_loc_m}
\left|\I_\chi(m)\right| \ \leq \ C_{k} \langle x\rangle^{k-2}.
\end{equation}
Now deal with the second part. Since $I_c(m) = \kappa^c*m - K^c m$, we just have to remark that
\begin{equation*}
\kappa^c*m\ \geq\ \kappa^c*(m(1)\mathds{1}_{B_1})\ \geq\ \frac{C}{(|x|+1)^{d+\alpha}}.
\end{equation*}
Then, by combining with \eqref{eq:i_loc_m}, we obtain
\begin{equation*}
\I(m) \ \geq \ \frac{C}{(|x|+1)^{d+\alpha}} - \left( K^c + \frac{C_k}{\weight{x}^2}\right)m.
\end{equation*}
what gives the result.
\end{demo}
\begin{lem}[Propagation of positivity]\label{lem:propag_positivity}
Let $f\in L^p((0,T),L^p(m\weight{x}^{{(\gamma-2)_+}/p}))$ be a solution to the \eqref{eq:FFP} equation such that $f^\mathrm{in}>\frac{1}{\weight{x}^a}$ with $a>d+\alpha+\gamma-2$. Then there exists $\lambda>0$ such that
\begin{eqnarray}\label{eq:propag_positivity}
f(t,x) & \geq & \frac{e^{-\lambda t}}{\weight{x}^a}.
\end{eqnarray}
\end{lem}
\begin{demo}[Lemma~\ref{lem:propag_positivity}]
Let $\beta := \gamma-2$. We prove that for $\lambda$ large enough, $g(t,x) := \mathfrak{m}(x)\psi(t)$ with $\psi(t) = e^{-\lambda t}$ and $\mathfrak{m}(x) = \weight{x}^k$ with $k<-(d+\alpha+\beta)$ is a subsolution. By Lemma~\ref{lem:I_m_below}, we have, indeed
\begin{equation*}
\I(\mathfrak{m}) \ \geq \ \left(C_k\weight{x}^{-(d+\alpha+k)}+\tilde{C}_k\right)\mathfrak{m}.
\end{equation*}
We deduce
\begin{align*}
(\partial_t - \Lambda)g \ & = \ -\lambda g - \I(\mathfrak{m})\psi(t) - \divg(E\mathfrak{m})\psi(t)
\\
& \leq \ \left(-\lambda - C_k\weight{x}^{-(d+\alpha+k)}+\tilde{C}_k - \divg(E) - E\frac{\nabla \mathfrak{m}}{\mathfrak{m}}\right)g
\\
& \leq \ \left(\tilde{C}_k-\lambda - C_k\weight{x}^{\beta+\eps} + C\,\weight{x}^\beta\right)g,
\end{align*}
where $\eps := -(k+d+\alpha+\beta)>0$. Therefore, by taking $\lambda$ sufficiently large we obtain $(\partial_t - \Lambda)g\leq 0$, i.e. $g$ is a subsolution to the equation. As $g\in L^p_{t,x}(\weight{x}^{\alpha+{\beta_+}/p})$, we can apply the weak parabolic maximum principle, Proposition~\ref{prop:wPMP}, to $f-g$ and we get that $f\geq g$.
\end{demo}
\begin{lem}[Creation of positivity]\label{lem:lower_bound} For $u\in L^1_+$ the following lower bound holds
\begin{equation}\label{eq:lower_bound_0}
\kappa^c * u \ \geq \ \frac{C}{\weight{x}^{d+\alpha}}\int_{B_R} u,
\end{equation}
where $C = (\sqrt{2}\,\max(r,R,1))^{-(d+\alpha)}$.
\end{lem}
\begin{demo}[Lemma~\ref{lem:lower_bound}]
If $y\in B_R$, then $|x-y|\leq |x|+R$. We deduce the following lower bound
\begin{equation*}
\kappa^c *u(x) \ \geq \ \int_{|y|<R} \mathds{1}_{|x-y|<r} \frac{u(y)}{|r|^{d+\alpha}} + \mathds{1}_{|x-y|>r} \frac{u(y)}{(|x|+R)^{d+\alpha}}\d y.
\end{equation*}
Let $r_1 := \max(r,R,1)$. As $|x|+R \leq |x|+ r_1$ and $r \leq |x|+ r_1$, we get
\begin{align*}
\kappa^c *u(x) \ & \geq \ \frac{1}{(|x|+r_1)^{d+\alpha}} \int_{|y|<R} u(y)\d y \ \geq \ \frac{C}{\weight{x}^{d+\alpha}}\int_{B_R} u,
\end{align*}
where $C = (\sqrt{2}\,r_1)^{-(d+\alpha)}$.
\end{demo}
Now we prove that $e^{tB}$ propagates the fact to have a positive mass in a compact set.
\begin{lem}\label{lem:mass_loc}
Let $u\in L^1(m)$ and $R>0$. Then for all $\delta > 0$, there exists $\lambda_\delta>0$ such that
\begin{equation}\label{eq:propag_mass_loc}
\int_{B_{R+\delta}} e^{tB} u \ \geq \ e^{-\lambda_\delta t}\int_{B_R} u.
\end{equation}
\end{lem}
\begin{demo}[Lemma \ref{lem:mass_loc}]
Let $\eta_0\in C^\infty_c$ be a radially decreasing function such that $\mathds{1}_{B_{\bar{R}}}\leq \eta_0 \leq \mathds{1}_{B_R}$ and $\eta_0>0$ on $B_R$. We also define for all $t>0$, $\eta_t := e^{-\lambda t}\eta_0$ for a given $\lambda > 0$. By construction, this is a subsolution of $\partial_t + E\cdot\nabla$ since
\begin{equation*}
\partial_t\eta + E\cdot\nabla\eta \ = \ -\lambda\eta - \left(E\cdot \frac{x}{|x|}\right) |\nabla \eta| \ \leq \ - \lambda \eta.
\end{equation*}
Our goal is to prove that for $\lambda$ sufficiently large, we even better have $\partial_t\eta + E\cdot\nabla\eta - I_\chi(\eta) \leq 0$. Therefore, we look at the behaviour of $I_\chi(\eta)$ where $\chi = \mathds{1}_{B_r}$. For $|x|>R$ we have
\begin{equation*}
I_\chi(\eta) \ = \ \int_{|x-y|<r} \frac{\eta(y)}{|x-y|^{d+\alpha}} \d y \ \geq \ \frac{1}{r^{d+\alpha}}\int_{B_r(x)} \eta \ \geq \ 0,
\end{equation*}
where $B_r(x)$ is the ball of center $x$ and radius $r$.
In particular, defining $j_R := I_\chi(\eta)(x)$ for $|x|=R$, we have $j_R > 0$. As $\eta\in C^\infty$, we easily deduce $I_\chi(\eta)\in C^\infty$ and the existence of $R' \in (\bar{R},R)$ such that for all $|x|\in[R',R]$, $I_\chi(\eta) \geq j_R/2 > 0$. Therefore, we obtain the following cases
\begin{align*}
|x|> R' \ & \implies \ I_\chi(\eta) + \lambda \eta \ \geq \ \lambda \eta \ \geq \ 0
\\
|x|< R' \ & \implies \ I_\chi(\eta) + \lambda \eta \ \geq \ \lambda \eta(R') - \|I_\chi(\eta)\|_{L^\infty},
\end{align*}
and the latter is positive for $\lambda$ sufficiently large. As $\eta \in C^\infty([0,T]\times B_R)$ all the estimates can easily be made uniform in time and we therefore obtain that
\begin{equation*}
(\partial_t-B^*)\eta \ \leq \ 0.
\end{equation*}
In particular, by application of the maximum principle (Proposition~\ref{prop:wPMP}) we obtain that $e^{tB^*}\mathds{1}_{B_R} \geq e^{tB^*}\eta_0 \geq \eta \geq e^{-\lambda t}\mathds{1}_{B_{\bar{R}}}$. By the dual definition of positivity, we obtain \eqref{eq:propag_mass_loc}.
\end{demo}
We can now prove the gain of positivity for the \eqref{eq:FFP} equation.
\begin{demo}[Proposition~\ref{prop:positivity}]
We combine \eqref{eq:lower_bound_0} and \eqref{eq:propag_mass_loc} to get
\begin{equation*}
Ae^{sB} f^\mathrm{in} \ \geq \ \frac{C_{R,\delta,\chi}e^{-\lambda_\delta s}}{\weight{x}^{d+\alpha}}\,\int_{B_R} f^\mathrm{in},
\end{equation*}
where $C = (2\sqrt{2})^{-(d+\alpha)}$. By propagation of the positivity (Lemma~\ref{lem:propag_positivity}), for any $a>d+\alpha+\gamma-2$,
\begin{equation*}
e^{(t-s)\Lambda}Ae^{sB} f^\mathrm{in} \ \geq \ \frac{C_{R,\delta,\chi}e^{-\lambda (t-s)}e^{-\lambda_\delta s}}{\weight{x}^a}\,\int_{B_R} f^\mathrm{in}.
\end{equation*}
In conclusion, by integrating on $s\in[0,t]$ and using the fact that $e^{tB}\geq 0$ and that by Duhamel's formula
\begin{equation*}
e^{t\Lambda} = e^{tB} + e^{t\Lambda}A\star e^{tB} \geq e^{t\Lambda}A\star e^{tB},
\end{equation*}
we obtain
\begin{equation*}
e^{t\Lambda} f^\mathrm{in} \ \geq \ \frac{\psi(t)}{\weight{x}^a}\,\int_{B_R} f^\mathrm{in}.
\end{equation*}
where $\psi(t) = C_{R,\delta,\chi}\frac{e^{-\lambda t} -e^{-\lambda_\delta t}}{\lambda-\lambda_\delta} \in C^0\cap L^\infty(\R_+^*,\R_+^*)$.
\end{demo}
\section{Existence and uniqueness of the steady state}\label{sec:steady}
\subsection{Splitting of $\Lambda$ as a bounded and a dissipative part}
This section uses a splitting of the operator $\Lambda$ in a dissipative part in $B\in\B(L^1(m),L^p(m^\theta))$ and a bounded part $A\in\B(L^1,L^1(m))$ in order to bound uniformly in time the solution to the \eqref{eq:FFP} equation and obtain the existence of a steady state. We define the new splitting as $\Lambda = A + B $ with
\begin{align*}
A :=M\chi_R \text{ and } B := \Lambda - M\chi_R,
\end{align*}
where $M>0$ is a large enough constant and $\mathds{1}_{B_R} \leq \chi_R \leq \mathds{1}_{B_{2R}}$ is a smooth cutoff function.
\begin{prop}\label{prop:estim_b}
Assume $\beta := \gamma - 2> -\alpha$ and let $k\in\left(0,\alpha\wedge1\right)$ and $p\in(1,p_\gamma)$. Then there exists $\omega:\R_+\to \R_+$ such that
\begin{equation}\label{eq:estim_b}
\|e^{tB}\|_{\B(L^p(m),L^p(m^\theta))} \ \lesssim \ \omega(t),
\end{equation}
where
\begin{itemize}
\item if $\beta \geq 0$, then $\theta=1$ and $\omega(t) = e^{-bt}$,
\item if $\beta \in (-\alpha,0)$, then $\theta$ is any number in $(0,1]$, $\omega(t) = \weight{t}^{-k(1-\theta)/|\beta|}$ and we require $p<\alpha/k$ if $k>\alpha+\beta$.
\end{itemize}
In particular, if $\beta > -\alpha$ and $p_\gamma \geq 1$, there exists $(p,\theta)$ such that $\omega \in L^1(\R_+)$. Moreover, the gain of integrability also holds for $B$ and writes
\begin{equation}\label{eq:gain_regu_b}
\|e^{tB}f\|_{\B(L^1(m),L^p(m))} \lesssim t^{-\frac{d}{q\alpha}},
\end{equation}
where we recall that $q=p'$.
\end{prop}
\begin{demo}[Proposition~\ref{prop:estim_b}]
By inequality~\eqref{eq:estim_a_priori_1}, if $0<kp<\alpha\wedge 1$, we have
\begin{equation*}
\frac{1}{p}\partial_t\left(\intd |f|^pm^p\right) \ \leq \ \frac{1}{p}\intd |f|^pm^p \left(\frac{C_k}{\weight{x}^\alpha} + \varphi_{m,p} - M\chi_R\right) - C\intd \Dp{p}(fm).
\end{equation*}
Or, by inequality~\eqref{eq:estim_a_priori_2}, we can also get for $k\in(0,\alpha)$,
\begin{equation*}
\frac{1}{p}\partial_t\left(\intd |f|^pm^p\right) \ \leq \ \frac{1}{p}\intd |f|^pm^p \left(\frac{C_k}{\weight{x}^{\alpha-k}} + \varphi_{m,p} - M\chi_R\right) - C\intd \Dp{p}(fm).
\end{equation*}
From \eqref{eq:strict_confinement}, for $p<p_\gamma$, we have $\varphi_{m,p}\leq b\mathds{1}_\Omega-a\weight{x}^\beta$. Therefore, since $\beta>-\alpha$, if $kp<\alpha$ or $k<\alpha+\beta$, for $M$ and $R$ large enough, we obtain
\begin{equation}\label{eq:quasi_gap_0}
\frac{1}{p}\partial_t\left(\intd |f|^pm^p\right) \ \leq \ - a \intd |f|^p \weight{x}^{kp+\beta} - C\intd \Dp{p}(fm),
\end{equation}
with $a>0$. In particular
\begin{equation}\label{eq:B_bound}
\|e^{tB}\|_{\B(L^p(m))} \ \leq \ 1,
\end{equation}
which proves inequality \eqref{eq:estim_b} for small times. If $\beta\geq 0$, since $m^p\leq\weight{x}^{kp+\beta}$, the result immediately follows be Grönwall's inequality. Assume now $\beta <0$ and let $\eps := pk(1-\theta)>0$. By Hölder's inequality, we have
\begin{equation*}
\intd |f|^pm^{\theta p} \ \leq \ \left(\intd |f|^p\weight{x}^{\theta kp+\beta}\right)^{\eps/(|\beta|+\eps)}\left(\intd |f|^pm^p\right)^{|\beta|/(|\beta|+\eps)}.
\end{equation*}
Combining it with \eqref{eq:quasi_gap_0} (where we replace $k$ by $\theta k$) and \eqref{eq:B_bound} leads to
\begin{equation}\label{eq:quasi_gap_1}
\partial_t\left(\intd |f|^p m^{\theta p}\right) \ \leq \ -\bar{a}\left(\intd |f|^pm^{\theta p}\right)^{1+|\beta|/\eps}\left(\intd |f^\mathrm{in}|^pm^p\right)^{-|\beta|/\eps}.
\end{equation}
By Grönwall's inequality, we obtain
\begin{equation*}
\intd |f|^p m^{\theta p} \ \lesssim \ \frac{1}{t^{\eps/|\beta|}} \intd |f^\mathrm{in}|^pm^p.
\end{equation*}
It proves inequality \eqref{eq:estim_b} for large times. Moreover, using this time the second term of the right-hand side in \eqref{eq:quasi_gap_0} and following the same proof as in Proposition~\ref{prop:regu}, we get
\begin{equation*}
\|e^{tB}f\|_{\B(L^1(m),L^p(m))} \lesssim t^{-\frac{d}{q\alpha}} e^{t\lambda_1}.
\end{equation*}
But using \eqref{eq:B_bound} for $p=1$ proves that we can take $\lambda_1 = 1$. It concludes the proof.
\end{demo}
\subsection{Existence of a unique steady state}
With this dissipative estimate, the gain of integrability property of Proposition \ref{prop:regu} and the properties of $A$, we obtain the following global in time estimates.
\begin{prop}[Global propagation of $L^p$ norms]\label{prop:asympt}
Assume $\gamma > 2-\alpha$ and let $f$ be a solution of the \eqref{eq:FFP} under the assumptions of Theorem~\ref{th:existence} with $f^\mathrm{in}\in L^1\cap L^p(m^\theta)$. Then, if $\gamma\geq 2$ and $p<p_\gamma$ or if $\gamma\in(2-\alpha,2)$ and $(p,k)$ is such that there exists $\theta\in (0,1)$ such that Proposition \ref{prop:estim_b} holds with $\omega\in L^1(\R_+)$, there exists $C>0$ such that
\begin{equation*}
\|e^{t\Lambda}f^\mathrm{in}\|_{L^p(m^\theta)} \ \leq \ C \left(\|f^\mathrm{in}\|_{L^1} + \|f^\mathrm{in}\|_{L^p(m^\theta)}\right).
\end{equation*}
\end{prop}
\begin{demo}[Proposition~\ref{prop:asympt}]
By noticing that $A\in \B(L^1,L^1(m))$, thanks to Proposition~\ref{prop:estim_b}, we obtain the following sequence of estimates
\begin{equation*}
L^1 \underset{e^{t\Lambda}}{\overset{1}{\longrightarrow}} L^1 \underset{A}{\overset{\left\|A\right\|}{\longrightarrow}} L^1(m) \underset{e^{tB/2}}{\overset{\omega_2(t)}{\longrightarrow}} L^p(m) \underset{e^{tB/2}}{\overset{\omega(t)}{\longrightarrow}} L^p(m^\theta),
\end{equation*}
where $\omega_2(t) = t^{-d/q\alpha}$ (which is integrable in $0$ since $q > d/\alpha$) and we have indicated the linear operator under the arrow and the corresponding growth rate above the arrows. Hence, by remarking that $\omega\omega_2\in L^1(\R_+)$ using the following Duhamel's Formula
\begin{equation*}
e^{t\Lambda} = e^{tB} + e^{tB/2}e^{tB/2} \star Ae^{t\Lambda},
\end{equation*}
and the global boundedness of $e^{tB}$ in $L^p(m^\theta)$ given by \eqref{eq:B_bound}, we deduce the announced result.
\end{demo}
This proposition together with the positivity properties of the semigroup are sufficient to prove existence and uniqueness of the steady state.
\begin{demo}[Theorem~\ref{th:unicite_equilibre}]
Since we have obtained a bound, uniform in time, in the weakly sequentially compact set $L^1_+\cap L^p(m)$, a fixed-point argument allows us to claim the existence of a stationary state. Following the same proof as in \cite[Lemma~3.6]{mischler_linear_2017} or \cite[Theorem 5.1]{kavian_fokker-planck_2015}, we obtain from the previous estimates the existence a stationary state $F\in L^1\cap L^p(m)$ to the \eqref{eq:FFP} equation.
Moreover, by the positivity results obtained in Proposition~\ref{prop:positivity} and since $1\in L^{p'}(m^{-1})\cap L^\infty$ for $p<\dfrac{d}{d-k}$, we obtained the following facts
\begin{list}{$\bullet$}{}
\item There exists $F\in L^p(m)\cap L^1_+$ such that $\Lambda F = 0$,
\item $\Lambda^*1=0$ and $1\in(L^p(m)\cap L^1)'_+$,
\item $\Lambda$ satisfies the strong and the weak maximum principle.
\end{list}
As a consequence of the Krein-Rutman Theorem (see e.g. \cite[Theorem 5.3]{mischler_spectral_2013}), we deduce the uniqueness of a stationary state $F\in L^p(m)\cap L^1_+$ of given mass $\|F\|_{L^1} = \|f^\mathrm{in}\|_{L^1}$. It finishes the proof of Theorem~\ref{th:unicite_equilibre}.
\end{demo}
\section{Polynomial Convergence to the equilibrium for $\gamma\in (2-\alpha,2)$}
When $\gamma\in (2-\alpha,2)$, the force field seems not confining enough to get exponential convergence since the derivatives of weighted Lebesgue norms let appear Lebesgue norms with smaller weights. Moreover, when $\gamma < 2-\alpha$, the effect of the force field at infinity is dominated by the effect of the fractional Laplacian, which prevent us from proving any explicit convergence result with our method.
\subsection{Generalized relative entropy}
In this section, we make a remark about the fact that we can already easily prove a non-quantitative version of the convergence toward equilibrium by generalized entropy method. Assume that there exists a steady state $F>0$ to the $\eqref{eq:FFP}$ equation and let $f$ be a solution of the equation of mass $0$. Then for $h := f/F$, by integration by parts, the following computation formally holds
\begin{equation*}
\frac{1}{p}\partial_t\left(\intd |h|^pF\right) \ = \ \intd (\I(h^{p-1})h - h E\cdot \nabla h^{p-1}) F.
\end{equation*}
Then, since by formula \eqref{eq:Gp_Dpp}
\begin{equation*}
\Dp{p}(h) \ \simeq \ \frac{1}{q}\I(|h|^p) - h\I(h^{p-1}),
\end{equation*}
we get
\begin{align*}
\frac{1}{p}\partial_t\left(\intd |h|^pF\right) \ & = \ \frac{1}{q}\intd (\I(|h|^p) - ph^{p-1} E\cdot \nabla h)F - \intd \Dp{p}(h)F
\\
& = \ \frac{1}{q}\intd (\I(|h|^p) - E\cdot \nabla |h|^p)F - \intd \Dp{p}(h)F
\\
& = \ \frac{1}{q}\intd |h|^p(\I(F)+\divg(EF)) - \intd \Dp{p}(h)F
\\
& = \ - \intd \Dp{p}(h)F.
\end{align*}
Thus, we obtain
\begin{eqnarray}\label{eq:dissip_entropy}
\partial_t\left(\intd |f|^pF^{1-p}\right)
& \leq & - p\intd \Dp{p}\left(\frac{f}{F}\right)F.
\end{eqnarray}
Since $\Dp{p}(h) \geq 0$ and $\Dp{p}(h) = 0 \ssi h$ is constant $\ssi f = F$ (by conservation of the mass), it implies that $\intd |h|^pF$ is a strict Lyapunov functional, which implies the convergence to the equilibrium in $L^p(F^{-1/q})$ (see for example \cite[Chapter 5]{mischler_introduction_2015} or \cite{haraux_systemes_1997}). However, we will prove that with other techniques we will get an explicit rate of convergence.
\subsection{Fractional Poincaré-Wirtinger inequality}
We prove in this section an inequality looking like a fractional Poincaré-Wirtinger inequality on a bounded set $\Omega$, but for the $p$-dissipation $\Dp{p}$ instead of a fractional gradient and for functions such that the mass is zero on the whole space (i.e. $v$ such that $\langle v\rangle_\mu = 0$).
We define the diameter of $\Omega$ as $\diam(\Omega) := \sup_{(x,y)\in\Omega^2}(|x-y|)$. Moreover, we introduce the following notation for the mass and the $L^p$ norm of a function $u$ for a measure $\mu$,
\begin{align*}
\langle u\rangle_{\mu,\Omega} &\ :=\ \frac{1}{\mu(\Omega)}\int_{\Omega} u\mu, & \|u\|_{L^p_\mu(\Omega)}^p &\ :=\ \int_\Omega |u|^p\mu,
\end{align*}
and we will use the shortcuts $\langle u\rangle_\mu := \langle u\rangle_{\mu,\R^d}$ and $\|u\|_{L^p_\mu}^p = \|u\|_{L^p_\mu(\R^d)}^p$.
\begin{prop}\label{prop:FPW_loc_p_dissip}
Let $\mu\in L^\infty_\mathrm{loc}\cap L^1_+$. Then for all $v\in L^p_\mu$ such that $\langle v\rangle_\mu = 0$, for all $\Omega\subset\R^d$ bounded, the following inequality holds
\begin{align*}
\int_\Omega \left|v\right|^p\mu \ \leq \ C_\mathrm{PW} \int_\Omega \Dp{p}(v)\mu + \eps_\Omega \left\|v\right\|_{L^p_\mu(\Omega)}^{p-1}\left\|v\right\|_{L^p_\mu(\Omega^c)},
\end{align*}
where $C_\mathrm{PW} = \diam(\Omega)^{d+\alpha} \left\|\mu\right\|_{L^\infty(\Omega)}$ and $\eps_\Omega = \frac{\mu(\Omega^c)}{\mu(\Omega)}$.
\end{prop}
It is a consequence of a the following more natural inequality where we control only the distance to the local mass $\langle u\rangle_{\mu,\Omega}$.
\begin{lem}\label{lem:FPW_loc_p_dissip}
Let $\Omega\subset\R^d$ be bounded and $\mu\in L^\infty_+(\Omega)$. Then for all $u\in L^p_\mu$,
\begin{equation*}
0\ \leq\ \int_\Omega u^{p-1}(u-\langle u\rangle_{\mu,\Omega})\mu \ \leq \ C_\mathrm{PW} \int_\Omega \Dp{p}(u)\mu,
\end{equation*}
where $C_\mathrm{PW} = \,\mathrm{diam}(\Omega)^{d+\alpha} \frac{\left\|\mu\right\|_{L^\infty(\Omega)}}{\mu(\Omega)}$.
\end{lem}
\begin{demo}[Lemma~\ref{lem:FPW_loc_p_dissip}]
We normalize $\mu$ to have $\mu\in\mathcal{P}(\Omega)$ (space of probability measures). For all $u\in L^p_\mu(\Omega)$ the following identity hold
\begin{align*}
0\ \leq\ \frac{1}{2}\iint_{\Omega^2} (u_*-u)(u_*^{p-1}-u^{p-1})\mu_*\mu \ & = \ \iint_{\Omega^2} u(u^{p-1}-u_*^{p-1})\mu_*\mu
\\
& = \ \iint_{\Omega^2}u^{p-1}(u-u_*)\mu_*\mu
\\
& = \ \int_\Omega u^{p-1}(u-\langle u\rangle_{\mu,\Omega})\mu.
\end{align*}
Hence, using that $|x-y|<2\,\mathrm{diam}(\Omega)$, we get
\begin{align*}
\int_\Omega u^{p-1}(u-\langle u\rangle_{\mu,\Omega})\mu \ & = \ \iint_{\Omega^2} \frac{(u_*-u)(u_*^{p-1}-u^{p-1})}{2|x-x_*|^{d+\alpha}}|x-x_*|^{d+\alpha}\mu_*\mu
\\
& \leq \ \mathrm{diam}(\Omega)^{d+\alpha}\|\mu\|_{L^\infty(\Omega)}\intd \Dp{p}(u)\mu.
\end{align*}
It concludes the proof.
\end{demo}
\begin{demo}[Proposition~\ref{prop:FPW_loc_p_dissip}]
Since $\langle v\rangle_{\mu} = 0$, we have
\begin{align*}
\int_\Omega |v|^p\mu \ & = \ \int_\Omega v^{p-1}(v-\langle v\rangle_{\mu,\Omega})\mu + \left(\int_\Omega v^{p-1}\mu\right) \frac{1}{\mu(\Omega)} \left(\int_\Omega v\mu\right)
\\
& = \ \int_\Omega v^{p-1}(v-\langle v\rangle_{\mu,\Omega})\mu - \frac{1}{\mu(\Omega)} \left(\int_\Omega v^{p-1}\mu\right) \left(\int_{\Omega^c} v\mu\right),
\end{align*}
and, by using Hölder's inequality, the second term can be bounded in the following way
\begin{align*}
\frac{1}{\mu(\Omega)}\left|\left(\int_\Omega v^{p-1}\mu\right)\left(\int_{\Omega^c} v\mu\right)\right| \ & \leq \ \frac{1}{\mu(\Omega)} \left\|v\right\|_{L^p_\mu(\Omega)}^{p-1}\mu(\Omega)^\frac{1}{p}\left\|v\right\|_{L^p_\mu(\Omega^c)}\mu(\Omega^c)^\frac{1}{q}
\\
& \leq \ \eps_\Omega \left\|v\right\|_{L^p_\mu(\Omega)}^{p-1}\left\|v\right\|_{L^p_\mu(\Omega^c)}.
\end{align*}
We apply Lemma~\ref{lem:FPW_loc_p_dissip} to conclude.
\end{demo}
\subsection{Lyapunov + Poincaré method}
The following proposition is nothing but the part of Theorem~\ref{th:cv} concerning $\gamma\in(2-\alpha,2)$, leading to polynomial convergence. It is inspired from \cite{bakry_rate_2008} where a Local Poincaré together with a Foster-Lyapunov condition are used in the case of the classical Laplacian to prove convergence in spaces of the form $L^2(F^{-1/2}M)$ where $M$ is an exponential or polynomial weight. As this technique strongly uses the formula for gradient of the product of two functions, which is not available for the fractional Laplacian, we work in spaces of the form $L^p((\lambda F^{1-p} + m^p)^{1/p})$ instead, and we use the fact that $F$ has polynomial decay at infinity.
\begin{prop}\label{prop:cv_poly}
Assume $\beta := \gamma-2 \in (-\alpha,0)$. Let $m = \weight{x}^k$ and $\bar{m} = \weight{x}^{\bar{k}}$ with $|\beta|<k<\bar{k}<\alpha\wedge1$ and $f\in L^p(\bar{m})$ be a solution to the $\eqref{eq:FFP}$ equation for $p < 1+\frac{k-|\beta|}{d+\alpha-k}$, $p<p_\gamma$ and $p<\frac{\alpha}{k}$ if $k>\alpha-\beta$. Then, the following polynomial convergence holds
\begin{equation*}
\|f-F\|_{L^p(m)} \ \lesssim \ \frac{1}{\weight{t}^{(\bar{k}-k)/|\beta|}} \|f^\mathrm{in}-F\|_{L^p(\bar{m})}.
\end{equation*}
\end{prop}
\begin{demo}[Proposition~\ref{prop:cv_poly}]
By replacing $f$ by $f-F$ and by conservation of the mass, we can assume $\langle f\rangle_{\R^d} = 0$. By Proposition~\ref{prop:positivity}, $F\gtrsim \frac{c}{\weight{x}^{d+\alpha}}$ and for $p\in(1,p_\alpha)$ with $p_\alpha' = \frac{d+\alpha}{k}$, we have $\eps_0 := kp - (p-1)(d+\alpha) > 0$. Therefore, we have
\begin{equation}\label{eq:sim}
F^{1-p} \ \lesssim \ \frac{m^p}{\weight{x}^{\eps_0}},
\end{equation}
and we deduce that $f\in L^p(F^{-1/q})$. Moreover $F\in L^1_+$ and $F\in L^\infty(m)$ from Proposition~\ref{prop:regu_L_infty}. Therefore, if $f\in L^p(F^{-1/q})$, by combining the fractional Poincaré-Wirtinger inequality (Proposition~\ref{prop:FPW_loc_p_dissip}) with \eqref{eq:dissip_entropy}, we get for a given $\Omega\subset\R^d$ bounded
\begin{equation}\label{eq:estim_F}
C_\mathrm{PW}\,\partial_t\left(\intd |f|^pF^{1-p}\right) \ \leq \ - \int_\Omega |f|^pF^{1-p} + \eps_\Omega \left\|f\right\|_{L^p_{F^{1-p}}(\Omega)}^{p-1}\left\|f\right\|_{L^p_{F^{1-p}}(\Omega^c)}.
\end{equation}
Moreover, from estimates \eqref{eq:estim_a_priori_E} and \eqref{eq:estim_a_priori_I_2}, for $kp<\alpha$, we have
\begin{equation*}
\frac{1}{p}\partial_t\left(\intd |f|^pm^p\right) \ \leq \ \intd |f|^pm^p \left(\frac{C}{\weight{x}^\alpha} + \varphi_{m,p}\right).
\end{equation*}
Or we can also use estimates \eqref{eq:estim_a_priori_E} and \eqref{eq:estim_a_priori_I} to deduce that for $k<\alpha$,
\begin{equation*}
\frac{1}{p}\partial_t\left(\intd |f|^pm^p\right) \ \leq \ \intd |f|^pm^p \left(\frac{C}{\weight{x}^{\alpha-k}} + \varphi_{m,p}\right).
\end{equation*}
From \eqref{eq:strict_confinement} and one of the two above estimates, if $kp<\alpha$ or $|\beta|<\alpha-k$, we get the Foster-Lyapunov like estimate
\begin{equation}\label{eq:estim_m}
\frac{1}{p}\partial_t\left(\intd |f|^pm^p\right) \ \leq \ \intd |f|^p \left(b\mathds{1}_\Omega - a \weight{x}^{kp+\beta}\right),
\end{equation}
for a given $(a,b)\in \R_+^2$. We define $M^p := m^p+\lambda C_\mathrm{PW} F^{1-p}$ in order to use the negative part of both estimates \eqref{eq:estim_F} and \eqref{eq:estim_m}. Adding the two expressions, we obtain
\begin{equation*}
\partial_t\left(\intd |f|^p M^p\right) \ \leq \ \intd |f|^p \left((b-\lambda F^{1-p})\mathds{1}_{\Omega} + \lambda\eps_\Omega F^{1-p} - a\weight{x}^{kp+\beta}\right).
\end{equation*}
Using the fact that $F\in L^\infty(m)$ and \eqref{eq:sim}, we obtain the existence of $c>0$ depending only on $F$ such that
\begin{equation*}
\partial_t\left(\intd |f|^p M^p\right) \ \leq \ \intd |f|^p \left((b-\lambda c )\mathds{1}_{\Omega} + \weight{x}^{kp}\left(\lambda\eps_\Omega c\weight{x}^{-\eps_0} - a\weight{x}^{-|\beta|}\right)\right).
\end{equation*}
Now we remark that since $|\beta| < k$, for $p < \frac{d+\alpha-|\beta|}{d+\alpha-k} = 1 + \frac{k-|\beta|}{d+\alpha-k}$, we obtain $-\eps_0 < \beta$. Taking also $\lambda > \frac{b}{c}$, we get
\begin{equation*}
\partial_t\left(\intd |f|^p M^p\right) \ \leq \ \intd |f|^p\weight{x}^{kp+\beta}\left(\lambda\eps_\Omega c - a\right).
\end{equation*}
Thus, by taking $\Omega$ large enough so that $\lambda\eps_\Omega c - a < 0$, we obtain the existence of $\bar{a}>0$ such that
\begin{equation}\label{eq:quasi_gap}
\partial_t\left(\intd |f|^p M^p\right) \ \leq \ -\bar{a}\intd |f|^p \weight{x}^{kp+\beta}.
\end{equation}
Let $\bar{k} \in (k,\alpha)$ and $\eps := p(\bar{k}-k)>0$. By Hölder's inequality, we have
\begin{equation*}
\intd |f|^p\weight{x}^{kp} \ \leq \ \left(\intd |f|^p\weight{x}^{kp+\beta}\right)^{\eps/(|\beta|+\eps)}\left(\intd |f|^p\weight{x}^{\bar{k}p}\right)^{|\beta|/(|\beta|+\eps)}.
\end{equation*}
Combining it with \eqref{eq:quasi_gap} and Proposition~\ref{prop:asympt} leads to
\begin{equation*}
\partial_t\left(\intd |f|^p M^p\right) \ \lesssim \ -\bar{a}\left(\intd |f|^pM^p\right)^{1+|\beta|/\eps}\left(\intd |f^\mathrm{in}|^p\weight{x}^{\bar{k}p}\right)^{-|\beta|/\eps},
\end{equation*}
where we used that by \eqref{eq:sim} and the positivity of $F$, we have $ \weight{x}^{kp} \leq M^p \lesssim \weight{x}^{kp}$. By Grönwall's inequality, we obtain
\begin{equation*}
\intd |f|^p m^p \ \leq \ \intd |f|^p M^p \ \lesssim \ \frac{1}{t^{\eps/|\beta|}} \intd |f^\mathrm{in}|^p\weight{x}^{\bar{k}p},
\end{equation*}
which gives the expected result.
\end{demo}
\section{Exponential Convergence to the equilibrium for $\gamma\geq 2$}
When $\gamma\geq 2$, the confinement is sufficiently strong to get an exponential time decay toward equilibrium for $|x|$ large. To get the local behavior, instead of using a local Poincaré inequality as in previous section, we will use the gain of positivity from Proposition~\ref{prop:positivity}.
\begin{prop}[Convergence in $L^1(m)$]\label{prop:cv_harris}
Assume $\gamma \geq 2$ and let $f$ be a solution of the \eqref{eq:FFP} equation with $f^\mathrm{in}\in L^1(m)$. Then, there exists $\bar{a}>0$ such that for any $t\in\R_+$
\begin{equation*}
\|f(t)-F\|_{L^1(m)} \ \leq \ e^{-\bar{a}t}\|f^\mathrm{in}-F\|_{L^1(m)}.
\end{equation*}
\end{prop}
\begin{demo}[Proposition~\ref{prop:cv_harris}]
We want here to use the strategy from Hairer and Mattingly in \cite{hairer_yet_2011} so that we use the following notations $P_t := e^{t\Lambda^*}$, $X := L^1(m)$ and $X' = L^\infty(m^{-1})$ where $m=\weight{x}^k$ with $k\in(0,\alpha\wedge1)$. We recall that from Theorem~\ref{th:existence} we immediately deduce by duality that $P : \R_+ \to \B(X')$ is a positive $C^0$-semigroup such that $P_t 1 = 1$. The strategy consists in proving the following Lyapunov and positivity conditions.
\step{1. Lyapunov condition} Since $E\cdot x \gtrsim |x|^2$, by Proposition~\ref{prop_I_m2}, we have
\begin{equation*}
\Lambda^* m \ = \ \I(m) - E\cdot\nabla m \ \leq \ b - a m.
\end{equation*}
Moreover, by using Duhamel's formula, we have
\begin{equation*}
e^{(\Lambda^*+a)t} \ = \ m + e^{(\Lambda^*+a)t} \star (\Lambda^*+a) m.
\end{equation*}
Therefore, we obtain
\begin{equation*}
e^{at}P_t m \ \leq \ m + \int_0^t e^{as}P_s b \d s\ \leq\ m + be^{at}/a,
\end{equation*}
from what we deduce
\begin{equation}\label{eq:markov_lyapunov}
P_t m \ \leq \ \gamma_tm + c,
\end{equation}
with $c = b/a$ and $\gamma_t = e^{-at}\in(0,1)$.
\step{2. Positivity condition} From Proposition~\ref{prop:positivity}, we know that there exists $\nu_t(x) = \nu(t,x)\in L^\infty(\R_+,L^1_+(m))$ strictly positive such that for any $f\in L^1_+(m)$, we have
\begin{equation*}
e^{t\Lambda}f \ \geq \ \nu_t\ \int_{B_R} f.
\end{equation*}
By duality, it implies that
\begin{equation}\label{eq:markov_strict_positivity}
P_t \geq \langle \nu_t,\cdot\rangle\mathds{1}_{m(x)<r}.
\end{equation}
where $r = m(R)$.
\step{3. Convergence in $L^1(m)$}
We define $m_\lambda := 1+\lambda m$ and the following seminorm on $L^\infty(m^{-1})$
\begin{equation*}
|\varphi|_{\dot{L}^\infty(m_\lambda^{-1})} \ := \ \sup_{(x,y)\R^{2d}} \left(\frac{|\varphi(x)-\varphi(y)|}{m_\lambda(x)+m_\lambda(y)}\right).
\end{equation*}
Then as proved in \cite[Lemma~2.1]{hairer_yet_2011}, we have
\begin{equation}\label{eq:seminorm_lambda}
|\varphi|_{\dot{L}^\infty(m_\lambda^{-1})} \ = \ \inf_{c\in\R}\|\varphi-c\|_{L^\infty(m_\lambda^{-1})}.
\end{equation}
Moreover, \cite[Theorem~3.1]{hairer_yet_2011} tells us that since \eqref{eq:markov_lyapunov} and \eqref{eq:markov_strict_positivity} imply that for any fixed time $t>0$ there exists a constant $\bar{\gamma}_t\in(0,1)$ such that
\begin{equation}\label{eq:markov_contraction}
|P_t\varphi|_{\dot{L}^\infty(m_\lambda^{-1})} \ \leq \ \bar{\gamma}_t|\varphi|_{\dot{L}^\infty(m_\lambda^{-1})}.
\end{equation}
By using the semigroup property, we obtain that the optimal $\bar{a}_t := -\ln(\bar{\gamma}_t) > 0$ verifies $\bar{a}_{t+s}\geq \bar{a}_t+\bar{a}_s$, from what we deduce the existence of $\bar{a}>0$ such that
\begin{equation*}
\inf_{c\in\R}\|P_t\varphi-c\|_{X'} \ \leq \ e^{-\bar{a}t}\|\varphi\|_{X'},
\end{equation*}
where we replaced $L^\infty(m_\lambda^{-1})$ by $X'$ by equivalence of the norms. Take a sequence $(c_n)_{n\in\N}$ converging to the minimizer. Then, we can write for $f^\mathrm{in}\in L^1(m)$ such that $\langle f^\mathrm{in}\rangle_{\R^d} = 0$
\begin{align*}
\langle e^{t\Lambda}f^\mathrm{in},\varphi\rangle_{X,X'} \ & = \ \langle f^\mathrm{in},P_t\varphi-c_n\rangle_{X,X'}
\\
& \leq \ \|f^\mathrm{in}\|_{L^1(m)} \|P_t\varphi-c_n\|_{X'}.
\end{align*}
Passing to the limit $n\to\infty$, for $f(t) := e^{t\Lambda}f^\mathrm{in}$, we get
\begin{equation*}
\|f\|_{L^1(m)} \ = \ \sup_{\|\varphi\|_{X'}\leq 1}\langle f,\varphi\rangle_{X,X'} \ \leq \ e^{-\bar{a}t}\|f^\mathrm{in}\|_{L^1(m)}.
\end{equation*}
Proposition~\ref{prop:cv_harris} follows by taking $f-F$ instead of $f$.
\end{demo}
\begin{demo}[Theorem~\ref{th:cv}]
The part concerning polynomial convergence when $\alpha\in(2-\alpha,2)$ was proved in Proposition~\ref{prop:cv_poly}. Therefore we just have to prove the part concerning exponential convergence when $\alpha\geq 2$. Thanks to the regularization property of the semigroup from $L^1(m)$ to $L^p(m)$ as proved in Proposition~\ref{prop:regu}, we know that
\begin{equation*}
\|f-F\|_{L^p(m)} \lesssim \left(c+t^{-\frac{d}{q\alpha}}\right) e^{t\lambda_1} \|f^\mathrm{in}-F\|_{L^1(m)}.
\end{equation*}
where $\lambda_1$ is exactly such that
\begin{equation*}
\|f-F\|_{L^1(m)} \lesssim e^{t\lambda_1} \|f^\mathrm{in}-F\|_{L^1(m)}.
\end{equation*}
From Proposition~\ref{prop:cv_harris}, we deduce that $\lambda_1 = -\bar{a} < 0$, which gives the result.
\end{demo}
\renewcommand{\bibname}{\centerline{Bibliography}}
\bibliographystyle{abbrv}
|
{
"timestamp": "2018-03-08T02:08:56",
"yymm": "1803",
"arxiv_id": "1803.02672",
"language": "en",
"url": "https://arxiv.org/abs/1803.02672"
}
|
\section{Introduction \label{sec:Introduction}}
The determination of collinear parton distribution functions (PDFs)
of the nucleon is becoming an increasingly precise discipline with
the advent of high-luminosity experiments at both colliders and fixed-target
facilities. Several research groups are involved in the rich research
domain of the modern PDF analysis \cite{Dulat:2015mca,Harland-Lang:2014zoa,Ball:2017nwa,Alekhin:2017kpj,Accardi:2016qay,Abramowicz:2015mha,Alekhin:2014irh}.
By quantifying the distribution of a parent hadron's longitudinal
momentum among its constituent quarks and gluons, PDFs offer both
a description of the hadronic structure and an essential ingredient
of perturbative QCD computations. PDFs enjoy a symbiotic relationship
with high-energy experimental data, in the sense that they are crucial
for understanding hadronic collisions in the Standard Model (SM) and
beyond, while reciprocally benefiting from a wealth of high-energy
data that constrain the PDFs. In fact, since the start of the Large
Hadron Collider Run II (LHC Run II), the volume of experimental data
pertinent to the PDFs is growing with such speed that keeping pace
with the rapidly expanding datasets and isolating measurements of
greatest impact presents a significant challenge for PDF fitters.
This paper intends to meet this challenge by presenting a method for
identifying high-value experiments which constrain the PDFs and the
resulting SM predictions that depend on them.
That such expansive datasets can constrain the PDFs is a consequence
of the latter's universality \textemdash{} a feature which relies
upon QCD factorization theorems to separate the inherently nonperturbative
PDFs (at long distances) from process-dependent, short-distance matrix
elements. For instance, the cross section for inclusive single-particle
hadroproduction (of, \textit{e.g.}, a weak gauge boson $W/Z$) in
proton-proton collisions at the LHC is directly sensitive to the nucleon
PDFs via an expression of the form
\begin{align}
& \sigma(AB\rightarrow W/Z\!+\!X)\ =\ \sum_{n}\,\alpha_{s}^{n}(\mu_{R}^{2})\,\sum_{a,b}\int dx_{a}dx_{b}\,\label{eq:fact}\\
& \times\,f_{a/A}(x_{a},\mu^{2})\,\hat{\sigma}_{ab\rightarrow W/Z+X}^{(n)}\big(\hat{s},\,\mu^{2},\mu_{R}^{2}\big)\,f_{b/B}(x_{b},\mu^{2})\ ,\nonumber
\end{align}
in which $f_{a/A}(x_{a},\mu^{2})$ represents the PDF for a parton
of flavor $f_{a}$ carrying a fraction $x_{a}$ of the 4-momentum
of proton $p_{A}$ at a factorization scale $\mu$; the $n^{\mathit{th}}$-order
hard matrix element is denoted by $\hat{\sigma}_{ab\rightarrow W/Z+X}^{(n)}\big(\hat{s},\,\mu^{2},\mu_{R}^{2}\big)$
and is dependent upon the partonic center-of-mass energy $\hat{s}=x_{a}x_{b}s$,
in which $s$ in the center-of-mass energy of the initial hadronic
system; and $\mu_{R}$ is the renormalization scale in the QCD coupling
strength $\alpha_{s}(\mu_{R})$. In Eq.~(\ref{eq:fact}), subleading
corrections $\sim\!\!\Lambda^{2}/M_{W/Z}^{4}$ have been omitted,
and we emphasize that factorization theorems like Eq.~(\ref{eq:fact})
have been proved to arbitrary order in $\alpha_{s}$ for essential
observables in the global PDF analysis, such as the inclusive cross
sections in DIS and Drell-Yan processes. For compactness and generality,
we shall refer henceforth to a PDF for the parton of flavor $f$ simply
as $f(x,\mu)$.
Given this formalism, one is confronted with the problem of finding
those experiments that provide reliable new information about the
PDF behavior. With the proliferation of potentially informative new
data, incorporating them all into a global QCD fit inevitably incurs
significant cost both in terms of computational resources and required
fitting time. Indeed, tremendous progress in the precision of PDFs
and robustness of SM predictions is driven by the technology for performing
global analysis that has vastly grown in complexity and sophistication.
Nowadays, the state-of-the-art in perturbative QCD (pQCD) treatments
are done at NNLO (and increasingly even N$^{3}$LO), and advanced
statistical techniques are commonly employed in PDF error estimation.
The magnitude of this subject is vast, and we refer the interested
reader to Refs.~\cite{Gao:2017yyd,Butterworth:2015oua} for comprehensive
reviews. The tradeoff of this progress is that the impact of an experiment
on the ultimate PDF uncertainty is often hard to foresee without doing
a complicated fit. Various publications claim sensitivity of
new experiments to the PDFs. In this paper, we look into these claims
using statistical techniques that bypass doing the fits,
and with an eye on theoretical, experimental, and methodological components
relevant at the NNLO precision.
The potential cost is steepened by the large size of the global datasets
usually involved. This point can be seen in Fig.~\ref{fig:data},
which plots the default dataset considered in the present analysis
in a space of partonic momentum fraction $x$ and factorization scale
$\mu$. We label these data as the ``CTEQ-TEA set,'' given that
it is an extension of the 3287 raw data points (given by the sum over
$N_{\mathit{pt}}$ in Tables~\ref{tab:EXP_1} and \ref{tab:EXP_2}) treated
in the NNLO CT14HERA2 analysis of Ref.~\cite{Hou:2016nqm}, now augmented
by the inclusion of 734 raw data points (given by the sum over $N_{\mathit{pt}}$
in Table~\ref{tab:EXP_3}) from more recent LHC data. These raw measurements
can ultimately be mapped to 5227 typical $\{x,\mu\}$ values
in Fig.~\ref{fig:data}, such that each symbol corresponds to a data point from
an experiment shown in the legend, at the approximate $x$ and $\mu$
values characterizing the data point as described in Appendix~\ref{sec:supp}.
The experiments are labeled by a short-hand name which includes the
year of final publication ({\it e.g.}, ``HERAI+II'15'' --- corresponding to the
2015 combined HERA Run I and Run II data),
following the translation key also given in Tables~\ref{tab:EXP_1}\textendash \ref{tab:EXP_3}
of App. \ref{sec:Tables}. The experiments included in the CT14HERA2
analysis are listed in the left column and upper part of the right column of the legend, while
the newer LHC data considered for the upcoming CTEQ-TEA analysis are the last 14
entries of the right column.
\begin{figure*}
\includegraphics[width=1\textwidth]{figs/xQbyexpt_xQ_replot.pdf}
\caption{A graphical representation of the space of $\{x,\mu\}$ points probed
by the full dataset treated in the present analysis, designated as
``CTEQ-TEA''. It represents an expansion to include newer LHC data of the CT14HERA2 dataset
\cite{Hou:2016nqm} fitted in the most recent CT14 framework \cite{Dulat:2015mca},
which involved measurements from Run II of HERA \cite{Abramowicz:2015mha}.
Details of the datasets corresponding to the short-hand names given in the legend
may be found in Tables~\ref{tab:EXP_1}--\ref{tab:EXP_3}.
}
\label{fig:data}
\end{figure*}
The growing complexity of PDF fitting stimulates development of
less computationally involved approaches to estimate the
impact of new experimental data on full global fits, such as Hessian
profiling techniques \cite{Camarda:2015zba} and Bayesian reweighting
\cite{Ball:2010gb,Ball:2011gg} of PDFs. Although these approaches
do simulate the expansion of a particular global fit
by including theretofore absent dataset(s), they are also limited in
that the interpretation of their outcomes is married to the specific
PDF parametrization and definition of PDF errors. For example, conclusions
obtained by PDF reweighting regarding the importance of a given data
set strongly depend on the assumed statistical tolerance or the choice
of reweighting factors \cite{Sato:2013ika,Paukkunen:2014zia}.
Parallel to these efforts, the notion of using correlations between
the PDF uncertainties of two physical observables was proposed in
Refs.~\cite{Pumplin:2001ct,Nadolsky:2001yg} as a means of quantifying
the degree to which these quantities were related based upon their
underlying PDFs. The PDF-mediated correlation $C_{f}$ in this case,
which we define in Sec.~\ref{sec:Correlations}, embodies
the Pearson correlation coefficient computed by a generalization of
the ``master formula'' \cite{Pumplin:2002vw} for the Hessian PDF
uncertainty. The Hessian correlation was deployed extensively in Ref.~\cite{Nadolsky:2008zw}
to explore implications of the CTEQ6.6 PDFs for envisioned LHC observables.
It proved to be instrumental for identifying the specific PDF flavors
and $x$ ranges most correlated with the PDF uncertainties for $W,$
$Z,$ $H$, and $t\bar{t}$ production cross sections as well as other
processes. The Pearson correlation coefficient has also
proven to be informative in the approach based on Monte-Carlo PDF replicas,
see, e.g., Refs.~\cite{Ball:2008by,Carrazza:2016htc}.
However, the PDF-mediated correlation with a theoretical
cross section is only partly indicative of the sensitivity of the
experiment. The constraining power of the experiment also depends
on the size of experimental errors that were not normally considered
in correlation studies, as well as on correlated systematic effects
that are increasingly important.
As a remedy to these limitations, we introduce a new format for the
output of CTEQ-TEA fits and a natural extension of the correlation
technique to quantify the sensitivity of any given experimental data
point to a PDF-dependent observable of the user's choice. In this
approach, we work with \emph{statistical residuals} quantifying the
goodness-of-fit to individual data points. We demonstrate that the
complete set of residuals computed for Hessian PDF sets characterizes
the CTEQ-TEA fit well enough to permit a means of gauging the influence
of empirical information on PDFs in a fashion that does not require
complete refits.
A generalization of the PDF-mediated correlations called
the \textit{sensitivity $S_{f}$} \textemdash{} to be characterized
in detail in Sec.~\ref{sec:Sensitivities} \textemdash{}
better identifies those experimental data points that tightly
constrain PDFs both by merit of their inherent precision and their
ability to discriminate among PDF error fluctuations. Such an approach
aids in identifying regions of $\{x,\mu\}$ for which PDFs are
particularly constrained by physical observables.
\begin{figure*}
\includegraphics[width=0.47\textwidth]{figs/corr_xQ+1_f8_samept_replot.pdf}\ \ \
\includegraphics[width=0.47\textwidth]{figs/corrdr_xQ+1_f8_samept_replot.pdf}
\caption{
For the full CTEQ-TEA dataset of Fig.~\ref{fig:data}, we show the
absolute correlation $|C_{f}|$ and sensitivity $|S_{f}|$ associated
with the 14 TeV Higgs production cross section $\sigma_{H^{0}}(14\,\mathrm{TeV})$.
310 input data points with most significant magnitudes of $|C_{f}|$
and $|S_{f}|$ are highlighted with color.
When only the $|C_{f}|$ plot is considered, only a very small
subpopulation of jet production data (diagonal open circles and closed squares
with $\mu\gtrsim100$ GeV) exhibits significant correlations with
$|C_f|>0.7$ (orange and red colors),
as well as some HERA DIS, high-$p_T$ $Z$ boson, and $t\bar{t}$ production data points.
Our novel definition for the sensitivity in the right panel, on the other
hand, reveals more points that have comparable potency for constraining
the Higgs cross section. In this case, a larger fraction of the jet production
points is important (especially CMS measurements of CMS8jets'17 and CMS7jets'14),
as well as a number of other processes at smaller $\mu$, particularly
DIS data from HERA, BCDMS, NMC, CDHSW, and CCFR (experiments HERAI+II'15, BCDMSd'90,
NMCrat'97, CDHSW-F2'91, CCFR-F2'01, CCFR-F3'97). Although its cumulative impact
is comparatively modest, ATLAS $t\bar{t}$
production data (ATL8ttb-pt'16, ATL8ttb-y\_ave'16, ATL8ttb-mtt'16, ATL8ttb-y\_ttb'16)
register significant per-point sensitivities, as do E866 $pp$ Drell-Yan pair production
(E866pp'03), LHCb $W, $Z production (LHCb7ZWrap'15, LHCb8WZ'16), and charge lepton
asymmetries at D0 and CMS (D02Masy'08, CMS7Masy2'14, CMS7Easy'12). Similarly,
some of the high-$p_T$ $Z$ production information (ATL7ZpT'14, ATL8ZpT'16)
from ATLAS provide modest constraints.
\label{fig:CorrSensH14}}
\end{figure*}
In fact, in the numerical approach presented in the forthcoming sections,
the user can quantify the sensitivity of data not only to individual
PDF flavors, but even to specific physical observables, including
the modifications due to correlated systematic uncertainties in every
experiment of the CT14HERA2 analysis. For example, for Higgs boson
production via gluon fusion ($gg\rightarrow H$) at the LHC 14 TeV, the short-distance
cross sections are known up to N$^{3}$LO with a scale uncertainty
of about 3\% \cite{Anastasiou:2016cez}. It has been suggested that
$t\bar{t}$ production and high-$p_{T}$ $Z$ boson production on their own
constrain the gluon PDF in the $x$ region
sensitive to the LHC Higgs production, and that these are comparable to the
constraints from LHC and Tevatron data \cite{Czakon:2016olj,Boughezal:2017nla}.
Verifying the degree to which this hypothesis is true has been difficult
without actually including all these data in a fit.
As an alternative to doing a full global fit, we can critically assess
this supposition in the context of the entire global dataset
of Fig.~\ref{fig:data} using the Hessian correlations and sensitivities,
$|C_{f}|$ and $|S_{f}|$.
The detailed procedure is explained in Secs.~\ref{sec:Correlations} and \ref{sec:Sensitivities}.
In the example at hand, we could rely on the established wisdom that
the theoretical cross sections that have an especially large correlation
with $\sigma_{H^{0}}$ may constrain the PDF dependence of
$\sigma_{H^{0}}$; say, when $|C_f|\gtrsim 0.7$ \cite{Nadolsky:2008zw}.
Along this reasoning, the left frame in Fig.~\ref{fig:CorrSensH14}
illustrates 310 experimental data points in $\{x,\mu\}$ space
that have the highest absolute correlation, $|C_{f}|$, between the point's
statistical residual defined in
Sec.~\ref{sec:Correlations} and the cross section
$\sigma_{H^{0}}$ at 14 TeV via the CT14HERA2 NNLO PDFs.
To locate such points in the figure,
we highlighted them with color according to
the convention shown on the color scale to the right.
The respective $|C_f|$ for the highlighted data points ranges between
0.42 and 1. The rest of the data points have smaller correlations and are
shown in gray.
We find that the 310 data points with the highest correlation
for $\sigma_{H^{0}}$ belong to 20 experiments. Nearly all of them
are contributed by HERA Neutral Current (NC) DIS,
LHC and Tevatron jet production, and HERA charm production.
Some of the data points with highest $|C_f|$ come from high-$p_T$ $Z$ boson
and even $t\bar t$ production.
The correlations $C_f$, however, do not reflect the experimental
uncertainties, which vary widely across the experiments.
In the left panel of Fig.~\ref{fig:CorrSensH14},
fewer than 30 points have a strong correlation of $0.7$ or more; but
more data points impose relevant constraints in the global fit.
To
include the information about the uncorrelated and correlated
experimental errors, in the right panel
of Fig.~\ref{fig:CorrSensH14}, we plot the distributions of 310 data
points with the highest sensitivity parameter $S_f$, which more faithfully
reproduces the actual constraints during the fitting.
In general, we find substantial differences between the
$C_f$ and $S_f$ distributions. Even the most significant correlations,
of order $|C_{f}| \sim 0.7$ and above, do not guarantee a significant
contribution of the experimental point to the log-likelihood
$\chi^2$ if the errors are large. On the other hand,
we argue that $|S_f|$ is closely related to a
contribution of the data point to $\chi^2$. According to the
distribution in the right figure, the 310 data points with the highest
sensitivity $|S_f|$ to $\sigma_{H^0}(\mbox{14 TeV})$ arise from 27
experiments. Among these data points, only some have
a large correlation $|C_f|$ with
$\sigma_{H^0}(\mbox{14 TeV})$. Nonetheless, they have medium-to-large
sensitivity, $|S_f| > 0.21$, according to the criterion developed
in Sec.~\ref{sec:Sensitivities}.
We stress that, while one might suggests plausible dynamical reasons why
certain experiments might be particularly sensitive to Higgs production via
the gluon PDF, ({\it e.g.}, via the leading-order $qg$ and $gg$
hard cross sections in jet production and DGLAP scaling violations in
inclusive DIS), this reasoning alone does not predict the actual
sensitivity revealed by $S_f$ in the presence of multiple experimental
constraints.
As one noticeable difference from the $|C_f|$ figure,
while inclusive DIS at HERA continues to
contribute a large number of data points (about 80) with a high
$|S_f|$, also the fixed-target DIS experiments (BCDMS, NMC, CDHSW,
CCFR) contribute about the same number of sensitive points in the
right panel that were not identified by large correlations. Other sensitive
points belong to the jet production
data sets from ATLAS and CMS and some vector boson production
experiments (muon charge asymmetries at D0, CMS;
E866 low-energy Drell-Yan production; LHCb 7 TeV $W$ and $Z$ cross
sections).
On the other
hand, HERA charm production, ATLAS 7/8 TeV high-$p_T$ $Z$ production,
have suppressed sensitivities despite their large correlations,
reflecting the larger experimental uncertainties in these measurements.
While the LHC $t\bar t$ production experiments have large per-point
sensitivities, they contribute relatively little to $\chi^2$ because
of their small total number of data points. From this comparison,
one finds, perhaps somewhat unexpectedly, that fixed-target DIS
experiments impose important constraints on $\sigma_{H^0}(\mbox{14
TeV})$, thus complementing the HERA inclusive DIS data.
One would conclude that efforts to constrain PDF-based SM predictions
for Higgs production by relying only on a few points of $t\overline{t}$
data, but to the neglect of high-energy jet production points, would
be significantly handicapped by the absence of the latter. We will
return to this example in Sec.~\ref{sec:CaseCTEQ-TEA}.
The discriminating power of a sensitivity-based analysis
therefore forms the primary motivation
for this work, and we present the attendant details below. To assess
information about the PDFs encapsulated in the residuals for large
collections of hadronic data implemented in the CTEQ-TEA global analysis,
we make available a new statistical package \textsc{PDFSense} to map
the regions of partonic momentum fractions $x$ and QCD factorization
scales $\mu$ where the experiments impose strong constraints on the
PDFs. In companion studies,
we have applied \textsc{PDFSense} to select new data sets
for the next generation of the
CTEQ-TEA global analysis, to quantitatively explore the physics
potential for constraining the PDFs at a future
Electron-Ion Collider (EIC) \cite{Accardi:2012qut,Boer:2011fh,Abeyratne:2012ah,Aschenauer:2014cki}
and Large Hadron-Electron Collider (LHeC) \cite{AbelleiraFernandez:2012cc}, and to investigate the
potential of high-energy data to inform lattice-calculable
quantities \cite{Lin:2017snn} like the Mellin moments of structure functions \cite{Gockeler:1995wg} and quark quasi-distributions \cite{Ji:2013dva}.
We reserve many instructive results for follow-up publications currently in preparation,
while presenting select calculations in this article to demonstrate the power of the method. We find that the
sensitivity technique generally agrees with the preliminary
CTEQ-TEA fits and Hessian reweighting realized in the \textsc{ePump} program
\cite{Schmidt:2018hvu}. However, assessing the sensitivity is much
simpler than doing the global fit. It does not require access to a fitting
program or the application of (potentially subtle) PDF reweighting techniques.
The remainder of the article proceeds as follows. Pertinent aspects
of the PDFs and their standard determination via QCD global analyses
are summarized in Sec.~\ref{sec:PDF-preliminaries}. Then,
we introduce {\it normalized residual variations} to extract,
visualize, and quantify statistical information about the global QCD
fit. In Sec.~\ref{sec:QuantifyingDistributionsOfResiduals}, we
construct a number of statistical quantities that
characterize the PDF constraints in the global analysis using the
residual variations. In Sec.\ \ref{sec:CaseCTEQ-TEA}, we apply
the thus constructed sensitivity parameter
to examine the impact of various CTEQ-TEA datasets
on extractions of the gluon PDF $g(x,\mu)$. In this section and
in the conclusion contained
in Sec.\ref{sec:Conclusions}, we emphasize a number of {\it physics insights}
that we obtained by applying our sensitivity analysis techniques. Additional
aspects of the technique and supplementary tables are reserved for
Apps.~\ref{sec:supp}, \ref{sec:Tables}, and \ref{sec:SM}.
\section{PDF preliminaries \label{sec:PDF-preliminaries}}
\subsection{Data residuals in a global QCD analysis \label{sec:Data-residuals}}
While various theoretical models exist for computing nucleon PDFs
\cite{Farrar:1975yb,Hobbs:2014lea,Hobbs:2013bia},
unambiguous evaluation of the PDFs entirely in terms of
QCD theory is not yet possible due to the fact that the PDFs can in
general receive substantial nonperturbative contributions at infrared
momenta. For this reason, precise PDF determination has proceeded
mainly through the technique of the QCD global analysis \textemdash{}
a method enabled by QCD factorization and PDF universality.
In this approach, a highly flexible parametric form is ascribed for
the various flavors in a given analysis at a relatively low scale
$Q_{0}^{2}$. For example, one might take the input PDF for a given
quark flavor $f$ to be a parametric form
\begin{equation}
f(x,\mu^{2}=Q_{0}^{2})=A_{f,0}\,x^{A_{f,1}}(1-x)^{A_{f,2}}\,F(x;\,A_{f,3},\dots)\ ,\label{eq:fitform}
\end{equation}
in which $F(x;\:A_{f,3},\dots)$ can be a suitable polynomial function,
\textit{e.g.}, a Chebyshev or Bernstein polynomial, or replaced with
a feed-forward neural network $\mathrm{NN}_{f}(x)$ as in the NNPDF
approach. While the full statistical theory for PDF determination
and error quantification is beyond the intended range of this analysis,
roughly speaking, a best fit is found for a vector $\vec{A}$ of $N$
PDF parameters $A_{l}$ by minimizing a goodness-of-fit function $\chi^{2}$
describing agreement of the QCD data and physical observables computed
in terms of the PDFs. Based on the behavior of $\chi^{2}$ in the
neighborhood of the global minimum, it is then possible to construct
an ensemble of error PDFs to quantify uncertainties of PDFs at a predetermined
probability level.
There are various ways to evaluate uncertainties on PDFs,
\emph{e.g.}, the Hessian \cite{Pumplin:2001ct,Pumplin:2002vw},
the Monte Carlo \cite{Giele:1998gw,Giele:2001mr}, and the
Lagrange Multiplier approaches \cite{Stump:2001gu}. In this analysis
our default PDF input set is CT14HERA2, which uses the Hessian method
to estimate uncertainties and is therefore based on the quadratic
assumption for $\chi^{2}(\vec{A}$) in the vicinity of the global
minimum. In the Hessian method, an orthonormal basis of PDF parameters
$\vec{a}$ is derived from the input PDF parameters $\vec{A}$ by
the diagonalization of a Hessian matrix $H$, which encodes the second-order
derivatives of $\chi^{2}$ with respect to $A_{l}$. The eigenvector
PDF combinations $\vec{a}_{l}^{\pm}$ are found for two extreme variations
from the best-fit vector $\vec{a}_{0}$ along the direction of the
$l^{th}$ eigenvector of $H$ allowed at a given probability level.
The uncertainty on a QCD observable $X$ can then be estimated with
one of the available ``master formulas'' \cite{Pumplin:2002vw,Nadolsky:2001yg},
the ``symmetric'' variety of which is
\begin{align}
\Delta X & =\frac{1}{2}\sqrt{\sum_{l=1}^{N}(X_{l}^{+}-X_{l}^{-})^{2}}\ .\label{DelX}
\end{align}
In the CTEQ-TEA global analysis, the $\chi^{2}$ function accounts
for multiple sources of experimental uncertainties, as well as for
some prior theoretical constraints on the $a_{l}$ parameters. Consequently,
the global $\chi^{2}$ function takes the form
\begin{equation}
\text{\ensuremath{\chi}}_{global}^{2}=\sum_{E}\chi_{E}^{2}+\chi_{th}^{2}\ ,
\label{eq:chi2glob}
\end{equation}
where the sum runs over all experimental datasets $(E);$ and $\chi_{th}^{2}$
imposes theoretical constraints. The complete formulas for $\chi_{E}^{2}$
and $\chi_{th}^{2}$ can be found in Ref.~\cite{Gao:2013xoa}. For
the purposes of this paper, we express $\chi_{E}^{2}$ for each experiment
$E$ in a compact form as a sum of squared\emph{ shifted residuals}
$r_{i}^{2}(\vec{a})$, which are summed over $N_{\mathit{pt}}$ individual
data points $i$ in this experiment, as well as the contributions
of $N_{\lambda}$ best-fit nuisance parameters $\overline{\lambda}_{\alpha}$
associated with correlated systematic errors:
\begin{align}
\chi_{E}^{2}(\vec{a}) & =\sum_{i=1}^{N_{\mathit{pt}}}\,r_{i}^{2}(\vec{a})+\sum_{\alpha=1}^{N_{\lambda}}\overline{\lambda}_{\alpha}^{2}(\vec{a})\ .\label{eq:chi2}
\end{align}
In turn, $r_{i}(\vec{a})$ for the $i^{th}$ data point is constructed
from the theoretical prediction $T_{i}(\vec{a})$ evaluated in terms
of PDFs, total uncorrelated uncertainty $s_{i}$, and the shifted
central data value $D_{i,sh}(\vec{a})$:
\begin{align}
r_{i}(\vec{a}) & =\frac{1}{s_{i}}\,\big(T_{i}(\vec{a})-D_{i,\mathit{sh}}(\vec{a})\big)\ .\label{eq:residual}
\end{align}
This representation arises in the Hessian formalism due to the presence
of correlated systematic errors in many experimental datasets, which
require $\chi_{E}^{2}$ to depend on nuisance parameters $\lambda_{\alpha}$.
This is in addition to the dependence of $\chi_{E}^{2}$ on the PDF
parameters $\vec{a}$ and theoretical parameters such as $\alpha_{s}(M_{Z})$
and particle masses. The $\lambda_{\alpha}$ parameters are optimized
for each $\vec{a}$ according to the analytic solution derived in
Appendix B of Ref.~\cite{Pumplin:2002vw}. Optimization effectively
shifts the central value $D_{i}$ of the data point by an amount determined
by the optimal nuisance parameters $\overline{\lambda}_{\alpha}(\vec{a})$
and the correlated systematic errors $\beta_{i\alpha}:$
\begin{equation}
D_{i}\rightarrow D_{i,\mathit{sh}}(\vec{a})=D_{i}-\sum_{\alpha=1}^{N_{\lambda}}\beta_{i\alpha}\overline{\lambda}_{\alpha}(\vec{a})\ .
\end{equation}
It should be noted that the contribution of the squared best-fit nuisance
parameters to $\chi_{E}^{2}$ in Eq.~(\ref{eq:chi2}) is dominated
in general by the first term involving the shifted residuals, which
tends to be much larger \textemdash{} especially for more sizable
datasets.
We point out also that some alternative representations for $\chi^{2}$
include the correlated systematic errors via a covariance matrix $\left(\mbox{cov}\right)_{ij}$,
rather than the above mentioned CTEQ-preferred form that explicitly
operates with $\lambda_{\alpha}$. Various $\chi^{2}$ definitions
in use are reviewed in \cite{Ball:2012wy}, as well as in \cite{Alekhin:2014irh}.
Crucially, however, the representations based upon operating with
$\lambda_{\alpha}$ and $\left(\mbox{cov}\right)_{ij}$ are derivable
from each other \cite{Gao:2013xoa}. From an extension of the derivation
in Ref.~\cite{Pumplin:2002vw}, we may relate the shifted residual
to the covariance matrix at an $i^{th}$ point and optimal nuisance
parameters as
\begin{align}
r_{i}(\vec{a})\ & =\ s_{i}\sum_{j=1}^{N_{\mathit{pt}}}(\mathrm{cov}^{-1})_{ij}\,\left(T_{j}(\vec{a})-D_{j}\right),\label{eq:res-cov}\\
\overline{\lambda}_{\alpha}(\vec{a}) & =\sum_{i,j=1}^{N_{\mathit{pt}}}(\mathrm{cov}^{-1})_{ij}\frac{\beta_{i\alpha}}{s_{i}}\frac{\left(T_{j}(\vec{a})-D_{j}\right)}{s_{j}},
\end{align}
where
\begin{equation}
(\mathrm{cov}^{-1})_{ij}\ =\ \left[\frac{\delta_{ij}}{s_{i}^{2}}\,-\,\sum_{\alpha,\beta=1}^{N_{\lambda}}\frac{\beta_{i\alpha}}{s_{i}^{2}}A_{\alpha\beta}^{-1}\frac{\beta_{j\beta}}{s_{j}^{2}}\right]\ ,\label{eq:covmat}
\end{equation}
and
\begin{equation}
A_{\alpha\beta}\ =\ \delta_{\alpha\beta}\,+\,\sum_{k=1}^{N_{\mathit{pt}}}\frac{\beta_{k\alpha}\beta_{k\beta}}{s_{k}^{2}}\ .
\end{equation}
Thus, even for those PDF analyses which operate with the covariance
matrix one is still able to determine the shifted residuals $r_{i}$
from $\left(\mbox{cov}^{-1}\right)_{ij}$ using Eq.~(\ref{eq:res-cov}).
In this article, we conveniently follow the CTEQ methodology and obtain
$r_{i}(\vec{a})$ directly from the CTEQ-TEA fitting program, together
with the optimal nuisance parameters $\overline{\lambda}_{\alpha}(\vec{a})$
and shifted central data values $D_{i,sh}(\vec{a}).$
\subsection{Visualization of the global fit with the help of residuals}
The shifted residuals $r_{i}$ draw our interest because, in consequence
of the definitions in Eqs.~(\ref{eq:chi2})-(\ref{eq:residual}),
they contain substantial low-level information about the agreement
of PDFs with every data point in the global QCD fit in the presence
of systematic shifts. The response of $r_{i}(\vec{a})$ to the variations
in PDFs depends on the experiment type and kinematic range associated
with the $i^{th}$ data point, and the totality of these responses
can be examined with modern data-analytical methods. The sum of squared
residuals over all points of the global dataset renders the bulk
of the log-likelihood, or experimental, component $\chi_{E}^{2}$
of the global $\chi^{2}$. In turn, the root-mean-squared residual
$\langle r_{0}\rangle_{E}$ for experiment $E$ and the central PDF
set $\vec{a}_{0}$ is tied to $\chi_{E}^{2}(\vec{a}_{0})/N_{\mathit{pt}},$
the standard measure of agreement with experiment $E$ at the best
fit:
\begin{equation}
\langle r_{0}\rangle_{E}\equiv\sqrt{\frac{1}{N_{\mathit{pt}}}\sum_{i=1}^{N_{\mathit{pt}}}r_{i}^{2}(\vec{a}_{0})}=\sqrt{\frac{1}{N_{\mathit{pt}}}\left(\chi_{E}^{2}(\vec{a}_{0})-\sum_{\alpha=1}^{N_{\lambda}}\overline{\lambda}_{\alpha}^{2}(\vec{a_{0}})\right)}\approx\sqrt{\frac{\chi_{E}^{2}(\vec{a}_{0})}{N_{\mathit{pt}}}}.\label{r0E}
\end{equation}
Notice that $\langle r_0 \rangle_E \approx 1$ when the fit to the experimental data set $E$ is good.
We will now invoke the Hessian formalism to first organize the analysis
of the PDF dependence of individual residuals, and then introduce
a framework to evaluate sensitivity of individual data points to PDF-dependent
physical observables. To test the effectiveness of the proposed method,
we study constraints using CT14HERA2 parton distributions \cite{Hou:2016nqm}
fitted to datasets from DIS processes, $Z\rightarrow l^{+}l^{-}$,
$d\sigma/dy_{l}$, $W\rightarrow l\nu$, and jet production ($p_{1}p_{2}\rightarrow jjX)$.
We include both the experiments that were used to construct the CT14HERA2
dataset, as well as a number of LHC experiments that may be fitted
in the future. The experimental data sets are summarized in
Tables~\ref{tab:EXP_1}-\ref{tab:EXP_3}.
Given the urgency in improving
constraints on the gluon PDF for investigations of the Higgs sector,
we focus attention on several candidate experiments that may probe
$g(x,\mu)$: high-$p_T$ $Z$-boson
production (ATL8ZpT'16, ATL7ZpT'14), $t\bar{t}$ production (ATL8ttb-pt'16, ATL8ttb-y\_ave'16, ATL8ttb-mtt'16, ATL8ttb-y\_ttb'16),
as well as high-luminosity or alternative data sets for jet production, such as
the high-luminosity ATLAS 7 TeV jet data (ATLAS7jets'15) that is to replace the
counterpart low-luminosity set ATL7jets'12, or the CMS 7 TeV jet data set (CMS7jets'14)
that extends to lower jet $p_T$ and higher rapidity,
$2.5<|y_{j}|<3$, than the previously fitted CMS 7 TeV
jet data set (CMS7jets'13).\footnote{As a result, a small number of data points
that contributes to both the data sets CMS7jets'14 and CMS7jets'13 is double-counted
in the histograms, without affecting the conclusions.
}
The dependence of such experiments on $g(x,\mu)$ is scrutinized in a
number of ways. We examine their statistical properties using both the
PDFs from the CT14HERA2 NNLO analysis, which already impose
significant constraints on the large-$x$ gluon using
the Tevatron inclusive jet data sets, CDF2jets'09 and D02jets'08; and in some comparisons using
a special version of the NNLO PDFs that are fitted to the same CT14HERA2
data set, except without including the above jet data sets. As yet
another aspect, we investigate a range of measurements of Drell-Yan
pair production cross sections and charge lepton asymmetries with the
goal to understand their sensitivity predominantly to the (anti)quark sector.
To parametrize the response of a residual $\vec{r}_{i}$, we evaluate
it for every eigenvector PDF $\vec{a}_{l}^{\pm}$ of the CT14HERA2
PDF set with $N=28$ PDF parameters. Then, given the
{\it normalized residual variations}
\begin{equation}
\delta_{i,l}^{\pm}\equiv\left(r_{i}(\vec{a}_{l}^{\pm})-r_{i}(\vec{a}_{0})\right)/\langle r_{0}\rangle_{E}\label{deltapmil}
\end{equation}
between the residuals for the PDF eigenvectors $\vec{a}_{l}^{\pm}$
and for the CT14HERA2 central PDF $\vec{a}_{0}$, we construct a $2N$-dimensional
vector
\begin{equation}
\vec{\delta}_{i}=\left\{ \delta_{i,1}^{+},\,\delta_{i,1}^{-},\,...,\delta_{i,N}^{+},\,\delta_{i,N}^{-}\right\} \label{deltail}
\end{equation}
for each data point of the global dataset.
The components of $\vec{\delta}_{i}$ parametrize responses of $r_{i}$
to PDF variations along the independent directions given by $\vec{a}_{l}^{\pm}$.
The differences are normalized to the central root-mean-square (r.m.s.)
residual $\langle r_{0}\rangle_{E}$ of experiment $E$ {[}see Eq.~(\ref{r0E}){]}
so that the normalized residual variations do not significantly depend on $\chi^{2}(\vec{a}_{0})/N_{\mathit{pt}},$
the quality of fit to experiment $E$. Recall that a substantial spread
over the fitted experiments is generally obtained for $\chi_{E}^{2}/N_{\mathit{pt}}$.
Moreover, it is reasonable to expect significantly larger values for
$\chi_{E}^{2}/N_{\mathit{pt}}$ for the experiments that have not been
yet fitted, but are included in the analysis of the residuals, \textit{e.g.},
the new LHC experiments shown in Fig.~\ref{fig:data}. With the definitions
in Eqs. (\ref{deltapmil}) and (\ref{deltail}), however, $\vec{\delta_{i}}$
is only weakly sensitive to $\chi_{E}^{2}/N_{\mathit{pt}}$.
Thus, we represent the PDF-driven variations of the residuals of a
global dataset by a bundle of vectors $\vec{\delta}_{i}$ in a $2N$-dimensional
space.\footnote{In this section, we consider separate variations along $\vec{a}_{l}$
in the positive and negative directions. Alternatively, it is possible
to work with a vector of $N$ symmetric differences $\delta_{i,l}\equiv\left(r_{i}(\vec{a}_{l}^{+})-r(\vec{a}_{l}^{-})\right)/\left(2\langle r_{0}\rangle_{E}\right)$
and arrive at similar conclusions. Symmetric differences will be employed
to construct correlations and sensitivities in Sec.~\ref{sec:QuantifyingDistributionsOfResiduals}.
\label{fn:sym-deltail}} This mapping opens the door to applying various data-analytical methods
for classification of the data points and identifying the data points
of the utmost utility for PDF fits. As the length of $\vec{\delta}_{i}$
is equal to the PDF-induced fractional error on $r_{i}$ as compared
to the average residual at the best fit, it can be argued that important
PDF constraints arise
from new data points that either have a large $|\vec{\delta}_{i}|$
or are otherwise distinct from the existing data points. Conversely,
new data points with a small $|\vec{\delta}_{i}|$, or the ones that
are embedded in the preexisting clusters of points, are not likely to
improve constraints on the PDFs.
\subsection{Manifold learning and dimensionality reduction \label{subsec:Manifold-learning}}
\subsubsection{\emph{PCA and t-SNE visualizations} \label{sec:embedding}}
We illustrate a possible analysis technique carried out with the help
of the TensorFlow Embedding Projector software for the visualization
of high-dimensional data \cite{EmbeddingProjector}. A table of 4021
vectors $\vec{\delta}_{i}$ for the CTEQ-TEA dataset (corresponding
to our total number of raw data points) is generated by our package
\textsc{PDFSense} and uploaded to the Embedding Projector website.
As variations along many eigenvector directions result only in small
changes to the PDFs, the 56-dimensional $\vec{\delta}_{i}$ vectors
can in fact be projected onto an effective manifold spanned by fewer
dimensions. Specifically, the Embedding Projector approximates the
56-dimensional manifold by a 10-dimensional manifold using principal
component analysis (PCA). In practice, this 10-dimensional manifold
is constructed out of the 10 components of greatest variance in the
effective space, such that the most variable combinations of $\delta_{i,l}$
are retained, while the remaining 46 components needed to fully reconstruct
the original 56-dimensional $\vec{\delta}_{i}$ are discarded. However,
because the 10 PCA-selected components describe the bulk of the variance
of $\delta_{i,l}$, the loss of these 46 components results in only
a minimal relinquishment of information, and in fact provides a more
efficient basis to study $\delta_{i,l}$ variations.
We encourage the reader to download the table of the normalized
residual variations $\vec\delta_i$ for
CT14HERA2 NNLO from the \textsc{PDFSense}
website \cite{PDFSenseWebsite} and explore it for themselves using the Embedding
Projector \cite{EmbeddingProjector} or another program for multidimensional data visualization
such as a tour \cite{Cook:2018mvr}. These tools help to understand
the detailed PDF dependence of individual data sets
{\it without doing the global fit}.
Performing such task has been challenging for non-experts,
if not for the PDF fitters themselves. With the proposed method, we
can visually examine the PDF dependence of the residuals from the diverse data
sets before quantitatively characterizing these distributions using
the estimators developed in the next sections.
In the future, a computer algorithm
can be written to select the experimental data for PDF fits,
based on the residual variations, and with minimal involvement from humans.
To offer an illustration, while grasping the full PDF dependence of the
data points in the original 56-parameter space is daunting,
in the 10-dimensional representation obtained via PCA,
some directions result in efficient separation of the data points
of different types according to their residual variations.
The left panel of Fig.~\ref{fig:PCA-TSNE} shows one such 3-dimensional
projection of $\vec{\delta}_{i}$ that separates clusters of residual
variations arising from data for DIS,
vector boson production, and jet/$t\bar{t}$ production.
In this example, the jet/$t\bar{t}$ cluster, shown in red, is roughly
orthogonal to the blue DIS cluster and intersects it. This separation
is quite remarkable, as it is based only on numerical properties of the
$\vec{\delta}_{i}$ vectors, and not on the meta-data about the types
of experiments that is entered only after the PCA is completed; in other
projections, the data types are not separated. The underlying
reasons for this separation, namely, dependence on independent PDF
combinations, will be quantified by the sensitivities in the next section.
As an alternative, the Embedding Projector can organize the $\vec{\delta}_{i}$
vectors into clusters according to their similarity using $t$-distributed
stochastic neighbor embedding (t-SNE) \cite{vanderMarten:2008}. A
representative 3-dimensional distribution of the vectors obtained
by t-SNE is displayed in the right panel of Fig.~\ref{fig:PCA-TSNE}.
In the figure, we show that the t-SNE method is able to identify and
separate the clusters of
data according to the experimental process (DIS, vector production, or
jet production). In fact, the reader can perform the t-SNE analysis
on the Embedding Projector website themselves and verify that it actually
sorts the $\vec\delta_i$ vectors into the clusters
according to their values of $x$ and
$\mu$, and even the experiment itself. This exercise demonstrates,
yet again, that the statistical residuals provided in \textsc{PDFSense}
reflect the key properties of the global fit.
Information can be extracted from them and examined in a number of ways.
The breakdown of the vectors over experiments in the PCA representation
is illustrated by Fig.~\ref{fig:PCA-CTExperiments}. Here, we see
that the bulk of the DIS cluster from the left Fig.~\ref{fig:PCA-TSNE}
originates with the combined HERA1+2 DIS data [HERAI+II'15].
The jet cluster in Fig.~\ref{fig:PCA-TSNE} will be dominated by
ATLAS and CMS inclusive jet datasets [CMS7jets'14, ATLAS7jets'15, and CMS8jets'17],
which add dramatically more points across a wider kinematical range
on top of the CDF Run-2 and D0 Run-2 jet production datasets (CDF2jets'09)
and (D02jets'08).
In contrast, although the $t\bar{t}$ production experiments (ATL8ttb-pt'16, ATL8ttb-y\_ave'16, ATL8ttb-mtt'16, ATL8ttb-y\_ttb'16)
are generally characterized by large $\vec{\delta}_{i}$
vectors, they contribute only a few data points lying within the jet
cluster of Fig.~\ref{fig:PCA-CTExperiments} and, by themselves, will not make much difference in a global
fit. The same conclusion applies to data from high-$p_{T}$ $Z$ production,
which has too few points to stand out in a fit with significant inclusive
jet data samples. We return to this point in the discussion of reciprocated
distances below.
It is also interesting to note that semi-inclusive charm production
at HERA [HERAc'13]
lies between, and partly overlaps with, the DIS and jet clusters. Finally,
CCFR/NuTeV dimuon semi-inclusive DIS [SIDIS] (CCFR-F2'01, CCFR-F3'97, NuTeV-nu'06, NuTeV-nub'06)
extends in an orthogonal direction, not well separated from the other datasets in the
selected three-dimensional projection.
\begin{figure*}
\centering{}\includegraphics[width=0.48\textwidth]{figs/pca.pdf}\ \ \includegraphics[width=0.48\textwidth]{figs/tsne.pdf}
\caption{Distributions of residual variations $\vec{\delta_{i}}$ from the
CTEQ-TEA analysis obtained by dimensionality reduction methods. Left:
a 3-dimensional projection of a 10-dimensional manifold constructed
by principal component analysis (PCA). Right: a distribution from
the 3-dimensional t-SNE clustering method. Blue, orange, and red colors
indicate data points from DIS, vector boson production, and jet/$t\bar{t}$
production processes. \label{fig:PCA-TSNE}}
\end{figure*}
\begin{figure*}[p]
\centering{}\includegraphics[width=0.9\textwidth,height=0.85\textheight]{figs/CTexperiments.pdf}
\caption{The PCA distribution from Fig.~\ref{fig:PCA-TSNE}, indicating distributions
of points from classes of experiments. In the numbering scheme used here, points labeled
1XX correspond to fixed-target measurements and 5XX to jet and $t\bar{t}$ production as given in Tables~\ref{tab:EXP_1}--\ref{tab:EXP_3}.
The specific experiments are noted in the plots.
}
\label{fig:PCA-CTExperiments}
\end{figure*}
\subsubsection{\emph{Reciprocated distances} \label{sec:Reciprocated-distances}}
As a complement to the visualization methods based on PCA and t-SNE
just presented, it is also possible to evaluate another similarity
measure based on the distances between the vectors of the
residual variations. For example,
rather than applying the PCA
to an ensemble of $\vec{\delta}_{i}$ vectors to perform dimensionality reduction,
we might instead compute over the vector space a pair-wise
\textit{reciprocated distance} measure, which we define as
\begin{equation}
\mathcal{D}_{i}\ \equiv\ \left(\sum_{j\neq i}^{N_{\mathit{all}}}\frac{1}{|\vec{\delta}_{j}-\vec{\delta}_{i}|}\right)^{-1}\ ,\label{eq:recip}
\end{equation}
and evaluate for the $i$ points in each experimental dataset. We
allow the sum over $j$ in Eq.~(\ref{eq:recip}) to run over all
the data points in the CTEQ-TEA set regardless of experiment
(denoted by $N_{\mathit{all}}$). The distances can be computed either
in the 56-dimensional space or in the reduced dimensionality space.\footnote{Alternative definitions for the reciprocated distance can be also
used, with qualitatively similar conclusions. For example, we could
sum over all experimental data, but excluding those points belonging
to the same experiment as point $i$, and normalizing $\mathcal{D}_{i}$
by $(N_{\mathit{pt}}-N_{\mathit{all}})/N_{\mathit{pt}}$ to compensate for different numbers of
points in the experiment. } We plot the result of applying Eq.~(\ref{eq:recip}) to the 56-dimensional
residual variations of the full CTEQ-TEA dataset computed using two PDF ensembles:
CT14HERA2 fitted to all data in the left panel, and CT14HERA2 fitted
only to the DIS and vector boson production data (excluding jet production
data) in the right panel. Fig. \ref{fig:recip} represents the distribution
of the reciprocated distances over individual experiments of the CTEQ-TEA
dataset. The CT Experiment ID \# is shown on the abscissa, and the $\mathcal{D}_{i}$
values for every point of the experiment are indicated by the scatter
points.
The advantage of the definition in Eq. (\ref{eq:recip}) is that it
enables a quantitative measure of the degree to which separate experiments
broadly differ in terms of their residual variations, and therefore
provides information analogous to that found in Figs. \ref{fig:PCA-TSNE}\textendash \ref{fig:PCA-CTExperiments}.
For example, by inspection of Eq. (\ref{eq:recip}) it can be seen
that those experimental measurements which are widely separated from
the rest of the CTEQ-TEA dataset in space of $\vec{\delta}_{i}$ vectors
will correspond to comparatively large values of $\mathcal{D}_{i}$,
and experiments that systematically differ from the rest of the total
dataset are thus expected to have especially tall distributions in
the panels of Fig. \ref{fig:recip}. On this basis, it can be seen
that information yielded by W asymmetry measurements (D02Masy'08, CMS7Masy2'14,
D02Easy2'15) are particularly distinct, as well as the combined HERA DIS data
(HERAI+II'15) and fixed-target Drell-Yan measurements, such as E605
(E605'91) and E866 data (E866rat'01 and E866pp'03). Similarly,
direct comparison of the $\mathcal{D}_{i}$ distributions in the panels
of Fig. \ref{fig:recip} allows one to compare constraints with and
without the jet data. We note that the 7 and 8 TeV ATLAS high-$p_{T}$
$Z$ production (ATL7ZpT'14 and ATL8ZpT'16) and $t\bar{t}$ production (ATL8ttb-pt'16)
provide a number of ``remote'' points and hence are potentially
useful in the fits sensitive to the gluon. On the other hand, new
jet production experiments (CMS7jets'14, ATLAS7jets'15, CMS8jets'17) all include large numbers
of points characterized by significant reciprocated distances.
\begin{figure*}
\centering{}\includegraphics[width=0.47\textwidth]{figs/rd54_CT14HERA2_new2.pdf}
\ \ \ \centering{}\includegraphics[width=0.47\textwidth]{figs/rd54_CT14HERA2_nojets_new2.pdf}
\caption{ A plot of the reciprocated distances $\mathcal{D}_{i}$ obtained
from the PDFs fitted to the full CT14HERA2 dataset {[}left{]} and
to the CT14HERA2 dataset without jet production experiments {[}right{]}.
The horizontal axis displays numerical experimental CT IDs of the constituent
CTEQ-TEA datasets, for each of which is shown a column of values of
the reciprocated distance. We highlight columns corresponding to Expt.~IDs
ATL7ZpT'14 [247], ATL8ZpT'16 [253], and ATL8ttb-pt'16 [565] as discussed in text. \label{fig:recip}}
\end{figure*}
\section{Quantifying distributions of residual variations\label{sec:QuantifyingDistributionsOfResiduals}}
We have demonstrated that the multi-dimensional distribution of
the $\vec{\delta}_i$ vectors reflects the PDF dependence of individual data points. In this section,
we will focus on numerical metrics to assess the emerging geometrical
picture associated with the $\vec{\delta}_i$ distribution, and to visualize the regions of partonic momentum fractions
$x$ and QCD factorization scales $\mu$ where the experiments impose
strong constraints on a given PDF-dependent observable $X$.
Gradients of $r_{i}$ in a space of Hessian eigenvector PDF parameters
$\vec{a}$ are naturally related to the PDF uncertainty. Recall that
in the Hessian method the PDF uncertainty on $X(\vec{a})$ is found
as
\begin{equation}
\Delta X(\vec{a})=X(\vec{a})-X(\vec{a}_{0})=\vec{\nabla}X|_{\vec{a}_{0}}\cdot\Delta\vec{a},
\end{equation}
where $\vec{a}_{0}$ is the best-fit combination of PDF parameters,
and $\Delta\vec{a}$ is the maximal displacement along the gradient
that is allowed within the tolerance hypersphere of radius $T$ centered
on the best fit \cite{Pumplin:2001ct,Pumplin:2002vw}. The standard
master formula
\begin{equation}
\Delta X=\left\vert \vec{\nabla}X\right\vert =\frac{1}{2}\sqrt{\sum_{l=1}^{N}\left(X_{l}^{+}-X_{i}^{-}\right)^{2}}\label{masterDX-1}
\end{equation}
is obtained by representing the components of $\vec{\nabla}X$ by
a finite-difference formula
\begin{equation}
\frac{\partial X}{\partial a_{i}}=\frac{1}{2}(X_{i}^{+}-X_{i}^{-}),\label{dXdzi-1-1}
\end{equation}
in terms of the values $X_{l}^{\pm}$ for extreme displacements of
$\vec{a}$ within the tolerance hypersphere along the $l$-th direction.
In this setup, a dot product between the gradients provides a convenient
measure of the degree of similarity between PDF dependence of two
quantities \cite{Nadolsky:2008zw}. A dot product $\vec{\nabla}r_{i}\cdot\vec{\nabla f}$
between the gradients of a shifted residual $r_{i}$ and another QCD
variable $f$, such as the PDF at some $\{x,\mu\}$ or a cross section,
can be cast in a number of useful forms.
\subsection{Correlation cosine }
\label{sec:Correlations}
The correlation for the $i^{th}$ $\{x,\mu\}$ point, which we define
following Refs.~\cite{Pumplin:2001ct,Nadolsky:2001yg,Nadolsky:2008zw,Gao:2017yyd}
as
\begin{equation}
C_{f}\,\equiv\,\mbox{Corr}[f,r_{i}]=\frac{\vec{\nabla} f\cdot\vec{\nabla} r_{i}}{\Delta f\,\Delta r_{i}},\label{eq:corr}
\end{equation}
can determine whether there \emph{may} exist a predictive relationship
between $f$ and goodness of fit to the $i^{th}$ point. The correlation
function $\mathrm{\mbox{Corr}}[X,Y]$ for the quantities $X,\,Y$
in Eq.~(\ref{eq:corr}) represents the realization in the Hessian
formalism of Pearson's correlation coefficient, which we express as
\begin{align}
\mathrm{\mbox{Corr}}[X,Y] & =\frac{1}{4\Delta X\Delta Y}\sum_{j=1}^{N}(X_{j}^{+}-X_{j}^{-})(Y_{j}^{+}-Y_{j}^{-})\ ,\label{eq:corr-def}
\end{align}
with the sum in these expressions being over the $j$ parameters of
the full PDF model space. Geometrically, $\mbox{Corr}[X,Y]$ represents
the cosine of the angle that determines the eccentricity of an ellipse
satisfying $\chi^{2}(\vec{a})<\chi^{2}(\vec{a}_{0})+T^{2}$ in the
$\{X,Y\}$ plane. This latter point follows from the fact that the
mapping of the tolerance hypersphere onto the $\{X,Y\}$ plane is
an ellipse with an eccentricity that depends on the correlation of
$X$ and $Y,$ which is given in turn by Eq.~(\ref{eq:corr-def})
above.
$\mbox{Corr}[f,r_{i}]$ does not indicate how constraining the residual
is, but it may indicate a predictive relation between $r_{i}$ and
$f$. On the basis of previous work \cite{Nadolsky:2008zw}, we say
that the (anti-)correlation between $X$ and $Y$ is significant roughly
if $\left|\mbox{Corr}[X,Y]\right|\gtrsim0.7$, while smaller (anti-)correlation
values are less robust or predictive. Following this rule-of-thumb,
correlations have been used successfully to identify PDF combinations
that dominate PDF uncertainties of complicated observables, for instance
to show that the gluon uncertainty dominates the total uncertainty
on LHC $W$ and $Z$ production, or that the uncertainty on the ratio
$\sigma_{W}/\sigma_{Z}$ of $W^{\pm}$ and $Z^{0}$ boson cross sections
at the LHC is dominated by the strangeness PDF, rather than $u$ and
$d$ (anti-)quark PDFs \cite{Nadolsky:2008zw}.
\subsection{Sensitivity in the Hessian method}
\label{sec:Sensitivities}
The correlation $C_{f}$ alone does not fully encode the potential
impact of separate or new measurements on improving PDF determinations
in terms of the uncertainty reduction. Rather, we employ $\vec{\nabla} f\cdot\vec{\nabla} r_{i}$
again to define the \textit{sensitivity} $S_{f}$ to $f$ of the $i^{th}$
point in experiment $E$:
\begin{equation}
S_{f}\equiv\frac{\vec{\nabla} f\cdot\vec{\nabla} r_{i}}{\Delta f\,\langle r_{0}\rangle_{E}}=\frac{\Delta r_{i}}{\langle r_{0}\rangle_{E}}\,C_{f}\ ,\label{eq:sens}
\end{equation}
where $\Delta r_{i}$ and $\langle r_{0}\rangle_{E}$ are computed
according to Eqs.~(\ref{DelX}) and (\ref{r0E}), respectively. In
other words, $\Delta r_{i}$ again represents the variation of the
residuals across the set of Hessian error PDFs, and we normalize it
to the r.m.s.\ residual for the whole dataset $E$ to reduce the impact
of random fluctuations in the data values $D_{i,\mathit{sh}}$. This definition
has the benefit of encoding not only the correlated relationship of
$f$ with $r_{i}$, but also the comparative size of the experimental
uncertainty with respect to the PDF uncertainty. In consequence, for
example, if new experimental data have reported uncertainties that
are much tighter than the present PDF errors, these data would then
register as high-sensitivity points by the definition in Eq.~(\ref{eq:sens}).
\begin{figure*}
\includegraphics[clip,width=0.48\textwidth]{figs/sf_projection1.pdf}
\quad\quad \includegraphics[clip,width=0.4\textwidth]{figs/sf_projection2.pdf}
\caption{Left: A PDF-dependent quantity $f$ defines a direction in
space of $(2)N$ PDF parameters. The direction is specified by the
gradient $\vec\nabla f$ in the symmetric convention.
Here, the Embedding
Projector \cite{EmbeddingProjector} visualizes the
vectors $\vec \delta_{907}$ and $\vec \delta_{914}$ for
NNLO cross sections for Higgs boson production at 7 and 14 TeV, and
vectors $\vec\delta_i$ for CT14HERA2 NNLO data points from
\cite{PDFSenseWebsite} (brown circles), showing only $\vec\delta_i$
with the smallest angular distances to $\vec\delta_{914}$. These
points impose the strongest constraints on the PDF dependence of the
Higgs cross sections in the CT14HERA2 analysis, if they have large enough
$|\vec \delta_i|$. Again, in the numbering scheme used here, points labeled
1XX correspond to fixed-target measurements, 2XX to Drell-Yan processes and boson production, and
5XX to jet and $t\bar{t}$ production as given in Tables~\ref{tab:EXP_1}--\ref{tab:EXP_3}. Right: the sensitivity $S_f$ of the $i$-th data
residual can be interpreted as the projection of $\vec \delta_i
\equiv \vec \nabla r_i/\langle r_0\rangle_E$ onto the direction of $\vec\nabla f$.}
\label{fig:sfprojection}
\end{figure*}
Geometrically, $S_{f}$ represents a projection onto the direction of
the gradient $\vec{\nabla}f$ of the residual variation $\vec{\delta_{i}}$, defined in Sec.~\ref{sec:QuantifyingDistributionsOfResiduals}
using the symmetrized formula for $\delta_{i,l}$ noted in footnote~\ref{fn:sym-deltail}, namely,
\begin{equation}
\delta_{i,l}\equiv\left(r_{i}(\vec{a}_{l}^{+})-r(\vec{a}_{l}^{-})\right)/\left(2\langle r_{0}\rangle_{E}\right)\ .
\end{equation}
Figure~\ref{fig:sfprojection} shows a pictorial illustration of this interpretation. This interpretation
suggests that the total strength of constraints along the direction
of $\vec{\nabla}f$ can be quantified by summing projections $S_{f}$
onto this direction of all individual vectors $\vec{\delta}_{i}$.
As with correlations, only a sufficiently large absolute magnitude
of $\left|S_{f}\right|$ is indicative of a predictive constraint
of the $i^{th}$ point on $f$. Recall that $r_{i}^{2}$ is the contribution
of the $i^{th}$ point to $\chi^{2},$ and that only residuals with
a large enough $\Delta r_{i}$ as compared to the r.m.s.\ residual
$\langle r_{0}\rangle_{E}$ are sensitive to PDF variations. The $S_{f}$
magnitude is of order $\Delta r_{i}/\langle r_{0}\rangle_{E},$ which
suggests an estimate of a minimal value of $S_{f}$ that would be
deemed sensitive according to the respective $\chi^{2}$ contribution.
For the numerical comparisons in this study, we assume that $\left|S_{f}\right|$
must be no less than 0.25 to indicate a predictive constraint, as
the PDF uncertainty of the $i^{th}$ residual contributes no less
than $r_{i}^{2}=$0.0625 to the variation in the global $\chi^{2}$.
The reader can choose a different minimal value in the
\textsc{PDFSense} figures depending
on the desired accuracy. The cumulative sensitivities that we obtain
in later sections are independent of this choice.
Yet another possible definition, which we list for completeness, is
to further normalize the sensitivity as
\begin{equation}
S_{f}^{\prime}\equiv\frac{\vec{\nabla} f\cdot\vec{\nabla} r_{i}}{f_{0}\,\langle r_{0}\rangle_{E}}=\frac{\Delta f}{f_{0}}\,S_{f}\ .\label{eq:sens-prime}
\end{equation}
For instance, if $f$ is the PDF $f(x_{i},\mu_{i})$ or parton luminosity
evaluated at the $\{x_{i},\mu_{i}\}$ points extracted according to
the data, the definition of $S_{f}^{\prime}$ in Eq. (\ref{eq:sens-prime})
de-emphasizes those points where the PDF uncertainty $\Delta f(x_{i},\mu_{i})$
is small compared to the best-fit PDF value $f_{0}(x_{i},\mu_{i})$
\textemdash{} analogously to how $S_{f}$ de-emphasizes (relative
to the correlation $C_{f}$) those data points whose normalized
residual variations $\Delta r_{i}/\langle r_{0}\rangle_{E}$
have already been more tightly constrained.
\subsection{Sensitivity in the Monte-Carlo method}
The above statistical measures are general enough and can be extended
to other representations for the PDF uncertainties, such as the
representation based on Monte-Carlo replica PDFs \cite{Giele:1998gw,Giele:2001mr,Ball:2008by}
of the kind employed, e.g., in the NNPDF framework.
A family of Monte-Carlo PDFs consists of $N_{\rm rep}$ member PDF sets
$q_a^{(k)}(x,\mu)\equiv \{ q^{(k)} \}$, with $k=1,\ ...,\ N_{\rm rep}$, and
those are used to determine an expectation value
$\langle X\rangle$ for a PDF-dependent quantity $X[\{ q \}]$ such as a
high-energy cross section:
\begin{equation}
\langle X \rangle = \frac{1}{N_{\rm rep}} \sum_{k=1}^{N_{\rm rep}}
X [ \{ q^{(k)} \}]\ .
\label{eq:NNPDF_masterave}
\end{equation}
The resulting Monte-Carlo uncertainty on $X$ can be extracted from the ensemble as
\begin{equation}
\Delta_{\rm MC} X\ =\ \left( \frac{1}{N_{\rm rep}-1}
\sum_{k=1}^{N_{\rm rep}}
\left( X [ \{ q^{(k)} \}]
- \langle X \rangle\right)^2
\right)^{1/2}\ .
\label{eq:NNPDF_error}
\end{equation}
In consequence of these definitions, the central value of a particular
PDF itself in the NNPDF framework is specified as
\begin{equation}
q_{(0)} \equiv \langle q \rangle = \frac{1}{N_{\rm rep}}
\sum_{k=1}^{N_{\rm rep}} q^{(k)} \ .
\label{eq:NNPDF_mcav}
\end{equation}
Akin to the Pearson correlation defined in Eq.~(\ref{eq:corr}) of
Sec.~\ref{sec:Correlations}, statistical correlations between two PDF-dependent
quantities $X[\{ q \}]$ and $Y[\{ q \}]$ can be constructed from the PDF
replica language above in terms of ensemble averages \cite{Ball:2008by}:
\begin{equation}
\mbox{Corr}_{\rm MC} \left[ X, Y \right]
=\frac{\langle X Y \rangle
- \langle X \rangle \langle Y \rangle}{\Delta_{\rm MC} X \Delta_{\rm MC} Y}\ .
\label{eq:NNPDF_corrPDF}
\end{equation}
Then, using our definitions in Eqs.~(\ref{eq:corr}) and
(\ref{eq:sens}), we immediately construct the realizations of the
correlation and sensitivity for a PDF-dependent quantity $f$
in the Monte-Carlo method:
\begin{eqnarray}
C_{f,\ {\rm MC}} &=& \mbox{Corr}_{\rm MC}[f,r_{i}]\ , \label{eq:corrMC}\\
S_{f,\ {\rm MC}} &=& \frac{\Delta_{\rm MC} r_{i}}{\langle
r_{0}\rangle_{E}}\,\mbox{Corr}_{\rm MC}\left[f, r_i \right].\label{eq:sensMC}
\end{eqnarray}
\section{Case study: CTEQ-TEA global data \label{sec:CaseCTEQ-TEA}}
\subsection{Maps of correlations and sensitivities}
\begin{figure*}
\includegraphics[clip,width=0.38\textwidth]{figs/ghist_C_fix_y.pdf}
\includegraphics[clip,width=0.60\textwidth]{figs/corr_xQ+1_f0_samept_nohigh.pdf}
\\
\includegraphics[clip,width=0.60\textwidth]{figs/corr_xQ+1_f0_samept_replot.pdf}
\caption{Representations of the correlation $|C_{g}|(x_{i},\mu_{i})$ of
the gluon PDF $g(x,\mu)$ with the point-wise residual $r_{i}$ of
the augmented CT14HERA2 analysis. In the first panel, we plot a histogram
showing the distribution of correlations for 4021 physical measurements.
In the second panel we show the 5227-point $\{x_{i},\mu_{i}\}$ map corresponding
to these data within the full dataset, generated as in Appendix~\ref{sec:supp}.
To adjust for the fact that some measurements of rapidity dependent quantities
match to two distinct points in $\{x_{i},\mu_{i}\}$ space using the rules of
Appendix~\ref{sec:supp}, we assign weights of $0.5$ to these complementary
$\{x_{i},\mu_{i}\}$ points in computing the $N_{\mathit{pt}}=4021$-count histogram
at left. The third figure is the same as the second one, but only the
data points satisfying $|C_f|>0.7$ are highlighted.
}
\label{fig:corr-main}
\end{figure*}
We will now discuss a number of practical examples of using $C_{f}$
or $S_{f}$ to quickly evaluate the impact of various hadronic data
sets upon the knowledge of the PDFs in a fashion that does not require
a full QCD analysis of the type described in Sec.~\ref{sec:PDF-preliminaries}.
For this demonstration, we will continue to study the dataset shown
in Fig.~\ref{fig:data} of the CT14HERA2 analysis~\cite{Hou:2016nqm}
augmented by the candidate LHC data.
We have already noted the extent of this dataset in the $\{x,\mu\}$
plane in Fig.~\ref{fig:data}, where it is decomposed into constituent
experiments labeled according to the conventions in Tables~\ref{tab:EXP_1}-\ref{tab:EXP_3}.
It is instructive to create similar maps in the $\{x,\mu\}$ plane
showing the $C_{f}$ or $S_{f}$ values for each data point.
Such maps are readily produced by the
\textsc{PDFSense} program for a variety of PDF flavors and for user-defined
observables, such as the Higgs cross section. For demonstration we
have collected a large number of these maps at the companion website
\cite{PDFSenseWebsite}. We invite the reader to review these additional
figures while reading the paper
to validate the conclusions that will be summarized below.
Thus, we obtain scatter plots of $C_{f}(x_{i},\mu_{i})$ or $S_{f}(x_{i},\mu_{i})$
for a given QCD observable $f=\sigma$, such as the LHC Higgs production
cross section shown in Fig.~\ref{fig:CorrSensH14}, or with a PDF
$f$ evaluated at the same $\{x_{i},\mu_{i}\}$ determined by the
data points, with examples shown for $g(x_{i},\mu_{i})$ in Figs.~\ref{fig:corr-main}
and \ref{fig:sens-main}. The typical $\{x_{i},\mu_{i}\}$ values
characterizing the data points are found according to Born-level approximations
appropriate for each scattering process included in the CTEQ-TEA dataset,
with the formulas to compute these kinematic matchings summarized
in App.~\ref{sec:supp}. Here and in general, we find it preferable to
consider the absolute
values $|C_f|$ and $|S_f|$ on the grounds that the
signs of $C_f$ and $S_f$ flip when the data points randomly overshoot
or undershoot their theory predictions.
Together with the map in the $\{x,\mu\}$ plane, \textsc{PDFSense
}also returns a histogram of the values for each quantity it plots.
An example is shown for $\left|C_{g}\right|(x_{i},\mu_{i})$ in the
first panel of Fig.~\ref{fig:corr-main}. One would judge that stronger
constraints are in general provided to those PDFs for which the $|C_{f}|$
histogram has many entries comparatively closely to $|C_{f}|\sim1$.
In the first panel of Fig.~\ref{fig:corr-main}, we can see that,
while the distribution peaks at low correlations,
$|C_{g}|\sim0$, the distribution has an extended
tail in the region $0.7\lesssim|C_{g}|\lesssim1$.
This feature shows that, of the 4021 experimental data
points within the augmented CT14HERA2 set in Fig.~\ref{fig:data},
nearly two-hundred --- specifically, 192 --- have especially strong
($|C_{f}|\ge0.7$) correlations (or anti-correlations) with the gluon
PDF. This region of such strong correlations within the histogram
is indicated by the horizontal blue bar that runs along the abscissa.
\begin{figure*}
\includegraphics[clip,width=0.38\textwidth]{figs/ghist_S_fix_y.pdf}
\includegraphics[clip,width=0.60\textwidth]{figs/corrdr_xQ+1_f0_samept_nohigh.pdf}\\
\includegraphics[clip,width=0.60\textwidth]{figs/corrdr_xQ+1_f0_samept_replot.pdf}
\caption{Like Fig.~\ref{fig:corr-main}, but for the gluon sensitivity $|S_{g}|(x_{i},\mu_{i})$
as defined in Eq.~(\ref{eq:sens}). In the third figure, only the
data points satisfying $|S_f|>0.25$ are highlighted.}
\label{fig:sens-main}
\end{figure*}
To identify these points, we plot complementary information in the
second panel of the same figure \textendash{} specifically, a map in
$\{x,\mu\}$ space of each of the data points shown in
Fig.~\ref{fig:data}.
As before, they are colorized according to the magnitude of $|C_{g}|$
following the color palette in the ``rainbow strip'' on the right.
``Cooler'' colors (green/yellow) correspond to weaker correlation
strengths, while ``hotter'' colors (orange/red) represent comparatively
stronger correlations, as indicated. To reveal the data points with
the highest correlations, we reproduce the same figure in the third
panel, but showing in color only the data points satisfying $|C_f|>0.7$.
Thus, we obtain two maps in the
$\{ x,\mu \}$ plane that look similar to the $|C_f|$ map in the left
panel of Fig.~\ref{fig:CorrSensH14}, apart from the differences that
(a) Fig.~\ref{fig:corr-main} shows the correlation $|C_g|$ for
$g(x_i,\mu_i)$ at the same typical values $\{x_i,\mu_i\}$ as in the
data, rather than $|C_{\sigma_{H^0}}|$ for Higgs production cross
section in Fig.~\ref{fig:CorrSensH14}; and (b)
Fig.~\ref{fig:CorrSensH14} highlights 310 points with the highest $|C_{\sigma_{H^0}}|$.
The correlations for the LHC Higgs production cross section trace those for
$g(x_i,\mu_i)$, but not entirely, as we will see in a moment.
Large magnitudes of $|C_{g}|$ in Fig.~\ref{fig:corr-main}
are found for inclusive jet production measurements, especially those
recently obtained by CMS at 8 TeV \cite{Khachatryan:2016mlc} (Expt.~CMS8jets'17,
inverted triangles) with $|C_{g}|(x_{i},\mu_{i})$ as high as
0.85, including at the highest values of $x$ and $\mu$. Beyond these,
a sizable cluster of HERA (HERAI+II'15) data points at the lowest values of
$x$ are also seen to have large correlations with the gluon PDF,
consistent with the common wisdom that HERA DIS constrains the gluon
PDF at small $x$ via DGLAP scaling violations. Under the jet production
cluster, high-$p_{T}$ $Z$ production (ATL7ZpT'14, ATL8ZpT'16) and $t\bar{t}$
production (ATL8ttb-pt'16, ATL8ttb-y\_ave'16, ATL8ttb-mtt'16,
ATL8ttb-y\_ttb'16) at the LHC show a high $|C_{g}|(x_{i},\mu_{i})$
correlation. At the same time, many other measurements, including
fixed-target data at large $x$ and $W$ asymmetry data near $\mu\!\sim\!100$
GeV, have feeble correlations with $g(x_i,\mu_i)$ and would therefore be less
emphasized by an analysis based solely upon the PDF-residual correlations.
We can also consider the analogous plots for the sensitivity
$\left|S_{g}\right|(x_{i},\mu_{i})$ as defined in Eq.~(\ref{eq:sens}), which we plot in Fig.~\ref{fig:sens-main}. In the first panel, we again consider
the histogram, here for the magnitudes of the gluon sensitivity $|S_{g}|(x_{i},\mu_{i})$,
in which the correlations $|C_{g}|$ are now weighted by the relative
size of the PDF uncertainty $\Delta r_{i}$ in the residual. As discussed
in Sec.~\ref{sec:Sensitivities}, this additional weighting
emphasizes those data points for which the PDF-driven fluctuations
in the residuals are comparatively large relatively to experimental
uncertainties. This leads to a redistribution of the data points
shown in the $|C_{g}|$ histogram of Fig.~\ref{fig:corr-main}, with
the result being a considerably longer-tailed histogram for $|S_{g}|$
such that, in this instance, there are 546 raw data points
with larger sensitivities,
$|S_{f}|\ge0.25$, indicated by the horizontal blue bar. Unlike the correlation, $|S_{g}|$ can be arbitrarily large,
depending on the $\Delta r_{i}$ value. It is suppressed at the data points with
large uncertainties or smeared over the regions of data points with correlated systematic
uncertainties.
In the second and third panels, we show the respective $\{x,\mu\}$
maps for $\left|S_{g}\right|$, with color highlighting given either for all points or only
those with high sensitivities $|S_f|>0.25$, respectively. $\left|S_{g}\right|$ places additional
emphasis on the combined HERA dataset (HERAI+II'15) constraining $g(x_{i},\mu_{i})$
at lowest $x$. In contrast to the $|C_{g}|$ plot, we observe increased
sensitivity in the precise fixed-target DIS data from BCDMS (BCDMSp'89, BCDMSd'90) and CCFR
(CCFR-F2'01, CCFR-F3'97), which are sensitive to the gluon via scaling violations
despite only moderate correlation values.
Similarly, we observe heightened sensitivities at
highest $x$ for the LHC (CMS7jets'14, ATLAS7jets'15, CMS8jets'17) and Tevatron (D02jets'08) jet production
data, which have both large correlations with $g(x_{i},\mu_{i})$
and small experimental uncertainties. Sensitivity $\left|S_{g}\right|$
of LHC jet experiments, CMS7jets'14, ATLAS7jets'15, CMS8jets'17, varies in a large range, and
can significantly improve, depending on the implementation of
experimental systematic uncertainties in the analysis, cf.\ the
discussion of the jet data in the next section.
We also observe enhanced
sensitivity for \emph{individual points} in a large number of experiments,
including CDHSW DIS (CDHSW-F2'91); HERA $F_{L}$ (HERA-FL'11); the Drell-Yan process
(E605'91, E866pp'03); CDF 8 TeV $W$ charge asymmetry (CMS7Masy2'14); HERA charm SIDIS
(HERAc'13); ATLAS high-$p_{T}$ $Z$ production (ATL7ZpT'14, ATL8ZpT'16); and especially
strongly sensitive points in $t\bar{t}$ production (ATL8ttb-pt'16, ATL8ttb-y\_ave'16,
ATL8ttb-mtt'16, ATL8ttb-y\_ttb'16). However, since the latter category includes fewer
points per each experiment, it constrains the gluon less than the high-statistics DIS
and jet production data.
These findings comport with the idea that the gluon PDF remains dominated
by substantial uncertainties at both $x\!\sim\!0$ and in the elastic
limit $x\!\rightarrow\!1$, a fact which has driven an intense focus upon production
of hadronic jets, $t\bar{t}$ pairs, and high-$p_{T}$ $Z$ bosons,
which themselves are measured at large center-of-mass energies $\sqrt{s}$
and are expected to be sensitive to the gluon PDF across a wide interval
of $x,$ including $x\!\sim\!0.01$ typical for Higgs boson production
via gluon fusion at the LHC. Turning back to the distributions
of $\left|C_{\sigma_{H}}\right|(x_{i},\mu_{i})$ and $\left|S_{\sigma_{H}}\right|(x_{i},\mu_{i})$
for the Higgs cross section $\sigma_{H}$ at $\sqrt{s}=14$ TeV in
Fig.~\ref{fig:CorrSensH14}, we notice that they largely reflect
the distributions
of $\left|C_{g}\right|(x_{i},\mu_{i})$ and $\left|S_{g}\right|(x_{i},\mu_{i})$
around $x \sim M_H/\sqrt{s}=125/14000=0.009$ and $\mu= M_H=125$ GeV.
We also see some differences: although the average $x$ and $\mu$
are fixed in $\sigma_{H}$, it is nonetheless sensitive to some constraints
at much lower $x$ values as a result of the momentum sum rule.
The reader is welcome to examine the plots of sensitivities and
correlations available on the \textsc{PDFSense} website for a large
collection of PDF flavors and PDF ratios, such as
$d/u$, $\overline{d}/\overline{u}$, and
$\left( s +
\overline{s}\right)\!/\!\left(\overline{u}+\overline{d}\right)$. Sensitivities
for other PDF combinations and hadronic cross sections can be
computed and plotted in a matter of minutes
using the \textsc{PDFSense} program. We will
now turn to another aspect of this analysis: summarizing the abundant
information contained in the sensitivity plots. For this purpose, we
will introduce
numerical indicators and propose a practical procedure to
rank the experimental data sets according to their
sensitivities to the PDFs or PDF-dependent observables of interest.
\subsection{Experiment rankings according to cumulative sensitivities \label{sec:Experiment-rankings-according}}
Being one-dimensional projections of normalized residual variations
$\vec\delta_i$ on a given direction in the PDF parameter space,
sensitivities can be linearly added to construct a number of useful
estimators. By summing absolute sensitivities $|S_f^{(i)}|$ over the
data points $i$ of a given data set $E$, we find the maximal cumulative
sensitivity of $E$ to the PDF dependence of a QCD observable $f$.
Alternatively, from the examination of multiple $\{x,\mu\}$ maps for $\left|S_{f}\right|$
of various PDF flavors collected on the website \cite{PDFSenseWebsite},
we find that the most precise experiments constrain several flavors
at the same time; most notably, the combined HERA
data. For the purpose of identifying such experiments, we can compute an
overall sensitivity statistic for each experiment $E$ to the parton
distributions $f_a(x_i,\mu_i)$ evaluated at the same kinematic parameters
$\{x_i, \mu_i\}$ as the data.
Furthermore, to obtain one overall ranking, we can add
up sensitivity measures as an unweighted sum over the ``basis PDF'' flavors,
such as the six light flavors ($\overline{d},\,\overline{u},\,g,\,u,\,d,\,s$).
To obtain these measures, we say that an experiment $E$ consisting of
$N_{\mathit{pt}}$ physical measurements can be characterized by its mean sensitivity per raw data point\footnote{
For those circumstances in which an individual measurement, {\it e.g.}, obtained
via the Drell-Yan process, maps to two sensitivity values in $\{x,\mu\}$ space, we
compute the average of these and assign the result to that specific measurement.
}
to a PDF of given flavor $f_a(x,\mu)$:
$\langle|S^E_f|\rangle \equiv (N_{\mathit{pt}})^{-1} \sum_{i=1}^{N_{\mathit{pt}}}\left|S_{f}\right|(x_{i},\mu_{i})$,
from which we derive several additional statistical measures of experimental sensitivity.
For each experiment and flavor we then determine a cumulative sensitivity measure, numerically adjusted to the size of each experimental dataset $E$,
according to $|S^E_{f}| \equiv N_{\mathit{pt}}\, \langle|S^E_f|\rangle$.
In addition, we also track cumulative, flavor-summed sensitivity
measures $\sum_{f}|S^E_{f}|$ and
$\langle\sum_{f}|S^E_f|\rangle$, with $f$ running
over $\overline{d},\,\overline{u},\,g,\,u,\,d,\,s$.
We list the corresponding values of these four types of sensitivities
for each experiment of the CTEQ-TEA dataset in summary tables
in App.~\ref{sec:Tables}
as well as extensive Supplementary Material in App.~\ref{sec:SM}. This is also
detailed for categories of experiments from the CTEQ-TEA dataset.
With the above estimators, we {\it quantify}
and {\it compare} the cumulative sensitivities of each experiment to the basis 6
parton flavors. In fact, based on the various trials that we
performed, we find that the cumulative
sensitivity to the 6 basic flavors is a good measure of the overall
sensitivity to a large range of PDF combinations.
Recall that the $N_f=5$ CT14HERA2 PDFs (with up to 11 independent
parton species) are obtained by DGLAP evolution of the 6
basic parton flavors from the initial scale of order 1 GeV.
There exist alternative approaches for measuring the importance of a given
experiment in a global fit, for example, by counting the numbers of eigenvector
parameters \cite{Pumplin:2009sc} or eigenvector directions
\cite{Harland-Lang:2014zoa} that the experiment constrains. Those
other methods, however, require access to the full machinery of the global fit,
while the sensitivities allow the reader to rank the experiments
according to much the same information, for a variety of PDF-dependent
observables, with the help of
\textsc{PDFSense}, and at a fraction of computational cost.
In fact, in a companion study we use the above sensitivity estimators to select the new LHC
experiments for the inclusion in the next generation of the CTEQ-TEA
PDF analysis. Full tables given in App.~\ref{sec:Tables} and in the Supplementary Material of App.~\ref{sec:SM}
provide detailed information about the PDF sensitivities of every
experiment of the CTEQ-TEA data set.
For a non-expert reader, along the full tables, we provide
their simplified versions in Tables~\ref{tab5}-\ref{tab6},
where we rank the experimental sensitivities
according to a reward system described in the caption of Table~\ref{tab5}.
In each table, experiments are listed in descending order according
to the cumulative sensitivity measure $\sum_{f}|S^E_{f}|$ to the
six light-parton flavors. For each PDF flavor, the experiments with
especially high overall flavor-specific sensitivities receive an ``\textbf{A}''
rating (shown in bold), per the convention in the caption of
Table~\ref{tab5}. Successively
weaker overall sensitivities receive marks of ``B'' and ``C,'' while
those falling below a lower limit $|S^E_{f}|=20$ are left unscored.
We similarly evaluate each experimental dataset based on its point-averaged
sensitivity, in this case scoring according to a complementary scheme
in which the highest score is ``\textbf{1}''. The short-hand names of
the candidate experiments that were {\it not} included in the
CT14HERA2 NNLO fit, that is, the new LHC experiments, are also shown
in bold to facilitate their recognition in the tables.
Not only do the sensitivity rankings confirm findings known
by applying other methods, they also provide new insights.
According to this ranking system in
Tables~\ref{tab5}-\ref{tab6}, we find that the expanded HERA dataset
(HERAI+II'15) tallies the highest overall sensitivity to the PDFs,
with enhanced sensitivity to the distributions of the $u$- and $\bar{u}$-quarks,
as well as that of the gluon.
On similar footing, but with slightly weaker overall sensitivities, are
a number of other fixed-target measurements, including structure
function measurements from BCDMS for $F^{p,d}_2$ (BCDMSp'89, BCDMSd'90) and CCFR extractions
of $xF^p_3$ (CCFR-F3'97) --- as well as several other DIS datasets.
Among the LHC experiments, the inclusive jet measurements have the
highest cumulative sensitivities, with CMS jets at 8 TeV (CMS8jets'17),
7 TeV (CMS7jets'13, CMS7jets'14), and ATLAS 7 TeV (ATLAS7jets'15)
occupying positions 10, 12/13, and 16 in the total sensitivity
rankings. They demonstrate the strongest sensitivities among the candidate LHC
experiments, and at the same time are not precise enough and fall behind the top
fixed-target DIS and Drell-Yan experiments: BCDMS, CCFR, E605, E866,
and NMC. The two versions CMS7jets'13 and CMS7jets'14 of the CMS 7 TeV jet
data that largely overlap have very
close sensitivities and rankings in
Tables~\ref{tab5}-\ref{tab6}. The set CMS7jets'13 that extends to higher $p_{Tj}$ has a
slightly better overall sensitivity, surpassing the larger data set
CMS7jets'14 that includes the extra data points at $p_{Tj}<100$ GeV or
$|y_{j}|>2.5$, yet cannot beat CMS7jets'13 except for in
the overall sensitivity to the Higgs cross section at 7 TeV.
Going beyond the rankings based upon overall sensitivities, which are
more closely tied to the impact of an entire experimental dataset
in aggregate, it is useful to consider the point-averaged sensitivity
as well, which quantifies how sensitive each individual point
is. [Some experiments with very high point-averaged sensitivity have a
small cumulative sensitivity because of a small number of points.]
Based on their high point-averaged sensitivity, CMS $\mu$ asymmetry
measurements at 8 and 7 TeV (CMS8Wasy'16 and CMS7Masy2'14) especially stand
out, despite their small number of individual points, $N_{\mathit{pt}}=11$);
this is especially true again for the gluon, $\overline{d}$-, and
$u$-quark PDFs, for which this set of measurements is particularly
highly rated in Table~\ref{tab5}. Another ``small-size'' data set with
the exceptional point-average sensitivity is the
$\sigma_{pd}/(2\sigma_{pp})$ ratio from the E866 lepton pair
production experiment (E866rat'01). The average sensitivity of this data set to
$\overline u$ and $\overline d$ PDFs is 0.8, making it extremely valuable
for constraining the ratio $\overline{d}/\overline{u}$ at $x\sim 0.1$,
in spite of its small size (15 data points).
Aside from the quark- and gluon-specific rankings of specific measurements,
we can also assess experiments based upon the constraints they impose
on various interesting flavor combinations and observables as presented
in Table~\ref{tab6}. As was the case with Table~\ref{tab5}, a
considerable amount of information resides in Table~\ref{tab6} of
which we only highlight several notable features here. Among these
features are the sharp sensitivities to the Higgs cross section (\textit{e.g.}, $|S|_{H7}$, $\langle|S_{H7}|\rangle$, \textit{etc.}) found for
Run I$+$II HERA data, as well as the tier-C overall sensitivities
of the BCDMS $F^{p,d}_2$ and CMS jet
production measurements, corresponding to Exps.~BCDMSd'90, BCDMSp'89, CMS8jets'17 and CMS7jets'14.
While their
overall sensitivity is small, the corresponding ATLAS $t\overline{t}$
data also possesses significant point-averaged sensitivity. On the other
hand, measurements of $p_{T}$-dependent $Z$ production (ATL7ZpT'14, ATL8ZpT'16)
appear to have somewhat less pronounced sensitivity to the gluon and other PDF flavor
combinations. The total and mean sensitivities of high-$p_T$ $Z$ boson
production experiment ATL8ZpT'16 at 8 TeV is on par with HERA
charm SIDIS data (HERAc'13) and provides comparable constraints to charm
DIS production, albeit in a different $\{x,\mu\}$ region.
For the light-quark PDF combinations like $u_{v},\,d_{v},\,d/u,$
and $\overline{d}/\overline{u}$, the various DIS datasets \textemdash{}
led by Run II of HERA and CCFR measurements of the proton structure
function \textemdash{} demonstrate the greatest sensitivity. At the
same time, however, Run-2 Tevatron data from D0 on the $\mu$ asymmetry (D02Easy2'15)
and Run-1 CDF measurements for the corresponding $A_e(\eta^e)$ asymmetry (CDF1Wasy'96)
also exhibit substantial point-wise sensitivity as well. We collect
a number of other observations in the conclusion below,
Sec.~\ref{sec:Conclusions}.
\subsection{Estimating the impact of LHC datasets on CTEQ-TEA fits}
\label{sec:CTEQfit}
The presented rankings suggest that including the candidate LHC data
sets will produce mild improvements in the uncertainties of
the CT14 HERA2 PDFs. This projection may appear underwhelming, but
keep in mind that the CT14HERA2 NNLO analysis
already includes significant experimental constraints, for example,
imposed on the gluon PDF at $x>0.01$ by the Tevatron and LHC jet
experiments, CDF2jets'09, D02jets'08, ATL7jets'12, CMS7jets'13. If all jet experiments are eliminated
from the PDF fit, as illustrated in the Supplementary Material tables of App.~\ref{sec:SM}, the candidate LHC
experiments will be promoted to higher rankings, with the CMS 8 and 7 TeV jet experiments
(CMS8jets'17 and CMS7jets'13/CMS7jets'14) elevated to positions 4 and 7/8 in the overall sensitivity rankings,
respectively.
Our investigations also find that the sensitivities of
CMS jet experiments may improve considerably if
the current correlated systematic effects are moderately reduced
compared to the published values.
For instance, by
requiring a full correlation of the JEC2 correlation error over all
rapidity bins in the CMS 7 TeV jet data set CMS7jets'14, instead of its
partial decorrelation implemented according to the CMS recommendation
\cite{Khachatryan:2014waa}, we obtain a very strong sensitivity of the
data set CMS7jets'14 to $g$ over the full $\{x,\mu\}$ region; but also strong
sensitivities to $\overline u, \overline d$, and even $\overline s$
PDFs.\footnote{With the fully correlated jet energy correction JEC2 source, the data set CMS7jets'14
would provide a strong overall constraint on $s(x,\mu)$ comparable to
one of the NuTeV or neutrino CCFR experimental data sets.}
The overall sensitivity of the data set CMS7jets'14 in this case is
elevated to the 4th position from the 13th position in the CT14HERA2
NNLO analysis in Tables~\ref{tab5} and \ref{tab6}. Similarly, for the CMS
8 TeV jet data set CMS8jets'17, the sensitivity to the above flavors can
increase under moderate reduction of systematic uncertainties, easily
surpassing the sensitivity of CMS7jets'14 because of the larger number of
points in CMS8jets'17.
\subsection{Comparing {\sc PDFSense} predictions to post-fit constraints from Lagrange Multiplier scans}
\label{sec:Validation}
\begin{figure}[p]
\hspace*{-0.2cm}\includegraphics[clip,width=0.46\textwidth]{figs/sens_LM_du_refit.pdf} \ \
\includegraphics[clip,width=0.53\textwidth]{figs/LM_du_scan2T2.pdf}
\caption{
Left: the \textsc{PDFSense} map for the sensitivity of the
fitted dataset of the CT18pre NNLO analysis to the $d/u$ PDF ratio,
$d/u(x\!=\!0.1,\mu\!=\!1.3\, \mbox{ GeV})$. Right:
Dependence of $\chi^2$ for the individual and all experiments
of the CT18pre dataset on the value of $d/u(x\!=\!0.1,\mu\!=\!1.3\, \mbox{ GeV})$
obtained with the LM scan technique. The curves show the deviations
$\Delta \chi^2_\mathrm{expt.}\equiv\chi^2_\mathrm{expt.}(\vec a)-\chi^2_\mathrm{expt.}(\vec a_0)$
from the best-fit values in $\chi^2$ for the indicated experiments, as well as for the totality of
all experiments.
}
\label{fig:valid_du}
\end{figure}
\begin{figure}[p]
\hspace*{-0.2cm}\includegraphics[clip,width=0.46\textwidth]{figs/sens_LM_gMH_refit.pdf} \ \
\includegraphics[clip,width=0.53\textwidth]{figs/LM_gMHT_scan2T.pdf}
\caption{
Like Fig.~\ref{fig:valid_du}, but comparing the \textsc{PDFSense} map (left)
and LM scan (right) for the gluon PDF $g(x\!=\!0.01,\mu\!=\!m_H)$ in the Higgs boson production region.
}
\label{fig:valid_higgs}
\end{figure}
How do the surveys based on \textsc{PDFSense} compare against the actual
fits? As we noted, the \textsc{PDFSense} method is designed to provide a
fast large-scope estimation of the impact of the existing and future
data sets in conjunction with other tools, such as the \textsc{ePump} \cite{Schmidt:2018hvu}
program for PDF reweighting. It works the best in the quadratic
(Hessian) approximation near the best fit, and when the new experiments
are compatible with the old ones. When detailed understanding of the
experimental constraints is necessary, the \textsc{PDFSense} approach
must be supplemented by other techniques, such as Lagrange
multiplier (LM) scans \cite{Stump:2001gu,Pumplin:2000vx,Brock:2000ud}.
As an illustration of the scope of the differences between the
\textsc{PDFSense} predictions before and after the fit,
the left panels in Figs.~\ref{fig:valid_du} and \ref{fig:valid_higgs}
show the \textsc{PDFSense} maps for
$d/u(x\!=\!0.1,\mu\!=\!1.3\,\mbox{ GeV})$ and
$g(x\!=\!0.01,\mu\!=\!125 \mbox{ GeV})$ evaluated using a
preliminary CT18 NNLO fit (designated as ``CT18pre'') that includes 11 new
LHC experimental data sets, namely CMS8jets'17, CMS7jets'14,
ATLAS7jets'15, LHCb8WZ'16, CMS8Wasy'16, LHCb8Zee'15, LHCb7ZWrap'15, ATL8ZpT'16, ATL8ttb-pt'16, ATL8ttb-mtt'16,
and 8 TeV $t\bar{t}$ production at CMS (`CMS8 ttb pTtyt') \cite{Sirunyan:2017azo} in
addition to the experiments included in the CT14HERA2 fit. The full
details of the CT18 fit will be presented in an upcoming
publication \cite{CT18}. Some modifications were made in the
methodology adopted in CT18, as compared to CT14HERA2; notably the PDF
parametrization forms and treatment of NNLO radiative contributions
have been changed, while some shown curves are also subject to a
theoretical uncertainty associated with the QCD scale
choices. In accord with the
\textsc{PDFSense} predictions based on the CT14HERA2 NNLO PDFs, we find
that including the above LHC experiments into the fit
produces only mild differences between the CT18pre and
CT14HERA2 NNLO PDFs. Consequently the \textsc{PDFSense} $\{x,\mu\}$ maps based on
CT18pre NNLO PDFs are similar to the CT14HERA2 ones
\cite{PDFSenseWebsite}. One noticeable difference is that the
sensitivity of the new experiments decreases after they are included
in the CT18pre fit, because the new information from the newly
added experiments suppresses PDF uncertainties of data residuals.
In the right panels of Figs.~\ref{fig:valid_du}
and \ref{fig:valid_higgs}, we illustrate the constraints on the same
quantities, $d/u(0.1,1.3\mbox{ GeV})$ and $g(0.01,125\mbox{ GeV})$
in the candidate CT18pre NNLO fit, now obtained with the help of
LM scans. A LM scan \cite{Stump:2001gu,Pumplin:2000vx,Brock:2000ud}
is a powerful technique that elicits detailed information about a
PDF-dependent quantity $X(\vec a)$, such as a PDF or cross section,
from a constrained global fit in which the value of $X(\vec a)$ is fixed by
an imposed condition. By minimizing a modified goodness-of-fit function
$\chi^2_{LM}(\lambda,\vec{a})$ that includes a `generalized-force'
term equal to $X(\vec{a})$ with weight $\lambda$, in addition to the
global $\chi_{global}^2$ in Eq.~(\ref{eq:chi2glob}), a LM scan reveals
the parametric relationship between $X(\vec a)$ and
$\chi^2_\mathrm{global}$ or $\chi^2_\mathrm{expt.}$ contributions from
individual experiments, including any non-Gaussian dependence.
In the LM scans at hand, the modified fitted function takes the form
\begin{equation}
\chi^2_{LM}(\lambda,\vec{a}) = \chi^2_\mathrm{global}(\vec{a}) + \lambda X(\vec{a}),\
\end{equation}
and $X(\vec a)$ are $d/u(x,\mu)$ or $g(x,\mu)$ at a specific location in
$\{x,\mu\}$ space. For the optimal parameter combination
$\vec{a} \equiv \vec{a}_0$ at which $\chi^2_\mathrm{global}(\vec{a})$ is minimized,
we find in Fig.~\ref{fig:valid_du} that $d/u(0.1,1.3\mbox{ GeV}) \approx 0.7$. The LM scan for the
$d/u$ then consists of a series of refits of the parameters
$\vec{a}_k$, as the multiplier parameter $\lambda$ is dialed along a set
of discrete values $\lambda_k$, effectively
pulling $d/u$ away from the value $\sim\! 0.7$ at $\vec{a} = \vec{a}_0$
preferred by the global fit. The right panel of Fig.~\ref{fig:valid_du}
shows the relationship between
$d/u(0.1,1.3\mbox{ GeV})$ and $\chi^2_\mathrm{global}$ that is quantified
this way; and similarly for $g(0.01,125\mbox{ GeV})$.
We can also examine how the $\chi^2$ changes for the individual
experiments. Figs.~\ref{fig:valid_du} and \ref{fig:valid_higgs} show
the curves for 11 experiments with the largest variations
$\mathrm{max}(\chi^2)-\mathrm{min}(\chi^2)$ in the shown ranges of $d/u$
and $g$, i.e., the most constraining experiments.
We notice that, while the $\Delta\chi^2$ dependence is
nearly Gaussian for the total $\chi^2$, it is sometimes less so for
the individual experiments. Some experiments may be inconsistent
when they have a large best-fit $\chi^2(\vec a_0)$ or
prefer an incompatible $X$ value. Figure~\ref{fig:valid_du} is an
example of a good agreement between the experiments, when the individual
$\Delta\chi^2_{expt.}$ curves are approximately quadratic and
minimized at about the same location. Figure~\ref{fig:valid_higgs}
shows more pronounced inconsistencies, notably in the case of the E866pp
and ATL8ZpT curves that prefer a significantly larger
$g(0.01, 125\mbox{ GeV})$ than in the rest of the
experiments.
The LM procedure thus allows a systematic exploration
of the exact constraints from the experiments on $X$
without relying on the Gaussian assumption
that is inherent to the \textsc{PDFSense} method.
Both \textsc{PDFSense} and LM scans successfully identify the experiments with the strongest sensitivity
to $X$, while their specific rankings of such experiments are not
strictly identical and reflect the chosen
ranking prescription and settings of the global fit.
We emphasize that, though informative, the LM scans are computationally intensive, with a typical
30-point scan at NNLO requiring $\sim\!\! 6500$ CPU core-hours on a
high-performance cluster. This is in contrast to the \textsc{PDFSense}
analysis, which can be run for our entire 4021-point dataset on a single CPU core
of a modern workstation in $\sim\!\! 5$ minutes, representing
a $\sim\! 0.8 \times 10^5$ savings in computational cost.
\begin{table}
\begin{tabular}{|c c| c || c c| c|}
\hline
\multicolumn{3}{|c||}{$d/u(x\!=\!0.1,\mu\!=\!1.3\,\mbox{ GeV})$}
&\multicolumn{3}{c|}{$g(x\!=\!0.01,\mu\!=\!125 \mbox{ GeV})$} \tabularnewline
\multicolumn{2}{|c|}{\textsc{PDFSense}} & LM scan & \multicolumn{2}{c|}{\textsc{PDFSense}} & LM scan \tabularnewline
\hspace{0.5cm} CT14HERA2 \ \ & \ \ CT18pre \ \ & \ \ CT18pre \ \ & \ \ CT14HERA2 \ \ & \ \ CT18pre \ \ & \ \ CT18pre \hspace{0.3cm} \tabularnewline
\hline
\hspace{0.5cm} HERAI+II'15 & NMCrat'97 & NMCrat'97 & HERAI+II'15 & HERAI+II'15 & HERAI+II'15 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} BCDMSp'89 & HERAI+II'15 & CCFR-F3'97 & CMS8jets'17 & CMS8jets'17 & CMS8jets'17 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} NMCrat'97 & BCDMSp'89 & HERAI+II'15 & CMS7jets'14 & CMS7jets'14 & ATL8ZpT'16 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} CCFR-F3'97 & CCFR-F3'97 & BCDMSd'90 & ATLAS7jets'15 & E866pp'03 & E866pp'03 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} E866pp'03 & BCDMSd'90 & BCDMSp'89 & E866pp'03 & ATLAS7jets'15 & ATLAS7jets'15 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} BCDMSd'90 & E605'91 & CDHSW-F3'91 & BCDMSd'90 & BCDMSd'90 & CCFR-F2'01 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} CDHSW-F3'91 & E866pp'03 & E866rat'01 & CCFR-F3'97 & BCDMSp'89 & D02jets'08 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} CMS8jets'17 & E866rat'01 & CMS7Masy2'14 & D02jets'08 & D02jets'08 & HERAc'13 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} E866rat'01 & CMS8jets'17 & NuTeV-nu'06 & NMCrat'97 & NMCrat'97 & NuTeV-nub'06 \hspace{0.3cm} \tabularnewline
\hspace{0.5cm} LHCb8WZ'16 & CDHSW-F3'91 & CMS8jets'17 & BCDMSp'89 & CDHSW-F2'91 & CCFR-F3'97 \hspace{0.3cm} \tabularnewline
\hline
\end{tabular}
\caption{
We list the top 10 experiments predicted to drive knowledge of the
$d/u$ PDF ratio and of the gluon distribution in the Higgs region
according to \textsc{PDFSense} and LM scans.
For both, we list the \textsc{PDFSense} evaluations based both
on the CT14HERA2 fit and on a preliminary CT18pre fit
in the first and second columns on either side of the
double-line partition.
}
\label{tab:valid}
\end{table}
Let us further illustrate these observations by referring again to
Figs.~\ref{fig:valid_du} and \ref{fig:valid_higgs}, as well as to
Table~\ref{tab:valid} that displays the top 10 experiments with the
largest cumulative sensitivity to $d/u(0.1,1.3 \mbox{ GeV})$ and
$g(0.01,\ 125 \mbox{ GeV})$ according to \textsc{ PDFSense} and LM
scans, with either CT14HERA2 or CT18pre PDFs used to
construct the \textsc{PDFSense} rankings.
In the \textsc{PDFSense} columns, the experiments are ranked in order
of descending cumulative sensitivities $\sum_{i=1}^{N_{pt}} |S_{f}|(x_i,\mu_i)$ according to the
same prescription as in Sec.~\ref{sec:Experiment-rankings-according}.
For the LM scans, the table shows the experiments that have the
largest variations $\mathrm{max}(\chi^2)-\mathrm{min}(\chi^2)$ in the
range of $X$ corresponding to $\Delta \chi^2_{global}\leq 100$, that
is, within approximately the 90\% probability level interval of the CT18pre NNLO
PDFs. As the residual uncertainties $\Delta r_i$ in the sensitivities $S_f$ are normalized to
the root-mean-squared residuals $\langle r_0\rangle_E$ at the best fit,
cf. Eq.~(\ref{eq:sens}), we similarly divide
$\mathrm{max}(\chi^2)-\mathrm{min}(\chi^2)$ by the best-fit
$\chi^2(\vec{a}_{0})/N_\mathit{pt}$ of the experiment in the rankings for the LM scans in Table~\ref{tab:valid}.
From the side-by-side examination of the figures and the table, we can draw a broad conclusion
that both the pre-fit \textsc{PDFSense} and post-fit LM scan
approaches agree in identifying the most constraining
experiments, even though they may result in different orderings of these
experiments. This agreement is especially impressive
in the instance of $d/u(x\!=\!0.1,\mu\!=\!1.3\,\mbox{ GeV})$, when
the rankings agree on 8 out of 10 leading experiments, confirming the
dominance of the NMC $p/d$ ratio, HERAI+II, CCFR $F_3$, and
BCDMS $p$ and $d$ measurements. For $g(x\!=\!0.01,\mu\!=\!m_H)$,
for which we see more tension and non-Gaussian behavior in Fig.~\ref{fig:valid_higgs},
both \textsc{PDFSense} and LM scans concur on the crucial role played by the top 5-6 experiments, namely,
HERAI+II, E866pp, and inclusive jet production data from CMS, ATLAS, and
D0 Run-2. The upward pull on $g$ from the incompatible ATL8ZpT data set seen
in Fig.~\ref{fig:valid_higgs} modifies the rankings of the trailing
experiments, such as CMS7 jets or BCDMS. Based upon an extended
battery of LM scans we have performed, including the two examples
presented here, we conclude that the \texttt{PDFSense} surveys perform
as intended.
Lastly, we reiterate that a number of subtleties exists in comparing
the results of LM scans and \textsc{PDFSense} sensitivity plots. Most importantly,
\textsc{PDFSense} is intended by conception as a tool to quantify the
anticipated {\it average} impact of potentially unfitted data
based upon their precision in comparison to the PDF
uncertainties. We discussed simplifying assumptions made in
\textsc{PDFSense} in order to bypass certain complexities of the full
fit and obtain quick estimates.
LM scans, on the other hand, provide post-fit assessments of the contributions of specific
data to the global $\chi^2$ function, as specific quantities predicted
by the QCD analysis are dialed away from
their optimal values. In the comparisons we made,
the detailed pictures produced by both \textsc{PDFSense} and the LM
scans depend on a variety of theoretical settings like pQCD scale
choices, as well as upon the specific implementation of correlated
experimental uncertainties [from up to $\sim\!\! 100$ different sources in
some experiments] and the parametric forms chosen for the nonperturbative
parametrizations at the starting scale $\mu = Q_0$. The inclusion of additional theory uncertainties
and decorrelation of some experimental correlated errors are
necessitated in a few experiments
by the relatively large $\chi^2$ values that would otherwise be obtained.
All these have some peripheral effect on the specific orderings of experiments shown in Table~\ref{tab:valid}.
Thus, rather than anticipating an exact point-to-point matching
between the \textsc{PDFSense} and LM methods, we instead expect, and
indeed find, the general congruity between the most important
experiments identified by the two approaches illustrated in this section.
\section{Conclusions }
\label{sec:Conclusions}
In the foregoing analysis, we have confronted the modern challenge
of a rapidly growing set of global QCD data with new statistical methodologies
for quantifying and exploring the impact of this information. These
novel methodologies are realized in a new analysis tool \textsc{PDFSense
\cite{PDFSenseWebsite},} which allows the rapid exploration of the
impact of both existing and potential data on PDF determinations,
thus providing a means of weighing the impact of measurements of QCD
processes in a way that allows meaningful conclusions to be drawn
without the cost of a full global analysis. We expect this approach
to guide future PDF fitting efforts by allowing fitters to examine
the world's data \textit{a priori,} so as to concentrate analyses on
the highest impact datasets. In particular, this work builds upon
the existing CT framework with its reliance on the Hessian formalism
and assumed quasi-Gaussianity, but these features do not impact the validity
of our analysis and conclusions.
Our approach provides a means to carry out a detailed study of
data residuals, for which we explored novel visualizations in several
ways, including the PCA, t-SNE, and reciprocated distance approaches
discussed in Sec. \ref{subsec:Manifold-learning}. These techniques
show promise for moving forward by providing useful insights into the
numerical relationships among datasets and experimental processes.
Crucial to this analysis is the leveraging of both the existing and
proposed statistical measures laid out in Secs.~\ref{sec:Correlations} and \ref{sec:Sensitivities}.
Of these, the flavor-specific sensitivity $S_{f}$ of Eq.~(\ref{eq:sens})
for a data point to the PDF serves as a particularly powerful discriminator,
and we deployed it and the correlation $C_{f}$ of Eq.~(\ref{eq:corr})
to map PDF constraints provided by data over a wide range in $\{x,\mu\}$.
This was facilitated by the fact that the sensitivity and correlation
are readily computable over the extent of the global dataset. The
companion website collects a large number of figures illustrating
the sensitivities to various flavors as a function of $x$ and $\mu$.
To quantify the abundant information contained in the maps of sensitivities,
in Sec.~\ref{sec:Experiment-rankings-according}
we presented statistical estimators to systematically rank and assess subsidiary
datasets within the world's data according to their potential to be influential
in constraining PDFs. We note that one
is allowed some freedom in choosing a specific ranking prescription, but we find our
conclusions to be stable against variations among these possible choices. In this context,
we reaffirmed the unique advantage of DIS and jet production for determination of the PDFs.
Many intriguing physics results can be established using our
sensitivity methods, and the specific results in the previous sections
are only illustrative examples. We stress that these results take
the complementary form of sensitivity tables (for example, Table \ref{tab5})
and $\{x,\mu\}$ plots (such as Fig. \ref{fig:CorrSensH14}), which
respectively offer global categorizations of the experimental landscape and
detailed mappings of the placements of PDF constraints in
$\{x,\mu\}$ space. In totality, the full range of physics insights from this method
is beyond the scope of the present article, but the interested user
can explore them using our \textsc{PDFSense} package at \cite{PDFSenseWebsite}.
We mention only a representative sample of these to motivate the reader:
\begin{itemize}
\item A wide range of experimental processes possess sensitivity to the
nucleon's quark sea distributions; for example, for the distribution
$\overline{d}(x,\mu)$, the $\sigma_{pd}$ DY measurements of E866
(E866rat'01) exhibit strong sensitivity, but so do DY data from E605 (E605'91)
as well as (at larger $\mu$) information on the $\mu$-production
asymmetry $A_{\mu}(\eta)$ from CMS at 7 TeV (CMS7Masy2'14); at high $x$ and
$\mu$, CMS inclusive jet data (CMS8jets'17, CMS7jets'14) also acquire some sensitivity
to $\bar u$ and $\bar d$.
Still, however, the recent HERA data (HERAI+II'15) registers the greatest
overall sensitivity.
\item Were they taken cumulatively together as a single dataset, CMS jet
production at 7 and 8 TeV (CMS7jets'14 and CMS8jets'17) would provide a total sensitivity
$|S^E_s| = 11.9 + 8.11$ to $s(x,\mu)$ that is comparable to
one of the NuTeV (NuTeV-nu'06) or CCFR (CCFR SI nu'01, CCFR SI nub'01) dimuon SIDIS experiments,
which have very strong average sensitivity to the strange
distribution. Still, the strongest constraint is contributed by
a mix of the DIS measurements, including $\nu\mu\mu$ data from NuTeV
(NuTeV-nu'06), data on $\nu(\overline{\nu})\mu\mu$ processes from SIDIS
at CCFR (CCFR SI nu'01 and CCFR SI nub'01), as well as the inclusive DIS data at lower $x$ from
HERA1+2 (HERAI+II'15) that actually has the strongest cumulative sensitivity.
Similarly, various vector boson production data sets have a rank-3
point-averaged sensitivity to the strangeness, including
the $A_{\mu}(\eta^\mu)$ data from D0 (D02Masy'08) and CMS (CMS8Wasy'16, CMS7Masy2'14), as well
ATLAS $W/Z$ production (ATL8DY2D'16, ATL7WZ'12) and high-$p_T$ $Z$ production (ATL8ZpT'16)
cross sections. Although each of the
individual vector boson production data set has a weak cumulative sensitivity
to $s(x,\mu)$ because of a small number of data points,
in totality a group of {\it mutually consistent} LHC experiments on
vector boson production can
provide a competing constraint on $s(x,\mu)$ that confronts the
low-energy CCFR/NuTeV constraints.
\item Knowledge of the charm distribution $c(x,\mu)$ is most influenced
by a number of datasets, with HERA (HERAI+II'15) at low $x$ especially important.
Fixed target measurements, particularly those of CDHSW on the proton's
$F_{2}^{p}$ structure function (CDHSW-F2'91) have strong sensitivity at slightly
higher $x\!\sim\!10^{-1}$, while a wide range of jet measurements, including
7 TeV data from ATLAS (ATLAS7jets'15) and CMS (CMS7jets'14), and 8 TeV CMS (CMS8jets'17) points
are also sensitive. This pattern of sensitive
measurements broadly follows the corresponding plot for $|S_{g}|(x_{i},\mu_{i})$
{[}as well as $|S_{b}|(x_{i},\mu_{i})${]} due to the dominance of
boson fusion graphs in heavy quark production. The datasets of
importance we identify are broadly consistent with the conclusions
of the recent CT14 analysis \cite{Hou:2017khm} of the nucleon's
intrinsic charm \cite{Hobbs:2013bia}.
\item One can also study the correlations and sensitivities for various
derived PDF combinations. For instance, for the $\overline{d}/\overline{u}$
ratio representing deviations from flavor symmetry in the nucleon
sea, the E866 experiment (E866rat'01) shows exceptional point-averaged
sensitivity, $\langle|S_{\bar{d}/\bar{u}}|\rangle=1.67$ such that
its ``C'' ranking for its overall sensitivity to $\bar{d}/\bar{u}$
places it in the company of only a few other DIS and DY experiments, despite
their much larger number of measurements, $N_{\mathit{pt}}=15$. At somewhat lower $x\gtrsim0.01$,
NMC data on the structure function ratio $F_{2}^{d}/F_{2}^{p}$ (NMCrat'97)
show sensitivity in the range $0.8<|S_{\overline{d}/\overline{u}}|<2$.
At still lower $x$, the CMS 8 and 7 TeV $A_{\mu}$ points (CMS8Wasy'16, CMS7Masy2'14) and
$W/Z$ data from LHCb (LHCb8WZ'16) show strong pull, corresponding to
point-averaged rankings of ``2,'' ``{\bf 1},'' and ``2,'' respectively.
\item We also consider the PDF ratio $d/u(x,\mu)$, which often serves as
a discriminant among various nucleon structure models, especially at
high $x$. For $x>0.1$ an amalgam of fixed-target experiments, including
the NMC $F_{2}^{d}/F_{2}^{p}$ data (NMCrat'97) particularly, but also $F_{2}^{p}$
measurements from BCDMS (BCDMSp'89) and CCFR (CCFR-F2'01) as well as $xF_{3}^{p}$ data
from CCFR drive the current status.
At higher $\mu$, however, the LHCb $W/Z$ data (LHCb8WZ'16) and $A_{e}(\eta)$
measurements from Run-2 of D0 (D02Easy2'15) also constrain the high $x$
behavior of $d/u$ together with $A_{\mu}(\eta)$ points
from CMS at 7 TeV (CMS7Masy2'14).
\item
More generally, we note that, among the new LHC experiments to be considered for future
global fits, the datasets for inclusive jet production are expected to have the greatest
impact, followed by a group of vector boson production experiments at ATLAS, CMS, and LHCb.
We find that the constraints from jet production at the LHC depend significantly on
the treatment of experimental systematic uncertainties --- especially the correlated systematic errors.
It is conceivable that, with the full implementation of NNLO theoretical cross sections and
modest reduction in the experimental systematic uncertainties, the constraints from the LHC jet
production will catch up in strength to the effect of adding a large fixed-target DIS dataset, such as
BCDMS $F^p_2$ (BCDMSp'89). Meanwhile, the magnitude of the constraint on the gluon PDF from high-$p_T$ $Z$
production (ATL8ZpT'16) is comparable to those from the combined HERA SIDIS charm dataset (HERAc'13) or
inclusive jet production from CDF Run-2 (CDF2jets'09); that is, the high-$p_T$ $Z$ data are significant
in the event that the jet datasets are not included, in overall consistency with the findings in
Ref.~\cite{Boughezal:2017nla}. The smaller ATLAS $t\overline{t}$ production data sets (ATL8ttb-pt'16,
ATL8ttb-y\_ave'16, ATL8ttb-mtt'16, ATL8ttb-y\_ttb'16) have strong point-by-point sensitivity to the gluon,
but will have a more diminished role when combined with other, larger data sets. HERA DIS (HERAI+II'15),
BCDMS $F_2^d$ (BCDMSd'90), and CMS inclusive jets at 8 TeV (CMS8jets'17) render the strongest overall
constraints on the Higgs production cross section at the LHC according to the rankings in Table~\ref{tab6}.
\end{itemize}
Quantifying correlations and sensitivities thus provides a comprehensive means of
evaluating the ability of a global dataset to constrain our knowledge
of nucleon structure. It must be emphasized,
however, that this analysis is not a substitute for actually performing
a QCD global analysis, which remains the single most robust means
of determining the nucleon PDFs themselves. Rather, the method presented
in the paper is a guiding tool to both supplement and direct fits
by gauging the potential for improving PDFs with the incorporation of new
datasets.
The essential ingredients of this study are the PDF-residual correlation
and sensitivity $|C_{f}|$ and $|S_{f}|$, with the latter representing
an extension of the correlation used elsewhere in the modern PDF literature.
These definitions are robust enough that we can exhaustively score
the data points in an arbitrary global dataset to construct and map
the resulting distributions, as shown in Figs.~\ref{fig:corr-main}
and \ref{fig:sens-main}. Accordingly, we found it possible to impose
cuts on these distributions to identify points of especially strong
correlation ($|C_{f}|>0.7$) or sensitivity ($|S_{f}|>0.25$); we
stress that these cuts are chosen as approximate indicators, and any
user can adjust them freely. On the other hand, the distributions
themselves, as shown in the second panels of Figs.~\ref{fig:corr-main} and \ref{fig:sens-main},
are not subject to such cut choices.
Although the conclusions of this analysis are resistant to alterations
in the basic approach, it is worth noting that other formats are possible
for evaluating experimental sensitivities and performing the rankings of
measurements. For example, one might use somewhat different matchings than
those outlined in App.~\ref{sec:supp} to extract $\{x,\mu\}$ points from the
experimental data, but we expect the resulting impact on
the overall picture to be minor. Similarly, while the ordering inside
ranking tables like Table~\ref{tab5} was decided according to the total
sensitivity to serve our specific goal of identifying the most valuable experiments
for the CTEQ-TEA fit, for other purposes one might produce alternative tables ranked
according to point-averaged sensitivities, or sensitivities to
specific flavors. Such alternate conventions would also yield important
information, and \textsc{PDFSense} allows the user to do this. It should
be stressed that these elections for the form of our presentation
can always be recovered from the more fundamental information ---
the numerical values of the sensitivities detailed in the Supplementary
Material of App.~\ref{sec:SM}.
While we have demonstrated these techniques in the context of the
CT14 family of global fits, they are of sufficient generality that
one could readily repeat our analysis using alternative PDF sets.
For the sake of testing this point and validating our predictions
for the most decisive experiments in the CTEQ-TEA dataset, we performed
a preliminary fit including the CT14HERA2 and the candidate LHC experiments
(`CT18pre'), and directly compared \textsc{PDFSense} predictions against
Lagrange multiplier scans quantifying the constraints these fitted measurements
imposed on select quantities. This provided a demonstration of the robustness
of our sensitivity-based analysis, which identified the same sets of high-impact
measurements {\it before fitting}.
The results of this study can be expected to vary somewhat depending
on the specifics of the PDF sets used to compute $|C_{f}|$ and $|S_{f}|$,
but we see this as an advantage of \textsc{PDFSense}. One
could imagine exploiting them to undertake a systematic analysis of the impact of various theoretical
assumptions implemented in competing global fits (\textit{e.g.}, the
choice of input PDF parametrization or the status of the perturbative
QCD treatment implemented in various processes). The sensitivity $S_f$
can be constructed either from the Hessian or Monte-Carlo PDF uncertainties,
as prescribed by Eqs.~(\ref{eq:sens}) and (\ref{eq:sensMC}), while the shifted
residuals that are crucial to our analysis can be recovered from any type of
covariance matrix, as argued in relation to Eq.~(\ref{eq:res-cov}).
In the same spirit but on
the side of the data, \textsc{PDFSense} empowers the user to evaluate
the combined impact of multiple experimental datasets \textemdash{}
for example, to evaluate the extent to which the impact of a proposed
experiment might be diminished by the constraints already imposed
by existing measurements. These various functions collectively suggest
a number of possible avenues to use the presented approach and the \textsc{PDFSense}
tool to advance PDF knowledge in the coming
years.
\subsection*{Acknowledgments}
We thank our CTEQ-TEA colleagues, Davison Soper, and Madeline Hamilton
for support and insightful discussions, and
appreciate helpful clarifications concerning the LHC
experimental data sets from Alexander Glazov, Uta Klein,
Bogdan Malescu, and Klaus Rabbertz. We also thank German Valencia,
Ursula Laa, and Dianne Cook for helpful discussions related to data visualizations based
on the PCA and t-SNE methods. This work was supported in part by the
U.S.~Department of Energy under Grant No.~DE-SC0010129 and by the
National Natural Science Foundation of China under the Grant No.~11465018.
T.J.~Hobbs acknowledges support from an EIC Center Fellowship.
The work of J.G. is sponsored by Shanghai Pujiang Program.
|
{
"timestamp": "2019-01-24T02:19:25",
"yymm": "1803",
"arxiv_id": "1803.02777",
"language": "en",
"url": "https://arxiv.org/abs/1803.02777"
}
|
\section*{Methods}
\label{(sec:methods)}
ADFMR devices were fabricated on commercially-available Y-Cut LiNbO$_3$ substrates (MTI Corp). Patterning was performed using standard g-line photolithography. Al contacts were deposited using thermal evaporation, and the nickel and cobalt films were deposited using e-beam evaporation and sputtering, respectively. Commercial nanodiamonds containing NV centers (Ad\'amas Nanotechnologies) were deposited using laser-pulled glass pipettes at multiple locations on top of the ferromagnetic pad. A signal generator was used to generate the microwave excitations, and electrical transmission measurements were performed both using both a time-gating technique in conjunction with a spectrum analyzer\cite{labanowski2016power, labanowski2017effect} as well as by using a lock-in amplifier and an RF power diode.
Optical PL measurements were performed using a home-built NV characterization setup. A 532 nm laser was used to excite the NV centers and the resulting PL was measured using a photodiode. A lock-in measurement was performed by modulating the amplitude of the acoustic waves in order to enable the measurement of small changes in the NV PL. Details of the measurement of FMR using NV centers can be found in existing literature\cite{wolfe2014off}. All optical measurements in this work were performed with the external bias field oriented at 45$^\circ$ in-plane from the SAW propagation direction.
\section*{Acknowledgements}
\label{(sec:acknowledgements)}
This research was supported by an appointment to the Intelligence Community Postdoctoral Research Fellowship Program at the University of California, Berkeley, administered by Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and the Office of the Director of National Intelligence. Funding for the research at Ohio State University was provided by the Army Research Office through Grant W911NF-16-1-0547. The authors would like to thank Dr. Charles Henry Lambert for assistance with sample preparation and Dr. Praveen Gowtham and Niklas Roschewsky for useful discussions.
\section*{Author Contributions}
\label{(sec:contributions)}
D.L., V.P.B., Q.G., C.M.P., and B.A.M. performed measurements and analyzed data. D.L. worked on device fabrication. V.P.B., B.A.M., Q.G., and C.M.P. built the measurement setup. P.C.H. and S.S. initiated the research. D.L. and V.P.B. wrote the manuscript. All authors read and commented on the manuscript.
|
{
"timestamp": "2018-03-13T01:01:10",
"yymm": "1803",
"arxiv_id": "1803.02863",
"language": "en",
"url": "https://arxiv.org/abs/1803.02863"
}
|
\section{Introduction}
While evidence suggests that the frequency of short period stellar and planetary-mass companions to main sequence stars is high, there appears to be a relative lack of companions in the brown dwarf regime \citep{2006ApJ...640.1051G}. These objects appear to exist in an overlap region of formation processes, as the low-mass tail of stellar binary formation, and as the high-mass end of the planetary distribution. Observations of brown dwarf companions to young stars then present an important opportunity to study these different formation pathways.
The processes that form companions of all masses appear to have a sensitive dependence on the host star mass, with evidence suggesting that high and intermediate mass stars have more of such companions than their lower mass counterparts \citep[e.g.][]{2010ApJ...709..396B,2010PASP..122..905J,2013ApJ...773..170J,2014A&A...566A.113J,2015ApJS..216....7B,2016A&A...596A..83L}.
However, our knowledge of such companions is limited by the small number of objects detected to date, covering a wide range of parameter space in terms of companion mass, primary mass, age and orbital semi-major axis. Only a few wide-separation objects have been detected around stars with masses $>2$\,M$_{\odot}$, such as HIP 78530B \citep{2011ApJ...730...42L}, $\kappa$ And b \citep{2013ApJ...763L..32C}, HR 3549B \citep{2015ApJ...811..103M} and HIP 77900B \citep{2013ApJ...773...63A}.
A new generation of dedicated planet-finding instruments such as SPHERE \citep{2008SPIE.7014E..18B} and GPI \citep{2008SPIE.7015E..18M} offer significant improvements in the detection capability for substellar companions, as well as for spectroscopic and astrometric follow-up.
The SHINE survey \citep{2017A&A...605L...9C} utilizes the SPHERE instrument at the VLT to search the close environments of 600 young, nearby stars for substellar and planetary companions. In this paper, we present the imaging discovery and follow-up spectroscopy of a young, low mass brown dwarf companion identified during the course of this survey.
\section{Stellar Properties}
HIP 64892 is a B9.5 star \citep{houk1993}, classified as a member of the Lower Centaurus Crux (hereafter LCC) association by \cite{dezeeuw1999} and \cite{rizzuto2011}, with membership probabilities of 99\% and 74\%, respectively. A recent update to the BANYAN tool by \citet{2018arXiv180109051G} gives a membership probability of 64\% to the LCC subgroup, 33\% to the younger Upper Centaurus Lupus subgroup, and a 3\% probability of HIP 64892 being a field star. Since the star is not included in GAIA DR1, we adopt the trigonometric parallax by \citet{vl2007}, yielding a distance of $125 \pm 9$ pc. Photometry of the system is collected in Table ~\ref{tab:stellar_params}. No significant variability is reported by Hipparcos (photometric scatter 0.007 mag).
To further characterize the system, the star was observed with the FEROS spectrograph at the 2.2m MPG telescope operated by the Max Planck Institute for Astronomy. Data were obtained on 2017-03-30 and were reduced with the CERES package \citep{ceres}\footnote{https://github.com/rabrahm/ceres}. A summary of the stellar properties derived from this analysis, and those compiled from the literature, is given in Table ~\ref{tab:stellar_params}. From the FEROS spectrum, we measured a Radial Velocity (RV) of 14.9 km/s and a projected rotational velocity of 178 km/s, using the cross-correlation function (CCF) procedure described in \citet{2017A&A...605L...9C}. This latter value is not unusual among stars of similar spectral type, unless significant projection effects are at work.
The FEROS CCF does not show any indication of additional components. This agrees with the conclusions of \citet{chini2012}, who found no evidence of binarity from five RV measurements. The observed RV values from that study were not published, preventing an assessment of possible long-term RV variability\footnote{The RV provided by SIMBAD originates from the study of \citet{madsen2002} and was not derived from spectroscopic measurements but rather from the velocity expected from kinematics.}.
The SPHERE data also do not provide any indication of the presence of bright stellar companions, ruling out an equal-luminosity binary down to a separation of about 40 mas.
By comparing the observed colours of HIP 64892 with those expected from the tables by \citet{pecaut2013}, we find they are consistent with the B9.5 spectral classification, and find a low reddening value of E(B-V)=0.01 consistent with the lack of V band extinction found by \citet{2012ApJ...756..133C}. In addition, the \citet{2012ApJ...756..133C} analysis showed no signs of an infrared excess. We explore the implications of this on the presence of dust in the system in Appendix \ref{appendix:dust_mass}.
The \citet{pecaut2013} tables predict an effective temperature of 10400\,K for a B9.5 star. For such a hot star, the pre-main sequence isochrones collapse on the zero-age main sequence (ZAMS) in less than 10\,Myr, and significant post-main sequence evolution is not expected for tens of Myr following this. When combined with the large uncertainty on the parallax of HIP 64892, this makes prediction of its age based on isochronal analysis difficult. The V magnitude and effective temperature of HIP 64892 are shown in Figure \ref{fig:isochrones}, relative to the \citet{bressan2012} isochrones at various ages between 5-200\,Myr. Within the uncertainties, the placement of HIP 64892 is close to the ZAMS, and therefore consistent with LCC membership.
The Sco-Cen sub-groups are known to show significant age spread. The recent age map by \citet{pecaut2016} yields a value of 16 Myr at the location of our target, equal to the commonly adopted age for the group. We adopt this value as the most likely age of HIP 64892, with the approximate ZAMS time as a lower limit. While an upper limit from comparison with the isochrones would be $\sim$100\,Myr, we instead use the nearby star TYC 7780-1467-1 to place a more precise bound on the age. This star has a well-determined age of 20\,Myr calculated from comparison with isochrones and supported by its lithium abundance \citep[EW Li=360 m\AA,][]{sacy} and fast rotation
\citep[rotation period 4.66d,][]{kiraga2012}. Given this value, we think it unlikely that HIP 64892 is older than $\sim$30\,Myr. We adopt an uncertainty of 15\,Myr, leading to an age estimate of $16^{+15}_{-7}$.
\begin{figure}
\includegraphics[width=0.98\columnwidth]{hip64892_isochrone.pdf}
\caption{The V-band absolute magnitude and effective temperature of HIP 64892 compared to predictions from the 5, 12, 20, 30, 70 and 200Myr isochrones of \citet{bressan2012}. The placement of HIP 64892 is consistent with the estimated local age of 16\,Myr within the measured uncertainties. }\label{fig:isochrones}
\end{figure}
The stellar mass and radius from the \citet{bressan2012} models are 2.35$\pm$0.09 $M_{\odot}$ and 1.79$\pm$0.10 $R_{\odot}$, respectively.
Due to its unknown rotational inclination and large measured $v\sin i$, HIP 64892 is expected to show significant oblateness. For this reason, any measurements of the stellar parameters may be influenced by its orientation. While a similar degeneracy between inclination and age caused issues for studies of other rapidly rotating stars such as $\kappa$ And \citep[e.g.][]{2013ApJ...779..153H,2016ApJ...822L...3J}, our age estimate relies on the firm Sco-Cen membership of HIP 64892, which is independent of these concerns. However, the effective temperature and mass in particular may be affected by the inclination of HIP 64892.
\begin{center}
\begin{table}
\caption{Stellar parameters of HIP 64892.}\label{tab:stellar_params}
\begin{tabular}{lcl}
\hline\hline
Parameter & Value & Ref \\
\hline
V (mag) & 6.799$\pm$0.008 & \citet{slawson1992} \\
U$-$B (mag) & -0.094$\pm$0.011 & \citet{slawson1992} \\
B$-$V (mag) & -0.018$\pm$0.006 & \citet{slawson1992} \\
V$-$I (mag) & 0.00 & Hipparcos \\
J (mag) & 6.809$\pm$0.023 & \citep{cutri20032mass} \\
H (mag) & 6.879$\pm$0.034 & \citep{cutri20032mass} \\
K (mag) & 6.832$\pm$0.018 & \citep{cutri20032mass} \\
E(B-V) & 0.01$^{+0.02}_{-0.01}$ & this paper \\
Parallax (mas) & 7.98$\pm$0.55 & \citet{vl2007} \\
Distance (pc) & $125\pm9$\,pc & from parallax\\
$\mu_{\alpha}$ (mas\,yr$^{-1}$) & -30.83$\pm$0.50 & \citet{vl2007} \\
$\mu_{\delta}$ (mas\,yr$^{-1}$) & -20.22$\pm$0.43 & \citet{vl2007} \\
RV (km\,s$^{-1}$) & 14.9 & this paper \\
SpT & B9.5V & \citet{houk1993}\\
$T_{\rm eff}$ (K) & 10400 & SpT+Pecaut calib. \\
$v \sin i $ (km\,s$^{-1}$) & 178 & this paper \\
Age (Myr) & $16^{+15}_{-7}$ & this paper \\
$M_{\text{star}} (M_{\odot})$ & 2.35$\pm$0.09 & this paper \\
$R_{\text{star}} (R_{\odot})$ & 1.79$\pm$0.10 $R_{\odot}$, & this paper \\
\hline\hline
\end{tabular}
\end{table}
\end{center}
\section{Observations and Data Reduction}
\subsection{SPHERE Imaging}
HIP 64892 was observed with SPHERE on 2016-04-01 as part of the SHINE exoplanet survey. These data were taken with the IRDIFS mode. After a bright companion candidate was discovered in these data, follow-up observations were taken on 2017-02-08 using the IRDIFS\_EXT mode to confirm its co-moving status and extend the spectral coverage.
These modes allow the Integral Field Spectrograph \citep[IFS;][]{2008SPIE.7014E..3EC} and Infra-Red Dual-band Imager and Spectrograph \citep[IRDIS;][]{2008SPIE.7014E..3LD} modules to be used simultaneously through the use of a dichroic. In these configurations, IFS provides a low resolution spectrum ($R\sim55$ across Y-J bands or $R\sim35$ across Y-H bands) while IRDIS operates in dual-band imaging mode \citep{2010MNRAS.407...71V}.
For each observing sequence, several calibration frames were taken at the beginning and end of the sequence. These consisted of unsaturated short exposure images to estimate the flux of the primary star and as a Point Spread Function (PSF) reference, followed by a sequence of images with a sinusoidal modulation introduced to the deformable mirror to generate satellite spots used to calculate the position of the star behind the coronagraph. The majority of the observing sequence consisted of long exposure (64\,s) coronagraphic imaging.
For the 2017 data, the satellite spots were used for the entirety of the coronagraphic imaging sequence rather than a separate set of frames at the beginning and end. This allowed us to correct for changes in flux and the star's position during the observations, at the cost of a small contrast loss at the separation of the satellite spots.
The data were reduced using the SPHERE Data Reduction and Handling (DRH) pipeline \citep{2008SPIE.7019E..39P} to perform the basic image cleaning steps (background subtraction, flat fielding, removal of bad pixels, calculation of the star's position behind the coronagraph, as well as extraction of the spectral data cubes for IFS). To process the IFS data, the DRH routines were augmented with additional routines from the SPHERE Data Center to reduce the spectral cross-talk and improve the wavelength calibration and bad pixel correction \citep{2015A&A...576A.121M}. Both IFS and IRDIS data were corrected using the astrometric calibration procedures described in \cite{2016arXiv160906681M}.
A bright companion candidate is clearly visible in the raw coronagraphic frames, at a separation of $1.270\pm0.002"$ in the 2016 dataset ($159\pm12$\,AU in projected physical separation). This separation places it outside of the field of view of the IFS module, and so we report the results from IRDIS only. To extract the astrometry and photometry of this object, a classical Angular Differential Imaging procedure \citep[ADI; ][]{2006ApJ...641..556M} was applied to remove the contribution from the primary star while minimising self-subtraction and other systematic effects that may be introduced by more aggressive PSF subtraction techniques. The ADI procedure and calculation of the relative astrometry and photometry of the companion were accomplished using the Specal pipeline developed for the SHINE survey (R. Galicher, 2018, in preparation). The final reduced images from this approach are shown in Fig.~\ref{fig:images}.
The astrometry was measured using the negative companion injection technique \citep{2010Sci...329...57L}, where the mean unsaturated PSF of the primary star was subtracted from the raw frames. The position and flux of the injected PSF was varied to minimise the standard deviation inside a 3 FWHM diameter region around the companion in the final ADI processed image. Once the best-fit values were found, each parameter was varied until the standard deviation increased by a factor of 1.15. This value was empirically calculated to correspond to $1\sigma$ uncertainties across a range of potential companion and observational parameters.
A systematic uncertainty of 2\,mas on the star position was adopted for the 2016 dataset, which dominates the astrometric uncertainty budget.
A second reduction of each dataset was performed with the aim of searching for companions at higher contrasts. For the IRDIS datasets, the TLOCI algorithm was used \citep{2014IAUS..299...48M}, while for the IFS datasets we used the PCA-based ASDI algorithm described in \citet{2015A&A...576A.121M}. The resulting detection limits are are shown in Figure \ref{fig:contrast_limits}. Apart from the bright companion at $1.27"$, we find no evidence of companions at smaller separations. Three additional objects were detected at much larger separations, and are discussed in Section \ref{sec:astrometry}.
\begin{figure*}
\includegraphics[width=0.95\textwidth]{images.png}
\caption{ADI-processed images from the two SPHERE-IRDIS datasets taken with the K1 and H2 filters, and the NACO data taken with the L' filter. The companion is detected with SNR > 1000 in the two SPHERE epochs, and SNR > 8 in the NACO data.}\label{fig:images}
\end{figure*}
\begin{figure}
\includegraphics[width=0.98\columnwidth]{contrast_curve_with_ifs.pdf}
\caption{5$\sigma$ detection limits for the two SPHERE datasets, after processing using the TLOCI algorithm. The grey shaded region indicates the separations partially or fully blocked by the coronagraph. }\label{fig:contrast_limits}
\end{figure}
\subsection{SPHERE Long Slit Spectroscopy}
HIP 64892 was also observed with SPHERE IRDIS Long Slit Spectroscopy \citep[LSS;][]{2008A&A...489.1345V} on 2017-03-18. These observations utilised the medium resolution spectroscopy mode, which covers wavelengths from 0.95-1.65\,$\mu$m with a spectral resolution of R$\sim 350$. The observing sequence consisted of a series of alternate images with the companion inside and outside of the slit. To move the companion outside of the slit, a small offset is applied on the derotator so as to keep the star centered on the coronagraph. This strategy has been demonstrated to be very efficient to build and subtract reference images of the speckles and stellar halo while minimising the self-subtraction effects on the spectrum of the companion \citep{2016SPIE.9912E..26V}. Additional sky backgrounds were also obtained at the end of the sequence along with an unsaturated spectrum of the star to serve as reference for the contrast.
The data were reduced using the SILSS pipeline \citep{2016ascl.soft03001V}. This pipeline combines recipes from the standard ESO pipeline with custom {\it IDL} routines. Briefly, data are background subtracted, flat fielded and corrected for bad pixels. The wavelength calibration is performed and the data are corrected for the slight tilt of the grism, which causes a change in the position of the PSF with wavelength. Finally, the speckles are subtracted using principal component analysis, with the modes constructed from the spectra obtained with the companion outside of the slit.
\subsection{Archival NACO Sparse Aperture Masking Data}
In addition to the SPHERE data, we utilised archival sparse aperture masking (SAM) data from the VLT-NACO instrument, taken on 2011-06-08 (Program 087.C-0790(A), PI: Ireland). While the main purpose of the SAM mode is to detect companions and resolve structures at small angular separations (typically $<$300\,mas), this does not preclude the detection of bright objects at larger separations. The data were taken with the $L'$ filter in pupil tracking mode, and were split into blocks of 1600 exposures of 0.2s each. Each of the 5 blocks was interspersed amongst observations of other targets.
Rather than process the data in an interferometric framework, we treated it as a traditional ADI dataset to focus on larger separations. The data were processed using the GRAPHIC pipeline \citep{2016MNRAS.455.2178H}. The data were sky subtracted, flat fielded, cleaned of bad pixels, centred on the primary star, and stacked by binning 200 frames at a time. A Gaussian fit was performed to each PSF to calculate the position of the star, since it provides a reasonable match to the core of the SAM PSF. A python implementation of the KLIP algorithm \citep{2012ApJ...755L..28S} was then applied, where the first 15 modes were removed, resulting in the redetection of the SPHERE companion. The final reduced image is shown in the right panel of Fig. \ref{fig:images}.
To calculate the position and flux of the companion, we used the negative companion injection technique applied to the data cube after binning. We chose to minimise the square of the residuals within a circle of radius $\lambda/D$ centred on the companion peak calculated from the first reduced image. To explore the likelihood function, we used {\it emcee} \citep{Foreman-Mackey2013emcee}, a python implementation of the affine-invariant MCMC ensemble sampler. Due to the lack of detailed study about potential systematic biases introduced into the companion astrometry by applying this technique to SAM data, we conservatively added an additional uncertainty to the companion position of 1 pixel (27.1\,mas). Of particular concern is the way in which the time-varying fine structure of the large SAM PSF may influence the calculation of the star's position.
\begin{table*}
\caption{Observing Log}
\label{tab:observing_log}
\center
\begin{tabular}{ccccccccccc}
\hline
UT Date & Instrument & Filter & DIT$^a$ & $\text{N}_{\text{exp}}^{a}$ & $\Delta \pi^{a}$ & True North correction & Plate Scale \\
& & & [s] & & [$^{\circ}$] & [$^{\circ}$] & [mas/pixel] \\
\hline
2011-06-08 & NACO & L' & 0.2 & $8000$ & 130.4 & $-0.5\pm0.1$ & $27.1\pm0.05$ \\
2016-04-01 & IRDIS & H2/H3 & $64$ & 64 & 45 & $-1.73\pm0.06$ & $12.255\pm0.009$ \\
2016-04-01 & IFS & YJ & $64$ & 64 & 45 & $-102.21\pm0.06$ & $7.46\pm 0.02$ \\
2017-02-08 & IRDIS & K1/K2 & 64 & 72 & 61 & $-1.71\pm0.06$ & $12.249\pm0.009$ \\
2017-02-08 & IFS & YH & $64$ & 72 & 61 & $-102.19\pm0.06$ & $7.46\pm 0.02$ \\
\hline
\end{tabular}
\tablefoot{$^a$DIT refers to the integration time of each image, $\text{N}_{\text{exp}}$ to the total number of images obtained, $\Delta \pi$ to the parallactic angle range during the
sequence}
\end{table*}
\section{Results}
\subsection{Astrometry}\label{sec:astrometry}
To confirm the co-moving status of HIP 64892B, we compared its position relative to HIP 64892A between the datasets. We used the astrometry from the 2016 SPHERE dataset with the parallax and proper motion of HIP 64892 to predict the position expected for a background object at the 2011 and 2017 epochs. The result is shown in Figure \ref{fig:proper_motion}.
For HIP 64892B, the observed positions in 2011 and 2017 differ from the prediction for a background object by 4$\sigma$ and 8$\sigma$ respectively. Instead we find a lack of significant relative motion. This clearly shows that the object is co-moving with HIP 64892A.
The angular separation of HIP 64892B corresponds to a projected separation of $159\pm12$\,AU. With the primary star mass of 2.35\,M$_{\odot}$ we would expect an orbital period of order $\sim10^3$\,yr. This is consistent with the lack of significant motion in the measured astrometry, and suggests that an additional epoch in 2018 or later may show clear orbital motion and help to constrain the orbital parameters of the companion.
In addition to HIP 64892B, three objects were detected at larger separations in the IRDIS field of view. Their astrometry and photometry are given in Table \ref{tab:background_astrophotometry}. Candidate 1 was detected at high significance in both epochs, while the other two were not detected in the 2017-02-08 K band data. The relative motion of candidate 1 between the two epochs is shown in Figure \ref{fig:proper_motion_cc1} and is similar to that expected from a background star. Its position differs from the prediction by 1.3$\sigma$, while it is inconsistent with a co-moving object at 2.8$\sigma$, showing that it is likely a background star.
The IRDIS H2 and H3 photometry of the remaining two objects indicates that they are also likely background stars. If located at the same distance as HIP 64892, their H2 magnitudes and H2-H3 colours would be inconsistent with those of other known objects. Their H2 magnitudes would be consistent with those of T dwarfs, but without significant methane absorption that would be measurable in their H2-H3 colours.
\begin{table*}
\caption{Observed Astrometry and Photometry of HIP 64892 B}
\label{tab:astrophotometry}
\center
\begin{tabular}{cccccccc}
\hline
UT Date & Instrument & Filter & $\rho$ (") & $\theta$ (\degr) & Contrast (mag) & Abs. mag & Mass (COND)\\
\hline
2011-06-08 & NACO & L' & $1.272\pm0.029$ & $310.0\pm1.3$ & $6.10\pm0.08$ & $7.61\pm0.17$ & $37\pm9$ \\
2016-04-01 & IRDIS & H2 & $1.2703\pm0.0023$ & $311.68\pm0.15$ & $7.23\pm0.08$ & $8.73\pm0.17$ & $29\pm4$ \\
2016-04-01 & IRDIS & H3 & $1.2704\pm0.0022$ & $311.69\pm0.15$ & $6.99\pm0.08$ & $8.46\pm0.17$ & $29\pm5$ \\
2017-02-08 & IRDIS & K1 & $1.2753\pm0.0010$ & $311.74\pm0.12$ & $6.80\pm0.08$ & $8.29\pm0.17$ & $34\pm7$ \\
2017-02-08 & IRDIS & K2 & $1.2734\pm0.0010$ & $311.77\pm0.12$ & $6.49\pm0.12$ & $7.97\pm0.19$ & $35\pm8$ \\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{Additional objects detected by SPHERE-IRDIS}
\label{tab:background_astrophotometry}
\center
\begin{tabular}{ccccccccc}
\hline
Candidate & UT Date & Filter & $\rho$ (mas) & $\theta$ (deg) & $\Delta$RA (mas) & $\Delta$Dec (mas) & Contrast (mag) \\
\hline
1 & 2016-04-01 & H2 & $6000\pm3$ & $201.86\pm0.12$ & $-2234\pm12$ & $-5568\pm5$ & $10.5\pm0.1$\\
1 & 2017-02-08 & K1 & $5978\pm4$ & $201.60\pm0.07$ & $-2201\pm7$ & $-5558\pm5$ & $10.7\pm0.1$\\
2 & 2016-04-01 & H2 & $5865\pm5$ & $202.86\pm0.12$ & $-2278\pm12$ & $-5404\pm7$ & $13.7\pm0.1$\\
3 & 2016-04-01 & H2 & $7003\pm6$ & $186.89\pm0.12$ & $-840\pm14$ & $-6952\pm6$ & $10.2\pm0.1$\\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\includegraphics[width=0.95\textwidth]{proper_motion_twofig}
\caption{The measured position of HIP 64892B relative to HIP 64892A. The 2011 NACO L', 2016 IRDIS H2 and 2017 IRDIS K1 positions are shown with blue triangles. The predicted motion for a stationary background object relative to the 2016 position is marked by the black line, with its uncertainty represented by the grey shaded region. The observed positions of HIP 64892B strongly conflict with the predictions for a background object, suggesting that it is co-moving. }\label{fig:proper_motion}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.95\textwidth]{proper_motion_twofig_cc1}
\caption{Same as Fig. \ref{fig:proper_motion}, showing the measured position of companion candidate 1 with respect to HIP 64892A. This object agrees with the prediction for a background object.} \label{fig:proper_motion_cc1}
\end{figure*}
\subsection{Mass}
To estimate the mass of HIP 64892B, we first converted each of the apparent photometric fluxes for the companion (IRDIS H2, H3, K1, K2 and NACO L') into absolute magnitudes using the distance of $125\pm9$\,pc. To calculate the photometric zeropoint of each filter, we used the spectrum of Vega from \cite{2007ASPC..364..315B} along with the filter transmission profiles from each instrument.
We then interpolated the COND evolutionary models \citep{2003A&A...402..701B} using the age and absolute magnitudes to yield estimates of the mass of HIP 64892B. The measurements, listed in Table \ref{tab:astrophotometry}, are consistent with masses of 29-37\,M$_{\text{J}}$. This implies a mass ratio between the brown dwarf and the primary of $q\sim0.014$.
\subsection{Spectral Properties}
\begin{figure*}
\includegraphics[width=0.95\textwidth]{spectrum4_log.png}
\caption{The observed spectrum of HIP 64892B. SPHERE IRDIS LSS data are shown in blue, while the SPHERE IRDIS and NACO photometric measurements are in red. The errorbars on the x-axis of the photometric measurements represent the FWHM of the filter used.}\label{fig:companion_spectrum}
\end{figure*}
We produced a spectral model for the primary by scaling a BT-Settl spectrum with $T_{\rm eff}=10400$\,K, $\log g=4$, [M/H]$=-0.5$ using photometric measurements compiled from 2MASS, Tycho-2 and WISE \citep{skrutskie2006two,2000AA...355L..27H,2010AJ....140.1868W}. This was then used to convert the contrast measurements from IRDIS and NACO to apparent fluxes. The resulting spectrum is shown in Figure \ref{fig:companion_spectrum}.
To estimate the spectral type and effective temperature of the companion, we compared the observed spectrum and photometry of HIP 64892B with a range of spectra compiled from the literature, using the goodness-of-fit statistic $G$ \citep[e.g.][]{2008ApJ...678.1372C}. We considered young L dwarfs from the Upper-Scorpius subgroup \citep{2008MNRAS.383.1385L} as well as companions of Upper Scorpius stars \citep{2008ApJ...689L.153L,2015ApJ...802...61L}. We also compared the companion spectrum to those of young free floating objects from the Montreal\footnote{https://jgagneastro.wordpress.com/the-montreal-spectral-library/} \citep{2016ApJ...830..144R,2014ApJ...785L..14G,2014ApJ...792L..17G,2015ApJ...808L..20G} and \citet{2013ApJ...772...79A} libraries, and to libraries of medium-resolution spectra of old MLT field dwarfs \citep{2003ApJ...596..561M,2005ApJ...623.1115C,2009ApJS..185..289R,2002ApJ...564..421B}.
The result is plotted in Fig. \ref{fig:spectral_type}, for a range of objects covering different ages, masses and spectral types. We find the best matches are given by young, low surface gravity objects with spectral types close to M9. From this, we adopt a spectral type of M9$\pm$1 for HIP 64892B.
In Fig. \ref{fig:spectral_comparison}, we show the comparison of the observed spectrum of HIP 64892B to those of young and old field dwarfs at the M/L transition. Several features in the spectrum are indicative of a young object. The doublets of gravity-sensitive potassium bands at 1.169/1.177\,$\mu$m and 1.243/1.253\,$\mu$m are reduced. The H-band has a triangular shape characteristic of low gravity atmospheres. Visually, the spectrum agrees well with those of M9$_{\gamma}$ candidate members of the $\beta$ Pictoris and Argus moving groups \citep[20-50 Myr;][]{2015ApJS..219...33G}, with 2MASS J20004841–7523070 giving the best fit.
As seen in Fig. \ref{fig:spectral_comparison}, the spectrum of HIP 64892B is also well reproduced by the spectrum of the 8 Myr-old M8 dwarf 2M1207A (TWA27) from the TW Hydrae association, which shows many of the same features seen in our data. When compared to the SPHERE-LSS data of the M7 companion PZ Tel B \citep{2016A&A...587A..56M}, we can see a clear difference in slope indicating a later spectral type for HIP 64892B.
Also shown are the objects UScoCTIO 108B \citep{2008ApJ...673L.185B}, HIP 78530B \citep{2011ApJ...730...42L} and several field dwarfs.
Using the empirical relation between spectral type and effective temperature for young objects with low surface gravity from \cite{2015ApJ...810..158F}, our spectral type constraints correspond to an effective temperature of $T_{\text{eff}}=2600\pm300$\,K.
From the flux-calibrated LSS spectrum, we derived a synthetic absolute magnitude of $M_{\text{J,2MASS}} = 9.37 \pm 0.15$\,mag. Combining this with the spectral type of M9$\pm$1, we can use the bolometric correction relations from \cite{2015ApJ...810..158F} to derive a bolometric luminosity of $\log(L/L_\odot)= -2.66\pm0.10$ dex. In addition, we converted the K1 flux measurement to a predicted $K_{\text{S,2MASS}}$ absolute magnitude using the SpeX spectrum of TWA 27A as an analog. The resulting value of $M_{\text{Ks,2MASS}} = 8.02 \pm 0.17$\,mag corresponds to a luminosity of $\log(L/L_\odot)= -2.51\pm0.11$ dex using the same method.
We also compared the observed spectrum to the BT-Settl model grid \citep{2015A&A...577A..42B} as a function of $T_{\text{eff}}$, $\log g$ and radius $R$. We find a best fit temperature and radius of $T_{\text{eff}}=2600 \pm 100$\,K and $R=2.3 \pm 0.14$\,R$_{\text{J}}$, similar to those predicted by the COND models for a $33$\,M$_{\text{J}}$\, object with an age of 16\,Myr. We find that the $\log g$ value is poorly constrained, with a best fit value $\log g=5.5$. When combined with the radius, this predicts a much larger-than-expected mass. However, gravity-sensitive features are generally narrow and fitting to the entire spectrum at once may complicate this measurement. To investigate this, we performed the same fit to the two sections of the LSS spectrum either side of the $1.4\mu$m telluric feature individually. The Y-J band spectrum gives an estimate of $\log g=4.0\pm0.5$, while the H band spectrum yields $\log g=3.5\pm1.0$, indicating that a lower value is likely.
\begin{figure}
\includegraphics[width=0.98\columnwidth]{G_vs_spt.pdf}
\caption{The $G$ goodness-of-fit statistic plotted as a function of spectral type for a number of objects in the range M0-T8. A clear $G$ minimum is seen at late M spectral types and the best matches are given by young, low surface gravity objects with spectral types of M9.}\label{fig:spectral_type}
\end{figure}
\begin{figure}
\includegraphics[width=0.98\columnwidth]{spectral_comparison.pdf}
\caption{Comparison of the observed spectrum (in black) with field dwarfs and similar young low-mass brown dwarf companions (coloured lines). The measured photometric points from the IRDIS filters are shown with grey circles. The M9 object 2MASS J20004841–7523070 (2M2000-75) provides the best match to the measured spectrum of HIP 64892B.}\label{fig:spectral_comparison}
\end{figure}
The spectral type estimate and low surface gravity are also supported by the position of HIP 64892B on colour-magnitude diagrams. In Figure \ref{fig:cmd_k1k2} we show its K1 magnitude and K1-K2 colour compared to a range of field objects assembled from the SpeX Prism Library \citep{2014ASInC..11....7B} and from \cite{2000ApJ...535..965L}. For these objects we generated synthetic photometry using the SPHERE filter bandpasses. In addition, we also show a range of young companions with SPHERE K1K2 photometry or K-band spectra from the literature \citep{2007ApJ...654..570L,2008ApJ...689L.153L,2010A&A...517A..76P,2014A&A...562A.127B,2015ApJ...802...61L,2016A&A...586L...8L,2016A&A...587A..56M,2016A&A...587A..57Z,2017A&A...605L...9C,2018arXiv180105850C}.
The position of HIP 64892B is similar to PZ Tel B and HIP 78530B, both young companions with late-M spectral types. All three lie close to the mid-M sequence of field dwarfs, showing a slight over-luminosity compared to late-M field objects that match their spectral types. This trend is also seen in young field brown dwarfs \citep{2012ApJ...752...56F,2013AN....334...85L}.
\begin{figure}
\includegraphics[width=0.98\columnwidth]{HIP64892b_K1_K1K2.pdf}
\caption{A colour-magnitude diagram comparing the K1 absolute magnitudes and K1-K2 colours of a range of low-mass stellar and substellar objects from the literature. Synthetic K1 and K2 fluxes were computed for each target from published spectra. A number of young and dusty companions observed with SPHERE or with published K band spectra were added to the diagram. HIP 64892B falls in a similar location to PZ Tel B and HIP 78530B, both young brown dwarfs with late-M spectral types. }\label{fig:cmd_k1k2}
\end{figure}
\section{Discussion and Conclusion}
HIP 64892 joins a growing number of high or intermediate mass stars with extreme mass ratio companions at large separations ($<10\%$, $>10$\,AU). While brown dwarf companions to solar-type stars from both RV and imaging surveys are inherently rare \citep[the so-called ``brown dwarf desert'';][]{2006ApJ...640.1051G,kraus08,2009ApJS..181...62M}, evidence suggests that the occurrence rates of companions around intermediate mass stars may be substantially higher \citep{2012A&A...544A...9V}. This trend is seen for companions with masses spanning the planetary to stellar regimes \citep[e.g.][]{2010PASP..122..905J,2013ApJ...773..170J}.
Despite this, HIP 64892 is one of the highest mass stars around which a substellar companion has been detected, due to the challenges associated with observing such stars and the tendency for large surveys to focus on solar-like or low-mass stars. Only HIP 78530B \citep{2011ApJ...730...42L}, $\kappa$ And b \citep{2013ApJ...763L..32C} and HIP 77900B \citep{2013ApJ...773...63A} orbit stars with a larger mass.
The large mass of the primary leads to a mass ratio for HIP 64892B that is particularly small ($q\sim0.014$). For low-mass and solar-like stars, such a value would correspond to objects at or below the deuterium burning limit, making HIP 64892B a valuable object for studying the overlapping processes of binary star and planet formation in the brown dwarf regime.
Indeed, the properties of HIP 64892B raise the possibility that it formed via gravitational instability, either through a binary-star mechanism or through disk instability in the circumstellar disk of the primary \citep{1997Sci...276.1836B}. We investigate the latter idea further in Appendix \ref{app:gi_models}, finding that the observed mass and separation of HIP 64892B are compatible with in-situ formation via disk instability. This process is one of the proposed pathways for giant planet formation, making HIP 64892B an important object for understanding this mechanism.
HIP 64892B also stands to be an important object for calibrating substellar formation and evolutionary models, with a well-known age tied to that of the LCC association, and its brightness allowing high SNR spectroscopic and photometric measurements.
While the spectroscopic and photometric measurements presented here have high SNR, the uncertainty on the distance is relatively large, and dominates the uncertainties on the absolute magnitudes, luminosity, and isochronal mass for HIP 64892B. A much more precise distance measurement will come from the GAIA parallax, which should be included in the second data release.
Many of the derived properties of the newly discovered companion match those of the planet-hosting brown dwarf TWA 27, with an excellent match found between their spectra. This object may prove to be a useful analogue, and a point of direct comparison between young brown dwarfs in single and multiple systems.
The HIP 64892 system as a whole has many parallels to that of $\kappa$ And. While the stellar hosts have a similar mass and spectral type, the companion HIP 64892B appears to be a hotter, younger and higher-mass analogue of $\kappa$ And B \citep{2013ApJ...779..153H,2014A&A...562A.111B,2016ApJ...822L...3J}. When combined with HIP 78530B \citep{2011ApJ...730...42L}, HR 3549B \citep{2015ApJ...811..103M}, HD 1160 \citep{2012ApJ...750...53N} and $\eta$ Tel B \citep{2000ApJ...541..390L}, they form a useful sample to explore the properties of low and intermediate-mass brown dwarf companions to young, 2-3\,M$_{\odot}$ stars.
\begin{acknowledgements}
This work has been carried out within the frame of the National Centre for Competence in Research ``PlanetS'' supported by the Swiss National Science Foundation (SNSF).\\
SPHERE is an instrument designed and built by a consortium consisting of IPAG (Grenoble, France), MPIA (Heidelberg, Germany), LAM (Marseille, France), LESIA (Paris, France), Laboratoire Lagrange (Nice, France), INAF - Osservatorio di Padova (Italy), Observatoire Astronomique de l'Université de Genève (Switzerland), ETH Zurich (Switzerland), NOVA (Netherlands), ONERA (France) and ASTRON (Netherlands) in collaboration with ESO. SPHERE was funded by ESO, with additional contributions from CNRS (France), MPIA (Germany), INAF (Italy), FINES (Switzerland) and NOVA (Netherlands). SPHERE also received funding from the European Commission Sixth and Seventh Framework Programmes as part of the Optical Infrared Coordination Network for Astronomy (OPTICON) under grant number RII3-Ct-2004-001566 for FP6 (2004-2008), grant number 226604 for FP7 (2009-2012) and grant number 312430 for FP7 (2013-2016). \\
This work has made use of the SPHERE Data Centre, jointly operated by OSUG/IPAG (Grenoble), PYTHEAS/LAM/CeSAM (Marseille), OCA/Lagrange (Nice) and Observatoire de Paris/LESIA (Paris) and supported by a grant from Labex OSUG@2020 (Investissements d’avenir – ANR10 LABX56).
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.\\
This publication makes use of VOSA, developed under the Spanish Virtual Observatory project supported from the Spanish MICINN through grant AyA2011-24052.\\
R.\,G., R.\,C., S.\,D. acknowledge support from the “Progetti Premiali” funding scheme of the Italian Ministry of Education, University, and Research.\\
J.\,O. acknowledges financial support from ICM N\'ucleo Milenio de Formaci\'on Planetaria, NPF.\\
Q.\,K. acknowledges funding from STFC via the Institute of Astronomy, Cambridge Consolidated Grant.
\end{acknowledgements}
\bibliographystyle{aa.bst}
|
{
"timestamp": "2018-03-08T02:10:08",
"yymm": "1803",
"arxiv_id": "1803.02725",
"language": "en",
"url": "https://arxiv.org/abs/1803.02725"
}
|
\section{Introduction}
\label{sec1} Let $q$ be a prime power, $\mathbb{F}_{q}$ be the
finite field with $q$ elements. A linear $[n,k,d]$ code
$\mathcal{C}$ over $\mathbb{F}_{q}$ is a $k$-dimensional subspace
of $\mathbb{F}^{n}_{q}$ with minimum (Hamming) distance $d$. A
linear code $\mathcal{C}$ is cyclic if
$(c_{0},c_{1},\cdots,c_{n-1}) \in \mathcal{C}$ implies
$(c_{n-1},c_{0},c_{1},\cdots,c_{n-2})$ $\in$ $\mathcal{C}$.
There is a bijective correspondence between vectors $
(c_{0},c_{1},\cdots,c_{n-1})$ $\in \mathbb{F}^{n}_{q}$ and
polynomials $ c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1}$ $\in \mathbb{F}_{q}[x]/(x^{n}-1),$
then a cyclic code $\mathcal{C}$ is identified with an ideal of
$\mathbb{F}_{q}[x]/(x^{n}-1)$. Since each ideal of
$\mathbb{F}_{q}[x]/(x^{n}-1)$ is principal, a cyclic code
$\mathcal{C}$ $=(g(x))$, where $g(x)$ is monic and has the smallest
degree among all the generators of $\mathcal{C}$, and $g(x)$ is
called the {\it generator polynomial} of $\mathcal{C}$.
If $\gcd(q,n)=1$, a {\it $q$-cyclotomic coset} modulo $n$ containing
$x$ is defined by
$$C_{x}=\{{x,xq,xq^{2},...,xq^{e-1}}\} \bmod n,$$
where $e$ is the smallest positive integer such that $q^{e}x\equiv x
\bmod n$, the cardinality of $C_{x}$ is denoted by $|C_{x}|=e$.
The smallest integer in $C_{x}$ is called the {\it coset leader} of
$C_{x}$ modulo $n$.
Let $\gcd(q,n)=1$. If $\xi$ is a primitive $n$-th root of unity in
some finite field containing $\mathbb{F}_{q}$, $T$$=\{i \mid
g(\xi^{i})=0\}$ is called the {\it defining set} of $\mathcal{C}$
$=(g(x))$. It is well known that $T$ is the union of some
$q$-cyclotomic cosets modulo $n$, the dimension $k$ of
$\mathcal{C}$ is determined by $k=n-|T|$, and the minimum distance
$d$ is also determined by $T$. If $T=C_{b}\cup C_{b+1}\cup \cdots
\cup C_{b+\delta-2}$, then $\mathcal{C}$ is called a BCH code of
designed distance $\delta$, and it can be denoted by
$\mathcal{C}(n,q,\delta,b)$ as \cite{Ding5}-\cite{Ding8}. If $b=1$,
$\mathcal{C}$ is called {\it narrow-sense}; otherwise, {\it
non-narrow-sense}. If $n=q^{m}-1$, $\mathcal{C}$ is called {\it
primitive};
otherwise {\it nonprimitive}.
Particularly, if $n=q^{m}+1$, it is called an {\it antiprimitive}
BCH code by Ding in \cite{Ding5}.
Since binary BCH codes were
discovered independently around 1960 by Hocquenghem
\cite{Hocquenghem}, Bose and Ray-Chaudhuri
\cite{Bose1},\cite{Bose2}, they have been studied and employed
widely in practice \cite{Ding5}. However, limited knowledge about
BCH codes is still limited. For example, for the given finite
field, the dimension and minimum distance of a BCH code are known
only for special code lengths and designed distances, while open in
general. As pointed out by Charpin in \cite{Char} and Ding in
\cite{Ding5}, it is notoriously hard to generally determine
the dimension and minimum distance of BCH codes. For details on
advancement about BCH codes, please see \cite{Mann}-\cite{Ding8}
and the references therein.
For general $n$ and $\delta$, little is known about the parameters
of $\mathcal{C}(n,q,\delta,b)$, only lower bounds on the dimension
and minimum distance have been developed, see
\cite{Mann}-\cite{Ding8} and the next section. Primitive BCH codes
are the most studied among all types of BCH codes. Ding {\it et al.}
deeply investigated parameters of primitive BCH codes, determined
their dimension and gave very good bounds on the minimum distance
of codes for
relatively small $\delta$ and special types of $\delta$
\cite{Ding1}-\cite{Ding8}. They also determined the first, the
second and third largest $q$-cyclotomic coset leaders $\delta_{1}$,
$\delta_{2}$, $\delta_{3}$ modulo $n=q^{m}-1$, and completely
presented the dimensions, minimum distances and weight distributions
of $\mathcal{C}(n,q,\delta,1)$ and $\mathcal{C}(n,q,\delta+1,0)$ for
$\delta=\delta_{i}$ with $i=1,2,3$. For more details, please see
\cite{Ding4} and the references therein.
Recently, scholars have paid much attention to antiprimitive BCH
codes $\mathcal{C}(n,q,\delta,b)$ for $n=q^m+1$. As pointed by
\cite{Ding7} \cite{Ding8}, antiprimitive BCH codes are all linear
codes with complementary dual (LCD) and contain many good codes. LCD
codes can be used against side-channel attacks and fault
noninvasive attacks \cite{Carlet}.
Ding, Li and Liu {\it et al.} acquired some achievements about the
parameters of BCH codes of length $n=q^{m}+1$ for designed
distance $\delta \leq q^{\lfloor \frac{m-1}{2} \rfloor}+3$ in
\cite{Ding7} and $\delta \leq q^{\lceil \frac{m}{2} \rceil}$ in
\cite{Ding8}, respectively. Ding pointed out that it is very
significative to find the second and third largest coset leaders
modulo $n=q^m+1$, which is helpful to deduce parameters of BCH codes
\cite{Ding5}.
The main objective of this paper is to determine the first five
largest 2-cyclotomic coset leaders modulo $n=2^m+1$ and calculate
the dimensions of $\mathcal{C}(2^m+1,2,\delta,1)$ and
$\mathcal{C}(2^m+1,2,\delta+1,0)$ for a larger range of $\delta$
when $m=2t+1, 4t+2, 8t+4$ and $m\geq 10$. In Section 2, basic
concepts on LCD codes, some known results on BCH codes are
reviewed. In Sections 3, 4 and 5, the parameters of BCH codes of
length $n=2^m+1$ with designed distance $\delta>2^{\lceil
\frac{m}{2} \rceil}$ are determined for $m=2t+1$, $m=4t+2$ and
$m=8t+4$, respectively. The final conclusion is drawn in Section 6.
\section{Preparation }
\label{sec2}
In this section, we recall the basic knowledge on BCH codes and LCD codes.
For more details, one can refer to Refs. \cite{mac} and \cite{huf}.
If $\mathcal{C}$ is a linear code of length $n$ over
$\mathbb{F}_{q}$, its Euclidean dual code is defined by
$$\mathcal{C}^{\perp }=\{ X \in
\mathbb{F}_{q}^n \mid(X, Y)=X Y^{T}=0 \ \hbox{for all}\ \ Y \in
\mathcal{C} \},$$ where $Y^{T}$ denotes the transposition of the
vector $Y=(y_{1},y_{2},..., y_{n})$.
A linear code $\mathcal{C}$ is called an {\it LCD} code if $\mathcal{C}^{\perp
}\bigcap\mathcal{C}=\{\bf 0\}$, which is equivalent to
$\mathcal{C}^{\perp }\bigoplus\mathcal{C}=\mathbb{F}_q^n$.
A linear code $\mathcal{C}$ is called {\it reversible} if
$(c_{0},c_{1},\cdots,c_{n-1}) \in \mathcal{C}$ implies that
$(c_{n-1},c_{n-2},\cdots,c_{0}) \in \mathcal{C}$. Hence, a cyclic
code $\mathcal{C}$ with generator polynomial $g(x)$ is LCD if and
only if $g(x)$ is self-reciprocal or $\mathcal{C}$ is reversible.
BCH codes have a long history and are the best linear codes known
when their lengths are moderate. However, it is challenging to
determine their parameters in general. The following lower bounds on
the dimension of BCH codes are well known, see \cite{mac},
\cite{huf}.
\begin{prop}\label{prop2.1}
Let $k$ be the dimension of
$\mathcal{C}(n,q,\delta,b)$. Then
(1) $k \geq n-ord_{n}(q)(\delta-1)$.
(2) If $q=2$, $b=1$ and $\delta$ is odd, then $k \geq
n-ord_{n}(q)(\delta-1)/2$.
\end{prop}
The above bounds are tight only when $\delta$ is very small, while
they will become negative when $\delta$ gets larger. In \cite{Aly},
a formula on $k$ $=\dim(\mathcal{C}(n,q,\delta,1))$ was obtained,
which generalized that of \cite{Yue}.
\begin{prop}\label{prop2.2}
Let $\gcd(q,n)=1$, $q^{\lfloor\frac{m}{2}\rfloor}\leq n\leq
q^{m}-1$ and $m=ord_{n}(q)$. If $2\leq \delta\leq
\min\{nq^{\lceil\frac{m}{2}\rceil}/(q^{m}-1),n \}$, then the
dimension $k$ of $\mathcal{C}(n,q,\delta,1)$ is
$k=n-m\lceil(\delta-1)(1-1/q)\rceil.$
\end{prop}
This result is useful in some
cases, such as primitive BCH codes, while useless for BCH codes with
length $n=q^{m}+1$. Cyclic codes with length $n=q^{m}+1$ are LCD
codes and contain many good linear codes, see \cite{Ding7},
\cite{Ding8} and \cite{Rao}. Ref. \cite{Rao} discussed binary LCD
cyclic codes of length $n \leq 1025$. Known results on the
dimensions of BCH codes $\mathcal{C}(q^{m}+1,q,\delta,1)$ and $
\mathcal{C}(q^{m}+1,q,\delta+1,0)$ in \cite{Ding7} and
\cite{Ding8} are as follows.
\begin{lemma}\label{lemm2.3}( Theorem 18 of \cite{Ding7} )
For any integer $\delta$ with $3 \leq \delta \leq q^{\lfloor
\frac{l-1}{2}\rfloor }+3$, the reversible code
$\mathcal{C}(q,n,\delta,0)$ has parameters
$[q^l+1,q^l-2l(\delta-2-\lfloor \frac{\delta-2}{q}\rfloor), d\geq
2(\delta-1)]$.
\end{lemma}
\begin{lemma}\label{lemm2.4} (Theorems 38 and
39 of \cite{Ding8}) Let $m\geq 3$ be an integer and $h=\lfloor
\frac{m-1}{2}\rfloor$. Then the following hold for $2 \leq \delta
\leq q^{h+1}$:
(1) If $m\geq 4$ is an even integer, then
$\mathcal{C}(n,q,\delta,1)$ has parameters
$[q^m+1,q^m+1-2m(\delta-1-\lfloor \frac{\delta-1}{q}\rfloor),d\geq
\delta]$ and $ \mathcal{C}(n,q,\delta+1,0)$ has parameters
$[q^m+1,q^m-2m(\delta-\lfloor \frac{\delta}{q}\rfloor),d\geq
2\delta]$.
(2) If $m\geq 3$ is an odd integer, $2 \leq \delta \leq q^{h+1}$,
$k=\dim(\mathcal{C}(n,q,\delta,1))$, then
$$k=\left\{
\begin{array}{lll}
q^m+1-2m(\delta-1-\lfloor \frac{\delta-1}{q}\rfloor) &\mbox {if $\delta\leq q^{h+1}-q$;}\\
q^m+1-2m(q^{h+1}-q-\lfloor \frac{\delta-1}{q}\rfloor) &\mbox {if $
q^{h+1}-q+1 \leq \delta\leq q^{h+1}$.}
\end{array}
\right.$$
\end{lemma}
In the sequel, we only consider BCH codes of length $n=2^m+1(m\geq10)$
with defining sets $T_{\delta}=C_{1}\cup C_{2}\cup \cdots \cup C_{\delta-1}$ and
$T_{\delta}'=C_{0}\cup C_{1}\cup \cdots \cup C_{\delta-1}$ for odd
$\delta$, which can be denoted by $\mathcal{C}(n,2,\delta,1)$ and
$\mathcal{C}(n,2,\delta+1,0)$ as Refs. \cite{Ding7} and
\cite{Ding8}. Hence, $\mathcal{C}(n,2,\delta,1)$ $=[n,k,d]=[n,
n-\mid T_{\delta}\mid,d\geq \delta]$ and
$\mathcal{C}(n,2,\delta+1,0)$ $=[n,k_{0},d_{0}]=[n, 2^m-\mid
T_{\delta}'\mid,d_{0}\geq 2\delta]$.
Our approach to the dimension of BCH codes is to find out all
coset leaders in $T_{\delta}$ and calculate their cardinalities.
While we find out coset leaders in $T_{\delta}$ or $T_{\delta}'$,
some new skills are developed, which are not employed in the
literature. For convenience, we first present three notations below.
{\bf Notation 1.} Let $x\equiv y$ denote $x\equiv y \bmod n$ unless
otherwise stated. We omit the words ``modulo $n$" when we mention
``cyclotomic coset", ``coset leader" and so on. ``$x$ is a coset
leader" means that $x$ is a coset leader of $C_x$ modulo $n$.
{\bf Notation 2.} Assume that $a,b,c$ are all nonnegative integers and $a\leq b$,
denote $[a,b]=\{a,a+1,\cdots,b\}$ and $[a,b]+c=[a+c,b+c]$.
{\bf Notation 3.} Suppose that $C_{x}$ is a $2$-cyclotomic coset containing
$x$.
$C_{x}$ can be given by $C_{x}=\{y_{_{x,0}},y_{_{x,1}},y_{_{x,2}},...,y_{_{x,2m-1}}\},$
where $$y_{_{x,k}}\equiv 2^k x~\hbox{and}~y_{_{x,k}}\in \mathbb{Z}_n=\{0,1,2,\cdots,n-1.\}$$
When $m\leq k\leq 2m-1$, $ y_{_{x,k}}\equiv2^k x=2^{k-m}(2^{m}+1-1)x \equiv -2^{k-m}x \equiv
n-y_{_{x,k-m}}$. Hence $C_{x}$ can be also denoted by $$\{y_{_{x,k}},n-y_{_{x,k}}| 0\leq k\leq m-1\}.$$
It is easy to
see that $x$ is a coset leader of $C_{x}$ if and only if
$y_{_{x,k}}-x \geq 0$ and $n-y_{_{x,k}}-x \geq 0$ for $0\leq k \leq
m-1$. Obviously, if $x$ is even, then $\frac{x}{2}\in C_x$ and $x$
is not a coset leader. Hence, to find out coset leaders in
$T_{\delta}$ or $T_{\delta}'$, it suffices to consider an odd $x$.
For example, if $n=2^3+1=9$, we have then
$C_{1}=\{y_{_{1,0}},y_{_{1,1}},y_{_{1,2}},y_{_{1,3}},y_{_{1,4}},y_{_{1,5}}\}=\{1,2,4,8,7,5\}$,
$C_{3}=\{y_{_{3,0}},y_{_{3,1}},y_{_{3,2}},y_{_{3,3}},y_{_{3,4}},y_{_{3,5}}\}=\{3,6,3,6,3,6\}=\{3,6\}$,
$C_{5}=\{y_{_{5,0}},y_{_{5,1}},y_{_{5,2}},y_{_{5,3}},y_{_{5,4}},y_{_{5,5}}\}=\{5,1,2,4,8,7\}$,\\
which imply that 1 and 3 are coset leaders, while 5 and 7 not.
Inspired by these latter work, by a detailed analysis of binary
cyclotomic cosets modulo $n=2^{m}+1$ with $m\not \equiv 0 \bmod 8$,
we deduce the first five largest coset leaders modulo $n$ and
determine the coset leader of $C_x$ for each
$$x\leq \left\{
\begin{array}{lll}
2^{t+2}+7
&\mbox {if $m=2t+1$};\\
2^{2t+2}+2^{2t+1}+3 &\mbox {if $m=4t+2$};\\
2^{4t+3}+2^{4t+2}+2^{4t+1}+1 &\mbox {if $m=8t+4$,}
\end{array}
\right.$$ along with their cardinalities.
Thus, parameters of some binary antiprimitive BCH codes with designed distance
$\delta>2^{\lceil \frac{m}{2} \rceil}$ are given, which widely
extend the range of $\delta$ in \cite{Ding7} and \cite{Ding8} for
$q=2$.
\section{ BCH Codes of length $n=2^{m}+1$ for $m=2t+1$}
\label{sec3} Throughout this section, we fix $n=2^{m}+1$ with
$m=2t+1\geq 11$.
It will be first determined when $x$ is a coset leader for $1\leq
x \leq 2^{t+2}+7$ in Theorem 3.1. Next, we show each $\delta_{i}$ is
a coset leader for $1\leq i \leq 5$ in Lemma 3.2. For verifying
the first five largest coset leaders in Theorem 3.5, we will
continue to give Lemmas 3.3-3.4 on basis of Lemma 3.2. The
cardinalities of relevant cyclotomic cosets will be calculated in
Lemma 3.6. At last, Theorem 3.7 can be further given, which lists
dimensions of some BCH codes of length $n$ with designed distance
$\delta$ for $2^{t+1}+3\leq \delta \leq 2^{t+2}+9$ and
$\delta_5+2=\frac{n-3}{6}-8 \leq \delta \leq n$.
For clarity, the proofs of Theorem 3.1,
Lemma 3.2 and Theorem 3.7 will be presented in {\bf Appendixes A, B \hbox{and} C},
respectively.
\begin{theorem}\label{ther3.1}Let $x$ be
odd. Then we have the following:
(1) If $1\leq x\leq 2^{t+1}-3$, then $x$ is a coset leader, see
Refs. \cite{Ding7} and \cite{Ding8}.
(2) If $2^{t+1}+3\leq x\leq 2^{t+1}+2^{t}-3$, then $x$ is a coset
leader.
(3) If $2^{t+1}+2^{t}+3\leq x\leq 2^{t+2}-9$, then $x$ is a coset
leader.
(4) If $x=2^{t+1}-1, 2^{t+1}+1$, $2^{t+1}+2^{t}-1, 2^{t+1}+2^{t}+1$
or $ 2^{t+2}-7\leq x\leq 2^{t+2}+7$, then $x$ is not a coset
leader.
\end{theorem}
\begin{lemma}\label{lemm3.2}
Suppose $\delta_{1}=\frac{n}{3}$, $\delta_{2}=\frac{n-3}{6}$,
$\delta_{3}=\delta_{2}-2$, $\delta_{4}=\delta_{2}-8$,
$\delta_{5}=\delta_{2}-10$. Then $\delta_{1}, \delta_{2},
\delta_{3}$, $\delta_{4}$ and $\delta_{5}$ are all coset leaders.
\end{lemma}
Next, we will show $\delta_i( i=1,2,3,4,5)$ are the first five
largest coset leaders modulo $n$, respectively. It is necessary to
verify
$x$ is not a coset leader if $\delta_{5}<x<n$ and $x\neq
\delta_{i}$ for $1\leq i \leq 4$.
To achieve this, we first introduce an iterative algorithm to
partition $ I^{(t)}=[1, 2^{2t-5}]$ into $2^{t-3}$ subintervals $$
I^{(t)}=[1, 2^{2t-5}] =I_1\bigcup I_2 \cdots \bigcup I_{2^{t-3}},
\hbox{for}~t\geq 5.$$
\noindent{\bf Iterative Algorithm 1 (IA 1):}
Suppose $I_i=[a_i,b_i]$, a partition of $I^{(t)}=[1, 2^{2t-5}]$ is
obtained as
$$I^{(t)}=I^{(t-1)}\bigcup (\bigcup \limits_{u=2^{t-4}+1}^{2^{t-3}} I_{u}), $$
$$\hbox{where} ~I_{u}=\left\{
\begin{array}{lll}
I_{j}+2^{2\times (t-4)+1} &\mbox {if $u=j+2^{t-4}, j \in[1, 2^{t-4}-1]$;}\\
\mbox{[}a_{2^{t-4}}+2^{2t-7},2^{2t-5}\mbox{]} &\mbox {if $
u=2^{t-3}$,}
\end{array}
\right.$$
with the initial conditions: $I_1=[1,2],
I_2=[a_1+2^1,2^{2\times1+1}]=[3,8]. $
For clarity, we give the partition in detail as follow.
1) If $t=5$, then $ I^{(5)}=[1, 2^{5}]$ can be partitioned into
$2^{2}$ subintervals as follows:
$I_1=I_{2^0}=[1,2^1]=[1,2]=[a_1,b_1]$,
$I_2=I_{2^1}=[a_1+2^1,2^{2\times1+1}]=[3,8]=[a_2,b_2]$,
$I_3=I_{1+2^1}=I_1+2^{2\times1+1}=I_1+2^3=[9,10]=[a_3,b_3]$,
$I_4=I_{2^2}=[a_{2^{1}}+2^{2\times 1+1},
2^{2\times 2+1}]=[a_2+2^3,2^5]=[11,32]=[a_4,b_4]$.
2) If $t=6$, then $ I^{(6)}=[1, 2^{7}]$ can be partitioned into
$2^{3}$ subintervals as follows:
Let $I_{s}=[a_s,b_s]$ for $1\leq
s\leq 4$ as be given in 1),
as for $2^2<s=j+2^2\leq 2^{2+1}$ with $1\leq j\leq 2^2-1$,
define
$I_s=I_j+2^{2\times 2+1}=I_j+2^{5}=[a_s,b_s]$ for $5\leq s\leq7$, and define
$I_8=I_{2^3}=[a_{2^{2}}+2^{2\times 2+1}, 2^{2\times 3+1}]=[a_4+2^5, 2^7]=[a_8,b_8]$.
3) Let $t\geq 7$. When a partition of $ I^{(t-1)}$ is given by
$I^{(t-1)}=[1, 2^{2t-7}]$ $=I_1\bigcup I_2 \cdots \bigcup
I_{2^{t-4}}=\bigcup \limits_{j=1}^{2^{t-4}} I_j$, where
$I_{j}=[a_j,b_j]$.
For $u=j+2^{t-4}$ with $1\leq j\leq 2^{t-4}-1$, define
$I_{u}=I_{j}+2^{2\times (t-4)+1}$
$=I_{j}+2^{2t-7}=[a_j+2^{2t-7},b_j+2^{2t-7}].$
For $u=2^{t-3}$, define
$I_{2^{t-3}}=[a_{2^{t-4}}+2^{2\times
(t-4)+1}, 2^{2\times (t-3)+1}]=[a_{2^{t-4}}+2^{2t-7},2^{2t-5}].$
Then, a partition of $I^{(t)}=[1, 2^{2t-5}]$ is iteratively obtained as\\
$I^{(t)}$ $=I^{(t-1)} \bigcup (\bigcup
\limits_{u=2^{t-4}+1}^{2^{t-3}} I_{u})$
$=(I_{1}\cup I_{2} \cdots
\cup I_{2^{t-4}})\cup I_{2^{t-4}+1} \cdots \cup I_{2^{t-3}}.$\\
\noindent {\bf Remark 1.} From the \textit{IA 1} above, we can
derive that:
(1) For $0\leq i \leq t-3$, $I_{2^i}$ has the form
$I_{2^i}= \left\{
\begin{array}{lll}
[1,2]=\hbox{[}1,2^{2i+1}\hbox{]} &\mbox {if $i=0$};\\
\hbox{[}1+(2+2^{3}+\cdots +2^{2i-1}),2^{2i+1}\hbox{]} &\mbox {if
$1\leq i \leq t-3$}.
\end{array}
\right.$
(2) Generally, for each $s\in [1, 2^{t-3}-1]$, let the 2-adic
expansion of $s$ be
$$s=a_02^0+a_12^1+a_2 2^2+\cdots+a_{t-4}2^{t-4}=(a_0a_1a_2 \cdots
a_{t-4})_{2}.$$
Define $i=i_{s}=\min\{j|a_j=1,0\leq j\leq t-4\}$. We have then
\begin{eqnarray*}
I_s&=&I_{2^i}+a_{i+1}2^{2i+3}+a_{i+2}2^{2i+5}+\cdots +a_{t-4}2^{2t-7}\\
&=&I_{2^i}+2^{2i+3}(a_{i+1}+a_{i+2}2^2+\cdots+a_{t-4}2^{2(t-i-5)})\\
&=&I_{2^i}+2^{2i+3}\lambda,
\end{eqnarray*}
$~\hbox{where}~\lambda=\lambda_{s}
=a_{i+1}+a_{i+2}2^2+\cdots+a_{t-4}2^{2(t-i-5)}.$
Notice
$0 \leq \lambda
\leq 1+2^2+\cdots+ 2^{2(t-i-5)} <2^{2(t-i-4)}$ for $s\leq 2^{t-3}-1$,
and $I_{2^{t-3}}=I_{2^{t-3}}+2^{2(t-3)+3}\times \lambda$ with
$\lambda=0$. It then follows from (1) and (2) that for any $s\in [1, 2^{t-3}]$,
there exists $0 \leq \lambda <2^{2(t-i-4)}$ such that
$$I_s=I_{i,\lambda}=I_{2^i}+2^{2i+3}\lambda,
~\hbox{where}~0\leq i \leq t-3.$$
\noindent {\bf Example 1}: When $t=8$, $I_{s}$ can be given as
follows:\\
If $s=8=(00010)_{2}$, then $i=3$ and $\lambda=0$, $I_{s}=I_{2^3}=I_{2^3}+2^9 \cdot
0$.\\
If $s=10=(01010)_{2}$, then $i=1$ and $\lambda=4$,
$I_{s}=I_{2^1}+2^7=I_{2^1}+2^5\cdot 4$.
If $s=15=(11110)_{2}$, then $i=0$ and $\lambda=21$,
$I_{s}=I_{2^0}+2^3+2^5+2^7=I_{2^0}+2^3\cdot 21$.
\begin{lemma}\label{lemm3.3}
Let $\delta_2$ be given as
above. If $x\in [\delta_2+2, \delta_2+2^{2t-4}]$ is odd, then $x$ is
not a coset leader.
\end{lemma}
\begin{proof}
Note that $x$ can be given by $x=\delta_2+2l$ with $l \in I^{(t)}=[1,
2^{2t-5}]$. Let $I^{(t)}=\bigcup \limits_{s=1}^{2^{t-3}} I_s$ be a
partition of $I^{(t)}$ defined as above. Then $l \in I_s$ for some
$s \in [1, 2^{t-3}]$.
From Remark 1, $ I_s=I_{2^i}+
2^{2i+3}\lambda$, where $0\leq i \leq t-3$ is determined by
$s$ and $0 \leq \lambda<2^{2(t-i-4)}$. Thus, one shall denote
$x=\delta_2+2l=\delta_2+2(l_0+2^{2i+3}\lambda)$ with
$$l_0\in
I_{2^i}= \left\{
\begin{array}{lll}
[1,2]=[1,2^{2i+1}] &\mbox {if $i=0$};\\
\hbox{[}1+(2+2^{3}+\cdots +2^{2i-1}),2^{2i+1}\hbox{]} &\mbox {if
$1\leq i \leq t-3$}.
\end{array}
\right.$$
Choose $k=2t-2i-3$. We then derive that
\begin{eqnarray*}
2^kx~ \equiv~ y_{_{x,k}}
&=& 2^k(\delta_2+2l_0+2^{2i+4}\lambda)-(\frac{2^{k-1}-1}{3}+\lambda)n\\
&=& 2^k(\delta_2+2l_0)-(\frac{2^{k-1}-1}{3})(6\delta_2+3)+(2^{2t+1}-n)\lambda\\
&=& 2\delta_2+2^{2(t-i-1)}l_0-2^{2(t-i-2)}+1-\lambda.
\end{eqnarray*}
Now, according to the value ranges of $i, l_0$ and $\lambda$ above,
when $k=2t-2i-3$, one can show
$\frac{n-\delta_2}{2}<y_{_{x,k}}<\frac{n+\delta_2}{2}$ as follows.
First, we study an upper bound of $y_{_{x,k}}$:
\begin{eqnarray*} y_{_{x,k}}&=&2\delta_2+2^{2(t-i-1)}l_0-2^{2(t-i-2)}+1-\lambda\\
&\leq& 2\delta_2+2^{2(t-i-1)}l_0-2^{2(t-i-2)}+1\\
&\leq &2\delta_2+2^{2(t-i-1)}\cdot 2^{2i+1}-2^{2(t-i-2)}+1\\
&= &2\delta_2+2^{2t-1}-2^{2(t-i-2)}+1.
\end{eqnarray*}
Next, we will investigate the lower bounds of $y_{_{x,k}}$:
If $i=0$, then $l_0\in[1,2]$, we get that \begin{eqnarray*}
y_{_{x,k}}&=& 2\delta_2+2^{2(t-1)}l_0-2^{2(t-2)}+1-\lambda\\
&>& 2\delta_2+2^{2(t-1)}l_0-2^{2(t-2)}+1-2^{2(t-4)}\\
&\geq &2\delta_2+2^{2(t-1)}\cdot 1-2^{2(t-2)}+1-2^{2(t-4)}.
\end{eqnarray*}
If $1\leq i \leq t-3$, then $l_0\in[1+(2+2^{3}+\cdots
+2^{2i-1}),2^{2i+1}]$, we have
\begin{eqnarray*}
y_{_{x,k}} &=& 2\delta_2+2^{2(t-i-1)}l_0-2^{2(t-i-2)}+1-\lambda\\
&>& 2\delta_2+2^{2(t-i-1)}l_0-2^{2(t-i-2)}+1-2^{2(t-i-4)}\\
&\geq &2\delta_2+2^{2(t-i-1)}[1+(2+2^{3}+\cdots +2^{2i-1})]-2^{2(t-i-2)}+1-2^{2(t-i-4)}\\
&=& \!\!2\delta_2+(2^{2t-3}\!\!+\!\!2^{2t-5}\!\!+\cdots+2^{2t-2i-1})\!\!+2^{2(t-i-1)}\!\!-\!\!2^{2(t-i-2)}+1-2^{2(t-i-4)}\\
&\geq &\!\!2\delta_2\!\!+(2^{2t-3}\!\!+\!\!\cdots+\!\!2^{2t-2(t-3)-1})\!\!+\!\!2^{2[t-(t-3)-1]}\!\!
-\!\!2^{2[t-(t-3)-2]}\!\!+\!\!1\!-\!\!2^{2[t-(t-3)-4]}\\
&= &2\delta_2+2^{2t-3} +2^{2t-5}+\cdots +2^5+2^4-2^2+1-2^{-2}.
\end{eqnarray*}
\begin{eqnarray*}
\hbox{Observe that}~ \frac{n+\delta_2}{2}&=&
\frac{7\delta_2+3}{2}=2\delta_2+2^{2t-1}+1,\\
\frac{n-\delta_2}{2}&=&\frac{5\delta_2+3}{2}=2\delta_2+(2^{2t-3}
+2^{2t-5}+\cdots +2^3+2)+2.
\end{eqnarray*}
Combining the previous bounds of $y_{_{x,k}}$, it is easy to check
$\frac{n-\delta_2}{2}<y_{_{x,k}}<\frac{n+\delta_2}{2}$.
If $\frac{n-\delta_2}{2}<y_{_{x,k}}\leq \frac{n-1}{2}$, then
$n-\delta_2<2y_{_{x,k}}\leq n-1$, one can infer there exists a
$j_{_{x,k}}=n-2y_{_{x,k}}\in C_{y_{_{x,k}}}$ such that $1\leq
j_{_{x,k}}<\delta_2$.
If $\frac{n+1}{2}\leq y_{_{x,k}}<\frac{n+\delta_2}{2}$, $n+1\leq
2y_{_{x,k}}<n+\delta_2$, one can derive there exists a $j_{_{x,k}}=
2y_{_{x,k}}-n\equiv 2y_{_{x,k}} \in C_{y_{_{x,k}}}$ achieving $1\leq
j_{_{x,k}}<\delta_2$.
Concluding the previous discussions, when $x\in [\delta_2+2,
\delta_2 +2^{2t-4}]$, there exists an integer $j_{_{x,k}}
\in[1,\delta_2)$ such that $j_{_{x,k}}\in C_x$, hence
$x$ is not a coset leader for $x\in [\delta_2+2, \delta_2 +2^{2t-4}]$.
\end{proof}
\begin{lemma}\label{lemm3.4}
Let $\delta_{1}$ and $\delta_{2}$ be given as
above, if an odd integer $x>\delta_{2}$ and $x\not=\delta_{1}$, then
$x$ is not a coset leader.
\end{lemma}
\begin{proof} According to the previous discussions, to attain
the desired conclusion, it suffices to verify there exists some
$k\in[0,m-1]$ such that $y_{_{x,k}}<x$ or $n-y_{_{x,k}}<x$ for
$x>\delta_2$ except $x=\delta_{1}=\frac{n}{3}$. We split into five
cases:
(1): If $x\in [\frac{n+1}{2},n-1]$, then $x<n<2x$ and
$n-y_{_{x,0}}=n-x<x$.
(2): If $x\in [\frac{n}{3}+1,\frac{n-1}{2}]$, then $2x<n<3x$ and
$n-y_{_{x,1}}=n-2x<x$.
(3): If $x\in [\lceil\frac{n}{4} \rceil,\frac{n}{3}-1]$, then $3x<n<4x$ and $y_{_{x,2}}=4x-n<x$.
(4): If $x\in [\lceil \frac{n}{5} \rceil,
\lfloor\frac{n}{4}\rfloor]$, then $4x<n<5x$ and
$n-y_{_{x,2}}=n-4x<x$.
(5): If $x\in [\lceil \frac{3n}{16} \rceil, \lfloor \frac{n}{5}
\rfloor]$, then $15x<3n<16x$ and $y_{_{x,4}}=16x-3n<x$.
Combining Lemma \ref{lemm3.3}, we can infer that $x$ is not a coset
leader for each
$x\in [\delta_2+2, \delta_2
+2^{2t-4}]\bigcup [\lceil \frac{3n}{16} \rceil, n-1] \setminus
\{\frac{n}{3}\}.$ Notice that $\lceil \frac{3n}{16} \rceil
=\delta_2+\lceil \frac{\delta_2}{8} \rceil<\delta_2+2^{2t-4}$. It
is easy to know if $x\in [\delta_2+2, n-1] \setminus \{\delta_1\}$,
$x$ is not a coset leader, which is equivalent to the desired
conclusion.
\end{proof}
\begin{theorem}\label{theo3.5}
Let $\delta_{1}, \delta_{2}, \delta_{3}$,
$\delta_{4}$ and $\delta_{5}$ be given as Lemma 3.2, then they are
the first, second, third, fourth and fifth largest coset leaders,
respectively.
\end{theorem}
\begin{proof} According to Lemmas 3.2-3.4, one can naturally infer $\delta_1$ and $\delta_2$
are the first and second largest coset leaders, respectively.
To prove
$\delta_{3}$, $\delta_{4}$ and $\delta_{5}$ are the third, fourth
and fifth largest coset leaders, respectively,
we need to
show neither $\delta_2-4$ nor $\delta_2-6$ is a coset leader.
It is not difficult to infer
\begin{eqnarray*}
2^{2t-3}(\delta_2-4) &\equiv&2^{2t-3}(\delta_2-4)-\frac{2^{2t-4}-1}{3}n\\
&=&2^{2t-3}+2^{2t-5}+\cdots+2+2^{2t-4}<\delta_2-4,\\
-2^{2t-3}(\delta_2-6) &\equiv&\frac{2^{2t-4}-1}{3}n-2^{2t-3}(\delta_2-6)\\
&=&\delta_2-2^{2t-2}+2^{2t-4}<\delta_2-6,
\end{eqnarray*}
which implies that $\delta_2-4$,
$\delta_2-6$ both are not coset leaders, then the desired conclusion can be derived.
\end{proof}
\begin{lemma}\label{lemm3.6}
If $1 \leq x \leq 2^{t+2}+7$ or $x\in \{\delta_2,
\delta_3, \delta_4, \delta_5\}$, then $|C_x|=2m$.
\end{lemma}
\begin{proof} Seeking a contradiction,
suppose $|C_x|=k$ with $k<2m$, it then follows that $x(2^{k}-1)
\equiv 0$. Since $k|2m$ and $m$ is odd, we have $k=m,
\frac{2m}{3}$ or $k \leq \frac{2m}{5}$.
(1) If $k=m$, then $(2^{k}-1, n)=(2^{m}-1, 2^{m}+1)=1$. From $1 \leq
x \leq 2^{t+2}+7<n$, one has $x(2^{k}-1) \not \equiv 0$.
(2) If $k=\frac{2m}{3}$, then $m\equiv 0(\bmod~3)$ and $(2^{k}-1,
n=2^{m}+1)=2^{\frac{m}{3}}+1$. Since $1 \leq x \leq
2^{t+2}+7<\frac{n}{2^{\frac{m}{3}}+1}$
$=2^{\frac{2m}{3}}-2^{\frac{m}{3}}+1$, $x(2^{k}-1) \not \equiv 0
$.
(3) If $k\leq \frac{2m}{5}$, we then obtain that $1\leq
x(2^{k}-1)\leq (2^{t+2}+7)(2^{\frac{2m}{5}}-1) <n$, so $x(2^{k}-1)
\not \equiv 0 $.
Collecting all the above cases, we then conclude that $x(2^{k}-1)
\not \equiv 0 $, a contradiction yields. Hence $|C_x|=2m$ for $1
\leq x \leq 2^{t+2}+7$.
In the proof of Lemma \ref{lemm3.2}, we have derived that when $x\in
\{\delta_2, \delta_3, \delta_4, \delta_5\}$, $y_{_{x,k}}>x$ for
$1\leq k \leq 2m-1$, obviously, $x(2^{k}-1) \not \equiv 0 $ and
$|C_x|=2m$.
\end{proof}
By the previous results, one can get the following theorem.
\begin{theorem}\label{theo3.7} Let $\delta_1$,
$\delta_2$, $\delta_3$, $\delta_4$ and $\delta_5$ be
given as Lemma 3.2. If $\delta$ is odd, then we have the
following:
(1) The narrow-sense BCH codes $\mathcal{C}(n,2,\delta,1)$ have
parameters
$$ \left\{
\begin{array}{lll}
\hbox{[}n, n-m\delta+5m, d \geq \delta\hbox{]} &\mbox {if $2^{t+1}+3\leq \delta \leq 2^{t+1}+2^{t}-3$;}\\
\hbox{[}n, n-m\delta+9m, d \geq \delta\hbox{]} &\mbox {if $2^{t+1}+2^{t}+3\leq \delta\leq 2^{t+2}-9 $;}\\
\hbox{[}n, n-2^{t+2}m+16m, d \geq 2^{t+2}+9\hbox{]} &\mbox {if $2^{t+2}-7\leq \delta\leq 2^{t+2}+9$;}\\
\hbox{[}n, 2m(i-1)+3, d \geq \delta_i \hbox{]} &\mbox {if $\delta_{i+1}+2 \leq \delta \leq \delta_{i}(i=1,2,3,4) $;}\\
\hbox{[}n, 1, n\hbox{]} &\mbox {if $\delta_1+2 \leq \delta \leq
n$.}
\end{array}
\right.$$
(2) The BCH codes $\mathcal{C}(n,2,\delta+1,0)$ have parameters
$$ \left\{
\begin{array}{lll}
\hbox{[}n, 2^m-m\delta+5m, d \geq 2\delta\hbox{]} &\mbox {if $2^{t+1}+3\leq \delta \leq 2^{t+1}+2^{t}-3$;}\\
\hbox{[}n, 2^m-m\delta+9m, d \geq 2\delta\hbox{]} &\mbox {if
$2^{t+1}+2^{t}+3\leq
\delta\leq 2^{t+2}-9 $;}\\
\hbox{[}n, 2^m-2^{t+2}m+16m, d \geq 2(2^{t+2}+9)\hbox{]} &\mbox {if $2^{t+2}-7\leq \delta\leq 2^{t+2}+9$;}\\
\hbox{[}n, 2m(i-1)+2, d \geq 2\delta_i \hbox{]} &\mbox {if
$\delta_{i+1}+2 \leq \delta \leq \delta_{i}(i=1,2,3,4) $.}
\end{array}
\right.$$
\end{theorem}
\section{ BCH codes of length $n=2^{m}+1$ with $m=4t+2$}
\label{sec4} In this section, we fix $m=4t+2 \geq 10$ and
$n=2^{m}+1$.
Similar to last Section 3, it will be determined when $x$ is a
coset leader for $1\leq x \leq 2^{2t+2}+2^{2t+1}+3$ in Theorem 4.1 in advance.
Next, we show each $\delta_{i}$ is a coset leader for $1\leq i
\leq 5$ in Lemma 4.2. For determining the first five largest coset
leaders in Theorem 4.5, we continue to give Lemmas 4.3-4.4 on the
basis of Lemma 4.2. The cardinalities of relevant cyclotomic cosets
will be calculated in Lemma 4.6. Finally, Theorem 4.7 can be
further given, which lists dimensions of some BCH codes of length
$n$ with designed distance $\delta$ for $2^{2t+1}+3\leq \delta \leq
2^{2t+2}+2^{2t+1}+5$ and $\delta_5+2\leq \delta \leq n$, where
$\delta_5$ is given as Lemma 4.2.
The proofs of Theorem 4.1,
Lemma 4.2 and Theorem 4.7 will be presented in {\bf Appendixes D, E \hbox{and} F},
respectively.
\begin{theorem}\label{theo4.1}
Suppose $x$ is odd, then we have the following:
(1) If $1\leq x\leq 2^{2t+1}-1$, then $x$ is a coset leader, see
Refs. \cite{Ding7} and \cite{Ding8}.
(2) If $2^{2t+1}+3\leq x\leq 2^{2t+2}-5$, then $x$ is a coset
leader.
(3) If $2^{2t+2}+5\leq x\leq 2^{2t+2}+2^{2t}-3$, then $x$ is a
coset leader.
(4) If $2^{2t+2}+2^{2t}+3\leq x\leq 2^{2t+2}+2^{2t+1}-3$, then $x$
is a coset leader.
(5) If $x=2^{2t+1}+1$, $2^{2t+2}-3\leq x \leq 2^{2t+2}+3$,
$2^{2t+2}+2^{2t}-1\leq x \leq 2^{2t+2}+2^{2t}+1$ or
$2^{2t+2}+2^{2t+1}-1\leq x \leq 2^{2t+2}+2^{2t+1}+3$, then $x$ is not a
coset leader.
\end{theorem}
\begin{lemma}\label{lemm4.2} Denote $\delta_{1}=\frac{n}{5}$,
$\delta_{2}=2^{4t-1}+\frac{2^{4t}-1}{5}$, $\delta_{3}=\delta_{2}-6$.
If $t=2$, let $\delta_{4}=\delta_{2}-8$ and
$\delta_{5}=\delta_{2}-24$;
if $t>2$, let $\delta_{4}=\delta_{2}-96$
and $\delta_{5}=\delta_{2}-102$.
Then $\delta_{1}$, $\delta_{2}$,
$\delta_{3}$, $\delta_{4}$ and $\delta_{5}$ are all coset leaders.
\end{lemma}
In the following, to verify the five largest coset leaders, we
first introduce an iterative algorithm similar to {\it IA 1} for
partitioning $ J^{(t)}=[1, 3\cdot2^{4t-6}]$ into $2^{t-2}$
subintervals $ J^{(t)}=J_1\bigcup J_2 \cdots \bigcup J_{2^{t-2}}$
for $t\geq 2$.
\noindent {\bf Iterative algorithm (IA 2):}
Suppose $J_i=[a_i,b_i]$, a partition of $J^{(t)}=[1,
3\cdot2^{4t-6}]$ is obtained as
$$J^{(t)}=J^{(t-1)}\bigcup (\bigcup \limits_{u=2^{t-3}+1}^{2^{t-2}} J_{u}),$$
$$\hbox{where} ~J_{u}=\left\{
\begin{array}{lll}
J_{u}=J_{j}+3 \cdot 2^{4\times (t-3)+2} &\mbox {if $u=j+2^{t-3}$, $1\leq j\leq 2^{t-3}-1$;}\\
\mbox{[}a_{2^{t-3}}+3 \cdot2^{4t-10}, 3 \cdot 2^{4t-6}\mbox{]}
&\mbox {if $ u=2^{t-2}$,}
\end{array}
\right.$$
with the initial conditions:
$ J_1=[1,3\cdot 2^2]=[1,12]$, $ J_2=[a_1+3\cdot 2^2, 3\cdot
2^6]=[13,192].
$
For clarity, we discuss the partition in detail as follows.
1) If $t=2$, let $J_1=J_{2^0}=[1,3\cdot 2^{4\times 2-6}]=[1,3\cdot
2^2]=[1,12]=[a_1,b_1]$;
2) If $t=3$, let $J_2=J_{2^1}=[a_1+3\cdot 2^{4\times 0+2}, 3\cdot
2^{4\times 3-6}]$=$[a_1+3\cdot 2^2, 3\cdot 2^6]=[a_2,b_2]$;
3) If $t=4$, $J_3=J_{1+2^1}=J_1+3 \cdot 2^{4\times 1+2}=[a_1+3 \cdot
2^{6}, b_1+3 \cdot 2^{6}]=[a_3,b_3]$,
$J_4=J_{2^2}=[a_2+3 \cdot 2^{4\times 1+2}, 3 \cdot 2^{10}]=[a_2+3
\cdot 2^6, 3 \cdot 2^{10}]=[a_4,b_4]$;
4) Let $t\geq 5$. Suppose a partition of $ J^{(t-1)}$ is given by
$J^{(t-1)}=[1, 2^{4t-10}]$ $=J_1\bigcup J_2 \cdots \bigcup
J_{2^{t-3}}=\bigcup \limits_{j=1}^{2^{t-3}} J_j$, where
$J_{j}=[a_j,b_j]$.\\
For $u=j+2^{t-3}$ with $1\leq j\leq 2^{t-3}-1$, define that
$J_{u}=J_{j}+3 \cdot 2^{4\times (t-3)+2}$ $=J_{j}+3
\cdot2^{4t-10}=[a_j+3 \cdot2^{4t-10},b_j+3 \cdot2^{4t-10}].$
For $u=2^{t-2}$, define that
$J_{2^{t-2}}=[a_{2^{t-3}}+3 \cdot 2^{4\times (t-3)+2}, 3 \cdot
2^{4\times (t-2)+2}]$=$[a_{2^{t-3}}+3 \cdot2^{4t-10}, 3 \cdot
2^{4t-6}].$
Then, a partition of $J^{(t)}=[1, 3 \cdot 2^{4t-6}]$ is obtained as\\
$J^{(t)}$ $=\bigcup
\limits_{s=1}^{2^{t-3}}\!\!J_s \bigcup (\!\!\bigcup
\limits_{u=2^{t-3}+1}^{2^{t-2}}\!\!J_{u})$
$=J_{1}\cup J_{2} \cdots
\cup J_{2^{t-3}}\cup J_{2^{t-3}+1} \cdots \cup J_{2^{t-2}}.$\\
\noindent {\bf Remark 2.} From the \textit{IA 2} above, one can
derive the following:
(1): $J_{2^i}= \left\{
\begin{array}{lll}
\hbox{[}1,12\hbox{]}=\hbox{[}1,3\cdot 2^{4i+2}\hbox{]} &\mbox {if $i=0$};\\
\hbox{[}1+3 \cdot (2^2+2^6+\cdots+2^{4i-2}),3 \cdot2^{4i+2}\hbox{]}
&\mbox {if $1\leq i \leq t-2$}.
\end{array}
\right.$
(2): Generally, for each $s\in [1, 2^{t-2}-1]$ with its 2-adic
expansion $s=(a_0a_1a_2 \cdots a_{t-3})_{2},$ denote
$i=i_{s}=\hbox{min}\{j|a_j=1,0\leq j\leq t-3\}$. Similar to $I_s$
in Section 3, $J_s$ can be given by
\begin{eqnarray*}
J_s&=&J_{2^i}+a_{i+1}2^{4i+6}+a_{i+2}2^{4i+10}+\cdots +a_{t-3}2^{4t-10}\\
&=&J_{2^i}+2^{4i+6}(a_{i+1}+a_{i+2}2^4+\cdots+a_{t-3}2^{4(t-i-4)})\\
&=&J_{2^i}+2^{4i+6}\lambda,
\end{eqnarray*}
$~\hbox{where}~\lambda
=a_{i+1}+a_{i+2}2^4+\cdots+a_{t-3}2^{4(t-i-4)}.$
Notice $0 \leq \lambda \leq 1+2^4+\cdots+ 2^{4(t-i-4)}
<2^{4(t-i-3)}$ for $s\leq 2^{t-2}-1$,
and $J_{2^{t-2}}=J_{2^{t-2}}+2^{4(t-2)+6}\times \lambda$ with
$\lambda=0$. Thus, from (1) and (2), one can then derive that for each $s\in [1, 2^{t-2}]$
there exists $0 \leq \lambda
<2^{4(t-i-3)}$ such that
$$J_s=J_{i,\lambda}=J_{2^i}+2^{4i+6}\lambda,
~\hbox{where}~0\leq i \leq t-2.$$
\noindent{\bf Example 2}: When $t=8$, then $J_{s}$ can be given
as follows:\\
if $s=8=(00010)_{2}$, then $i=3$ and $\lambda=0$, $J_{s}=J_{2^3}=J_{2^3}+2^{18} \cdot0$.\\
if $s=10=(01010)_{2}$, then $i=1$ and $\lambda=2^{4}$,
$J_{s}=J_{2^1}+2^{14}=J_{2^1}+2^{10}\cdot 2^{4}$.\\
if $s=15=(11110)_{2}$, then $i=0$ and $\lambda=273$,
$J_{s}\!=\!J_{2^0}+2^6+2^{10}+2^{14}\!=\!J_{2^0}+2^6\cdot273$.
\begin{lemma}\label{lemm4.3} Let $\delta_2$ be given as Lemma 4.2. If $x\in
[\delta_2+2, \delta_2+3 \cdot 2^{4t-5}]$ is odd, then $x$ is not a
coset leader.
\end{lemma}
\begin{proof} Similar to the proof of Lemma 3.3,
one can denote
$x=\delta_2+2(l_0+2^{4i+6}\lambda)$ with
$ l_0\in J_{2^i}.$
Choose $k=4t-4i-5$. We then have that
\begin{eqnarray*}
2^kx\equiv y_{_{x,k}}
&=& 2^k(\delta_2+2l_0+2^{4i+7}\lambda)-(2^{k-3}+\frac{2^{k-2}-2}{5}+\lambda)n\\
&=& 2^k(\delta_2+2l_0)-(\frac{7\cdot 2^{k-3}-2}{5})(\frac{40\delta_2+15}{7})+(2^{4t+2}-n)\lambda\\
&=& \frac{16\delta_2+6}{7}+2^{4(t-i-1)}l_0-3\cdot
2^{4(t-i-2)}-\lambda.
\end{eqnarray*}
Now, according to the value ranges of $i, l_0$ and $\lambda$ above,
one can show $\frac{n-\delta_2}{2}<y_{_{x,k}}<\frac{n+\delta_2}{2}$
for $k=4t-4i-5$.
First, we study an upper bound of $y_{_{x,k}}$:
\begin{eqnarray*}
y_{_{x,k}}&=& \frac{16\delta_2+6}{7}+2^{4(t-i-1)}l_0-3\cdot2^{4(t-i-2)}-\lambda\\
&\leq&
\frac{16\delta_2+6}{7}+2^{4(t-i-1)}l_0-3\cdot
2^{4(t-i-2)}\\
&\leq & \frac{16\delta_2+6}{7}+2^{4(t-i-1)}(3 \cdot2^{4i+2})-3\cdot
2^{4(t-i-2)}\\
&= &
\frac{16\delta_2+6}{7}+3 \cdot 2^{4t-2}-3 \cdot2^{4(t-i-2)}.
\end{eqnarray*}
Then, we will provide the lower bounds of $y_{_{x,k}}$:
If $i=0$, then $l_0\in [1,12]$, one can obtain that
\begin{eqnarray*}
y_{_{x,k}}&=& \frac{16\delta_2+6}{7}+2^{4(t-1)}l_0-3\cdot 2^{4(t-2)}-\lambda\\
&>& \frac{16\delta_2+6}{7}+2^{4(t-1)}l_0-3\cdot
2^{4(t-2)}-2^{4(t-3)}\\
&\geq & \frac{16\delta_2+6}{7}+2^{4(t-1)}\cdot 1-3\cdot
2^{4(t-2)}-2^{4(t-3)}.
\end{eqnarray*}
If $1\leq i \leq t-2$, thus $l_0\in [1+3 \cdot
(2^2+2^6+\cdots+2^{4i-2}),3 \cdot2^{4i+2}]$, we get that
\begin{eqnarray*}
y_{_{x,k}}
&=& \frac{16\delta_2+6}{7}+2^{4(t-i-1)}l_0-3\cdot 2^{4(t-i-2)}-\lambda\\
&>&
\frac{16\delta_2+6}{7}+2^{4(t-i-1)}l_0-3\cdot
2^{4(t-i-2)}-2^{4(t-i-3)}\\
&\geq &\!\!\!\!\frac{16\delta_2+6}{7}+\!\!2^{4(t-i-1)}[1+3 \cdot (2^2+2^6+\cdots+2^{4i-2})]-\!\!3\cdot
2^{4(t-i-2)}-\!\!2^{4(t-i-3)}\\
&= &\frac{16\delta_2+6}{7}+3\cdot(2^{4t-6}+2^{4t-10}+\!\!\cdots
+2^{4t-4i-2})+\!\!13\cdot2^{4(t-i-2)}-\!\!2^{4(t-i-3)}\\
&>&\!\!\!\!\frac{16\delta_2+6}{7}+\!\!3\cdot(2^{4t-6}
+\!\!\cdots+\!\!2^{4t-4(t-2)-2})\!\!+\!\!13\cdot2^{4[t-(t-2)-2]}\!\!-\!\!2^{4[t-(t-2)-3]}\\
&= &
\frac{16\delta_2+6}{7}+3\cdot(2^{4t-6} +2^{4t-10}+\cdots +2^6)+13-2^{-4}.
\end{eqnarray*}
\begin{eqnarray*}
\hbox{Observe that}~ \frac{n+\delta_2}{2}&=&
\frac{16\delta_2+6}{7}+\frac{15\delta_2+3}{14}=\frac{16\delta_2+6}{7}+3\cdot
2^{4t-2},\\
\frac{n-\delta_2}{2}&=& \frac{16\delta_2+6}{7}+3(2^{4t-6}
+2^{4t-10}+\cdots+2^6)+2^2+1.
\end{eqnarray*}
Combining the previous bounds of $y_{_{x,k}}$, it is easy to check
$\frac{n-\delta_2}{2}<y_{_{x,k}}<\frac{n+\delta_2}{2}$.
Similar to the proof of Lemma \ref{lemm3.3},
one can infer that there exists an integer $j_{_{x,k}}
\in[1,\delta_2)$ such that $j_{_{x,k}}\in C_{y_{_{x,k}}}$ for
$\frac{n-\delta_2}{2}\leq y_{_{x,k}}<\frac{n+\delta_2}{2}$.
Summarizing the previous discussions, when $x\in [\delta_2+2,
\delta_2+3 \cdot 2^{4t-5}]$, there exists an integer $j_{_{x,k}}
\in [1,\delta_2)$ such that $j_{_{x,k}}\in C_x$, it then follows
that
$x$ is not a coset leader for all $x\in [\delta_2+2, \delta_2+3 \cdot 2^{4t-5}]$,
this completes the proof. \end{proof}
\begin{lemma}\label{lemm4.4} Let $\delta_{1}$ and $\delta_{2}$ be given as
Lemma 4.2. If an odd integer $x>\delta_{2}$ and $x\not=\delta_{1}$,
then $x$ is not a coset leader.
\end{lemma}
\begin{proof}
According to previous discussions, to attain
the desired conclusion, it suffices to verify there exists
$y_{_{x,k}}<x$ or $n-y_{_{x,k}}<x$ for some $k \in [0, m-1]$ and
$x\in [\delta_2+2,n-1] \setminus \{\delta_1=\frac{n}{5}\}$. We give
our discussions in following cases:
(1): If $x\in [\frac{n+1}{2},n-1]$, then $x<n<2x$ and
$n-y_{_{x,0}}=n-x<x$;
(2): If $x\in [\frac{n+1}{3},\frac{n-1}{2}]$, then $2x<n<3x$ and
$n-y_{_{x,1}}=n-2x<x$;
(3): If $x\in [\frac{n+3}{4},\frac{n-2}{3}]$, then $3x<n<4x$ and
$y_{_{x,2}}=4x-n<x$;
(4): If $x\in [\frac{n}{5}+1,\frac{n-1}{4}]$, then $4x<n<5x$ and
$n-y_{_{x,2}}=n-4x<x$;
(5): If $x\in [\frac{3(n-1)}{16}+1,\frac{n}{5}-1]$, then
$15x<3n<16x$ and $y_{_{x,4}}=16x-3n<x$.
Combining Lemma \ref{lemm4.3}, one can infer that $x$ is not a
coset leader for all $x\in [\delta_2+2, \delta_2 +3 \cdot
2^{4t-5}]\bigcup [\frac{3(n-1)}{16}+1, n-1] \setminus
\{\frac{n}{5}\}.$ Notice that $\frac{3(n-1)}{16}+1 =\delta_2+
\frac{2^{4t-2}+1}{5}+1 <\delta_2+3 \cdot 2^{4t-5}$, the conclusion
can be easily derived from the discussions above.\end{proof}
\begin{theorem}\label{theo4.5}
Let $\delta_{1}, \delta_{2}, \delta_{3}$,
$\delta_{4}$ and $\delta_{5}$ be given as Lemma 4.2, then they are
the first, second, third, fourth and fifth largest coset leaders,
respectively.
\end{theorem}
\begin{proof} According to Lemmas 4.2-4.4, one can naturally infer that $\delta_1$ and $\delta_2$
are the first and second largest coset leaders, respectively.
To verify that $\delta_3$
is the third largest coset leader, it suffices to show neither
$\delta_2-4$ nor $\delta_2-2$ is not a coset leader. It is not
difficult to derive that
\begin{eqnarray*}
-2^{4t-1}(\delta_2-4) &\equiv&(2^{4t-4}+\frac{2^{4t-3}-2}{5})n-2^{4t-1}(\delta_2-4)\\
&=&\delta_2-2^{4t-4}-\frac{2^{4t-2}+1}{5}<\delta_2-4,\\
2^{4t-1}(\delta_2-2) &\equiv&2^{4t-1}(\delta_2-2)-(2^{4t-4}+\frac{2^{4t-3}-2}{5})n\\
&=&2^{4t-1}-\frac{2^{4t-3}-2}{5}-2^{4t-4}<\delta_2-2,
\end{eqnarray*}
which implies that $\delta_2-4$ and $\delta_2-2$ both are not
coset leaders.
Similarly, it shall be obtained that $x$ is not a coset leader for
$\delta_5 < x <\delta_4$ and $\delta_4 <x< \delta_3$, the detailed
proof is omitted. As thus, it is easy to know $\delta_4$(resp.
$\delta_5$) is the fourth (resp. fifth) largest coset leader,
which completes the proof.\end{proof}
\begin{lemma}\label{lemm4.6} Let $\delta_{1}, \delta_{2}, \delta_{3}$,
$\delta_{4}$ and $\delta_{5}$ be given as Lemma 4.2. If $1 \leq x
\leq 2^{2t+2}+2^{2t+1}+3$ or $x\in \{\delta_2, \delta_3, \delta_4,
\delta_5\}$, then $|C_x|=2m$.
\end{lemma}
\begin{proof} Seeking a contradiction, if $|C_x|=k$ with $k<2m$,
we have $x(2^{k}-1) \equiv 0$. From $k|2m$, one can deduce
$k=m, \frac{2m}{3}, \frac{2m}{4}$ or $k \leq \frac{2m}{5}$.
(1): If $k=\frac{2m}{2}=m$ or $k=\frac{2m}{4}=\frac{m}{2}$, then
$(2^{k}-1, n)=(2^{m}-1, 2^{m}+1)=(2^{\frac{m}{2}}-1, 2^{m}+1)=1$ and
$1 \leq x \leq 2^{2t+2}+2^{2t+1}+3<n$, $x(2^{k}-1)\not \equiv 0$.
(2): If $k=\frac{2m}{3}$, then $m\equiv 0 (\bmod ~3)$ and
$(2^{k}-1, n=2^{m}+1)=2^{\frac{m}{3}}+1$. Since $1 \leq x \leq
2^{2t+2}+2^{2t+1}+3<\frac{n}{2^{\frac{m}{3}}+1}$
$=2^{\frac{2m}{3}}-2^{\frac{m}{3}}+1$, $x(2^{k}-1) \not \equiv 0$.
(3): If $k\leq \frac{2m}{5}$, then $1\leq x(2^{k}-1)\leq
(2^{2t+2}+2^{2t+1}+3)(2^{k}-1) <n$, $x(2^{k}-1) \not \equiv 0 $.
Collecting the discussions above, we then conclude that $x(2^{k}-1)
\not \equiv 0 $, a contradiction yields. Hence $|C_x|=2m$ for $1
\leq x \leq 2^{2t+2}+2^{2t+1}+3$.
In the proof of Lemma \ref{lemm4.2}, we have derived that when $x\in
\{\delta_2, \delta_3, \delta_4, \delta_5\}$,
$y_{_{x,k}}>x$
for $1\leq k \leq 2m-1$, it then follows that $x(2^{k}-1) \not
\equiv 0 $ and $|C_x|=2m$. \end{proof}
According to the results
above, one can get the following theorem.
\begin{theorem}\label{theo4.7} Denote that $a=2^{2t+2}+2^{2t+1}$.
Let $\delta_1$, $\delta_2$, $\delta_3$, $\delta_4$ and $\delta_5$ be
given as Lemma 4.2. If $\delta$ is odd, then we have the
following:
(1) The narrow-sense BCH codes $\mathcal{C}(n,2,\delta,1)$ have
parameters
$$ \left\{
\begin{array}{lll}
\hbox{[}n, n-m\delta+3m, d \geq \delta\hbox{]} &\mbox {if $2^{2t+1}+3\leq \delta\leq 2^{2t+2}-5$;}\\
\hbox{[}n, n-m\delta+11m, d \geq \delta\hbox{]} &\mbox {if $2^{2t+2}+5\leq \delta\leq 2^{2t+2}+2^{2t}-3$;}\\
\hbox{[}n, n-m\delta+15m, d \geq \delta\hbox{]} &\mbox {if
$2^{2t+2}+2^{2t}+3\leq \delta \leq a-3$;}\\
\hbox{[}n, n-ma+16m, d \geq a+5 \hbox{]} &\mbox {if $a-1 \leq \delta\leq a+5$;}\\
\hbox{[}n, 2m(i-1)+5, d \geq \delta_i\hbox{]} &\mbox {if $\delta_{i+1}+2 \leq \delta \leq \delta_{i}(i=1,2,3,4) $;}\\
\hbox{[}n, 1, n\hbox{]} &\mbox {if $\delta_1+2 \leq \delta \leq
n$.}
\end{array}
\right.$$
(2) The BCH codes $\mathcal{C}(n,2,\delta+1,0)$ have parameters
$$ \left\{
\begin{array}{lll}
\hbox{[}n, 2^m-m\delta+3m, d \geq 2\delta\hbox{]} &\mbox {if $2^{2t+1}+3\leq \delta\leq 2^{2t+2}-5$;}\\
\hbox{[}n, 2^m-m\delta+11m, d \geq 2\delta\hbox{]} &\mbox {if $2^{2t+2}+5\leq \delta\leq 2^{2t+2}+2^{2t}-3$;}\\
\hbox{[}n, 2^m-m\delta+15m, d \geq 2\delta\hbox{]} &\mbox {if
$2^{2t+2}+2^{2t}+3\leq\delta \leq a-3$;}\\
\hbox{[}n, 2^m-ma+16m, d \geq 2(a+5)\hbox{]} &\mbox {if $a-1 \leq \delta\leq a+5$;}\\
\hbox{[}n, 2m(i-1)+4, d \geq 2\delta_i\hbox{]} &\mbox {if
$\delta_{i}+2 \leq \delta \leq \delta_{i}(i=1,2,3,4)$.}
\end{array}
\right.$$
\end{theorem}
\section{BCH codes of length $n=2^{m}+1$ with $m=8t+4$}
\label{sec5} In this section, suppose that $n=2^{m}+1$ with $m=8t+4
\geq 12$, we only list the main results, omitting their proofs
because of the similarity with ones in the two sections above.
\begin{theorem}\label{theo5.1}
Suppose that $x$ is odd, then we have the following:
(1) If $1\leq x\leq 2^{4t+2}-1$, then $x$ is a coset leader.
(2) If $2^{4t+2}+3\leq x\leq 2^{4t+3}-5$, then $x$ is a coset
leader.
(3) If $2^{4t+3}+5\leq x\leq 2^{4t+3}+2^{4t+1}-3$, then $x$ is a
coset leader.
(4) If $2^{4t+3}+2^{4t+1}+3\leq x\leq 2^{4t+3}+2^{4t+2}-3$, then
$x$ is a coset leader.
(5) If $2^{4t+3}+2^{4t+2}+5\leq x\leq
2^{4t+3}+2^{4t+2}+2^{4t+1}-3$, then $x$ is a coset leader.
(6) If $x=2^{4t+2}+1$,
$2^{4t+3}-3\leq x \leq 2^{4t+3}+3$,
$2^{4t+3}+2^{4t+1}-1\leq x \leq 2^{4t+3}+2^{4t+1}+1$,
$2^{4t+3}+2^{4t+2}-1\leq x \leq 2^{4t+3}+2^{4t+2}+3$ or
$2^{4t+3}+2^{4t+2}+2^{4t+1}-1\leq x \leq
2^{4t+3}+2^{4t+2}+2^{4t+1}+1$,
then $x$ is not a coset leader.
\end{theorem}
\begin{theorem}\label{theo5.2} Let $\delta_{1}=\frac{3n}{17}$.
If $t=1$, let $\delta_{2}=\delta_{1}-6$,
$\delta_{3}=\delta_{2}-24$, $\delta_{4}=\delta_{3}-2$
and $\delta_{5}=\delta_{4}-38$;
If $t=2$, let $\delta_{2}=\delta_{1}-\frac{\delta_1+45}{128}$ and
$\delta_{3}=\delta_{2}-90$, $\delta_{4}=\delta_{3}-6$,
$\delta_{5}=\delta_{4}-384$;
If $t\geq 3$, let $\delta_{2}=\delta_{1}-\frac{\delta_1+45}{128}$
and $\delta_{3}=\delta_{2}-90$, $\delta_{4}=\delta_{3}-22950$,
$\delta_{5}=\delta_{4}-90$.
Then they are the first, second, third, fourth and fifth largest
coset leaders, respectively.
\end{theorem}
\begin{lemma}\label{lemm5.3}
Let ${\delta_1, \delta_2, \delta_3, \delta_4,
\delta_5}$ be given as Theorem 5.2. If $1 \leq x \leq
2^{4t+3}+2^{4t+2}+2^{4t+1}+1$ or $x\in \{\delta_2, \delta_3,
\delta_4, \delta_5\}$, then $|C_x|=2m$ and $|C_{\delta_1}|=8$.
\end{lemma}
\begin{theorem}\label{theo5.4}
Suppose that $b=2^{4t+3}+2^{4t+2}+2^{4t+1}$. Let $\delta_1$,
$\delta_2$, $\delta_3$, $\delta_4$ and $\delta_5$ be given as
Theorem 5.2. If $\delta$ is odd, then we have the following:
(1) The narrow-sense BCH codes $\mathcal{C}(n,2,\delta,1)$ have
parameters
$$ \left\{
\begin{array}{lll}
\hbox{[}n, n-m\delta+3m, d \geq \delta\hbox{]} &\mbox {if $2^{4t+2}+3\leq \delta\leq 2^{4t+3}-5$;}\\
\hbox{[}n, n-m\delta+11m, d \geq \delta\hbox{]} &\mbox {if $2^{4t+3}+5\leq \delta\leq 2^{4t+3}+2^{4t+1}-3$;}\\
\hbox{[}n, n-m\delta+15m, d \geq \delta\hbox{]} &\mbox {if $2^{4t+3}+2^{4t+1}+3\leq \delta \leq 2^{4t+3}+2^{4t+2}-3$;}\\
\hbox{[}n, n-m\delta+21m, d \geq \delta\hbox{]} &\mbox {if $2^{4t+3}+2^{4t+2}+5\leq \delta \leq b-3$;}\\
\hbox{[}n, n-mb+22m, d \geq b+3\hbox{]} &\mbox {if $b-1 \leq \delta \leq b+3$;}\\
\hbox{[}n, 2m(i-1)+9, d \geq \delta_i\hbox{]} &\mbox {if $\delta_{i+1}+2 \leq \delta \leq \delta_i(i=1,2,3,4)$;}\\
\hbox{[}n, 1, n\hbox{]} &\mbox {if $\delta_1+2 \leq \delta \leq
n$.}
\end{array}
\right.$$
(2) The BCH codes $\mathcal{C}(n,2,\delta+1,0)$ have parameters
$$ \left\{
\begin{array}{lll}
[n, 2^m-m\delta+3m, d \geq 2\delta] &\mbox {if $2^{4t+2}+3\leq \delta\leq 2^{4t+3}-5$;}\\
\hbox{[}n, 2^m-m\delta+11m, d \geq 2\delta\hbox{]} &\mbox {if $2^{4t+3}+5\leq \delta\leq 2^{4t+3}+2^{4t+1}-5$;}\\
\hbox{[}n, 2^m-m\delta+15m, d \geq 2\delta\hbox{]} &\mbox {if
$2^{4t+3}+2^{4t+1} \!\!+3\leq \delta \leq \!\!2^{4t+3}+2^{4t+2} \!\!-3$;}\\
\hbox{[}n, 2^m-m\delta+21m, d \geq 2\delta\hbox{]} &\mbox {if $2^{4t+3}+2^{4t+2}+5\leq \delta \leq b-3$;}\\
\hbox{[}n, 2^m-mb+22m, d \geq 2(b+3)\hbox{]} &\mbox {if $b-1 \leq \delta \leq b+3$;}\\
\hbox{[}n, 2m(i-1)+8, d \geq 2\delta_i\hbox{]} &\mbox {if
$\delta_{i+1}+2 \leq \delta \leq \delta_i(i=1,2,3,4)$.}
\end{array}
\right.$$
\end{theorem}
\section{Conclusion}
We have discussed the dimension of some binary BCH codes of length
$n=2^m+1$ with designed distance $\delta>2^{\lceil \frac{m}{2}
\rceil}$ for $m=2t+1$, $m=4t+2$, $m=8t+4$ and $m\geq10$. The
main contributions of this paper are summarized as follows:
1) Some new techniques to find out coset leaders were presented.
2) The first five largest coset leaders were determined.
3) The dimensions of some classes of BCH codes of length $n$ with
designed distance $\delta>2^{\lceil \frac{m}{2} \rceil}$ were
determined. The Bose distances of these codes were also obtained.
Though it is not easy to study parameters of antiprimitive BCH
codes as pointed out in \cite{Ding5}, fortunately, we have gone one
step further on the basis of Refs. \cite{Ding7} and \cite{Ding8}. It
is worthwhile and expectant to develop more techniques and
achievements on antiprimitive LCD BCH codes over finite fields. We
believe that our results will shed light on BCH codes and cyclic
codes. It is also hoped that these results will work to discuss
constructions of LCD codes from negacyclic codes over finite fields
as did in \cite{Shixin}.
\section*{Acknowledgements}
This work is a revised edition on the basis of its original
edition, which was submitted to the journal "Finite Fields and Their
Applications" on May 4, 2017. Here, we are greatly indebted to two
anonymous reviewers and the Associate Editor, Prof. Pascale Charpin,
for their comments and suggestions that much improved the
presentation and quality of this paper.
The first author would like to express his gratitude to Prof.
Cunsheng Ding for his helpful suggestions by e-mail communication
and inspiring discussions during National Conference on Coding
theory and Cryptography in Hangzhou, China.
This work is supported by National Natural Science Foundation
of China under Grant No.11471011 and Natural Science Foundation of
Shaanxi under Grant No.2017JQ1032.
\section*{Appendix}
First of all, we present the following Lemma
to get the minimum values in the proofs of Theorems 3.1 and 4.1.
\noindent{\bf Lemma 0.1} Let $f(k)=2^{-k}a+2^{k}b$, where $a, b$
and $k$ are positive real numbers. If $k_2\geq k_1\geq
\frac{\log_{2}a-\log_{2}b}{2}$, then $f(k_2)\geq f(k_1)$.
\begin{proof} Since $f'(k)=(-2^{-k}a+2^{k}b)\ln 2$, one can
easily deduce that if $k\geq \frac{\log_{2}a-\log_{2}b}{2}$ , then
$ f'(k)\geq 0$. It follows that when $k\geq
\frac{\log_{2}a-\log_{2}b}{2}$, $f(k)$ is monotonically increasing,
this completes the proof.
\end{proof}
\subsection*{Appendix A: The proof of Theorem 3.1}
\begin{proof} To prove this theorem, it suffices to vertify the items (2)-(4).
(2): Since $x$ is odd, $x$ can be denoted by
$$x=2^{t+1}+1+2l, \hbox{where}~l \in I=[1, 2^{t-1}-2].$$
To verify (2), it suffices to prove $y_{_{x,k}}-x \geq 0$ and
$n-y_{_{x,k}}-x \geq 0$ for $k\in[0,m-1=2t]$. We first determine
$y_{_{x,k}}$, then show $y_{_{x,k}}-x \geq 0$ and $n-y_{_{x,k}}-x
\geq 0$ according to different $k$.
(2.1): When $k=0,1,2,\cdots,t-1$, from $2^{t+1}+3\leq x\leq 2^{t+1}+2^{t}-3$,
we have $x\leq 2^kx <n$, hence, $y_{_{x,k}}=2^kx\geq
x$, and
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&2^{2t+1}+1-(2^k+1)x\\
&\geq&2^{2t+1}+1-(2^{t-1}+1)x\\
&\geq&2^{2t+1}+1-(2^{t-1}+1)(2^{t+1}+2^{t}-3)\\
&= & 2^{2t-1}-2^{t-1}-2^t+4>0.
\end{eqnarray*}
(2.2): When $k=t, t+1$, we have $y_{_{x,k}}=2^kx-2^{k-t}n$, thus
\begin{eqnarray*}
y_{_{x,k}}-x &=&(2^k-1)x-2^{k-t}n\\
&=& (2^k-1)(2^{t+1}+1+2l)-2^{k-t}(2^{2t+1}+1)\\
&=&(2^{k+1}-2)l+2^k-2^{t+1}-1-2^{k-t}\\
&\geq&(2^{k+1}-2)\cdot1+2^k-2^{t+1}-1-2^{k-t}\\
&=&(3-2^{-t})\cdot2^k-2^{t+1}-3\\
&\geq&(3-2^{-t})\cdot2^t-2^{t+1}-3= 2^{t}-4>0,
\end{eqnarray*}
\begin{eqnarray*}
n-y_{_{x,k}}-x &=&(2^{k-t}+1)n-(2^k+1)x\\
&=& (2^{k-t}+1)(2^{2t+1}+1)-(2^k+1)(2^{t+1}+1+2l)\\
&=& 2^{k-t}+2^{2t+1}-2^{t+1}-2^{k}-(2^{k+1}+2)l\\
&\geq& 2^{k-t}+2^{2t+1}-2^{t+1}-2^{k}-(2^{k+1}+2)(2^{t-1}-2)\\
&=& 2^{2t+1}-2^{t}-2^{t+1}+4-(2^t-2^{-t}-3)2^{k}\\
&\geq& 2^{2t+1}-2^{t}-2^{t+1}+4-(2^t-2^{-t}-3)2^{t+1}\\
&=&3(2^{t}+2)>0.
\end{eqnarray*}
(2.3): When $k=t+2,t+3,\cdots,2t-1$, it is a little complex to
determine $y_{_{x,k}}$ and check $y_{_{x,k}}-x \geq 0$ along
with $n-y_{_{x,k}}-x \geq 0$. To achieve this, for each $k$, we
divide the value range $I=[1, 2^{t-1}-2]$ of $l$ into $2^{k-1-t}$
subintervals as follows:
$I_{_{\lambda,k}}=[1,\lambda 2^{2t-k}-1]~\hbox{for}~ \lambda =1$,
$I_{_{\lambda,k}}=[(\lambda-1)2^{2t-k},
\lambda2^{2t-k}-1] ~\hbox{for}~ \lambda \in [2, 2^{k-t-1}-1]$,
$I_{_{\lambda,k}}=[(\lambda-1)2^{2t-k},2^{t-1}-2] ~\hbox{for}~ \lambda = 2^{k-t-1}$.
Fix $k$, for each $\lambda \in [1, 2^{k-t-1}]$, $\lambda$ is
called the {\it identity tag} of the subinterval
$I_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$. Thus, for
$x=2^{t+1}+1+2l$ and given $\lambda$, it is not difficult to
derive $y_{_{x,k}}=2^kx-(2^{k-t}+\lambda-1)n$ if $l\in
I_{_{\lambda,k}}$.
{\bf Case 2.3.1:} Firstly, we show $y_{_{x,k}}-x\geq0$.
For general $\lambda$,
\begin{eqnarray*}
y_{_{x,k}}-x &=&(2^k-1)x-(2^{k-t}+\lambda-1)n\\
&=& (2^k-1)(2^{t+1}+1+2l)-(2^{k-t}+\lambda-1)(2^{2t+1}+1)\\
&=&(2^k-1)(1+2l)-2^{t+1}-2^{k-t}-(\lambda-1)(2^{2t+1}+1).
\end{eqnarray*}
When $\lambda=1$, from $l\geq 1$ and $k\geq t+2$, we have
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)(1+2l)-2^{t+1}-2^{k-t}\\
&\geq& (2^k-1)(1+2\cdot1)-2^{t+1}-2^{k-t}\\
&=& (3-2^{-t})\cdot 2^k-2^{t+1}-3\\
&\geq& (3-2^{-t})\cdot 2^{t+2}-2^{t+1}-3\\
&=&5\cdot 2^{t+2}-7>0.
\end{eqnarray*}
For given $\lambda \in [2, 2^{k-t-1}]$, from $l\in
I_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{2t-k},l_{_{\lambda,e}}]$,
we have
\begin{eqnarray*}
y_{_{x,k}}-x
&=&(2^k-1)(1+2l)-2^{t+1}-2^{k-t}-(\lambda-1)(2^{2t+1}+1)\\
&\geq&(2^k-1)(1+2\cdot
l_{_{\lambda,b}})-2^{t+1}-2^{k-t}-(\lambda-1)(2^{2t+1}+1)\\
&=& 2^{2t+1-k}+2^k-2^{t+1}-2^{k-t}-(2^{2t-k+1}+1)\lambda\\
&\geq& 2^{2t+1-k}+2^k-2^{t+1}-2^{k-t}-(2^{2t-k+1}+1)\cdot2^{k-t-1}\\
&=& 2^{2t+1-k}+2^k(1-2^{-t}-2^{-t-1})-2^{t+1}-2^{t}\\
&\geq& \!2^{2t+1-(t+2)}+\!\!2^{t+2}(1-2^{-t}-2^{-t-1})\!\!-2^{t+1}-\!2^{t}(\hbox{see Lemma 0.1}) \\
&=& 2^{t}+2^{t-1}-6>0.
\end{eqnarray*}
As thus, we have shown $y_{_{x,k}}-x\geq0$ for
$k=t+2,t+3,\cdots,2t-1$.
{\bf Case 2.3.2:} Secondly, we will show $n-y_{_{x,k}}-x\geq0$.
For general $\lambda$,
\begin{eqnarray*}
n-y_{_{x,k}}-x
&=& (2^{k-t}+\lambda)(2^{2t+1}+1)-(2^k+1)(2^{t+1}+1+2l).
\end{eqnarray*}
If $\lambda$ satisfies $1\leq\lambda \leq 2^{k-t-1}-1$ and $l\in
I_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}=\lambda2^{2t-k}-1]$,
then
\begin{eqnarray*}
&&n-y_{_{x,k}}-x\\
&\geq& (2^{k-t}+\lambda)(2^{2t+1}+1)-(2^k+1)(2^{t+1}+1+2\cdot l_{_{\lambda,e}})\\
&=&2^{k-t}-2^{t+1}+2^{k}+1-(2^{2t+1-k}-1)\lambda \\
&\geq&2^{k-t}-2^{t+1}+2^{k}+1-(2^{2t+1-k}-1)\cdot(2^{k-t-1}-1) \\
&=& 2^{2t+1-k}+2^k(1+2^{-t}+2^{-t-1})-2^{t+1}-2^{t}\\
&\geq& 2^{2t+1-(t+2)}+2^{t+2}(1+2^{-t}+2^{-t-1})-2^{t+1}-2^{t}(\hbox{see Lemma 0.1})\\
&=& 3(2^{t-1}+2)>0.
\end{eqnarray*}
If $\lambda= 2^{k-t-1}$, let $l\in
I_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{2t-k},
l_{_{\lambda,e}}=2^{t-1}-2]$, we get then
\begin{eqnarray*}
n-y_{_{x,k}}-x
&=&(2^{k-t}+2^{k-t-1})(2^{2t+1}+1)-(2^k+1)(2^{t+1}+1+2l)\\
&\geq&(2^{k-t}+\lambda)(2^{2t+1}+1)-(2^k+1)(2^{t+1}+1+2\cdot l_{_{\lambda,e}})\\
&=&2^k(3+2^{-t}+2^{-t-1})-2^{t+1}-2^{t}+3\\
&\geq&2^{t+2}(3+2^{-t}+2^{-t-1})-2^{t+1}-2^{t}+3\\
&=& 9(2^{t}+1)>0.
\end{eqnarray*}
(2.4): When $k=2t$, for $x=2^{t+1}+1+2l$ with $l\in
I=[1,2^{t-1}-2]$, we have
\begin{eqnarray*}
y_{_{x,k}}&=&2^kx-(2^{k-t}+l)n=2^{2t}-2^{t}-l.
\end{eqnarray*}
Thus, we check that \begin{eqnarray*}
y_{_{x,k}}-x
&=&2^{2t}-2^{t}-l-(2^{t+1}+1+2l)\\
&=&2^{2t}-2^{t+1}-2^t-1-3l\\
&\geq&2^{2t}-2^{t+1}-2^t-1-3\cdot(2^{t-1}-2)\\
&=& 2^{2t}-2^{t+2}-2^{t-1}+5>0,\\
n-y_{_{x,k}}-x
&=&2^{2t}-2^{t+1}+2^t-l\\
&\geq&2^{2t}-2^{t+1}+2^t-(2^{t-1}-2)\\
&=& 2^{2t}-2^{t+1}+2^{t-1}+2>0.
\end{eqnarray*}
Summarizing the four cases above, we then conclude that
$y_{_{x,k}}-x\geq0$ and $n-y_{_{x,k}}-x\geq0$ for $2^{t+1}+3\leq
x\leq 2^{t+1}+2^{t}-3$
and $k\in [0,m-1=2t]$, (2) follows.\\
(3) For an odd $x \in [2^{t+1}+2^{t}+3,2^{t+2}-9]$, let
$x=2^{t+1}+2^{t}+1+2l$ with $l \in J=[1, 2^{t-1}-5]$. To verify (3)
holds, we will first determine $y_{_{x,k}}$ and then show
$y_{_{x,k}}-x \geq 0$ and $n-y_{_{x,k}}-x \geq 0$ for all
$k\in[0,m-1=2t]$. Similar to (2) above, we split into following
cases according to different $k$.
(3.1) When $k=0,1,2,\cdots,t-1$, it is clear that $x\leq 2^kx<n$,
then we have $y_{_{x,k}}=2^kx\geq x$ and
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&2^{2t+1}+1-(2^k+1)x\\
&\geq&2^{2t+1}+1-(2^k+1)(2^{t+2}-9)\\
&= & 2^{t-1}+10>0.
\end{eqnarray*}
(3.2): When $k=t$, we have $y_{_{x,k}}=2^kx-n$, hence
\begin{eqnarray*}
y_{_{x,k}}-x &=& (2^t-1)x-n\\
&\geq& (2^t-1)(2^{t+1}+2^{t}+3)-n\\
&=&2^{2t}-4>0,\\
& &\\
n-y_{_{x,k}}-x &=&2n-(2^t+1)x\\
&\geq&2n-(2^t+1)(2^{t+2}-9)\\
&=&5\cdot2^t+11>0.
\end{eqnarray*}
(3.3): When $k=t+1$, we have $y_{_{x,k}}=2^kx-3n$,
it follows that
\begin{eqnarray*}
y_{_{x,k}}-x &=& (2^{t+1}-1)x-3n\\
&\geq& (2^{t+1}-1)(2^{t+1}+2^{t}+3)-3n\\
&=&3(2^{t}-2)>0,\\
& &\\
n-y_{_{x,k}}-x &=&4n-(2^{t+1}+1)x\\
&\geq&4n-(2^{t+1}+1)(2^{t+2}-9)\\
&=&7 \cdot 2^{t+1}+13>0.
\end{eqnarray*}
(3.4): For each $k=t+2,t+3,\cdots, 2t-3$, to determine $y_{_{x,k}}$
and show $y_{_{x,k}}-x\geq 0$ and $n-y_{_{x,k}}-x\geq0$, we divide
the value range $J=[1, 2^{t-1}-5]$ of $l$ into $2^{k-1-t}$
subintervals as follows:
$J_{_{\lambda,k}}=[1,\lambda2^{2t-k}-1]~\hbox{for}~ \lambda=1$,
$J_{_{\lambda,k}}=[(\lambda-1)2^{2t-k},
\lambda2^{2t-k}-1] ~\hbox{for}~ \lambda \in [2,2^{k-t-1}-1]$,
$J_{_{\lambda,k}}=[(\lambda-1)2^{2t-k},2^{t}-5]~\hbox{for}~ \lambda=2^{k-t-1}$.
For given $k$, we can define $\lambda \in [1, 2^{k-t-1}]$ as the
{\it identity tag} of the subinterval
$J_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$. Fix
$\lambda$, if $x=2^{t+1}+2^{t}+1+2l$ with $l\in J_{_{\lambda,k}}$,
it follows that $$y_{_{x,k}}=2^kx-(2^{k-t}+2^{k-t-1}+\lambda-1)n.$$
As so, we can further verify $y_{_{x,k}}-x>0$ and
$n-y_{_{x,k}}-x>0$.
{\bf Case 3.4.1:} Firstly, we show $y_{_{x,k}}-x>0$
\begin{eqnarray*}
y_{_{x,k}}-x &=&(2^k-1)x-(2^{k-t}+2^{k-t-1}+\lambda-1)n\\
&=&(2^k-1)(2^{t+1}+2^{t}+1+2l)-(2^{k-t}+2^{k-t-1}+\lambda-1)(2^{2t+1}+1)\\
&=&(2^k-1)(1+2l)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(\lambda-1)(2^{2t+1}+1).
\end{eqnarray*}
When $\lambda=1$, notice that $l\in J_{_{\lambda,k}}=[1,2^{2t-k}-1]$
and $k\geq t+2$, we get that
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)(1+2l)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}\\
&\geq&(2^k-1)(1+2\cdot 1)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}\\
&=& (3-2^{-t}-2^{-t-1})\cdot 2^k-2^{t+1}-2^{t}-3\\
&\geq& (3-2^{-t}-2^{-t-1})\cdot 2^{t+2}-2^{t+1}-2^{t}-3\\
&=&3\cdot 2^{t+2}-2^{t+1}-2^{t}-9>0.
\end{eqnarray*}
For given $\lambda \in [2, 2^{k-t-1}]$, when $l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{2t-k},l_{_{\lambda,e}}]$,
we have a similar derivation process:
\begin{eqnarray*}
&&y_{_{x,k}}-x\\
&=&(2^k-1)(1+2l)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(\lambda-1)(2^{2t+1}+1)\\
&\geq&(2^k-1)(1+2\cdot l_{_{\lambda,b}})-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(\lambda-1)(2^{2t+1}+1)\\
&=& 2^{2t+1-k}+2^k-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(2^{2t-k+1}+1)\lambda\\
&\geq& 2^{2t+1-k}+2^k-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(2^{2t-k+1}+1)\cdot2^{k-t-1}\\
&=& 2^{2t+1-k}+2^k(1-2^{1-t})-2^{t+2}\\
&\geq& 2^{2t+1-(t+2)}+2^{t+2}(1-2^{1-t})-2^{t+2}(\hbox{see Lemma 0.1})\\
&=& 2^{t-1}-8>0.
\end{eqnarray*}
{\bf Case 3.4.2:} Secondly, we show $n-y_{_{x,k}}-x>0$
\begin{eqnarray*}
n-y_{_{x,k}}-x &=&(2^{k-t}+2^{k-t-1}+\lambda)n-(2^k+1)x\\
&=&
(2^{k-t}+2^{k-t-1}+\lambda)(2^{2t+1}+1)-(2^k+1)(2^{t+1}+2^{t}+1+2l).
\end{eqnarray*}
For given $\lambda \in [1, 2^{k-t-1}-1]$, let $l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}=\lambda2^{2t-k}-1]$.
Then one can check that
\begin{eqnarray*}
&&n-y_{_{x,k}}-x\\
&\geq&(2^{k-t}+2^{k-t-1}+\lambda)(2^{2t+1}+1)-(2^k+1)(2^{t+1}+2^{t}+1+2\cdot
l_{_{\lambda,e}})\\
&=&2^{k-t}+2^{k-t-1}-2^{t+1}-2^{t}+2^{k}+1-(2^{2t+1-k}-1)\lambda\\
&\geq&2^{k-t}+2^{k-t-1}-2^{t+1}-2^{t}+2^{k}+1-(2^{2t+1-k}-1)(2^{k-t-1}-1)\\
&=& 2^{2t+1-k}+2^k(1+2^{1-t})-2^{t+2}\\
&\geq& 2^{2t+1-(t+2)}+2^{t+2}(1+2^{1-t})-2^{t+2}(\hbox{see Lemma 0.1})\\
&=& 2^{t-1}+8>0.
\end{eqnarray*}
When $\lambda= 2^{k-t-1}$, since $l\in
I_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{2t-k},l_{_{\lambda,e}}=2^{t}-5]$,
it is easy to deduce that
\begin{eqnarray*}
& &n-y_{_{x,k}}-x\\
&=&(2^{k-t}+2^{k-t-1}+2^{k-t-1})(2^{2t+1}+1)-(2^k+1)(2^{t+1}+2^{t}+1+2l)\\
&\geq&(2^{k-t}+2^{k-t-1}+2^{k-t-1})(2^{2t+1}+1)-(2^k+1)(2^{t+1}+2^{t}+1+2l_{_{\lambda,e}})\\
&=&2^k(2^{1-t}+9)-2^{t+2}+9\\
&\geq&2^{t+2}(2^{1-t}+9)-2^{t+2}+9\\
&=& 2^{t+5}+17>0.
\end{eqnarray*}
(3.5): For each $k=2t-2, 2t-1$, similar to (3.3) and (3.4), we
divide the value range $J=[1, 2^{t-1}-5]$ of $l$ into
$2^{k-1-t}-2^{k+2-2t}$ subintervals as follows:
$J_{_{\lambda,k}}=[1,\lambda2^{2t-k}-1]~\hbox{for}~ \lambda=1$,
$J_{_{\lambda,k}}=[(\lambda-1)2^{2t-k},
\lambda2^{2t-k}-1] ~\hbox{for}~ \lambda \in [2,2^{k-1-t}-2^{k+2-2t}]$.
Let $\lambda \in [1,2^{k-1-t}-2^{k+2-2t}]$, we can define
$\lambda$ as the {\it identity tag} of the subinterval
$J_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$.
For $x=2^{t+1}+2^{t}+1+2l$ with $l\in J_{_{\lambda,k}}$, one can
check that $$y_{_{x,k}}=2^kx-(2^{k-t}+2^{k-t-1}+\lambda-1)n,$$
then we split into following two cases to deduce the desired result.
{\bf Case 3.5.1:} Firstly, we show $y_{_{x,k}}-x>0$
\begin{eqnarray*}
y_{_{x,k}}-x &=&(2^k-1)x-(2^{k-t}+2^{k-t-1}+\lambda-1)n\\
&=&(2^k-1)(2^{t+1}+2^{t}+1+2l)-(2^{k-t}+2^{k-t-1}+\lambda-1)(2^{2t+1}+1)\\
&=&(2^k-1)(1+2l)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(\lambda-1)(2^{2t+1}+1).
\end{eqnarray*}
If $\lambda=1$, it then follows from $l\geq1$ and $k\geq2t-2$ that
\begin{eqnarray*}
y_{_{x,k}}-x &=&(2^k-1)(1+2l)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}\\
&\geq&(2^k-1)(1+2\cdot1)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}\\
&=&(3-2^{-t}-2^{-t-1})\cdot 2^k-2^{t+1}-2^{t}-3\\
&\geq&(3-2^{-t}-2^{-t-1})\cdot\!\! 2^{2t-2}-\!\! 2^{t+1}-2^{t}-3(\hbox{see Lemma 0.1})\\
&=&3\cdot 2^{2t-2}-2^{t+1}-2^{t}-2^{t-2}-2^{t-3}-3>0.
\end{eqnarray*}
For given $\lambda \in [2, 2^{k-t-1}-2^{k+2-2t}]$, notice that
$l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{2t-k},l_{_{\lambda,e}}]$,
thus we have
\begin{eqnarray*}
& &y_{_{x,k}}-x\\
&=&(2^k-1)(1+2l)-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(\lambda-1)(2^{2t+1}+1)\\
&\geq&(2^k-1)(1+2\cdot l_{_{\lambda,b}})-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(\lambda-1)(2^{2t+1}+1)\\
&=& 2^{2t+1-k}+2^k-2^{t+1}-2^{t}-2^{k-t}-2^{k-t-1}-(2^{2t-k+1}+1)\lambda\\
&\geq& \!\! 2^{2t+1-k}+2^k-2^{t+1}-2^{t}-2^{k-t}\!-\!2^{k-t-1}\!-\!(2^{2t-k+1}+1)(2^{k-t-1}-2^{k+2-2t})\\
&=& 2^{2t+1-k}+2^k(1+ 2^{2-2t}-2^{1-t})-2^{t+2}+8\\
&\geq& 2^{2t+1- (2t-2)}\!\!+\!\! 2^{2t-2}(1+ 2^{2-2t}-2^{1-t})\!\! -2^{t+2}+8(\hbox{see Lemma 0.1})\\
&=& 2^{2t-2}-2^{t+2}-2^{t-1}+17>0.
\end{eqnarray*}
{\bf Case 3.5.2:} Secondly, we show $n-y_{_{x,k}}-x>0$
\begin{eqnarray*}
n-y_{_{x,k}}-x &=&(2^{k-t}+2^{k-t-1}+\lambda)n-(2^k+1)x\\
&=&(2^{k-t}+2^{k-t-1}+\lambda)(2^{2t+1}+1)\!\!-(2^k+1)(2^{t+1}+2^{t}+1+2l).
\end{eqnarray*}
For given $\lambda \in [1, 2^{k-t-1}-2^{k+2-2t}]$, since $l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}=\lambda2^{2t-k}-1]$,
thus one can deduce that
\begin{eqnarray*}
&&n-y_{_{x,k}}-x\\
&\geq&(2^{k-t}+2^{k-t-1}+\lambda)(2^{2t+1}+1)-(2^k+1)(2^{t+1}+2^{t}+1+2l_{_{\lambda,e}})\\
&=&2^{k-t}+2^{k-t-1}-2^{t+1}-2^{t}+2^{k}+1-(2^{2t+1-k}-1)\lambda\\
&\geq&2^{k-t}+2^{k-t-1}-2^{t+1}-2^{t}+2^{k}+1-(2^{2t+1-k}-1)(2^{k-t-1}-2^{k+2-2t})\\
&=& 2^k(1+2^{1-t}-2^{2-2t})-2^{t+2}+9\\
&\geq& 2^{2t-2}(1+2^{1-t}-2^{2-2t})-2^{t+2}+9\\
&=& 2^{2t-2}+2^{t-1}-2^{t+2}+8>0.
\end{eqnarray*}
(3.6): When $k=2t$, according to $x=2^{t+1}+2^{t}+1+2l$ with $l\in
J=[1, 2^{t-1}-5]$, we have
\begin{eqnarray*}
y_{_{x,k}}&=&2^kx-(2^{k-t}+l)n\\
&=&2^{2t}(2^{t+1}+2^{t}+1+2l)-(2^{t}+l)(2^{2t+1}+1)\\
&=&2^{2t}-2^{t}-2^{t-1}-l,
\end{eqnarray*}
thus, one can easily derive from $l\in J=[1, 2^{t-1}-5]$ that
\begin{eqnarray*}
y_{_{x,k}}-x&=&2^{2t}-2^{t}-2^{t-1}-l-(2^{t+1}+2^{t}+1+2l)\\
&=&2^{2t}-2^{t+2}-2^{t-1}-1-3l\\
&\geq&2^{2t}-2^{t+2}-2^{t-1}-1-3(2^{t-1}-5)\\
&=& 2^{2t}-2^{t+2}-2^{t+1}+14>0,\\
& &\\
n-y_{_{x,k}}-x&=&(2^{2t+1}+1)-(2^{2t}-2^{t}-2^{t-1}-l)-(2^{t+1}+2^{t}+1+2l)\\
&=&2^{2t}-2^{t+1}+2^{t-1}-l\\
&\geq&2^{2t}-2^{t+1}+2^{t-1}-(2^{t-1}-5)\\
&=& 2^{2t}-2^{t+1}+5>0.
\end{eqnarray*}
Concluding the previous six cases (3.1)-(3.6), we then conclude that
$y_{_{x,k}}-x\geq0$ and $n-y_{_{x,k}}-x\geq0$ for
$2^{t+1}+2^{t}+3\leq x\leq 2^{t+2}-9$ and each $k\in [0,m-1=2t]$,
which implies that (3) holds.\\
(4) It is easy to check that:
$(2^{t+1}-1)2^{3t+1}=(2^{t+1}-1)2^{t}\cdot
2^{2t+1}=(n-1-2^t)(n-1)\equiv 2^{t}+1$;
$(2^{t+1}+1)2^{t}=2^{m}+2^{t}\equiv 2^{t}-1$;
$(2^{t+1}+2^{t}-1)2^{3t+2}=(3n-3-2^{t+1})(n-1)\equiv 2^{t+1}+3$;
$(2^{t+1}+2^{t}+1)2^{t+1}=2^{m+1}+2^{m}+2^{t+1}\equiv 2^{t+1}-3$;
If $i=1,3,5 \hbox{~or~}7$, then
$(2^{t+2}-i)2^{3t}=(2^{t+2}-i)2^{t-1}\cdot 2^{2t+1}=(n-1-i\cdot
2^{t-1})(n-1)\equiv i \cdot 2^{t-1}+1$;
$ (2^{t+2}+i)2^{t-1} =2^{2t+1}+i\cdot2^{t-1}\equiv i\cdot
2^{t-1}-1$.
Combining the definition of a cyclotomic coset, the above congruence
expressions imply that there exists an odd integer $y\in[1,x-1]$
satisfying $y\in C_{x}$ for each $x$ in (4), hence $x$ is not a
coset leader, (4) follows.\end{proof}
\subsection*{Appendix B: The proof of Lemma 3.2}
\begin{proof} From $\delta_{1}=\frac{n}{3}=2^{2t}-2^{2t-1}+\cdots+4-2+1$,
$\delta_{2}=\frac{n-3}{6}=2^{2t-2}+2^{2t-4}+\cdots+4+1$, we can
obviously know $\delta_{1}, \delta_{2}, \delta_{3}, \delta_{4},
\delta_{5}$ are odd integers. It is easy to derive
$C_{\delta_{1}}=\{\delta_{1}, 2\delta_{1}\}$,
which implies that $|C_{\delta_{1}}|=2$ and $\delta_{1}$ is a coset leader.
We then show that $\delta_{2}$, $\delta_{3}$, $\delta_{4}$ and
$\delta_{5}$ are all also coset leaders.
{\it Step 1:} We show $y_{_{\delta_2,k}}-\delta_2\geq 0$ and
$n-y_{_{\delta_2,k}}-\delta_2\geq 0$ in three cases:
(1.1): If $k=0,1,2$, it is clear that
$y_{_{\delta_2,k}}=2^k\delta_2\geq \delta_2$,
$n-y_{_{\delta_2,k}}-\delta_2=n-2^k\delta_2-\delta_2=(5-2^k)\delta_2+3
>0$.
(1.2): If $k=3,5,\cdots,2t-1$, we then get that
\begin{eqnarray*}
y_{_{\delta_2,k}}&=&2^k\delta_2-\frac{2^{k-1}-1}{3}n=2\delta_2-2^{k-1}+1,\\
y_{_{\delta_2,k}}-\delta_2&=&\delta_2-2^{k-1}+1
\geq\delta_2-2^{(2t-1)-1}+1
=\delta_2-2^{2t-2}+1>0,\\
n-y_{_{\delta_2,k}}-\delta_2&=&3\delta_2+2^{k-1}+2>0.
\end{eqnarray*}
(1.3): If $k=4,6,\cdots,2t$, we have then
\begin{eqnarray*}
y_{_{\delta_2,k}}&=&2^k\delta_2-\frac{2^{k-1}-2}{3}n=4\delta_2-2^{k-1}+2,\\
y_{_{\delta_2,k}}-\delta_2&=&3\delta_2-2^{k-1}+2
\geq3\delta_2-2^{2t-1}+2>0,\\
n-y_{_{\delta_2,k}}-\delta_2&=&\delta_2+2^{k-1}+1>0.
\end{eqnarray*}
From the three cases above, we then can conclude $\delta_2$ is a
coset leader.
{\it Step 2:} Now, we show $y_{_{\delta_3,k}}-\delta_3\geq 0$ and
$n-y_{_{\delta_3,k}}-\delta_3\geq 0$.
(2.1): If $k=0,1,2$, it is clear that
$y_{_{\delta_3,k}}=2^k\delta_3\geq\delta_3$, then
$n-y_{_{\delta_3,k}}-\delta_3=n-(2^k+1)\delta_3\geq
\frac{n+15}{6}+10>0$.
(2.2): If $k=3,5,\cdots,2t-3$, then we can obtain that
\begin{eqnarray*}
y_{_{\delta_3,k}}&=&2^k(\delta_2-2)-\frac{2^{k-1}-1}{3}n=2\delta_2-2^{k+1}-2^{k-1}+1,\\
y_{_{\delta_3,k}}-\delta_3&=&\delta_2-2^{k+1}-2^{k-1}+3\\
&\geq& \delta_2-2^{(2t-3)+1}-2^{(2t-3)-1}+3=
\delta_2-2^{2t-2}-2^{2t-4}+3>0,\\
n-y_{_{\delta_3,k}}-\delta_3&=&3\delta_2+2^{k+1}+2^{k-1}+4>0.
\end{eqnarray*}
(2.3): If $k=4,6,\cdots,2t-2$, we can deduce that
\begin{eqnarray*}
y_{_{\delta_3,k}}&=&2^k(\delta_2-2)-\frac{2^{k-1}-2}{3}n=4\delta_2-2^{k+1}-2^{k-1}+2,\\
y_{_{\delta_3,k}}-\delta_3&=&3\delta_2-2^{k+1}-2^{k-1}+4\\
&\geq&\!\!3\delta_2-2^{(2t-2)+1}-2^{(2t-2)-1}+4=3\delta_2-2^{2t-1}-2^{2t-3}+4>0,\\
n-y_{_{\delta_3,k}}-\delta_3&=&\delta_2+2^{k+1}+2^{k-1}+3>0.
\end{eqnarray*}
(2.4): If $k=2t-1$, it is easy to know
\begin{eqnarray*}
y_{_{\delta_3,k}}&=&2^k(\delta_2-2)-(\frac{2^{k-1}-1}{3}-1)n=n+2\delta_2-2^{k+1}-2^{k-1}+1,\\
y_{_{\delta_3,k}}-\delta_3&=&n+\delta_2-2^{k+1}-2^{k-1}+3=\delta_2+2^{2t}-2^{2t-2}+4>0,\\
n-y_{_{\delta_3,k}}-\delta_3&=&2^{k+1}+2^{k-1}+1-3\delta_2=2^{2t-2}+2>0.
\end{eqnarray*}
(2.5): If $k=2t$, then we can check that
\begin{eqnarray*}
y_{_{\delta_3,k}}&=&2^k(\delta_2-2)-(\frac{2^{k-1}-2}{3}-1)n=4\delta_2-2^{k-1}+3,\\
y_{_{\delta_3,k}}-\delta_3&=&3\delta_2-2^{k-1}+5=2^{2t-1}+4>0,\\
n-y_{_{\delta_3,k}}-\delta_3&=&\delta_2+2^{k-1}+2>0.
\end{eqnarray*}
Summarizing the previous five cases, we know
$\delta_3$ is a coset leader.\\
{\it Step 3:} We show $y_{_{\delta_4,k}}-\delta_4\geq 0$ and
$n-y_{_{\delta_4,k}}-\delta_4\geq 0$ in seven cases:
(3.1): If $k=0,1,2$, it is clear that
$y_{_{\delta_4,k}}=2^k\delta_4\geq \delta_4$, then
$n-y_{_{\delta_4,k}}=n-2^k(\delta_2-8)=(6-2^k)\delta_2+2^{k+3}+3
>\delta_4$.
(3.2): If $k=3,5,\cdots,2t-5$, we have then
\begin{eqnarray*}
y_{_{\delta_4,k}}&=&2^k(\delta_2-8)-\frac{2^{k-1}-1}{3}n=2\delta_2-2^{k+3}-2^{k-1}+1,\\
y_{_{\delta_4,k}}-\delta_4&=&\delta_2-2^{k+3}-2^{k-1}+9\\
&\geq&\delta_2-2^{(2t-5)+3}-2^{(2t-5)-1}+9=\delta_2-2^{2t-2}-2^{2t-6}+9>0,\\
n-y_{_{\delta_4,k}}-\delta_4&=&3\delta_2+2^{k+3}+2^{k-1}+10>0.
\end{eqnarray*}
(3.3): If $k=4,6,\cdots,2t-4$, we can check that
\begin{eqnarray*}
y_{_{\delta_4,k}}&=&2^k(\delta_2-8)-\frac{2^{k-1}-2}{3}n=4\delta_2-2^{k+3}-2^{k-1}+2,\\
y_{_{\delta_4,k}}-\delta_4&=&3\delta_2-2^{k+3}-2^{k-1}+10\\
&\geq&3\delta_2-2^{(2t-4)+3}-2^{(2t-4)-1}+10=2^{2t-1}-2^{2t-5}+9>0,\\
n-y_{_{\delta_4,k}}-\delta_4&=&\delta_2+2^{k+3}+2^{k-1}+9>0.
\end{eqnarray*}
(3.4): If $k=2t-3$, then we deduce that
\begin{eqnarray*}
y_{_{\delta_4,k}}&=&2^k(\delta_2-8)-(\frac{2^{k-1}-1}{3}-1)n=5\delta_2-2^{k-1}+3,\\
y_{_{\delta_4,k}}-\delta_4&=&4\delta_2-2^{k-1}+11=\delta_2+2^{2t}-2^{2t-4}+10>0,\\
n-y_{_{\delta_4,k}}-\delta_4&=&2^{k-1}+8>0.
\end{eqnarray*}
(3.5): If $k=2t-2$, we obtain that
\begin{eqnarray*}
y_{_{\delta_4,k}}&=&2^k(\delta_2-8)-(\frac{2^{k-1}-2}{3}-1)n=4\delta_2-2^{k-1}+3,\\
y_{_{\delta_4,k}}-\delta_4&=&3\delta_2-2^{k-1}+11=2^{2t}-2^{2t-3}+10>0,\\
n-y_{_{\delta_4,k}}-\delta_4&=&\delta_2+2^{k-1}+8>0.
\end{eqnarray*}
(3.6): If $k=2t-1$, then we get
\begin{eqnarray*}
y_{_{\delta_4,k}}&=&2^k(\delta_2-8)-(\frac{2^{k-1}-1}{3}-2)n=2\delta_2-2^{k-1}+3,\\
y_{_{\delta_4,k}}-\delta_4&=&\delta_2-2^{k-1}+11=\delta_2-2^{2t-2}+11>0,\\
n-y_{_{\delta_4,k}}-\delta_4&=&3\delta_2+2^{k-1}+8>0.
\end{eqnarray*}
(3.7): If $k=2t$, it is easy to derive that
\begin{eqnarray*}
y_{_{\delta_4,k}}&=&2^k(\delta_2-8)-(\frac{2^{k-1}-2}{3}-4)n=4\delta_2-2^{k-1}+6,\\
y_{_{\delta_4,k}}-\delta_4&=&3\delta_2-2^{k-1}+14=2^{2t-1}+13>0,\\
n-y_{_{\delta_4,k}}-\delta_4&=&\delta_2+2^{k-1}+5>0.
\end{eqnarray*}
It follows from the seven cases above that $\delta_4$ is a coset leader.\\
{\it Step 4:} We show $y_{_{\delta_5,k}}-\delta_5\geq 0$ and
$n-y_{_{\delta_5,k}}-\delta_5\geq 0$ in seven cases:
(4.1): If $k=0,1,2$, it is clear that
$y_{_{\delta_5,k}}=2^k\delta_5\geq \delta_5$, then
$n-y_{_{\delta_5,k}}=n-2^k(\delta_2-10)=(6-2^k)\delta_2+2^{k+3}+2^{k}+3>\delta_5$.
(4.2): If $k=3,5,\cdots,2t-5$, then we can check that
\begin{eqnarray*}
y_{_{\delta_5,k}}&=&2^k(\delta_2-10)-\frac{2^{k-1}-1}{3}n=2\delta_2-2^{k+3}-2^{k+1}-2^{k-1}+1,\\
y_{_{\delta_5,k}}-\delta_5&=&\delta_2-2^{k+3}-2^{k+1}-2^{k-1}+11\\
&\geq&\delta_2-2^{(2t-5)+3}-2^{(2t-5)+1}-2^{(2t-5)-1}+11\\
&=&\delta_2-2^{2t-2}-2^{2t-4}-2^{2t-6}+11>0,\\
n-y_{_{\delta_5,k}}-\delta_5&=&3\delta_2+2^{k+3}+2^{k+1}+2^{k-1}+12
>0.
\end{eqnarray*}
(4.3): If $k=4,6,\cdots,2t-4$, then we deduce
\begin{eqnarray*}
y_{_{\delta_5,k}}&=&2^k(\delta_2-10)-\frac{2^{k-1}-2}{3}n=4\delta_2-2^{k+3}-2^{k+1}-2^{k-1}+2,\\
y_{_{\delta_5,k}}-\delta_5&=&3\delta_2-2^{k+3}-2^{k+1}-2^{k-1}+12\\
&\geq&3\delta_2-2^{(2t-4)+3}-2^{(2t-4)+1}-2^{(2t-4)-1}+12\\
&=&2^{2t-1}-2^{2t-3}-2^{2t-5}+11>0,\\
n-y_{_{\delta_5,k}}-\delta_5&=&\delta_2+2^{k+3}+2^{k+1}+2^{k-1}+11>0.
\end{eqnarray*}
(4.4): If $k=2t-3$, it is not difficult to get that
\begin{eqnarray*}
y_{_{\delta_5,k}}&=&2^k(\delta_2-10)-(\frac{2^{k-1}-1}{3}-1)n=5\delta_2-2^{k+1}-2^{k-1}+3,\\
y_{_{\delta_5,k}}-\delta_5&=&4\delta_2-2^{k+1}-2^{k-1}+13\\
&=&\delta_2+2^{2t}-2^{2t-2}-2^{2t-4}+12>0,\\
n-y_{_{\delta_5,k}}-\delta_5&=&2^{k+1}+2^{k-1}+10>0.
\end{eqnarray*}
(4.5): If $k=2t-2$, we can easily obtain
\begin{eqnarray*}
y_{_{\delta_5,k}}&=&2^k(\delta_2-10)-(\frac{2^{k-1}-2}{3}-1)n=4\delta_2-2^{k+1}-2^{k-1}+3,\\
y_{_{\delta_5,k}}-\delta_5&=&3\delta_2-2^{k+1}-2^{k-1}+13=2^{2t-1}-2^{2t-3}+12>0,\\
n-y_{_{\delta_5,k}}-\delta_5&=&\delta_2+2^{k+1}+2^{k-1}+10>0.
\end{eqnarray*}
(4.6): If $k=2t-1$, then we get
\begin{eqnarray*}
y_{_{\delta_5,k}}&=&2^k(\delta_2-10)-(\frac{2^{k-1}-1}{3}-3)n=5\delta_2-2^{k-1}+5,\\
y_{_{\delta_5,k}}-\delta_5&=&4\delta_2-2^{k-1}+15=\delta_2+2^{2t}-2^{2t-2}+14>0,\\
n-y_{_{\delta_5,k}}-\delta_5&=&2^{k-1}+8>0.
\end{eqnarray*}
(4.7): If $k=2t$, then we deduce
\begin{eqnarray*}
y_{_{\delta_5,k}}&=&2^k(\delta_2-10)-(\frac{2^{k-1}-2}{3}-5)n=4\delta_2-2^{k-1}+7,\\
y_{_{\delta_5,k}}-\delta_5&=&3\delta_2-2^{k-1}+17=2^{2t-1}+16>0,\\
n-y_{_{\delta_5,k}}-\delta_5&=&\delta_2+2^{k-1}+6>0.
\end{eqnarray*}
Then we can conclude that $\delta_5$ is a coset leader from the
previous seven cases.
\end{proof}
\subsection*{Appendix C: The proof of Theorem 3.7}
\begin{proof}(1) Let $T_{\delta}$ be the defining set of
$\mathcal{C}(n,2,\delta,1)$ and $T_{\delta}=\bigcup\limits_{i\in
S_{\delta}}C_{i}$, where
$S_{\delta}=\{x|x~\hbox{is a coset leader,
} C_x\subseteq T_{\delta}\}$, then
$\mathcal{C}(n,2,\delta,1)$ has
dimension $k=n-|T_{\delta}|=n-\sum\limits_{i\in S_{\delta}}|C_{i}|$.
(i): When $2^{t+1}+3\leq \delta \leq 2^{t+1}+2^{t}-3$, from Theorem
3.1, we have $$S_{\delta}=\{x|x~\hbox{is odd and}~x\in
[1,\delta-1]\setminus\{2^{t+1}\pm1\}\},$$ thus
$|S_{\delta}|=\frac{\delta-1}{2}-2$. According to Lemma 3.6, all
cyclotomic cosets in $T_{\delta}$ have cardinality $2m$, it then
follows that $$k=n-\sum\limits_{i\in S_{\delta}}|C_{i}|=n-2m\cdot
(\frac{\delta-1}{2}-2)=n-m\delta+5m.$$ It is obvious that there
exist $\delta-1$ consecutive integers, according to the BCH bound,
the minimum distance $d\geq \delta$ .
(ii): Similar to (i),
when $2^{t+1}+2^{t}+3\leq \delta\leq 2^{t+2}-9$, from Theorem 3.1 and Lemma 3.6,
$$S_{\delta}=\{x|x~\hbox{is odd and}~x\in
[1,\delta-1]\setminus\{2^{t+1}\pm1,2^{t+1}+2^{t}\pm1\}\},$$ then we
have $d\geq \delta$ and $k=n-2m\cdot
(\frac{\delta-1}{2}-4)=n-m\delta+9m.$
(iii): Similar to (i),
when $2^{t+2}-7\leq\delta\leq 2^{t+2}+9$, from Theorem 3.1 and Lemma 3.6,
$$S_{\delta}=\{x|x~\hbox{is odd and}~x\in
[1,2^{t+2}-9]\setminus\{2^{t+1}\pm1,2^{t+1}+2^{t}\pm1\}\},$$ then
we have $d\geq 2^{t+2}+9$ and $k=n-2m\cdot
(2^{t+1}-8)=n-2^{t+2}m+16m.$
(iv): When $\delta_{i+1}+2\leq\delta\leq \delta_{i}(i=1,2,3,4)$, we
can infer from Theorem 3.5 that
thus $T_{\delta}=\bigcup\limits_{i\in
S_{\delta}}C_{i}=\{1,2,\cdots,n-1\}\setminus\bigcup
\limits_{j=1}^{i}C_{\delta_j}$.
Since $|C_{\delta_{1}}|=2$ from the proof of Lemma 3.2, combining
Lemma 3.6, every $C_{\delta_{i}}(i=2,3,4,5)$ has cardinality $2m$,
it then follows that
$$k=n-|T_{\delta}|=n-[n-1-2m (i-1)-2]=2m (i-1)+3.$$
On the other hand, there exist $\delta_{i}-1$ consecutive integers
in $T_{\delta}$, the minimum distance $d\geq \delta_{i}$.
(v): When $\delta_{1}+2\leq\delta\leq n$, it is easy to infer from
Theorem 3.5 that $T_{\delta}=\bigcup\limits_{i\in
S_{\delta}}C_{i}=\{1,2,\cdots,n-1\}$, then
$k=n-|T_{\delta}|=n-(n-1)=1.$ Obviously, the minimum distance
$d=n$ by the Singleton bound.
(2) On the basis of the proof of (1), (2) can be
easily given.
\end{proof}
\subsection*{Appendix D: The proof of Theorem 4.1}
\begin{proof} Similar to the proof of Theorem 3.1, it suffices
to prove items (2)-(5).
(2) Since $x$ is odd, $x$ can be denoted by
$$x=2^{2t+1}+1+2l, \hbox{where}~l \in I=[1, 2^{2t}-3].$$
To verify (2), we will first determine $y_{_{x,k}}$, then
show $y_{_{x,k}}-x \geq 0$ and $n-y_{_{x,k}}-x \geq 0$ according to
different $k$.
(2.1): If $k=0,1,2,\cdots,2t$, note that $2^{2t+1}+3\leq x\leq
2^{2t+2}-5$, then $x\leq 2^kx <n$, it then follows that
$y_{_{x,k}}=2^kx\geq x$, and
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&n-(2^k+1)x\\
&\geq&n-(2^{2t}+1)x\\
&\geq&n-(2^{2t}+1)(2^{2t+2}-5)=2^{2t}+6>0.
\end{eqnarray*}
(2.2): If $k=2t+1$, we check that $y_{_{x,k}}=2^kx-n$, it is easy
to obtain
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-n\\
&\geq &(2^k-1)(2^{2t+1}+3)-n=2^{2t+2}-4>0,\\
n-y_{_{x,k}}-x&=& 2n-(2^k+1)x\\
&\geq& 2n-(2^k+1)(2^{2t+2}-5)= 3\cdot2^{2t+1}+7>0.
\end{eqnarray*}
(2.3): For each $k=2t+2, 2t+3,\cdots,4t-1$, it is a little difficult
to determine $y_{_{x,k}}$ and then check $y_{_{x,k}}-x \geq 0$
along with $n-y_{_{x,k}}-x \geq 0$. To achieve this, we divide
$I=[1, 2^{2t}-3]$ into $2^{k-2t-1}$ subintervals as follows:
$I_{_{\lambda,k}}=[1,\lambda2^{4t+1-k}-1]$ for $\lambda=1$,
$I_{_{\lambda,k}}=[(\lambda-1)2^{4t+1-k},
\lambda2^{4t+1-k}-1]$ for $2\leq \lambda \leq 2^{k-2t-1}-1$,
$I_{_{\lambda,k}}=[(\lambda-1)2^{4t+1-k}, 2^{2t}-3]$ for $\lambda=2^{k-2t-1}$.
Fix $k$, for each $\lambda \in [1, 2^{k-2t-1}]$, we can define
$\lambda$ as the {\it identity tag} of the subinterval
$I_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$.
For given $\lambda$, $x=2^{2t+1}+1+2l$ with $l\in
I_{_{\lambda,k}}$, we can easily calculate that
$y_{_{x,k}}=2^kx-(2^{k-2t-1}+\lambda-1)n$, then we split into
following two subcases.
{\bf Case 2.3.1:} Firstly, we show $y_{_{x,k}}-x>0$. For general
$\lambda$, we have
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-(2^{k-2t-1}+\lambda-1)n\\
&=&(2^k-1)(2^{2t+1}+1+2l)-(2^{k-2t-1}+\lambda-1)(2^{4t+2}+1)\\
&=&(2^k-1)(1+2l)-2^{2t+1}-2^{k-2t-1}-(\lambda-1)(2^{4t+2}+1).
\end{eqnarray*}
If $\lambda=1$, then $l\in I_{_{\lambda,k}}=[1,2^{4t+1-k}-1]$, it
follows from $k\geq 2t+2$ that
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)(1+2l)-2^{2t+1}-2^{k-2t-1}\\
&\geq&(2^k-1)(1+2\cdot1)-2^{2t+1}-2^{k-2t-1}\\
&=&(3-2^{-2t-1})\cdot2^k-2^{2t+1}-3\\
&\geq&(3-2^{-2t-1})\cdot2^{2t+2}-2^{2t+1}-3\\
&=&3\cdot2^{2t+2}-2^{2t+1}-5>0.
\end{eqnarray*}
If $\lambda\in[2,2^{k-2t-1}]$, since $l\in
I_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{4t+1-k},
l_{_{\lambda,e}}]$, we can similarly obtain that
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)(1+2l)-2^{2t+1}-2^{k-2t-1}-(\lambda-1)(2^{4t+2}+1)\\
&\geq&(2^k-1)(1+2\cdot l_{_{\lambda,b}})-2^{2t+1}-2^{k-2t-1}-(\lambda-1)(2^{4t+2}+1)\\
&=&2^{4t+2-k}+2^{k}-2^{2t+1}-2^{k-2t-1}-(2^{4t+2-k}+1)\lambda\\
&\geq&2^{4t+2-k}+2^{k}-2^{2t+1}-2^{k-2t-1}-(2^{4t+2-k}+1)\cdot2^{k-2t-1}\\
&=&2^{4t+2-k}+(1-2^{-2t})2^{k}-2^{2t+2}\\
&\geq&2^{4t+2-(2t+2)}+(1-2^{-2t})2^{2t+2}-2^{2t+2}(\hbox{see Lemma 0.1})\\
&=&2^{2t}-4>0.
\end{eqnarray*}
{\bf Case 2.3.2:} Secondly, we show $n-y_{_{x,k}}-x>0$. For general
$\lambda$, we have
\begin{eqnarray*}
n-y_{_{x,k}}-x&=& (2^{k-2t-1}+\lambda)(2^{4t+2}+1)-(2^k+1)(2^{2t+1}+1+2l)\\
& =&\lambda(2^{4t+2}+1)+2^{k-2t-1}-2^{2t+1}-(2^k+1)(1+2l).
\end{eqnarray*}
If $\lambda\in[1,2^{k-2t-1}-1]$ and $l\in [l_{_{\lambda,b}},
l_{_{\lambda,e}}=\lambda2^{4t+1-k}-1]$, we can derive from $k\geq
2t+2$ that
\begin{eqnarray*}
n-y_{_{x,k}}-x
&\geq&\lambda(2^{4t+2}+1)+2^{k-2t-1}-2^{2t+1}-(2^k+1)(1+2l_{_{\lambda,e}})\\
&=&2^{k-2t-1}+2^k-2^{2t+1}+1-2^{4t+2-k}\lambda\\
&\geq&2^{k-2t-1}+2^k-2^{2t+1}+1-2^{4t+2-k}(2^{k-2t-1}-1)\\
&=&2^{4t+2-k}+(1+2^{-2t})2^k-2^{2t+2}\\
&\geq&2^{4t+2-(2t+2)}+(1+2^{-2t})2^{2t+2}-2^{2t+2}(\hbox{see Lemma 0.1})\\
&=&2^{2t}+4>0.
\end{eqnarray*}
If $\lambda=2^{k-2t-1}$ and $l\in
I_{_{2^{k-2t-1},k}}=[l_{_{\lambda,b}}=(2^{k-2t-1}-1)2^{4t+1-k},l_{_{\lambda,e}}=
2^{2t}-3]$, then we similarly get that
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&2^{k-2t-1}(2^{4t+2}+1)+2^{k-2t-1}-2^{2t+1}-(2^k+1)(1+2l)\\
&\geq&2^{k-2t-1}(2^{4t+2}+1)+2^{k-2t-1}-2^{2t+1}-(2^k+1)(1+2l_{_{\lambda,e}})\\
&=&(5+2^{-2t})2^k-2^{2t+2}+5\\
&\geq&(5+2^{-2t})2^{2t+2}-2^{2t+2}+5=2^{2t+4}+9>0.
\end{eqnarray*}
(2.4): If $k=4t$, similarly, we partition $I=[1, 2^{2t}-3]$ into
$2^{2t-1}-1$ subintervals as follows.
$I_{_{\lambda,k}}=[1,1=2\lambda-1]$, where $\lambda=1$,
$I_{_{\lambda,k}}=[2(\lambda-1),
2\lambda-1]$, where $2\leq \lambda \leq 2^{2t-1}-1$.
Let $\lambda \in [1,2^{2t-1}-1]$, we can define $\lambda$ as the
{\it identity tag} of the subinterval
$I_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$.
For given $\lambda$, if $x=2^{2t+1}+1+2l$ with $l\in
I_{_{\lambda,k}}$, we can easily check that
$y_{_{x,k}}=2^kx-(2^{2t-1}+\lambda-1)n$, then we split into
following two subcases.
{\bf Case 2.4.1:} First, we show $y_{_{x,k}}-x>0$. For general
$\lambda$,
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^{4t}-1)x-(2^{2t-1}+\lambda-1)n\\
&=&(2^{4t}-1)(2^{2t+1}+1+2l)-(2^{2t-1}+\lambda-1)(2^{4t+2}+1)\\
&=&(2^{4t}-1)(1+2l)-2^{2t+1}-2^{2t-1}-(\lambda-1)(2^{4t+2}+1).
\end{eqnarray*}
If $\lambda=1$, we have $l\in I_{_{1,k}}=[1,1]$, that is $l=1$, we
can easily obtain that $$
y_{_{x,k}}-x=2^{4t}-2^{2t-1}+2^{2t+1}-3>0.$$
If $\lambda\in[2,2^{2t-1}-1]$ and $l\in
[l_{_{\lambda,b}}=2(\lambda-1),
2\lambda-1]$, we get
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^{4t}-1)(1+2l)-2^{2t+1}-2^{2t-1}-(\lambda-1)(2^{4t+2}+1)\\
&\geq&(2^{4t}-1)(1+2l_{_{\lambda,b}})-2^{2t+1}-2^{2t-1}-(\lambda-1)(2^{4t+2}+1)\\
&=&2^{4t}-2^{t+1}-2^{t-1}+4-5\lambda\\
&\geq&2^{4t}-2^{t+1}-2^{t-1}+4-5(2^{2t-1}-1)\\
&=&2^{4t}-5\cdot2^{2t}+9>0.
\end{eqnarray*}
{\bf Case 2.4.2:} Secondly, we show $n-y_{_{x,k}}-x>0$. For general
$\lambda$,
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&(2^{2t-1}+\lambda)(2^{4t+2}+1)-(2^{4t}+1)(2^{2t+1}+1+2l)\\
& =&\lambda(2^{4t+2}+1)+2^{2t-1}-2^{2t+1}-(2^{4t}+1)(1+2l)\\
&\geq&\lambda(2^{4t+2}+1)+2^{2t-1}-2^{2t+1}-(2^{4t}+1)[1+2(2\lambda-1)]\\
&=&2^{4t}+2^{2t-1}-2^{2t+1}+1-3\lambda\\
&\geq&2^{4t}+2^{2t-1}-2^{2t+1}+1-3(2^{2t-1}-1)\\
&=&2^{4t}-3\cdot 2^{2t}+4>0.
\end{eqnarray*}
(2.5): If $k=4t+1$, for given $l\in I=[1, 2^{2t}-3]$, we have
$y_{_{x,k}}=2^kx-(2^{2t}+l)n$, thus
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-(2^{2t}+l)n\\
&=&2^{4t+1}-2^{2t+1}-2^{2t}-1-3l\\
&\geq &2^{4t+1}-2^{2t+1}-2^{2t}-1-3(2^{2t}-3)\\
&=&2^{4t+1}-3\cdot2^{2t+1}+8>0,\\
n-y_{_{x,k}}-
&=& 2^{4t+1}+2^{2t}-2^{2t+1}-l\\
&\geq&2^{4t+1}+2^{2t}-2^{2t+1}-(2^{2t}-3)\\
&=& 2^{4t+1}-2^{2t+1}+3>0.
\end{eqnarray*}
Concluding the previous five cases (2.1)-(2.5), (2) holds.
(3) Since $x$ is odd and $2^{2t+2}+5\leq x\leq 2^{2t+2}+2^{2t}-3$,
then $x$ can be denoted as
$$x=2^{2t+2}+1+2l, \hbox{where}~l\in J=[2,2^{2t-1}-2].$$
To verify that (3) holds, it is necessary to show $y_{_{x,k}}-x \geq 0$ and
$n-y_{_{x,k}}-x \geq 0$. Similar to (2), we split into following
cases according to different $k$.
(3.1): If $k=0,1,2,\cdots,2t-1$, obviously, $x\leq 2^kx <n$, then
$y_{_{x,k}}=2^kx\geq x$, we get that
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&n-(2^k+1)x\\
&\geq&n-(2^{2t-1}+1)x\\
&\geq&n-(2^{2t-1}+1)(2^{2t+2}+2^{2t}-3)\\
&=&3\cdot2^{4t-1}-7\cdot2^{2t-1}+4>0.
\end{eqnarray*}
(3.2): If $k=2t,2t+1,2t+2$, we check that
$y_{_{x,k}}=2^kx-2^{k-2t}n$, thus
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-2^{k-2t}n\\
&=&2^k-2^{k-2t}-2^{2t+2}-1+(2^{k+1}-2)l\\
&\geq&2^k-2^{k-2t}-2^{2t+2}-1+(2^{k+1}-2)\cdot 2\\
&=&(5-2^{-2t})\cdot2^k-2^{2t+2}-5\\
&\geq&(5-2^{-2t})\cdot2^{2t}-2^{2t+2}-5\\
&=&2^{2t}-6>0,\\
n-y_{_{x,k}}-x&=&(2^{k-2t}+1)n-(2^k+1)(2^{2t+2}+1+2l)\\
&\geq&(2^{k-2t}+1)n-(2^k+1)[2^{2t+2}+1+2(2^{2t-1}-2)]\\
&=&2^{4t+2}+2^{2t+2}+2^{2t}-2-(2^{2t}+2^{-2t}-3)2^k\\
&\geq&2^{4t+2}+2^{2t+2}+2^{2t}-2-(2^{2t}+2^{-2t}-3)2^{2t+2}\\
&=&7\cdot2^{2t}+8>0.
\end{eqnarray*}
(3.3): For each $k=2t+3, 2t+4,\cdots,4t-1$, it is not easy to
determine $y_{_{x,k}}$ and check $y_{_{x,k}}-x \geq 0$ along
with $n-y_{_{x,k}}-x \geq 0$. To achieve this, we divide the value
range $J=[2,2^{2t-1}-2]$ of $l$ into $2^{k-2t-2}$ subintervals as
follows:
$J_{_{\lambda,k}}=[2,\lambda2^{4t+1-k}-1]$ for $\lambda=1$,
$J_{_{\lambda,k}}=[(\lambda-1)2^{4t+1-k},
\lambda2^{4t+1-k}-1]$ for $2\leq \lambda \leq 2^{k-2t-2}-1$
$J_{_{\lambda,k}}=[(\lambda-1)2^{4t+1-k}, 2^{2t-1}-2]$ for
$\lambda=2^{k-2t-2}$.
Let $\lambda \in [1, 2^{k-2t-2}]$, where $\lambda$ is called the
{\it identity tag} of subinterval
$I_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$.
For given $\lambda$, if $x=2^{2t+1}+1+2l$ with $l\in
J_{_{\lambda,k}}$, we can check
$y_{_{x,k}}=2^kx-(2^{k-2t}+\lambda-1)n$, then we split into
following two subcases:
{\bf Case 3.3.1:} Firstly, we show
$y_{_{x,k}}-x>0$:
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-(2^{k-2t}+\lambda-1)n\\
&=&(2^k-1)(2^{2t+2}+1+2l)-(2^{k-2t}+\lambda-1)(2^{4t+2}+1)\\
&=&(2^k-1)(1+2l)-2^{2t+2}-2^{k-2t}-(\lambda-1)(2^{4t+2}+1).
\end{eqnarray*}
If $\lambda=1$, from $l\in J_{_{1,k}}=[2, 2^{4t+1-k}-1]$ and $k\geq
2t+3$, we get
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)(1+2l)-2^{2t+2}-2^{k-2t}\\
&\geq&(2^k-1)(1+2\cdot2)-2^{2t+2}-2^{k-2t}\\
&=&(5-2^{-2t})\cdot2^k-2^{2t+2}-5\\
&\geq&(5-2^{-2t})\cdot2^{2t+3}-2^{2t+2}-5\\
&=&9\cdot2^{2t+2}-13>0.
\end{eqnarray*}
If $\lambda\in[2,2^{k-2t-2}]$, thus $l\in J_{_{\lambda,k}}=
[l_{_{\lambda,b}}=(\lambda-1)2^{4t+1-k}, l_{_{\lambda,e}}]$, we can
similarly deduce that
\begin{eqnarray*}
&&y_{_{x,k}}-x\\
&=&(2^k-1)(1+2l)-2^{2t+2}-2^{k-2t}-(\lambda-1)(2^{4t+2}+1)\\
&\geq&(2^k-1)(1+2l_{_{\lambda,b}})-2^{2t+2}-2^{k-2t}-(\lambda-1)(2^{4t+2}+1)\\
&=&2^{4t+2-k}+2^{k}-2^{2t+2}-2^{k-2t}-(2^{4t+2-k}+1)\lambda\\
&\geq&2^{4t+2-k}+2^{k}-2^{2t+2}-2^{k-2t}-(2^{4t+2-k}+1)2^{k-2t-2}\\
&=&2^{4t+2-k}+(1-2^{-2t}-2^{-2t-2})2^{k}-2^{2t+2}-2^{2t}\\
&\geq&2^{4t+2-(2t+3)}+\!\!(1-\!\!2^{-2t}-\!\!2^{-2t-2})2^{2t+3}-2^{2t+2}-2^{2t}(\hbox{see Lemma 0.1})\\
&=&7\cdot2^{2t-1}-10>0.
\end{eqnarray*}
{\bf Case 3.3.2:} Secondly, we show $n-y_{_{x,k}}-x>0$:
\begin{eqnarray*}
n-y_{_{x,k}}-x&=& (2^{k-2t}+\lambda)n-(2^k+1)x\\
&=& (2^{k-2t}+\lambda)(2^{4t+2}+1)-(2^k+1)(2^{2t+2}+1+2l)\\
& =&\lambda(2^{4t+2}+1)+2^{k-2t}-2^{2t+2}-(2^k+1)(1+2l).
\end{eqnarray*}
If $\lambda\in[1,2^{k-2t-2}-1]$, we have $l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}},
l_{_{\lambda,e}}=\lambda2^{4t+1-k}-1]$, then
\begin{eqnarray*}
&&n-y_{_{x,k}}-x\\
&\geq&\lambda(2^{4t+2}+1)+2^{k-2t}-2^{2t+2}-(2^k+1)(1+2l_{_{\lambda,e}})\\
&=&2^{k-2t}+2^k-2^{2t+2}+1-2^{4t+2-k}\lambda\\
&\geq&2^{k-2t}+2^k-2^{2t+2}+1-2^{4t+2-k}\cdot(2^{k-2t-2}-1)\\
&=&2^{4t+2-k}+(1+2^{-2t}+2^{-2t-2})2^k-2^{2t+2}-2^{2t}\\
&\geq&2^{4t+2-(2t+3)}+(1+2^{-2t}+2^{-2t-2})2^{2t+3}-2^{2t+2}-2^{2t}(\hbox{see Lemma 0.1})\\
&=&7\cdot2^{2t-1}-10>0;
\end{eqnarray*}
If $\lambda=2^{k-2t-2}$, we have $l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{4t+1-k},
l_{_{\lambda,e}}= 2^{2t-1}-2]$, it is easy
to obtain that
\begin{eqnarray*}
n-y_{_{x,k}}-x&
=&2^{k-2t-2}(2^{4t+2}+1)+2^{k-2t}-2^{2t+2}-(2^k+1)(1+2l)\\
&\geq&2^{k-2t-2}(2^{4t+2}+1)+2^{k-2t}-2^{2t+2}-(2^k+1)(1+2 l_{_{\lambda,e}})\\
&=&(3+2^{-2t}+2^{-2t-2})2^k-2^{2t+2}-2^{2t}+3\\
&\geq&(3+2^{-2t}+2^{-2t-2})2^{2t+3}-2^{2t+2}-2^{2t}+3\\
&=&19\cdot2^{2t}+13>0.
\end{eqnarray*}
(3.4): If $k=4t$, we divide $J=[2,2^{2t-1}-2]$ into the following
$2^{2t-2}-1$ subintervals:
$J_{_{\lambda,k}}=[2\lambda, 2\lambda+1]$, where $1\leq \lambda \leq 2^{2t-2}-2$,
$J_{_{\lambda =2^{2t-2}-1,k}}= [2^{2t-1}-2=2\lambda,2^{2t-1}-2=2\lambda]$.
Let $\lambda \in [1,2^{2t-2}-1]$, we can define $\lambda$ as the
{\it identity tag} of the subinterval $J_{_{\lambda,k}}$.
For $x=2^{2t+1}+1+2l$, if $l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}}=2\lambda,l_{_{\lambda,e}}]$, we
have $y_{_{x,k}}=2^kx-(2^{2t}+\lambda)n$, then we split into
following two subcases.
{\bf Case 3.4.1:} Firstly, we show $y_{_{x,k}}-x>0$:
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^{4t}-1)x-(2^{2t}+\lambda)n\\
&=&(2^{4t}-1)(2^{2t+2}+1+2l)-(2^{2t}+\lambda)(2^{4t+2}+1)\\
&=&(2^{4t}-1)(1+2l)-2^{2t+2}-2^{2t}-(2^{4t+2}+1)\lambda\\
&\geq&(2^{4t}-1)(1+2l_{_{\lambda,b}})-2^{2t+2}-2^{2t}-(2^{4t+2}+1)\lambda\\
&=&2^{4t}-2^{2t+2}-2^{2t}-1-5\lambda\\
&\geq&2^{4t}-2^{2t+2}-2^{2t}-1-5(2^{2t-1}-1)\\
&=&2^{4t}-25\cdot2^{2t-2}+4>0.
\end{eqnarray*}
{\bf Case 3.4.2:} Secondly, we show $n-y_{_{x,k}}-x>0$:
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&(2^{2t}+\lambda+1)n-(2^{4t}+1)x\\
&=&(2^{2t}+\lambda+1)(2^{4t+2}+1)-(2^{4t}+1)(2^{2t+2}+1+2l).
\end{eqnarray*}
When $\lambda \in [1,2^{2t-2}-2]$, we have $l\in
J_{_{\lambda,k}}=[l_{_{\lambda,b}}=2\lambda,
l_{_{\lambda,e}}2\lambda+1]$, then
\begin{eqnarray*}
n-y_{_{x,k}}-x&\geq&(2^{2t}+\lambda+1)(2^{4t+2}+1)-(2^{4t}+1)(2^{2t+2}+1+2l_{_{\lambda,e}})\\
&=&2^{4t}-3\cdot2^{2t}-2-3\lambda\\
&\geq&2^{4t}-3\cdot2^{2t}-2-3(2^{2t-2}-2)\\
&=&2^{4t}-15\cdot 2^{2t-2}+4>0.
\end{eqnarray*}
When $\lambda=2^{2t-2}-1$ and $l\in
J_{_{\lambda,k}}=[2^{2t-1}-2,2^{2t-1}-2]$, we easily deduce that
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&(2^{2t}+\lambda+1)(2^{4t+2}+1)-(2^{4t}+1)(2^{2t+2}+1+2l)\\
&\geq&(2^{2t}+2^{2t-2})(2^{4t+2}+1)-(2^{4t}+1)[2^{2t+2}+1+2(2^{2t-1}-2)]\\
&=&3\cdot 2^{4t}-15\cdot 2^{2t-2}+3>0.
\end{eqnarray*}
(3.5): If $k=4t+1$, for all $x=2^{2t+2}+1+2l$ with $l\in
J=[2,2^{2t-1}-2]$, we have $y_{_{x,k}}=2^kx-(2^{k-2t}+l)n$, it
follows that
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-(2^{k-2t}+l)n\\
&=&2^{4t+1}-3\cdot2^{2t+1}-1-3l\\
&\geq &2^{4t+1}-3\cdot2^{2t+1}-1-3(2^{2t-1}-2)\\
&=&2^{4t+1}-15\cdot2^{2t-1}+5>0,\\
&&\\
n-y_{_{x,k}}-x&=& (2^{2t}+l+1)n-(2^k+1)x\\
&=& 2^{4t+1}-2^{2t+1}-l\\
&\geq &2^{4t+1}-2^{2t+1}-(2^{2t-1}-2)\\
&=&2^{4t+1}-5\cdot2^{2t+1}+2>0.
\end{eqnarray*}
To conclude the five cases (3.1)-(3.5), (3) holds.
(4) Similarly, for odd $x$ with $2^{2t+2}+2^{2t}+3 \leq x\leq
2^{2t+2}+2^{2t+1}-3$, $x$ can be denoted as
$$x=2^{2t+2}+2^{2t}+1+2l, \hbox{where}~l \in S=[1,2^{2t-1}-2].$$
To verify that (4) holds, it is necessary to show $y_{_{x,k}}-x \geq 0$ and
$n-y_{_{x,k}}-x \geq 0$. We now split into following cases
according to different $k$.
(4.1): If $k=0,1,2,\cdots, 2t-1$, it follows from
$2^{2t+2}+2^{2t}+3\leq x\leq 2^{2t+2}+2^{2t+1}-3$ that $x\leq 2^kx
<n$, then we easily know $y_{_{x,k}}=2^kx\geq x$ and
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&n-(2^k+1)x\\
&\geq&n-(2^{2t-1}+1)x\\
&\geq&n-(2^{2t-1}+1)(2^{2t+2}+2^{2t+1}-3)\\
&=&2^{4t}-9\cdot2^{2t-1}+4>0.
\end{eqnarray*}
(4.2): If $k=2t,2t+1$, we check that $y_{_{x,k}}=2^kx-2^{k-2t}n$,
then
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-2^{k-2t}n\\
&=&2^k+2^{k+2t}-2^{2t}-2^{k-2t}-2^{2t+2}-1+(2^{k+1}-2)l\\
&\geq&2^k+2^{k+2t}-2^{2t}-2^{k-2t}-2^{2t+2}-1+(2^{k+1}-2)\cdot1\\
&=&(3+2^{2t}-2^{-2t})\cdot2^k-2^{2t+2}-2^{2t}-3\\
&\geq&(3+2^{2t}-2^{-2t})\cdot2^{2t}-2^{2t+2}-2^{2t}-3\\
&=&2^{4t}-2^{2t+1}-4>0,
\end{eqnarray*}
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&(2^{k-2t}+1)n-(2^k+1)(2^{2t+2}+2^{2t}+1+2l)\\
&\geq&(2^{k-2t}+1)n-(2^k+1)[2^{2t+2}+2^{2t}+1+2(2^{2t-1}-2)]\\
&=&2^{4t+2}-2^{2t+2}-2^{2t+1}-(2^{2t+1}-2^{-2t}-3)2^k+4\\
&\geq&2^{4t+2}-2^{2t+2}-2^{2t+1}-(2^{2t+1}-2^{-2t}-3)2^{2t+1}+4\\
&=& 6>0.
\end{eqnarray*}
(4.3): If $k=2t+2$, we can deduce $y_{_{x,k}}=2^kx-5n$, it follows
that \begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-5n\\
&\geq&(2^k-1)(2^{2t+2}+2^{2t}+3)-5n\\
&=& 7\cdot2^{2t}-8>0,\\
&&\\
n-y_{_{x,k}}-x&=&6n-(2^k+1)x\\
&\geq&6n-(2^k+1)(2^{2t+2}+2^{2t+1}-3)\\
&=& 3(2^{2t+1}+3)>0.
\end{eqnarray*}
(4.4): For each $k=2t+3, 2t+4,\cdots,4t-1$, it is a little
intractable to determine $y_{_{x,k}}$ and check $y_{_{x,k}}-x
\geq 0$ along with $n-y_{_{x,k}}-x \geq 0$. To complete this, we
first divide $S=[1,2^{2t-1}-2]$ into $2^{k-2t-2}$ intervals as
follows:
$S_{_{\lambda,k}}=[1,2^{4t+1-k}-1]$ for $\lambda=1$,
$S_{_{\lambda,k}}=[(\lambda-1)2^{4t+1-k},
\lambda2^{4t+1-k}-1]$ for $2\leq \lambda \leq 2^{k-2t-2}-1$,
$S_{_{\lambda,k}}=[(\lambda-1)2^{4t+1-k}, 2^{2t-1}-2]$ for $\lambda=2^{k-2t-2}$.
Let $\lambda \in [1,2^{k-2t-2}]$, we can define $\lambda$ as the
{\it identity tag} of the subinterval
$S_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$.
For $x=2^{2t+2}+2^{2t}+1+2l$, for given $\lambda$, if $l\in
S_{_{\lambda,k}}$, it is not difficult to check
$y_{_{x,k}}=2^kx-(2^{k-2t}+2^{k-2t-2}+\lambda-1)n$, then we can
split into following two subcases to verify the desired conclusion.
{\bf Case 4.4.1:} Firstly, we show $y_{_{x,k}}-x>0$:
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-(2^{k-2t}+2^{k-2t-2}+\lambda-1)n\\
&=&(2^k-1)(2^{2t+2}+2^{2t}+1+2l)-(2^{k-2t}+2^{k-2t-2}+\lambda-1)n\\
&=&(2^k-1)(1+2l)-2^{2t+2}-2^{2t}-2^{k-2t}-2^{k-2t-2}-(\lambda-1)n.
\end{eqnarray*}
If $\lambda=1$, we have $l\in S_{_{1,k}}=[1,2^{4t+1-k}-1]$, it
follows from $k\geq 2t+3$ that
\begin{eqnarray*}y_{_{x,k}}-x
&=&(2^k-1)(1+2l)-2^{2t+2}-2^{2t}-2^{k-2t}-2^{k-2t-2}\\
&\geq&(2^k-1)(1+2cdot1)-2^{2t+2}-2^{2t}-2^{k-2t}-2^{k-2t-2}\\
&=&(3-2^{-2t}-2^{-2t-2})\cdot2^k-2^{2t+2}-2^{2t}-3\\
&\geq&(3-2^{-2t}-2^{-2t-2})\cdot2^{2t+3}-2^{2t+2}-2^{2t}-3\\
&=&19\cdot2^{2t}-13>0.
\end{eqnarray*}
If $\lambda\in[2,2^{k-2t-2}]$, since $l\in
S_{_{\lambda,k}}=[l_{_{\lambda,b}}=(\lambda-1)2^{4t+1-k},l_{_{\lambda,e}}]$,
we can similarly deduce from $k\geq 2t+3$ that
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)(1+2l)-2^{2t+2}-2^{2t}-2^{k-2t}-2^{k-2t-2}-(\lambda-1)n\\
&\geq&(2^k-1)(1+2l_{_{\lambda,b}})-2^{2t+2}-2^{2t}-2^{k-2t}-2^{k-2t-2}-(\lambda-1)n\\
&=&2^{4t+2-k}+2^{k}-2^{2t+2}-2^{2t}-2^{k-2t}-2^{k-2t-2}-(2^{4t+2-k}+1)\lambda\\
&\geq&\!\!2^{4t+2-k}+2^{k}-2^{2t+2}-2^{2t}-2^{k-2t}-\!2^{k-2t-2}\!\!-(2^{4t+2-k}+1)2^{k-2t-2}\\
&=&2^{4t+2-k}+(1-2^{-2t}-2^{-2t-1})2^{k}-2^{2t+2}-2^{2t+1}(\hbox{see Lemma 0.1})\\
&\geq&2^{4t+2-(2t+3)}+(1-2^{-2t}-2^{-2t-1})2^{2t+3}-2^{2t+2}-2^{2t+1}\\
&=&5\cdot2^{2t-1}-12>0.
\end{eqnarray*}
{\bf Case 4.4.2:} Secondly, we show $n-y_{_{x,k}}-x>0$:
\begin{eqnarray*}
&&n-y_{_{x,k}}-x\\
&=& (2^{k-2t}+2^{k-2t-2}+\lambda)n-(2^k+1)x\\
&=&\lambda(2^{4t+2}+1)+2^{k-2t}+2^{k-2t-2}-\!\!2^{2t+2}-2^{2t}-\!\!(2^k+1)(1+2l).
\end{eqnarray*}
If $\lambda\in[1,2^{k-2t-2}-1]$, we have $l\in
S_{_{\lambda,k}}=[l_{_{\lambda,b}},
l_{_{\lambda,e}}=\lambda2^{4t+1-k}-1]$, then
\begin{eqnarray*}
&&n-y_{_{x,k}}-x\\
&\geq&\lambda(2^{4t+2}+1)+2^{k-2t}+2^{k-2t-2}-2^{2t+2}-2^{2t}-(2^k+1)(1+2l_{_{\lambda,e}})\\
&=&2^{k-2t}+2^{k-2t-2}+2^k-2^{2t+2}-2^{2t}+1-(2^{4t+2-k}-1)\lambda\\
&\geq&2^{k-2t}+2^{k-2t-2}+2^k-2^{2t+2}-2^{2t}+1-(2^{4t+2-k}-1)(2^{k-2t-2}-1)\\
&=&2^{4t+2-k}+(1+2^{-2t}+2^{-2t-1})2^k-2^{2t+2}-2^{2t+1}\\
&\geq&2^{4t+2-(2t+3)}+(1+2^{-2t}+2^{-2t-1})2^{2t+3}-2^{2t+2}-2^{2t+1}(\hbox{see Lemma 0.1})\\
&=&5\cdot2^{2t-1}+12>0.
\end{eqnarray*}
If $\lambda=2^{k-2t-2}$, it follows that $l\in
S_{_{\lambda,k}}=[l_{_{\lambda,b}}=(2^{k-2t-2}-1)2^{4t+1-k},
l_{_{\lambda,e}}=2^{2t-1}-2]$, then we obtain
\begin{eqnarray*}
&& n-y_{_{x,k}}-x\\
&=&2^{k-2t-2}(2^{4t+2}+1)+2^{k-2t}+\!\!2^{k-2t-2}-\!\!2^{2t+2}-\!\!2^{2t}-(2^k+1)(1+2l)\\
&\geq&2^{k-2t-2}(2^{4t+2}+1)+2^{k-2t}+\!\!2^{k-2t-2}-\!\!2^{2t+2}-\!\!2^{2t}-\!\!(2^k+1)(1+2l_{_{\lambda,e}})\\
&=&2^{k-2t-1}+2^{k-2t}+2^{k+1}-2^{2t+2}-2^{2t+1}+3\\
&\geq&2^{(2t+3)-2t-1}+2^{(2t+3)-2t}+2^{(2t+3)+1}-2^{2t+2}-2^{2t+1}+3\\
&=&5(2^{2t+1}+3)>0.
\end{eqnarray*}
(4.5): If $k=4t$, similar to (4.4) above, to determine $y_{_{x,k}}$
and check $y_{_{x,k}}-x \geq 0$ along with $n-y_{_{x,k}}-x \geq
0$, we first divide $S=[1,2^{2t-1}-2]$ into $2^{2t-2}$ intervals
as follows:
$S_{_{\lambda,k}}=[1,1=2\lambda-1]$ for $\lambda=1$,
$S_{_{\lambda,k}}=[2(\lambda-1),
2\lambda-1] ~ \mbox{for} ~ 2\leq \lambda \leq 2^{2t-2}-1 $,
$S_{_{\lambda,k}}=[2^{2t-1}-2=2(\lambda-1),2^{2t-1}-2]$ for $\lambda=2^{2t-2}$.
Let $\lambda \in [1,2^{2t-2}]$, we can define $\lambda$ as the
{\it identity tag} of the subinterval
$S_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}]$.
For give $\lambda$, if $x=2^{2t+2}+2^{2t}+1+2l$ with $l\in
S_{_{\lambda,k}}$, it is not difficult to check that
$y_{_{x,k}}=2^kx-(2^{k-2t}+2^{k-2t-2}+\lambda-1)n$, then we split
into following two subcases.
{\bf Case 4.5.1:} Firstly, we show $y_{_{x,k}}-x>0$:
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^{4t}-1)x-(2^{2t}+2^{2t-2}+\lambda-1)n\\
&=&(2^{4t}-1)(2^{2t+2}+2^{2t}+1+2l)-(2^{2t}+2^{2t-2}+\lambda-1)(2^{4t+2}+1)\\
&=&(2^{4t}-1)(1+2l)-2^{2t+2}-2^{2t+1}-2^{2t-2}-(2^{4t+2}+1)(\lambda-1).
\end{eqnarray*}
If $\lambda=1$, we have $l\in S_{_{\lambda,k}}=[1,1]$, that is,
$l=1$, we can easily deduce
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^{4t}-1)(1+2l)-2^{2t+2}-2^{2t+1}-2^{2t-2}\\
&=&(2^{4t}-1)(1+2\cdot1)-2^{2t+2}-2^{2t+1}-2^{2t-2}\\
&\geq& 3\cdot2^{4t}-25\cdot 2^{2t-2}-3>0.
\end{eqnarray*}
If $\lambda\in[2,2^{2t-2}]$, we know $l\in
S_{_{\lambda,k}}=[l_{_{\lambda,b}}=2\lambda-2,l_{_{\lambda,e}}]$, it
follows that
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^{4t}-1)(1+2l)-2^{2t+2}-2^{2t+1}-2^{2t-2}-(2^{4t+2}+1)\\
&\geq&(2^{4t}-1)(1+2l_{_{\lambda,b}})-2^{2t+2}-2^{2t+1}-2^{2t-2}-(2^{4t+2}+1)\\
&=&2^{4t}-25\cdot2^{2t-2}+4-5\lambda\\
&\geq&2^{4t}-25\cdot2^{2t-2}+4-5\cdot2^{2t-2}\\
&=&2^{4t}-15\cdot2^{2t-1}+4>0.
\end{eqnarray*}
{\bf Case 4.5.2:} Secondly, we show $n-y_{_{x,k}}-x>0$:
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&(2^{2t}+2^{2t-2}+\lambda)n-(2^{4t}+1)x\\
&=&(2^{2t}+2^{2t-2}+\lambda)(2^{4t+2}+1)-(2^{4t}+1)(2^{2t+2}+2^{2t}+1+2l).
\end{eqnarray*}
If $1\leq \lambda \leq 2^{2t-2}-1$ and $l\in
S_{_{\lambda,k}}=[l_{_{\lambda,b}},l_{_{\lambda,e}}=2\lambda-1]$,
then we get that
\begin{eqnarray*}
n-y_{_{x,k}}-x&\geq&(2^{2t}+2^{2t-2}+\lambda)(2^{4t+2}+1)-(2^{4t}+1)(2^{2t+2}+2^{2t}+1+2l_{_{\lambda,e}})\\
&=&2^{4t}+2^{2t-2}-2^{2t+2}+1-3\lambda\\
&\geq&2^{4t}+2^{2t-2}-2^{2t+2}+1-3(2^{2t-2}-1)\\
&=&2^{4t}-9\cdot2^{2t-1}+4>0.
\end{eqnarray*}
If $\lambda=2^{2t-2}$, we have $x=2^{2t+2}+2^{2t}+1+2l$ with $l\in
S_{_{\lambda,k}}$, it is easy to deduce that
\begin{eqnarray*}
n-y_{_{x,k}}-x&=&(2^{2t}+2^{2t-2}+2^{2t-2})(2^{4t+2}+1)-(2^{4t}+1)x\\
&\geq&(2^{2t}+2^{2t-2}+2^{2t-2})(2^{4t+2}+1)-(2^{4t}+1)(2^{2t+2}+2^{2t+1}-3)\\
&=&3\cdot 2^{4t}-9\cdot 2^{2t-1}+3>0.
\end{eqnarray*}
(4.6): If $k=4t+1$, for all $x=2^{2t+2}+2^{2t}+1+2l$ with $l\in
S=[1,2^{2t-1}-2]$, we have
$y_{_{x,k}}=2^kx-(2^{k-2t}+2^{k-2t-2}+l)n$, then
\begin{eqnarray*}
y_{_{x,k}}-x&=&(2^k-1)x-(2^{k-2t}+l)n\\
&=&2^{4t+1}-3\cdot2^{2t+1}-1-3l\\
&\geq &2^{4t+1}-3\cdot2^{2t+1}-1-3(2^{2t-1}-2)\\
&=&2^{4t+1}-15\cdot2^{2t-1}+5>0,\\
&&\\
n-y_{_{x,k}}-x&=& (2^{2t}+l+1)n-(2^k+1)x\\
&=& 2^{4t+1}-2^{2t+1}-l\\
&\geq& 2^{4t+1}-2^{2t+1}-(2^{2t-1}-2)\\
&=&2^{4t+1}-5\cdot2^{2t+1}+2>0.
\end{eqnarray*}
Summarizing the six cases (4.1)-(4.6), one can derive that (4)
holds.
(5) For $i=1\hbox{~or~} 3$, it is easy to check the following:
$(2^{2t+1}+1)2^{2t+1}=2^{m}+2^{t+1}\equiv 2^{2t+1}-1 $;
$(2^{2t+2}-i)2^{6t+2}=(2^{2t+2}-i)2^{2t}\cdot
2^{4t+2}\equiv2^{2t}i+1$;
$(2^{2t+2}+i)2^{2t}=2^{m}+2^{2t}i\equiv 2^{2t}i-1 $;
$(2^{2t+2}+2^{2t}-1)2^{6t+4}=(2^{2t+2}+2^{2t}-1)2^{2t}\cdot 2^{4t+2}
\equiv 2^{2t+2}+5$;
$(2^{2t+2}+2^{2t}+1)2^{2t+2}=2^{m+2}+2^{m}+2^{2t+2}\equiv
2^{t+2}-5$;
$(2^{2t+2}+2^{2t+1}-1)2^{6t+3}=(3\cdot2^{2t+1}-1)2^{2t+1}\cdot
2^{4t+2}\equiv 2^{2t+1}+3$;
$(2^{2t+2}+2^{2t+1}+i)2^{2t+1}=2^{m+1}+2^{m}+2^{2t+1}i\equiv
2^{2t+1}i-3 $.
Combining the definition of a cyclotomic coset, these congruence
expressions above imply that there exists some odd integer
$y\in[1,x-1]$ such that $y\in C_{x}$ for $x$ in (5), hence $x$ is
not a coset leader.
\end{proof}
\subsection*{Appendix E: The proof of Lemma 4.2}
\begin{proof}
From $\delta_{1}=\frac{n}{5}=2^{4t}-2^{4t-2}+\cdots-2^{2}+1$,
$\delta_{2}=2^{4t-1}+\frac{2^{4t}-1}{5}=2^{4t-1}+3(2^{4t-4}+2^{4t-8}+\cdots+2^{4}+1)$,
we can deduce $\delta_{1}$, $\delta_{2}$, $\delta_{3}$, $\delta_{4}$
and $\delta_{5}$ are all odd. It is easy to derive
$C_{\delta_{1}}=\{ \delta_{1}, 2\delta_{1}, 4\delta_{1}, 3\delta_{1}
\}$, which implies that $|C_{\delta_{1}}|=4$ and $\delta_{1}$ is a coset leader.
Hence, we will prove that $\delta_{2}$, $\delta_{3}$, $\delta_{4}$ and $\delta_{5}$
are all coset leaders.
{\it Step 1:} We show $y_{_{\delta_2,k}}-\delta_2\geq 0$ and
$n-y_{_{\delta_2,k}}-\delta_2\geq 0$ in three cases:
(1.1): If $k=0,1,2$, it is clear that
$y_{_{\delta_2,k}}=2^k\delta_2>\delta_2$,
$n-y_{_{\delta_2,k}}-\delta_2=n-(2^k+1)\delta_2 \geq 2^{4t-1}+2>0.$
(1.2): If $k=3,4$, we have
\begin {eqnarray*}
y_{_{\delta_2,k}}&=&2^k\delta_2-2^{k-3}n,\\
5(y_{_{\delta_2,k}}-\delta_2)&=&2^{4t+k}-13\cdot2^{k-3}-7\cdot2^{4t-1}+1\\
&\geq&2^{4t+3}-13\cdot2^{3-3}-7\cdot2^{4t-1}+1=9\cdot2^{4t-1}-12>0,\\
5(n-y_{_{\delta_2,k}}-\delta_2)&=&33\cdot2^{4t-1}+13\cdot2^{k-3}-2^{4t+k}+6\\
&\geq&33\cdot2^{4t-1}+13\cdot2^{4-3}-2^{4t+4}+6=2^{4t-1}+32>0.
\end {eqnarray*}
(1.3): If $5\leq k\leq 4t+1$, the proof can be split into following
two subcases.
\quad (1.3.1): When $k\equiv 1 \bmod 4$ (i.e., $k=5, 9,\cdots,
4t+1$), we have
\begin {eqnarray*}
y_{_{\delta_2,k}}&=&2^k\delta_2-(2^{k-3}+\frac{2^{k-2}-3}{5})n,\\
5(y_{_{\delta_2,k}}-\delta_2)&=&17\cdot2^{4t-1}-2^k-7\cdot2^{k-3}+4\\
&\geq&17\cdot2^{4t-1}-2^{4t+1}-7\cdot2^{(4t+1)-3}+4=
19\cdot2^{4t-2}+4>0,\\
5(n-y_{_{\delta_2,k}}-\delta_2)&=&9\cdot2^{4t-1}\!\!+2^k+7\cdot2^{k-3}+3\\
&\geq&9\cdot2^{4t-1}\!\!+2^5+7\cdot2^{5-3}+3= 9\cdot2^{4t-1}+63>0.
\end {eqnarray*}
\quad (1.3.2): When $k\equiv \gamma \bmod 4$ ($\gamma=2,3,4$),
i.e., $k=4\times1+\gamma, 4\times2+\gamma,\cdots, 4(t-1)+\gamma$, it
then follows that
\begin {eqnarray*}
y_{_{\delta_2,k}}&=&2^k\delta_2-(2^{k-3}+\frac{2^{k-2}-2^{\gamma-2}}{5})n,\\
5(y_{_{\delta_2,k}}-\delta_2)&=&(2^{\gamma+1}-7)2^{4t-1}+1+2^{\gamma-2}-15\cdot2^{k-3}\\
&\geq&(2^{\gamma+1}-7)2^{4t-1}+1+2^{\gamma-2}-15\cdot2^{[4(t-1)+\gamma]-3}\\
&=&113\cdot 2^{4t+\gamma-7}+2^{\gamma-1}-7\cdot2^{4t-1}+1\\
&\geq&113\cdot 2^{4t+2-7}+2^{2-1}-7\cdot2^{4t-1}+1=2^{4t-5}+2>0,\\
5(n-y_{_{\delta_2,k}}-\delta_2)&=&(33-2^{\gamma+1})2^{4t-1}+6-2^{\gamma-2}+15\cdot2^{k-3}\\
&\geq&(33-2^{\gamma+1})2^{4t-1}+6-2^{\gamma-2}+15\cdot2^{(4\times1+\gamma)-3}\\
&=&33\cdot2^{4t-1}+6-(2^{4t+2}-119)\cdot2^{\gamma-2}\\
&\geq&33\cdot2^{4t-1}+6-(2^{4t+2}-119)\cdot2^{4-2}=2^{4t-1}+482>0.
\end {eqnarray*}
From the three cases above, we then conclude that $\delta_2$ is a
coset leader.
{\it Step 2:} Now, we show $y_{_{\delta_3,k}}-\delta_3$ and
$n-y_{_{\delta_3,k}}-\delta_3$ for $0\leq k\leq 4t+1$ as follows:
(2.1): If $k=1,2$, it is clear that
$y_{_{\delta_3,k}}=2^k\delta_3>\delta_3$,
$n-y_{_{\delta_3,k}}-\delta_3=n-(2^k+1)\delta_3\geq 2^{4t-1}+32>0$.
(2.2): If $k=3,4$, we obtain
\begin {eqnarray*}
y_{_{\delta_3,k}}&=&2^k\delta_3-2^{k-3}n,\\
5(y_{_{\delta_3,k}}-\delta_3)&=&2^{4t+k}-253\cdot2^{k-3}-7\cdot2^{4t-1}+31\\
&\geq&2^{4t+3}-253\cdot2^{3-3}-7\cdot2^{4t-1}+31\\
& =&9\cdot 2^{4t-1}-222\geq 930,\\
5(n-y_{_{\delta_3,k}}-\delta_3)&=&33\cdot2^{4t-1}+1+253\cdot2^{k-3}-2^{4t+k}+35\\
&\geq&33\cdot2^{4t-1}+1+253\cdot2^{4-3}-2^{4t+4}+35\\
&=& 2^{4t-1} +542>0.
\end {eqnarray*}
(2.3): If $5\leq k\leq 4t-3$, the discussion can be given in two
subcases.
\quad (2.3.1): When $k\equiv 1\bmod4$ (i.e., $k=5, 9,\cdots,
4t-3$), we can get
\begin
{eqnarray*}
y_{_{\delta_3,k}}&=&2^k\delta_3-(2^{k-3}+\frac{2^{k-2}-3}{5})n,\\
5(y_{_{\delta_3,k}}-\delta_3)&=&17\cdot2^{4t-1}+4-255\cdot2^{k-3}+30\\
&\geq&17\cdot2^{4t-1}+4-255\cdot2^{(4t-3)-3}+30\\
&=& 289\cdot2^{4t-6}+34>0,\\
5(n-y_{_{\delta_3,k}}-\delta_3)&=&9\cdot2^{4t-1}+255\cdot2^{k-3}+33>0.
\end {eqnarray*}
\quad (2.3.2): When $t\geq3$ and $k\equiv \gamma\bmod4$
($\gamma=2,3,4$), i.e., $k=4\times1+\gamma, 4\times2+\gamma,\cdots,
4(t-2)+\gamma$, we obtain
\begin {eqnarray*}
y_{_{\delta_3,k}}&=&2^k\delta_3-(2^{k-3}+\frac{2^{k-2}-2^{\gamma-2}}{5})n,\\
5(y_{_{\delta_3,k}}\!\!\!-\delta_3)&=&(2^{\gamma+1}-7)\cdot2^{4t-1}+2^{\gamma-2}+1-255\cdot2^{k-3}+30\\
&\geq&(2^{\gamma+1}-7)\cdot2^{4t-1}+2^{\gamma-2}+1-255\cdot2^{[4(t-2)+\gamma]-3}+30\\
&=&1793\cdot2^{4t+\gamma-11}+2^{\gamma}-7\cdot2^{4t-1}+1\\
&\geq&1793\cdot2^{4t+2-11}+2^{2}-7\cdot2^{4t-1}+1\\
&=&2^{4t-9}+32>0,\\
5(n-y_{_{\delta_3,k}}\!\!\!-\delta_3)&=&(33-2^{\gamma+1})2^{4t-1}-2^{\gamma-2}+1+255\cdot2^{k-3}+35\\
&\geq&(33-2^{\gamma+1})2^{4t-1}-2^{\gamma-2}+1+255\cdot2^{(4\times1+\gamma)-3}+35\\
&=&33\cdot2^{4t-1}+1-(2^{4t+2}-2039)\cdot2^{\gamma-2}+35\\
&\geq&33\cdot2^{4t-1}+1-(2^{4t+2}-2039)\cdot2^{4-2}+35\\
&=&2^{4t-1}+8192>0.
\end {eqnarray*}
(2.4): If $4t-2\leq k\leq 4t+1$, the discussion can be given in two
subcases according to different $\gamma$.
\quad (2.4.1): When $k=4(t-1)+\gamma$ ($\gamma=2,3,4$), we can
deduce
\begin {eqnarray*}
y_{_{\delta_3,k}}&=&2^k\delta_3-(2^{k-3}+\frac{2^{k-2}-2^{\gamma-2}}{5}-1)n,\\
5(y_{_{\delta_3,k}}-\delta_3)&=&(2^{\gamma+1}+33)\cdot2^{4t-1}+2^{\gamma-2}+1-255\cdot2^{k-3}+35\\
&=&33\cdot2^{4t}+1-(127\cdot2^{4t+\gamma-7}-2^{\gamma-2})+35\\
&\geq&33\cdot2^{4t}+1-(127\cdot2^{4t+4-7}-2^{4-2})+35\\
&=&5(2^{4t-3}+8)>0,\\
5(n-y_{_{\delta_3,k}}-\delta_3\!\!\!)&=&\!\!\!255\cdot2^{k-3}-(2^{\gamma+1}+7)2^{4t-1}-2^{\gamma-2}+31\\
&=&\!\!\!127\cdot2^{4t+\gamma-7}-2^{\gamma-2}-7\cdot2^{4t-1}+31\\
&\geq&127\cdot2^{4t+2-7}-2^{2-2}-7\cdot2^{4t-1}+31\\
&=&15(2^{4t-5}+2)>0.
\end {eqnarray*}
\quad (2.4.2): When $k=4t+1$, we have
{ \begin {eqnarray*}
y_{_{\delta_3,k}}&=&2^k\delta_3-(2^{k-3}+\frac{2^{k-2}-3}{5}-3)n,\\
5(y_{_{\delta_3,k}}-\delta_3)&=&137\cdot2^{4t-1}-255\cdot2^{k-3}+49= 19\cdot2^{4t-2}+49>0,\\
5(n-y_{_{\delta_3,k}}-\delta_3)&=&255\cdot2^{k-3}-111\cdot2^{4t-1}
+18= 33\cdot2^{4t-2}+18>0.
\end {eqnarray*}}
It is easy to know $\delta_1$ is a coset leader, by Steps 1 and 2
above, we have shown $\delta_{2}$ and $\delta_{3}$ are coset leaders
respectively. Similar to Steps 1 and 2, we can also attain
$\delta_4$ and $\delta_5$ are both coset leaders, the detailed
proofs are omitted.
\end{proof}
\subsection*{Appendix F: The proof of Theorem 4.7}
\begin{proof} Let $\mathcal{C}(n,2,\delta,1)$, $T_{\delta}$ and $S_{\delta}$ be given similar to the proof of Theorem 3.7, then
$\mathcal{C}(n,2,\delta,1)$ has dimension $k=n-|T_{\delta}|=n-\sum\limits_{i\in S_{\delta}}|C_{i}|$.
(i): When $2^{2t+1}+3\leq \delta\leq 2^{2t+2}-5$, from Theorem 4.1,
we check that $S_{\delta}=\{x|x~\hbox{is odd and}~x\in
[1,\delta-1]\setminus\{2^{2t+1}+1\}\},$ thus
$|S_{\delta}|=\frac{\delta-1}{2}-1$. By Lemma 4.6, all cyclotomic
cosets in $T_{\delta}$ have cardinality $2m$, it follows that $k=
n-2m\cdot (\frac{\delta-1}{2}-1)=n-m\delta+3m $ and $d\geq \delta$.
(ii): Similar to (i), when $2^{2t+2}+5\leq \delta\leq
2^{2t+2}+2^{2t}-3$, we can derive from Theorem 4.1 and Lemma 4.6
that
$$S_{\delta}=\{x|x~\hbox{is odd and}~x\in
[1,\delta-1]\setminus\{2^{2t+1}+1,2^{2t+2}\pm1,2^{2t+2}\pm3\}\},$$
it follows that $d\geq \delta$ and $k=n-\sum\limits_{i\in
S_{\delta}}|C_{i}|=n-2m\cdot (\frac{\delta-1}{2}-5)=n-m\delta+11m.$
(iii): Similar to (i), when $2^{2t+2}+2^{2t}+3\leq \delta \leq a-3$, we can derive from Theorem 4.1 and Lemma 4.6
that $$S_{\delta}=\{x|x~\hbox{is odd and}~x\in
[1,\delta-1]\setminus\{2^{2t+1}+1,2^{2t+2}\pm1,2^{2t+2}\pm3,2^{2t+2}+2^{2t}\pm1\}\},$$
thus we have $d\geq \delta$ and
$k=n-\sum\limits_{i\in S_{\delta}}|C_{i}|=n-2m\cdot
(\frac{\delta-1}{2}-7)=n-m\delta+15m.$
(iv): Similar to (i), when $a-1 \leq \delta\leq a+5$, we can infer
from Theorem 4.5 and Lemma 4.6 that$$S_{\delta}=\{x|x~\hbox{is odd
and}~x\in
[1,a-3]\setminus\{2^{2t+1}+1,2^{2t+2}\pm1,2^{2t+2}\pm3,2^{2t+2}+2^{2t}\pm1\}\},$$
thus we get $d\geq
a+5$ and $k=n-2m\cdot (2^{2t+1}+2^{2t}-8)=n-am+16m.$
(v): When $\delta_{i+1}+2\leq\delta\leq \delta_{i}(i=1,2,3,4)$, from
Theorem 4.5, we get that
$T_{\delta}=\bigcup\limits_{i\in
S_{\delta}}C_{i}=\{1,2,\cdots,n-1\}\setminus\bigcup
\limits_{j=1}^{i}C_{\delta_j}$.
We have known $|C_{\delta_{1}}|=4$ from the proof of Lemma 4.2,
according to Lemma 4.6, $C_{\delta_{i}}(i=2,3,4,5)$ has cardinality
$2m$, it then follows that
$$k=n-|T_{\delta}|=n-[n-1-2m(i-1)-4]=2m(i-1)+5.$$
On the other hand, there exist $\delta_{i}-1$ consecutive integers
in $T_{\delta}$, the minimum distance $d\geq \delta_{i} $.
(vi): When $\delta_{1}+2\leq\delta\leq n$, we check from Theorem 4.5
that $T_{\delta}=\bigcup\limits_{i\in
S_{\delta}}C_{i}=\{1,2,\cdots,n-1\}$, then
$k=n-|T_{\delta}|=n-(n-1)=1.$ It follows from Singleton bound that
the minimum distance $d=n$.
(2) On the basis of the proof of (1), (2) can be
easily obtained.
\end{proof}
|
{
"timestamp": "2018-03-08T02:10:16",
"yymm": "1803",
"arxiv_id": "1803.02731",
"language": "en",
"url": "https://arxiv.org/abs/1803.02731"
}
|
\section{Introduction}
Algorithms designed to learn distributed sentence representations
have been shown to be transferable across a range of tasks \cite{mou2016transferable} and languages \cite{jorg2018emerging}.
For example, \citet{guu2017generating} proposed to represent sentences as vectors that encode a notion similarity between sentence pairs, and showed that vector manipulations of the representation can result in meaningful change in semantics. The question we would like to explore is whether the semantic relationship between sentence pairs can be modeled in a more explicit manner.
More specifically, we want to model the \emph{logical relationship} between sentences.
Controlling the logical relationship between sentences has many direct applications.
First of all, we can use it to provide a more clear definition of paraphrasing.
To do so, we require two simultaneous conditions: (i) that the input sentence \emph{entails} the output sentence; and (ii) that the output sentence \emph{entails} the input sentence. \begin{equation}
\begin{split}
&(\textsc{Sentence1} \models \textsc{Sentence2}) \,\wedge \\
&\qquad(\textsc{Sentence2} \models \textsc{Sentence1})
\end{split}
\end{equation}
The first requirement ensures the output sentence cannot be false if the input sentence is true, so that the output sentence can be considered a fact expressed by the input sentence.
The second requirement ensures that the output contains at least the input's information.
The two requirements together can be used to define semantic equivalence between sentence.
Another interesting application is multi-document summarization.
Traditionally, to summarize multiple documents, one would expect the model to abstract the most important part of the source documents, and this is usually measured by the amount of overlap that the output document has with the inputs.
Informally, one finds the maximal amount of information that has the highest precision with each source document.
Alternatively, if one wants to automate news aggregation,
the ideal summary would need to contain the same number of facts as are contained in the union of all source documents.
We can think of this second objective as requiring that the output document entail every single sentence across all source documents.
In this paper, we propose an approach to generating sentences, conditioned on an input sentence and a logical inference label.
We do this by modeling the different possibilities for the output sentence as a distribution over the latent representation, which we train using an adversarial objective.
In particular, we differ from the usual adversarial training on text by using a differentiable global representation. Architecture-wise, we also propose a Memory Operation Selection Module (MOSM) for encoding a sentence into a vector representation. Finally, we evaluate the quality and the diversity of our samples.
The rest of the paper is organized as follows: Sec. \ref{sec:related_work} will cover the related literature. Sec. \ref{sec:method} will detail the proposed model architecture, and Sec. \ref{sec:experiments} will describe and analyze the experiments run. Sec. \ref{sec:discussion} will then discuss the implications of being able to solve this task well, and the future research directions relating to this work. Finally, we conclude in Sec. \ref{sec:conclusion}.
\section{Related Work} \label{sec:related_work}
Many natural language tasks require reasoning capabiliities. The Recognising Textual Entailment (RTE) task requires the system to determine if the \emph{premise} and \emph{hypothesis} pair are (i) an entailment, (ii) contradicting each other or (iii) neutral to each other. The Natural language Inference (NLI) Task from \citet{bowman2015large} introduces a large dataset with labeled pairs of sentences and their corresponding logical relationship. This dataset allows us to quantify how well current systems are able to be trained to recognise sentences with those relationships. Examples of the current state-of-the-art for this task include \citet{chen2017enhanced} and \citet{gong2017natural}.
Here we are interested in generating natural language that satisfies the given textual entailment class.
\citet{kolesnyk2016generating} has attempted this using only sentences from the entailment class, and focusing on generating a hypothesis given the premise. Going in this direction results in removal of information from the premise sentence.
In this paper, we focus on going in the other direciton: generating a premise from a hypothesis. This requires adding additional details to the premise which have to make sense in context.
In order to produce sentences with extra details and without some other details, we suggest that a natural way to model this kind of structure is to impose a distribution over an intermediate distribution representing the semantic space of the premise sentence.
In the realm of learning representations for sentences, \citet{kiros2015skip} has a popular method for learning representations called ``skip-thought'' vectors. These are trained by using the encoded sentence to predict the previous and next sentence in a passage.
\citet{conneau2017supervised} specifically learned sentence representations from the SNLI dataset. They claim that using the supervised data from SNLI can outperform ``skip-thought'' representations on different tasks. There have also been several efforts towards learning a distribution over sentence embeddings.
\citet{bowman2015generating} used Variational Autoencoders (VAEs) to learn Gaussian distributed word embeddings.
\citet{hu2017toward} use a combined VAE/GAN objective to produce a disentangled representation that can be used to modify some attributes like sentiment and tense.
There have also been forays into conditional distributions for sentences -- which is what is required here.
Both \citet{gupta2017deep} and \citet{guu2017generating} introduce models of the form $p(\mathbf{x}|\mathbf{z}, \mathbf{x}')$, where $\mathbf{x}$ is a paraphrase of $\mathbf{x}'$, and $\mathbf{z}$ represents the variability in the output sentence. \citet{guu2017generating} introduces $\mathbf{z}$ as an edit vector.
However, because $\mathbf{z}$ has to be paired with $\mathbf{x}'$ in order to generate the sentence, $\mathbf{z}$ serves a very different purpose, and cannot be considered a sentence embedding in its own right.
Ideally, what we want is a distribution over sentence representations, each one mapping to a set of semantically similar sentences.
This is important if we want the distribution to model the possibilities of concepts that correspond to the right textual entailment with the hypothesis.
\section{Method} \label{sec:method}
Some approaches map a sentence to a distribution in the embedding space \cite{bowman2015generating}. The assumption when doing this is that there is some uncertainty over the latent space when mapping from the sentence. Some approaches, like \citet{hu2017toward} attempt to disentangle factors in the learnt latent variable space, so that modifying each dimension in the latent representation modifies sentiment or tense in the original sentence.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node[obs] (h) {${\boldsymbol \eta}$};
\node[obs, below=of h] (l) {$\ell$};
\node[latent, right=of h] (z) {$\mathbf{z}$};
\node[obs, right=of z] (p) {${\boldsymbol \phi}$};
\edge {h} {z} ; %
\edge {l} {z} ; %
\edge {z} {p} ; %
\draw [->,red] (p) to [out=150,in=30] (z);
\end{tikzpicture}
\end{center}
\caption{The conceptual graphical model behind the formulation of our model. The red arrow represents the inference path from $\phi$ to $\mathbf{z}$.} \label{fig:graphicalmodel}
\end{figure}
If we consider plausible premise sentences ${\boldsymbol \phi}$ given a hypothesis ${\boldsymbol \eta}$ and an inference label $\ell$, there are many possible solutions, of varying likelihoods.
We can model this probabilistically as $p({\boldsymbol \phi}|{\boldsymbol \eta},\ell)$.
In our model, we assume an underlying latent variable $\mathbf{z}$ that accounts for the variation in possible output sentences,
$$p({\boldsymbol \phi}|{\boldsymbol \eta},\ell) = \int p({\boldsymbol \phi}|\mathbf{z})p(\mathbf{z}|{\boldsymbol \eta},\ell) \mathrm{d}\mathbf{z}$$
Another assumption we make is that given ${\boldsymbol \phi}$, $\mathbf{z}$ is independent of ${\boldsymbol \eta}$ and $\ell$. The resulting graphical model associated with the above dependency assumptions are depicted in Figure \ref{fig:graphicalmodel}.
In our proposed model, we take inspiration from the Adversarial Autoencoder \cite{makhzani2015adversarial}, however our prior is conditioned on ${\boldsymbol \eta}$ and $\ell$. \citet{zhang2017age} also proposed a Conditional Adversarial Autoencoder for age progression prediction.
In addition to the adversarial discriminator, our model includes a classifier on the representation and the hypothesis and label. A similar framework is also discussed in \citet{salimans2016improved}.
\subsection{Architecture}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/Architecture}
\caption{The architecture of the model. The autoencoder maps given premise ${\boldsymbol \phi}$ to a sentence representation $\mathbf{z}$, and reconstructs ${\boldsymbol \phi}$ from $\mathbf{z}$.
Samples are drawn from the prior conditioned on ${\boldsymbol \eta}$ and $\ell$.
The classifier takes $\mathbf{z}$ and ${\boldsymbol \eta}$ as input, and outputs probability of $l$.
The discriminator takes $\mathbf{z}$, ${\boldsymbol \eta}$ and $l$ as input, and predicts whether $\mathbf{z}$ is given by the autoencoder or the prior.}
\label{fig:architecture}
\end{figure}
The model consists of an encoder $q(\mathbf{z}|{\boldsymbol \phi})$, a conditional prior, $p(\mathbf{z}|{\boldsymbol \eta},\ell)$, a decoder $p({\boldsymbol \phi}| \mathbf{z})$, and a discriminator $\mathrm{D}(\mathbf{z},{\boldsymbol \eta},\ell)$.
\paragraph{Autoencoder} The autoencoder comprises of two parts. An encoder that maps the given premise ${\boldsymbol \phi}$ to a sentence representation $\mathbf{z}$, and a decoder that reconstructs ${\boldsymbol \phi}$ from a given $\mathbf{z}$. In our model, the encoder reads the input premise ${\boldsymbol \phi}=(x^{\boldsymbol \phi}_1,...,x^{\boldsymbol \phi}_{|{\boldsymbol \phi}|})$ using an RNN network:
\begin{equation}
h^{\boldsymbol \phi}_1,...,h^{\boldsymbol \phi}_{|{\boldsymbol \phi}|} = \mathrm{RNN}_\mathrm{enc}(x^{\boldsymbol \phi}_1, ..., x^{\boldsymbol \phi}_{|{\boldsymbol \phi}|}) \label{eq:prem_encoder_rnn}
\end{equation}
and
\begin{equation}
\mathbf{z} = f_\mathrm{compress}(h^{\boldsymbol \phi}_1, ..., h^{\boldsymbol \phi}_{|{\boldsymbol \phi}|}) \label{eq:encoder_compress}
\end{equation}
where $h_t \in \mathcal{R}^n$ is a hidden state at time $t$. $\mathbf{z}$ is a vector generated from sequence of the hidden states. We will call $f_\mathrm{compress}(\cdot)$ the compression function.
The decoder is trained to predict the next word $x'_t$ given the sentence representation $\mathbf{z}$ and all the previously predicted words $(x'_1, ..., x'_{t-1})$. With an RNN, the conditional probability distribution of $x'_t$ is modeled as:
\begin{equation}
p(x'_t|x'_1,...,x'_{t-1},\mathbf{z})=g(s_t,c_t) \label{eq:conditional_word_prob}
\end{equation}
and
\begin{eqnarray}
s_1,...,s_{|{\boldsymbol \phi}|} = \mathrm{RNN}_\mathrm{dec}( x'_1, ..., x'_{|{\boldsymbol \phi}|}) \\
c_t = f_\mathrm{retrieve}(\mathbf{z}, s_t) \label{eq:decoder_retrieve}
\end{eqnarray}
where $g(\cdot)$ is a nonlinear, potentially multi-layered, function that outputs the probability of $x'_t$, $s_t$ is the hidden state of decoder RNN, and $f_\mathrm{retrieval}$ takes $s_t$ as the key to retrieve related information from $\mathbf{z}$. We note that other architectures such as a CNN or a transformer \cite{vaswani2017attention} can be used in place of the RNN. The details of the compression function and retrieval function will be discussed in Sec. \ref{sec:compress_and_retrieve}.
\paragraph{Prior}
We draw a sample, conditioned on $({\boldsymbol \eta}, \ell)$, through the prior, which is described using following equations:
\begin{eqnarray}
h^{\boldsymbol \eta}_1,...,h^{\boldsymbol \eta}_{|{\boldsymbol \eta}|} &=& \mathrm{RNN}_\mathrm{enc}(x^{\boldsymbol \eta}_1, ..., x^{\boldsymbol \eta}_{|{\boldsymbol \eta}|}) \label{eq:hypo_encoder_rnn} \\
\tilde{h}_t &=& \mathrm{MLP}([h^{\boldsymbol \eta}_t, e_\ell, \epsilon]) \\
\hat{h}_1,...,\hat{h}_{|{\boldsymbol \eta}|} &=& \mathrm{RNN}_\mathrm{refine}(\tilde{h}_1, ..., \tilde{h}_{|{\boldsymbol \eta}|}) \\
\mathbf{z} &=& f_\mathrm{compress} ( \hat{h}_1,...,\hat{h}_{|{\boldsymbol \eta}|} )
\end{eqnarray}
where $\epsilon$ is a random vector, $\epsilon_i \sim \mathcal{N}(0,1)$; $e_\ell$ is the label embedding and $[\cdot, \cdot]$ represents the concatenation of input vectors.
\paragraph{Classifier}
This outputs the probability distribution over labels, taking as input the tuple $(\mathbf{z},{\boldsymbol \eta})$, and is described using the following equations:
\begin{eqnarray}
h^{\boldsymbol \eta}_1,...,h^{\boldsymbol \eta}_{|{\boldsymbol \eta}|} &=& \mathrm{RNN}_\mathrm{enc}(x^{\boldsymbol \eta}_1, ..., x^{\boldsymbol \eta}_{|{\boldsymbol \eta}|}) \label{eq:hypo_encoder_rnn} \\
c_t &=& f_\mathrm{retrieve}(\mathbf{z}, h^{\boldsymbol \eta}_t) \label{eq:cls_retrieve} \\
\tilde{h}_t &=& \mathrm{MLP}([h^{\boldsymbol \eta}_t, c_t, || h^{\boldsymbol \eta}_t - c_t ||, h^{\boldsymbol \eta}_t \odot c_t]) \label{eq:classifier_combine}\\
\hat{h}_1,...,\hat{h}_{|{\boldsymbol \eta}|} &=& \mathrm{RNN}_\mathrm{refine}(\tilde{h}_1, ..., \tilde{h}_{|{\boldsymbol \eta}|}) \\
\hat{h}_\mathrm{max} &=& \mathrm{Pooling}_\mathrm{max}(\hat{h}_1,...,\hat{h}_{|{\boldsymbol \eta}|}) \\
\hat{h}_\mathrm{mean} &=& \mathrm{Pooling}_\mathrm{mean}(\hat{h}_1,...,\hat{h}_{|{\boldsymbol \eta}|}) \\
p(\ell|\mathbf{z},{\boldsymbol \eta}) &=& \sigma(\mathrm{MLP}([\hat{h}_\mathrm{max}, \hat{h}_\mathrm{mean}]))
\end{eqnarray}
where $\mathrm{Pooling}(\cdot)$ refers to an element-wise pooling operator, and the activation function $\sigma$ for output layer is the softmax function. The architecture of the classifier is inspired by \citep{chen2017enhanced}. Instead of doing attention over the sequence of hidden states for the premise, we use the retrieval function in Equation \ref{eq:cls_retrieve} to retrieve related information $c_t$ in $\mathbf{z}$ for $h^{\boldsymbol \eta}_t$.
\paragraph{Discriminator}
The discriminator takes as input $(\mathbf{z}, {\boldsymbol \eta},\ell)$, and tries to determine if the $\mathbf{z}$ in question comes from the encoder or prior. The architecture of the discriminator is similar to that of the classifier, with the exception that Equation \ref{eq:classifier_combine} is replaced by:
\begin{eqnarray}
\tilde{h}_t &=& \mathrm{MLP}([h^{\boldsymbol \eta}_t, c_t, e_\ell])
\end{eqnarray}
to pass label information to the discriminator. The sigmoid function is used as the activation for the output layer.
In our model the autoencoder, prior and classifier share the same $\mathrm{RNN}_\mathrm{enc}(\cdot)$ parameters. The prior and the autoencoder share the same $f_\mathrm{compress}(\cdot)$ parameters. The classifier and the autoencoder share the same $f_\mathrm{retrieve}(\cdot)$ parameters. The discriminator does not share any parameters with the rest of model.
\subsection{Compression and Retrieval Functions} \label{sec:compress_and_retrieve}
The compression (Equation \ref{eq:encoder_compress}) and retrieval (Equation \ref{eq:decoder_retrieve}) functions can be modeled through many different mechanisms. Here, we introduce two different methods:
\paragraph{Mean Pooling} can be used to compress the sequence of the hidden states:
\begin{equation}
f_\mathrm{compress}(h_1, ..., h_T) = \frac{1}{T}\sum_{t=1}^{T}h_t
\end{equation}
and its retrieve counterpart directly returns $\mathbf{z}$:
\begin{equation}
f_\mathrm{retrieve}(\mathbf{z}, s_t)=\mathbf{z}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{figures/memory}
\caption{\textit{Memory Operation Selection Module} takes a pair of vector $(\mathbf{k}, \mathbf{v})$ as input, output a vector $\mathbf{o}$. $\mathbf{k}$ provide the control signal for the layer to compute a weighted sum of candidate weight matrices. The obtained matrix is used as the weight matrix in a normal feedforward layer, that takes $\mathbf{v}$ as input and outputs $\mathbf{o}$.}
\label{fig:map_function}
\end{figure}
\paragraph{Memory Operation Selection Module (MOSM)}
As an alternative to mean pooling, we use the architecture shown in Figure \ref{fig:map_function}. A layer is defined as:
\begin{eqnarray}
\gamma &=& \mathrm{softmax}(\mathbf{\Omega}\mathbf{k}) \\
\tilde{\mathbf{W}} &=& \sum_{i=1}^{N_\mathbf{W}} \gamma_i\mathbf{W}_i \\
\mathbf{o} &=& \sigma(\tilde{\mathbf{W}}\mathbf{v})
\end{eqnarray}
where $\sigma$ can be any activation function, $\mathbf{v}$ is the input vector, $\mathbf{k}$ is the control vector, $\{ \mathbf{W}_i \}$ are $N_\mathbf{W}$ candidate weight matrices.
For convenience, we denote the MOSM function as $f_\mathrm{MOSM}(\mathbf{v},\mathbf{k})$.
Thus, we can define the MOSM compression method as:
\begin{equation}
f_\mathrm{compress}(h_1, ..., h_T) = \tanh \left( \frac{1}{T}\sum_{t=1}^{T} f_\mathrm{MOSM} \left( h_t,h_t \right) \right)
\end{equation}
The compression function uses $\{h_t\}$ as both control and input vector, to write themselves into $\mathbf{z}$. Because different $h_t$s select different combinations of candidate matrices, we can have different mapping function each different $h_t$ at each time step.
\begin{equation}
f_\mathrm{retrieve}(\mathbf{z}, s_t) = f_\mathrm{MOSM}(\mathbf{z},s_t)
\end{equation}
Retrieval functions use $\{s_t\}$ as control vectors to retrieve information from $\mathbf{z}$. Since the layer generates a different weight matrix for the feedforward path for different $s_t$, we can output different $\mathbf{o}$ for the same $\mathbf{z}$.
\subsection{Model Learning}
Like most adversarial networks, the conditional adversarial autoencoder is trained with a gradient descent based method in two phases: the \textit{generative} phase and the \textit{discriminative} phase.
In the \textit{generative} phase, the autoencoder is updated to minimize the reconstruction error of the premise. The classifier and the encoder are updated to minimize the classification error of the premise-hypothesis pair. The prior is also updated to optimize the classification error of $p(\ell|\mathbf{z},{\boldsymbol \eta})$, where $\mathbf{z}$ is draw from the prior. The encoder and the prior are updated to confuse the discriminator.
In our initial experiments, we found that the samples from just the adversarial training alone results in wildly varied output sentences. To ameliorate this, we propose an \textit{auxiliary loss}:
\begin{equation}
\mathcal{L}_\mathrm{auxiliary} = \min_{i\in (1,...,N)} \{ \mathrm{NLL}({\boldsymbol \phi}|\mathbf{z}_i) \}\ , \quad \mathbf{z}_i \sim p(\mathbf{z}|{\boldsymbol \eta},\ell) \label{eqn:aux_loss}
\end{equation}
where $N$ is the number of samples that are drawn from prior. The auxiliary loss measures how far our generated premises are from the true premise when conditioned on the hypothesis and label.
As shown in experiment the model has better generating diversity, while more samples were drawn during training.
One can view this auxiliary loss as a `hard' version of taking the log average of the probability of $N$ Monte-Carlo samples,
\begin{align}
&-\log \E_{p(\mathbf{z}|{\boldsymbol \eta},\ell)}\left[p({\boldsymbol \phi}|\mathbf{z})\right] \\
&~\approx -\log \frac{1}{N} \sum_i^N p({\boldsymbol \phi}|\mathbf{z}_i) , \qquad \mathbf{z}_i \sim p(\mathbf{z}|{\boldsymbol \eta},\ell) \\
&~= -\log \frac{1}{N} - \log \sum_i^N \exp \log p({\boldsymbol \phi}|\mathbf{z}_i) \\
&\leq -\log \frac{1}{N} + \min_{i\in (1,...,N)} -\log p({\boldsymbol \phi}|\mathbf{z}_i) \label{eqn:approx_loss}
\end{align}
Since $\log \frac{1}{N}$ is a constant, minimizing over the Equation \ref{eqn:approx_loss} is the same as minimizing Equation \ref{eqn:aux_loss}.
In the \textit{discriminative} phase, the discriminator is updated to tell apart the true $\mathbf{z}$ (generated using the prior) from the generated samples (given by autoencoder).
\section{Experiments} \label{sec:experiments}
We use the Stanford Natural Language Inference (SNLI) corpus \cite{bowman2015large} to train and evaluate our models.
From our experiments, we want to determine two things.
First, do the sentences produced by the model form the correct textual entailment class on which it was conditioned on? Second, is there diversity among the sentences that are generated?
\subsection{Baseline Methods} \label{sec:baseline}
For comparison, we use a normal RNN encoder-decoder as a baseline method. The model uses a bidirectional LSTM network as encoder. The encoder reads the input hypothesis into a sequence of hidden states $\{ h_t \}$:
\begin{eqnarray}
h^{\boldsymbol \eta}_1,..,h^{\boldsymbol \eta}_{|{\boldsymbol \eta}|} &=& \mathrm{RNN}_\mathrm{enc}(x^{\boldsymbol \eta}_1,..,x^{\boldsymbol \eta}_{|{\boldsymbol \eta}|}) \\
z_{\boldsymbol \eta} &=& f_\mathrm{compress}(h^{\boldsymbol \eta}_1,..,h^{\boldsymbol \eta}_{|{\boldsymbol \eta}|})
\end{eqnarray}
Where $f_\mathrm{compress}(\cdot)$ can be the mean method or an MOSM.
The distributed representation of label $e_\ell$ and $z_{\boldsymbol \eta}$ are concatenated together to feed into a normal MLP network, which output the sentence representation $\mathbf{z}$:
\begin{eqnarray}
\mathbf{z} = \mathrm{MLP}([ z_{\boldsymbol \eta}, e_\ell ])
\end{eqnarray}
The decoder compute the conditional probability distribution with equations:
\begin{eqnarray}
p(x'_t|x'_1,...,x'_{t-1})=g([s_t,\mathbf{z}]) \\
s_t = \mathrm{RNN}_\mathrm{dec}(x'_{t-1},s_{t-1})
\end{eqnarray}
Thus, the baseline model share a similar architecture with prior and decoder in our model, while the randomness been toke out.
\subsection{Experiment Settings} \label{sec:settings}
For all models, $\mathrm{RNN}_\mathrm{enc}$ and $\mathrm{RNN}_\mathrm{refine}$ are 2-layers bi-directional LSTM \citep{hochreiter1997long}, $\mathrm{RNN}_\mathrm{dec}$ are 2-layers uni-directional LSTM. The dimension of hidden state, embeddings and latent representation $\mathbf{z}$ are 300.
When training, optimization is performed with Adam using learning rate $lr = 0.001$, $\beta_1 = 0$, $\beta_2 = 0.999$ and $\sigma = 10^{-8}$. We carry out gradient clipping with maximum norm $1.0$. We train each model for 30 epoch. For each iteration, we randomly choose to run the generative phase or discriminative phase with probability $0.5:0.5$. Since we didn't observe significant benefit from using Beam Search, all premises are generated using greedy search.
\subsection{Quality Evaluation} \label{sec:quality}
In order to evaluate the quality of the samples from our model, we trained two state-of-the-art models for SNLI: (1) Densely Interactive Inference Network (DIIN)\footnote{https://github.com/YichenGong/Densely-Interactive-Inference-Network} \cite{gong2017natural}, (2) Enhanced Sequential Inference Model (ESIM)\footnote{https://github.com/lukecq1231/nli} \cite{chen2017enhanced}.
In our experiments, we found that it is possible to achieve an accuracy of 68\% on SNLI label prediction by training a classifier using \emph{only} the hypothesis as input.
This calls into question how much the classification models rely on just the hypothesis for performing its task.
To investigate this phenomena further, we randomly permuted the premises of the original test set and passed these new (random) permis-hypothesis pairs to the classifiers. The results are shown in the row labelled \textsc{Random} in Table \ref{table:rte_score}. We were satisfied that at 42.7\% and 41.1\%, the classification models (both DIIN and ESIM) were not relying entirely on the hypothesis for prediction.
\begin{table}[t]
\caption{Classification accuracies for different state-of-the-art models on our samples. The row labeled \textsc{Random} we randomly permuted the premises of the original test set and ran them through the classifiers to test for the models' reliance on just the hypothesis for classification.}
\label{table:rte_score}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccr}
\hline
\abovespace\belowspace
Model & DIIN & ESIM \\
\hline
\abovespace
Random & 42.7\% & 41.1\% \\
\hline
\abovespace
Baseline (Mean) & 59.6\% & 59.6\% \\
Baseline (MOSM) & 62.7\% & 62.6\% \\
\hline
\abovespace
MOSM (N=1, -classifier) & 67.2\% & 67.3\% \\
MOSM (-auxiliary loss) & 63.2\% & 60.6\% \\
\hline
\abovespace
Mean (N=1) & 64.4\% & 62.4\% \\
Mean (N=10) & 64.3\% & 62.3\% \\
MOSM (N=1) & 76.1\% & 75.9\% \\
MOSM (N=10) & 72.6\% & 71.8\%\\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
We sampled 9845 hypotheses from the test set, and produced ${\boldsymbol \phi}$ for each example with the given $\ell$. The $({\boldsymbol \eta}, {\boldsymbol \phi}, \ell)$ triplet was then passed to the classifiers and evaluated for accuracy. Both classification models perform at $\sim$88\% accuracy, but, while they were not perfect, they provided a good probe for how well our models were generating the required sentences. Table \ref{table:rte_score} shows the accuracy of prediction on the respective models.
Both the DIIN and ESIM models give similar results.
Our results show that using the MOSM gives an improvement over just taking the mean.
Using the adversarial training also results in some gains, which suggests that training the model with the `awareness' of the distribution over the representation space results in better quality samples.
Using the adversarial training in conjunction with the MOSM layer gives us the model with the best performance.
We also performed ablation tests, removing certain components of the model from the training to see how it affects the quality of samples.
The difference between our best model against \textsc{MOSM ($N=1$, -classifier)} suggests that the classifier plays in important role in ensuring $\mathbf{z}$ is a representation in the right class. In our experiment removing the auxiliary loss, we still achieve an accuracy $\sim$61\%.
However, looking at the samples for this iteration of the model, while having some concepts in common with the hypothesis, the sentences in general are more nonsensical in comparison to those trained with the auxiliary loss (See an example in Figure \ref{fig:samples}).
\begin{table}[t]
\caption{The confusion matrix for the samples from the best model \textsc{MOSM} ($N = 1$)}
\vskip 0.15in
\label{table:conf_mat}
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lccc}
\hline
\abovespace\belowspace
Label \textbackslash Pred. & Ent. & Neut. & Cont. \\
\hline
\abovespace
Entailment & 67.8\% & 20.9\% & 11.4\% \\
Neutral & 6.6\% & 76.7\% & 16.7\% \\
Contradiction & 2.9\% & 12.8\% & 84.4\% \\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
The confusion matrix produced when evaluating our best model ({\textsc{MOSM}, $N=1$}) on DIIN shows us where the classification model and our generative model agree (See Table \ref{table:conf_mat}).
In our \textsc{Random} experiments, we find that the model has a bias towards predicting contradictions. This is observed here as well, with contradictions being the category with the highest agreement.
We therefore cannot conclude that contradictions are easier for our model to generate.
Also, using the original test set, the category in which DIIN performs the best is entailment, with a precision of 89.1\% compared to 84.3\% for neutral and 88.4\% for contradiction. This suggests that generating suitable premises that entail the hypothesis is the hardest task for the model.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/prec}
\caption{Different classification precisions given by our classifier in our model (MOSM, N=10) during training. Sample Precision shows the probability that classifier predicts correct label for generated premise and related real hypothesis. Valid precision shows the probability that classifier predicts correct label for real premise and real hypothesis. Z precision shows the probability that the feedforward network $f_\mathrm{classifier}(\mathbf{z},z_{\boldsymbol \eta})$ predicts correct label $\ell$, for given ${\boldsymbol \eta}$, $\ell$ and $\mathbf{z}$ drawn from prior $p(\mathbf{z}|{\boldsymbol \eta},\ell)$.}
\label{fig:precision}
\end{figure}
We also want to study how the classifier component of our model affects the generation of good samples. As shown in Figure \ref{fig:precision}, ``Z precision'' is higher then $0.9$. This suggests that the classifier provides a strong regularization signal to the sentence representation $\mathbf{z}$. Because the autoencoder is not perfect, we do not observe the the same sample classification precision after $\mathbf{z}$ is decoded. However, we still observe a synchronous improvement of both sample and valid precision. It is therefore reasonable to expect that a better classifier and a better autoencoder would result in better generated premises.
\subsection{Diversity Evaluation} \label{sec:diversity}
In order to evaluate the diversity of samples given by our model, we compute the BLEU score between to premises generated conditioned on the same hypothesis and label. In other words, given a triple $({\boldsymbol \phi}_i, {\boldsymbol \eta}_i, \ell_i)$ from test set, we draw two different samples $(\mathbf{z}_{i1}, \mathbf{z}_{i2})$ from the prior distribution $p(\mathbf{z}|{\boldsymbol \eta}_i,\ell_i)$. Then the decoder generates two premises $({\boldsymbol \phi}_{i1}, {\boldsymbol \phi}_{i2})$ using greedy search conditioned on $(\mathbf{z}_{i1}, \mathbf{z}_{i2})$ respectively. The similarity score between generated premises is then estimated by:
\begin{equation}
\mathrm{BLEU}_i = \frac{1}{2} \left( \mathrm{BLEU}(\mathbf{z}_{i1}, \mathbf{z}_{i2}) + \mathrm{BLEU}(\mathbf{z}_{i2}, \mathbf{z}_{i1}) \right).
\end{equation}
For comparison, we also compute the BLEU score between real premise and generated premise $({\boldsymbol \phi}_i, {\boldsymbol \phi}_{i1})$. The average of diversity score between two generated premises is noted as $\mathrm{BLEU}_\mathrm{SS}$, the one between real and generated premises is noted as $\mathrm{BLEU}_\mathrm{RS}$. Since it is not necessary have n-gram match between premises, BLEU score can be inaccurate on some data points. We employ the Smoothing technique 2 described in \citet{chen2014systematic}.
\begin{table}[t]
\caption{BLEU score for different models}
\label{table:div_score}
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcc}
\hline
\abovespace\belowspace
Model & $\mathrm{BLEU}_\mathrm{RS}$ & $\mathrm{BLEU}_\mathrm{SS}$ \\
\hline
\abovespace
Baseline (Mean) & 14.4 & N/A \\
Baseline (MOSM) & 14.7 & N/A \\
\hline
\abovespace
MOSM (N=1, -classifier) & 14.4 & 46.7 \\
MOSM (-auxiliary loss) & 10.3 & 14.8 \\
\hline
\abovespace
Mean (N=1) & 11.9 & 27.9 \\
Mean (N=10) & 11.3 & 17.3 \\
MOSM (N=1) & 14.2 & 38.9 \\
MOSM (N=10) & 13.2 & 22.5 \\
\hline
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table}
As shown in Table \ref{table:div_score}, when we increase the number of samples $N$ in the auxiliary loss, the diversity of samples increases for both mean pooling and \textsc{MOSM}. This can serve as empirical evidence that the diversity of our model can be controlled by choosing a different hyper-parameter $N$.
The higher $\mathrm{BLEU}_\mathrm{RS}$ given by MOSM method could be interpreted as real premise is more close to the center of mass of prior distribution.
We also observe a gap between $\mathrm{BLEU}_\mathrm{RS}$ and $\mathrm{BLEU}_\mathrm{SS}$. The gap shows that the sampled premise is still relatively similar between themselves.
After removing the classifier, we observe an increase in $\mathrm{BLEU}_\mathrm{SS}$. One possible explanation is that classifier prevents the prior from overfitting the training data.
We observe an decrease in both BLEU scores, after removing the auxiliary loss. However, Table \ref{table:rte_score} and Figure \ref{fig:samples} shows that removing auxiliary loss give low quality samples.
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/z}
\caption{Visualization of the effect of auxiliary loss with multiple samples. For a pair of $({\boldsymbol \phi}, {\boldsymbol \eta})$, we repeat 100 times the process of compute auxiliary loss (N=10) in Equation \ref{eqn:aux_loss}. Blue points represent $\mathbf{z}_i$ selected by minimum function, green points represent $\mathbf{z}_i$ that are not selected. Our model (\textsc{MOSM}, N=10) is used for computing $\mathbf{z}$ and perplexities. t-SNE is used to visualize high-dimensional data \citep{maaten2008visualizing}.}
\label{fig:z}
\end{figure}
While the auxiliary loss is essential for the prior and the decoder to learn to cooperate, using an auxiliary loss where $(N=1)$ will collapse the prior distribution; instead of a distribution, the prior will learn to ignore the random input and deterministically predict $\mathbf{z}$.
As shown in Figure \ref{fig:z}, the auxiliary loss $(N=10)$ only passes gradients to the $\mathbf{z}$s in the left region of the distribution.
As a result, samples drawn from right region have a significant lower chance receive gradient from decoder, while the entire region receives gradients from the discriminator and classifier.
Therefore, the prior distribution can expand to more regions, but only those regulated by discriminator and classifier. This will increase the diversity of samples.
However, we also observe that the precision slightly decreases in Table \ref{table:rte_score}. This suggests that the discriminator and classifier in our model are not perfect for regularizing the prior distribution.
\subsection{Samples}
\begin{figure}[t]
\scriptsize
\begin{framed}
\fontsize{8}{9}\selectfont
\textsc{\textbf{Samples from MOSM (N=10)}}\\
\\
\textbf{H:} a worker stands over a bread display . \\
\textbf{L:} Entailment \\
\textbf{S1:} a man in a blue shirt is preparing food in a kitchen . \\
\textbf{S2:} a man in a blue shirt is washing a window . \\
\textbf{H:} there is a jockey riding a horse . \\
\textbf{L:} Entailment \\
\textbf{S1:} a horse rider on a bucking horse . \\
\textbf{S2:} a jockey riding a horse in a rodeo . \\
\textbf{H:} a man sitting on the couch reading a book . \\
\textbf{L:} Contradiction \\
\textbf{S1:} a man is sitting on a bench with his hands in his pockets . \\
\textbf{S2:} a man in a blue shirt is standing in front of a store . \\
\textbf{H:} a baby in his stroller outside . \\
\textbf{L:} Contradiction \\
\textbf{S1:} a woman is sitting on a bench next to a baby . \\
\textbf{S2:} a woman is sitting on a bench in a park . \\
\textbf{H:} the man is being watched . \\
\textbf{L:} Neutral \\
\textbf{S1:} a man jumps from a bridge for an elderly couple at a beach . \\
\textbf{S2:} a man in a blue shirt is standing in front of a building .\\
\textbf{H:} there is a human selling hot dogs . \\
\textbf{L:} Neutral \\
\textbf{S1:} a person is standing in front of a food cart . \\
\textbf{S2:} a woman in a white shirt is standing in front of a counter selling food .
\noindent\rule{7.5cm}{0.4pt}
\textsc{\textbf{Samples from MOSM (-auxiliary loss)}}\\
\\
\textbf{H:} a restaurant prepares for a busy day .\\
\textbf{L:} Neutral\\
\textbf{S1:} a pink teenager prepares on a tune on the roots .\\
\textbf{S2:} a UNK restaurant dryer for a canvas .
\end{framed}
\vspace*{-0.5cm}
\caption{\label{examples} Example sentence generated by our model (\textsc{MOSM}, N=10). \textbf{H} is the hypothesis, \textbf{L} is the label, \textbf{S1} is the first sample, and \textbf{S2} is the second sample. The samples shown below the line are drawn from a model trained without the auxiliary loss.}
\label{fig:samples}
\end{figure}
Figure \ref{fig:samples} shows several examples generated by our model (\textsc{MOSM}, N=10). These example shows that our model can generate a variety of different premise while keep the correct semantic relation. Some of subjects in hypothesis are correctly replace by synonyms (e.g. ``jocky'' is replaced by ``horse rider'', ``human'' is replaced by ``person'' and ``woman'').
The model also get some potential logical relation correct (e.g. ``reading a book'' is contradicted by ``with his hands in his pockets'', ``stands over a bread display'' can either means ``washing a window'' or ``preparing food in a kitchen'').
However, we also observe that the model tries to add ``a blue shirt'' for most ``man''s in the sentences, which is one of the easiest way to add extra information into the model. The phenomenon aligned with well-known \textit{model collapse} failure case for most adversarial training based method. This observation give an explanation for the relatively higher BLEU between sample.
The model also have some bias while generating premise (e.g. when hypothesis mention ``a baby'', the premise automatically mention ``a woman''), which aligns with the recent discovery that visual recognition tasks model tend to output biased predictions \citep{zhao2017men}.
\section{Discussion} \label{sec:discussion}
The broader vision of our project is to attain logical control for language, which we believe will allow us to perform better across many natural language applications.
This is most easily achieved at the word-level, by adding or removing specific words to a sentence, using word generation rules based on language-specific grammars. However, just as distributed word representations can be meaningfully combined \cite{mikolov2013distributed} with good outcomes, we believe that sentence-level representations are the way forward for manipulation of text.
The kind of control we seek to model, specifically, is characterized by the logical relationships between sentence pairs.
Controlling semantic representation by modeling logical relationship between the input and output sentences has many potential use cases.
Returning to the task of multi-document summarization discussed in the introduction,
operating in the semantic space allows one to abstract the information of a document.
Controlling the logical relationships among sentences provides a new way to think about what a summary is.
Ideally, when multiple sources of information are given, we would like the output summary ${\boldsymbol \phi}$ generated by a machine to be entailable by the union of inputs $(\cup_{j\in\mathcal{J}} {\boldsymbol \eta}_j) \models {\boldsymbol \phi}$ \footnote{Here we assume there are no conflicting details.}.
This addresses the problem of precision: the resulting summary now has a subset of the information available in the union of all the given hypotheses.
To address the problem of recall,
we need the resulting summary to entail each one of the individual hypotheses: $\wedge_i({\boldsymbol \phi} \models {\boldsymbol \eta}_j)$
Together, these two criteria form a clear formal definition for multi-document summarization,
$$ \{\,{\boldsymbol \phi} \,: \,\wedge_i({\boldsymbol \phi} \models {\boldsymbol \eta}_j)\,\wedge\,\,(\cup_{j\in\mathcal{J}} {\boldsymbol \eta}_j) \models {\boldsymbol \phi} \,\}$$
which represents the set of all possible ${\boldsymbol \phi}$ that fit the criteria.
In our paper, we toyed with the possibility of modeling the set
$ \{\,{\boldsymbol \phi} \,: \,{\boldsymbol \phi} \models {\boldsymbol \eta}\,\}$ by training a model with a distribution over different premises in the latent space $\mathbf{z}$.
A good subsequent step would be modelling the first part of our logical description of multi-document summarisation,
$$ \{\,{\boldsymbol \phi} \,: \,\wedge_{i \in \mathcal{J}}({\boldsymbol \phi} \models {\boldsymbol \eta}_j)\,\} = \cap_{i \in \mathcal{J}} \{\,{\boldsymbol \phi} \,: \,{\boldsymbol \phi} \models {\boldsymbol \eta}_j\,\}$$
This suggests a possible avenue for producing such a premise is finding the intersection of the distribution over $\mathbf{z}$ for two given hypotheses that are likely enough to occur.
Future work can explore the possibility of this and determining the union of the hypotheses entailing the given premise.
\section{Conclusion} \label{sec:conclusion}
We have proposed a model that generates premises from hypotheses with an intermediate latent space, which we interpret as different possible premises for a given hypothesis. This was trained using a Conditional Adversarial Autoencoder.
This paper also proposed the Memory Operation Selection Module for encoding sentences to a distributed representation that uses attention over different operations in order to encode the input.
The model was evaluated for quality and diversity. In terms of quality, we used two state-of-the-art models for the RTE task on SNLI, and the samples generated by our best model were able to achieve an accuracy of 76.1\%. For diversity, we compared the BLEU scores between the real premises and the generated premises, and the BLEU scores between the generated premises. In this regard, while our model is able to generate different premises for each hypothesis, there is still a gap between when compared to the similarities to the real premises. Looking at the samples, we note that the additional details that our model generates tend to repeat, and correspond to some type of mode collapse.
The task of performing reasoning well with natural language still remains a challenging problem. Our experiments demonstrate that while we can generate sentences with the logical entailment properties we desire, there is still much to be done in this direction. We hope with the new lens on some NLP tasks as natural language manipulation with logical control, new perspectives and methods will emerge to improve the field.
|
{
"timestamp": "2018-03-08T02:09:45",
"yymm": "1803",
"arxiv_id": "1803.02710",
"language": "en",
"url": "https://arxiv.org/abs/1803.02710"
}
|
\section{Introduction}
\IEEEPARstart{I}{n} this paper, we study the distributed linear quadratic optimal control problem for multi-agent networks.
In this problem, we are given a number of identical agents represented by a finite dimensional linear input-state system, and an undirected graph representing the communication between these agents.
Given is also a quadratic cost functional that penalizes the differences between the states of neighboring agents and the size of the local control inputs.
The distributed linear quadratic problem is then to find a distributed diffusive control law that, for given initial states of the agents, minimizes the cost functional, while achieving consensus for the controlled network.
This problem is non-convex and difficult to solve, and it is unclear whether a solution exists in general \cite{lunze_conf_2014}.
Therefore, in this paper, instead of addressing the version formulated above, we will study a {\em suboptimal} version of the distributed optimal control problem.
Our aim is to design suboptimal distributed diffusive control laws that guarantee the controlled network to reach consensus and the associated cost to be smaller than an a priori given upper bound.
In the past, there has been work on the distributed optimal control problem before.
In \cite{jan_mm2015}, \cite{shen_zeng2017} and \cite{7862732}, it is shown that diffusive couplings are necessary for minimizing a cost functional that integrates a quadratic form involving state differences and inputs.
However, these papers do not provide a design method for finding an optimal distributed control law.
On the other hand, there has been some work on the design of distributed diffusive control laws.
It is shown in \cite{tuna2008lqr} and \cite{hongwei_zhang2011} that, using the distributed control law derived from the solution of a local algebraic Riccati equation, synchronization is achieved with sufficiently large coupling gain. However, no cost functionals were taken into consideration.
In \cite{tamas2008}, a design method was introduced for computing distributed suboptimal controllers, which requires the solution of a single LQR problem whose size depends on the maximum node degree of the communication graph.
In \cite{wei_ren2010}, the authors consider a distributed optimal control problem for multi-agent systems with single integrator agent dynamics, and obtain an expression for the optimal distributed diffusive control law.
In addition, a distributed optimal control problem was considered from the perspective of cooperative game theory in \cite{Semsar2009}. The problem there is then solved by transforming it into a maximization problem for LMI's, taking into consideration the structure of the graph Laplacian.
For related work we also mention \cite{lunze_ecc_2013}, \cite{guaranteed_cost}, \cite{Lunze_ijc_2014} and \cite{Fazelnia2017}, to name a few.
Also, in \cite{Nguyen2015}, a hierarchical control approach was introduced for linear leader-follower multi-agent systems.
For the case that the weighting matrices in the cost functional are chosen to be of a special form, two suboptimal controller design methods are given.
In addition, in \cite{kristian2014}, an inverse optimal control problem was addressed both for leader-follower and leaderless multi-agent systems.
For a class of digraphs, the authors show that distributed optimal controllers exist and can be obtained if the weighting matrices are assumed to be of a special form, capturing the graph information.
For other papers related to distributed inverse optimal control, see also \cite{huaguang_zhang2015}, \cite{nguyen_2017}.
As announced before, in this paper our objective is to design distributed diffusive control laws that guarantee the controlled network to reach consensus and the associated cost to be smaller than an a priori given upper bound.
The main contributions of the paper are the following:
\begin{enumerate}
\item
We present two design methods for computing suboptimal distributed diffusive control laws, both based on computing a positive semi-definite solution of a single Riccati inequality of dimension equal to the dimension of the agent dynamics.
%
In the computation of the local control gain, the smallest nonzero eigenvalue and the largest eigenvalue of the graph Laplacian are involved.
\item
For the case that exact information on the smallest nonzero eigenvalue and the largest eigenvalue of the graph Laplacian is not available, we establish a design method using only lower and upper bounds on these Laplacian eigenvalues.
\end{enumerate}
The remainder of this paper is organized as follows.
In Section \ref{sec_notation}, we introduce the basic notation and formulate the suboptimal distributed linear quadratic control problem.
Section \ref{sec_single_sys} presents the analysis and design of suboptimal linear quadratic control for linear systems,
collecting preliminary results for treating the actual suboptimal distributed control problem for multi-agent systems.
Then, in Section \ref{sec_mas}, we study the suboptimal distributed control problem for linear multi-agent systems.
In addition, a simulation example is provided in Section \ref{sec_simulation} to illustrate our results.
Finally, Section \ref{sec_conclusion} concludes this paper.
\section{Notation and Problem Formulation}\label{sec_notation}
\subsection{Notation}
We denote by $\mathbb{R}$ the field of real numbers, by $\mathbb{R}^{n\times m}$ the set of $n\times m$ real matrices.
For a given matrix $A$, its transpose and inverse (if it exists) are denoted by $A^{\top}$ and $A^{-1}$, respectively.
The identity matrix of dimension $n \times n$ is denoted by $I_n$.
We denote the Kronecker product of two matrices $A$ and $B$ by $A\otimes B$, which has the property that $(A_1\otimes B_1)(A_2\otimes B_2) = A_1 A_2\otimes B_1 B_2 $.
For a given symmetric matrix $P$ we denote $P>0$ if it is positive definite and $P\geq 0$ if it is positive semi-definite.
By $\text{diag} ( a_1, a_2, \ldots, a_n )$,
we denote the $n\times n$ diagonal matrix with $a_1, a_2, \ldots, a_n $ on the diagonal.
The column vector $\mathbf{1}_n\in \mathbb{R}^n$ denotes the vector whose components are all $1$.
Throughout this paper, an undirected graph is denoted by $\mathscr{G} = (\mathscr{V},\mathscr{E})$ with nonempty finite set of $N$ nodes $\mathscr{V} = \{ v_1, v_2, \ldots, v_N \}$ and edge set $\mathscr{E} = \{ e_1, e_2, \ldots,e_M \}$.
A pair $(v_i, v_j) \in \mathscr{E}$, with $v_i, v_j\in \mathscr{V}$ and $i \neq j$, represents an edge from node $i$ to node $j$.
The graph is called undirected if $(v_i, v_j) \in \mathscr{E}$ implies $(v_j, v_i) \in \mathscr{E}$.
The neighbor set of node $i$ is denoted by $\mathscr{N}_i =\{ v_j \in \mathscr{V}:(v_i, v_j)\in \mathscr{E} \}$.
The Laplacian matrix $L$ of an undirected graph is symmetric and consequently has real eigenvalues.
For an undirected graph, all eigenvalues of Laplacian are nonnegative and it always has $0$ as an eigenvalue.
The graph is connected if and only if $0$ is a simple eigenvalue of $L$.
In the sequel, assume that $\mathscr{G}$ is connected. In that case the eigenvalues of $L$ can be ordered in increasing order as $0=\lambda_1 < \lambda_2 \leq \cdots \leq \lambda_N$ and there exists an orthogonal matrix $U$ such that
$U^{\top}LU = \text{diag} ( 0, \lambda_2, \ldots, \lambda_N )$.
Moreover, there holds that $U = \left( \frac{1}{\sqrt{N}}\mathbf{1}_N\quad U_2 \right)$ and $U_2 U^{\top}_2 = I_N - \frac{1}{N}\mathbf{1}_N\mathbf{1}_N^{\top}$.
\subsection{Problem Formulation}
In this paper, we consider a multi-agent system consisting of $N$ identical agents.
The underlying graph is assumed to be undirected and connected, and the corresponding Laplacian matrix is denoted by $L$.
The dynamics of the identical agents is represented by
the continuous-time linear time-invariant
(LTI) system given by
\begin{equation}\label{sys_i}
\dot{x}_i(t) = Ax_i(t) + Bu_i(t),\quad x_i(0) = x_{i0}, \quad i=1, 2, \ldots,N
\end{equation}
where $A\in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}$, and $x_i\in \mathbb{R}^{n}, u_i \in \mathbb{R}^{m}$ are the state and input of the $i$-th agent, respectively.
Throughout this paper, we assume that the pair $(A,B)$ is stabilizable.
We consider the infinite horizon distributed linear quadratic optimal control problem for multi-agent system (\ref{sys_i}), where the global cost functional integrates the weighted quadratic difference of states between every agent and its neighbors, and also penalizes the inputs in a quadratic form.
Thus, the cost functional considered in this paper is given by
\begin{equation}\label{cost_i}
J(u) = \int_{0}^{\infty}\frac{1}{2}\sum_{i=1}^{N}\sum_{j\in \mathscr{N}_i}(x_i-x_j)^{\top} Q (x_i-x_j) + \sum_{i=1}^{N}u_i^{\top}R u_i \ dt
\end{equation}
where $Q\geq 0$ and $R > 0$ are given real weighting matrices.
We can rewrite multi-agent system (\ref{sys_i}) in compact form as
\begin{equation}\label{net_sys_1}
\dot{x} = (I_N\otimes A) x + (I_N\otimes B)u,\quad x(0) =x_0
\end{equation}
with $x = \left( x_1^{\top},\ldots,x_N^{\top} \right)^{\top}$, $u = \left( u_1^{\top},\ldots,u_N^{\top} \right)^{\top}$, where $x\in \mathbb{R}^{nN}$, $u\in \mathbb{R}^{mN}$ contain the states and inputs of all agents, respectively.
Note that, although the agents have identical dynamics, we allow the initial states of the individual agents to differ.
These initial states are collected in the joint vector of initial states $x_0$.
Moreover, we can also write the cost functional (\ref{cost_i}) in compact form as
\begin{equation}\label{cost_all}
J(u) = \int_{0}^{\infty} x^{\top}(L\otimes Q)x +u^{\top}(I_N\otimes R)u \ dt.
\end{equation}
The distributed linear quadratic problem is the problem of minimizing the cost functional (\ref{cost_all}) over all distributed control laws that achieve consensus.
By a distributed control law we mean a control law of the form
\begin{equation}\label{controller}
u = (L\otimes K) x,
\end{equation}
where
$K \in \mathbb{R}^{m\times n}$ is an identical feedback gain for all agents.
By interconnecting the agents using this control law, we obtain the overall network dynamics
\begin{equation}\label{sys_closed}
\dot{x} = (I_N\otimes A + L \otimes BK) x.
\end{equation}
Foremost, we want the control law to achieve consensus:
\begin{defn}\label{def_consensus}
We say the network reaches consensus using control law (\ref{controller}) if for all $i,j =1,2,\ldots,N$ and for all initial conditions on $x_i$ and $x_j$, we have $$x_i(t)-x_j(t) \rightarrow 0
\text{ as } t \rightarrow \infty.$$
\end{defn}
As a function of the to be designed feedback gain $K$, the cost functional (\ref{cost_all}) can be rewritten as
\begin{equation}\label{cost_all_new}
J(K) = \int_{0}^{\infty}x^{\top}\left(L\otimes Q + L^2\otimes K^{\top} R K\right)x \ dt.
\end{equation}
In other words, the distributed linear quadratic control problem is to minimize (\ref{cost_all_new}) over all $K \in \mathbb{R}^{m\times n}$
such that the controlled network (\ref{sys_closed}) reaches consensus.
Due to the distributed nature of the control law (\ref{controller}) as imposed by the network topology, the distributed linear quadratic problem is a non-convex optimization problem.
It is therefore difficult, if not impossible, to find a closed form solution for an optimal controller, or such optimal controller may not even exist.
Therefore, as already announced in the introduction, in this paper we will study and resolve a version of this problem involving the design of suboptimal distributed control laws.
Specifically, we want to design distributed suboptimal controllers of the form (\ref{controller}) for system (\ref{net_sys_1}) such that consensus is achieved and the associated cost functional (\ref{cost_all_new}) is smaller than an a priori given upper bound.
More concretely, we will consider the following problem:
\begin{prob}\label{prob1}
Consider multi-agent system (\ref{net_sys_1}) with identical linear agent dynamics and given initial state $x(0) = x_0$.
Assume the network graph is a connected undirected graph with Laplacian $L$.
Consider the associated cost functional given by (\ref{cost_all}).
Let $\gamma >0$ be an a priori given upper bound for the cost to be achieved.
The problem is to find a distributed controller of the form (\ref{controller}) so that the controlled network (\ref{sys_closed}) reaches consensus and the cost (\ref{cost_all_new}) associated with this controller is smaller than the given upper bound, i.e., $J(K) < \gamma$.
\end{prob}
Before we address Problem \ref{prob1}, we first briefly discuss the suboptimal linear quadratic optimal problem for a single linear system.
This will be the subject of the next section.
\section{Suboptimal Control for Linear Systems}\label{sec_single_sys}
In this section, we consider the linear quadratic suboptimal problem for a single linear system.
We will first analyze the quadratic performance of a given autonomous system.
Subsequently, we will discuss how to design suboptimal control laws for a linear system with inputs.
\subsection{Suboptimality analysis for autonomous systems}
Consider the autonomous system
\begin{equation}\label{sys_auto}
\dot{x}(t) =\bar{A}x(t),\quad x(0) = x_0
\end{equation}
where $\bar{A}\in \mathbb{R}^{n\times n}$ and $x\in \mathbb{R}^{n}$ is the state.
We consider the quadratic performance of system (\ref{sys_auto}), given by
\begin{equation}\label{cost_auto}
J = \int_{0}^{\infty} x^{\top}\bar{Q} x \ dt
\end{equation}
where $\bar{Q}\geq 0$ is a given real weighting matrix.
Note that the performance $J$ is finite if system (\ref{sys_auto}) is stable, i.e., $\bar{A}$ is Hurwitz.
We are interested in finding conditions such that the performance (\ref{cost_auto}) of system (\ref{sys_auto}) is smaller than a given upper bound.
For this, we have the following lemma (see also \cite{algebraic_1997}, \cite{harry_book}):
\begin{lem}\label{lem_autonomous}
Consider system (\ref{sys_auto}) with the corresponding quadratic performance (\ref{cost_auto}).
%
The performance is finite if system (\ref{sys_auto}) is stable, i.e., $\bar{A}$ is Hurwitz.
%
In this case, it is given by
\begin{equation}\label{quad_perf}
J = x_0^{\top} Yx_0,
\end{equation}
where $Y$ is the unique positive semi-definite solution of
\begin{equation}\label{lyap_eq}
\bar{A}^{\top} Y + Y\bar{A} + \bar{Q} = 0.
\end{equation}
%
Alternatively,
\begin{equation}\label{lyap_ineq}
\begin{split}
J =\inf \{ x_0^{\top} Px_0 \ | \ P \geq 0 \text{ and } \bar{A}^{\top} P + P\bar{A} + \bar{Q} < 0 \}.
\end{split}
\end{equation}
\end{lem}
\begin{proof}
The fact that the quadratic performance (\ref{cost_auto}) is given by the quadratic expression (\ref{quad_perf}) involving the Lyapunov equation (\ref{lyap_eq}) is well-known.
We will now prove (\ref{lyap_ineq}).
Let $Y$ be the solution to Lyapunov equation (\ref{lyap_eq}) and let $P$ be a positive semi-definite solution to the Lyapunov inequality in (\ref{lyap_ineq}). Define $ X := P - Y$. Then we have
\begin{equation*}
\bar{A}^{\top} (X + Y) + (X + Y) \bar{A} + \bar{Q} < 0.
\end{equation*}
So consequently,
\begin{equation*}
\bar{A}^{\top} X + X \bar{A} <0.
\end{equation*}
Since $\bar{A}$ is Hurwitz, it follows that $X>0$.
Thus, we have $P > Y$ and hence $J \leq x_0^{\top}Px_0$ for any positive semi-definite solution $P$ to the Lyapunov inequality.
Next we will show that for any $\epsilon >0$ there exists a positive semi-definite matrix $P_{\epsilon}$ satisfying the Lyapunov inequality such that $P_{\epsilon} < Y+\epsilon I$, and consequently $x_0^{\top}P_{\epsilon}x_0 \leq J +\epsilon \|x_0 \|^2$.
Indeed, for given $\epsilon$, take $P_{\epsilon}$ equal to the unique positive semi-definite solution of
\begin{equation}\label{new_lya}
\bar{A}^{\top} P + P\bar{A} + \bar{Q} +\epsilon I= 0.
\end{equation}
Clearly then, $P_{\epsilon} = \int_{0}^{\infty} e^{\bar{A}^{\top}t} (\bar{Q} +\epsilon I)e^{\bar{A}t}\ dt$,
so $P_{\epsilon} \downarrow Y$ as $\epsilon \downarrow 0$. This proves our claim.
\end{proof}
The following theorem now yields {\em necessary} and {\em sufficient} conditions such that, for a given upper bound $\gamma >0$, the quadratic performance (\ref{cost_auto}) satisfies $J < \gamma$.
\begin{thm}\label{thm_autonomous}
Consider system (\ref{sys_auto}) with the associated quadratic performance (\ref{cost_auto}).
%
For given $\gamma > 0$, we have that $\bar{A}$ is Hurwitz and $J < \gamma$ if and only if there exists a positive semi-definite solution $P$ satisfying
\begin{align}
\bar{A}^{\top} P + P\bar{A} + \bar{Q} &< 0,\label{lya_ineq_3} \\
x_0^{\top} Px _0 &< \gamma. \label{conditions_p}
\end{align}
\end{thm}
\begin{proof}
(if)
Since there exists a positive semi-definite solution to the Lyapunov inequality (\ref{lya_ineq_3}), it follows that $\bar{A}$ is Hurwitz.
Take a positive semi-definite matrix $P$ satisfying
the inequalities
(\ref{lya_ineq_3}) and (\ref{conditions_p}).
By Lemma \ref{lem_autonomous}, we then immediately have $J \leq x_0^{\top}Px_0 <\gamma$.
(only if)
If $\bar{A}$ is Hurwitz and $J <\gamma$, then, again by Lemma \ref{lem_autonomous}, there exists a positive semi-definite solution $P$ to the Lyapunov inequality (\ref{lya_ineq_3}) such that $J \leq x^{\top}_0 P x_0 < \gamma$.
\end{proof}
\begin{rem}\label{rem_autonomous}
Theorem \ref{thm_autonomous} provides a necessary and sufficient condition for the performance of (\ref{sys_auto}) to be less than a given upper bound.
%
Given initial condition $x_0$, we can either solve equation (\ref{lyap_eq}) and compute $J$ to check whether $J < \gamma$.
%
Alternatively, there exists a positive semi-definite solution to the linear matrix inequalities (\ref{lya_ineq_3}) and (\ref{conditions_p}) if and only if $J <\gamma$.
\end{rem}
In the next subsection, we will discuss the suboptimal control problem for a linear system with inputs.
\subsection{Suboptimal control design for linear systems with inputs}
In this section, we consider the finite dimensional LTI system given by
\begin{equation}\label{sys_input}
\dot{x}(t) = Ax(t) + Bu(t),\quad x(0) =x_0
\end{equation}
where $A\in \mathbb{R}^{n\times n}, B\in \mathbb{R}^{n\times m}$, and $x\in \mathbb{R}^{n}$, $u\in \mathbb{R}^{m}$ are state and input, respectively.
Assume that the pair $(A,B)$ is stabilizable.
The associated cost functional is given by
\begin{equation}\label{cost_input}
J(u) = \int_{0}^{\infty} x^{\top}Q x + u^{\top}R u \ dt
\end{equation}
where $Q \geq 0$ and $R > 0$ are given weighting matrices that penalize the state and input, respectively.
We want to find a state feedback control law $u=Kx$ such that the closed system
\begin{equation}\label{close_sys}
\dot{x}(t) = (A + BK)x(t),\quad x(0) =x_0
\end{equation}
is stable and, for a given upper bound $\gamma >0$, the corresponding cost
\begin{equation}\label{cost_k}
J(K) = \int_{0}^{\infty} x^{\top}(Q + K^{\top} R K) x \ dt
\end{equation}
satisfies $J(K) < \gamma$.
The following theorem gives us a sufficient condition for the existence of such control law.
\begin{thm}\label{thm_single_sys}
Consider system (\ref{sys_input}) with the associated cost functional (\ref{cost_input}).
Assume that the pair $(A,B)$ is stabilizable. Let $\gamma >0$.
%
Suppose that there exists
a positive semi-definite $P$ satisfying
\begin{align}
A^{\top} P + PA - PBR^{-1}B^{\top}P + Q &< 0,\label{ineq_p} \\
x_0^{\top} P x_0 &< \gamma. \label{ini_gamma}
\end{align}
%
Let $K:=- R^{-1}B^{\top}P$. Then the controlled system (\ref{close_sys}) is stable and the control law $u = Kx$ is suboptimal, i.e., $J(K) <\gamma$.
\end{thm}
\begin{proof}
Substituting $K:= -R^{-1}B^{\top}P$ into (\ref{close_sys}) yields
\begin{equation}\label{closed_sys_pp}
\dot{x}(t) = (A-BR^{-1}B^{\top}P)x(t) ,\quad x(0) =x_0.
\end{equation}
Since $P$ satisfies (\ref{ineq_p}), it should also satisfy
\begin{equation*}
(A-BR^{-1}B^{\top}P)^{\top}P + P(A-BR^{-1}B^{\top}P) + Q + PBR^{-1}B^{\top}P < 0,
\end{equation*}
which implies that $A-BR^{-1}B^{\top}P$ is Hurwitz,
i.e., the closed system (\ref{closed_sys_pp}) is stable.
Consequently, the corresponding cost is finite and equal to
\begin{equation*}
J(K) = \int_{0}^{\infty} x^{\top} (Q + PBR^{-1}B^{\top}P) x \ dt.
\end{equation*}
Since (\ref{ini_gamma}) holds, by taking $\bar{A} = A-BR^{-1}B^{\top}P$ and $\bar{Q} = Q + PBR^{-1}B^{\top}P$ in Theorem \ref{thm_autonomous}, we immediately have $J(K) < \gamma$.
\end{proof}
\begin{rem}
Theorem \ref{thm_single_sys} provides a method to find a class of suboptimal control laws satisfying $J(K)<\gamma$.
%
Choosing the feedback gain as $K := -R^{-1}B^{\top}P$ is one possible choice for such suboptimal control laws.
%
For other design methods see also \cite{guaranteed_cost}.
\end{rem}
\begin{rem}
Note that the suboptimal control design given in Theorem \ref{thm_single_sys} is more flexible than the optimal control design. Any positive semi-definite matrix $P$ satisfying inequalities (\ref{ineq_p}) and (\ref{ini_gamma}) makes control law $u=Kx$ with $K = -R^{-1}B^{\top}P$ suboptimal with respect to $J(K) <\gamma$.
\end{rem}
In the next section, we will show how to apply the above design method for suboptimal control to the distributed linear quadratic control problem for multi-agent systems.
\section{Suboptimal Control Design for Linear Multi-Agent Systems}\label{sec_mas}
In this section, we consider the distributed linear quadratic control problem for a multi-agent system consisting of $N$ agents with identical finite dimensional LTI system.
As in Section II, the dynamics of the identical agents is represented by
\begin{equation}
\dot{x}_i(t) = Ax_i(t) + Bu_i(t), \quad x_{i}(0) =x_{i0}, \quad i = 1, 2, \ldots,N
\end{equation}
where $A\in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}$, and $x_i\in \mathbb{R}^{n}, u_i \in \mathbb{R}^{m}$ are the state and input of $i$-th agent, respectively.
Assume that the pair $(A,B)$ is stabilizable, and the underlying graph is an undirected connected graph with corresponding Laplacian denoted by $L$.
Denoting $x = \left( x_1^{\top},\ldots,x_N^{\top} \right)^{\top}$, $u = \left( u_1^{\top},\ldots,u_N^{\top} \right)^{\top}$,
we can rewrite the multi-agent system in compact form as
\begin{equation}\label{net_sys}
\dot{x} = (I_N\otimes A) x + (I_N\otimes B)u,\quad x(0) =x_0.
\end{equation}
The cost functional we consider was already introduced in (\ref{cost_all}). We repeat it here for convenience:
\begin{equation}\label{cost_all_2}
J(u) = \int_{0}^{\infty}x^{\top}(L\otimes Q)x +u^{\top}(I_N\otimes R)u \ dt
\end{equation}
where $Q\geq 0$ and $R > 0$ are given real weighting matrices.
As already formulated in Problem \ref{prob1}, given a desired upper bound $\gamma >0$ for multi-agent system (\ref{net_sys}) with given initial state $x(0) = x_0$, we want to design a control law of the form
\begin{equation}\label{control_dis}
u=(L\otimes K)x
\end{equation}
where $K \in \mathbb{R}^{m\times n}$ is an identical feedback gain for all agents,
such that the controlled network
\begin{equation}\label{net_closed}
\dot{x} = (I_N\otimes A + L \otimes BK) x
\end{equation}
reaches consensus and, moreover, the associated cost
\begin{equation}\label{cost_kk}
J(K) = \int_{0}^{\infty}x^{\top}\left(L\otimes Q +L^2\otimes K^{\top} R K\right)x \ dt
\end{equation}
is smaller than the given upper bound, i.e., $J(K)<\gamma$.
Let the matrix $U\in \mathbb{R}^{N\times N}$ be an orthogonal matrix that diagonalizes the Laplacian $L$.
Define $\Lambda : = U^{\top}L U = \text{diag} ( 0, \lambda_2,\ldots,\lambda_N )$.
To simplify the problem given above, by applying the state and input transformations $\bar{x} =(U^{\top}\otimes I_n)x$ and $\bar{u} =(U^{\top}\otimes I_m)u$ with $\bar{x} = \left( \bar{x}_1^{\top},\ldots,\bar{x}_N^{\top} \right)^{\top}$, $\bar{u} = \left( \bar{u}_1^{\top},\ldots,\bar{u}_N^{\top} \right)^{\top}$, system (\ref{net_sys}) becomes
\begin{equation}\label{new_newtwork_sys}
\dot{\bar{x}} = (I_N\otimes A) \bar{x} + (I_N\otimes B)\bar{u},\quad \bar{x}(0) =\bar{x}_{0},
\end{equation}
with $\bar{x}_0 = (U^{\top} \otimes I_n)x_0$.
Clearly, (\ref{control_dis})
is transformed to
\begin{equation}\label{control_law}
\bar{u} = (\Lambda \otimes K)\bar{x},
\end{equation}
and the controlled network (\ref{net_closed}) transforms to
\begin{equation}\label{clsed_sys}
\dot{\bar{x}} = \left(I_N\otimes A + \Lambda \otimes BK\right) \bar{x} .
\end{equation}
In terms of the transformed variables, the cost (\ref{cost_kk}) is given by
\begin{equation}\label{new_newtwork_cost}
{J}(K) = \int_{0}^{\infty}\sum_{i=1}^{N} \bar{x}_i^{\top} (\lambda_i Q + \lambda_i^2 K^{\top} R K) \bar{x}_i\ dt.
\end{equation}
Note that the transformed states $\bar{x}_i$ and inputs $\bar{u}_i$, $i = 2,3,\ldots, N$ appearing in system (\ref{clsed_sys}) and cost (\ref{new_newtwork_cost}) are decoupled from each other, so that we can write system (\ref{clsed_sys}) and cost (\ref{new_newtwork_cost}) as
\begin{align}
\dot{\bar{x}}_1 &= A\bar{x}_1,\label{sys1} \\
\dot{\bar{x}}_i &= (A +\lambda_i BK)\bar{x}_i ,\quad i = 2,3,\ldots,N, \label{closed_loop}
\end{align}
and
\begin{equation}\label{cost_cost}
{J}(K)= \sum_{i=2}^{N}{J}_i(K)
\end{equation}
with
\begin{equation}\label{coco}
{J}_i(K) = \int_{0}^{\infty} \bar{x}_i^{\top} (\lambda_i Q + \lambda_i^2 K^{\top}RK) \bar{x}_i\ dt, \quad i = 2, 3, \ldots,N.
\end{equation}
Note that $\lambda_1 = 0$, and that therefore (\ref{sys1}) does not contribute to the cost $J(K)$.
We first record a well-known fact (see \cite{zhongkui_li_unified_2010}, \cite{harry_2013}) that we will use later:
\begin{lem}\label{lem_stable_consensus}
Consider the multi-agent system with identical agent dynamics (\ref{net_sys}). Assume that the network graph is undirected and connected. Then the controlled network reaches consensus with control law (\ref{control_dis}) if and only if, for $i=2,3,\ldots,N$, systems (\ref{closed_loop}) are stable.
\end{lem}
Thus, we have transformed the problem of distributed suboptimal control for system (\ref{net_sys}) into the problem of finding a feedback gain $K\in \mathbb{R}^{m\times n}$ such that the systems (\ref{closed_loop}) are stable and $J(K) < \gamma$.
Moreover, since the pair $(A, B)$ is stabilizable, there exists such a feedback gain $K$ \cite{harry_2013}.
The following lemma gives a {\em necessary} and {\em sufficient} condition for a given feedback gain $K$ to make all systems (\ref{closed_loop}) stable and to satisfy $J(K) < \gamma$.
\begin{lem}\label{lem_n_riccati}
Let $K$ be a feedback gain.
%
Consider the systems (\ref{closed_loop}) with associated cost functionals (\ref{cost_cost}) and (\ref{coco}).
%
Let $\gamma>0$.
%
Then all systems (\ref{closed_loop}) are stable and ${J}(K)<\gamma$ if and only if
there exist positive semi-definite matrices $P_i$ satisfying
\begin{align}
(A + \lambda_i B K)^{\top}P_i + P_i (A + \lambda_i B K) + \lambda_i Q +\lambda_i^2 K^{\top}R K &<0, \label{are_ineq}\\
\sum_{i=2}^{N} \bar{x}_{i0}^{\top} P_i \bar{x}_{i0} & <\gamma, \label{initial_condition_n_1}
\end{align}
for $i= 2, 3, \ldots, N$, respectively.
\end{lem}
\begin{proof}
(if)
Since (\ref{initial_condition_n_1}) holds, there exist sufficiently small $\epsilon_i > 0, i=2,\ldots,N$ such that $\sum_{i=2}^{N} \gamma_i < \gamma$ where $\gamma_i := \bar{x}_{i0}^{\top} P_i \bar{x}_{i0} +\epsilon_i$.
%
Because there exists $P_i$ such that (\ref{are_ineq}) and $\bar{x}_{i0}^{\top} P_i \bar{x}_{i0} < \gamma_i$ holds for all $i =2,\ldots,N$, by taking $\bar{A} = A + \lambda_i BK$ and $\bar{Q} = \lambda_i Q +\lambda_i^2 K^{\top}R K$, it follows from Theorem \ref{thm_autonomous} that all systems (\ref{closed_loop}) are stable and $J_i(K) <\gamma_i$ for $i = 2,\ldots,N$.
%
Since $J(K) = \sum_{i=2}^{N} J_i(K)$, this implies that $J(K) < \sum_{i=2}^{N} \gamma_i < \gamma$.
(only if)
Since $J(K)<\gamma$ and $J(K) = \sum_{i=2}^{N} J_i(K)$, there exist sufficiently small $\epsilon_i > 0, i=2,\ldots,N$ such that $\sum_{i=2}^{N} \gamma_i < \gamma$ where $\gamma_i := J_i(K)+\epsilon_i$.
%
Because all systems (\ref{closed_loop}) are stable and $J_i(K) <\gamma_i$ for $i=2,\ldots, N$, by taking $\bar{A} = A + \lambda_i BK$ and $\bar{Q} = \lambda_i Q +\lambda_i^2 K^{\top}R K$, it follows from Theorem \ref{thm_autonomous} that there exist positive semi-definite $P_i$ such that (\ref{are_ineq}) and $\bar{x}_{i0}^{\top} P_i \bar{x}_{i0} < \gamma_i$ hold for all $i =2,\ldots,N$.
%
Since $\sum_{i=2}^{N} \gamma_i < \gamma$, this implies that $\sum_{i=2}^{N} \bar{x}_{i0}^{\top} P_i \bar{x}_{i0} <\sum_{i=2}^{N} \gamma_i <\gamma$.
\end{proof}
Lemma \ref{lem_n_riccati} establishes a {\em necessary} and {\em sufficient} condition for a given feedback gain $K$ to stabilize all systems (\ref{closed_loop}) and to satisfy $J(K)<\gamma$.
However, Lemma \ref{lem_n_riccati} does yet not provide a method to compute such $K$.
To this end, the following two theorems present two design methods for $K$ and, correspondingly, two suboptimal distributed control laws for multi-agent system (\ref{net_sys}) with cost functional (\ref{cost_kk}).
\begin{thm} \label{Main1}
Consider multi-agent system (\ref{net_sys}) with associated cost functional (\ref{cost_kk}). Assume that the underlying graph is undirected and connected. Let $\gamma>0$.
Choose $c$ such that
\begin{equation}\label{c1}
\frac{2}{\lambda_2+\lambda_N} \leq c < \frac{2}{\lambda_N}.
\end{equation}
Then there exists a positive semi-definite matrix $P$ satisfying the Riccati inequality
\begin{equation} \label{one_are1}
A^{\top}P + PA +(c^2\lambda_N^2-2c\lambda_N)PBR^{-1}B^{\top}P +\lambda_N Q < 0.
\end{equation}
Assume, moreover, that $P$ can be found such that
\begin{equation}\label{p_N1}
x_0^{\top}\left(\left(I_N - \frac{1}{N}\mathbf{1}_N\mathbf{1}_N^{\top}\right)\otimes P\right) x_0 < \gamma.
\end{equation}
Let $K := -cR^{-1}B^{\top}P$. Then the controlled network (\ref{net_closed}) reaches consensus and the control law (\ref{control_dis}) is suboptimal, i.e., $J(K) <\gamma$.
\end{thm}
\begin{proof}
Using the upper and lower bounds on $c$ given by (\ref{c1}), it can be verified that $c^2\lambda_i^2-2c\lambda_i \leq c^2\lambda_N^2-2c\lambda_N <0$ for $i=2,3,\ldots,N$.
Since also $\lambda_i \leq \lambda_N$, $P$ is a solution to the $N-1$ Riccati inequalities
\begin{equation}\label{n_are1}
A^{\top}P + PA +(c^2\lambda_i^2-2c\lambda_i)PBR^{-1}B^{\top}P +\lambda_i Q < 0,\quad i=2,\ldots,N.
\end{equation}
Equivalently, $P$ also satisfies
the Lyapunov inequalities
\begin{equation}\label{lya_ineq}
\begin{aligned}
&(A-c\lambda_i BR^{-1}B^{\top}P)^{\top}P + P(A- c\lambda_iBR^{-1}B^{\top}P) \\
&\qquad + \lambda_i Q + c^2\lambda_i^2 PBR^{-1}B^{\top}P < 0,\quad i=2,\ldots,N.
\end{aligned}
\end{equation}
Next, recall that $\bar{x} = (U^{\top}\otimes I_n)x$ with $U = \left( \frac{1}{\sqrt{N}}\mathbf{1}_N\quad U_2 \right)$.
From this it is easily seen that $(\bar{x}_{20}^{\top}, \bar{x}_{30}^{\top}, \cdots , \bar{x}_{N0}^{\top})^{\top} = (U_2^{\top} \otimes I_n)x_0 $.
Also, $U_2 U^{\top}_2 = I_N - \frac{1}{N}\mathbf{1}_N\mathbf{1}_N^{\top}$.
Since (\ref{p_N1}) holds, we have
\begin{align*}
x_0^{\top} \left( U_2 U_2^{\top} \otimes P \right)x_0 &<\gamma \\
\Leftrightarrow \quad
((U_2^{\top} \otimes I_n)x_0)^{\top} \left( I_{N-1} \otimes P \right) ((U_2^{\top} \otimes I_n)x_0) &<\gamma \\
\Leftrightarrow \quad
(\bar{x}_{20}^{\top}, \bar{x}_{30}^{\top}, \cdots , \bar{x}_{N0}^{\top})
\left(I_{N-1}\otimes P\right)
(\bar{x}_{20}^{\top}, \bar{x}_{30}^{\top}, \cdots , \bar{x}_{N0}^{\top})^{\top} &<\gamma,
\end{align*}
which is equivalent to
\begin{equation}\label{xbar}
\sum_{i=2}^{N} \bar{x}_{i0}^{\top} P \bar{x}_{i0} <\gamma.
\end{equation}
Taking $P_i = P$ for $i = 2,3,\ldots,N$ and $K := -cR^{-1}B^{\top}P$ in inequalities (\ref{are_ineq}) and (\ref{initial_condition_n_1}) immediately gives us inequalities (\ref{lya_ineq}) and (\ref{xbar}).
Then it follows from Lemma \ref{lem_n_riccati} that all systems (\ref{closed_loop}) are stable and $J(K)<\gamma$. Furthermore, it follows from Lemma \ref{lem_stable_consensus} that the controlled network (\ref{net_closed}) reaches consensus and $J(K)<\gamma$.
\end{proof}
\begin{rem}\label{rem_main1}
Theorem \ref{Main1} states that that after choosing $c$ satisfying (\ref{c1}) and positive semi-definite $P$ satisfying (\ref{one_are1}), the distributed control law with local gain $K = -c R^{-1}B^{\top}P$ is suboptimal for all initial states of the network that satisfy the inequality (\ref{p_N1}).
%
The question then arises: how should we choose $c$ and $P$ such that this local gain is suboptimal for as many initial states as possible?
%
By writing $x_0 = ( x_{10},x_{20}, \ldots,x_{N0} )$, it is easily seen that (\ref{p_N1}) is equivalent to
\begin{equation}\label{xixj}
\frac{1}{N}\sum_{i=1}^{N}\sum_{j>i}^{N}(x_{i0}-x_{j0})^{\top}P(x_{i0}-x_{j0}) <\gamma.
\end{equation}
%
In other words, the smaller $P$, the bigger the differences between the local initial states are allowed to be, while still leading to suboptimality with respect to $\gamma$.
%
In other words, we should try to find $P$ as small as possible.
In fact, one can find a positive definite solution $P(c,\epsilon)$ to (\ref{one_are1}) by solving the Riccati equation
\begin{equation}\label{are}
A^{\top}P + PA -PB\bar{R}^{-1}B^{\top}P +\bar{Q}= 0
\end{equation}
with $\bar{R}(c) = \frac{1}{-c^2\lambda_N^2+2c\lambda_N}R$ and $\bar{Q}(\epsilon) =\lambda_N Q +\epsilon I_n $ where $c$ is chosen as in (\ref{c1}) and $\epsilon >0$.
%
If $c_1$ and $c_2$ as in (\ref{c1}) satisfy $c_1 \leq c_2$, then we have
$\bar{R}(c_1) \leq \bar{R}(c_2)$,
so, clearly, $P(c_1,\epsilon) \leq P(c_2,\epsilon)$.
%
Similarly, if $0 < \epsilon_1 \leq \epsilon_2$, we immediately have $\bar{Q}(\epsilon_1) \leq \bar{Q}(\epsilon_2)$. Again, it follows that $P(c,\epsilon_1) \leq P(c,\epsilon_2)$.
%
Therefore, if we choose $\epsilon>0$ very close to $0$ and $c = \frac{2}{\lambda_2+\lambda_N}$, we find the `best' solution to the Riccati inequality (\ref{one_are1}) in the sense explained above.
\end{rem}
Theorem \ref{Main1} provides a method to find a suboptimal distributed control law for particular choices of the parameter $c$.
In fact, such $c$ can be also chosen in another way, which is shown in the next theorem:
\begin{thm} \label{Main2}
Consider multi-agent system (\ref{net_sys}) with associated cost functional (\ref{cost_kk}).
%
Assume that the underlying graph is undirected and connected.
%
Let $\gamma >0$.
%
Choose $c$ such that
\begin{equation}\label{c2}
0 < c <\frac{2}{\lambda_2+\lambda_N}.
\end{equation}
%
Then there exists a positive semi-definite $P$ satisfying Riccati inequality
\begin{equation} \label{one_are}
A^{\top}P + PA +(c^2\lambda_2^2-2c\lambda_2)PBR^{-1}B^{\top}P +\lambda_N Q < 0.
\end{equation}
%
Assume, moreover, that $P$ can be found such that
\begin{equation}\label{p_N}
x_0^{\top}\left(\left(I_N - \frac{1}{N}\mathbf{1}_N\mathbf{1}_N^{\top}\right)\otimes P\right) x_0 < \gamma.
\end{equation}
%
Let $K := -cR^{-1}B^{\top}P$. Then the controlled network (\ref{net_closed}) reaches consensus and the control law (\ref{control_dis}) is suboptimal, i.e., $J(K)<\gamma$.
\end{thm}
\begin{proof}
The proof is similar to the proof of Theorem \ref{Main1} and hence is omitted here.
\end{proof}
\begin{rem}\label{rem_main2}
Theorem \ref{Main2} states that that after choosing $c$ satisfying (\ref{c2}) and positive semi-definite $P$ satisfying (\ref{one_are}), the distributed control law with local gain $K = -c R^{-1}B^{\top}P$ is suboptimal for all initial states of the network that satisfy the inequality (\ref{p_N}).
%
Again, the question then arises: how should we choose $c$ and $P$ such that this local gain is suboptimal for as many initial states as possible?
%
Following the idea in Remark \ref{rem_main1}, if we choose $\epsilon>0$ very close to $0$ and $c>0$ very close to $\frac{2}{\lambda_2+\lambda_N}$, we find the `best' solution to the Riccati inequality (\ref{one_are}) in the sense as explained in Remark \ref{rem_main1}.
\end{rem}
Note that, in Theorem \ref{Main1} and Theorem \ref{Main2}, in order to compute a suitable feedback gain $K$, one needs to know $\lambda_2$ and $\lambda_N$, the smallest nonzero eigenvalue (the algebraic connectivity) and the largest eigenvalue of the graph Laplacian, exactly.
This requires so-called global information on the network graph which might not always be available.
There exist algorithms to estimate $\lambda_2$ in a distributed way, yielding lower and upper bounds, see e.g. \cite{ARAGUES20143253}.
Moreover, also an upper bound for $\lambda_N$ can be obtained in terms of the maximal node degree of the graph, see \cite{eigenvaluesL}.
Then the question arises: can we still find a suboptimal controller reaching consensus, using as information only a lower bound for $\lambda_2$ and an upper bound for $\lambda_N$?
The answer to this question is affirmative, as shown in the following theorem.
\begin{thm}\label{Main3}
Let a lower bound for $\lambda_2$ be given by $l_2 > 0$ and an upper bound for $\lambda_N$ be given by $L_N$.
Let $\gamma > 0$. Choose $c$ such that
\begin{equation}\label{c3}
\frac{2}{l_2+L_N} \leq c < \frac{2}{L_N}.
\end{equation}
Define $K=-cR^{-1}B^{\top}P$,
with $P\geq 0$ satisfying
\begin{equation}\label{pp}
A^{\top}P + PA +(c^2L_N^2-2cL_N)PBR^{-1}B^{\top}P +L_N Q < 0.
\end{equation}
Then the controlled network reaches consensus and the distributed control law $L \otimes K$ is suboptimal, i.e. $J(K) <\gamma$, for all initial states $x_0$ that satisfy \eqref{p_N1}, equivalently \eqref{xixj}.
Furthermore, if we choose $c$ such that
\begin{equation}\label{c4}
0 < c <\frac{2}{l_2+L_N}.
\end{equation}
and define $K=-cR^{-1}B^{\top}P$
with $P \geq 0$ satisfying
\begin{equation}\label{p}
A^{\top}P + PA +(c^2 l_2^2-2c l_2)PBR^{-1}B^{\top}P +L_N Q < 0,
\end{equation}
then, still, the controlled network reaches consensus and the distributed control law $L \otimes K$ is suboptimal for all $x_0$ that satisfy \eqref{p_N}, equivalently \eqref{xixj}.
\end{thm}
\begin{proof}
A proof can be given along the lines of the proofs of Theorem \ref{Main1} and Theorem \ref{Main2}.
\end{proof}
\begin{rem}
%
Note that also in Theorem \ref{Main3} the question arises how to choose $c>0$ and $P\geq 0$ such that the local gain is suboptimal for as many initial states $x_0$ as possible.
%
Following the same idea as in Remark \ref{rem_main1} and Remark \ref{rem_main2}, if we choose $\epsilon >0$ very close to $0$ and $c>0$ equal to $\frac{2}{l_2 +L_N}$ in \eqref{pp} (respectively very close to $\frac{2}{l_2 +L_N}$ in \eqref{p}), we find the `best' solution to the Riccati inequalities (\ref{pp}) and (\ref{p}).
Moreover, one may also ask the question: can we compare, with the same choice for $c$, solutions to (\ref{pp}) with solutions to (\ref{one_are1}), and also solutions to (\ref{p}) with solutions to (\ref{one_are})?
%
The answer is affirmative.
%
Choose $c$ that satisfies both conditions (\ref{c1}) and (\ref{c3}). One can then check that the computed positive semi-definite solution to (\ref{pp}) is indeed `larger' than that to (\ref{one_are1}) as explained in Remark \ref{rem_main1}. A similar remark holds for the positive semi-definite solutions to (\ref{p}) and corresponding solutions to (\ref{one_are}) if $c$ satisfies both (\ref{c2}) and (\ref{c4}).
We conclude that if, instead of using the exact values $\lambda_2$ and $\lambda_N$, we use a lower bound, respectively upper bound for these eigenvalues, then the computed distributed control law is suboptimal for `less' initial values of the agents.
\end{rem}
\section{Illustrative Example}\label{sec_simulation}
In this section we use a simulation example borrowed from \cite{Nguyen2015} to illustrate the proposed design method for suboptimal distributed controllers.
Consider a group of 8 linear oscillators with identical dynamics
\begin{equation}\label{oscillators}
\dot{x}_i = A x_i +B u_i, \quad x_i(0) =x_{i0}, \quad i = 1,\ldots ,8
\end{equation}
with
\begin{equation*}
A =
\begin{pmatrix}
0 & 1\\
-1 & 0
\end{pmatrix},
\quad
B =
\begin{pmatrix}
0 \\ 1
\end{pmatrix}.
\end{equation*}
Assume the underlying graph is the undirected line graph with Laplacian matrix
\begin{equation*}
L = \begin{pmatrix}
1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\
0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 1
\end{pmatrix}.
\end{equation*}
We consider the cost functional
\begin{equation}\label{cost_exam}
J(u) = \int_{0}^{\infty}x^{\top}(L\otimes Q)x +u^{\top}(I_{8}\otimes R)u \ dt
\end{equation}
where the matrices $Q$ and $R$ are chosen to be
\begin{equation*}
Q =\begin{pmatrix}
2 & 0 \\ 0 & 1
\end{pmatrix},\quad
R = 1.
\end{equation*}
Let the desired upper bound for the cost functional (\ref{cost_exam}) be given as $\gamma =3$.
Our goal is to design a control law $u =(L\otimes K)x$ such that the controlled network reaches consensus and the associated cost is less than $\gamma = 3$.
In this example, we adopt the control design method given in Theorem \ref{Main1}.
The smallest nonzero and largest eigenvalue of $L$ are $\lambda_2 = 0.0979$ and $\lambda_{8} = 3.8478$.
First, we compute a positive semi-definite solution $P$ to (\ref{one_are1}) by solving the Riccati equation
\begin{equation}\label{are8}
A^{\top}P + PA +(c^2\lambda_8^2-2c\lambda_8)PBR^{-1}B^{\top}P +\lambda_{8} Q + \epsilon I_2 = 0
\end{equation}
with $\epsilon >0$ chosen small as mentioned in Remark \ref{rem_main1}.
Here we choose $\epsilon = 0.001$.
Moreover, we choose $c = \frac{2}{\lambda_2 +\lambda_8} = 0.5$, which is the `best' choice as mentioned in Remark \ref{rem_main1}.
Then, by solving (\ref{are8}) in Matlab, we obtain
\begin{equation*}
P = \begin{pmatrix}
12.1168 & 3.1303\\
3.1303 & 8.3081
\end{pmatrix}.
\end{equation*}
Correspondingly, the local feedback gain is then equal to $K = \begin{pmatrix} -1.5652 & -4.1541 \end{pmatrix}$.
The corresponding distributed diffusive control law is now suboptimal (with respect to the given $\gamma$) for all initial conditions $x_0$ that satisfy the inequality
\begin{equation}
x_0^{\top}\left(\left(I_{8} - \frac{1}{{8}}\mathbf{1}_{8}\mathbf{1}_{8}^{\top}\right)\otimes P\right) x_0 < 3,
\end{equation}
which is equivalent to
\begin{equation*}
\frac{1}{8}\sum_{i=1}^{8}\sum_{j>i}^{8}(x_{i0}-x_{j0})^{\top}P(x_{i0}-x_{j0}) <3,
\end{equation*}
which, for example, is satisfied by the inital conditions
$x_{10}^{\top} =
\begin{pmatrix}
-0.08 & 0.11
\end{pmatrix}$,
$x_{20}^{\top} =
\begin{pmatrix}
0.12 & -0.08
\end{pmatrix}$,
$x_{30}^{\top} =
\begin{pmatrix}
-0.09 & -0.14
\end{pmatrix}$,
$x_{40}^{\top} =
\begin{pmatrix}
-0.12 & 0.04
\end{pmatrix}$,
$x_{50}^{\top} =
\begin{pmatrix}
0.07 & -0.16
\end{pmatrix}$,
$x_{60}^{\top} =
\begin{pmatrix}
-0.21 & 0.12
\end{pmatrix}$,
$x_{70}^{\top} =
\begin{pmatrix}
0.15 & -0.22
\end{pmatrix}$,
$x_{80}^{\top} =
\begin{pmatrix}
-0.17 & -0.14
\end{pmatrix}$.
The plots of the eight decoupled oscillators without control are shown in Figure \ref{decoupled}.
\begin{figure}[t!]
\centering
\includegraphics[width=8.5cm]{decoupled1.eps}
\caption{Plots of the state vector $x^1 = (x_{1,1},\ldots, x_{8,1})$ (upper plot) and $x^2 = (x_{1,2},\ldots, x_{8,2})$ (lower plot) of the 8 decoupled oscillators without control} \label{decoupled}
\end{figure}
Figure \ref{consensus} shows that the controlled network of oscillators reaches consensus.
\begin{figure}[t!]
\centering
\includegraphics[width=8.5cm]{8_agents1.eps}
\caption{Plots of the state vector $x^1 = (x_{1,1},\ldots, x_{8,1})$ (upper plot) and $x^2 = (x_{1,2},\ldots, x_{8,2})$ (lower plot) of the controlled oscillator network} \label{consensus}
\end{figure}
\balance
\section{Conclusion}\label{sec_conclusion}
In this paper, we have studied a suboptimal distributed linear quadratic control problem for undirected linear multi-agent networks.
Given a multi-agent system with identical linear agent dynamics and an associated global quadratic cost functional, we provide two design methods for computing suboptimal distributed diffusive control laws such that the controlled network is guaranteed to reach consensus and the associated cost is smaller than a given upper bound for suitable initial conditions.
The computation of the local gain involves finding solutions of a single Riccati inequality, whose dimension is equal to the dimension of the agent dynamics, and also involves the smallest nonzero and largest eigenvalue of the graph Laplacian.
As an extension, we remove the requirement of having exact knowledge on the smallest nonzero and largest eigenvalue of the graph Laplacian by, instead, using only lower and upper bounds for these eigenvalues.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{ieeetran}
|
{
"timestamp": "2018-03-08T02:09:11",
"yymm": "1803",
"arxiv_id": "1803.02682",
"language": "en",
"url": "https://arxiv.org/abs/1803.02682"
}
|
\section{Introduction}
\IEEEPARstart{R}{adio} astronomy continues to face the problem of radio frequency interference (RFI). As instruments become more sensitive, so the impact of existing RFI sources becomes more significant. New technologies that make use of the RF spectrum become more widely adopted over time. To counter the growing problem of RFI, a variety of approaches have been developed and refined.
Most commonly, RFI is detected in data directly from radio telescopes. Such approaches typically distinguish only between RFI and astronomical signals, making no attempt to determine the identity of the sources of RFI signals. A wide variety of algorithms have been developed, mostly for application with 2D time-frequency data \cite{mitigation2, mitigation1}.
An additional approach, one being employed at the Square Kilometer Array (SKA) site in South Africa, is to develop independent RFI monitoring stations \cite{monitor2, monitor1, monitor3}. These stations will continuously monitor almost the full bandwidth of the radio telescope, simultaneously and in all directions. They also have the ability to capture time-domain transient RFI signals. Such monitoring stations make it easy to detect nearby sources of RFI, so that they may be removed or replaced.
In the case of intentional transmissions (for example, telecommunications) it is usually easy to identify their sources since they adhere to government-allocated frequency bands. Transient RFI signals are much harder to identify, however, since they are intermittent and broadband. Typically, they are produced as a byproduct of the normal operation of devices such as mechanical relays, fluorescent lights, AC machines etc.
There are few prior attempts to identify the sources of transient RFI in a radio-astronomy context. Unsupervised clustering via the k-means algorithm was applied to transient RFI at the Parkes radio telescope \cite{doran2013}, but individual sources were not classified. In other work \cite{wolfaardt2016} a variety of supervised learning techniques were employed to classify sources of RFI in labelled data recorded at the MeerKAT construction site in South Africa. Gaussian mixture model and k-nearest neighbours classifiers were applied to the data. High classification accuracy was achieved, however the number of samples per class was very small (in some cases less than 10).
In our own prior work, we looked at classifying RFI events using nonlinear principal components analysis techniques \cite{czech2017a} as well as a dictionary-based approach in conjunction with hidden Markov models \cite{czech2017b}. Signals were recorded from a number of common sources of transient RFI under controlled conditions, using a custom capturing system very similar to those which are installed in RFI monitoring stations at the MeerKAT/SKA site in South Africa.
Attempts have been made in other fields to classify similar types of transient RF signal. In one such approach, basic neural networks were used to classify the makes of different vehicles based on their transient RF emissions \cite{d2006}.
In this paper, we propose a novel approach to classifying the sources of transient RFI. Recurrent neural networks, in particular, long short-term memory (LSTM) networks \cite{h1997} have proven highly effective at modeling time-dependent signals in a variety of applications, for example phoneme classification in automated speech recognition \cite{graves2005} and acoustic modeling \cite{peddinti2017}.
While they are known best for their use in visual processing, convolutional neural networks (CNNs) have also shown success in dealing with temporal sequence data, for example human speech \cite{sainath2013} and wireless interference identification for coexistence management \cite{schmidt2017}. A CNN-based approach has been used to identify sources of interference in WiFi signals \cite{longi2017}, although most of the sources dealt with were intentional continuous transmitters. Recordings were limited to the WiFi band and were recorded as time-frequency data. In a radio astronomy context, CNNs have been used to flag RFI in data from radio telescope arrays in recent work \cite{akeret2017}, but not to classify the flagged RFI by source. In addition, they were applied to data represented in the 2D time-frequency domain.
The main novelties of our work consist of the following: As far as we are aware, this is the first time that either CNNs or LSTMs have been employed to identify the sources of time-domain transient RFI. In addition, we believe our approach combining 1D-CNNs and bidirectional LSTMs has not been attempted elsewhere for the classification of RFI signals (of any type) by their sources. This work constitutes one of the relatively few attempts available in the open literature thus far to identify the sources of transient RFI signals.
The rest of this paper is organised as follows: The experimental data and associated preprocessing steps are described in Section~\ref{data}. The models and and their application to the data are discussed in Section~\ref{model}. In Section~\ref{results}, results are presented. Finally, conclusions are drawn in Section~\ref{conclusion}.
\section{Data and Preprocessing}
\label{data}
The data used in this analysis are derived from our prior work \cite{czech2017a}. In the original dataset, full RFI recordings consist of a sequences of individual transients. In other prior work \cite{czech2017b} we presented an algorithm for extracting these individual transients from full RFI recordings. In this paper, we use these individual transients, extracted using this algorithm. This dataset consists of 63130 individual transients from 8 different sources. An example of a transient RFI signal from each class is given in Fig.~\ref{example_signals}. Transients are aligned by their largest peaks, and padded with zeros where necessary (since their lengths vary).
\begin{figure*}[t]
\centering
\includegraphics[width=15cm]{EXAMPLES.eps}
\caption{Examples of individual transients from each class as extracted by the automated algorithm given in \cite{czech2017b}. The lengths of the transients differ to an extent.}
\label{example_signals}
\end{figure*}
This data format, time-domain captures of short transient RFI signals, is one of which independent RFI monitoring stations at the SKA site in South Africa will record. The ability to identify the sources of transient RFI, as recorded by such monitoring stations, would be highly valuable.
\subsection{Preprocessing}
Prior to classification, limited preprocessing steps are carried out. Each transient is limited in length to 5000 raw samples for two reasons: One, the majority of transients are shorter than 5000 samples, and two, unnecessary computational overhead is avoided. The amplitude range of each individual transient (for all train, test and validation sets) is scaled to range between -1 and 1: for each transient $t$, $t_{scaled} = 2\frac{t - min(t)}{max(t)-min(t)}-1$. This scaling is applied because ideally, an RFI classification system would be capable of handling variations in amplitude. For example, in the field, the amplitudes of RFI signals will vary according to the distance from their sources. Transients are also standardised across each feature (time step). For each feature vector $x_j$ containing one value from each sample in a training set, $x_{j(scaled)} = \frac{x_j - \mu_j}{\sigma_j}$. This ensures that each feature is no more influential than the next. The standardisation parameters are determined from the training data alone; these predetermined parameters are used when standardising validation and test data.
The data are split into training, validation and testing sets. The training set consists of 60\% of the available data, while the others account for 20\% each. The data is stratified by class (each set contains an equal proportion of samples from each class) and shuffled (so the order of the samples in each set is random). The validation set is used for hyperparameter tuning, while the test set is kept separate until final evaluation, where the training set consists of both the original training and validation sets combined.
\subsection{Class Imbalance}
\label{imbalance}
Certain RFI sources (such as the mechanical relay) produced many more transients in a single event sequence than others. As a result, the number of individual transients is significantly imbalanced by class. The number of transients (equivalently, samples) per class is given in Table~\ref{devices_table}. Due to the class imbalance, we perform two separate analyses. In the first approach, we balance the classes by limiting the number of samples per class to the number of samples in the smallest class. For the larger classes, a subset of their samples is drawn at random. In the second approach, rather discarding data, samples are weighted by class in the cost function. Samples from rarer classes are weighted higher than samples from common classes, ensuring that each class has an equivalent influence on the model during training.
\begin{table}[h]
\renewcommand{\arraystretch}{1.4}
\centering
\caption[]{RFI sources}
\vspace{4mm}
\begin{tabular}{>{\centering\arraybackslash}m{6mm}m{45mm} m{18mm}}
\hline
\textbf{Class} & \textbf{Description} & \textbf{No. Samples}\\
\hline
1 & Compact fluorescent lamp & 662\\
2 & Power tool & 543 \\
3 & Step-down transformer & 5523 \\
4 & Cable & 264 \\
5 & Mechanical relay (700W resistive load) & 16006 \\
6 & Mechanical relay (without load) & 35932 \\
7 & AC motor (approximately 1 kW) & 3675 \\
8 & Small switching power supply & 525 \\
\hline
\end{tabular}
\label{devices_table}
\end{table}
\section{Model Architecture}
\label{model}
The architecture we selected is relatively uncomplicated - consisting of a 1D convolutional layer, followed by a bidirectional LSTM layer and finally a fully-connected layer, presenting the output in a 1-hot configuration. Fig.~\ref{architecture} illustrates the chosen model. The 1-D CNN layer serves both to identify salient features in the transient signals, and to reduce the length of the time-dependent input sequence to the LSTM layer. We chose to use LSTMs since they have proven highly effective at modeling temporal sequences in a wide variety of fields. In particular, bidirectional LSTMs have in some cases proven superior to other architectures in applications such as automated speech recognition, for example \cite{graves2005, zeyer2017}.
\begin{figure}[t]
\centering
\includegraphics[width=4.6cm]{ARCHITECTURE2.eps}
\caption{The architecture of the chosen model. In the bidirectional LSTM layer, the outputs of each LSTM are concatenated. The particular values given here apply for the balanced subset of the full dataset. Parameters were changed in some cases when the full unbalanced dataset was used: The CNN's pre-training batch size was increased to 256, while the kernel size was reduced to 160 time-steps.}
\label{architecture}
\end{figure}
The hyperparameters for the CNN and LSTM layers were selected by training different configurations on the training set, and evaluating them on the validation set. Hyperopt \cite{hyperopt}, a Python library, was used to automate the hyperparameter selection process. Model training was carried out using the Python library Keras \cite{keras} with Tensorflow \cite{tensorflow}. Computations were performed using an Amazon p2.xlarge instance (2.7 GHz Broadwell CPU; 61 GiB RAM; 12 GiB NVIDIA Kepler K80 GPU).
Model training was accomplished in two stages. Firstly, the CNN was pre-trained by replacing the LSTM layer with a temporary fully-connected classification layer. Next, the weights and filters obtained were kept fixed, and the temporary fully-connected layer replaced with the LSTM layer and a new, final fully-connected classification layer. Fig.~\ref{filters} shows 6 of the CNN's 64 filters. Some of the filters, such as those labelled 1, 2 and 4 suggest sinusoids of differing frequency and phase, while others such as 5 and 64 approximate other structural features.
\begin{figure}[t]
\centering
\includegraphics[width=6.5cm]{FILTERS.eps}
\caption{Several of the CNN's 64 filters and examples of their outputs when applied to a single preprocessed transient signal.}
\label{filters}
\end{figure}
As discussed in Section~\ref{imbalance}, two approaches were taken when dealing with the dataset's class imbalance. In the first approach, each class was cut down to the same size, selecting (at random) an equal number of samples for each. This reduced the total number of samples considerably. In the second approach, the full dataset was used, balancing classes by increasing the weighting of samples from rarer classes accordingly in the cost function. If $L$ is the vector containing the number of samples $L_i$ in each class $i$ then the vector of class weights $C$ is calculated as follows:
$$C = \frac{max(L)}{L}$$
\noindent
To account for these weights using categorical cross-entropy, the cost function is altered as follows: For a batch size $N$ of 1-hot output vectors of length $M$:
$$\mathcal{L}(y, \hat{y}) = -\frac{1}{N} \sum_{i=1}^{N}\sum_{j=1}^{M} y_{ij} \log(\hat{y}_{ij}) C_{j}$$
\noindent
and since we are using the softmax function, the partial derivative with respect to the output of the final layer is simply:
$$\frac{\partial \mathcal{L}}{\partial o_{i}} = C\odot(\hat{y}_i - y_i)$$
\noindent
where $\odot$ indicates element-wise multiplication and $o_i$ is the activation output. The gradients of the rarer classes are promoted relative to those of the more common classes due to the class weighting.
\section{Results}
\label{results}
Results are presented for both approaches to the problem of class imbalance in the dataset. For evaluation, each model was trained with both the training and validation sets together (amounting to 80\% of the data) and tested on the as-yet unseen test set. In the first approach, the number of samples per class is limited to the number of samples in the smallest class. A confusion matrix is given in Table~\ref{confusion1} and other accuracy metrics in Table~\ref{metrics}. Precision and recall are calculated for each class and the mean of each metric reported. For $M$ classes, precision $ = \frac{1}{M}\sum_{i = 1}^{M}\frac{\text{tp}_i}{\text{tp}_i + \text{fp}_i}$ and recall $ = \frac{1}{M}\sum_{i = 1}^{M}\frac{\text{tp}_i}{\text{tp}_i + \text{fn}_i}$ where $\text{tp}_i = $ true positives, $\text{fp}_i = $ false positives and $\text{fn}_i$ = false negatives for class $i$.
In the second approach, no samples are discarded. Rather, samples are weighted in the training cost function according to the rarity of their class. A second confusion matrix is provided in Table~\ref{confusion2} and accuracy metrics given in Table~\ref{metrics}. Despite class imbalance, even the smallest classes are well classified. For example, the smallest class is correctly classified 96.15\% of the time. The single class with the worst classification accuracy was still classified correctly 77.14\% of the time.
\begin{table}[h]
\renewcommand{\arraystretch}{1.4}
\centering
\caption[]{Evaluation of Results}
\vspace{4mm}
\begin{tabular}{m{11mm}m{22mm} m{40mm}}
\hline
\textbf{Metric} &\textbf{Approach 1}& \textbf{Approach 2}\\
&(Limited class size)& (Classes weighted in cost-function)\\
\hline
Accuracy & 0.8413 & 0.9636\\
Precision & 0.8475 & 0.8467 \\
Recall & 0.8413 & 0.9138 \\
\hline
\end{tabular}
\label{metrics}
\end{table}
\begin{table}
\renewcommand{\arraystretch}{1.9}
\caption[]{Confusion matrix for the unseen test-set when classes are balanced by discarding data.}
\vspace{3mm}
\begin{tabular}{ r|p{1.85mm}|p{1.85mm}|p{1.85mm}|p{1.85mm}|p{1.85mm}|p{1.85mm}|p{1.85mm}|p{1.85mm}| }
\multicolumn{1}{r}{}
& \multicolumn{1}{c} {\rotatebox[origin=l]{90}{CFL}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{power tool}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{transformer}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{cable}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{relay (load)}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{relay}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{AC motor}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{PSU}}\\
\cline{2-9}
{\small CFL} & \textbf{\scriptsize 44} &\scriptsize 0 &\scriptsize 2 &\scriptsize 0 &\scriptsize 2&\scriptsize 0 &\scriptsize 2 &\scriptsize 2\\
\cline{2-9}
\small power tool &\scriptsize 2 & \textbf{\scriptsize38} &\scriptsize 1 &\scriptsize 1 &\scriptsize 0 &\scriptsize 1 &\scriptsize 0 &\scriptsize 9 \\
\cline{2-9}
\small transformer &\scriptsize 3&\scriptsize 0 &\textbf{\scriptsize36} &\scriptsize 1 &\scriptsize 0 &\scriptsize 4 &\scriptsize 1 &\scriptsize 7\\
\cline{2-9}
\small cable &\scriptsize 2 &\scriptsize 0 &\scriptsize 0& \textbf{\scriptsize49} &\scriptsize 0 &\scriptsize 1 &\scriptsize 0 &\scriptsize 0\\
\cline{2-9}
\small relay (load) &\scriptsize4 &\scriptsize 0 &\scriptsize 0 &\scriptsize 1 & \textbf{\scriptsize45}&\scriptsize 1 &\scriptsize 1 & \scriptsize0\\
\cline{2-9}
\small relay &\scriptsize 1 &\scriptsize 0 &\scriptsize 4 &\scriptsize 0 &\scriptsize 0&\textbf{\scriptsize47} &\scriptsize 0 &\scriptsize 0 \\
\cline{2-9}
\small AC motor &\scriptsize1 & \scriptsize0 &\scriptsize0 &\scriptsize 1 &\scriptsize 1 &\scriptsize 1 &\textbf{\scriptsize48} &\scriptsize 0\\
\cline{2-9}
\small PSU &\scriptsize 1 &\scriptsize 3 &\scriptsize 3 &\scriptsize 1 &\scriptsize 0 &\scriptsize 1 & \scriptsize0 &\textbf{\scriptsize43} \\
\cline{2-9}
\end{tabular}
\label{confusion1}
\end{table}
\begin{table}
\renewcommand{\arraystretch}{1.9}
\caption[]{Confusion matrix for the unseen test-set when classes weighted in the loss function.}
\vspace{3mm}
\begin{tabular}{r|@{}p{2.5mm}|@{}p{2.5mm}|@{}p{2.5mm}|@{}p{2.5mm}|@{}p{2.5mm}|@{}p{2.5mm}|@{}p{2.5mm}|@{}p{2.5mm}| }
\multicolumn{1}{r}{}
& \multicolumn{1}{c} {\rotatebox[origin=l]{90}{CFL}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{power tool}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{transformer}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{cable}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{relay (load)}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{relay}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{AC motor}}
& \multicolumn{1}{c}{\rotatebox[origin=l]{90}{PSU}}\\
\cline{2-9}
{\small CFL} & \textbf{\scriptsize~~~120} &\scriptsize~~~1 &\scriptsize~~~7 &\scriptsize~~~0 &\scriptsize~~~2&\scriptsize~~~0 &\scriptsize~~~0 &\scriptsize~~~2\\
\cline{2-9}
\small power tool & \scriptsize~~~1 & \textbf{\scriptsize~~~88} &\scriptsize~~~6 &\scriptsize~~~2 &\scriptsize~~~2 &\scriptsize~~~3 &\scriptsize~~~0 &\scriptsize~~~6 \\
\cline{2-9}
\small transformer &\scriptsize~~~9&\scriptsize~~~4 &\textbf{\scriptsize~1035} &\scriptsize~~~0 &\scriptsize~~~1 &\scriptsize~~~19 &\scriptsize~~~5 &\scriptsize~~~31\\
\cline{2-9}
\small cable &\scriptsize~~~0 &\scriptsize~~~1 &\scriptsize~~~1& \textbf{\scriptsize~~~50} &\scriptsize~~~0 &\scriptsize~~~0 &\scriptsize~~~0 &\scriptsize~~~0\\
\cline{2-9}
\small relay (load) &\scriptsize~~~43 &\scriptsize~~~4 &\scriptsize~~~9 &\scriptsize~~~2 & \textbf{\scriptsize~3117}&\scriptsize~~~13 &\scriptsize~~~13 &\scriptsize~~~0\\
\cline{2-9}
\small relay &\scriptsize~~~2 &\scriptsize~~~9 &\scriptsize~~166 &\scriptsize~~~0 &\scriptsize~~~18& \textbf{\scriptsize~6957} &\scriptsize~~~22 &\scriptsize~~~12 \\
\cline{2-9}
\small AC motor &\scriptsize~~~1 &\scriptsize~~~0 &\scriptsize~~~11 &\scriptsize~~~0 &\scriptsize~~~4 &\scriptsize~~~3 &\textbf{\scriptsize~~716} &\scriptsize~~~0\\
\cline{2-9}
\small PSU &\scriptsize~~~2 &\scriptsize~~~3 &\scriptsize~~~16 &\scriptsize~~~0 &\scriptsize~~~0 &\scriptsize~~~3 &\scriptsize~~~0 &\textbf{\scriptsize~~~81} \\
\cline{2-9}
\end{tabular}
\label{confusion2}
\end{table}
\section{Conclusion}
\label{conclusion}
RFI is a significant concern for modern radio astronomy, so the ability to identify the sources of RFI near radio telescope arrays is highly desirable. Once identified, RFI sources can be removed or replaced. Transient RFI as generated unintentionally by devices such as mechanical relays or fluorescent lights is especially difficult to identify, but once identified, potentially easy to mitigate. In this paper, we have demonstrated a novel approach to identifying the sources of transient RFI in the time domain. Our proposed approach is the first to make use of CNNs and bidirectional LSTMs to classify transient RFI by source. Applied to an existing dataset of 63130 individual transient signals recorded from 8 common sources of RFI, good classification accuracy is achieved.
Our approach is well suited for future use with independent RFI monitoring stations at radio telescope arrays such as MeerKAT. In particular, since it only requires short recordings of individual transients, it is unaffected by limited recording time, a problem faced by some RFI recording systems \cite{wolfaardt2016}.
In future work, rather than identifying specific devices, we aim to classify sources by their physical components to permit more general source identification. For example, it may be possible to identify physical features such as mechanical contacts, brushes and inductive coils, among others. From their presence, the nature of an unknown device may be inferred.
\section*{Acknowledgment}
The financial assistance of the South African SKA project (SKA SA) is hereby acknowledged. Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the SKA SA (www.ska.ac.za).
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\balance
\input{bibliography2.bbl}
\end{document}
|
{
"timestamp": "2018-03-08T02:09:13",
"yymm": "1803",
"arxiv_id": "1803.02684",
"language": "en",
"url": "https://arxiv.org/abs/1803.02684"
}
|
\section{Results}
We have identified a number of choices that must be made to compute a compactness score. In addition to the choice of
(1)~compactness definition,
it is also important to consider how to handle
(2)~non-contiguous districts,
(3)~districts with holes,
(4)~political superunit boundaries,
(5)~map projections,
(6)~topography,
(7)~data resolution,
(8)~floating-point uncertainty,
and
(9)~whether alternative choies were possible in drawing a district's boundaries.
These are considered independently below.
In combination, these choices provide potentially undesirable implementation flexibility. This flexibility can be abused: Different implementation choices applied to what is nominally the same data lead to very different conclusions about fairness of a districting plan.
To demonstrate this effect, we have selected ten
U.S.\ Congressional Districts widely considered to be gerrymandered. Using an optimizer, we apply the full flexibility detailed in this paper and are able to find sets of implementation decisions for which these districts' compactness scores are outliers when compared against the full distribution of districts' scores. We are also able to find sets of decisions which make these districts appear reasonable. That is, we can exploit implementation flexibility to build a seemingly reasonable argument that these districts are both gerrymandered and not.
\autoref{fig:evil} shows the effects of this adversarial choice of parameters. In the case of NC01, IL04, and PA07, it was possible to move the districts from obvious outliers to middle-of-the-pack status. In other cases, such as NC12, NC04, and TX35, this was not possible, but the districts can still be moved considerably closer to the mean, countering arguments that they are outliers.
The optimizer does not need to use extreme settings to produce the desired results.
For example, TX33 appears most gerrymandered using the CvxHullPTB score (scores are defined below) at a 500\,m simplification tolerance in a locally-optimized Lambert conformal conic projection all districts included in the distribution; it appears least gerrymandered using the ReockPT score with a 500\,m tolerance in a Gall projection with districts comprising an entire state excluded.
\subsection{Open Source Tools}
Of the many compactness scores discussed in the literature, some are better able to cope with the complexities discussed here than others. Many of the more robust metrics, however, are also difficult or impossible to calculate using commonly-available software. For instance, QGIS~\citep{qgis} includes the area of multipolygons as a built-in display field, convex hulls as a function three menu levels deep, and has no functionality to calculate the minimum bounding circles needed for Reock scores. Other scores, such as bizarreness~\citep{chambers2010} have mathematical descriptions of a complex calculation, but no associated source code.
To address this situation, we have released a family of open source packages which share a common library designed to efficiently, reproducibly, and correctly calculate a variety of compactness scores. The basis of this ecosystem is \texttt{compactnesslib},\footnote{\url{https://github.com/gerrymandr/compactnesslib}} a C++ library and associated command-line interface which ingests bulk or single data in a variety of formats and calculates compactness scores. The \texttt{python-mander} Python package\footnote{\url{https://github.com/gerrymandr/python-mander}} (available via pip\footnote{\url{https://pypi.python.org/pypi/mander}}) and the \texttt{mandeR} R package\footnote{\url{https://github.com/gerrymandr/mandeR}}
provide high-level interfaces to this library. In addition, a QGIS plugin\footnote{\url{https://github.com/gerrymandr/qgis-compactness}} provides GIS users an easy means of calculating scores~\cite{compactnesslib,python-mander,mandeR,qgismander}. This stack was utilized to produce the calculations in this paper: The complete source code for generating all the diagrams presented here is available at \href{https://github.com/r-barnes/Barnes2018-compactness-implementation}{github.com/r-barnes/Barnes2018-compactness-implementation}.
\subsection{Coda}
The measurement of compactness can be used as a tool to help detect and quantify gerrymandering. Numerous engineering and implementation decisions, however, must be made to calculate a score. Whether used unintentionally or maliciously, this flexibility has strong bearing on the quality of compactness measurements and can be leveraged to shape conclusions about the quality of a districting plan.
Beyond providing ``best practices'' for implementations of compactness standards, we intend the open source software accompanying this paper as a first step toward fair and accurate measurement of compactness,
allowing scientists, politicians, and the public to
evaluate aspects of their democracy using
reproducible, mathematically well-founded, and computationally stable tools.
Finally, we remind the reader that the goal of all of this is to help governments represent their people. Compactness, while attractive as a quantitative metric, is a tool, not the end-game.
\section{Discussion}
\subsection{Best Practices}
The foregoing highlights the importance of being clear about how a score is calculated. In general, a mathematical definition alone is not sufficient: Attention must be paid to data and algorithmic quality. Here, we suggest best practices for the calculation of compactness scores:
\begin{itemize}
\item \textbf{Scores.} Be explicit about what each variable in a compactness score means. Does area include holes? Is it constrained by political superunits? How should non-contiguous districts be handled? Score names should be distinct and informative. Appending a clarifying suffix to the name of a score (e.g.\ \textit{PTSHp}) informs readers as to what is being done. See our Methods for examples.
\item \textbf{Projections.} Scale distortion should be limited to less than 1.25\% throughout the region of interest. Reasonable choices of national or local projections usually suffice.
\item \textbf{Resolution.} Use the best available resolution from a trusted source. Simplified or down-scaled data give altered results. Alternatively, choose a score which is robust to changes in resolution: hull-based scores seem to do well in this regard. The U.S.\ Census Bureau produces reasonable data designed such that all borders that are at the same resolution align. Ideally, districting data should be drawn from a common, trusted, non-partisan source, regardless of who is performing an analysis.
\item \textbf{Border constraints.} Scores which do not explicitly account for constraints imposed by superunit boundaries leave out valuable information about what was possible in drawing a district. That is, they may unfairly penalize a district for having an odd shape when no other shape was possible. Use a score that accounts for superunit borders. Be sure that borders are cropped to features such as major coastlines.
\item \textbf{Choice.} Before doing statistics on a set of district plans, eliminate those districts which encompass an entire political superunit, as no other choices of shape were possible.
\item \textbf{Topography.} We have not found including topography in the calculation of area to be a significant source of error, assuming the use of acceptable map projections.
\item \textbf{Border coalignment.} Coalignment of borders is a concern, though the effect was small in our data. To avoid problems, datasets used in an analysis should always be at the same resolution and carefully coaligned during their creation. In the U.S., Census data satisfies these requirements.
\item \textbf{Floating-point considerations.} We have not found the choice of single- or double-precision floating-point representations to be important in our calculations.
\item \textbf{Transparency.} A compactness score should not be accepted and cannot be interpreted without knowing the steps that went into creating it. From a scientific standpoint this relates strongly to reproducibility: We cannot trust what we cannot reproduce. Therefore, documentation is needed down to the equation level, and the release of source code is critical~\citep{barnes2010,merali2010,ince2012}.
\end{itemize}
\subsection{Policy Implications}
While the U.S.\ court system has declared that egregious gerrymandering is unconstitutional~\citep{scotus1986,usfed2016,scopenn2018}, the courts have thus far declined to adopt a quantitative standard by which gerrymandering can be judged; however, they have left open the possibility that a ``workable standard'' exists.~\cite{scotus2004} This paper demonstrates that any standard must be specified precisely and carefully, since differences in interpretation can have large effects on scores. Furthermore, this paper demonstrates that even a well-specified standard may judge unreasonable districts as being reasonable (see \autoref{fig:evil}). Therefore, any legally-mandated standard of compactness should leave open the possibility of challenges. Additionally, given the implementation flexibility discussed here and its potential for abuse, courts should not accept quantitative arguments unless the code used to build those arguments is made publicly accessible.
\section{Materials and Methods}
\subsection{Definitions of Compactness}
There are over 24 different measures of compactness in the literature, and no doubt many others exist. The measures break down into roughly five categories: (1)~length vs.\ width, (2)~area ratios, (3)~perimeter-to-area ratios, (4)~other geometric measures (moment of inertia, interior angles, \&c.), and (5)~measures incorporating population or other such information.~\cite{altman1998,niemi1990}
In this paper, we consider three widely-used compactness scores and their variants (\autoref{fig:interps} provides a depiction):
\begin{enumerate}
\item Polsby-Popper~\citep{polsby1991}: $4\pi A/P^2$ where $A$ is the area of a district and $P$ its perimeter
\item Reock~\citep{reock1961}: the ratio of a district's area to the area of its minimum bounding circle. Note that finding this circle is harder than locating the two most distant points of a district; an efficient algorithm and associated implementation is described in \cite{gartner1999}.
\item Convex Hull~\citep{niemi1990}: the ratio of a district's area to the area of its convex a hull (the minimum convex shape that completely contains the district).
\end{enumerate}
All of the above scores are in the range $[0,1]$ with higher values indicating greater compactness. Districts with relatively low values might be suspected of having been gerrymandered.
Note that these scores are purely geometric; it may be that scores incorporating population densities or other demographic data provide a better means of measuring gerrymandering, but we do not pursue this direction in our discussion. It is likely that incorporating such additional data would exacerbate the issues we discuss.
\begin{figure}
\centering
\begin{tabular}{cc}
\raisebox{-.5\height}{\includegraphics[width=0.35\columnwidth]{imgs/la_bc.png}} &
\raisebox{-.5\height}{\includegraphics[width=0.35\columnwidth]{imgs/la_ch.png}} \\ \vspace{0.2cm}
\footnotesize ReockPT=0.26 & \footnotesize CvxHullPT=0.44 \\
\raisebox{-.5\height}{\includegraphics[width=0.35\columnwidth]{imgs/la_sbc.png}} &
\raisebox{-.5\height}{\includegraphics[width=0.35\columnwidth]{imgs/la_sch.png}} \\
\footnotesize ReockPS=0.26 & \footnotesize CvxHullPS=0.58
\end{tabular}
\caption{Reock and Convex Hull scores for Louisiana 01 shown with both the \textit{PS} and \textit{PT} interpretation depicted. It is coincidental that ReockPT and ReockPS are the same here. Note that for both of the ReockPT and CvxHullPT scores the hull polygons overlap; this is potentially problematic since it could be considered double-counting. \label{fig:interps} }
\end{figure}
\subsection{Nomenclature}
All of these measures are under-defined: They assume that an electoral district is described by a single polygon without any holes. In reality, districts, such as those with islands (see \autoref{fig:interps}), often are comprised of many polygons. While holes in districts are rarer, they do occur. To resolve these difficulties, we suggest methods be defined with specific reference to multiple polygons and holes.
Here, whether or not contiguity is accounted for in a score will be indicated by the suffixes \textit{PT} (polygons together) and \textit{PS} (polygons separate). Whether or not holes are accounted for will be indicated by the suffixes \textit{AH} (add holes) and \textit{SH} (subtract holes). If there is ambiguity regarding whether area, perimeter, or some other quantity is being treated in this way, then terms such as \textit{PTaSHp} (treat the area of the polygons together, subtract the perimeter of holes) may be used. The suffix \textit{B} indicates that a score accounts for constraints imposed by the boundaries of political superunits.
\subsection{Non-contiguous Districts}
There is no federal requirement that districts be contiguous, nor do many states require it. Indeed, the presence of islands (e.g.\ Hawaii) can make contiguity an impossibility. Non-contiguity may arise in other ways. Civil rights considerations have given Louisiana 01, depicted in \autoref{fig:interps}, two large portions separated by Louisiana 02; Louisiana 02 was drawn as a majority-minority district following the passage of the Voting Rights Act of 1965. Wisconsin's 61st Assembly District (\autoref{fig:wisconsin}) exemplifies a different situation. The city of Racine, WI, had a non-contiguous boundary as a by-product of annexation, yet Wisconsin required that its districts be composed entirely of wards. As a result, the district itself is non-contiguous and could not legally be drawn in any other way~\cite{altman2011}. For the 114th Congress 1:500,000 resolution data, 84 of 441 districts are non-contiguous. Of the non-contiguous districts the largest number of subdivisions was 580 (in Alaska) and the median was 5.
The question then is whether a district should be treated as a single unit or several independent units. Treating the district as a single unit by, e.g., enclosing it in a single hull, will tend to result in lower compactness scores indicative of gerrymandering.
Treating the district as a separate units and summing the areas of the units' enclosing hulls will result in higher compactness scores indicating less gerrymandering.
Mathematically speaking, although Polsby-Popper is usually calculated as being proportional to $\frac{A}{P^2}$, there are at least two possibilities for extending this formula to non-contiguous districts, in particular $\sum_i \frac{A_i}{P_i^2}$ and $(\sum_i A_i)/(\sum_i P_i^2)$, where $i$ enumerates the non-contiguous subregions of the district. Note that although the original Polsby-Popper score is bound to the range $[0,1]$, this is not true of the first of these alternatives. Here, we use the latter alternative.
Ultimately, special attention should be given to non-contiguous districts to determine whether they result from natural features, legitimate legal requirements, or electoral engineering. \autoref{fig:definitiondiff} shows the effect the foregoing interpretations can have on compactness scores. The wide gap between different interpretations of what is nominally the same score supports the need for exactitude in both language and implementation.
\subsection{Holes}
Holes are relatively rare in districting, but many of the same considerations apply. Wisconsin 61, discussed previously, has a legally-mandated hole (\autoref{fig:wisconsin}). Texas 18 very nearly surrounds the urban core of Houston and could, in a low-resolution dataset, be assigned a hole. Holes also appear as artifacts of the digitization process (\autoref{fig:border-artifacts}). For the 114th Congress 1:500,000 resolution data, four of 441 districts have holes as artifacts.
\subsection{Borders}
Districts are constrained by borders imposed by higher geopolitical units as well as by nature. Compactness scores that do not account for such constraints may assign inappropriately low scores to a district. The panhandles of Florida and Oklahoma, as well as Kentucky's border with the Ohio River (see \autoref{fig:polysimp}), contain electoral districts whose shape, at least in part, cannot be dictated by politics. The same is true of almost any coastal district since islands and peninsulas must be included, but lengthen their perimeters. Louisiana (see \autoref{fig:polysimp}) exmplifies this.
Some scores can be modified to account for this issue. They can be marked with the suffix \textit{B} (borders accounted for). For example, in the case of the convex hull and Reock scores, if the hull or minimum bounding circle is intersected with a state polygon, the result is a better representation of what was possible and, therefore, a better indicator of whether gerrymandering took place. Taking this into account can have a considerable impact on compactness scores (\autoref{fig:borders}). Those scores, such as the Polsby-Popper, which cannot be modified to account for borders, are calculated as described elsewhere without consideration of borders.
\begin{figure}
\centering
\includegraphics[width=0.75\columnwidth]{imgs/tl_greatlakes.png}
\caption{Electoral districts of the 114th Congress including maritime regions. Two datasets of electoral districts are overlaid. The gray area depicts electoral district boundaries cropped to coastlines whereas the dashed red line indicates the full extent of the electoral districts. Note the growth of the district's areas and the relative smoothness of the perimeters.
Data was drawn from the US Census Bureau~\citep{shapefiles}: cropped data is from the Cartographic Boundaries dataset, e.g.\ \href{https://www.census.gov/geo/maps-data/data/cbf/cbf_cds.html}{\textit{cb\_2015\_us\_cd114\_rr.zip}}, whereas uncropped data is from the TIGER/Line dataset, e.g., \href{https://www.census.gov/geo/maps-data/data/tiger-line.html}{\textit{tl\_2015\_us\_cd114.shp}}. \label{fig:waterboundaries}}
\end{figure}
The boundaries of electoral districts, states, and countries may include large maritime regions, as shown in \autoref{fig:waterboundaries}. Insofar as these regions generally cannot be populated, save for areas immediately adjacent to the shore, their inclusion in compactness calculations may serve to hide the effects of gerrymandering. Input data should be cropped to major coastlines to account for this, though, doing so is not a panacea: coastlines tend to be fractal (see \autoref{fig:koch}).
As \autoref{fig:border-misalign} shows, border data, especially when drawn from disparate sources, may not always co-align. We attempted to quantify this effect by overlaying high-resolution district data with medium-resolution state data and found that the impact was usually small (see \autoref{fig:misalign-effect} for details). Problems can be avoided entirely by using data which is co-aligned, such as is available from the U.S.\ Census.
\subsection{Choice}
\label{sec:choice}
If only one possible plan exists for a district, that district cannot be gerrymandered and should be excluded from analysis. In the Census Bureau data~\citep{shapefiles} used here, 13 congressional districts, including Alaska, Delaware, and Vermont, had only one congressional district. No matter how oddly shaped these districts are, they are not gerrymandered.
\subsection{Projections}
Although scores are often defined as though districts exist on a plane, in reality districts are wrapped around the curvature of the Earth and local topographical features. Several interpretations of scores are possible: districts could be mapped to the plane using a projection designed to minimize distortion across an entire country, a subdivision of a country such as a state, or even the district itself. Alternatively, variables could be calculated on the sphere or WGS84 ellipsoid. As \autoref{fig:projections} shows, despite all the possibilities, compactness measures appear to be stable to \textit{reasonable} choices among \textit{localized} (country-scale) map projections used in practice. Alaska demonstrates what happens when an unreasonable choice is made: its score in a projection suitable for the conterminous United States differs that in an Alaska-specific projection by up to 20\%.
Clearly, using a global projection such as the standard Mercator induces too much distortion. This implies that Web Mercator (EPSG:3857) should never be used for compactness calculations, despite its ubiquitous use on the internet. Across all districts, scores, and projections, the absolute score difference between a district as measured in a locally-optimal projection versus a conterminous projection was less than 0.009 in 99\% of cases. The other 1\% of cases comprise districts such as Alaska and American Somoa, which are outside the region of interest for the conterminous projections. Given this, nation-sized projections---excluding outlying states and territories---are likely reasonable choices. Quantitatively, the conterminous Albers Equal Area (EPSG:102003) projection has a maximum scale distortion of 1.25\%~\citep{deetz1945}: this value hence can be taken as an upper limit on what is acceptable for any projection and is our recommended choice for districts in the conterminous United States.
\subsection{Topography}
A different effect of mapping electoral districts to a plane is that topography, such as mountains, is left out of quantities such as area and perimeter. As a result, the true land area and overland distance between points is under-estimated. Using the 30\,m USGS National Elevation Dataset~\citep{ned}, we calculated the surface area of districts using RichDEM's implementation~\citep{richdem} of an algorithm by Jenness~\cite{jenness2004} and modeled perimeter as the summed length of all the raster elevation cells at the edge of a district.
The difference in Polsby-Popper scores between the topographic and non-topographic data was less than 0.03 for all districts, with 75\% of districts having deviations less than 0.005. This should be expected given that Kansas (and every other state) is provably flatter than a pancake.~\citep{fonstad2003}
\subsection{Resolution}
\begin{figure*}
\centering
\begin{tabular}{cccccc}
\raisebox{-.5\height}{\includegraphics[width=2.2cm]{imgs/img_st_2103_0.png}} &
\raisebox{-.5\height}{\includegraphics[width=2.2cm]{imgs/img_st_2103_1.png}} &
\raisebox{-.5\height}{\includegraphics[width=2.2cm]{imgs/img_st_2103_2.png}} &
\raisebox{-.5\height}{\includegraphics[width=2.2cm]{imgs/img_st_2201_0.png}} &
\raisebox{-.5\height}{\includegraphics[width=2.2cm]{imgs/img_st_2201_1.png}} &
\raisebox{-.5\height}{\includegraphics[width=2.2cm]{imgs/img_st_2201_2.png}} \\
500k & 5m & 20m & 500k & 5m & 20m \\
\pscore{0.379992} & \pscore{0.476327} & \pscore{0.54611} & \pscore{0.032985} & \pscore{0.0553361} & \pscore{0.150796} \\
\cscore{0.780833} & \cscore{0.828378} & \cscore{0.789624} & \cscore{0.575963} & \cscore{0.598142} & \cscore{0.6684} \\
& Kentucky 03 & & & Louisiana 01 &
\end{tabular}
\caption{Effect of polygon simplification on districts and their compactness scores. Districts from the 114th Congress are shown at 1:500,000 (500k), 1:5,000,000 (5m), and 1:20,000,000 (20m) resolution. Simplification was performed by the US Census Bureau using in-house algorithms that ensure border alignment. Here, PP stands for PolsbyPTAH while CH stands for CvxHullPT; note how these scores change with resolution. Kentucky 03 encompasses metropolitan Louisville and is bounded on the north by Kentucky's state border and the Ohio River. Louisiana 01 is bounded by the Mississippi Delta, divided by Louisiana 02, and includes unexpected parts of New Orleans. Note that under simplification the rough edges of Kentucky 03 disappear, as do entire bays and islands in Louisiana 01.
\label{fig:polysimp}}
\end{figure*}
Resolution can be thought of as the density of points describing a boundary. \autoref{fig:polysimp} shows the same district at several resolutions; lower resolutions lead to simpler shapes usually, but not always, by reducing the length of the perimeter. The U.S.\ Census Bureau releases boundary data of Congressional Districts in four resolutions: full, 1:500k, 1:5M, and 1:20M~\citep{shapefiles}. The full-resolution data is available as ``TIGER/Line'' data whereas the other resolutions are available as ``Cartographic Boundary Shapefiles.'' At these resolutions the perimeters of the districts of the 114th Congress are defined by an average of 8914, 1531, 322, and 70 points, respectively.
As services move online and onto mobile devices with constrained processing,
it will be tempting for practitioners to introduce lower-resolution or simplified data into compactness measurements. Even in the high-performance environments used for automated redistricting efforts~\cite{tam2016}, low-resolution data is tempting as it may yield substantial savings on compute time.
Ultimately, we find that the choice of resolution has a substantial impact on compactness scores (\autoref{fig:simp_together} and \ref{fig:simp_indiv}) with the Polsby-Popper score especially affected. This adds to a growing list of criticisms of the Polsby-Popper score.~\citep{Alexeev2017,chambers2010}
Since data may be supplied to users by outside sources, adversarial inputs are possible: A high-frequency wave applied to the boundary of a district may be visually imperceptible while introducing substantial alterations to a district's score. The Koch snowflake is an extreme example of this: It has an arbitrarily-long perimeter surrounding a finite area (\autoref{fig:koch}). More practically, data may contain digitization or simplification artifacts that only become apparent under significant magnification, as shown in \autoref{fig:border-artifacts}.
\section{Ordering}
The foregoing considerations change not only what the values of the calculated scores are, but also the relative ordering of the scores (\autoref{fig:order}). If this is quantified using Spearman's rank correlation coefficient (\autoref{fig:order_spear}), it is apparent that different scores give markedly different rankings. Thus, any ranking of districts by compactness is thoroughly tied to and arises from choices made in developing the scores. \autoref{fig:evil} explores this issue further.
\section{Floating-point Issues}
Computers generally store fractional values based on the IEEE754 specification using either the 32-bit single-precision type, which gives about 7 decimal places of precision, or the 64-bit double-precision type, which gives about 15 decimal places of precision. In terms of decimal degrees, the former provides approximately centimeter accuracy while the latter provides nanometer accuracy; thus, single-precision is sufficient for storing geographic coordinates. However, performing mathematics on fractional numbers, especially 32-bit types, is known to give potentially erroneous results~\citep{goldberg1991}.
We tested for this by computing all of the scores mentioned here using both 32-bit and 64-bit IEE754 compliant types, with the latter taken as the ``true'' value. No score had a percent difference between the two of more than 0.027\%.
\acknow{
The open source software described here had its genesis in the \textit{Geometry of Redistricting} workshop held at Tufts University August 7--11, 02017. John Connors helped develop the \texttt{mandeR} package. Max Gardner, Aaron Dennis, Daniel McGlone, and Ariel M'ndange-Pfupfu helped develop the \texttt{python-mander} package. Ariel M'ndange-Pfupfu and Vanessa Archambault helped develop the QGIS plugin.
Computation and data utilized XSEDE's Comet supercomputer~\citep{xsede}.
Travel funding for RB and research support for JS was provided by a Prof.\ Amar G.\ Bose Research Grant and an Amazon Research Award. In-kind support was provided by Isaac B., Hannah J., Kelly K., Vivian L., and Jerry W.}
\showacknow{}
|
{
"timestamp": "2018-03-09T02:00:58",
"yymm": "1803",
"arxiv_id": "1803.02857",
"language": "en",
"url": "https://arxiv.org/abs/1803.02857"
}
|
\section{Introduction}
Advances in next-generation sequencing (NGS) techniques have enabled researchers to produce
millions of relatively short reads for genome-scale bioinformatics research \citep{encode2012integrated,wang2009rna,mortazavi2008mapping}. Transcriptome analyses,
including gene expression profiling and transcript quantification through RNA-sequencing (RNA-seq), can help better understand biological processes of interest. RNA-seq count data are highly over-dispersed with large dynamic ranges \citep{anders2015htseq}. A large number of statistical tools have been developed for differential gene expression analysis of RNA-seq data \citep{anders2010differential,dadaneh2017bnp,robinson2010edger,love2014moderated,law2014voom,hardcastle2010bayseq,leng2013ebseq}, which mostly have adopted the negative binomial (NB) distribution to account for over-dispersion as well as high uncertainty inherent in RNA-seq data due to the small number of replicate samples in typical differential expression experiments~\citep{love2014moderated}.
Living systems are complex and dynamic. There has been significant interest in analyzing temporal RNA-seq count data \citep{bar2012studying}. For example, in cell biology or drug discovery research, monitoring molecular expression changes in response to specific stimuli can help better understand cellular mechanisms at the transcriptional and post-transcriptional regulatory levels under different conditions. One important task is to identify the genes that are differentially expressed over time across different conditions, which is more challenging compared to static RNA-seq data analysis due to potential temporal dependencies \citep{lienau2009insight}.
Recently, several dynamic differential RNA-seq analysis methods have been developed to better capture temporal dependency. For example, EBSeq-HMM \citep{leng2015ebseq} takes an empirical Bayesian mixture modeling approach to compare the expression change across consecutive time points to identify genes that display significant transcription changes over time under one treatment condition. Across different conditions, it is desirable to identify genes that have different dynamic patterns. For this purpose, next-maSigPro \citep{nueda2014next} has extended a generalized linear model (GLM) \citep{mccullagh1984generalized} based dynamic differential expression analysis for microarray data from multiple time points to analyze temporal RNA-seq data. However, modeling RNA-seq counts by real values may lead to information loss and GLM may not be able to capture complicated dynamic changes in expression. An autoregressive time-lagged $AR(1)$ model with Markov Chain Monte Carlo (MCMC) inference~\citep{oh2013time} has also been proposed to identify genes with different temporal expression changes. But the posterior estimates of model parameters through Metropolis-Hastings inference lead to high computational complexity. DyNB \citep{aijo2014methods} has been proposed recently to model the temporal RNA-seq counts by NB distributions with their temporal expected values modeled by non-parametric Gaussian Processes (GP). DyNB can detect the genes with differential dynamic patterns that static differential expression analysis, which consider individual time points, fail to discover. In addition to high computational complexity due to MCMC inference \citep{spies2015dynamics,sun2016statistical}, DyNB may fail to model potential abrupt expression changes due to its inherent smoothness assumptions~\citep{rasmussen2006gaussian}.
We present a new dynamic differential expression analysis method for temporal RNA-seq data, GMNB (gamma Markov negative binomial), which is a hierarchical model to introduce a gamma Markov chain \citep{acharya2015nonparametric,schein2016poisson} to model the potential dynamic transitions of the model parameters in NB distributions. With this new model for temporal RNA-seq data and an efficient inference algorithm, GMNB is expected to provide the following advantages over existing methods: 1) GMNB can model more general dynamic expression patterns than DyNB, especially for abrupt expression changes across consecutive time points; 2) The closed-form Gibbs sampling can be derived to infer the model parameters in GMNB, which is computationally more efficient than the existing methods; 3) For dynamic differential expression, genes are ranked based on the Bayes factor (BF), which is very general especially when considering differential expression under multiple factors; 4) Last but not least, GMNB avoids the normalization preprocessing step due to the explicit modeling of the sequencing depth in NB distributions, as described in \citet{dadaneh2017bnp}, %
and we expect similar superior performance of GMNB compared to existing methods requiring such heuristic preprocessing steps.
The remainder of the paper is organized as follows. Section 2 introduces the GMNB model, inference algorithm, and dynamic differential expression analysis. Section 3 compares the experimental
results from both synthetic and real-world benchmark data using GMNB and other state-of-the-art dynamic differential expression methods for temporal RNA-seq data. We conclude the paper in Section 4.
\section{Methods}
\subsection{Notation}
Throughout this paper, we use the NB distribution to model RNA-seq read counts. We parameterize a NB random variable as $n \sim \text{NB}(r,p)$, where $r$ is the nonnegative dispersion and $p$ is the probability parameter. The probability mass function (pmf) of $n$ is expressed as $f_N(n)=\frac{\Gamma(n+r)}{n!\Gamma(r)}p^n(1-p)^r$, where $\Gamma(\cdot)$ is the gamma function. The NB random variable $n \sim \text{NB}(r,p)$ can be generated from a compound Poisson distribution:
\begin{equation}
n = \sum_{t=1}^{\ell} u_t, \;\; u_t \sim \text{Log}(p), \;\; \ell \sim \text{Pois}(-r\ln (1-p)), \nonumber
\end{equation}
where $u \sim \text{Log}(p)$ corresponds to the logarithmic random variable \citep{johnson2005univariate}, with the pmf $f_U(u) = -\frac{p^u}{u\ln(1-p)}$, $u=1,2,...$. As shown in \citet{zhou2015negative}, given $n$ and $r$, the distribution of $\ell$ is a Chinese Restaurant Table (CRT) distribution, $(\ell | n,r) \sim \text{CRT}(n,r)$, a random variable from which can be generated as %
$\ell = \sum_{t=1}^{n} b_t$, with $b_t \sim \text{Bernoulli}(\frac{r}{r+t-1})$.
\subsection{GMNB model}
We model the dynamic gene expression changes in a temporal RNA-seq dataset by constructing a Markov chain where the expression of a gene at time $t$ only depends on that of time $t-1$. Specifically, for the RNA-seq reads mapped to gene $k$ in a given sample $j$ under different conditions, the read count at time $t$ follows:
\begin{equation} \label{eq:model}
n_{kj}^{(t)} \sim \text{NB}(r_k^{(t)}, p_j^{(t)}),
\end{equation}
where to impose the dependence between consecutive time points, we model the dispersion parameters dynamically by introducing a gamma Markov chain, in which $r_k^{(t)}$ is distributed according to:
\begin{equation}
\label{eq:gammamodel}
r_k^{(t)} \sim \text{Gamma}(r_k^{(t-1)}, \frac{1}{c_k}).
\end{equation}
As previously shown in \citet{dadaneh2017bnp}, the probability parameter $p_j^{(t)}$ accounts for the effect of varying sequencing depth of sample $j$ at time point $t$. More precisely, the expected expression of gene $k$ in sample $j$ and time $t$ is $r_k^{(t)} \frac{p_j^{(t)}}{1-p_j^{(t)}}$, and hence the dispersion parameter $r_k^{(t)}$ can be viewed as the true abundance of gene $k$ at time $t$, after removing the effects of sequencing depth. Thus the differential expression analysis of temporal RNA-seq data can be performed without any normalization preprocessing steps.
Note that the scale parameter ${1}/{c_k}$ of the Gamma distribution in (\ref{eq:gammamodel}) is shared between different time points, thereby making statistical inference more robust by borrowing information from various samples at multiple time points. To complete the model we sample the dispersion parameter at the first time point as $r_k^{(0)} \sim \text{Gamma}(e^{(0)}, \frac{1}{f_0})$, and use conjugate priors as $c_k \sim \text{Gamma}(c_0, \frac{1}{d_0})$ and $p_j^{(t)} \sim \text{Beta}(a_0, b_0)$. %
In addition to the flexibility of modeling temporal RNA-seq data, this GMNB model enables an efficient inference procedure by taking advantage of unique data augmentation and marginalization techniques for the NB distribution \citep{zhou2015negative}, as described in detail below.
\subsection{Gibbs sampling inference}
By exploiting novel data augmentation techniques in \citet{zhou2015negative}, we implement an efficient Gibbs sampling algorithm with closed-form updating steps. More specifically, we infer the dispersion parameter of the NB distribution by first drawing latent random counts from the CRT distribution, and then update the dispersion by employing the gamma-Poisson conjugacy. Furthermore, due to the Markovian construction of the model, it is necessary to consider both backward and forward flow of information for the inference of $r_k^{(t)}$. First, in the backward stage, starting from the last time point $t = T$, we draw two sets of auxiliary random variables as
\begin{eqnarray}
l_{kj}^{(t)} &\sim& \text{CRT} (n_{kj}^{(t)}, r_k^{(t)}) \nonumber\\
l_{k.}^{(t)} &=& \sum_j l_{kj}^{(t)} \nonumber\\
u_{k}^{(t-1)(t)} &\sim& \text{CRT} (u_{k}^{(t)(t+1)}+l_{k.}^{(t)}, r_k^{(t-1)}),
\end{eqnarray}
for $t=T,T-1,\ldots,1$. %
For the last time point, we assume $u_{k}^{(T)(T+1)}=0$. Next, in the forward stage of Gibbs sampling, we sample $r_{k}^{(t)}$ starting from $t = 0$ to $t = T$ as
\begin{equation}
(r_{k}^{(t)} | - ) \sim \text{Gamma} \big( r_{k}^{(t-1)} + u_{k}^{(t)(t+1)} + l_{k.}^{(t)}, \frac{1}{\theta_{k}^{(t)}} \big),
\end{equation}
where $\theta_{k}^{(t)} = c_k - \sum_j \ln(1-p_j^{(t)}) - \ln{(1-q_k^{(t)})}$. For $t=0,...,T-1$, $q_{k}^{(t)}$ is defined as
\begin{equation}
q_{k}^{(t)} = \frac{- \sum_j \ln(1-p_j^{(t+1)})-\ln(1-q_{k}^{(t+1)})}{c_k - \sum_j \ln(1-p_j^{(t+1)})-\ln(1-q_{k}^{(t+1)})},
\end{equation}
and $q_{k}^{(T)}=0$. Finally, by taking advantage of %
conjugate priors,
in each iteration of Gibbs sampling, $c_k$ and $p_j^{(t)}$ can be drawn as
\begin{eqnarray}
(c_k | - ) &\sim& \text{Gamma}(e_0 + \sum_{t=0}^{T-1} r_{k}^{(t)}, 1 / (f_0 + \sum_{t=1}^{T} r_k^{(t)})), \nonumber\\
(p_j^{(t)} | - ) &\sim& \text{Beta}(a_0 + \sum_k n_{kj}^{(t)}, b_0 + \sum_k r_k^{(t)}).
\end{eqnarray}
The efficient augmentation technique employed in our Gibbs sampling inference removes the need for specifying a suitable proposal distribution, as in the Metropolis-Hastings inference of both DyNB~\citep{aijo2014methods} and NB-AR(1) methods~\citep{oh2013time}. Our experiments in the next section demonstrate that the Gibbs sampling algorithm of GMNB has fast convergence.
\subsection{Dynamic differential expression using Bayes factors}
\label{sec:bf}
The main goal of differential expression analysis is to identify the genes whose expressions demonstrate significant variations across conditions. In the classic static RNA-seq data analysis, this goal is usually obtained via the comparison of expression averages across groups. In dynamic RNA-seq measurement settings, however, this task becomes more challenging as any change of temporal expression patterns between groups may reflect interesting biological mechanisms. Hence, as in \citet{aijo2014methods}, we adopt the Bayes Factor (BF) as a measure that exploits information collectively from all time points to detect the genes with significant variations in temporal expression patterns across conditions.
To compute the Bayes Factor, we first consider the null hypothesis $\mbox{H}_0$ that the genes are not differentially expressed across conditions, and thus the same set of parameters govern the temporal gene expressions. In this case, we aggregate the counts $\mathcal{D}$ of both experimental conditions to fit the GMNB model $\mbox{M}_0$. On the other hand, under the alternative hypothesis $\mbox{H}_1$, the differentially expressed genes possess different model parameters in each group. Hence, GMNB models $\mbox{M}_1$ and $\mbox{M}_2$ are independently fitted to the counts in conditions 1 ($\mathcal{D}_1$) and 2 ($\mathcal{D}_2$), respectively. Then, the BF can be calculated~as
\begin{eqnarray}
\mbox{BF} &=& \frac{P(\mathcal{D} | \mbox{H}_1)}{P(\mathcal{D} | \mbox{H}_0)} = \frac{P(\mathcal{D}_1 | \mbox{M}_1) P(\mathcal{D}_2 | \mbox{M}_2)}{P(\mathcal{D} | \mbox{M}_0)}, \nonumber
\end{eqnarray}
where we have assumed equal prior probabilities for both hypotheses. The BF computation requires marginalizing out model parameters, which we conduct through Monte Carlo %
integration using posterior samples collected in the Gibbs sampling procedure. %
\section{Experimental Results}
We evaluate the proposed GMNB model and compare its performance on both synthetic and real-world temporal RNA-seq data with DyNB \citep{aijo2014methods}. We also consider DESeq2 \citep{love2014moderated}, which is a popular tool for differential expression analysis, however, not specifically designed for temporal RNA-seq data. We first consider synthetic RNA-seq data generated by different temporal models, and show that GMNB consistently provides outstanding performance in terms of the area under the curves (AUCs) of receiver operating characteristic (ROC) and precision-recall (PR) curves. Furthermore, we present two case studies on human Th17 cell differentiation \citep{tuomela2016comparative,chan2016subpopulation,aijo2014methods}, and explain the biomedical implications based on differential expression analysis over time by GMNB.
Throughout the experimental studies for synthetic and real-world data, for GMNB, in each run of Gibbs sampling inference $1000$ MCMC samples of parameters are collected after $1000$ burn-in iterations. We use the collected MCMC samples to calculate the BF for each gene as explained in Section~\ref{sec:bf}, and rank the genes according to these BFs. For DyNB, we follow the settings provided in \citet{aijo2014methods} and rank the genes using the computed BFs. We consider three different setups for differential expression analysis of temporal RNA-seq data using DESeq2. In the first setup, denoted by DESeq2-GLM in the experiments, time information is incorporated as a covariate of the generalized linear model in DESeq2 in differential expression analysis to determine temporal data in one model. In the second and third setups, we apply DESeq2 to the data at different time points independently, and use the average and minimum computed p-values from the respective differential expression analyses as an overall measure of differential expression across conditions, denoted by DESeq2-avg and DESeq2-min in the experiments, respectively.
It is worth mentioning that the use of an efficient closed-form Gibbs sampling makes GMNB, on average, 10 times faster than DyNB for both simulated and real-world temporal RNA-seq datasets by reducing the number of iterations required to converge. This is due to the low acceptance rate of the Metropolis-Hastings step of DyNB inference. Thus, to ensure the convergence of its MCMC inference, we consider performing $100,000$ iterations in DyNB for each dataset. On the other hand, our experiments show that as few as $2000$ iterations %
are sufficient for the proposed Gibbs sampling algorithm of GMNB.
\iffalse
For the analysis of the first real-world dataset
(around 20K genes with 5 time points) using 20 Matlab workers on a single
cluster node with Intel Xeon 2.5GHz E5-2670 v2 processor, it tooks around 295494 s (CPU time) and 545584 s for
the normalized GMNB and GMNB-Beta methods with $2000$ MCMC iterations, respectively, 4234441 s
for the DyNB method. For the second real time-course RNA-seq data with around 15K genes and 10 time points,
it took 557852 s and 5721046 s for GMNB-Beta and DyNB methods, respectively.
The extensive computational cost of DyNB is one of its drawback,
thereby making it less accessible to a broad range of users.
While the close form inference of the proposed method reduces the number of MCMC iterations,
DyNB considers $100000$ MCMC iterations due to $17\%$ acceptance rate.
\fi
\subsection{Synthetic data}
We first perform a comprehensive evaluation of GMNB with the synthetic data generated under different temporal RNA-seq models. More precisely, we simulat the data under the following three different setups: the proposed GMNB generative model, the DyNB generative model \citep{aijo2014methods}, and the auto-regressive (AR) based procedure \citep{oh2013time}. In all setups 10\% of genes are randomly set to be truly differentially expressed, with the procedure described in detail for each setup in the following subsections. For each specific generative model, we change the corresponding model parameters to ensure that the expected expression changes of truly differentially expressed genes are different across two conditions.
The impact of sequencing depth variation is simulated by drawing the corresponding size factors from the interval $[0.8,1.2]$ uniformly at random.
%
\subsubsection{Comparison based on GMNB generative model}
In the first simulation study, we generate the synthetic RNA-seq count data for $1000$ genes under two conditions according to the GMNB model (\ref{eq:model}) with the gamma-Markov temporal dependencies (\ref{eq:gammamodel}) between dispersion parameters. The gene-wise scale parameters $c_k$ are drawn from the uniform distribution in the interval $[0.8,2]$. To simulate 10\% differentially expressed genes, the scale parameter in the second condition are modified to be $c_k + b$, where
$$b =
\begin{cases}
0.02 & \quad \text{if } c_k < 1\\
-0.02 & \quad \text{if } c_k \geq 1
\end{cases}
$$ determines the significance of differential expression across conditions. The dispersion parameter at the initial time point, $r_k^{(0)}$, is generated for both conditions according to $\mbox{Gamma}(e_0,10)$ where $e_0=\mbox{Uniform}(30,50)$. To simulate the effect of potential varying sequencing depths, the size factors are drawn uniformly at random from the interval $[0.8, 1.2]$. For each condition and each time point, 4 replicates are generated based on the explained procedure.
Figure~\ref{gmnb} illustrates the performance of different methods evaluated based on the simulated data. The proposed GMNB model outperforms the other methods with a significant margin for both ROC and PR curves. The AUCs of both curves are also significantly higher than those by the other methods (in the legends of Figure~\ref{gmnb} and Table~\ref{Tab:auc}). On the other hand, as shown in this figure, DyNB performs close to DESeq2-GLM and worse than DESeq2-min, indicating its limitations to analyze temporal RNA-seq data from this GMNB generative model. This is due to the reason that the smooth assumption of DyNB may not always hold for the data generated by this gamma-Markov-chain based generative model.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.85 \textwidth]{simNB.pdf}
\caption{{\bf Left column:} PR Curve, {\bf Right column:} ROC Curve. Performance comparison of different methods for differential gene expression over time based on the GMNB generative model. AUCs are given in the corresponding legends in the plots.}
\label{gmnb}
\end{center}
\end{figure}
\subsubsection{Comparison based on DyNB generative model}
In the second simulation study, data is generated according to the DyNB model assumptions. More specifically, we draw the true mean values $\mu_k$, for $1000$ genes from a Gaussian process with the mean $m_k$ and the covariance matrix $Cov(t_i, t_j) = \theta_k \text{exp}(-\frac{1}{2\alpha_k}|t_i - t_j|)$, where $m_k$, $\theta_k$ and $\alpha_k$ are uniformly distributed in the intervals $[1000, 2000]$, $[100, 10000]$ and $[0.5, 1]$, respectively. We consider five time points at 0, 12, 24, 48 and 72 hours, similar to the real-world dataset \citep{aijo2014methods}. 10\% of genes are set to be truly differentially expressed across conditions by changing their mean values $m_k$ and covariance function parameters $\{ \theta_k, \alpha_k \}$ to $\{ b m_k, c \theta_k, \alpha_k \pm d \}$, where $b = 1.5$, $c = 10$, and $d = 0.25$ determine the significance of expected expression changes across conditions. Similar to the previous simulation setup, 4 replicates are generated for each time point in the corresponding condition. %
Figure~\ref{gp} demonstrates the performances of different methods applied to the data generated according to the above procedure. GMNB still clearly outperforms the other methods based on both ROC and PR curves. The inferior performance of DyNB may be due to the small number of replicates for each time point, leading to poor estimates of both $\theta_k$ (heuristically estimated by the data dependent value $10 \times \mbox{stdev}(\mathbf{Y})$ based on the observed replicates $\mathbf{Y}=\{\mathbf{y_1, \dots, y_J}\}$ in \citet{aijo2014methods}) and $\mu_k$ (heuristically estimated by $\frac{\mbox{min}(\mathbf{Y}) + \mbox{max}(\mathbf{Y})}{2}$ in \citet{aijo2014methods}). On the contrary, the fully Bayesian nature of GMNB makes its performance robust to the number of replicates as well as potential noise at each time point. Both variates of static DESeq2 under-perform the dynamic approaches remarkably, as they neglect the correlation between samples across time points. In addition, DESeq2-GLM under-performs both GMNB and DyNB substantially, as it neglects the inherent dynamics of RNA-seq experiments specifically. %
To further demonstrate the potential influence of gene expression levels, reflected by the expected read counts, on the detection power of differentially expressed genes,
additional simulations are performed with the same parameters as above, except for the mean parameter $m_k$, for which three sampling uniform distributions are tested with the intervals $[1000, 2000]$, $[200, 1200]$, and $[50, 200]$, leading to three different datasets with different expected overall counts. We compare the differential expression analysis results for GMNB, DyNB, and DESeq2-min as they are top performing methods in this set of experiments for 20 randomly generated synthetic datasets for each setup. As shown in Figure~\ref{barplot}, according to both AUC-ROC and AUC-PR, GMNB consistently outperforms the other two methods no matter when we have low or high level counts. It is also noticeable that DyNB and DESeq2-min are more sensitive with variable performances across 20 randomly generated datasets. This indicates that GMNB better borrows signal strengths across time points compared to DyNB and DESeq2-min. %
\begin{figure}[t]
\begin{center}
\includegraphics[width=.85 \textwidth]{simGP.pdf}
\caption{{\bf Left column:} PR Curve, {\bf Right column:} ROC Curve. Performance comparison of different methods for differential gene expression over time based on the DyNB generative model. AUCs are given in the corresponding legends in the plots.}
\label{gp}
\end{center}
\end{figure}
\subsubsection{Comparison based on NB-AR(1) generative model}
In addition to synthetic data based on the GMNB and DyNB models, we evaluate these methods with the simulated data based on the NB-AR(1) model~\citep{oh2013time}. More precisely, the count for gene $k$ at time $t$ is distributed according to a NB distribution whose mean parameter satisfies $\text{log}(\mu_k^{(t)}) = \omega_k^{(t)} + \beta_k$. Here $\beta_k$ follows the uniform distribution in $[4.5, 5.5]$ to test the temporal differential expression performance with low read counts. The parameter $\omega_k^{(t)}$ is obtained through an auto-regressive process $\phi_k \omega_k^{(t-1)} + \epsilon^{(t)}$, where $\phi_k$ is randomly generated from the uniform distribution in $[0.1, 0.9]$, and $\epsilon^{(t)}$ is a standard zero-mean white noise process. Similar to the previous two simulation models, read counts are generated for $1000$ genes and 10\% of them selected to be differentially expressed by changing the parameter $\phi_k$ to $b \phi_k$ for the second condition, where
$$b =
\begin{cases}
3/2 & \quad \text{if } \phi_k \leq 0.5\\
2/3 & \quad \text{if } \phi_k > 0.5
\end{cases}
$$ determines the significance of differential expression across conditions.
Figure~\ref{ar} demonstrates the performances of different methods applied to the NB-AR(1) data. GMNB again outperforms DyNB and DESeq2 with a remarkable margin in both the ROC and PR curves. %
This is due to the state-space nature of the NB-AR(1) simulation setup, in which differential expression is defined through the model parameter $\phi_k$ that controls the temporal dependence of gene expression. However, the temporal correlation assumptions of the Gaussian process, %
different from this generative model, makes it less powerful to identify all differentially expressed genes. The results in Figure~\ref{ar} demonstrate the higher power of GMNB in detecting temporal differential expression patterns, especially with low expression levels (read counts are approximately 150 here). Similar to the DyNB generative model, we compare the performance of GMNB with DyNB and DESeq2-min (top performing methods) with different expected counts. In Figure~\ref{barplot}, additional simulations are performed with the same parameters as above, except the uniform distribution for the $\beta_k$ with three varying intervals: $[4.5, 5.5]$, $[4.5, 6.5]$, and $[5.5, 6.5]$, leading to corresponding expected counts from 150 to 450. Figure~\ref{barplot} shows again that for low counts, GMNB outperforms both DyNB and DESeq2, especially in AUC-PR. Note that, with increasing expression levels, DESeq2-min's performance improves because of high signal strengths at individual time points.
As shown by the ROC and PR curves in both the GMNB and AR generative models, DESeq2-min outperforms DyNB. This indicates that the temporal correlation assumptions in DyNB may not fully capture the dynamic changes in these two state-space generative models, which can have abrupt non-smooth changes. In addition, the heuristic estimation of model parameters adopted in DyNB \citep{aijo2014methods} when the number of replicates is low can be the other reason for the degraded performance.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.85 \textwidth]{simAR.pdf}
\caption{{\bf Left column:} PR Curve, {\bf Right column:} ROC Curve. Performance comparison of different methods for differential gene expression over time based on the NB-AR(1) generative model. AUCs are given in the corresponding legends in the plots.}
\label{ar}
\end{center}
\end{figure}
In summary, on synthetic RNA-seq count data from different generative models, comparison of both the ROC and PR curves shows that GMNB outperforms both the recently proposed temporal (DyNB) and static differential analysis methods that aggregate differential statistics in heuristic ways (DESeq2 with different setups). Table~\ref{Tab:auc} summarizes the average AUCs and their standard deviation values of both ROC and PR curves for 20 randomly generated synthetic datasets by the top three performing methods (GMNB, DyNB, and DESeq2-min).
GMNB improves the performances of DyNB and DESeq2-min, in terms of AUC-PR, at least by $23\%$ and $17\%$, respectively. In the best case scenario, GMNB improves the AUC-PR performances of DyNB and DESeq2-min up to $48\%$ and $71\%$, respectively. In terms of AUC-ROC, GMNB improves the best case performances of DyNB and DESeq2-min by $12\%$ and $10\%$, respectively.
Even with the data from the DyNB generative model, the fully Bayesian method GMNB outperforms DyNB, which estimates some of its model parameters in a heuristic manner \citep{aijo2014methods}. In addition, GMNB achieves robust performance in both state-space (GMNB and NB-AR(1)) and functional (DyNB) generative models. We demonstrate the superior power of GMNB in low count situations by collective information across time points. For these three different types of synthetic data, as shown in Figures~\ref{gmnb} - \ref{barplot}, and Table~\ref{Tab:auc}, measured by
both AUC-ROC and AUC-PR, it is interesting to notice that DyNB works better than DESeq2-min only when the synthetic data are generated based on its model assumption.
\begin{figure}[h]
\begin{center}
\includegraphics[width=.85 \textwidth]{barplot.pdf}
\caption{AUC comparison of different methods for differential gene expression analysis over time in low counts.}
\label{barplot}
\end{center}
\end{figure}
\begin{table}[!t]
\centering
\caption{Comparison of AUCs based on 20 runs for each method.\label{Tab:auc}}
{\begin{tabular}{@{}l|l|lll@{}}
\toprule {\bf AUC} & {\bf Generative Model} & {\bf GMNB } & {\bf DyNB} & {\bf DESeq2-min} \\\hline \hline
{} & {\bf GMNB} & {\bf 0.84 $\pm$ 0.02} & {0.75 $\pm$ 0.05} & {0.80 $\pm$ 0.07}\\
{\bf ROC} &{\bf DyNB} & {\bf 0.94 $\pm$ 0.01} & {0.86 $\pm$ 0.08} & {0.85 $\pm$ 0.03}\\
{} & {\bf NB-AR(1)} & {\bf 0.81 $\pm$ 0.03} & {0.73 $\pm$ 0.07} & {0.77 $\pm$ 0.07}\\\hline
{} & {\bf GMNB} & {\bf 0.61 $\pm$ 0.04} & {0.41 $\pm$ 0.08} & {0.52 $\pm$ 0.06}\\
{\bf PR} & {\bf DyNB} & {\bf 0.79 $\pm$ 0.03} & {0.64 $\pm$ 0.20} & {0.46 $\pm$ 0.06}\\
{} & {\bf NB-AR(1)} & {\bf 0.51 $\pm$ 0.06} & {0.39 $\pm$ 0.07} & {0.43 $\pm$ 0.10}\\\hline
\end{tabular}}{}
\end{table}
\subsection{Human Th17 cell induction}
To further illustrate how GMNB may help identify differentially expressed genes from temporal RNA-seq data for biologically significant results, we provide such a case study consisting of $57$ human samples during the priming of T helper 17 (Th17) cell differentiation \citep{tuomela2012identification}. The main goal of designing this case study is to gain insights into the differentiation process by unraveling dependency between different genetic factors in various pathways, which may serve as potential biomarkers of immunological diseases for therapeutic intervention design. In this dataset \citep{tuomela2016comparative},
at $0, 0.5, 1, 2, 4, 6, 12, 24, 48$, and $72$ hours of Th17 polarized cells and control Th0 cells, three biological replicates were collected for transcript profiling by RNA-seq. The data were downloaded from Gene Omnibus with the accession number GSE52260~\citep{tuomela2016comparative,chan2016subpopulation}. %
When checking the 10 %
most
differentially expressed genes based on their BFs by GMNB, all of them have been reported to be differentially expressed in other studies investigating Th17 cell differentiation. Among them, the top differentially expressed gene is thrombospondin-1 (TSP1), whose encoded protein participates in the differentiation of Th17 cells by activating transforming growth factor beta (TGF-$\beta$) and enhancing the inflammatory response in experimental autoimmune encephalomyelitis (EAE) \citep{yang2009deficiency}.
The second gene in the list is Lymphotoxin $\alpha$ (LTA), a member of the tumor necrosis factor (TNF) superfamily
that is both secreted and expressed on the cell surface of activated Th17 cells \citep{chiang2009targeted}.
The third gene, COL6A3, contributes to adipose tissue inflammation \citep{pasarica2009adipose}
and responds quickly to Th17 cell polarizing stimulation \citep{tripathi2017genome}.
The gene Cathepsin L (CTSL1) is ranked as the fourth in the list and is linked to the regulation of immune responses at the level of MHC complex maturation and Ag presentation influencing differentiation of CD4+ cells and autoimmune reactions \citep{reiser2010specialized}. The fifth gene, FURIN, has been reported as a T cell activation gene that regulates the T helper cell balance of the immune system \citep{pesu2008t}. The sixth gene lamin A (LMNA) has been identified as one of the immune response regulators \citep{gonzalez2014nuclear}. The seventh gene, Filamin A (FLNA), is required for T cell activation \citep{hayashi2006filamin}. The eighth gene, SBNO2, has been reported to influence Th17 cell differentiation \citep{tripathi2017genome}. \citet{zhao2014comparison} observed significant changes in the expression of the ninth gene ACTB in activated T cells.
Finally, the tenth gene Notch1 is activated in both mouse and human in vitro-polarized Th17 cells and also in Th17 polarized cells as compared with Th0 control cells \citep{keerthivasan2011notch}.
We then investigate how the results of DyNB differ from those of GMNB. The majority of the above genes are indeed ranked relatively high by DyNB as differentially expressed, except two genes: FLNA and ACTB. For these two genes, their expression levels change abruptly after 12 hours of T17 differentiation. These two genes demonstrate that the DyNB method may fail to detect temporal differential expression when the temporal gene expression trends are not smooth. As an instance, Figure~\ref{g9} illustrates that DyNB is not able to capture the temporal expression changes of gene FLNA accurately. More precisely, Figure~\ref{g9}(a) shows the posterior means of expected gene expression $\mu_k$ based on DyNB and their corresponding confidence intervals, where circles and diamonds represent the normalized counts from Th0 and Th17 lineages, respectively. To further assess the power of the models in reproducing the observed gene counts, for each model, we generate 1000 gene counts per sample and time point based on the inferred parameters, and then calculate the 99\% confidence interval using these synthetically generated counts. Figure~\ref{g9}(b) demonstrates the means and confidence intervals of the counts generated via this procedure for DyNB, where the circles and diamonds represent the observed raw counts from Th0 and Th17 lineages, respectively. Similar to plots in Figures~\ref{g9}(a) and ~\ref{g9}(b), we perform the same examinations on expression pattern of FLNA by the GMNB model. To demonstrate the expression levels of the $k$th gene between two groups, the DyNB compares the posterior NB mean parameters $\mu_k$, whereas the GMNB compares the posterior NB shape parameters $r_k$.
One may consider that the expression level of gene $k$ is assumed to roughly follow a function of the shape parameter $r_k$ in the GMNB, but the observed counts should be demonstrated in a same scale as the shape parameter. The difference between the posterior shape parameters $r_k$ explains the differences between the means, since if $n_{kj}^{(t)} \sim \mbox{NB}(r_k^{(t)}, p_j^{(t)} )$, then $\mathop{\mathbb{E}} [n_{kj}^{(t)}] = r_k^{(t)}p_j^{(t)}/(1 - p_j^{(t)})$.
Therefore, Figure~\ref{g9}(c) shows the posterior means of $r_k$ based on GMNB and their corresponding confidence intervals, where the circles and diamonds are obtained by dividing the observed counts by the parameter $p_j^{(t)}/(1-p_j^{(t)})$ representing the sequencing depth in the proposed model. Additionally, Figure~\ref{g9}(d) demonstrates the means and confidence intervals for synthetically generated gene counts based on the inferred parameters of GMNB, where the read counts on the y-axis are observed read counts. Not only GMNB improves the model fitting over 24h to 72h, but also it has more robust estimation of expression patterns for the starting time points with lower counts (Figures~\ref{g9}(c) and~\ref{g9}(d)). The calculated BFs for the gene FLNA are 2.3461 and $1.60 \times 10^{308}$ by DyNB and GMNB, respectively.
GMNB also identifies ACTB as a gene with significant differential temporal expression (BF $>$ 10) but DyNB again fails to capture the abrupt expression changes and thereby associates low BF (supplement materials). The corresponding temporal expression plots are depicted in Figure~S1 of the supplement materials.%
\begin{figure}[htp]
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{FLNA_DyNB_scale.pdf}
\caption{}
\label{g9_norm}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{FLNA_DyNB_raw.pdf}
\caption{}
\label{g9_raw}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{FLNA_Beta.pdf}
\caption{}
\label{g9_gmnb_norm}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{FLNA_Beta_raw.pdf}
\caption{}
\label{g9_gmnb_raw}
\end{subfigure}
\caption{\textbf{Differentially expressed gene FLNA detected by GMNB but not by DyNB.} (a) The normalized gene expression profile of {\bf FLNA} over time estimated by DyNB model. The normalization of read counts on the y-axis are obtained by using the normalization method of DESeq. The solid blue and red curves are the posterior means under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas). (b) The gene expression profile of \textbf{FLNA} over time estimated by DyNB. The read counts on the y-axis are observed read count. The solid blue and red curves are the means of the generated samples based on the inferred parameters by DyNB under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas around means). (c) The normalized gene expression profile of {\bf FLNA} over time estimated by the proposed GMNB model. The normalization of read counts on the y-axis are obtained by dividing the observed counts by the parameter ${p_j^{(t)}}/({1-p_j^{(t)})}$ representing the sequencing depth in the model. The solid blue and red curves are posterior means of $r_k$ under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas). (d) The gene expression profile of \textbf{FLNA} over time estimated by GMNB. The read counts on the y-axis are observed read count. The solid blue and red curves are the means of the generated samples based on the inferred parameters by GMNB under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas around means).}
\label{g9}
\end{figure}
On the other hand, LGALS1, SEPT5, BATF3, COL1A2, and ENO2 are five genes out of 90 differentially expressed genes detected by DyNB with BFs $2.59 \times 10^{7}$, $472.34$, $2.90 \times 10^{4}$, $404.34$, and $398.43$, whereas they are associated with BFs lower than 10 by GMNB. %
Figure~\ref{lgals} illustrates the expression profile of the gene LGALS1 inferred by DyNB and GMNB, indicating that DyNB is not able to filter out those low count genes for which the replicated Th0 and Th17 lineages are seemingly similar and leads to this potential false positive. Figure~\ref{lgals}(a) shows the posterior means of expected gene expression $\mu_k$ based on DyNB under Th0 and Th17 lineages with their corresponding confidence intervals. Figure~\ref{lgals}(b) shows the means and confidence intervals for 1000 generated samples based on the inferred parameters of DyNB model. While the normalized counts are plotted in Figure~\ref{lgals}(a), the circles and diamonds mark Th0 and Th17 lineages, respectively, for the observed raw counts in Figure~\ref{lgals}(b). On the contrary, GMNB considers this gene not significantly differentially expressed with similar inferred temporal expression profiles across conditions, as demonstrated in Figures~\ref{lgals}(c) and~\ref{lgals}(d). This may be explained by the fact that GMNB employs a fully generative model of gene expressions, including the sequencing depth, while DyNB uses a deterministic ad-hoc procedure to normalize gene counts, and thus neglecting the uncertainty over the sequencing depth when computing the BF, leading to potential false positives.
Figures~\ref{lgals}(c) and~\ref{lgals}(d) demonstrate the posterior means of $r_k$ based on GMNB and the means of synthetically generated samples based on the inferred parameters of the proposed model, respectively. %
Similar to the plots for LGALS1, Figures~S8,~S9,~S10, and~S11 in the supplement materials show the similar trends for the genes SEPT5, BATF3, COL1A2, and ENO2 based on the results by DyNB and GMNB.
\begin{figure}[htp]
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{LGALS_DyNB.pdf}
\caption{}
\label{lgals_norm}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{LGALS_DyNB_raw.pdf}
\caption{}
\label{lgals_raw}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{LGLAS_Beta.pdf}
\caption{}
\label{lgals_gmnb_norm}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\textwidth]{LGLAS_Beta_raw.pdf}
\caption{}
\label{lgals_gmnb_raw}
\end{subfigure}
\caption{\textbf{Example of genes detected as differentially expressed by DyNB but not by GMNB: \textbf{LGALS1}.} (a) The normalized gene expression profile of {\bf LGALS1} over time estimated by DyNB model. The normalization of read counts on the y-axis are obtained by using the normalization method of DESeq. The solid blue and red curves are the posterior means under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas). (b) The gene expression profile of \textbf{LGALS1} over time estimated by DyNB. The read counts on the y-axis are observed read count. The solid blue and red curves are the means of the generated samples based on the inferred parameters by DyNB under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas around means). (c) The normalized gene expression profile of {\bf LGALS1} over time estimated by the proposed GMNB model. The normalization of read counts on the y-axis are obtained by dividing the observed counts by the parameter ${p_j^{(t)}}/({1-p_j^{(t)})}$ representing the sequencing depth in the model. The solid blue and red curves are posterior means of $r_k$ under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas). (d) The gene expression profile of \textbf{LGALS1} over time estimated by GMNB. The read counts on the y-axis are observed read count. The solid blue and red curves are the means of the generated samples based on the inferred parameters by GMNB under Th0 and Th17 lineages, respectively, with corresponding $99\%$ CIs (shaded areas around means).}
\label{lgals}
\end{figure}
In order to further demonstrate the advantages of GMNB, the overlap of three approaches (GMNB, DyNB and DESeq2-min), for 100 top differentially expressed genes identified by GMNB, is depicted as a Venn diagram in Figure~\ref{venn}. A gene is differentially expressed based on DESeq2-min if the corresponding $\text{p-value} < 0.05$ at any time point. Out of top 100 differentially expressed genes identified by GMNB ($\mbox{log(BF)} > 100$), 16 genes are identified only by GMNB. The temporal expression plots for six of them, i.e. the genes EGR1, NR4A1, MYC, PKM2, EGR2, and IL6ST, are depicted in %
Figures~S2,~S3,~S4,~S5,~S6, and~S7, indicating the differential dynamic patterns identified by GMNB.
Among these genes, EGR1 is a transcription factor known to inhibit the expression of GFI1, a negative regulator of Th17 differentiation, by directly binding to its promoter and its expression is detected only in the early phase of Th17 differentiation \citep{kurebayashi2012pi3k}. The gene NR4A1 plays critical roles in T cell apoptosis during the thymocyte development \citep{doi2008orphan}. Not only this gene is a proapoptotic transcription factor, but also it is reported as a survival factor and activator of metabolic pathways. Both facets show the NR4A1's role in T-cell differentiation as a balancing molecule in the fate determination \citep{fassett2012nuclear}.
The gene MYC has been reported as one of the key transcript factors for Th17 differentiation \citep{yosef2013dynamic,sawcer2011genetic,gnanaprakasam2017myc}. PKM2 is induced and interacts with and promotes the function of HIF1$\alpha$ that is critical to drive Th17 differentiation \citep{corcoran2016hif1alpha}. EGR2 has been identified as an important transcription factor in the development and function of Th17 cells \citep{zhang2015roles,zhu2008early}. IL6ST is known as a signature transcript of Th17 cells \citep{ghoreschi2010generation}.
This again illustrates the benefits of GMNB on better modeling temporal dynamic changes to detect biologically meaningful genes who show significant difference in temporal changes but do not show significant differential expression when studying them at individual time points.
\begin{figure}
\begin{center}
\includegraphics[width=.5 \textwidth]{venn.pdf}
\caption{A Venn diagram representing the overlaps of the top 100 differentially expressed genes detected by GMNB with DyNB and DESeq2-min.}
\label{venn}
\end{center}
\end{figure}
\subsubsection{Gene Ontology (GO) analysis}
To further demonstrate the biological relevance of the detected genes by GMNB,
GO analysis of top 100 differentially expressed genes ($\mbox{log(BF)} > 100$) has been performed using Fisher's exact test. Enriched GO terms (Table S1 in the supplement materials) by these genes agree
with the current biological understanding of the Th17 differentiation process.
The most significantly enriched GO terms are related to the organ development ($\text{p-value} < 2 \times 10^{-23}$), immune system process ($\text{p-value} < 6 \times 10^{-21}$), immune response ($\text{p-value} < 1 \times 10^{-19}$), response to stimulus ($\text{p-value} < 3 \time 10^{-19}$), cell differentiation ($\text{p-value} < 3 \times 10^{-18}$), and defense response ($\text{p-value} < 2 \times 10^{-16}$).
In particular, $38\%$ and $74\%$ of these 100 genes are annotated to immune response and response to stimulus, respectively, supported by the central role of Th17 cells in the pathogenesis of autoimmune and inflammatory diseases \citep{waite2011th17}. %
\subsection{RNA-seq data in \citet{aijo2014methods}: Human-activated T- and Th17 cells}
\label{sec:dynbds}
We further analyze the second temporal RNA-seq dataset, for which DyNB was implemented for studying Th17 cell lineage \citep{aijo2014methods}.
In this dataset, CD4+ T cells were activated and polarized as described in \citet{tuomela2012identification} and RNA-seq data were collected at 0, 12, 24, 48 and 72 hours of both the activation (Th0) and differentiation (Th17).
At each time point, there are 3 biological replicates for both cell lineages.
The original paper \citep{aijo2014methods} performed DyNB to quantify Th17-specific gene expression dynamics.
The authors in~\citet{aijo2014methods} first normalized the RNA-seq counts by the DESeq pipeline \citep{anders2010differential}. Then, DyNB was applied to the normalized expression values to identify differentially expressed genes between the Th0 and Th17 lineages. Genes were considered differentially expressed if (i) $\mbox{BF} > 10$, and (ii) $\mbox{fold-change} > 2$ for at least one time point. Out of 698 differentially expressed genes identified by DyNB, three genes were investigated and discussed in~\citet{aijo2014methods} with the qRT-PCR validation: $IL17A$, $IL17F$, and $RORC$.
We apply GMNB to analyze the same Th17 cell lineage dataset to identify differentially expressed genes. To compare the ranked lists of genes by GMNB and by DyNB respectively, Table 1 gives the ranks as well as the computed BF values by GMNB and DyNB for these reported genes in \citet{aijo2014methods}. These qRT-PCR validated genes are in fact ranked higher by GMNB, indicating more promising potential for marker gene identification.
\begin{table}[!t]
\centering
\caption{Comparison of BF ranks for the reported genes by DyNB\label{Tab:01}}
{\begin{tabular}{@{}l||ll@{}}
\toprule {\bf Genes} & {\bf DyNB } & {\bf GMNB} \\\hline \hline
{\bf RORC} & 37 (BF = $2.26 \times 10^{93}$) & {\bf 26} (BF = $2.98 \times 10^{48}$) \\
{\bf IL17F} & 352 (BF = $1.74 \times 10^{15}$) & {\bf 175} (BF = $2.53 \times 10^{9}$)\\
{\bf IL17A} & 755 (BF = $6.96 \times 10^{8}$) & {\bf 345} (BF = $3.90 \times 10^{4}$) \\\hline
\end{tabular}}{}
\end{table}
\section{Conclusions}
GMNB offers a comprehensive and fully Bayesian solution to study temporal RNA-seq data. The most notable advantage is the capacity to capture a broad range of gene expression patterns over time by the integration of a gamma Markov chain into a negative binomial distribution model. This allows GMNB to offer consistent performance over different generative models and makes it be robust for studies with different numbers of replicates by borrowing the statistical strength across both genes and samples. Another critical advantage is the efficient closed-form Gibbs sampling inference of the model parameters, which improves the computational complexity compared to the state-of-the-art methods. This is achieved by using a statistically well-founded data augmentation solution. In addition, GMNB explicitly models the potential sequencing depth heterogeneity so that no heuristic preprocessing step is required. Experimental results on both synthetic and real-world RNA-seq data demonstrate the state-of-the-art performance of the GMNB method for temporal differential expression analysis of RNA-seq data.
\section{Acknowledgments}
This research was supported in part by the NSF Awards CCF-1553281, CCF-1718513, and the USDA NIFA Award 06-505570-01006 to X. Qian and the NSF Award DBI-1532188 and the funding support from QNRF (NPRP9-001-2-001, NPRP7-1634-2-604), CSTR (2017-01), and CONACYT to P. de Figueiredo.
\begin{spacing}{1}
\bibliographystyle{abbrvnat} %
|
{
"timestamp": "2018-03-08T02:04:34",
"yymm": "1803",
"arxiv_id": "1803.02527",
"language": "en",
"url": "https://arxiv.org/abs/1803.02527"
}
|
\section{Introduction}
\label{sec:intro}
As the popularity of big data increases and more data is being gathered, the importance of sequential models that are able to continuously update with new data has increased. These models are particularly crucial in high throughput real-time applications such as speech or streaming text classification. To this end, we propose a sequential framework to update the probabilistic maximum margin classifier built from the Maximum Entropy Discrimination (MED) principle of \cite{NIPS1999_1733}.
The proposed sequential MED framework can be cast as recursive Bayesian estimation where the likelihood function is a log-linear model formed from a series of constraints and weighted by Lagrange multipliers. In the Gaussian case it shares similarities with the problem of Gaussian process classification, which has been previously studied \cite{wahba1999support, jaakkola1999probabilistic, smola1998connection, opper1999gaussian, Sollich2002, Rasmussen:2005:GPM:1162254}, but to the best of our knowledge, a method to recursively update the Gaussian process classifier has not been developed. In the single time point case, sequential MED can be specialized to the support vector machine \cite{smola1998connection} and Laplacian support vector machine \cite{Belkin:2006:MRG:1248547.1248632} as previously discussed in \cite{NIPS1999_1733} and \cite{hou}.
We are interested in situations where we receive a stream of data $ \bm{X}_{(1)}, \bm{X}_{(2)}, \ldots$ over time $t$ where each $X_{(t)}$ is a matrix of dimension $p \times n$, with $p$ denoting the number of feature variables and $n$ denoting the number of i.i.d. samples, where $n=n_{(t)}$ may vary with time. In the fully labeled scenario, the data has corresponding labels $y_i = [1, -1] \, \forall i \text{ and } t$; however in the partially labeled scenario, at each time point $t$, only $ l_{(t)} < n_{(t)} $ of the samples have labels. We define the observed data at any time point $t$ as $\mathcal{D}_{(t)} = \{\bm{X}_{(t)}, \bm{y}_{(t)} \}$ and all observed data up to time $\tau$ as $ \{ \mathcal{D}_{(t)} \}_{t=1}^\tau$. Such scenarios would arise in a variety of domains such as a satellite that only transmits its data daily or a government agency that only releases its data quarterly with their corresponding reports. The rest of the paper is organized as follows: Section 2 and Section 3 will discuss how to sequentially update the corresponding MED models for supervised and semi-supervised classification. Section 4 validates the method by simulation and we present an application to a dataset of spoken letters of the English alphabet.
\section{Sequential MED} \label{sec:SeqMED}
Constrained relative entropy minimization is used to estimate the closest distribution to a given prior distribution subject to a set of moment constraints. The authors of \cite{koyejo2013representation} show that, if the prior distribution is from the exponential family, then the density that optimizes the constrained relative entropy problem is also a member of the exponential family. Similar to Bayesian conjugate priors, there exist relative entropy conjugate priors that facilitate evaluation of the closest distribution. These produce optimal constrained relative entropy densities, which can be thought of as posteriors, from the same parametric family as the prior. Maximum entropy discrimination (MED) \cite{NIPS1999_1733} also admits conjugate priors as it a special case of constrained relative entropy minimization where one of the constraints is over a parametric family of discriminant functions $ \mathcal{L}(\bm{X} | \bm{\Theta}) $.
\subsection{Review of MED for Maximum Margin Classification}
In this paper, we are interested in maximum margin binary classifiers. In this case the discriminant function $ \mathcal{L}(\bm{X} | \bm{\theta}, b) = f(\bm{X}) \bm{\theta} + b $ is linear for some feature transformation $f(\cdot)$, feature weights vector $\bm{\theta}$, and bias term $b$. Slack variables $\gamma_i$ are used to create a margin in the constraints $\text{E}(y_i ( f(\bm{X}_i) \bm{\theta} + b) - \gamma_i)$, the expected hinge loss with slack variables. The MED objective function is
\begin{flalign} \label{MED_obj}
&\underset{\P(\bm{\Theta}, \bm{\gamma} | \mathcal{D} )}{\min} \text{KL}\left(\P(\bm{\Theta}, \bm{\gamma} | \mathcal{D} || \P_0(\bm{\Theta}, \bm{\gamma}) \right) \qquad \text{subject to} \\
&\iint \P(\bm{\Theta}, \bm{\gamma}| \mathcal{D} ) \, (y_i ( f(\bm{X}_i) \bm{\theta} + b) - \gamma_i) \, d\bm{\Theta} d\bm{\gamma} \ge 0 \,\, \forall i = 1, \dots, n \notag
\end{flalign}
whose solution $\P(\bm{\Theta}, \bm{\gamma} | \mathcal{D}) $ is the constrained minimum relative entropy posterior. The associated MED decision rule $ \hat{y}_{i'} = \text{sgn}( \iint \P(\bm{\Theta}|\mathcal{D}) (f(\bm{x}_{i'}) \bm{\theta} + b) \, d\bm{\Theta} ) $ is a weighted combination of discriminant functions. The minimum relative entropy posterior has the form
$$ \P(\bm{\Theta}, \bm{\gamma}| \mathcal{D}) = \frac{\P_0(\bm{\Theta}, \bm{\gamma})}{Z(\bm{\alpha})} \exp \left\{ \sum_{i=1}^n \alpha_i\left (y_i (f(\bm{X}) \bm{\theta} + b) - \gamma_i \right) \right\} $$
where $ \bm{\alpha} = [ \alpha_1, ..., \alpha_n ]^T \ge 0 $ are Lagrange multipliers that minimize the partition function $Z(\bm{\alpha})$. It is common to set the initial prior distribution to the separable form: \\ $ \P_0(\bm{\Theta}, \bm{\gamma}) = \P_0(\bm{\theta}) \P_0(b) \prod_{i=1}^n \P_0(\gamma_i ) $. If in addition, we specify that $ \P_0(\gamma_i) = C e^{-C(1-\gamma_i)} \mathcal{I}(\gamma_i \le 1) $, $ \P_0(\bm{\theta}) $ is $ N(\bm{0}, \bm{\mathrm{I}}) $, and $ \P_0(b) $ is a zero mean Bayesian non-informative (diffuse) prior, denoted $N(0, \infty)$, then the Lagrange multipliers can be obtained as the solution $\hat{\bm{\alpha}}$ to the constrained optimization
\begin{flalign*}
& \underset{\bm{\alpha}}{\max} -\frac{1}{2} \bm{\alpha}^T \bm{Y} f(\bm{X}) f(\bm{X})^T \bm{Y} \bm{\alpha} + \sum_{i=1}^n \alpha_i + \log(1-\alpha_i /C) \\
& \text{subject to } \sum_{i=1}^n y_i \alpha_i = 0 \text{ and } \alpha_1, \dots, \alpha_n \ge 0
\end{flalign*}
where $\bm{Y} = \text{diag}(\bm{y})$. This objective function has a log barrier term $ \log(1-\alpha_i/C) $ instead of the inequality constraints $ \alpha_i \leq C $ commonly found in the dual form of the SVM. Except in some ill-defined cases where the maximum lies near the boundary of the feasible set, the $ \hat{\alpha}_i $ will be identical to the optimal support vectors that maximize the SVM objective. The authors in \cite{NIPS1999_1733, hou} show that the \textit{maximum a posteriori} (MAP) estimator for $\bm{\theta}$ of the MED posterior is related to the Lagrange multipliers by $ \hat{\bm{\theta}} = f(\bm{X})^T \hat{\bm{\alpha}} $, so the MED posterior mode is equivalent to a maximum margin classifier.
\subsection{Updating MED}
Under the separable prior assumptions above, the MED posterior $ \P(\bm{\Theta}, \bm{\gamma}| \mathcal{D}) $ will take the factored form $\P(\bm{\theta}| \mathcal{D}) \P(b| \mathcal{D}) \P(\bm{\gamma}) $. Due to the fact that the slack parameters $\gamma_i$ do not depend on the data $\mathcal{D}$, the density $\P(\bm{\gamma})$ does not affect the MED decision rule given after \eqref{MED_obj}. Hence only $\P(\bm{\theta}| \mathcal{D}) $ and $\P(b| \mathcal{D}$ are important. This remaining part of the MED posterior has the form: $ \P(\bm{\theta}| \mathcal{D}) \P(b| \mathcal{D}) = N( f(\bm{X})^T \bm{Y} \bm{\alpha} , \bm{\mathrm{I}}) N(0, \infty)$, which is a conjugate distribution. Due to this conjugacy the posterior distribution optimizing the objective in \eqref{MED_obj} can be propagated forward in time in a recursive manner. The updating procedure is given in the following theorem and corollaries.
\begin{theorem} \label{thm:SeqMED}
Let the MED prior at $t = 1$ be $ \bm{\theta} \sim N(\bm{0}, \bm{\mathrm{I}}), b \sim N(0, \infty)$, and $\emph{P}_0(\gamma_i) = C_{(1)} e^{-C_{(1)} (1-\gamma_i)} \mathcal{I}(\gamma_i \le 1) $. Then given data $ \mathcal{D}_{(\tau)}$ at time point $\tau$, the relative entropy conjugate priors are
\begin{flalign*}
& \emph{P}_0 \left(\bm{\theta} | \{ \mathcal{D}_{(t)} \}_{t=1}^{\tau-1} \right) = N\left( \sum_{t=1}^{\tau-1} f(\bm{X}_{(t)})^T \bm{Y}_{(t)} \hat{\bm{\alpha}}_{(t)} , \bm{\mathrm{I}}\right) \\
& \emph{P}_0\left(b | \{ \mathcal{D}_{(t)} \}_{t=1}^{\tau-1} \right) = N(0, \infty) \\
& \emph{P}_0(\bm{\gamma}) = \prod_{i=1}^{n_{(\tau)}} C_{(\tau)} \exp \left\{-C_{(\tau)} (1-\gamma_i) \right\} \mathcal{I}(\gamma_i \le 1)
\end{flalign*}
and the MED posterior $ \emph{P}(\bm{\Theta}| \left\{ \mathcal{D} \right\}_{t=1}^{\tau} ) $ can represented as
$$ \emph{P} \left(\bm{\theta} | \left\{ \mathcal{D} \right\}_{t=1}^{\tau} \right) = N\left( \bm{\mu}_0 + f(\bm{X}_{(\tau)})^T \bm{Y}_{(\tau)} \hat{\bm{\alpha}}_{(\tau)}, \bm{\mathrm{I}} \right) $$
where $ \bm{\mu}_0 = \sum_{t=1}^{\tau-1} f(\bm{X}_{(t)})^T \bm{Y}_{(t)} \hat{\bm{\alpha}}_{(t)} $ is the prior mean and $ \emph{P}(b | \left\{ \mathcal{D} \right\}_{t=1}^{\tau} ) $ is the same as the Bayes non-informative prior.
\end{theorem}
Introducing the kernel function $ k(\bm{x}, \bm{x}') = \langle f(\bm{x}), f(\bm{x}')\rangle $ and the parameter transformation $ \bm{\omega} = f(\bm{X}) \bm{\theta} $, the posterior at time $\tau>0$ can be represented in terms of this kernel.
\begin{corollary} \label{coll:kern}
The equivalent prior at $t = 1$ for the transformed parameter is $ \bm{\omega} \sim N(\bm{0}, \bm{K}_{(1)})$ where $\bm{K}_{(1)} = f(\bm{X}_{(1)}) f(\bm{X}_{(1)})^T $. Furthermore, the posterior at time $\tau$ is of Gaussian form \\ $ \emph{P}(\bm{\omega} | \{\mathcal{D}_{(t)}\}_{t=1}^\tau) = N(\bm{\mu}_{(\tau)}, \bm{K}_{(\tau)} ) $ where the mean parameter satisfies the recursions
$\bm{\mu}_{(\tau)} = \bm{\mu}_{(\tau-1)} + \bm{K}_{(\tau)} \bm{Y}_{(\tau)} \hat{\bm{\alpha}}_{(\tau)} $.
\end{corollary}
Since $\P(\bm{\theta}| \{ \mathcal{D}_{(t)} \}_{t=1}^{\tau}) $ is Gaussian, the MAP estimator is simply the mean parameter $\bm{\mu}_{(\tau)}$ given in the Corollary \ref{coll:kern}. Thus the decision rule reduces to $ \hat{y}_{i'} = \text{sgn}(f(\bm{x}_{i'}) \hat{\bm{\theta}} + \hat{b}) $ where the MAP estimator $ \hat{\bm{\theta}} $ is a function of the previously estimated Lagrange multipliers $\hat{\bm{\alpha}}_{(1)}, \dots, \hat{\bm{\alpha}}_{(\tau-1)}$ and the maximizing values $\hat{\bm{\alpha}}_{(\tau)}$ and $\hat{b}$ for the current time point $\tau$.
\begin{corollary} \label{coll:Opt_Lagrange}
Given all previous $\hat{\bm{\alpha}}_{(1)}, \dots, \hat{\bm{\alpha}}_{(\tau-1)}$, the current optimal Lagrange multipliers $\hat{\bm{\alpha}}_{(\tau)} $ are the solution to
\begin{flalign*}
& \underset{\bm{\alpha}_{(\tau)}}{\max} \, -\frac{1}{2} \bm{\alpha}_{(\tau)}^T \bm{Y}_{(\tau)} \bm{K}_{(\tau)} \bm{Y}_{(\tau)} \bm{\alpha}_{(\tau)} + \sum_{i=1}^{n_{(\tau)}} \log ( 1-\alpha_{(\tau) i}/ C_{(\tau)}) \\
& \hspace{34pt} + \bm{\alpha}_{(\tau)}^T \left(\bm{1} - \bm{Y}_{(\tau)} \sum_{t=1}^{\tau-1} k(\bm{X}_{(\tau)} , \bm{X}_{(t)}) \bm{Y}_{(t)} \hat{\bm{\alpha}}_{(t)} \right) \\
& \text{subject to }\bm{y}_{(\tau)}^T \bm{\alpha}_{(\tau)} = 0 \text{ and } \alpha_{(\tau) i} \, \ge 0 \text{ for all } i = 1, \dots, n_{(\tau)}
\end{flalign*}
and, holding the Lagrange multipliers fixed, the optimal bias $\hat{b} = $
$$
\underset{b}{\arg\min} \hspace{-6pt} \sum_{s \in \{i | \hat{\alpha}_{(\tau) i} \neq 0 \} } \left\vert \left(y_{(\tau) s} - \sum_{t=1}^{\tau} k(\bm{X}_{(\tau) s} , \bm{X}_{(t)} ) \bm{Y}_{(t)} \hat{\bm{\alpha}}_{(t)} \right) - b \right\vert
$$
ensures that the expectation constraints in the objective hold.
\end{corollary}
The above dual formulation for the Lagrange multipliers $ \bm{\alpha}_{(\tau)}$ has some interesting implications. Since the Lagrange multipliers from the previous time points are fixed at time step $\tau$, the factor $ {\bm{1} - \bm{Y}_{(\tau)} \sum_{t=1}^{\tau-1} k(\bm{X}_{(\tau)} , \bm{X}_{(t)}) \bm{Y}_{(t)} \hat{\bm{\alpha}}_{(t)} }$ are constants and can be thought of as (unnormalized) weights for $ \bm{\alpha}_{(\tau)}$, the Lagrange multipliers from the current time point. Thus the corresponding Lagrange multipliers for samples that are easily predicted using only the prior information will have lower weight than the Lagrange multipliers for samples that are difficult or incorrect.
\section{Manifold Regularization}
Next we consider the case wheres some of the labels are missing. Without loss of generality we will assume the first $l$ points are labeled and the latter $n-l$ points are unlabeled.
We will adopt the semi-supervised MED classification framework of \cite{hou}, called Laplacian MED (LapMED). LapMED introduces an additional ``geometric" constraint
\begin{flalign} \label{laplace_const}
\hspace{-5pt} \iint \P(\bm{\theta}, \lambda) \left( \int_{x \in \mathcal{M}} \hspace{-2pt} \bm{\theta}^T \hspace{-2pt} f(\bm{x}) \Delta_{\mathcal{M}} f(\bm{x}) \bm{\theta} \, d\mathcal{P}_x - \lambda \right) d\bm{\theta} d\lambda \leq 0
\end{flalign}
to \eqref{MED_obj} where $ \mathcal{M} = \text{supp}(\mathcal{P}_X ) \subset \mathbb{R}^n $ is a compact submanifold, $ \Delta_{\mathcal{M}} $ is the Laplace-Beltrami operator on $\mathcal{M}$, and $ \lambda $ controls the complexity of the decision boundary in the intrinsic geometry of $ \mathcal{P}_X $. This constraint was motivated by the semi-supervised framework of \cite{Belkin:2006:MRG:1248547.1248632} to encourage the function $f(x)$ to be smooth over the support set of the feature distribution $\mathcal{P}_X$, inducing a geometric interpolation of unlabeled points. Since the marginal distribution is unknown, from \cite{grigor2006heat}
$$
f(\bm{X})^T \bm{L} f(\bm{X}) \rightarrow \int_{x \in \mathcal{M}} f(\bm{x}) \Delta_{\mathcal{M}} f(\bm{x}) \, d\mathcal{P}_x, \, \text{ as } n\rightarrow \infty
$$
where $ \bm{L} $ is the normalized graph Laplacian formed with a heat kernel. The LapMED posterior can be approximated as
\begin{flalign*}
& \P(\bm{\theta}, b, \bm{\gamma}, \lambda | \mathcal{D}) = \frac{\P_0(\bm{\theta}, b, \bm{\gamma}, \lambda)}{Z(\bm{\alpha}, \beta)} \exp \Bigg\{ \\
& \sum_{i=1}^l \alpha_i \left( y_i ( f(\bm{X}) \bm{\theta} + b) - \gamma_i \right) + \beta \left(\lambda - \bm{\theta}^T f(\bm{X})^T \bm{L} f(\bm{X}) \bm{\theta} \right) \Bigg\}
\end{flalign*}
where $ \beta \ge 0 $ is a Lagrange multiplier for the smoothness constraint.
\subsection{Sequential Laplacian MED}
The distribution $ \P(\bm{\Theta}, \bm{\gamma}, \lambda| \mathcal{D}) $ that minimizes the objective with the additional constraint \eqref{laplace_const} can similarly be factorized and, like the distribution of slack parameters considered in Section 2, the distribution of the smoothness parameter $\lambda$ is also independent of the data $\mathcal{D}$. Likewise, the distribution of the decision rule coefficients $ \P(\bm{\Theta} | \mathcal{D} )$ are conjugate distributions with their priors. Thus the updating procedure for the LapMED problem is similar to the updating procedure in Section \ref{sec:SeqMED}.
\begin{theorem} \label{thm:SeqLapMED}
At $t = 0$, the MED priors for $\bm{\theta}$ (or $\bm{\omega}$), $b$, and $\gamma_i$ are the same as in Theorem 1, and the prior for $\lambda$ is a Bayesian zero mean point prior, denoted $Exp.(\infty)$.
Then given data $\mathcal{D}_{(\tau)}$ at time point $\tau$, the MED conjugate prior and posterior are still $Exp.(\infty)$ for $\lambda$, the same as in Theorem 1 for $b$ and $\gamma_i$, and Gaussian of form $N \left(\bm{\mu}_{(\tau)}, \bm{\Sigma}_{(\tau)} \right)$ for $\bm{\theta}$ (or $\bm{\omega}$). Define a $ l \times n $ expansion matrix as $\bm{J} = [ \bm{\mathrm{I}} \,\, \bm{0} ] $. Then the mean and covariance parameters for the distribution of $\bm{\theta}$ are
$$
\bm{\mu}_{(\tau)} = \bm{G}_{(\tau)}^{-1} \sum_{t=1}^{\tau} f(\bm{X}_{(t)})^T \bm{J}^T \bm{Y}_{(t)} \hat{\bm{\alpha}}_{(t)}, \quad \bm{\Sigma}_{(\tau)} = \bm{G}_{(\tau)}^{-1},
$$
where $ \bm{G}_{(\tau)} = \bm{G}_{(\tau-1)} + 2\beta_{(\tau)} f(\bm{X}_{(\tau)})^T \bm{L}_{(\tau)} f(\bm{X}_{(\tau)}) $ is a recursive graph of vertex disjoint subgraphs, and for the distribution of $\bm{\omega}$ are
$$
\bm{\mu}_{(\tau)} = \hspace{-2pt} \sum_{t=1}^{\tau} k_{(\tau)} \hspace{-2pt} \left( \bm{X}_{(\tau)}, \bm{X}_{(t)} \right) \hspace{-1pt} \bm{J}^T \bm{Y}_{(t)} \hat{\bm{\alpha}}_{(t)}, \, \bm{\Sigma}_{(\tau)} = k_{(\tau)} \hspace{-2pt} \left( \bm{X}_{(\tau)}, \bm{X}_{(\tau)} \right)
$$
where $k_{(\tau)}( \bm{x}, \bm{x}') = \langle f(\bm{x}), \bm{G}_{(\tau)}^{-1} f(\bm{x}') \rangle$ is a kernel function that can be recursively defined as
\begin{flalign} \label{kern_func}
& k_{(\tau)}( \bm{x}, \bm{x}') = k_{(\tau-1)}(\bm{x}, \bm{x}') - k_{(\tau-1)}(\bm{x}, \bm{X}_{(\tau)}) \bigg( \hspace{-2pt} \left(2 \beta_{(\tau)}\bm{L}_{(\tau)}\right)^{-1} \notag \\
& \hspace{42pt} + k_{(\tau-1)} \left(\bm{X}_{(\tau)}, \bm{X}_{(\tau)} \right) \hspace{-2pt} \bigg)^{-1} \hspace{-2pt} k_{(\tau-1)}( \bm{X}_{(\tau)}, \bm{x}') .
\end{flalign}
\end{theorem}
Theorem 2 gives the posterior distribution for semi-supervised classification whose form is comparable to the form given in Corollary \ref{coll:kern} for the supervised case. Indeed the forms are identical except for the presence of the precision matrix term $G_{(\tau)}$ in the semi-supervised case. As the sparsity of $G_{(\tau)}$ is associated with the graph Laplacian, the kernel function of the semi-supervised case is a regularized version of the kernel function that appears in Corallary \ref{coll:kern}. If we let $ \beta_{(t)} $ be a fixed parameter, then $\hat{\bm{\alpha}}_{(t)}$ and $\hat{b}$ optimize an objective of the same form as in Corollary \ref{coll:Opt_Lagrange}, but with kernel function $k_{(\tau)}( \bm{x}, \bm{x}')$. If $ \beta_{(t)} $ is chosen to be 0, the sequential LapMED simply ignores the unlabeled data of time point $t$, and if all $\beta_{(i)}$'s are $0$, then the unlabeled data is always ignored and the updating procedure is exactly the same as in the supervised scenario. These parameters are functions of the $\gamma_A$ and $\gamma_I$, which are identical to the penalty parameters in the Laplacian SVM \cite{Belkin:2006:MRG:1248547.1248632}, associated with the reproducing kernel Hilbert space and data distribution respectively: $ C_{(t)} = \frac{1}{2 l_{(t)} \gamma_A } $ and $ \beta_{(t)} = \frac{\gamma_I }{2 \gamma_A n_{(t)}^2} $.
\subsection{Approximating the Kernel Function} \label{approx_k_func}
Because the kernel function in \eqref{kern_func} is a function of the previous kernel functions, calculating a map to its associated Hilbert space $ \mathcal{H}_{(\tau)} $ can be computationally expensive. Thus in this subsection, we derive an approximation to the map to $ \langle \bm{x}, \bm{x}' \rangle_{\mathcal{H}_{(\tau)}} $, which is computationally easier than direct recursive calculation.
Recall that we approximate the constraint in \eqref{laplace_const}, at any time point $t$, empirically with the graph Laplacian $\bm{L}_{(t)}$ formed using the data from that time point $\bm{X}_{(t)}$. However, the non-empirical constraint using the Laplace-Beltrami operator over the unknown marginal distribution $\mathcal{P}_x$, is actually the same at every time point. Thus as $n_{(\tau-1)} \rightarrow \infty$, the prior graph $\bm{G}_{(\tau-1)}$ converges to
\begin{flalign} \label{decomp_Lap}
B \int_{x \in \mathcal{M}} f(\bm{x}) \Delta_{\mathcal{M}} f(\bm{x}) \, d\mathcal{P}_x \approx B \sum_{i=1}^{\infty} \delta_i \xi_i^2 \upsilon_i(z) \upsilon_i(z)
\end{flalign}
where $B = 2 \sum_{t=1}^{\tau} \beta_{(t)} $, $\delta_i $ are the eigenvalues of the Laplace-Beltrami operator, and $\upsilon_i(z)$ and $\xi_i$ are the infinite sequence of right singular functions and singular values of $ f(x) = \int k(x, z) f(z) \, dz $. The approximate decomposition arises since the left singular functions of $f$ are the eigenfunctions of the Laplace-Beltrami operator \cite{Lederman} and \cite{Belkin:2006:MRG:1248547.1248632}. Thus instead of empirically approximating the Laplacian as a sum of subgraphs \\ $\bm{G}_{(\tau-1)} = \bm{\mathrm{I}} + \sum_{t=1}^{\tau-1} 2\beta_{(t)} f(\bm{X}_{(t)})^T \bm{L}_{(t)} f(\bm{X}_{(t)}) $, we can instead implement approximations to the eigen/singular values and singular functions in \eqref{decomp_Lap}.
Assuming that the sample size $n$ is large enough, the average eigenvalues of the $\tau-1$ graph Laplacians would be a good estimator for the eigenvalues of the Laplace-Beltrami operator. Additionally the rows of the matix $\bm{V}^T$ from the singular value decomposition of $\bm{X}$ will contain the basis for its row space. Thus because the right singular functions form an orthonormal basis for the coimage of $f$, if the mapping approximately preserves the basis, the mapped average singular vectors $f(\bar{\bm{V}}_i)$ would be good estimators for the right singular functions $\upsilon_i(z)$ and correspondingly so for the singular values.
The posterior kernel function $k_{(\tau)}( \bm{x}, \bm{x}')$ using an approximation to the decomposition in \eqref{decomp_Lap} will no longer be a recursive function of prior kernel functions $k_{(\tau-1)}( \bm{x}, \bm{x}')$ that have the same form, like in \eqref{kern_func}. Instead for $\tau > 2$, it uses a prior kernel function
\begin{flalign*}
& \tilde{k}_{(\tau-1)}(\bm{x}, \bm{x}') = k(\bm{x}, \bm{x}') - k(\bm{x}, \bar{\bm{V}}_{(\tau-1)}) \bigg( \hspace{-2pt} \frac{\text{ diag}(\bar{\bm{s}}_{(\tau-1)}^{\,2} \bar{\bm{d}}_{(\tau-1)})^{-1}}{B} \\
& \hspace{52pt} + k (\bar{\bm{V}}_{(\tau-1)}, \bar{\bm{V}}_{(\tau-1)}) \bigg)^{-1} \hspace{-2pt} k( \bar{\bm{V}}_{(\tau-1)}, \bm{x}') .
\end{flalign*}
where $ k(\bm{x}, \bm{x}') = \langle f(\bm{x}), f(\bm{x}')\rangle $ is the non-regularized kernel function. So at time $\tau$, the singular vectors of $\bm{X}_{(\tau-1)}$ are used to update the average singular vectors, in the above function, through
$$ \bar{\bm{V}}_{(\tau-1)} = \bar{\bm{V}}_{(\tau-2)} + \frac{\bm{V}_{(\tau-1)} - \bar{\bm{V}}_{(\tau-2)}}{\tau-1} $$ and similarly so for the average corresponding singular values $ \bar{\bm{s}}_{(\tau-1)}$ and the average eigenvalues of the graph Laplacians $ \bar{\bm{d}}_{(\tau-1)}$.
\section{Experiments}
In this section, we compare the proposed sequential maximum margin classifiers to popular supervised and semi-supervised maximum margin classifiers (SVM \cite{smola1998connection} and LapSVM \cite{Belkin:2006:MRG:1248547.1248632}) where the model is trained using just the current time points data and where the model has been re-trained on all previous data. The former type of model is a lower bound on performance since it ignores all previous data and the latter type of model is an upper bound since it is re-trained on all previous data at every time point. Note the MED and SVM models only differ by a weak log-barrier term in the objective function making their performance identical, and similarly so for LapMED and LapSVM. Thus their performance curves will referred to as Full SVM/MED and Full LapSVM/LapMED.
\subsection{Simulations}
In both of the following simulations, the models receive roughly 100 samples ($n_{(t)} = [97, 103] $) at every time point, the parameters are empirically chosen with a validation set, and then the models are tested on an independent data set of 1000 test points. The test accuracy $\frac{TP + TN}{1000}$ is the average accuracy over 100 trials of simulation.
In the first simulation, we generate data from 200 categorical distributions where 100 of the variables are sparse so they have high probability of being 0, another 50 of the variables have lower probability of being 0, and the final 50 variables are used to distinguish between the two classes. We use the term frequency - inverse document frequency (TF-IDF) kernel of \cite{elkan2005deriving}, which is used in document processing and topic models. Figure \ref{fig:super} shows that the accuracy of the sequential model (SeqMED) improves as the model is updated with more training data and has much better results even after one model update versus the independent model (SVM) that ignores previous training data. Of course the sequential model does not improve as rapidly as the model that is re-trained on all the data (Full SVM/MED), but this is the price paid for lower computational complexity. For example, at $t = 30$, SeqMED updates and fits 100 coefficients for the new data whereas Full SVM/MED fits 3,000 coefficients for all the data.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.5cm]{words_Acc.png}}
\end{minipage}
\vspace{-20pt}
\caption{Accuracy of prediction for categorical fully labeled simulated data. The proposed sequential MED (SeqMED) classifier performs almost as well as the full batch implementation of the SVM/MED (Full SVM/MED). }
\label{fig:super}
\end{figure}
In the second simulation, we generate data from the interior of a 3-dimensional sphere where one class is roughly at the center of the sphere and the other class is on the shell, but only 10\% of the samples are labeled. We use a rbf kernel with width 1 for the kernel function and a heat kernel with width 0.01 and a 20 nearest neighbors graph for the graph Laplacian. Figure \ref{fig:semisuper} shows improvement in performance of the sequential model similar to in Figure \ref{fig:super}. We use the approximate kernel function of Subsection \ref{approx_k_func} to perform each update, establishing that the approximation is adequate.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.5cm]{semi_sphere_Acc.png}}
\end{minipage}
\vspace{-20pt}
\caption{Accuracy of prediction for continuous simulated data with 10\% labeled.}
\label{fig:semisuper}
\end{figure}
\subsection{Data}
We compare the proposed algorithms on the Isolet speech database from the UCI machine learning repository \cite{Lichman:2013} following the experimental framework used in \cite{Belkin:2006:MRG:1248547.1248632}. To train the models, we take the entire training set of 120 speakers (isolet1 - isolet4) and break them into 24 groups (time points) of 5 speakers where only the first speaker is labeled. At each time point, the models train on 260 samples ($t=21$ and $23$ only have 259) where 52 of the samples are labeled. The parameters are set in the same way as in \cite{Belkin:2006:MRG:1248547.1248632} and the test set is similarly composed of the 1,559 samples from isolet5. Figure \ref{fig:data} shows that, after two time points, the sequential model always performs better than the model that ignores previous data, and comes close to performing as well as the fully re-trained model as time progresses.
\begin{figure}[htb]
\begin{minipage}[b]{1.0\linewidth}
\centering
\centerline{\includegraphics[width=8.5cm]{isolet_Acc.png}}
\end{minipage}
\vspace{-20pt}
\caption{Accuracy of prediction on isolet5 for models trained on partially labeled speech isolets 1-4. The proposed semi-supervised sequential Laplacian MED classifier (SeqLapMED) comes close to the full Laplacian SVM \cite{Belkin:2006:MRG:1248547.1248632} as time progresses. }
\label{fig:data}
\end{figure}
\section{Conclusions}
We have proposed recursive versions of supervised and semi-supervised maximum margin classifiers in the minimum entropy discrimination (MED) classification framework. The proposed sequential maximum margin classifiers perform nearly as well as a much more computationally expensive fully re-trained maximum margin classifiers and significantly better than a classifier that ignores previous data.
|
{
"timestamp": "2018-03-08T02:04:09",
"yymm": "1803",
"arxiv_id": "1803.02517",
"language": "en",
"url": "https://arxiv.org/abs/1803.02517"
}
|
\subsection{Multiplicity and Jacobi fields} \label{subsec:multiplicity.jacobi.fields}
In this section we prove that uniform bounds on the Morse index generically prevent multiplicity from occurring in the Allen-Cahn setting. Specifically:
\begin{theo} \label{theo:bounded.index}
Suppose $(M^3, g)$ is a compact Riemannian 3-manifold possibly with $\partial M \neq \emptyset$, and that $u_i \in C^\infty(M; [-1,1])$, $\varepsilon_i > 0$, where each $u_i$ is a critical point of $E_{\varepsilon_i}$, and
\begin{equation} \label{eq:bounded.index.i}
E_{\varepsilon_i}[u_i] \leq E_0, \; \ind(u_i) \leq I_0 \text{ for all } i = 1, 2, \ldots
\end{equation}
Suppose $\lim_i \varepsilon_i = 0$. Passing to a subsequence, write $V \triangleq \lim_i h_0^{-1} V_{\varepsilon_i}[u_i]$ for the limit $2$-varifold. Then $V$ is a stationary integral varifold, $\support \Vert V \Vert$ is smooth in the interior of $M$, and if $\Sigma$ denotes a connected component of $\support \Vert V \Vert$ that is a compact submanifold without boundary, then one of the following is true:
\begin{enumerate}
\item $\Sigma$ is two-sided and $\Theta^2(V, \cdot) = 1$ on $\Sigma$ (i.e., $\Sigma$ has multiplicity $1$);
\item $\Sigma$ is two-sided, $\Theta^2(V, \cdot) \geq 2$ on $\Sigma$ (i.e., multiple interfaces have converged), it is stable (see \eqref{eq:stable.min.surf}) and carries a smooth positive Jacobi field; or
\item $\Sigma$ is one-sided, and the two-sided double cover of $\Sigma$ is stable and carries a smooth positive Jacobi field.
\end{enumerate}
\end{theo}
\begin{proof}
For $p \in M$, $i = 1, 2, \ldots$, define the index concentration scale by
\begin{equation} \label{eq:bounded.index.instability.radius}
\mathcal{R}(p, i) \triangleq \inf \{ r > 0 : \ind(u_i; B_r(p)) \geq 1 \},
\end{equation}
and then further let
\[ \mathring{\Sigma} \triangleq \{ p \in M : \liminf_{i \to \infty} \mathcal{R}(p, i) > 0 \}. \]
By passing to an appropriate subsequence at the beginning of the proof, an elementary covering argument allows us to assume that $\mathcal{H}^0(\Sigma \setminus \mathring{\Sigma}) \leq I_0$.
The curvature estimates from Theorem \ref{theo:curvature.estimate} combined with the varifold convergence of $V_{\varepsilon_{i}}[u_{i}]$ (from\footnote{See Remark \ref{rema:HT.simplified}.} \cite[Theorem 1]{HutchinsonTonegawa00}, and \cite[Appendix B]{Guaraco}) show that along $\mathring \Sigma$, the limit varifold is supported with integer multiplicity (possibly greater than one) on a $C^{1,1}$ (and thus smooth) minimal surface. At this point, we may argue that $\Sigma$ extends smoothly across the index concentration set $\Sigma \setminus \mathring \Sigma$ exactly as in \cite[Proposition 3.10]{Guaraco}. We emphasize here that by using to our curvature estimates, we give an proof of the regularity of $\Sigma$ that does not rely on the deep works of Wickramasekera \cite{Wickramasekera14} and Tonegawa--Wickramasekera \cite{TonegawaWickramasekera12} (cf.\ \cite{Guaraco,Hiesmayr}).
We now assume that $\Sigma$ is connected (in general, one can apply the following argument to each component of the support of the limit varifold $V$).
First, suppose $\Sigma$ is two-sided and denote
\begin{equation*}
U \triangleq \text{ tubular neighborhood of } \Sigma \text{ such that } (\Sigma \cup \partial M) \cap U = \emptyset.
\end{equation*}
We may suppose that $U = Z_\Sigma(\Sigma \times (-1, 1))$.
By the Constancy Theorem \cite[Theorem 41.1]{Simon83}, $\Theta^{2}(V, \cdot)$ is constant on $\Sigma$. If $\Theta^{2}(V, \cdot) = 1$ somewhere on $\Sigma$, then $\Sigma$ occurs entirely with multiplicity one as claimed.
In what follows we may assume, then, that $\Theta^{2}(V, \cdot) \equiv m \in \{2, 3, \ldots\}$ on $\Sigma$.
Let us assume, for the time being, that $I_0 = 0$, i.e., that the critical points $u_i$ are all stable. The general case will be dealt with later.
It follows from \eqref{eq:bounded.index.i}, Corollary \ref{coro:curvature.estimates}, and the two-sidedness of $\Sigma$, that the level sets $\{ u_i = 0 \} \cap U$ converge graphically in $C^{2,\theta}$ to $\Sigma$. In the case that $\{u_{i}=0\} \cap U$ were minimal surfaces, it is standard to produce a positive Jacobi field on $\Sigma$ out of this setup. We recall the argument here, with the necessary modifications for our lower regularity situation.
Since $\Sigma$ is two-sided, the level sets $\{ u_i = 0 \} \cap U$ (which are \emph{smoothly} embedded) can be ordered by their signed distance to $\Sigma$ in a fashion that is consistent across $\Sigma$. Without loss of generality, we may assume that there are $Q = 2$ level sets\footnote{Otherwise, we apply the same argument verbatim to the \emph{top} and \emph{bottom} level sets, ignoring all intermediate ones.}. To stay consistent with Section \ref{sec:jacobi.toda.reduction}, let's label the level sets
\[ \Gamma_{i,1}, \Gamma_{i,2} \subset \{ u_i = 0 \} \cap U. \]
Denote their corresponding height functions (over $\Sigma$) as $f_{i,1}$, $f_{i,2} : \Sigma \to \mathbf{R}$, $\ell \in \{1,2\}$, so that $f_{i,1} < f_{i,2}$ on $\Sigma$. We recall \eqref{eq:mean.curv.graphical} from Appendix \ref{app:mean.curvature.graphs}, which tells us that:
\begin{multline} \label{eq:bounded.index.h}
H_{\Gamma_{i,\ell}} = - \divg_{g_{f_{i,\ell}}} \left( \frac{\nabla_{g_{f_{i,\ell}}} f_{i,\ell}}{(1 + g^{pq}_{f_{i,\ell}} (f_{i,\ell})_p (f_{i,\ell})_q)^{1/2}} \right)
- \frac{\sff_{f_{i,\ell}}^{pq} (f_{i,\ell})_p (f_{i,\ell})_q}{(1 + g^{pq}_{f_{i,\ell}} (f_{i,\ell})_p (f_{i,\ell})_q)^{1/2}} \\
+ (1 + g^{pq}_{f_{i,\ell}} (f_{i,\ell})_p (f_{i,\ell})_q)^{1/2} H_{f_{i,\ell}}
\end{multline}
for $\ell = 1$, $2$. Here we're using notation from the appendix, where, e.g., $g = g_z + dz^2$ on $U$.
We now claim that $H_{\Gamma_{i,2}} - H_{\Gamma_{i,1}}$ satisfies a \emph{linear} uniformly elliptic equation in $f_{i,2} - f_{i,1}$, whose parameters (obviously) depend on $f_{i,1}$, $f_{i,2}$. Indeed, \eqref{eq:bounded.index.h} tells us that
\begin{equation} \label{eq:bounded.index.h.abstract.pde}
H_{\Gamma_{i,\ell}} = - A(f_{i,\ell}) \divg_{\Sigma} \left( \mathscr{B}(f_{i,\ell}, \nabla_{\Sigma} f_{i,\ell}) \nabla_{\Sigma} f_{i,\ell} \right) + C(f_{i,\ell}, \nabla_\Sigma f_{i,\ell})
\end{equation}
for \emph{smooth} functions (for each $p \in \Sigma$)
\begin{align*} A & = A_p : \mathbf{R} \to \mathbf{R}, \\
\mathscr{B} & = \mathscr{B}_p : \mathbf{R} \times T_p \Sigma \to \operatorname{End}(T_p \Sigma),\\
C & = C_p : \mathbf{R} \times T_p \Sigma \to \mathbf{R},
\end{align*}
which, additionally, satisfy: $A > 0$, $\mathscr{B}$ is positive definite. More specifically, at each point $p \in \Sigma$:
\begin{align*}
A(z) & \triangleq \frac{\sqrt{g_0}}{\sqrt{g_z}}, \; z \in \mathbf{R},\\
\mathscr{B}(z, \mathbf{v}) \mathbf{w} & \triangleq \frac{\sqrt{g_z}}{\sqrt{g_0}} \frac{g_z^{ij} g^0_{jk} \mathbf{w}^j \partial_{y_k}}{(1 + g_z^{pq} g^0_{pk} g^0_{q\ell} \mathbf{v}^k \mathbf{v}^\ell)^{1/2}}, \; z \in \mathbf{R}, \; \mathbf{v}, \mathbf{w} \in T_p \Sigma,\\
C(z, \mathbf{v}) & \triangleq - \frac{\sff^{pq}_z g^0_{ip} g^0_{jq} \mathbf{v}^i \mathbf{v}^j}{(1 + g_z^{pq} g^0_{pk} g^0_{q\ell} \mathbf{v}^k \mathbf{v}^\ell)^{1/2}} + (1 + g_z^{pq} g^0_{pk} g^0_{q\ell} \mathbf{v}^k \mathbf{v}^\ell)^{1/2} H_z, \; z \in \mathbf{R}, \; \mathbf{v} \in T_p \Sigma.
\end{align*}
From the fundamental theorem of calculus, as well as the fact that the two divergences (for the two cases $\ell = 1$, $2$) are \emph{pointwise} bounded (because the two mean curvatures are bounded), it follows that
\begin{equation} \label{eq:bounded.index.h.diff.abstract.pde}
H_{\Gamma_{i,2}} - H_{\Gamma_{i,1}} = - A \divg_\Sigma( \widehat{\mathscr{B}} \nabla_\Sigma f_i + f_i \widehat{\mathbf{C}})
+ \langle \mathbf{\widehat{D}}, \nabla_\Sigma f_i \rangle_{\Sigma} + \widehat{E}f_i \text{ on } \Sigma,
\end{equation}
where $f_i \triangleq f_{i,2} - f_{i,1} > 0$ on $\Sigma$, with coefficients
\begin{align*} \widehat{\mathscr{B}} & = \widehat{\mathscr{B}}_p : \mathbf{R}^2 \times (T_p \Sigma)^2 \to \operatorname{End}(T_p \Sigma), \\
\widehat{\mathbf{C}} & = \widehat{\mathbf{C}}_p, \widehat{\mathbf{D}} = \widehat{\mathbf{D}}_p : \mathbf{R}^2 \times (T_p \Sigma)^2 \to T_p \Sigma, \\
\widehat{E} & = \widehat{E}_p : \mathbf{R}^2 \times (T_p \Sigma)^2 \to \mathbf{R},
\end{align*}
whose arguments are $(f_{i,1}, f_{i,2}, \nabla_\Sigma f_{i,1}, \nabla_\Sigma f_{i,2}) \in \mathbf{R}^2 \times (T_p \Sigma)^2$. These coefficients are \emph{uniformly bounded} and satisfy
\[ A \geq \mu, \; \langle B\mathbf{v}, \mathbf{v}\rangle_\Sigma \geq \mu \Vert \mathbf{v} \Vert_\Sigma^2, \; \mathbf{v} \in T_p \Sigma, \]
for a fixed $\mu > 0$, provided
\[ \limsup_{i \to \infty} \Vert f_{i,1} \Vert_{C^1(\Sigma)} + \Vert f_{i,2} \Vert_{C^1(\Sigma)} < \infty. \]
It will be convenient to carry out the exact computation, as that will allow us to study a particular rescaled limit as $i \to \infty$. It will also be convenient to denote
\[ \zeta_i^{(t)} \triangleq f_{i,1} + t (f_{i,2} - f_{i,1}) \equiv f_{i,2} + tf_i, \; t \in [0,1]. \]
Note that
\[ \zeta_i^{(0)} \equiv f_{i,1}, \; \zeta_i^{(1)} \equiv f_{i,2}, \text{ and } \tfrac{\partial}{\partial t} \zeta_i^{(t)} \equiv f_i \text{ on } \Sigma. \]
Let us define $\widehat{\mathscr{B}}$, $\widehat{\mathbf{C}}$, $\widehat{\mathbf{D}}$, $\widehat{E}$. The easiest term to deal with in \eqref{eq:bounded.index.h.abstract.pde} is the low order term, $C$. Indeed
\begin{align*}
& C(f_{i,2}, \nabla_\Sigma f_{i,2}) - C(f_{i,1}, \nabla_\Sigma f_{i,1}) = \underbrace{\left[ \int_0^1 D_z C(\zeta_i^{(t)}, \nabla_\Sigma \zeta_i^{(t)}) \, dt \right]}_{\widehat{E} \text{, term 1 out of 2}} f_i + \left\langle \underbrace{\int_0^1 D_{\mathbf{v}} C(\zeta_i^{(t)}, \nabla_\Sigma \zeta_i^{(t)}) \, dt}_{\widehat{\mathbf{D}}}, \nabla_\Sigma f_i \right\rangle.
\end{align*}
We study the higher order term in two steps. First:
\begin{align*}
& \mathscr{B}(f_{i,2}, \nabla_\Sigma f_{i,2}) \nabla_\Sigma f_{i,2} - \mathscr{B}(f_{i,1}, \nabla_\Sigma f_{i,1}) \nabla_\Sigma f_{i,1} \\
& = \underbrace{\left( \left[ \int_0^1 D_z \mathscr{B}(\zeta_i^{(t)}, \nabla_\Sigma \zeta_i^{(t)}) \nabla_\Sigma \zeta_i^{(t)} \, dt \right] \right)}_{\widehat{\mathbf{C}}} f_i + \left\langle \underbrace{\left[ \int_0^1 D_{\mathbf{v}} \mathscr{B}(\zeta_i^{(t)}, \nabla_\Sigma \zeta_i^{(t)}) \nabla_\Sigma \zeta_i^{(t)} \, dt \right]}_{\widehat{\mathscr{B}} \text{, term 1 out of 2}}, \nabla_\Sigma f_i \right\rangle \\
& \qquad + \underbrace{\left[ \int_0^1 \mathscr{B}(\zeta_i^{(t)}, \nabla_\Sigma \zeta_i^{(t)}) \, dt \right]}_{\widehat{\mathscr{B}} \text{, term 2 out of 2}} \nabla_\Sigma f_i.
\end{align*}
Second,
\begin{align*}
& (A(f_{i,2}) - A(f_{i,1})) \divg_\Sigma (\mathscr{B}(f_{i,2}, \nabla_\Sigma f_{i,2}) \nabla_\Sigma f_{i,2}) \\
& \qquad = \underbrace{\left( \left[ \int_0^1 D_z A(\zeta_i^{(t)}) \, dt \right] \divg_\Sigma (\mathscr{B}(f_{i,1}, \nabla_\Sigma f_{i,1}) \nabla_\Sigma f_{i,1}) \right)}_{\widehat{E} \text{, term 2 out of 2}} f_i.
\end{align*}
We now return to the qualitative study of $f_i$. Applying the Harnack inequality in divergence form to \eqref{eq:bounded.index.h.diff.abstract.pde} (after multiplying through by $A^{-1}$), we get
\begin{equation} \label{eq:bounded.index.harnack}
\sup_{\Sigma} f_i \leq c \inf_{\Sigma} f_i \text{ for } i = 1, 2, \ldots
\end{equation}
with a constant $c > 0$ that doesn't depend on $i$. From Proposition \ref{prop:ultimate.stable.estimates} and Corollary \ref{coro:ultimate.stable.estimates}, we know that
\begin{align}
\lim_{i \to \infty} \frac{\Vert H_{\Gamma_{i,\ell}} \Vert_{C^0(\Gamma_{i,\ell})}}{\varepsilon_i |\log \varepsilon_i|} & = 0 \text{ for } \ell = 1, 2, \label{eq:bounded.index.estimate.meancurv} \\
\liminf_{i \to \infty} \frac{\inf_{\Sigma} f_i}{\varepsilon_i |\log \varepsilon_i|} & > 0. \label{eq:bounded.index.estimate.height}
\end{align}
Define the normalizations
\begin{equation} \label{eq:bounded.index.normalized.function}
\widehat{f}_i \triangleq (\sup_\Sigma f_i)^{-1} f_i : \Sigma \to [\tfrac{1}{c}, 1],
\end{equation}
where $c$ is as in \eqref{eq:bounded.index.harnack}. In view of \eqref{eq:bounded.index.h.diff.abstract.pde}, ${\widehat{f}_i}$ satisfies the linear, uniformly elliptic equation (note that we've multiplied through by $A^{-1}$, which is uniformly bounded):
\begin{equation} \label{eq:bounded.index.h.diff.norm.abstract.pde}
\frac{H_{\Gamma_{i,2}} - H_{\Gamma_{i,1}}}{A \cdot \sup_\Sigma f_i} = - \divg_\Sigma( \widehat{\mathscr{B}} \nabla_\Sigma \widehat{f}_i + \widehat{f}_i \widehat{\mathbf{C}})
+ \langle A^{-1} \mathbf{\widehat{D}}, \nabla_\Sigma \widehat{f}_i \rangle_{\Sigma} + A^{-1} \widehat{E}\widehat{f}_i \text{ on } \Sigma.
\end{equation}
We will \emph{test} this PDE by multiplying through by some $\zeta \in C^\infty(\Sigma)$ and integrating by parts. By testing, first, with $\zeta = \widehat{f}_i$, we get uniform energy estimates
\[ \limsup_{i \to \infty} \int_\Sigma |\nabla_\Sigma \widehat{f}_i|^2 < \infty. \]
Moreover, since $\widehat{f}_i$ is (trivially) bounded, it follows from Rellich's theorem that there exist $\widehat{f} \in W^{1,2}(\Sigma)$ and a subsequence such that
\[ \widehat{f}_i \rightharpoonup \widehat{f} \text{ in } W^{1,2}(\Sigma), \; \widehat{f}_i \to \widehat{f} \text{ in } L^2(\Sigma). \]
Therefore, since the coefficients in \eqref{eq:bounded.index.h.diff.norm.abstract.pde} are all uniformly bounded as $i \to \infty$, it follows that we can test \eqref{eq:bounded.index.h.diff.norm.abstract.pde} with arbitrary $\zeta \in C^\infty(\Sigma)$, and pass to a subsequential limit $i \to \infty$.
The left hand side of \eqref{eq:bounded.index.h.diff.norm.abstract.pde} converges to zero uniformly as $i \to \infty$ because of \eqref{eq:bounded.index.estimate.meancurv}-\eqref{eq:bounded.index.estimate.height} above. Thus, $\widehat{f}$ is a $W^{1,2}$-weak solution of
\begin{equation} \label{eq:bounded.index.h.diff.norm.limit.pde}
-\divg_\Sigma (\widehat{\mathscr{B}}_\infty \nabla_\Sigma \widehat{f} + \widehat{f} \widehat{\mathbf{C}}_\infty) + \langle A^{-1}_\infty \widehat{\mathbf{D}}_\infty, \nabla_\Sigma \widehat{f} \rangle + A^{-1}_\infty \widehat{E}_\infty \widehat{f} = 0 \text{ on } \Sigma,
\end{equation}
where $A_\infty$, $\widehat{\mathscr{B}}_\infty$, $\widehat{\mathbf{C}}_\infty$, $\widehat{\mathbf{D}}_\infty$, $\widehat{E}_\infty$ are just the same coefficients, except now they are evaluated at the limiting configuration of $(0, 0, \mathbf{0}, \mathbf{0})$. It is not hard to see, using the evolutions in Appendix \ref{app:mean.curvature.graphs}, that
\begin{equation*}
A_\infty \equiv 1, \; \widehat{\mathscr{B}}_\infty \equiv \operatorname{Id}, \; \widehat{\mathbf{C}}_\infty \equiv \widehat{\mathbf{D}}_\infty \equiv \mathbf{0},
\text{ and } \widehat{E}_\infty \equiv - (|\sff_\Sigma|^2 + \ricc_g(\nu_\Sigma, \nu_\Sigma)).
\end{equation*}
Thus, $\widehat{f}$ is $W^{1,2}$-weak solution of the Jacobi equation,
\begin{equation} \label{eq:bounded.index.h.jacobi.equation}
(\Delta_\Sigma + |\sff_\Sigma|^2 + \ricc_g(\nu_\Sigma, \nu_\Sigma)|_\Sigma) h = 0 \text{ on } \Sigma.
\end{equation}
It must be smooth---and thus classically a solution---by elliptic regularity. Moreover
\[ \tfrac{1}{c} \leq \widehat{f}_i \leq 1 \text{ for all } i = 1, 2, \ldots \implies \tfrac{1}{c} \leq \widehat{f} \leq 1. \]
In particular, the function is positive. It follows that the principal eigenvalue of the Jacobi operator is zero, so $\Sigma$ is stable.{\footnote{The fact that $\ind(u_i) = 0$ for all $i = 1, 2, \ldots$ implies the stability of $\Sigma$ is not new: see \cite{Tonegawa05, TonegawaWickramasekera12, Hiesmayr, Gaspar}. Nonetheless, by appropriately generalizing the argument given here, we are going to be able to extend the conclusion that $\Sigma$ is stable in the case where $\ind(u_i) \leq I_0$ for $i = 1, 2, \ldots$, $I_0 \in \{0, 1, \ldots \}$ and convergence occurs with multiplicity $\geq 2$.}} The result follows.
We now drop the stability assumption and proceed to the general case of $I_0 \in \{0, 1, 2, \ldots\}$. We continue to assume that $\Sigma$ is two-sided. Without loss of generality, we'll assume $I_0 = 1$ from this point on. The general case is similar.
The index concentration set is either empty (in which case, we can argue as in the previous case) or satisfies $\mathring{\Sigma} = \Sigma \setminus \{ P_\star \}$ for some $P_\star \in \Sigma$, and the convergence of $\{ u_i = 0 \} \cap U$ to $\mathring{\Sigma}$ is graphical $C^{2,\theta}_{\loc}$ on $\Sigma \setminus \{ P_\star \}$. Notice that, by definition, for every $r > 0$ there exists a subsequence along which
\begin{equation} \label{eq:bounded.index.concentration.puncture}
\ind(u_i; M \setminus B_{r/2}(P_\star)) = 0.
\end{equation}
Our previous discussion regarding the stable case applies verbatim to $M \setminus C_{r}(P_\star)$, where, in exponential normal coordinates,
\[ C_{\rho}(P_\star) \triangleq B^2_{\rho}(P_\star) \times (-1,1), \]
and yields functions $f_{i,1}$, $f_{i,2} : \Sigma \setminus B_{r}^2(P_\star) \to \mathbf{R}$ representing the \emph{incomplete} properly embedded surfaces\emph{-with-boundary}
\begin{equation} \label{eq:bounded.index.graphing.punctured.sheets}
\Gamma_{i,1}, \Gamma_{i,2} \subset \{ u_i = 0 \} \cap U \setminus C_{r}(P_\star).
\end{equation}
\begin{rema} \label{rema:bounded.index.notation.i}
Recall that we assumed $U$ is the image of the normal exponential map of $\Sigma$ restricted to $\Sigma \times (-1, 1)$. Then, $\partial C_\rho(P_\star) \cap U = \partial B_\rho^2(P_\star) \times (-1, 1)$ for every sufficiently small $\rho > 0$.
\end{rema}
All of \eqref{eq:bounded.index.h}-\eqref{eq:bounded.index.estimate.height} continue to hold over $\Sigma \setminus B_{r}^2(P_\star)$ instead of $\Sigma$.
All the constants inevitably depend on our choice of $r > 0$, which is yet to be determined. We note that, trivially, the energy estimate
\[ \limsup_{i \to \infty} \int_{\Sigma \setminus B_{2r}^2(P_\star)} |\nabla_\Sigma \widehat{f}_i|^2 < \infty \]
holds true for any fixed $r > 0$ by our previous discussion. In fact, because $\Gamma_{i,1} \setminus C_{r}(P_\star)$, $\Gamma_{i,2} \setminus C_{r}(P_\star)$ converge in $C^{2,\theta}$ to $\Sigma \setminus B^2_{r}(P_\star)$ as $i \to \infty$, a subset of the fixed surface $\Sigma$, the coefficients of \eqref{eq:bounded.index.h.diff.abstract.pde} will satisfy
\begin{multline*}
\limsup_{r \to 0} \Big[ \limsup_{i \to \infty} \Vert A \Vert_{C^0(\Sigma \setminus B_{3r/2}^2(P_\star))} + \Vert \widehat{\mathscr{B}} \Vert_{C^0(\Sigma \setminus B_{3r/2}^2(P_\star))} \\
+ \Vert \widehat{\mathbf{C}} \Vert_{C^0(\Sigma \setminus B_{3r/2}^2(P_\star))} + \Vert \widehat{\mathbf{D}} \Vert_{C^0(\Sigma \setminus B_{3r/2}^2(P_\star))}
+ \Vert \widehat{E} \Vert_{C^0(\Sigma \setminus B_{3r/2}^2(P_\star))} \Big] < \infty,
\end{multline*}
and, therefore, we'll have the \emph{uniform} energy estimate
\[ \limsup_{r \to 0} \left[ \limsup_{i \to \infty} \int_{\Sigma \setminus B_{2r}^2(P_\star)} |\widehat{f}_i|^2 \right] < \infty. \]
This means we can pass to a limiting $\widehat{f}$ in the following sense:
\begin{equation} \label{eq:bounded.index.renormalized.h}
\widehat{f}_i \rightharpoonup \widehat{f} \text{ in } W^{1,2}_{\loc}(\mathring{\Sigma}), \; \widehat{f}_i \to \widehat{f} \text{ in } L^2_{\loc}(\mathring{\Sigma}).
\end{equation}
Now, \eqref{eq:bounded.index.harnack}-\eqref{eq:bounded.index.estimate.height} also hold for each fixed $r > 0$, with the $\sup$ and $\inf$ taken over $\Sigma \setminus B^2_r(P_\star)$, the $C^0$ norm of $H_{\Gamma_{i,\ell}}$ taken over $\Gamma_{i,\ell} \setminus C_r(P_\star)$; the constant $c$ and rates of convergence of the limits, a priori, depend on $r$. Nonetheless, $\widehat{f} \in W^{1,2}_{\loc}(\mathring{\Sigma})$ is a weak solution of \eqref{eq:bounded.index.h.jacobi.equation} on $\mathring{\Sigma}$. By elliptic regularity, $\widehat{f}$ is smooth and solves \eqref{eq:bounded.index.h.jacobi.equation} classically on $\mathring{\Sigma}$.
\begin{prop} \label{prop:bounded.index.nontrivial.limit}
$\widehat{f} \in L^\infty(\mathring{\Sigma})$, $\widehat{f} \not \equiv 0$ a.e. on $\mathring{\Sigma}$.
\end{prop}
We defer the proof of Proposition \ref{prop:bounded.index.nontrivial.limit} to the next section, since the argument is of independent interest.
This proposition, once verified, completes the proof of Theorem \ref{theo:bounded.index}: by standard removable singularity results for elliptic PDE, $\widehat{f}$ must extend to a smooth nonnegative solution of \eqref{eq:bounded.index.h.jacobi.equation} on $\Sigma$, which is not identically zero, and the result follows as it did in the stable setting.
Finally, we explain the necessary modifications when $\Sigma$ is one-sided. Assume, as above, that $I_{0}=1$ (the general case is similar). As before, we can define $\mathring \Sigma$ to be the complement of the index concentration set. Considering a tubular neighborhood $U$, of $\Sigma$, we can use the normal exponential map to lift $\Sigma$ and $u : U\to \mathbf{R}$ to $\check \Sigma \subset \check U$, where $\check\Sigma$ is the orientable double cover of $\Sigma$ and $\check U$ is the associated lift of $U$. We can assume that $\check U$ is diffeomorphic (via the normal exponential map) to $\check \Sigma \times (-1,1)$. Let $\check{\mathring{\Sigma}}$ be the lift of $\mathring{\Sigma}$. Observe that $\check{\Sigma} \setminus \check{\mathring{\Sigma}}$ contains at most two points (more generally $2I_{0}$ points).
Note that the covering map $\pi : \check U \to U$ admits an deck transformation $\tau: \check U \to \check U$ with $\tau^{2}$ equal to the identity. Define $\check u \triangleq u \circ \pi$, which is still a critical point of $E_{\varepsilon_{i}}$. Clearly $\check u \circ \tau = \check u$.
We claim that the convergence of $\check u$ to $\check \Sigma$ occurs with even multiplicity. If not, (up to switching the normal vector) we can assume that $\check u \to -1$ on $\check\Sigma \times (-1,0)$ and $\check u \to 1$ on $\check\Sigma \times (0,1)$ (this is clear on $\check{\mathring{\Sigma}}$, which then implies that it holds for all $p\in\check\Sigma$). Note, however, that $\tau(\{p\} \times (0,1)) = \{\tau(p) \} \times (-1,0)$ (otherwise, we would find that $\Sigma$ was two-sided). This contradicts the fact that $\check u$ is invariant under $\tau$.
Thus, the convergence of $\check u$ occurs with even multiplicity (and thus multiplicity at least two). Now, the argument above can be applied verbatim to $\check \Sigma$ and $\check u$ to produce a smooth positive Jacobi field on $\check \Sigma$ (we emphasize that it is not clear what the index of $\check u$ is; here, we rely on the index bounds of $u$ to bound the cardinality of $\check{\Sigma} \setminus \check{\mathring{\Sigma}}$; after this step, the definition of $\check{\mathring \Sigma}$, rather than the index bounds is all that is used). As above, this implies that $\check\Sigma$ is stable.
\end{proof}
\subsection{Sliding heteroclinic barriers} \label{subsec:bounded.index.height.bounds}
For the reader's convenience we start by recalling the following important result of White on local foliations by minimal surfaces.
\begin{prop}[{\cite[Appendix]{White:curvature}}] \label{prop:white.foliation}
Let $\Phi$ be an even elliptic integrand, where $\Phi$ and $D_2 \Phi$ are $C^{2,\theta}$. Let $\Phi_r$ be the integrand defined by $\Phi_r(x, v) = \Phi(rx, v)$. There is an $\eta > 0$ such that if $r < \eta$ and if
\[ w : B_1 \subset \mathbf{R}^2 \to \mathbf{R}, \; \Vert w \Vert_{C^{2,\theta}} < \eta, \]
then for each $t \in [-1,1]$, there is a $C^{2,\theta}$ function $v_t : B_1 \to \mathbf{R}$ whose graph is $\Phi_r$-stationary and such that
\[ v_t(x) = w(x) + t \; \text{ if } x \in \partial B_1. \]
Furthermore, $v_t$ depends in a $C^1$ way on $t$ so that the graphs of the $v_t$ foliate a region of $\mathbf{R}^3$. If $M$ is a $C^1$ properly immersed $\Phi_r$-stationary surface in $B_{1/2}(0)$ with $\partial M \subset \graph v_t$, then $M \subset \graph v_t$.
\end{prop}
We will use the minimal disks constructed by this proposition to construct barriers (using Theorem \ref{theo:dirichlet.data.construction}) that will allow us to control the height of the top and bottom $\{ u_i = 0 \}$ sheets near $P_\star$. This can be thought of as a variant of the moving planes method adapted to the Riemannian Allen--Cahn setting.
\begin{proof}[Proof of Proposition \ref{prop:bounded.index.nontrivial.limit}]
We continue with the same notation as in the previous section. Let $\rho > 2r$, with still being such that \eqref{eq:bounded.index.concentration.puncture}-\eqref{eq:bounded.index.graphing.punctured.sheets} apply.
Let $w_i : B^2_{2\rho}(P_\star) \to \mathbf{R}$ be a harmonic function (defined on $B^2_{2\rho}(P_\star) \subset \Sigma$) with boundary data
\[ w_i = f_{i,2} \text{ on } \partial B^2_{\rho}(P_\star). \]
Recalling, from Corollary \ref{coro:curvature.estimates}, that $f_{i,2} \to 0$ in $C^{2,\theta}(B^2_{2\rho}(P_\star) \setminus B^2_{\rho/2}(P_\star))$, it follows that (by potentially going farther down the sequence of $i = 1, 2, \ldots$) $\Vert w_i \Vert_{C^{2,\theta}}$ --suitably scaled-- is small enough for Proposition \ref{prop:white.foliation} to apply. Once we're in that regime, Proposition \ref{prop:white.foliation} guarantees a foliation
\[ t \mapsto D_{i,\rho}(t), \; t \in [-\delta, \delta], \]
consisting of minimal disks that all project to $B^2_{\rho}(P_\star) \subset \Sigma$. Without loss of generality, we may suppose that the foliated region $\cup_{|t| < \delta} D_{i,\rho}(t)$ lies entirely within $U$. Note that:
\begin{enumerate}
\item the curves $t \mapsto \partial D_{i,\rho}(t)$ move at unit vertical speed in $\partial C_{\rho}(P_\star)$;
\item the second fundamental form of the disks $D_{i,\rho}(t)$ is bounded in $C^{0,\theta}$ uniformly over $i = 1, 2, \ldots$, $t \in [-\delta, \delta]$,
\begin{equation} \label{eq:bounded.index.disk.c2alpha}
|\sff_{D_{i,\rho}(t)}| + [\sff_{D_{i,\rho}(t)}]_{\theta} \leq \eta,
\end{equation}
and $\eta > 0$ can be made arbitrarily small.
\end{enumerate}
As a consequence of \eqref{eq:bounded.index.disk.c2alpha}, \eqref{eq:sheets.enhanced.sff.grad.bound}, and minimal surface curvature estimates, the disks also satisfy the following weaker $C^{3,\theta}$ bound uniformly over $i = 1, 2, \ldots$, $t \in [-\delta, \delta]$,
\begin{equation} \label{eq:bounded.index.disk.c3alpha}
\varepsilon |\nabla_{D_{i,\rho}(t)} \sff_{D_{i,\rho}(t)}| + \varepsilon^{1+\theta} [\nabla_{D_{i,\rho}(t)} \sff_{D_{i,\rho}(t)}]_{\theta} \leq \eta,
\end{equation}
after possibly relaxing $\eta > 0$, which can still nevertheless be made arbitrarily small.
We'll now use a sliding/moving planes argument that relies on the barrier construction in Section \ref{sec:dirichlet.data}, adopting relevant notation from therein. We assume, without loss of generality, that
\begin{equation} \label{eq:bounded.index.sign.u}
u > 0 \text{ above } \Gamma_{i,2} \text{ in } U \setminus C_{\rho/2}(P_\star).
\end{equation}
Our constructions below will take place for $\varepsilon = \varepsilon_i$, $i = 1, 2, \ldots$, but we suppress the dependence on $i$ for the sake of notational brevity.
Define $\hat{\chi} : \mathbf{R} \to [0,1]$ to be a cutoff function such that
\begin{equation} \label{eq:bounded.index.chi.hat}
\hat{\chi}(s) = \begin{cases} 1 & |s| \leq B \varepsilon |\log \varepsilon| \\ 0 & |s| \geq 2B \varepsilon |\log \varepsilon|, \end{cases}
\end{equation}
where $B \gg 1$ is to be chosen later. This can be constructed so that
\[ |\hat{\chi}^{(k)}| = O((\varepsilon |\log \varepsilon|)^{-k}) \text{ for } k \geq 1, \; \varepsilon \to 0. \]
Let's very briefly run through some notation which is introduced later, in Section \ref{sec:dirichlet.data}; we will need to use some of it here in invoking that section's main theorem. In Section \ref{sec:dirichlet.data} we consider $\delta_* \in (0, 1)$ fixed and a H\"older exponent $\alpha \in (0, 1)$, $\alpha \leq \theta$, where $\theta$ is as in \eqref{eq:bounded.index.disk.c2alpha}, \eqref{eq:bounded.index.disk.c3alpha}. (We will eventually choose $\alpha$ near $0$ and $\theta$ near $1$.) In \eqref{eq:dirichlet.data.cutoff}, cutoff functions $\chi_j$ are introduced that are supported on strips of width $O(\varepsilon^{\delta_*})$ (while $\hat{\chi}$ is supported on a thinner strip of size $O(\varepsilon |\log \varepsilon|)$). Finally, in \eqref{eq:dirichlet.data.approximate.heteroclinic}, $\chi_1$ is used to define a suitably truncated approximate heteroclinic solution $\widetilde{\mathbb{H}}_\varepsilon$ that is constant outside a strip of size $O(\varepsilon^{\delta_*})$. (See Remark \ref{rema:dirichlet.data.trivialization.window}.)
Given this notation, let's set:
\[ \hat{v}^\sharp(s) \triangleq \gamma \hat{\chi}(s) \mathbb{H}'(\varepsilon^{-1} s) + (1-\hat{\chi}(s)) \begin{cases} 1 - \varepsilon^3 - \widetilde{\mathbb{H}}_\varepsilon(s) & s > 0 \\ - 1 - \widetilde{\mathbb{H}}_\varepsilon(s) & s < 0, \end{cases} \]
where $\gamma \in \mathbf{R}$ is chosen so that the orthogonality constraint
\begin{equation} \label{eq:bounded.index.v.sharp.orthogonality}
\int_{-\infty}^\infty \hat{v}^\sharp(s) \mathbb{H}'(\varepsilon^{-1} s) \, ds = 0
\end{equation}
holds. Recalling \eqref{eq:heteroclinic.expansion.i}-\eqref{eq:heteroclinic.expansion.ii}, and that $\delta_* \in (0, 1)$, \eqref{eq:bounded.index.v.sharp.orthogonality} is equivalent to
\begin{align}
\gamma(h_0 - o(1)) & = O(\varepsilon^{-1}) \int_{B \varepsilon |\log \varepsilon|}^\infty (1-\mathbb{H}(\varepsilon^{-1} s)) \mathbb{H}'(\varepsilon^{-1} s) \, ds + O(\varepsilon^2) \int_{2B \varepsilon |\log \varepsilon|}^{\tfrac{47}{50} \varepsilon^{\delta_*}} |\mathbb{H}'(\varepsilon^{-1} s)| \, ds \nonumber \\
& = O(1) \int_{B |\log \varepsilon|}^\infty (1-\mathbb{H}(s)) \mathbb{H}'(s) \, ds + O(\varepsilon^3) \int_{2B |\log \varepsilon|}^{\tfrac{47}{50} \varepsilon^{\delta_*-1}} |\mathbb{H}'(s)| \, ds \nonumber \\
& = O(1) \exp(-2\sqrt{2} B |\log \varepsilon|) + O(\varepsilon^3) \exp(-2\sqrt{2} B |\log \varepsilon|) = O(\varepsilon^{2\sqrt{2} B}). \label{eq:bounded.index.v.sharp.estimate.i}
\end{align}
Also,
\begin{equation} \label{eq:bounded.index.v.sharp.estimate.ii}
\Vert \hat{\chi}(s) \mathbb{H}'(\varepsilon^{-1} s) \Vert_{C^{2,\alpha}_\varepsilon(\mathbf{R})} = O(1) \text{ as } \varepsilon \to 0.
\end{equation}
Taking $B$ sufficiently large, \eqref{eq:bounded.index.v.sharp.estimate.i}-\eqref{eq:bounded.index.v.sharp.estimate.ii} together imply
\begin{equation} \label{eq:bounded.index.v.sharp.estimate}
\Vert \hat{v}^\sharp(s) \Vert_{C^{2,\alpha}_\varepsilon(\mathbf{R})} = O(\varepsilon^3).
\end{equation}
Next, for $(y, s) \in \partial (B_\rho^2(P_\star) \times [-\tfrac12, \tfrac12])$, define
\[ \hat{v}^\flat(y,s) \triangleq (1-\chi_4(s)) \begin{cases} 1 - \varepsilon^3 - \widetilde{\mathbb{H}}_\varepsilon(s) & s > 0 \\ -1 - \widetilde{\mathbb{H}}_\varepsilon(s) & s < 0. \end{cases} \]
Recall that $\chi_{4}$ is defined in \eqref{eq:dirichlet.data.cutoff}. It is easy to see that $\Vert \hat{v}^\flat \Vert_{C^{2,\alpha}_\varepsilon} = O(\varepsilon^3)$. In fact, $\chi_5 \hat{v}^\flat = 0$, so
\begin{equation} \label{eq:bounded.index.v.flat.estimate}
\Vert \hat{v}^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon} = O(\varepsilon^3)
\end{equation}
as well (see \eqref{eq:dirichlet.data.ckalpha.eps.modified} for the definition of $\widetilde{C}^{2,\alpha}_\varepsilon$).
We emphasize that everything from \eqref{eq:bounded.index.chi.hat} to \eqref{eq:bounded.index.v.flat.estimate} above is \emph{agnostic} of our particular solutions with bounded Morse index. They will serve as prescribed boundary data for solutions of the Allen-Cahn equation on the fixed product manifold $B_{\rho}^2 \times [-\tfrac12, \tfrac12]$, albeit with varying interior metric that will depend on $g$, $i = 1, 2, \ldots$, $\rho$, and $t \in [-\delta, \delta]$. Indeed, we let
\begin{multline} \label{eq:bounded.index.pullback.metric}
g_{i,\rho}(t) \triangleq \text{ pullback metric from } Z_{D_{i,\rho}(t)}(D_{i,\rho}(t) \times [-\tfrac12, \tfrac12]) \subset U \\
\text{ to } B_{\rho}^2 \times [-\tfrac12, \tfrac12] \text{ under Fermi coordinates } (y,s) \text{ with respect to } D_{i,\rho}(t).
\end{multline}
We may apply Theorem \ref{theo:dirichlet.data.construction} with $\hat{v}^\sharp$, $\hat{v}^\flat$ as above, $\hat{\zeta} \equiv 0$, and with the H\"older exponents $\alpha$ near $0$ and $\theta$ near $1$ per the theorem, to $\Omega \triangleq B_{\rho}^2 \times [-\tfrac12, \tfrac12]$ and the nonconstant Riemannian metric $g_{i,\rho}(t)$. Note that the conditions of the theorem are met trivially for $\hat{\zeta}$, and are also met for $\hat{v}^\sharp$, $\hat{v}^\flat$ by \eqref{eq:bounded.index.v.sharp.estimate}-\eqref{eq:bounded.index.v.flat.estimate}.
The theorem yields $\mathfrak{b}_{i,\rho,t} : \Omega \to \mathbf{R}$ such that
\begin{equation} \label{eq:bounded.index.barrier.pde}
\varepsilon_i^2 \Delta_{g_{i,\rho}(t)} \mathfrak{b}_{i,\rho,t} = W'(\mathfrak{b}_{i,\rho,t})
\end{equation}
and, for all $(y, s) \in \partial \Omega$,
\begin{equation} \label{eq:bounded.index.barrier.boundary.data}
\mathfrak{b}_{i,\rho,t}(y,s) = \widetilde{\mathbb{H}}_\varepsilon(s) + \chi_4(s) \hat{v}^\sharp(s) + \hat{v}^\flat(y,s).
\end{equation}
We constructed $\hat{v}^\sharp$, $\hat{v}^\flat$ specifically so that:
\begin{equation} \label{eq:bounded.index.barrier.boundary.data.explicit}
\mathfrak{b}_{i,\rho,t}(y,s) = \begin{cases} 1-\varepsilon_i^3 & (y,s) \in \partial \Omega, \; s \geq 2B \varepsilon_i |\log \varepsilon_i| \\ -1 & (y, s) \in \partial \Omega, \; s \leq -2B \varepsilon_i |\log \varepsilon_i|. \end{cases}
\end{equation}
\begin{clai}
For every $\beta > 0$, $\varepsilon_i \leq 1$, we have
\begin{equation} \label{eq:bounded.index.barrier.strip}
|\mathfrak{b}_{i,\rho,t}(y,s)| \leq 1-\beta \implies |s| \leq c' \varepsilon_i,
\end{equation}
where $c' = c'(W, \beta, \eta, c_0) > 0$.
\end{clai}
\begin{proof}[Proof of Claim]
This is a straightforward consequence of the ansatz $\mathfrak{b}_{i,\rho,t} = (\widetilde{\mathbb{H}}_\epsilon + \chi_4 v^\sharp + v^\flat) \circ D_\zeta$, $\Vert v^\sharp \Vert_{C^0}$, $\Vert v^\flat \Vert_{C^0} = o(1)$, $\Vert \zeta \Vert = O(\varepsilon_i^{2-2\alpha})$, and \eqref{eq:heteroclinic.expansion.i}, at least provided we take $\alpha$ small enough.
\end{proof}
\begin{clai}
For sufficiently large $i$,
\begin{equation} \label{eq:bounded.index.barrier.start}
\mathfrak{b}_{i,\rho,\delta} < (Z_{D_{i,\rho}(\delta)})^* u_i \text{ on } \Omega.
\end{equation}
Recall that $\delta > 0$ represents the ``top'' of the foliation $D_{i,\rho}(\delta)$.
\end{clai}
\begin{proof}[Proof of Claim]
Let's agree, for the remainder of the proof of this claim, to write $u_i$ instead of $(Z_{D_{i,\rho}(\delta)})^* u_i$. We seek to show that $\mathcal{G} \triangleq \{ x \in \Omega : \mathfrak{b}_{i,\rho,\delta}(x) < u_i(x) \}$ coincides with $\Omega$. (Recall: we're assuming \eqref{eq:bounded.index.sign.u}.)
Fix $\beta \in (0,1)$ so that $W'' \geq 2\kappa^2 > 0$ on $[-1,-1+\beta] \cup [1-\beta, 1]$ for some $\kappa > 0$.
From \eqref{eq:bounded.index.barrier.strip}, $\{ |\mathfrak{b}_{i,\rho,\delta}| \leq 1-\beta \}$ is contained in an $O(\varepsilon_i)$-neighborhood of $D_{i,\rho}(\delta)$. From \cite[Theorem 1]{HutchinsonTonegawa00}, $\{ |u_i| \leq 1-\beta \}$ converges, in the Hausdorff sense, to $\Sigma$. In particular, for sufficiently large $i$,
\[ (\Omega \cap \{ |u_i| \leq 1-\beta \}) \cup \{ |\mathfrak{b}_{i,\rho,\delta}| \leq 1-\beta \} \subset \mathcal{G}. \]
Note that
\begin{equation*}
\varepsilon_i^2 \Delta_g (1-u_i) = - W'(u_i) = \tfrac{W'(1)-W'(u_i)}{1-u_i} (1-u_i)
\geq 2\kappa^2 (1-u_i) \text{ on } \{ u_i > 1-\beta \},
\end{equation*}
\begin{equation*}
\varepsilon_i^2 \Delta_g (1+u_i) = W'(u_i) = \tfrac{W'(u_i)-W'(-1)}{u_i-(-1)} (1+u_i)
\geq 2\kappa^2 (1+u_i) \text{ on } \{ u_i < -1+\beta \},
\end{equation*}
so by an application of the barrier principle together with the saddle property of $W$ at zero (see \cite[Lemma 4.1]{KowalczykLiuPacard12}) we get:
\begin{equation} \label{eq:bounded.index.exp.decay}
|u_i^2 - 1| = O\big( \exp(- \kappa \varepsilon_i^{-1} \dist_g(\cdot, \{ u_i = 0 \})) \big).
\end{equation}
Combined with \eqref{eq:bounded.index.barrier.boundary.data.explicit}, this shows $\partial \Omega \subset \mathcal{G}$ for sufficiently large $i$. Thus:
\begin{equation} \label{eq:bounded.index.barrier.strip.i}
\Omega \setminus \mathcal{G} \subset \Omega \setminus (\partial \Omega \cup \{ |u_i| \leq 1-\beta \} \cup \{ |\mathfrak{b}_{i,\rho,\delta}| \leq 1-\beta \}).
\end{equation}
Subtracting from \eqref{eq:bounded.index.barrier.pde} the PDE satisfied by $u_i$, we see that
\[ \varepsilon_i^2 \Delta_g (\mathfrak{b}_{i,\rho,\tau} - u_i) = c(x) (\mathfrak{b}_{i,\rho,\tau} - u_i) \]
for $c(x) \triangleq (W'(\mathfrak{b}_{i,\rho,t}(x)) - W'(u_i(x)))/(\mathfrak{b}_{i,\rho,\tau}(x) - u_i(x))$. This is \emph{negative} on $\Omega \setminus \mathcal{G}$ by \eqref{eq:bounded.index.barrier.strip.i}, and this violates the maximum principle unless $\mathcal{G} = \Omega$. The claim follows.
\end{proof}
Next, since:
\begin{enumerate}
\item $\mathfrak{b}_{i,\rho,t}$ and $(Z_{D_{i,\rho}(t)})^* u_i$ both vary continuously in $t \in [-\delta, \delta]$ by Theorem \ref{theo:dirichlet.data.construction},
\item \eqref{eq:bounded.index.barrier.start} holds true, and
\item $\mathfrak{b}_{i,\rho,-\delta} \not \leq (Z_{D_{i,\rho}(-\delta)})^* u_i$,
\end{enumerate}
there will exist exactly one $\tau_i \in (-\delta,\delta)$, and at least one $Q_i^\star \in \Omega$, such that
\begin{equation} \label{eq:bounded.index.tau.definition}
\mathfrak{b}_{i,\rho,t} < (Z_{D_{i,\rho}(t)})^* u_i \text{ on } \Omega \text{ for all } t \in (\tau_i, \delta],
\text{ and } \mathfrak{b}_{i,\rho,\tau_i}(Q_i^\star) = [(Z_{D_{i,\rho}(\tau_i)})^* u](Q_i^\star).
\end{equation}
Our goal is to estimate $\tau_i$. Abusing notation again, we'll write $u_i$ instead of $(Z_{D_{i,\rho}(\tau_i)})^* u_i$, and $g$ instead of $g_{i,\rho}(\tau_i)$. Thus:
\begin{equation} \label{eq:bounded.index.max.principle.conditions}
u_i - \mathfrak{b}_{i,\rho,\tau_i} \geq 0 \text{ on } \Omega, \; (u_i - \mathfrak{b}_{i,\rho,\tau_i})(Q_i^\star) = 0.
\end{equation}
Subtracting \eqref{eq:bounded.index.barrier.pde} from the PDE satisfied by $u$, we see that
\[ \varepsilon_i^2 \Delta_g (u_i - \mathfrak{b}_{i,\rho,\tau_i}) = c(x) (u_i - \mathfrak{b}_{i,\rho,\tau_i}) \]
for $c(x) \triangleq (W'(u_i(x)) - W'(\mathfrak{b}_{i,\rho,\tau_i}(x)))/(u_i(x) - \mathfrak{b}_{i,\rho,\tau_i}(x))$. The maximum principle, then, tells us that
\begin{enumerate}
\item either $Q_i^\star \in \partial \Omega$, or
\item $u_i \equiv \mathfrak{b}_{i,\rho,\tau_i}$ on $\Omega$.
\end{enumerate}
We only consider the first case here, since the second reduces to it by replacing $Q_i^\star$ with another point on $\partial \Omega$. Note that \eqref{eq:bounded.index.tau.definition}, the fact that $\mathfrak{b}_{i,\rho,0}|_{\partial D_{i,\rho}(0)} \equiv 0$, and the uniqueness of $\tau_i$ give a lower bound on $\tau_i$:
\begin{equation} \label{eq:bounded.index.tau.lower.bound}
\tau_i \geq 0.
\end{equation}
The upper bound is more subtle. We claim that
\begin{equation} \label{eq:bounded.index.tau.upper.bound}
\tau_i < 7B \varepsilon_i |\log \varepsilon_i|,
\end{equation}
provided $B$ is chosen (independently of $i$) such that
\begin{equation} \label{eq:bounded.index.B.requirement}
\dist_g(x, \{ u_i = 0 \}) > 3B \varepsilon_i |\log \varepsilon_i| \implies |u_i(x)| > 1-\varepsilon_i^3.
\end{equation}
The existence of a $B$ that satisfies \eqref{eq:bounded.index.B.requirement} is guaranteed by \eqref{eq:bounded.index.exp.decay}.
It will be convenient to introduce the notation (here, $\lambda \geq 0$ is some parameter):
\begin{align*}
\overline{\partial \Omega}[\lambda] & \triangleq \{ (y, s) \in \partial \Omega : s \in [\lambda, \tfrac12] \}, \\
\underline{\partial \Omega}[\lambda] & \triangleq \{ (y,s) \in \partial \Omega : s \in [-\tfrac12, -\lambda] \}.
\end{align*}
To start, let's estimate the height of $Q_i^\star$ from below. We have
\[ u_i > -1 \text{ on } (M^n, g) \implies \mathfrak{b}_{i,\rho,\tau_i}(Q_i^\star) = u_i(Q_i^\star) > -1, \]
so, from \eqref{eq:bounded.index.barrier.boundary.data.explicit}:
\[ Q_i^\star \in \partial \Omega \setminus \underline{\partial \Omega}[2B\varepsilon_i |\log \varepsilon_i|]. \]
Equivalently, the image $\widetilde{Q}_i^\star$ of $Q_i^\star$ to $(M^n, g)$ under $Z_{D_{i,\rho}(\tau_i)}$ satisfies
\[ \widetilde{Q}_i^\star \in Z_{D_{i,\rho}(\tau_i)}(\partial \Omega \setminus \underline{\partial \Omega}[2B \varepsilon_i |\log \varepsilon|]). \]
In particular, $\widetilde{Q}_i^\star$ belongs to the open tubular neighborhood of the image $Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[0]) \subset (M^n, g)$ with radius $3B \varepsilon_i |\log \varepsilon_i|$:
\begin{equation} \label{eq:bounded.index.height.Q.est.i}
\widetilde{Q}_i^\star \in B_{3 B \varepsilon_i |\log \varepsilon_i|}\big( Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[0]) \big).
\end{equation}
We now prove \eqref{eq:bounded.index.tau.upper.bound} by contradiction. We'll show that
\begin{equation} \label{eq:bounded.index.height.Q.est.ii}
\dist_g(Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[0]), \{u_i=0\}) > 6B \varepsilon_i |\log \varepsilon_i|
\end{equation}
when \eqref{eq:bounded.index.tau.upper.bound} fails, i.e., when $\tau_i \geq 7B\varepsilon_i |\log \varepsilon_i|$.
Since $D_{i,\rho}(\tau_i)$ is an $o(1)$-Lipschitz graph over $\Sigma$ (note that the argument used to prove \eqref{eq:bounded.index.barrier.start} shows that $\tau_{i}\to 0$), and $Z_{D_{i,\rho}(\tau_i)}(\partial \Omega) \perp D_{i,\rho}(\tau_i)$, there will exist $\eta > 0$ (independent of $i$) such that
and
\[ Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[0] \setminus \overline{\partial \Omega}[\eta]) \subset C_{3\rho/2}(P_\star) \setminus C_{\rho/2}(P_\star) \]
for sufficiently large $i$. Moreover $\lim_{i \to \infty} \{ u_i = 0 \} = \Sigma$ in the Hausdorff topology (\cite[Theorem 1]{HutchinsonTonegawa00}, \cite[Appendix B]{Guaraco}), so
\[ \liminf_{i \to \infty} \dist_g(Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[\eta]), \{ u_i = 0 \}) > 0 \]
because $\tau_i \geq 0$ by \eqref{eq:bounded.index.tau.lower.bound}. Thus, \eqref{eq:bounded.index.height.Q.est.ii} will follow from
\begin{equation*}
\dist_g(Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[0] \setminus \overline{\partial \Omega}[\eta]), \{ u_i = 0 \} \cap C_{2\rho}(P_\star) \setminus C_{\rho/2}(P_\star)) > 6B \varepsilon_i |\log \varepsilon_i|.
\end{equation*}
when $\tau_i \geq 7B \varepsilon_i |\log \varepsilon_i|$. Since the components of $\{ u_i = 0 \} \cap C_{2\rho}(P_\star) \setminus C_{\rho/2}(P_\star)$ are well-ordered $o(1)$-Lipschitz graphs over $\Sigma$, with $\Gamma_{i,2}$ being the topmost, we may equivalently show
\begin{equation*}
\dist_g(Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[0] \setminus \overline{\partial \Omega}[\eta]), \Gamma_{i,2}) > 6B \varepsilon_i |\log \varepsilon_i|.
\end{equation*}
Because $D_{i,\rho}(t)$, $t \in [0,\tau_i]$, are all $o(1)$-Lipschitz graphs over $\Sigma$ as well, we have
\begin{equation*}
\nabla_g (\dist^\pm_g(\cdot; \Gamma_{i,2})), \nabla_g (\dist^\pm_g(\cdot; D_{i,\rho}(t)) \rangle \geq 1-o(1), \; t \in [0,\tau_i]
\end{equation*}
in a small (but definite) neighborhood of $\Sigma$ in $C_{2\rho}(P_\star) \setminus C_{\rho/2}(P_\star)$. Here $\dist_g^\pm$ denotes the signed distance. From it follows that for every $P \in Z_{D_{i,\rho}(\tau_i)}(\overline{\partial \Omega}[0] \setminus \overline{\partial \Omega}[\eta])$,
\begin{multline*}
\dist^\pm_g(P; \Gamma_{i,2}) \geq (1-o(1)) \dist_g^\pm(P; D_{i,\rho}(0)) \geq (1-o(1))(\tau_i + \dist_g^\pm(P; D_{i,\rho}(\tau_i))) \\
\geq (1-o(1)) \tau_i > (1-o(1)) 7B \varepsilon_i |\log \varepsilon_i| > 6B \varepsilon_i |\log \varepsilon_i|,
\end{multline*}
as claimed, and \eqref{eq:bounded.index.height.Q.est.ii} follows. It is now an automatic consequence of \eqref{eq:bounded.index.height.Q.est.i}-\eqref{eq:bounded.index.height.Q.est.ii} that:
\[ \dist_g(\widetilde{Q}_i^\star, \{ u_i = 0 \}) > 3B \varepsilon_i |\log \varepsilon_i|. \]
Recalling \eqref{eq:bounded.index.B.requirement}, we find: $|u_i(Q_i^\star)| > 1-\varepsilon_i^3$. Combined with $\dist_g^\pm(\widetilde{Q}_i^\star; \Gamma_{i,2}) > 6B \varepsilon_i |\log \varepsilon_i| > 0$, which guarantees that $u_i(Q_i^\star) > 0$, we conclude $u_i(Q_i^\star) > 1-\varepsilon_i^3$. This contradicts \eqref{eq:bounded.index.barrier.boundary.data.explicit}. Thus, \eqref{eq:bounded.index.tau.upper.bound} is true.
Summarizing \eqref{eq:bounded.index.tau.lower.bound}-\eqref{eq:bounded.index.tau.upper.bound}: $0 \leq \tau_i < 7B \varepsilon_i |\log \varepsilon_i|$. Combined with the defining property \eqref{eq:bounded.index.max.principle.conditions} of $\tau_i$, we get the following height estimate over $\Sigma$:
\[ f_{i,2} \leq h^{D_{i,\rho}(\tau_i)} \leq h^{D_{i,\rho}(7B \varepsilon_i |\log \varepsilon_i|)} \text{ on } \Sigma \cap B^2_{2\rho}(P_\star) \setminus B^2_r(P_\star), \]
where:
\begin{enumerate}
\item $f_{i,2} : \Sigma \setminus B_r^2(P_\star) \to \mathbf{R}$ is the height of $\Gamma_{i,2}$ over $\Sigma$, with $r \in (0,\rho/2)$ as in \eqref{eq:bounded.index.concentration.puncture}-\eqref{eq:bounded.index.graphing.punctured.sheets}, and
\item $h^{D_{i,\rho}(t)}$ denotes the height of the minimal disk $D_{i,\rho}(t)$ over $\Sigma$.
\end{enumerate}
The same sliding argument, carried out below the bottom-most sheet $\Gamma_{i,1}$ of $\{ u_i = 0 \}$, similarly gives:
\[ f_{i,1} \geq h^{D'_{i,\rho}(-7B \varepsilon_i |\log \varepsilon_i|)} \text{ on } \Sigma \cap B^2_{2\rho}(P_\star) \setminus B^2_{r}(P_\star). \]
Notice that we're denoting the disks by $D'_{i,\rho}(-7B\varepsilon_i |\log \varepsilon_i|)$, since they come from a different foliation, namely, the one generated by applying Proposition \ref{prop:white.foliation} to $w_i = f_{i,1}$. Therefore, by the regularity of the foliation guaranteed by Proposition \ref{prop:white.foliation}:
\begin{align*}
f_i = f_{i,2} - f_{i,1} & \leq h^{D_{i,\rho}(7B \varepsilon_i |\log \varepsilon_i|)} - h^{D'_{i,\rho}(-7B\varepsilon_i |\log \varepsilon_i|)} \leq c \left( 7B \varepsilon_i |\log \varepsilon_i| + h^{D_{i,\rho}(0)} - h^{D'_{i,\rho}(0)} \right) \\
& \leq c' \left( \varepsilon_i |\log \varepsilon_i| + \max_{B^2(\rho)(P_\star)} (h^{D_{i,\rho}} - h^{D'_{i,\rho}}) \right) \leq c' \left( \varepsilon_i |\log \varepsilon_i| + \max_{\partial B^2_\rho(P_\star)} (f_{i,2} - f_{i,1}) \right)
\end{align*}
on $\Sigma \cap B^2_{2\rho}(P_\star) \setminus B^2_{r}(P_\star)$. The last inequality follows from the maximum principle. We emphasize that $c'$ is independent of $i$ and $r$.
The proof of Proposition \ref{prop:bounded.index.nontrivial.limit} is essentially done. Indeed, fix $0 < r < \rho/2$. By what we've shown so far, we have
\[ \sup_{\Sigma \setminus B^2_r(P_\star)} f_i \leq c' \left( \varepsilon_i |\log \varepsilon_i| + \sup_{\Sigma \setminus B^2_{\rho}(P_\star)} f_i \right). \]
By the Harnack inequality \eqref{eq:bounded.index.harnack} and sheet separation lower bound \eqref{eq:bounded.index.estimate.height} on $\Sigma \setminus B^2_{2\rho}(P_\star)$:
\[ \sup_{\Sigma \setminus B^2_r(P_\star)} f_i \leq c'' \inf_{\Sigma \setminus B^2_{\rho}(P_\star)} f_i. \]
This holds independently of $i$, $r$, so the renormalized limit $\widehat{f}$ taken in \eqref{eq:bounded.index.renormalized.h} (first with $i \to \infty$ and then with $r \to 0$) is nontrivial. This completes the proof of Proposition \ref{prop:bounded.index.nontrivial.limit}.
\end{proof}
\subsection{Setup} \label{subsec:dirichlet.data.setup}
The heteroclinic solution from Section \ref{subsec:heteroclinic.solution} lifts trivially to a solution of the Allen-Cahn PDE, \eqref{eq:ac.pde}, on $\mathbf{R}^n$, for any $n \geq 1$; indeed, one may just take $u(x^1, \ldots, x^n) \triangleq \mathbb{H}_\varepsilon(x^n)$. Notice that this solution is ``centered'' on the $\{x^n = 0\}$ hyperplane. One may just as easily center it on any hyperplane in $\mathbf{R}^n$ by a suitable translation and rotation.
The question of centering approximate heteroclinic solutions on arbitrary minimal $\Sigma^{n-1} \subset (M^n, g)$ has been well-studied in the compact setting; see, e.g., \cite{PacardRitore03} for the boundary-less case and the geometrically natural case of Neumann conditions at the boundary when $\partial M$, $\partial \Sigma \neq \emptyset$, or see \cite{Pacard12} for a more general survey with a faster construction than \cite{PacardRitore03}, albeit only presented in the boundary-less case.
In this section we establish a corresponding existence theorem similar in spirit to those in \cite{PacardRitore03, Pacard12}, except we prescribe Dirichlet data. This theorem provides the barriers that were a crucial ingredient in the final ``sliding'' argument of Section \ref{sec:bounded.index}.
The setup is as follows. Define $C^{k,\alpha}_\varepsilon$, $\alpha \in (0,1)$, $\varepsilon > 0$, to be the standard H\"older space after rescaling by $\varepsilon$, i.e., whose Banach norm is
\begin{equation} \label{eq:dirichlet.data.ckalpha.eps}
\Vert v \Vert_{C^{k,\alpha}_\varepsilon} \triangleq \sum_{j=0}^k \varepsilon^j \Vert \nabla^j v \Vert_{L^\infty} + \varepsilon^{k+\alpha} [\nabla^k v]_\alpha.
\end{equation}
Various choices of domain and metric will be specified below. See Remarks \ref{rema:dirichlet.data.regularity.product.vs.omega}, \ref{rema:dirichlet.data.various.norms}.
Next, suppose that $D^{n-1}$ is an $(n-1)$-dimensional manifold with nonempty boundary, over which we take a topological cylinder $\Omega \triangleq D \times [-1,1]$, whose coordinates we label $X = (y, z) \in D \times [-1,1]$. Let $g$ be a smooth metric on $\Omega$, given in $(y, z)$ coordinates (Fermi coordinates) by
\[ g = g_z + dz^2. \]
We require that
\begin{equation} \label{eq:dirichlet.data.sigma.minimal}
\Sigma \triangleq D \times \{0\} \subset (\Omega, g) \text{ is a minimal surface}
\end{equation}
whose second fundamental form is uniformly bounded in $C^{0,\theta}$, for some $\theta \in (0, 1)$ that will be eventually chosen to be near $1$ (see Theorem \ref{theo:dirichlet.data.construction}):
\begin{equation} \label{eq:dirichlet.data.sigma.c2alpha}
|\sff_\Sigma| + [\sff_\Sigma]_{\theta} \leq \eta,
\end{equation}
and also\footnote{It is crucial for Section \ref{sec:bounded.index} that we only work with the weaker bounds on derivatives of $\sff$ given in \eqref{eq:dirichlet.data.sigma.c2alpha}, \eqref{eq:dirichlet.data.sigma.c3alpha}, which are precisely the types of estimates we derived in Section \ref{sec:stable.solutions}.} in $C^{1,\theta}_\varepsilon$:
\begin{equation} \label{eq:dirichlet.data.sigma.c3alpha}
\varepsilon |\nabla_{\Sigma} \sff_\Sigma| + \varepsilon^{1+\theta} [\nabla_{\Sigma} \sff_\Sigma]_{\theta} \leq \eta,
\end{equation}
with $\eta > 0$ small. We furthermore assume that there are $C^{2,\theta}$-coordinate charts on $\Sigma$ so that the induced metric $g_{0}$ is $C^{0,\theta}$ and $C^{1,\theta}_{\varepsilon}$-close to the Euclidean metric in the sense that
\begin{equation}\label{eq:dirichlet.data.sigma.induced.c2alpha}
|(g_{0})_{ij} - \delta_{ij}| + [(g_{0})_{ij}]_{\theta} \leq \eta,
\end{equation}
\begin{equation}\label{eq:dirichlet.data.sigma.induced.c3alpha}
\varepsilon|\partial_{k}(g_{0})_{ij}| + \varepsilon^{1+\theta} [\partial_{k}(g_{0})_{ij}]_\theta \leq \eta,
\end{equation}
where $i$, $j$, $k$ run through the coordinates $(y^{1},\dots,y^{n-1})$ on $\Sigma$ in the given coordinate chart.
Note that \eqref{eq:dirichlet.data.sigma.c2alpha} implies that Fermi coordinates $(y, z)$ with respect to $\Sigma$ are a diffeomorphism which is $C^{1,\theta}$-close to the identity, so in particular, together with \eqref{eq:dirichlet.data.sigma.induced.c2alpha}, it follows that the metric $g$ is $C^{0,\theta}$-close to being Euclidean in Fermi coordinates:
\begin{equation} \label{eq:dirichlet.data.g.c0alpha}
|g_{\kappa \lambda} - \delta_{\kappa \lambda}| + [g_{\kappa \lambda}]_{\theta} \leq \eta',
\end{equation}
for small $\eta' = \eta'(\eta, n) > 0$. Here, $\kappa$, $\lambda$ run through all $n$ Fermi coordinates $(y^1, \ldots, y^{n-1}, z)$.
Likewise, \eqref{eq:dirichlet.data.sigma.c3alpha} and \eqref{eq:dirichlet.data.sigma.induced.c3alpha} imply that Fermi coordinates are $C^{2,\theta}_\varepsilon$-close to the identity and
\begin{equation} \label{eq:dirichlet.data.g.c1alpha}
\varepsilon |\partial_{\mu} g_{\kappa \lambda}| + \varepsilon^{1+\theta} [\partial_{\mu} g_{\kappa \lambda}]_{\theta} \leq \eta'.
\end{equation}
Here, $\kappa$, $\lambda$, $\mu$ run through all $n$ Fermi coordinates.
We also require that $\Sigma$ carries no nontrivial Jacobi fields with Dirichlet boundary conditions in the following quantitative sense:
\begin{equation} \label{eq:dirichlet.data.sigma.nondegenerate}
\int_\Sigma (J_\Sigma f)^2 \, d\mu_{g_0} \geq \eta \int_\Sigma f^2 \, d\mu_{g_0} \text{ for every } f \in C^\infty_c(\Sigma \setminus \partial \Sigma).
\end{equation}
where
\begin{equation} \label{eq:dirichlet.data.sigma.jacobi.operator}
J_\Sigma f \triangleq -\Delta_{g_0} f - (|\sff_\Sigma|^2 + \ricc_{g}(\partial_z, \partial_z)|_{\Sigma})f
\end{equation}
denotes the Jacobi operator on $\Sigma$. (Note that our sign convention for the Jacobi operator differs from the one in \cite{Pacard12}.)
Let's also fix $\delta_* \in (0, 1)$, and define cutoff functions $\chi_j : \mathbf{R} \to [0,1]$, with $\chi_j' \geq 0$ on $[0, \infty)$, so that
\begin{equation} \label{eq:dirichlet.data.cutoff}
\chi_j(t) = \begin{cases} 1 & |t| \leq \varepsilon^{\delta_*} \Big( 1 - \tfrac{2j-1}{100} \Big) \\ 0 & |t| \geq \varepsilon^{\delta_*} \Big( 1 - \tfrac{2j-2}{100} \Big). \end{cases}
\end{equation}
as well as $\Vert \chi_j \Vert_{C^{3}_{\varepsilon^{\delta_*}}(\mathbf{R})} \leq 200$. We further require that the $\chi_j$ be even functions.
For $\varepsilon > 0$, set
\begin{equation} \label{eq:dirichlet.data.approximate.heteroclinic}
\widetilde{\mathbb{H}}_\varepsilon(t) \triangleq \chi_1(t) \mathbb{H}_\varepsilon(t) \pm (1 - \chi_1(t)),
\end{equation}
where the $\pm$ corresponds to $t > 0$, $t < 0$, respectively, and $\mathbb{H}_\varepsilon$ is as in \eqref{eq:heteroclinic.eps}. This is a truncation of the one-dimensional solution, $\mathbb{H}_\varepsilon$, which coincides with $\mathbb{H}_\varepsilon$ near $\Sigma$ and with $\pm 1$ away from $\Sigma$.
The functions $\chi_j$, $\mathbb{H}_\varepsilon$, $\widetilde{\mathbb{H}}_\varepsilon$ lift trivially to $\Sigma \times \mathbf{R}$. We also set
\[ \Omega_j \triangleq \{ (y, z) \in \Sigma \times \mathbf{R} : z \in \support \chi_j \}. \]
Using the Fermi coordinates $(y, z)$, $\chi_j$, $\mathbb{H}_\varepsilon$, $\widetilde{\mathbb{H}}_\varepsilon$ also give functions on $\Omega$ that depend only on $z$. By \eqref{eq:dirichlet.data.sigma.induced.c3alpha}, \eqref{eq:dirichlet.data.cutoff}, these functions are uniformly $C^{2,\theta}_\varepsilon$ in $\Sigma \times \mathbf{R}$ with respect to the product metric $g_0 + dz^2$ and also in $\Omega$ with respect to the metric $g$. Likewise, by \eqref{eq:dirichlet.data.sigma.c2alpha}, \eqref{eq:dirichlet.data.sigma.c3alpha}, the slab $\Omega_j$ can also be viewed as a subset of $(\Omega, g)$ whose boundary is $C^{1,\theta}$ and $C^{2,\theta}_\varepsilon$-close to being totally geodesic.
\begin{rema} \label{rema:dirichlet.data.regularity.product.vs.omega}
By \eqref{eq:dirichlet.data.sigma.induced.c3alpha}, \eqref{eq:dirichlet.data.g.c1alpha}, there exists a constant $C = C(\eta)$ such that
\[ C^{-1} \Vert f \Vert_{C^{k,\alpha}_\varepsilon(\Omega)} \leq \Vert f \Vert_{C^{k,\alpha}_{\varepsilon}(\Sigma \times \mathbf{R})} \leq C \Vert f \Vert_{C^{k,\alpha}_\varepsilon(\Omega)}, \; k = 0, 1, 2, \; \alpha \in (0, \theta], \]
for any function $f : \Omega \to \mathbf{R}$ with support in the interior of $\Omega$. The norms above are taken with respect to the product metric $g_0 + dz^2$ on $\Sigma \times \mathbf{R}$ and the metric $g$ on $\Omega$.
\end{rema}
\begin{rema} \label{rema:dirichlet.data.trivialization.window}
We cannot reuse the truncation from Section \ref{sec:jacobi.toda.reduction}, because we now need a truncation that trivializes outside a polynomial window instead of a logarithmic window.
\end{rema}
For subsets $S \subset \Sigma$, let's define
\[ \Pi_\varepsilon : L^2(S \times \mathbf{R}) \to L^2(S), \; \Pi_\varepsilon^\perp : L^2(S \times \mathbf{R}) \to L^2(S \times \mathbf{R}) \]
to be given by
\begin{align}
\Pi_\varepsilon(f)(y) & \triangleq \varepsilon^{-1} h_0^{-1} \int_{-\infty}^\infty f(y, z) \cdot \mathbb{H}'(\varepsilon^{-1} z) \, dz, \label{eq:dirichlet.data.proj} \\
\Pi_\varepsilon^\perp(f)(y,z) & \triangleq f(y,z) - \Pi_\varepsilon(f)(y) \mathbb{H}'(\varepsilon^{-1} z). \label{eq:dirichlet.data.proj.perp}
\end{align}
We note two things:
\begin{enumerate}
\item $S$ does not appear in the projection notation, but it will clear from the context when it is relevant.
\item Our normalization is such that $\Pi_\varepsilon( \{ z \mapsto \mathbb{H}'(\varepsilon^{-1} z) \} ) = \varepsilon \Pi_\varepsilon (\mathbb{H}_\varepsilon') = 1$.
\end{enumerate}
From this point forward we also consider another H\"older exponent, $\alpha \in (0, 1)$, which is such that
\[ \alpha \leq \theta \]
(with $\theta$ is as in \eqref{eq:dirichlet.data.sigma.c2alpha}-\eqref{eq:dirichlet.data.sigma.induced.c3alpha}). The exponent $\alpha$ will be eventually taken to be near $0$ (see Theorem \ref{theo:dirichlet.data.construction}).
We point out the following trivial lemma:
\begin{lemm} \label{lemm:dirichlet.data.proj.holder.norms}
Both $\Pi_\varepsilon$ and $\Pi_\varepsilon^\perp$ lift to linear maps
\[ \Pi_\varepsilon : C^{0,\alpha}_\varepsilon(S \times \mathbf{R}) \to C^{0,\alpha}_\varepsilon(S), \; \Pi_\varepsilon^\perp : C^{0,\alpha}_\varepsilon(S \times \mathbf{R}) \to C^{0,\alpha}_\varepsilon(S \times \mathbf{R}). \]
The $C^{0,\alpha}_\varepsilon(S \times \mathbf{R})$ norm is taken with respect to the product metric $g_{0} + dz^{2}$. Viewed as linear maps over these H\"older spaces, we have $\sup_{\varepsilon > 0} \big( \Vert \Pi_\varepsilon \Vert + \Vert \Pi_\varepsilon^\perp \Vert \big) < \infty$.
\end{lemm}
For $\zeta \in C^{2,\alpha}(\Sigma)$, we define $D_\zeta$ to be the map
\begin{equation} \label{eq:dirichlet.data.offset.map}
D_\zeta(y, t) \triangleq (y, t - \chi_2(t) \zeta(y)).
\end{equation}
Finally, we introduce the modified H\"older norm:
\begin{equation} \label{eq:dirichlet.data.ckalpha.eps.modified}
\Vert v \Vert_{\widetilde{C}^{k,\alpha}_\varepsilon(\Omega)} \triangleq \varepsilon^{-2} \Vert \chi_5 v \Vert_{C^{k,\alpha}_\varepsilon(\Omega)} + \Vert v \Vert_{C^{k,\alpha}_\varepsilon(\Omega)}.
\end{equation}
Recall that $\Vert \cdot \Vert_{C^{k,\alpha}_\varepsilon}$ is as in \eqref{eq:dirichlet.data.ckalpha.eps}. As with Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, the $C^{k,\alpha}_{\varepsilon}(\Omega)$ norm is taken with respect to $g$.
The main result of this section is:
\begin{theo} \label{theo:dirichlet.data.construction}
If $\alpha \leq \alpha_0$, $\varepsilon \leq \varepsilon_0$ and we're given boundary data
\begin{enumerate}
\item $\widehat{v}^\flat \in \widetilde{C}^{2,\alpha}_\varepsilon(\partial \Omega)$, $\Vert \widehat{v}^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\partial \Omega)} \leq \mu \varepsilon^2$, $\widehat{v}^\flat = 0$ on $\{ \chi_4 = 1 \} \cap \partial\Omega$,
\item $\widehat{v}^\sharp \in C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R})$, $\Vert \widehat{v}^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R})} \leq \mu \varepsilon^2$, $\Pi_\varepsilon(\widehat{v}^\sharp) \equiv 0$ on $\partial \Sigma$,
\item $\widehat{\zeta} \in C^{2,\alpha}(\partial \Sigma)$, $\varepsilon^{2\alpha} \Vert \widehat{\zeta} \Vert_{C^{2,\alpha}(\partial \Sigma)} \leq \mu \varepsilon^2$,
\end{enumerate}
and a metric $g$ for which \eqref{eq:dirichlet.data.sigma.minimal}-\eqref{eq:dirichlet.data.sigma.induced.c3alpha} hold with $\theta \geq \theta_0 \geq \alpha_0$, there exist
\begin{enumerate}
\item $v^\flat \in \widetilde{C}^{2,\alpha}(\Omega)$, $v^\flat|_{\partial \Omega} = \widehat{v}^\flat$, $\Vert v^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} \leq C \varepsilon^2$,
\item $v^\sharp \in C^{2,\alpha}(\Sigma \times \mathbf{R})$, $v^\sharp|_{\partial \Sigma \times \mathbf{R}} = \widehat{v}^\sharp$, $\Pi_\varepsilon v^\sharp \equiv 0$, $\Vert v^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \leq C \varepsilon^2$,
\item $\zeta \in C^{2,\alpha}(\Sigma)$, $\zeta|_{\partial \Gamma} = \widehat{\zeta}$, $\varepsilon^{2\alpha} \Vert \zeta \Vert_{C^{2,\alpha}(\Sigma)} \leq C \varepsilon^2$,
\end{enumerate}
so that $\mathfrak{u} = (\widetilde{\mathbb{H}}_{\varepsilon} + \chi_4 v^\sharp + v^\flat) \circ D_\zeta$ satisfies
\begin{equation} \label{eq:dirichlet.data.pde}
\varepsilon^2 \Delta_g \mathfrak{u} = W'(\mathfrak{u}) \text{ on } \Omega.
\end{equation}
The solution map $(\widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}, g) \mapsto (v^\flat, v^\sharp, \zeta)$ is Lipschitz continuous, with Lipschitz constant $L$, as a map
\begin{equation*}
\widetilde{C}^{2,\alpha}_\varepsilon(\partial \Omega) \times C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R}) \times C^{2,\alpha}(\partial \Sigma) \times \operatorname{Met}_{\varepsilon,\eta}(\Omega)
\to \widetilde{C}^{2,\alpha}_\varepsilon(\Omega) \times C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R}) \times C^{2,\alpha}(\Sigma)
\end{equation*}
where $\operatorname{Met}_{\varepsilon,\eta}(\Omega)$ denotes the set of metrics satisfying \eqref{eq:dirichlet.data.g.c0alpha}-\eqref{eq:dirichlet.data.g.c1alpha} with the obvious topology. The spaces $\widetilde{C}^{2,\alpha}_\varepsilon(\Omega) \times C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R}) \times C^{2,\alpha}(\Sigma)$, $\widetilde{C}^{2,\alpha}(\partial \Omega) \times C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R}) \times C^{2,\alpha}(\partial \Sigma)$ are topologized using the norms in \eqref{eq:dirichlet.data.interior.product.norm}, \eqref{eq:dirichlet.data.boundary.product.norm}, respectively. Here, $\varepsilon_0 = \varepsilon_0(n, \eta, W, \delta_*, \mu, \alpha)$, $\alpha_0 = \alpha_0(n, \eta, W, \delta_*, \mu)$, $\theta_0 = \theta_0(\delta_*)$, $C = C(n, \eta, W, \delta_*, \mu, \alpha)$, $L = L(n, \eta, W, \delta_*, \mu, \alpha, \theta)$.
\end{theo}
This follows along the lines of \cite[Section 3]{Pacard12}, provided one makes the necessary modifications to account for (possibly nonzero, but small) Dirichlet data as well as the important fact that our Fermi coordinate regularity is constrained by the weaker assumptions \eqref{eq:dirichlet.data.sigma.c2alpha}-\eqref{eq:dirichlet.data.sigma.c3alpha}. This lower regularity situation makes certain aspects of Theorem \ref{theo:dirichlet.data.construction} delicate, so we describe the proof in detail below.
\subsection{Linear scheme} \label{subsec:dirichlet.data.linear.scheme}
In this section we generalize linear estimates found in \cite[Section 3]{Pacard12} to allow Dirichlet boundary conditions, possibly with nonzero data. The operators we'll study are:
\begin{align}
L_* & \triangleq \Delta_{\mathbf{R}^n} + \partial_z^2 - W''(\mathbb{H}) \text{ on } \mathbf{R}_+^n \times \mathbf{R}, \label{eq:dirichlet.data.l.star} \\
L_\varepsilon & \triangleq \varepsilon^2 (\Delta_{g_0} + \partial_z^2) - W''(\mathbb{H}_\varepsilon) \text{ on } \Sigma \times \mathbf{R}, \label{eq:dirichlet.data.l.eps} \\
\mathcal{L}_\varepsilon & \triangleq \varepsilon^2 \Delta_g - W''(\pm 1) \text{ on } \Omega. \label{eq:dirichlet.data.cl.eps}
\end{align}
\begin{lemm}[cf. {\cite[Lemma 3.7]{Pacard12}}] \label{lemm:dirichlet.data.lemm.3.7}
Assume that $w \in L^\infty(\mathbf{R}^n_+ \times \mathbf{R})$ satisfies $L_* w = 0$ and $w \equiv 0$ on $\partial \mathbf{R}^n_+ \times \mathbf{R}$. Then $w \equiv 0$.
\end{lemm}
\begin{proof}
The result follows from \cite[Lemma 3.7]{Pacard12} after an odd reflection of $w$ across $\partial \mathbf{R}^{n}_{+}$.
\end{proof}
The next results that need to be adapted pertain to $L_\varepsilon$ and functions $\varphi \in L^\infty(\Sigma \times \mathbf{R})$ satisfying $\Pi_\varepsilon(\varphi) \equiv 0$ on $\Sigma$, where $\Pi_\varepsilon$ is as in \eqref{eq:dirichlet.data.proj}.
\begin{lemm}[cf. {\cite[Proposition 3.1]{Pacard12}}] \label{lemm:dirichlet.data.prop.3.1}
If $\varepsilon \leq \varepsilon_0$, $w \in C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})$, and $\Pi_\varepsilon(w) \equiv 0$ on $\Sigma$, then
\[ \Vert w \Vert_{C^{2,\alpha}_\varepsilon(\Sigma\times \mathbf{R})} \leq C( \Vert L_\varepsilon w \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert w|_{\partial \Sigma \times \mathbf{R}} \Vert_{C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R})}). \]
Here, $\varepsilon_0 = \varepsilon_0(n, \eta, W)$, $C = C(n, \eta, W, \alpha)$.
\end{lemm}
\begin{proof}
This follows from the $C^{1,\alpha}_\varepsilon$ control of $g_0$ by way of \eqref{eq:dirichlet.data.sigma.induced.c3alpha}, \cite[Proposition 3.1]{Pacard12}, Lemma \ref{lemm:dirichlet.data.lemm.3.7}, and boundary Schauder estimates (e.g., \cite[Theorem 5]{Simon97}).
\end{proof}
\begin{lemm}[cf. {\cite[Proposition 3.2]{Pacard12}}] \label{lemm:dirichlet.data.prop.3.2}
There exists $\varepsilon_0 > 0$ depending on $n$, $\eta > 0$, $W$, such that for all $\varepsilon \in (0, \varepsilon_0)$, all $f \in C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})$ with $\Pi_\varepsilon(f) \equiv 0$ on $\Sigma$, and all $\hat f \in C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R})$ with $\Pi_\varepsilon(\hat f) \equiv 0$ on $\partial \Sigma$, there exists a unique function $w \in C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})$, also with $\Pi_\varepsilon(w) \equiv 0$ on $\Sigma$, such that
\[ L_\varepsilon w = f \text{ in } \Sigma \times \mathbf{R}, \; w = \hat f \text{ on } \partial \Sigma \times \mathbf{R}. \]
\end{lemm}
\begin{proof}
When $\hat f \equiv 0$ this follows from the functional analytic argument already found in \cite[Proposition 3.2]{Pacard12} applied, instead, to $W^{1,2}_0(\Sigma \times \mathbf{R})$.
When $\hat f \not \equiv 0$, this follows by extending $\hat f$ to $C^{2,\alpha}(\Sigma \times \mathbf{R})$, $\Pi_\varepsilon(\hat f) \equiv 0$, and applying the previous existence result with zero boundary data to solve $L_\varepsilon w = f - L_\varepsilon \hat f$.
\end{proof}
Finally, \cite{Pacard12} deals with $\mathcal{L}_\varepsilon$.
\begin{lemm}[cf. {\cite[Proposition 3.3]{Pacard12}}] \label{lemm:dirichlet.data.prop.3.3}
If $\varepsilon \in (0, 1)$, then
\[ \Vert w \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} \leq C(\Vert \mathcal{L}_\varepsilon w \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} + \Vert w|_{\partial \Omega} \Vert_{C^{2,\alpha}_\varepsilon(\partial \Omega)}). \]
Here, $C = C(n, \eta, W, \alpha)$.
\end{lemm}
\begin{proof}
The interior estimate follows from interior Schauder theory, since $g$ is $C^{1,\alpha}_\varepsilon$ by \eqref{eq:dirichlet.data.g.c1alpha}. The boundary estimate on the regular portion of $\partial \Omega$ follows from boundary Schauder theory, because $\partial \Omega$ is $C^{2,\alpha}_\varepsilon$ at those points by \eqref{eq:dirichlet.data.sigma.c3alpha}. Finally, the estimate at the corners of $\partial \Omega$ follows from the boundary theory as well. This is because we can carry out odd reflections across $D \times \{\pm 1\}$ since the angles at the corners are all $\pi/2$.
\end{proof}
We also derive an \emph{improved} estimate for functions satisfying $\mathcal{L}_\varepsilon w = 0$ on a strip of height $O(\varepsilon^{\delta_*})$, and $w = 0$ on its lateral boundary. Recall the definition of the norm $\widetilde{C}^{2,\alpha}_{\varepsilon}$ in \eqref{eq:dirichlet.data.ckalpha.eps.modified}.
\begin{lemm}[cf. {\cite[(3.26)]{Pacard12}}] \label{lemm:dirichlet.data.eq.3.26}
If $\varepsilon \leq \varepsilon_0$, $w \in C^{2,\alpha}_\varepsilon(\Omega)$, and
\[ \mathcal{L}_\varepsilon w = 0 \text{ on } \Omega_4, \text{ and } w = 0 \text{ on } \partial \Omega_4 \cap \partial \Omega, \]
then
\[ \Vert w \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} \leq C ( \Vert \mathcal{L}_\varepsilon w \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} + \Vert w|_{\partial \Omega} \Vert_{C^{2,\alpha}_\varepsilon(\partial \Omega)}). \]
Here, $\varepsilon_0 = \varepsilon_0(n, \eta, W, \delta_*)$, $C = C(n, \eta, W, \delta_*, \alpha)$.
\end{lemm}
\begin{proof}
Considering Lemma \ref{lemm:dirichlet.data.prop.3.3}, it suffices to check that
\begin{equation} \label{eq:dirichlet.data.eq.3.26.i}
\Vert \chi_5 w \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} \leq C \varepsilon^2 (\Vert \mathcal{L}_\varepsilon w \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} + \Vert w|_{\partial \Omega} \Vert_{C^{2,\alpha}_\varepsilon(\partial \Omega)}).
\end{equation}
Since $\mathcal{L}_\varepsilon = 0$ on $\Omega_4$, $w = 0$ on $\partial \Omega_4 \cap \partial \Omega$, and $\delta_* \in (0,1)$, Schauder's \emph{interior} estimates estimates on $\partial\Omega_5 \setminus \partial \Omega$, Schauder's \emph{boundary} estimates near $\partial \Omega_5 \cap \partial \Omega$, \eqref{eq:dirichlet.data.sigma.c3alpha}, and \eqref{eq:dirichlet.data.g.c1alpha}, imply:
\[ \Vert w \Vert_{C^{2,\alpha}_\varepsilon(\Omega_5)} \leq C \Vert w \Vert_{L^\infty(\{ \chi_4 = 1 \})}. \]
In particular, given the decay of the first and second derivatives of $\chi_j$ from \eqref{eq:dirichlet.data.cutoff} and $\delta_* \in (0,1)$, \eqref{eq:dirichlet.data.eq.3.26.i} will follow as long as
\begin{equation} \label{eq:dirichlet.data.eq.3.26.ii}
\Vert w \Vert_{L^\infty(\{ \chi_4 = 1 \})} \leq C \varepsilon^2 \Vert w \Vert_{L^\infty(\Omega)}
\end{equation}
We use the same barrier argument as in \cite[Remark 3.2]{Pacard12}, paying closer attention to the boundary and to the regularity. Define
\[ \varphi_{z_0}(z) \triangleq \cosh (\gamma \varepsilon^{-1} (z-z_0)) \]
with $|z_0| \leq \varepsilon^{\delta_*}$ and $\gamma \in (0, (W''(\pm 1))^{\tfrac{1}{2}})$. If $H_z$ denotes the mean curvature of of a $z$-level set in Fermi coordinates, then:
\begin{align*}
\varepsilon^2 \Delta_g \varphi_{z_0}(z)
& = \gamma^2 \varphi_{z_0}(z) + H_z \gamma \varepsilon \sinh( \gamma \varepsilon^{-1} (z-z_0)) \leq (\gamma^2 + \gamma \varepsilon |H_z|) \varphi_{z_0}(z).
\end{align*}
It follows from \eqref{eq:dirichlet.data.sigma.c2alpha} and \eqref{eq:mean.curv.ddt.sff}-\eqref{eq:mean.curv.ddt.h} that $|H_z|$ is uniformly bounded. In particular, for sufficiently small $\varepsilon$, depending on $\gamma$, $\eta$, $n$, we have
\[ \varepsilon^2 \Delta_g \varphi_{z_0}(z) \leq W''(\pm 1) \varphi_{z_0}(z), \]
so $\varphi_{z_0}$ is a barrier, as it was in \cite{Pacard12}. It therefore follows from the maximum principle applied to $w - t \varphi_{z_0}$ that, for $(y, z_0) \in \Omega_4$,
\[ |w(y,z_0)| \leq \left( \inf_{\Omega \setminus \Omega_4} \varphi_{z_0} \right)^{-1} \max_{\partial \Omega_4} |w|, \]
which is trivially bounded by $ c \varepsilon^2 \Vert w \Vert_{L^\infty(\Omega)}$ whenever $(y,z_0) \in \{ \chi_4 = 1 \}$, and $\varepsilon > 0$ is small. This implies \eqref{eq:dirichlet.data.eq.3.26.ii} and, in turn, \eqref{eq:dirichlet.data.eq.3.26.i}.
\end{proof}
\subsection{Nonlinear scheme} \label{subsec:dirichlet.data.nonlinear.scheme}
We consider the following nonlinear functionals, originally defined in \cite[Section 3]{Pacard12}:
\begin{align}
\mathscr{E}_\varepsilon(\zeta)
& \triangleq \varepsilon^2 \Delta_g (\widetilde{\mathbb{H}}_\varepsilon \circ D_\zeta) \circ D_\zeta^{-1} - W'(\widetilde{\mathbb{H}}_\varepsilon), \label{eq:dirichlet.data.E.eps} \\
Q_\varepsilon(v)
& \triangleq W'(\widetilde{\mathbb{H}}_\varepsilon + v) - W'(\widetilde{\mathbb{H}}_\varepsilon) - W''(\widetilde{\mathbb{H}}_\varepsilon) v, \label{eq:dirichlet.data.Q.eps} \\
M_\varepsilon(v^\flat, v^\sharp, \zeta)
& \triangleq \chi_3 \Big[ L_\varepsilon v^\sharp - \varepsilon^2 \Delta_g (v^\sharp \circ D_\zeta) \circ D_\zeta^{-1} + W''(\mathbb{H}_\varepsilon) v^\sharp \label{eq:dirichlet.data.M.eps} \\
& \qquad - \varepsilon^2 (\Delta_g (v^\flat \circ D_\zeta) \circ D_\zeta^{-1} - \Delta_g v^\flat) - \mathscr{E}_\varepsilon(\zeta) + \varepsilon^2 (J_\Sigma \zeta) \partial_z \mathbb{H}_\varepsilon \nonumber \\
& \qquad - Q_\varepsilon(\chi_4 v^\sharp + v^\flat) + (W''(\mathbb{H}_\varepsilon) - W''(\pm 1)) v^\flat \Big], \nonumber \\
N_\varepsilon(v^\flat, v^\sharp, \zeta)
& \triangleq (\chi_4-1) \Big[ \varepsilon^2( \Delta_g( v^\flat \circ D_\zeta) \circ D_\zeta^{-1} - \Delta_g v^\flat) \label{eq:dirichlet.data.N.eps} \\
& \qquad \qquad + (W''(\widetilde{\mathbb{H}}_\varepsilon) - W''(\pm 1)) v^\flat - \mathscr{E}_\varepsilon(\zeta) - Q_\varepsilon(\chi_4 v^\sharp + v^\flat) \Big] \nonumber \\
& \qquad \qquad - \varepsilon^2 (\Delta_g ((\chi_4 v^\sharp) \circ D_\zeta) - \chi_4 \Delta_g(v^\sharp \circ D_\zeta)) \circ D_\zeta^{-1} . \nonumber
\end{align}
These functionals allow us to pose \eqref{eq:dirichlet.data.pde} as a fixed point problem:
\begin{align}
\mathcal{L}_\varepsilon v^\flat & = N_\varepsilon(v^\flat, v^\sharp, \zeta) \label{eq:dirichlet.data.pde.vsharp} \\
L_\varepsilon v^\sharp & = \Pi_\varepsilon^\perp M_\varepsilon(v^\flat, v^\sharp, \zeta) \label{eq:dirichlet.data.pde.vflat} \\
J_\Sigma \zeta & = \varepsilon^{-1} \Pi_\varepsilon M_\varepsilon(v^\flat, v^\sharp, \zeta) \label{eq:dirichlet.data.pde.zeta},
\end{align}
(cf. \cite[(3.31), (3.32), (3.33)]{Pacard12}). We impose, as does \cite[Section 3]{Pacard12}, the additional constraint:
\[ \Pi_\varepsilon v^\sharp \equiv 0 \text{ on } \Sigma. \]
\begin{lemm}[cf. {\cite[Lemma 3.8]{Pacard12}}] \label{lemm:dirichlet.data.lemm.3.8}
The following estimates hold:
\[ \Vert N_\varepsilon(0, 0, 0) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} + \Vert \Pi_\varepsilon^\perp M_\varepsilon(0, 0, 0) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \varepsilon^{-1} \Vert \Pi_\varepsilon M_\varepsilon(0, 0, 0) \Vert_{C^{0,\alpha}(\Sigma)} \leq c_0 \varepsilon^2. \]
Here, $\varepsilon \in (0, \tfrac12)$, $c_0 = c_0(n, \eta, W, \delta_*, \alpha)$.
\end{lemm}
\begin{proof}
Note that
\[ M_\varepsilon(0, 0, 0) = - \chi_3 \mathscr{E}_\varepsilon(0), \; N_\varepsilon(0, 0, 0) = (1-\chi_4) \mathscr{E}_\varepsilon(0). \]
Straightforward computation shows $\mathscr{E}_\varepsilon(0) = \varepsilon^2 \Delta_g \widetilde{\mathbb{H}}_\varepsilon - W'(\widetilde{\mathbb{H}}_\varepsilon)$. From \eqref{eq:dirichlet.data.approximate.heteroclinic}:
\begin{equation} \label{eq:dirichlet.data.lemm.3.8.Htilde.minus.H}
\widetilde{\mathbb{H}}_\varepsilon - \mathbb{H}_\varepsilon = (1-\chi_1)(\pm 1 - \mathbb{H}_\varepsilon),
\end{equation}
($\pm$ depends on $z > 0$ or $z < 0$), a quantity that decays exponentially to all orders with $\varepsilon \to 0$. Since $\mathbb{H}_\varepsilon$ does too on $\support (1-\chi_4)$, we in fact get
\[ \Vert N_\varepsilon(0, 0, 0) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \leq C_m \varepsilon^m \]
for all $m \in \mathbf{N}$. (Taking $m=2$ will suffice.)
To estimate $M_\varepsilon(0, 0, 0)$, we proceed to further rewrite:
\begin{align*}
\mathscr{E}_\varepsilon(0)
& = \varepsilon^2 \Delta_g \widetilde{\mathbb{H}}_\varepsilon - W'(\widetilde{\mathbb{H}}_\varepsilon) \\
& = \varepsilon^2 \Delta_g \mathbb{H}_\varepsilon - W'(\mathbb{H}_\varepsilon) + \varepsilon^2 \Delta_g (\widetilde{\mathbb{H}}_\varepsilon - \mathbb{H}_\varepsilon) - (W'(\widetilde{\mathbb{H}}_\varepsilon) - W'(\mathbb{H}_\varepsilon)) \\
& = \varepsilon^2 H_z \partial_z \mathbb{H}_\varepsilon - (\varepsilon^2 \Delta_g - W''(\widetilde{\mathbb{H}}_\varepsilon) - Q_\varepsilon) ({\mathbb{H}}_\varepsilon - \widetilde{\mathbb{H}}_\varepsilon).
\end{align*}
Note that
\[ \widetilde{\mathbb{H}}_\varepsilon \equiv 1 \text{ on } \Omega \setminus \Omega_1 \implies \mathscr{E}_\varepsilon(0) \equiv 0 \text{ on } \Omega \setminus \Omega_1. \]
If $\chi : \Omega \to [0,1]$ is the cutoff function $\chi(z) = \chi_1(z/2)$, then note that $\chi \equiv 1$ on $\support \mathscr{E}_\varepsilon(0)$ so that
\[ \mathscr{E}_\varepsilon(0) = \chi \cdot \varepsilon^2 H_z \partial_z \mathbb{H}_\varepsilon - \chi \cdot (\varepsilon^2 \Delta_g - W''(\widetilde{\mathbb{H}}_\varepsilon) - Q_\varepsilon)({\mathbb{H}}_\varepsilon - \widetilde{\mathbb{H}}_\varepsilon). \]
It follows from \eqref{eq:dirichlet.data.g.c1alpha}, \eqref{eq:dirichlet.data.cutoff}, and \eqref{eq:dirichlet.data.lemm.3.8.Htilde.minus.H} that
\begin{equation} \label{eq:dirichlet.data.lemm.3.8.i}
\Vert \chi \cdot (\varepsilon^2 \Delta_g - W''(\widetilde{\mathbb{H}}_\varepsilon) - Q_\varepsilon)({\mathbb{H}}_\varepsilon - \widetilde{\mathbb{H}}_\varepsilon) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \leq C_m \varepsilon^m,
\end{equation}
for $m \in \mathbf{N}$. (Taking $m = 4$ will suffice.)
Recalling \eqref{eq:mean.curv.ddt.h}:
\begin{equation} \label{eq:dirichlet.data.lemm.3.8.ddt.h}
\partial_z H_z = - |\sff_z|^2 + \ricc_g(\partial_z, \partial_z)|_{D \times \{z\}}, \; z \in [-1,1].
\end{equation}
Certainly, this already implies, since $\alpha \leq \theta$,
\[ \sup_{|z| \leq 1} \Vert y \mapsto \partial_z H_z \Vert_{C^{0,\alpha}(\Sigma)} \leq C. \]
Combining \eqref{eq:dirichlet.data.lemm.3.8.ddt.h} with \eqref{eq:dirichlet.data.sigma.c2alpha}, $\alpha \leq \theta$, \eqref{eq:mean.curv.ddt.metric}, and \eqref{eq:mean.curv.ddt.sff},
we even find that
\begin{equation} \label{eq:dirichlet.data.lemm.3.8.ddt.sq.h}
\sup_{|z| \leq 1} \Vert y \mapsto \partial_z^2 H_z(y,z) \Vert_{C^{0,\alpha}(\Sigma)} \leq C.
\end{equation}
In particular, \eqref{eq:dirichlet.data.sigma.minimal}, \eqref{eq:dirichlet.data.lemm.3.8.ddt.sq.h} and Taylor's theorem imply
\begin{equation} \label{eq:dirichlet.data.lemm.3.8.taylor}
H_z = - (|\sff_0|^2 + \ricc_g(\partial_z, \partial_z)|_{\Sigma}) z + \mathcal{R}(y,z) z^2,
\end{equation}
where
\begin{equation} \label{eq:dirichlet.data.3.8.proj.remainder}
\sup_{|z|\leq 1} \Vert y \mapsto \mathcal{R}(y,z) \Vert_{C^{0,\alpha}(\Sigma)} \leq C.
\end{equation}
From the trivial estimate $|z| \partial_z \mathbb{H}_\varepsilon \leq C$, \eqref{eq:dirichlet.data.cutoff}, and \eqref{eq:dirichlet.data.lemm.3.8.taylor}, we find that
\begin{equation} \label{eq:dirichlet.data.lemm.3.8.ii}
\Vert \chi \cdot \varepsilon^2 H_z \partial_z \mathbb{H}_\varepsilon \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \leq C \varepsilon^2.
\end{equation}
Put together, \eqref{eq:dirichlet.data.lemm.3.8.i}, \eqref{eq:dirichlet.data.lemm.3.8.ii}, and Lemma \ref{lemm:dirichlet.data.proj.holder.norms} imply:
\[ \Vert \Pi_\varepsilon^\perp M_\varepsilon(0,0,0) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \leq C \varepsilon^2. \]
Finally, by \eqref{eq:dirichlet.data.lemm.3.8.taylor},
\[ \Pi_\varepsilon (\chi \cdot \varepsilon^2 H_z \partial_z \mathbb{H}_\varepsilon) = h_0^{-1} \int_{-\infty}^{\infty} \chi(z) (\partial_z H_z(y,0) \cdot z + \mathcal{R}(y,z) z^2) (\mathbb{H}'(\varepsilon^{-1} z))^2 \, dz. \]
Recalling that, from parity, (since $\chi(z)$ is even)
\[ \int_{-\infty}^\infty \chi(z) z (\mathbb{H}'(\varepsilon^{-1} z))^2 \, dz = 0 \]
it follows that
\[ \Pi_\varepsilon(\chi \cdot \varepsilon^2 H_z \partial_z \mathbb{H}_\varepsilon) = h_0^{-1} \int_{-\infty}^\infty \chi(z) \mathcal{R}(y,z) z^2 (\mathbb{H}'(\varepsilon^{-1} z))^2 \, dz, \]
at which point we can directly estimate using \eqref{eq:heteroclinic.expansion.ii}, \eqref{eq:heteroclinic.eps}, and \eqref{eq:dirichlet.data.3.8.proj.remainder}, and get:
\[ \Vert \Pi_\varepsilon (\chi \cdot \varepsilon^2 H_z \partial_z \mathbb{H}_\varepsilon) \Vert_{C^{0,\alpha}(\Sigma)} \leq C \varepsilon^3, \]
Together with \eqref{eq:dirichlet.data.lemm.3.8.i} (with $m=4$), this implies
\[ \Vert \Pi_\varepsilon M_\varepsilon(0, 0, 0) \Vert_{C^{0,\alpha}(\Sigma)} \leq C \varepsilon^3. \]
This completes the proof.
\end{proof}
\begin{lemm}[cf. {\cite[Lemma 3.9]{Pacard12}}] \label{lemm:dirichlet.data.lemm.3.9}
For $\alpha \leq \alpha_0$, $\varepsilon \leq \varepsilon_0$:
\begin{align}
& \Vert N_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - N_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \label{eq:dirichlet.data.lemm.3.9.n} \\
& \qquad \leq c_1 \varepsilon^{\delta} \Big( \Vert v_2^\flat - v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)} \Big), \nonumber \\
& \Vert \Pi_\varepsilon^\perp(M_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - M_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1)) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \label{eq:dirichlet.data.lemm.3.9.m.perp} \\
& \qquad \leq c_1 \varepsilon^\delta \Big( \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)} \Big), \nonumber \\
& \Vert \Pi_\varepsilon( M_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - M_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) ) \Vert_{C^{0,\alpha}(\Sigma)} \label{eq:dirichlet.data.lemm.3.9.m} \\
& \qquad \leq c_1 \varepsilon^{1 + \delta} \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + c_1 \varepsilon^{1-\alpha} \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + c_1 \varepsilon^{1 + \delta} \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}, \nonumber
\end{align}
provided \eqref{eq:dirichlet.data.sigma.c2alpha}-\eqref{eq:dirichlet.data.sigma.induced.c3alpha} hold with $\theta \geq \theta_0 \geq \alpha_0$, and
\[ \sum_{j=1,2} \Vert v_j^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + \Vert v_j^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \varepsilon^{2\alpha} \Vert \zeta_j \Vert_{C^{2,\alpha}(\Sigma)} \leq C' \varepsilon^2. \]
Here, $\varepsilon_0 = \varepsilon_0(n, \eta, W, \delta_*)$, $\delta = \delta(\delta_*)$, $\theta_0 = \theta_0(\delta_*)$, $\alpha_0 = \alpha_0(\delta_*)$, $c_1 = c_1(n, \eta, W, \delta_*, C', \alpha)$.
\end{lemm}
\begin{rema} \label{rema:dirichlet.data.various.norms}
We emphasize that three different norms are used:
\begin{enumerate}
\item On $v^\flat$, we use the \emph{modified} weighted H\"older norm
\[ \Vert w \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} = \Vert w \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + \varepsilon^{-2} \Vert \chi_5 w \Vert_{C^{2,\alpha}_\varepsilon(\Omega)}. \]
Here, the H\"older norms are measured with respect to the metric $g$.
\item On $v^\sharp$, we use the standard weighted H\"older norm $C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})$. Here, the H\"older norms are measured with respect to the product metric $g_{0}+dz^{2}$.
\item On $\zeta$, we use the \emph{unweighted} H\"older norm $C^{2,\alpha}(\Sigma)$, which strictly dominates $C^{2,\alpha}_\varepsilon(\Sigma)$:
\[ \Vert \zeta \Vert_{C^{2,\alpha}_\varepsilon(\Sigma)} \leq \Vert \zeta \Vert_{C^{2,\alpha}(\Sigma)}. \]
Here, the H\"older norms are measured with respect to the metric $g_{0}$ induced on $\Sigma$.
\end{enumerate}
\end{rema}
\begin{proof}[Proof of Lemma {\ref{lemm:dirichlet.data.lemm.3.9}}]
In what follows we may assume that $\alpha_0 \leq \tfrac14$.
Note, from \eqref{eq:dirichlet.data.cutoff}, \eqref{eq:dirichlet.data.N.eps}, that
\[ N_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \equiv N_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) \equiv 0 \text{ on } \{ \chi_4 = 1 \}. \]
Therefore, since $\delta_* \in (0,1)$,
\begin{align*}
\Vert N_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - N_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} & = \Vert N_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - N_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\{ \chi_4 \neq 1 \})} \\
& \leq \Vert N_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - N_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega \setminus \Omega_5)}.
\end{align*}
We'll estimate this by pairing up the terms, making sure to use use the fact that our H\"older norm is taken over $\Omega \setminus \Omega_5$ instead of over $\Omega$, in order to gain a factor of $\varepsilon^\delta$, for some $\delta > 0$ that depends on $\delta_*$.
In all that follows, we'll repeatedly (and implicitly) use that our Fermi coordinates (and thus also $D_\zeta$, $D_\zeta^{-1}$) are $C^{2,\alpha}_\varepsilon$ close to the identity, and that our metric $g$ in Fermi coordinates is $C^{1,\alpha}_\varepsilon$ close to Euclidean.
We start by estimating
\begin{equation*}
\Vert \varepsilon^2 (\Delta_g (v_2^\flat \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g v_2^\flat)
- \varepsilon^{2} (\Delta_g (v_1^\flat \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1} - \Delta_g v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}.
\end{equation*}
(We can deduce a good estimate on all of $\Omega$, not just on $\Omega \setminus \Omega_5$.) By working in Fermi coordinates in scale $O(\varepsilon)$, we see that
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F1}
\mathcal{F}_1(v, \zeta) \triangleq \varepsilon^2 \Delta_g (v \circ D_\zeta) \circ D_\zeta^{-1}
\end{equation}
is a \emph{smooth} nonlinear Banach space functional $\mathcal{F}_1 : C^{2,\alpha}_\varepsilon(\Omega) \times C^{2,\alpha}_\varepsilon(\Sigma) \to C^{0,\alpha}_\varepsilon(\Omega)$, and is \emph{linear} in $v$. In particular,
\begin{align*}
& \varepsilon^2 \big[ (\Delta_g (v_2^\flat \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g v_2^\flat) - (\Delta_g (v_1^\flat \circ D_{\zeta_1}) \circ D_{\zeta_1} - \Delta_g v_1^\flat) \big] \\
& \qquad = (\mathcal{F}_1(v_2^\flat, \zeta_2) - \mathcal{F}_1(v_1^\flat, \zeta_1)) - (\mathcal{F}_1(v_2^\flat, 0) - \mathcal{F}_1(v_1^\flat, 0)) \\
& \qquad = \int_0^1 \langle D_v \mathcal{F}_1(v_1^\flat + t(v_2^\flat - v_1^\flat), \zeta_1 + t(\zeta_2 - \zeta_1)), v_2^\flat - v_1^\flat \rangle \\
& \qquad \qquad + \langle D_\zeta \mathcal{F}_1(v_1^\flat + t(v_2^\flat - v_1^\flat), \zeta_1 + t(\zeta_2 - \zeta_1)), \zeta_2 - \zeta_1 \rangle \, dt \\
& \qquad - \int_0^1 \langle D_v \mathcal{F}_1(v_1^\flat + t(v_2^\flat - v_1^\flat), 0), v_2^\flat - v_1^\flat \rangle dt \\
& \qquad = \int_0^1 \int_0^1 \langle D_\zeta D_v \mathcal{F}_1(v_1^\flat + t(v_2^\flat - v_1^\flat), s \zeta_1 + st (\zeta_2-\zeta_1)), (\zeta_1 + t(\zeta_2 - \zeta_1)) \otimes (v_2^\flat - v_1^\flat) \rangle \, ds \, dt \\
& \qquad + \int_0^1 \langle D_\zeta \mathcal{F}_1(v_1^\flat + t(v_2^\flat - v_1^\flat), \zeta_1 + t(\zeta_2 - \zeta_1)), \zeta_2 - \zeta_1 \rangle \, dt.
\end{align*}
Seeing as to how $\Vert v_j^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} \leq C' \varepsilon^2$, $\Vert \zeta_j \Vert_{C^{2,\alpha}(\Sigma)} \leq C' \varepsilon^{2-2\alpha}$, and using the linearity in $v$ of $\mathcal{F}_1$ (and thus of $D_\zeta \mathcal{F}_1$), we can directly estimate:
\begin{align}
& \Vert \varepsilon^2((\Delta_g (v_2^\flat \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g v_2^\flat) - (\Delta_g (v_1^\flat \circ D_{\zeta_1}) \circ D_{\zeta_1} - \Delta_g v_1^\flat)) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \nonumber \\
& \qquad \leq C (\Vert \zeta_1 \Vert_{C^{2,\alpha}_\varepsilon(\Sigma)} + \Vert \zeta_2 \Vert_{C^{2,\alpha}_\varepsilon(\Sigma)}) \Vert v_2^\flat - v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + C (\Vert v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + \Vert v_2^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)}) \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}_\varepsilon(\Sigma)} \nonumber \\
& \qquad \leq C (\Vert \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)} + \Vert \zeta_2 \Vert_{C^{2,\alpha}(\Sigma)}) \Vert v_2^\flat - v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + C (\Vert v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + \Vert v_2^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)}) \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)} \nonumber \\
& \qquad \leq C \varepsilon^{2-2\alpha} \Vert v_2^\flat - v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + C \varepsilon^2 \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}. \label{eq:dirichlet.data.lemm.3.9.laplace.zeta.commutator}
\end{align}
This estimate is of the desired form.
Next, we estimate
\[ \Vert (W''(\widetilde{\mathbb{H}}_\varepsilon) - W''(\pm 1))(v_2^\flat - v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega \setminus\Omega_5)} \]
The desired estimate is a simple consequence of Remark \ref{rema:dirichlet.data.regularity.product.vs.omega} and how, on $\Omega \setminus \Omega_5$, we have
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.H.decay}
\Vert W''(\widetilde{\mathbb{H}}_\varepsilon) - W''(\pm 1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega \setminus \Omega_5)} \leq C_m \varepsilon^m,
\end{equation}
for all $m \in \mathbf{N}$; thus, any $\delta > 0$ will do.
Next, we estimate
\[ \Vert \mathscr{E}_\varepsilon(\zeta_2) - \mathscr{E}_\varepsilon(\zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega \setminus \Omega_5)}. \]
We have
\begin{align*}
& \mathscr{E}_\varepsilon(\zeta_2) - \mathscr{E}_\varepsilon(\zeta_1) = \varepsilon^2 ( \Delta_g (\widetilde{\mathbb{H}}_\varepsilon \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g (\widetilde{\mathbb{H}}_\varepsilon \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1}) = \mathcal{F}_1'(\widetilde{\mathbb{H}}_\varepsilon, \zeta_2) - \mathcal{F}_1'(\widetilde{\mathbb{H}}_\varepsilon, \zeta_1),
\end{align*}
where $\mathcal{F}_1' : C^{2,\alpha}_\varepsilon(\Omega \setminus \Omega_5) \times C^{2,\alpha}_\varepsilon(\Sigma) \to C^{0,\alpha}_\varepsilon(\Omega \setminus \Omega_5)$ is the restriction of $\mathcal{F}_1$ from \eqref{eq:dirichlet.data.lemm.3.9.F1}. Arguing as before, we get
\begin{align}
& \Vert \mathscr{E}_\varepsilon(\zeta_2) - \mathscr{E}_\varepsilon(\zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega \setminus \Omega_5)} \leq C \Vert \widetilde{\mathbb{H}}_\varepsilon \Vert_{C^{2,\alpha}_\varepsilon(\Omega \setminus \Omega_5)} \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}_\varepsilon(\Sigma)} \leq C_m \varepsilon^m \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}, \label{eq:dirichlet.data.lemm.3.9.Eeps.n}
\end{align}
for all $m \in \mathbf{N}$, which implies what we want, for any $\delta > 0$.
Next, we estimate
\[ \Vert Q_\varepsilon(\chi_4 v_2^\sharp + v_2^\flat) - Q_\varepsilon(\chi_4 v_1^\sharp + v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}. \]
Note that
\begin{multline*}
Q_\varepsilon(\chi_4 v_2^\sharp + v_2^\flat) - Q_\varepsilon(\chi_4 v_1^\sharp + v_1^\flat) \\
= W'(\widetilde{\mathbb{H}}_\varepsilon + \chi_4 v_2^\sharp + v_2^\flat) - W'(\widetilde{\mathbb{H}}_\varepsilon + \chi_4 v_1^\sharp + v_1^\flat)
- W''(\widetilde{\mathbb{H}}_\varepsilon)(\chi_4 (v_2^\sharp - v_1^\sharp) + (v_2^\flat - v_1^\flat)).
\end{multline*}
Define
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F2}
\mathcal{F}_2(v) \triangleq W'(\widetilde{\mathbb{H}}_\varepsilon + v),
\end{equation}
viewed as a \emph{smooth} nonlinear Banach space functional $\mathcal{F}_2 : C^{0,\alpha}_\varepsilon(\Omega) \to C^{0,\alpha}_\varepsilon(\Omega)$. Note that
\[ \langle D_v \mathcal{F}_2(v), w \rangle = W''(\widetilde{\mathbb{H}}_\varepsilon + v) w, \; \langle D_v D_v \mathcal{F}_2(v), w \otimes w' \rangle = W''(\widetilde{\mathbb{H}}_\varepsilon + v) ww', \]
for $w$, $w' \in C^{0,\alpha}_\varepsilon(\Omega)$. In particular, the expression we're trying to bound equals
\begin{align*}
& = \mathcal{F}_2(\chi_4 v_2^\sharp + v_2^\flat) - \mathcal{F}_2(\chi_4 v_1^\sharp + v_1^\flat) - \langle D_v \mathcal{F}_2(0), \chi_4 (v_2^\sharp-v_1^\sharp) + v_2^\flat - v_1^\flat \rangle \\
& = \int_0^1 \langle D_v \mathcal{F}_2(\chi_4 v_1^\sharp + v_1^\flat + t(\chi_4(v_2^\sharp - v_1^\sharp) + v_2^\flat - v_1^\flat)), \chi_4 (v_2^\sharp - v_1^\sharp) + v_2^\flat - v_1^\flat \rangle \, dt \\
& \qquad - \langle D_v \mathcal{F}_2(0), \chi_4 (v_2^\sharp-v_1^\sharp) + v_2^\flat - v_1^\flat \rangle \\
& = \int_0^1 \int_0^1 \langle D_v D_v \mathcal{F}_2(s(\chi_4 v_1^\sharp + v_1^\flat + t(\chi_4(v_2^\sharp - v_1^\sharp) + v_2^\flat - v_1^\flat)), \\
& \qquad \qquad \qquad (\chi_4 v_1^\sharp + v_1^\flat + t(\chi_4(v_2^\sharp - v_1^\sharp) + v_2^\flat - v_1^\flat)) \otimes (\chi_4(v_2^\sharp - v_1^\sharp) + v_2^\flat - v_1^\flat)) \rangle \, ds \, dt.
\end{align*}
Recalling Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, \eqref{eq:dirichlet.data.cutoff}, and $\delta_* \in (0,1)$, we can estimate
\begin{align*}
\Vert Q_\varepsilon(\chi_4 v_2^\sharp + v_2^\flat) - Q_\varepsilon(\chi_4 v_1^\sharp + v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} & \leq C (\Vert v_1^\sharp \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert v_1^\flat \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} + \Vert v_2^\sharp \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert v_2^\flat \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}) \\
& \qquad \qquad \cdot (\Vert v_2^\sharp - v_1^\sharp \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert v_2^\flat - v_1^\flat \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}).
\end{align*}
This gives
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.Qeps}
\Vert Q_\varepsilon(\chi_4 v_2^\sharp + v_2^\flat) - Q_\varepsilon(\chi_4 v_1^\sharp + v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}
\leq C \varepsilon^2 (\Vert v_2^\sharp - v_1^\sharp \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert v_2^\flat - v_1^\flat \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}),
\end{equation}
using $\Vert v_j^\sharp \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})}, \Vert v_j^\flat \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \leq C' \varepsilon^2$.
Next, we consider
\begin{multline*}
\Vert \varepsilon^2 ((\Delta_g ((\chi_4 v_2^\sharp) \circ D_{\zeta_2}) - \chi_4 \Delta_g(v_2^\sharp \circ D_{\zeta_2})) \circ D_{\zeta_2}^{-1}) \\
- (\Delta_g ((\chi_4 v_1^\sharp) \circ D_{\zeta_1}) - \chi_4 \Delta_g(v_1^\sharp \circ D_{\zeta_1})) \circ D_{\zeta_1}^{-1}) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}
\end{multline*}
Define
\[ \mathcal{F}_3(v, \zeta) \triangleq \varepsilon^2 (\Delta_g ((\chi_4 v) \circ D_\zeta) - \chi_4 \Delta_g (v \circ D_\zeta)) \circ D_{\zeta}^{-1}, \]
which is, once again, viewed as a map $\mathcal{F}_3 : C^{2,\alpha}_\varepsilon(\Omega) \times C^{2,\alpha}_\varepsilon(\Sigma) \to C^{0,\alpha}_\varepsilon(\Omega)$, is a smooth nonlinear Banach space functional. We can then write
\begin{align*}
& \varepsilon^2 ((\Delta_g ((\chi_4 v_2^\sharp) \circ D_{\zeta_2}) - \chi_4 \Delta_g(v_2^\sharp \circ D_{\zeta_2})) \circ D_{\zeta_2}^{-1}) - (\Delta_g ((\chi_4 v_1^\sharp) \circ D_{\zeta_1}) - \chi_4 \Delta_g(v_1^\sharp \circ D_{\zeta_1})) \circ D_{\zeta_1}^{-1}) \\
& \qquad = \mathcal{F}_3(v_2^\sharp, \zeta_2) - \mathcal{F}_3(v_1^\sharp, \zeta_1) \\
& \qquad = \int_0^1 \langle D_v \mathcal{F}_3(v_1^\sharp + t(v_2^\sharp - v_1^\sharp), \zeta_1 + t(\zeta_2 - \zeta_1)), v_2^\sharp - v_1^\sharp \rangle \\
& \qquad \qquad \qquad + \langle D_\zeta \mathcal{F}_3(v_1^\sharp + t(v_2^\sharp - v_1^\sharp), \zeta_1 + t(\zeta_2 - \zeta_1)), \zeta_2 - \zeta_1 \rangle \, dt.
\end{align*}
The second term can be estimated by using the linearity in $v$ of $\mathcal{F}_3$ (and thus of $D_\zeta \mathcal{F}_3$), and Remark \ref{rema:dirichlet.data.regularity.product.vs.omega} to give:
\begin{align*}
& \Vert \langle D_\zeta \mathcal{F}_3(v_1^\sharp + t(v_2^\sharp - v_1^\sharp), \zeta_1 + t(\zeta_2 - \zeta_1)), \zeta_2 - \zeta_1 \rangle \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \\
& \qquad \leq C (\Vert v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert v_2^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})}) \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}_\varepsilon(\Sigma)} \leq C \varepsilon^2 \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)},
\end{align*}
which is of the desired form with $\delta = 2$.
The first term instead requires that we use the product rule on $\mathcal{F}_3$ to recast it as
\begin{equation*}
\mathcal{F}_3(v, \zeta) = \varepsilon^2 (2 \langle \nabla_g (\chi_4 \circ D_\zeta), \nabla_g (v \circ D_\zeta) \rangle
+ (\Delta_g(\chi_4 \circ D_\zeta)) (v \circ D_\zeta)) \circ D_{\zeta}^{-1},
\end{equation*}
which can, in turn, be differentiated in $v$ to give
\begin{equation*}
\langle D_v \mathcal{F}_3(v, \zeta), w \rangle = \varepsilon^2 (2 \langle \nabla_g (\chi_4 \circ D_\zeta), \nabla_g (w \circ D_\zeta) \rangle
+ (\Delta_g(\chi_4 \circ D_\zeta)) (w \circ D_\zeta)) \circ D_{\zeta}^{-1}.
\end{equation*}
At this point, we note that there are no zero-order $\chi_4$'s remaining, so we use Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, \eqref{eq:dirichlet.data.cutoff}, $\delta_* \in (0,1)$ to get
\begin{align*}
& \Vert \langle D_v \mathcal{F}_3(v_1^\sharp + t(v_2^\sharp - v_1^\sharp), \zeta_1 + t(\zeta_2 - \zeta_1)), v_2^\sharp - v_1^\sharp \rangle \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}\\
& \qquad \leq C \varepsilon^{1-\delta_*} \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{1,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \leq C \varepsilon^{1-\delta_*} \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})}.
\end{align*}
Summarizing, we have shown that
\begin{multline} \label{eq:dirichlet.data.lemm.3.9.laplace.chi.commutator}
\Vert \varepsilon^2 ((\Delta_g ((\chi_4 v_2^\sharp) \circ D_{\zeta_2}) - \chi_4 \Delta_g(v_2^\sharp \circ D_{\zeta_2})) \circ D_{\zeta_2}^{-1})
- (\Delta_g ((\chi_4 v_1^\sharp) \circ D_{\zeta_1}) - \chi_4 \Delta_g(v_1^\sharp \circ D_{\zeta_1})) \circ D_{\zeta_1}^{-1}) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \\
\leq C \varepsilon^{1-\delta_*} ( \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}).
\end{multline}
The contraction estimate on $N_\varepsilon$, \eqref{eq:dirichlet.data.lemm.3.9.n}, now follows from \eqref{eq:dirichlet.data.lemm.3.9.laplace.zeta.commutator}, \eqref{eq:dirichlet.data.lemm.3.9.H.decay}, \eqref{eq:dirichlet.data.lemm.3.9.Eeps.n}, \eqref{eq:dirichlet.data.lemm.3.9.Qeps}, and \eqref{eq:dirichlet.data.lemm.3.9.laplace.chi.commutator}.
We move on to the contraction estimates on $M_\varepsilon$, \eqref{eq:dirichlet.data.lemm.3.9.m.perp} and \eqref{eq:dirichlet.data.lemm.3.9.m}. Before we derive those two precise estimates, we investigate several of the easier terms in $M_{\varepsilon}(v_{2}^{\flat},v_{2}^{\sharp},\zeta_{2})-M_{\varepsilon}(v_{1}^{\flat},v_{1}^{\sharp},\zeta_{1})$.
We note, right away, that we've already shown in \eqref{eq:dirichlet.data.lemm.3.9.laplace.zeta.commutator}:
\begin{multline*}
\Vert \varepsilon^2 (\Delta_g (v_2^\flat \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g v_2^\flat)
- (\Delta_g (v_1^\flat \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1} - \Delta_g v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \\
\leq C \varepsilon^{2-2\alpha} \Vert v_2^\flat - v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + C \varepsilon^2 \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}.
\end{multline*}
In particular, Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, Lemma \ref{lemm:dirichlet.data.proj.holder.norms}, and $\Vert \cdot \Vert_{C^{0,\alpha}(\Sigma)} \leq \varepsilon^{-\alpha} \Vert \cdot \Vert_{C^{0,\alpha}_\varepsilon(\Sigma)}$ imply
\begin{align} \label{eq:dirichlet.data.lemm.3.9.zeta.commutator.m}
& \varepsilon^{\alpha} \Big\Vert \Pi_\varepsilon \Big[ \varepsilon^2 (\Delta_g (v_2^\flat \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g v_2^\flat) - (\Delta_g (v_1^\flat \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1} - \Delta_g v_1^\flat) \Big] \Big\Vert_{C^{0,\alpha}(\Sigma)} \nonumber \\
& + \Big\Vert \Pi_\varepsilon^\perp \Big[ \varepsilon^2 (\Delta_g (v_2^\flat \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g v_2^\flat) - (\Delta_g (v_1^\flat \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1} - \Delta_g v_1^\flat) \Big] \Big\Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \nonumber \\
& \qquad \leq C \varepsilon^{2-2\alpha} \Vert v_2^\flat - v_1^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)} + C \varepsilon^2 \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}.
\end{align}
Next, from Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, \eqref{eq:dirichlet.data.lemm.3.9.Qeps}, we conclude
\begin{align} \label{eq:dirichlet.data.lemm.3.9.Qeps.m}
& \varepsilon^{\alpha} \Big\Vert \Pi_\varepsilon \big[ Q_\varepsilon(\chi_4 v_2^\sharp + v_2^\flat) - Q_\varepsilon(\chi_4 v_1^\sharp + v_1^\flat) \big] \Big\Vert_{C^{0,\alpha}(\Sigma)} + \Big\Vert \Pi_\varepsilon^\perp \big[ Q_\varepsilon(\chi_4 v_2^\sharp + v_2^\flat) - Q_\varepsilon(\chi_4 v_1^\sharp + v_1^\flat) \big] \Big\Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \nonumber \\
& \qquad \qquad \leq C \varepsilon^2 (\Vert v_2^\sharp - v_1^\sharp \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert v_2^\flat - v_1^\flat \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}).
\end{align}
Next, we estimate
\[ \Vert (W''(\mathbb{H}_\varepsilon) - W''(\pm 1))(v_2^\flat - v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)}. \]
This is the only time we will use $\Vert \cdot \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)}$ for the purposes of \eqref{eq:dirichlet.data.lemm.3.9.m.perp}. We have
\begin{align*}
& \Vert (W''(\mathbb{H}_\varepsilon) - W''(\pm 1))(v_2^\flat - v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \\
& \qquad \leq \Vert (W''(\mathbb{H}_\varepsilon) - W''(\pm 1)) \chi_5(v_2^\flat - v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} + \Vert (W''(\mathbb{H}_\varepsilon) - W''(\pm 1)) (1-\chi_5) (v_2^\flat - v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \\
& \qquad \leq \varepsilon^2 \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + \Vert (W''(\mathbb{H}_\varepsilon) - W''(\pm 1)) (1-\chi_5) (v_2^\flat - v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega \setminus \Omega_5)}.
\end{align*}
Recalling $\Vert W''(\mathbb{H}_\varepsilon) - W''(\pm 1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega \setminus \Omega_5)} \leq C_m \varepsilon^m$ for all $m \in \mathbf{N}$, e.g., as in \eqref{eq:dirichlet.data.lemm.3.9.H.decay}, we deduce
\[ \Vert (W''(\mathbb{H}_\varepsilon) - W''(\pm 1))(v_2^\flat - v_1^\flat) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^2 \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)}, \]
so, combined with Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, Lemma \ref{lemm:dirichlet.data.proj.holder.norms}, \eqref{eq:dirichlet.data.cutoff}, $\delta_* \in (0, 1)$, this gives:
\begin{multline} \label{eq:dirichlet.data.lemm.3.9.WH}
\varepsilon^{\alpha} \Vert \Pi_\varepsilon \big[ \chi_3 (W''(\mathbb{H}_\varepsilon) - W''(\pm 1))(v_2^\flat - v_1^\flat) \big] \Vert_{C^{0,\alpha}(\Sigma)}
+ \Vert \Pi_\varepsilon^\perp \big[ \chi_3 (W''(\mathbb{H}_\varepsilon) - W''(\pm 1))(v_2^\flat - v_1^\flat) \big] \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \\
\leq C \varepsilon^2 \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)}.
\end{multline}
We now proceed to the more involved contraction estimates pertaining to $M_\varepsilon$. We will estimate:
\begin{multline} \label{eq:dirichlet.data.lemm.3.9.Leps}
\Vert (L_\varepsilon v_2^\sharp - \varepsilon^2 \Delta_g (v_2^\sharp \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} + W''(\mathbb{H}_\varepsilon) v_2^\sharp) \\
- (L_\varepsilon v_1^\sharp - \varepsilon^2 \Delta_g (v_1^\sharp \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1} + W''(\mathbb{H}_\varepsilon) v_1^\sharp) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)}.
\end{multline}
Note that, by repeating the argument carried out to obtain \eqref{eq:dirichlet.data.lemm.3.9.laplace.zeta.commutator}, except with $v_j^\sharp$ in place of $v_j^\flat$, and also using Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, we get
\begin{align}
& \Vert \varepsilon^2 ((\Delta_g (v_2^\sharp \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1} - \Delta_g v_2^\sharp) - (\Delta_g (v_1^\sharp \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1} - \Delta_g v_1^\sharp)) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \nonumber \\
& \qquad \leq C \varepsilon^{2-2\alpha} \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + C \varepsilon^2 \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}. \label{eq:dirichlet.data.lemm.3.9.laplace.zeta.commutator.m}
\end{align}
In view of Remark \ref{rema:dirichlet.data.regularity.product.vs.omega} and Lemma \ref{lemm:dirichlet.data.proj.holder.norms}, this allows us to estimate
\begin{align*}
& (L_\varepsilon v_2^\sharp - \varepsilon^2 \Delta_g v_2^\sharp + W''(\mathbb{H}_\varepsilon) v_2^\sharp) - (L_\varepsilon v_1^\sharp - \varepsilon^2 \Delta_g v_1^\sharp + W''(\mathbb{H}_\varepsilon) v_1^\sharp) \\
& \qquad = L_\varepsilon (v_2^\sharp - v_1^\sharp) - \varepsilon^2 \Delta_g (v_2^\sharp - v_1^\sharp) + W''(\mathbb{H}_\varepsilon)(v_2^\sharp - v_1^\sharp) \\
& \qquad = \varepsilon^2 (\Delta_{g_0} + \partial_z^2 - \Delta_g) (v_2^\sharp - v_1^\sharp)
\end{align*}
instead of \eqref{eq:dirichlet.data.lemm.3.9.Leps} in both \eqref{eq:dirichlet.data.lemm.3.9.m.perp} and \eqref{eq:dirichlet.data.lemm.3.9.m}. Let's denote
\[ \mathcal{F}_4(v) \triangleq \varepsilon^2 (\Delta_g - \Delta_{g_0} - \partial_z^2) v, \]
which is evidently a linear functional $\mathcal{F}_4 : C^{2,\alpha}_\varepsilon(\Omega_3) \to C^{0,\alpha}_\varepsilon(\Omega_3)$. Because $\Delta_g = \Delta_{g_z} + \partial_z^2 + H_z \partial_z$ in Fermi coordinates, we can rewrite
\[ \mathcal{F}_4(v) = \varepsilon^2 (\Delta_{g_z} - \Delta_{g_0}) v + \varepsilon^2 H_z \partial_z v. \]
We now make use of \eqref{eq:mean.curv.ddt.laplace} to write:
\[ \mathcal{F}_4(v) = \Big[ - \varepsilon^2 \int_0^z (2 \langle \sff_t, \nabla^2_{g_t} v \rangle_{g_t} + \langle \nabla_{g_t} H_t, \nabla_{g_t} v \rangle_{g_t}) \, dt \Big] + \varepsilon^2 H_z \partial_z v. \]
First, let's derive $C^0$ bounds. Let $(y, z) \in \Omega_3$. It follows from \eqref{eq:dirichlet.data.sigma.c2alpha}, \eqref{eq:mean.curv.ddt.metric}, \eqref{eq:mean.curv.ddt.sff}, and \eqref{eq:mean.curv.ddt.hess} that
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.c0.i}
\left| 2 \varepsilon^2 \int_0^z \langle \sff_t, \nabla^2_{g_t} v \rangle_{g_t} \, dt \right| \leq C |z| \Vert v \Vert_{C^2_\varepsilon(\Omega_3)}.
\end{equation}
It follows from \eqref{eq:dirichlet.data.sigma.c3alpha}, \eqref{eq:mean.curv.ddt.metric}, \eqref{eq:mean.curv.ddt.sff}, \eqref{eq:mean.curv.ddt.grad}, and \eqref{eq:mean.curv.ddt.gradsff} that
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.c0.ii}
\left| \varepsilon^2 \int_0^z \langle \nabla_{g_t} H_t, \nabla_{g_t} v \rangle_{g_t} \, dt \right| \leq C |z| \Vert v \Vert_{C^1_\varepsilon(\Omega_3)}.
\end{equation}
It follows from \eqref{eq:dirichlet.data.sigma.minimal}, \eqref{eq:dirichlet.data.sigma.c2alpha}, \eqref{eq:mean.curv.ddt.sff} that
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.c0.iii}
|\varepsilon^2 H_z \partial_z v| \leq C \varepsilon |z| \Vert v \Vert_{C^1_\varepsilon(\Omega_3)}.
\end{equation}
Altogether, \eqref{eq:dirichlet.data.lemm.3.9.F4.c0.i}-\eqref{eq:dirichlet.data.lemm.3.9.F4.c0.iii},
show:
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.c0}
|\mathcal{F}_4(v)| \leq C |z| \Vert v \Vert_{C^{2,\alpha}_\varepsilon(\Omega_3)} \text{ on } \Omega_3.
\end{equation}
Next, let's derive H\"older bounds. For fixed $z \in \Omega_3$, an analogous argument gives
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.calpha.y}
\varepsilon^\alpha [ y \mapsto \mathcal{F}_4(v)(y,z) ]_\alpha \leq C |z| \Vert v \Vert_{C^{2,\alpha}_\varepsilon(\Omega_3)}.
\end{equation}
Now fix $y$. By \eqref{eq:dirichlet.data.sigma.c2alpha}, \eqref{eq:mean.curv.ddt.metric}, \eqref{eq:mean.curv.ddt.sff}, and \eqref{eq:mean.curv.ddt.hess}, we have the \emph{Lipschitz} bound
\[ \left| \frac{\partial}{\partial z} \left( 2 \varepsilon^2 \int_0^z \langle \sff_t, \nabla^2_{g_t} v \rangle_{g_t} \, dt \right) \right| \leq C \Vert v \Vert_{C^2_\varepsilon(\Omega_3)} \]
In view of the a priori height bound $|z| \leq \varepsilon^{\delta_*}$, this trivially implies the H\"older bound
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.calpha.z.i}
\varepsilon^\alpha \left[ z \mapsto \varepsilon^2 \int_0^z \langle \sff_t, \nabla^2_{g_t} v \rangle_{g_t} \, dt \right]_\alpha \leq C \varepsilon^\alpha \varepsilon^{\delta_*(1-\alpha)} \Vert v \Vert_{C^2_\varepsilon(\Omega_3)}.
\end{equation}
By \eqref{eq:dirichlet.data.sigma.c3alpha}, \eqref{eq:mean.curv.ddt.metric}, \eqref{eq:mean.curv.ddt.sff}, \eqref{eq:mean.curv.ddt.grad}, and \eqref{eq:mean.curv.ddt.gradsff}, we have another Lipschitz bound:
\begin{equation}
\left| \frac{\partial}{\partial z} \left( \varepsilon^2 \int_0^z \langle \nabla_{g_t} H_t, \nabla_{g_t} v \rangle_{g_t} \, dt \right) \right| \leq C \Vert v \Vert_{C^1_\varepsilon(\Omega_3)},
\end{equation}
which, again by $|z| \leq \varepsilon^{\delta_*}$, implies
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.calpha.z.ii}
\varepsilon^\alpha \left[ z \mapsto \varepsilon^2 \int_0^z \langle \nabla_{g_t} H_t, \nabla_{g_t} v \rangle_{g_t} \, dt \right]_\alpha \leq C \varepsilon^\alpha \varepsilon^{\delta_*(1-\alpha)} \Vert v \Vert_{C^1_\varepsilon(\Omega_3)}.
\end{equation}
Finally, from \eqref{eq:mean.curv.ddt.h} we have the Lipschitz bound
\[ \left| \frac{\partial}{\partial z} (\varepsilon^2 H_z \partial_z v) \right| \leq C \Vert v \Vert_{C^2_\varepsilon(\Omega_3)}, \]
which, again by $|z| \leq \varepsilon^{\delta_*}$, improves to
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.calpha.z.iii}
\varepsilon^{\alpha} [ z \mapsto \varepsilon^2 H_z \partial_z v ]_\alpha \leq C \varepsilon^\alpha \varepsilon^{\delta_*(1-\alpha)} \Vert v \Vert_{C^2_\varepsilon(\Omega_3)}.
\end{equation}
Altogether, \eqref{eq:dirichlet.data.lemm.3.9.F4.calpha.z.i}-\eqref{eq:dirichlet.data.lemm.3.9.F4.calpha.z.iii} imply
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.calpha.z}
\varepsilon^\alpha [ z \mapsto \mathcal{F}_4(v)(y,z) ]_\alpha \leq C \varepsilon^{\delta_* + \alpha(1-\delta_*)} \Vert v \Vert_{C^2_\varepsilon(\Omega_3)}.
\end{equation}
Together, {\eqref{eq:dirichlet.data.lemm.3.9.F4.c0},} \eqref{eq:dirichlet.data.lemm.3.9.F4.calpha.y}, \eqref{eq:dirichlet.data.lemm.3.9.F4.calpha.z} imply
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4}
\Vert \mathcal{F}_4(v) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^{\delta_*} \Vert v \Vert_{C^{2,\alpha}_\varepsilon(\Omega_3)}.
\end{equation}
Together with Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, Lemma \ref{lemm:dirichlet.data.proj.holder.norms}, this gives:
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.proj.perp}
\Vert \Pi_\varepsilon^\perp \mathcal{F}_4(v) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^{\delta_*} \Vert v \Vert_{C^{2,\alpha}_\varepsilon(\Omega_3)}.
\end{equation}
It remains to estimate $\Pi_\varepsilon \mathcal{F}_4 (v)$. Note the obvious inequality (which follows from \eqref{eq:heteroclinic.expansion.ii}, \eqref{eq:heteroclinic.eps})
\[ \int_{-\infty}^\infty |z| \partial_z \mathbb{H}_\varepsilon(z) \, dz = \varepsilon \int_{-\infty}^\infty |t| \mathbb{H}'(t) \, dt \leq C \varepsilon \]
combined with \eqref{eq:dirichlet.data.lemm.3.9.F4.c0} and \eqref{eq:dirichlet.data.lemm.3.9.F4.calpha.y}, readily implies:
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F4.proj}
\Vert \Pi_\varepsilon \mathcal{F}_4(v) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma)} \leq C \varepsilon \Vert v \Vert_{C^{2,\alpha}_\varepsilon(\Omega_3)}
\implies \Vert \Pi_\varepsilon \mathcal{F}_4(v) \Vert_{C^{0,\alpha}(\Sigma)} \leq C \varepsilon^{1-\alpha} \Vert v \Vert_{C^{2,\alpha}_\varepsilon(\Omega_3)}.
\end{equation}
This completes our study of $\mathcal{F}_4$, as we have the desired estimates in view of Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}.
We proceed to the final contraction estimate pertaining to $M_\varepsilon$, which involves $\Pi_\varepsilon$, $\Pi_\varepsilon^\perp$ of
\[ \chi_3(\mathscr{E}_\varepsilon(\zeta_2) - \mathscr{E}_\varepsilon(\zeta_1) - \varepsilon^2 J_\Sigma (\zeta_2 - \zeta_2) \partial_z \mathbb{H}_\varepsilon). \]
By \eqref{eq:dirichlet.data.cutoff}, $\delta_* \in (0,1)$, and Lemma \ref{lemm:dirichlet.data.proj.holder.norms}, we may just estimate $\mathscr{E}_\varepsilon(\zeta_2) - \mathscr{E}_\varepsilon(\zeta_1) - \varepsilon^2 J_\Sigma (\zeta_2 - \zeta_2) \partial_z \mathbb{H}_\varepsilon$ on $\Omega_3$.
Fix $(y, z) \in \Omega_3$. Recall the definition of $D_\zeta$ in \eqref{eq:dirichlet.data.offset.map}, and the estimate
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.buffer}
\dist_g( \Omega_3, \{ \chi_2 \neq 1 \} ) = O(\varepsilon^{\delta_*}) \gg \Vert \zeta_2 \Vert_{C^0(\Sigma)} + \Vert \zeta_1 \Vert_{C^0(\Sigma)}
\end{equation}
that follows from the a priori bound on $\zeta_1$, $\zeta_2$. Also recall that $\mathbb{H}_\varepsilon \equiv \widetilde{\mathbb{H}}_\varepsilon$ on $\Omega_3$. Then, in Fermi coordinates $(y,z)$, we have:
\begin{align} \label{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.expr}
& \mathscr{E}_\varepsilon(\zeta_2)(y,z) - \mathscr{E}_\varepsilon(\zeta_1)(y,z) - \varepsilon^2 J_\Sigma (\zeta_2 - \zeta_1)(y) \cdot \partial_z \mathbb{H}_\varepsilon(z) \\
& \qquad = \varepsilon^2 \Delta_g (\mathbb{H}_\varepsilon \circ D_{\zeta_2}) \circ D_{\zeta_2}^{-1}(y,z) - \varepsilon^2 \Delta_g (\mathbb{H}_\varepsilon \circ D_{\zeta_1}) \circ D_{\zeta_1}^{-1}(y,z) \nonumber \\
& \qquad \qquad - \varepsilon^2 J_\Sigma(\zeta_2 - \zeta_1)(y) \cdot \partial_z \mathbb{H}_\varepsilon(z) \nonumber \\
& \qquad = \varepsilon^2 \Big[ \partial_z^2 \mathbb{H}_\varepsilon(z) \big( |\nabla_{g_{z+\zeta_2(y)}} \zeta_2(y)|^2 - |\nabla_{g_{z+\zeta_1(y)}} \zeta_1(y)|^2 \big) \nonumber \\
& \qquad \qquad - \partial_z \mathbb{H}_\varepsilon(z) \big( (\Delta_{g_{z+\zeta_2(y)}} \zeta_2(y) - H_{z+\zeta_2(y)}(y)) - (\Delta_{g_{z+\zeta_1(y)}} \zeta_1(y) - H_{z+\zeta_1(y)}(y)) + J_\Sigma (\zeta_2 - \zeta_1)(y) \big) \Big]. \nonumber
\end{align}
Denote, for $\zeta \in C^{1,\alpha}(\Sigma)$,
\[ \mathcal{F}_5(\zeta)(y,z) \triangleq |\nabla_{g_{z + \zeta(y)}} \zeta(y)|^2 = g_{z + \zeta(y)}^{ij} \zeta_i(y) \zeta_j(y) \]
to be the smooth nonlinear functional, $\mathcal{F}_5 : C^{1,\alpha}(\Sigma) \to C^{0,\alpha}(\Omega_3)$. By virtue of \eqref{eq:mean.curv.ddt.metric}, we know that:
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F5.derivative}
\langle D_\zeta \mathcal{F}_5(\zeta), w \rangle(y,z) = - 2 \sff_{z+\zeta(y)}^{ij} \zeta_i(y) \zeta_j(y) w(y) + 2 g^{ij}_{z + \zeta(y)} w_i(y) \zeta_j(y).
\end{equation}
By the fundamental theorem of calculus,
\[ \mathcal{F}_5(\zeta_2) - \mathcal{F}_5(\zeta_1) = \int_0^1 \langle D_\zeta \mathcal{F}_5( \zeta_1 + t (\zeta_2 - \zeta_1)), \zeta_2 - \zeta_1 \rangle \, dt, \]
so together with \eqref{eq:dirichlet.data.sigma.c2alpha}, the a priori estimates on $\zeta_1$, $\zeta_2$, \eqref{eq:dirichlet.data.lemm.3.9.F5.derivative}, \eqref{eq:mean.curv.ddt.metric}, and \eqref{eq:mean.curv.ddt.sff}:
\[ \Vert \mathcal{F}_5(\zeta_2) - \mathcal{F}_5(\zeta_1) \Vert_{C^{0,\alpha}(\Omega_3)} \leq C \varepsilon^{2-2\alpha} \Vert \zeta_2 - \zeta_1 \Vert_{C^{1,\alpha}(\Sigma)}. \]
Alongside \eqref{eq:heteroclinic.expansion.iii}, Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, Lemma \ref{lemm:dirichlet.data.proj.holder.norms}, \eqref{eq:dirichlet.data.cutoff}, $\delta_* \in (0,1)$, this implies:
\begin{multline} \label{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.i}
\varepsilon^{\alpha} \left\Vert \Pi_\varepsilon \big( \chi_3 \varepsilon^2 (\partial_z^2 \mathbb{H}_\varepsilon) (\mathcal{F}_5(\zeta_2) - \mathcal{F}_5(\zeta_1)) \big) \right\Vert_{C^{0,\alpha}(\Sigma)} + \left\Vert \Pi_\varepsilon^\perp \big( \chi_3 \varepsilon^2 (\partial_z^2 \mathbb{H}_\varepsilon) (\mathcal{F}_5(\zeta_2) - \mathcal{F}_5(\zeta_1)) \big) \right\Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \\
\leq C \varepsilon^{2-2\alpha} \Vert \zeta_2 - \zeta_1 \Vert_{C^{1,\alpha}(\Sigma)}.
\end{multline}
Finally, let's denote
\[ \mathcal{F}_6(\zeta)(y,z) \triangleq \varepsilon \Big( \Delta_{z + \zeta(y)} \zeta(y) - H_{z + \zeta(y)}(y) + J_\Sigma \zeta(y) \Big) \]
to be the smooth nonlinear Banach space functional $\mathcal{F}_6 : C^{2,\alpha}(\Sigma) \to C^{0,\alpha}_\varepsilon(\Omega_3)$. By \eqref{eq:mean.curv.ddt.h} and \eqref{eq:mean.curv.ddt.laplace},
\begin{align*}
\langle D_\zeta \mathcal{F}_6(\zeta), w \rangle & = \varepsilon \Big( \Delta_{z + \zeta} w + \big( -2 \langle \sff_{z+\zeta}, \nabla^2_{g_{z+\zeta}} \zeta \rangle_{g_{z+\zeta}} - \langle \nabla_{g_{z+\zeta}} H_{z+\zeta}, \nabla_{g_{z+\zeta}} \zeta \rangle_{g_{z+\zeta}} \big) w \\
& \qquad + (|\sff_{z+\zeta}|^2 + \ricc_g(\partial_z, \partial_z)|_{D \times \{z+\zeta\}}) w + J_\Sigma w \Big) \\
& = \varepsilon \Big( \big( -2 \langle \sff_{z+\zeta}, \nabla^2_{g_{z+\zeta}} \zeta \rangle_{g_{z+\zeta}} - \langle \nabla_{g_{z+\zeta}} H_{z+\zeta}, \nabla_{g_{z+\zeta}} \zeta \rangle_{g_{z+\zeta}} \big) w \\
& \qquad - \int_{0}^{z+\zeta} ( 2 \langle \sff_t, \nabla^2_{g_t} w \rangle_{g_t} + \langle \nabla_{g_t} H_t, \nabla_{g_t} w \rangle_{g_t} ) \, dt \\
& \qquad + \Big( \int_{0}^{z+\zeta} \tfrac{\partial}{\partial t} (|\sff_t|^2 + \ricc_g(\partial_z, \partial_z)|_{D \times \{t\}}) \, dt \Big) w\Big).
\end{align*}
By the fundamental theorem of calculus,
\[ \mathcal{F}_6(\zeta_2) - \mathcal{F}_6(\zeta_1) = \int_0^1 \langle D_\zeta \mathcal{F}_6(\zeta_1 + t(\zeta_2 - \zeta_1)), \zeta_2 - \zeta_1 \rangle \, dt. \]
We now estimate $\langle D_\zeta \mathcal{F}_6(\zeta), w \rangle$ for $\zeta = \zeta_1 + t(\zeta_2 - \zeta_1)$ and $w = \zeta_2 - \zeta_1$. We will make repeated use of \eqref{eq:dirichlet.data.sigma.c2alpha}, \eqref{eq:dirichlet.data.sigma.c3alpha}, \eqref{eq:mean.curv.ddt.metric}, \eqref{eq:mean.curv.ddt.sff}, \eqref{eq:mean.curv.ddt.grad}, \eqref{eq:mean.curv.ddt.christoffel}, \eqref{eq:mean.curv.ddt.hess}, $\Vert \zeta \Vert_{C^{2,\alpha}(\Sigma)} \leq C' \varepsilon^{2-2\alpha}$, and $\Vert \cdot \Vert_{C^{0,\alpha}_\varepsilon(\Sigma)} \leq \Vert \cdot \Vert_{C^{2,\alpha}(\Sigma)}$. First,
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F6.i}
\left\Vert \varepsilon \big( 2 \langle \sff_{z+\zeta}, \nabla^2_{g_{z+\zeta}} \zeta \rangle_{g_{z+\zeta}} + \langle \nabla_{g_{z+\zeta}} H_{z+\zeta}, \nabla_{g_{z+\zeta}} \zeta \rangle_{g_{z+\zeta}} \big) w \right\Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^{2-2\alpha} \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}.
\end{equation}
Additionally using the $O(\varepsilon^{\delta_*})$ height bound on $\Omega_3$, we also have:
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F6.ii}
\left\Vert \varepsilon \int_0^{z+\zeta} \langle \sff_{z+\zeta}, \nabla^2_{g_{z+\zeta}} w \rangle_{g_{z+\zeta}} \right\Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^{1+\delta_*} \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}.
\end{equation}
Likewise:
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F6.iii}
\left\Vert \varepsilon \Big( \int_{0}^{z+\zeta} \tfrac{\partial}{\partial t} (|\sff_t|^2 + \ricc_g(\partial_z, \partial_z)|_{D \times \{t\}}) \, dt \Big) w \right\Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^{1+\delta_*} \Vert \zeta_2 - \zeta_1 \Vert_{C^{0,\alpha}(\Sigma)}.
\end{equation}
It remains to estimate:
\[ \left\Vert \varepsilon \int_0^{z+\zeta} \langle \nabla_{g_t} H_t, \nabla_{g_t} w \rangle_{g_t} \, dt \right\Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)}. \]
Now is the only place in the proof where we need to distinguish the H\"older exponents $\alpha \leq \theta$, taking the prior to be small and the latter to be large. From \eqref{eq:dirichlet.data.sigma.c2alpha}, \eqref{eq:dirichlet.data.sigma.c3alpha}, \eqref{eq:mean.curv.ddt.metric}, \eqref{eq:mean.curv.ddt.sff}, \eqref{eq:mean.curv.ddt.grad} and the interpolation of (unweighted) H\"older spaces $C^{1,\theta} \hookrightarrow C^{1,\alpha} \hookrightarrow C^{0,\theta}$ (Lemma \ref{lemm:holder.space.interpolation}), we have
\[ \Vert \nabla_{g_z} H_z \Vert_{C^{0,\alpha}(\Omega_3)} \leq C \Vert H_z \Vert_{C^{0,\theta}(\Omega)}^{\theta-\alpha} \Vert H_z \Vert_{C^{1,\theta}(\Omega)}^{1+\alpha-\theta} \leq C \varepsilon^{-2 (1+\alpha-\theta)} \leq C \varepsilon^{-\frac12 \delta_*}, \]
as long as $\alpha_0$, $\theta_0$ are chosen sufficiently close to $0$ and to $1$, respectively, depending on $\delta_*$. It is now easy to see, as before, that
\begin{equation} \label{eq:dirichlet.data.lemm.3.9.F6.iv}
\left \Vert \varepsilon \int_0^{z+\zeta} \langle \nabla_{g_t} H_t, \nabla_{g_t} w \rangle_{g_t} \, dt \right\Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^{1+\frac12 \delta_*} \Vert \zeta_2 - \zeta_1 \Vert_{C^{1,\alpha}(\Sigma)}.
\end{equation}
Altogether, \eqref{eq:dirichlet.data.lemm.3.9.F6.i}, \eqref{eq:dirichlet.data.lemm.3.9.F6.ii}, \eqref{eq:dirichlet.data.lemm.3.9.F6.iii}, \eqref{eq:dirichlet.data.lemm.3.9.F6.iv} imply:
\[ \Vert \mathcal{F}_6(\zeta_2) - \mathcal{F}_6(\zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega_3)} \leq C \varepsilon^{1+\frac12 \delta_*} \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}. \]
Alongside \eqref{eq:heteroclinic.expansion.ii}, Remark \ref{rema:dirichlet.data.regularity.product.vs.omega}, Lemma \ref{lemm:dirichlet.data.proj.holder.norms}, \eqref{eq:dirichlet.data.cutoff}, $\delta_* \in (0,1)$, this implies:
\begin{multline} \label{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.ii}
\varepsilon^{\alpha} \left\Vert \Pi_\varepsilon \big( \chi_3 \varepsilon (\partial_z \mathbb{H}_\varepsilon) (\mathcal{F}_6(\zeta_2) - \mathcal{F}_6(\zeta_1)) \big) \right\Vert_{C^{0,\alpha}(\Sigma)} + \left\Vert \Pi_\varepsilon^\perp \big( \chi_3 (\varepsilon \partial_z \mathbb{H}_\varepsilon) (\mathcal{F}_6(\zeta_2) - \mathcal{F}_6(\zeta_1)) \big) \right\Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \\
\leq C (\varepsilon^{2-2\alpha} + \varepsilon^{1+\frac12 \delta_*}) \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)}.
\end{multline}
Together, \eqref{eq:dirichlet.data.lemm.3.9.zeta.commutator.m}, \eqref{eq:dirichlet.data.lemm.3.9.Qeps.m}, \eqref{eq:dirichlet.data.lemm.3.9.WH}, \eqref{eq:dirichlet.data.lemm.3.9.laplace.zeta.commutator.m}, \eqref{eq:dirichlet.data.lemm.3.9.F4.proj.perp}, \eqref{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.expr}, \eqref{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.i}, and \eqref{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.ii} imply \eqref{eq:dirichlet.data.lemm.3.9.m.perp} for $\alpha_0$, $\theta_0$ depending on $\delta_*$.
Likewise, \eqref{eq:dirichlet.data.lemm.3.9.zeta.commutator.m}, \eqref{eq:dirichlet.data.lemm.3.9.Qeps.m}, \eqref{eq:dirichlet.data.lemm.3.9.WH}, \eqref{eq:dirichlet.data.lemm.3.9.laplace.zeta.commutator.m}, \eqref{eq:dirichlet.data.lemm.3.9.F4.proj}, \eqref{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.expr}, \eqref{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.i}, \eqref{eq:dirichlet.data.lemm.3.9.M.eps.E.eps.ii} imply \eqref{eq:dirichlet.data.lemm.3.9.m} for $\alpha_0$, $\theta_0$ depending on $\delta_*$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{theo:dirichlet.data.construction}]
As was already pointed out, we can rewrite \eqref{eq:dirichlet.data.pde} as the nonlinear fixed point problem \eqref{eq:dirichlet.data.pde.vsharp}-\eqref{eq:dirichlet.data.pde.zeta}. We'll take $\alpha$, $\theta$, $\delta$ as in Lemma \ref{lemm:dirichlet.data.lemm.3.9}, and $M \geq 1$.
Consider $g$ as in Section \ref{sec:dirichlet.data}, and also define
\begin{multline} \label{eq:dirichlet.data.contraction.interior.ball}
\mathcal{U}(\varepsilon; M) \triangleq \Big\{ (v^\flat, v^\sharp, \zeta) \in \widetilde{C}^{2,\alpha}_\varepsilon(\Omega) \times C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R}) \times C^{2,\alpha}(\Sigma) : \\
\Vert v^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + \Vert v^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \varepsilon^{2\alpha} \Vert \zeta \Vert_{C^{2,\alpha}(\Sigma)} \leq M \varepsilon^2 \Big\}.
\end{multline}
\begin{multline} \label{eq:dirichlet.data.contraction.boundary.ball}
\mathcal{B}(\varepsilon; \mu) \triangleq \Big\{ (\widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}) \in C^{2,\alpha}_\varepsilon(\partial \Omega) \times C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R}) \times C^{2,\alpha}(\partial \Sigma) : \\
\widehat{v}^\flat \equiv 0 \text{ on } \{ \chi_4 = 1 \}, \; \Pi_\varepsilon(\widehat{v}^\sharp) \equiv 0 \text{ on } \partial \Sigma, \\
\Vert \widehat{v}^\flat \Vert_{C^{2,\alpha}_\varepsilon(\partial \Omega)} + \Vert \widehat{v}^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R})} + \Vert \widehat{\zeta} \Vert_{C^{2,\alpha}(\partial \Sigma)} \leq \mu \varepsilon^2 \Big\}.
\end{multline}
Lemmas \ref{lemm:dirichlet.data.lemm.3.8}, \ref{lemm:dirichlet.data.lemm.3.9}, guarantee that for every $(v^\flat, v^\sharp, \zeta) \in \mathcal{U}(\varepsilon; M)$,
\begin{align}
\Vert N_\varepsilon(v^\flat, v^\sharp, \zeta) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)}
& \leq c_1' \varepsilon^{2+\delta-2\alpha} + c_0 \varepsilon^2, \label{eq:dirichlet.data.thm.est.n} \\
\Vert \Pi_\varepsilon^\perp M_\varepsilon (v^\flat, v^\sharp, \zeta) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})}
& \leq c_1' \varepsilon^{2+\delta-2\alpha} + c_0 \varepsilon^2, \label{eq:dirichlet.data.thm.est.m.perp} \\
\Vert \varepsilon^{-1} \Pi_\varepsilon M_\varepsilon(v^\flat, v^\sharp, \zeta) \Vert_{C^{0,\alpha}(\Sigma)}
& \leq c_1' \varepsilon^{2+\delta-2\alpha} + c_1' \varepsilon^{2-\alpha} + c_0 \varepsilon^2, \label{eq:dirichlet.data.thm.est.m}
\end{align}
with $c_0$ as in Lemma \ref{lemm:dirichlet.data.lemm.3.8}, and with $c_1' = M \cdot c_1$, $\varepsilon \leq \varepsilon_0$ as in Lemma \ref{lemm:dirichlet.data.lemm.3.9}.
Let
\[ \Phi : \mathcal{U}(\varepsilon; M) \times \mathcal{B}(\varepsilon; \mu) \times \operatorname{Met}_{\varepsilon,\eta}(\Omega) \to \widetilde{C}^{2,\alpha}_\varepsilon(\Omega) \times C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R}) \times C^{2,\alpha}(\Sigma), \]
be the solution map $\Phi : (v^\flat, v^\sharp, \zeta, \widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}, g) \mapsto (V^\flat, V^\sharp, Z)$ for the \emph{linear} system
\begin{equation} \label{eq:dirichlet.data.thm.pde.vflat}
\mathcal{L}_\varepsilon V^\flat = N_\varepsilon(v^\flat, v^\sharp, \zeta) \text{ on } \Omega, \; V^\flat|_{\partial \Omega} = \widehat{v}^\flat,
\end{equation}
\begin{equation} \label{eq:dirichlet.data.thm.pde.vsharp}
L_\varepsilon V^\sharp = \Pi_\varepsilon^\perp M_\varepsilon(v^\flat, v^\sharp, \zeta) \text{ on } \Sigma \times \mathbf{R}, \; V^\sharp|_{\partial \Sigma \times \mathbf{R}} = \widehat{v}^\sharp,
\end{equation}
\begin{equation} \label{eq:dirichlet.data.thm.pde.zeta}
J_\Sigma Z = \varepsilon^{-1} \Pi_\varepsilon M_\varepsilon(v^\flat, v^\sharp, \zeta) \text{ on } \Sigma, \; Z|_{\partial \Sigma} = \widehat{\zeta}.
\end{equation}
The existence of $V^\flat$ follows from Fredholm theory. In fact, together with Lemma \ref{lemm:dirichlet.data.eq.3.26}, \eqref{eq:dirichlet.data.thm.est.n}, we have
\begin{align}
\Vert V^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)}
& \leq C ( \Vert N_\varepsilon (v^\flat, v^\sharp, \zeta) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} + \Vert \widehat{v}^\flat \Vert_{C^{2,\alpha}_\varepsilon(\Omega)}) \leq C c_1' \varepsilon^{2+\delta-2\alpha} + C (c_0 + \mu) \varepsilon^2. \label{eq:dirichlet.data.thm.est.vflat}
\end{align}
The existence of $V^\sharp$ follows from Lemma \ref{lemm:dirichlet.data.prop.3.1}. In fact, together with Lemma \ref{lemm:dirichlet.data.prop.3.2}, \eqref{eq:dirichlet.data.thm.est.m.perp}, we have
\begin{equation} \label{eq:dirichlet.data.thm.est.vsharp}
\Vert V^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})}
\leq C ( \Vert \Pi_\varepsilon^\perp M_\varepsilon(v^\flat, v^\sharp, \zeta) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert \widehat{v}^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R})} \leq C c_1' \varepsilon^{2+\delta-2\alpha} + C (c_0 + \mu) \varepsilon^2.
\end{equation}
Finally, the existence of $Z$ follows from Fredholm theory and \eqref{eq:dirichlet.data.sigma.nondegenerate}. In fact, by Schauder theory on the elliptic operator $J_\Sigma$ on $\Sigma$, and \eqref{eq:dirichlet.data.thm.est.m}, we find:
\begin{align}
\Vert Z \Vert_{C^{2,\alpha}(\Sigma)}
& \leq C ( \Vert \varepsilon^{-1} \Pi_\varepsilon M_\varepsilon (v^\flat, v^\sharp, \zeta) \Vert_{C^{0,\alpha}(\Sigma)} + \Vert \widehat{\zeta} \Vert_{C^{2,\alpha}(\partial \Sigma)}) \nonumber \\
& \leq C c_1' \varepsilon^{2+\delta-2\alpha} + C c_1' \varepsilon^{2-\alpha} + C c_0 \varepsilon^2 + C \mu \varepsilon^{2-2\alpha}, \nonumber \\
\implies \varepsilon^{2\alpha} \Vert Z \Vert_{C^{2,\alpha}(\Sigma)} & \leq C c_1' \varepsilon^{2+\delta} + Cc_1' \varepsilon^{2+\alpha} + C c_0 \varepsilon^{2+2\alpha} + C \mu \varepsilon^2. \label{eq:dirichlet.data.thm.est.zeta}
\end{align}
We emphasize that the constant $C$ in \eqref{eq:dirichlet.data.thm.est.vflat}, \eqref{eq:dirichlet.data.thm.est.vsharp}, and \eqref{eq:dirichlet.data.thm.est.zeta} depends only on $n$, $\eta > 0$, and $W$.
The expressions in \eqref{eq:dirichlet.data.thm.est.vflat}, \eqref{eq:dirichlet.data.thm.est.vsharp}, and \eqref{eq:dirichlet.data.thm.est.zeta} can all be made to be $\leq \tfrac{1}{3} M \varepsilon^2$ as follows:
\begin{enumerate}
\item Choose $M$ large, depending on $c_0$, $C$, $\mu$, so that $C(c_0 + \mu) \leq \tfrac{1}{6} M$.
\item Then, choose $\varepsilon \leq \varepsilon_0$ small depending on $C$, $c_1'$, $M$, so that
\begin{equation} \label{eq:dirichlet.data.thm.contraction.constant}
C c_1' \varepsilon^{\alpha} \ll 1;
\end{equation}
note that, since $M \geq 1$, the left hand side is also $\leq \tfrac{1}{12} M$.
\item Using $\alpha \in (0, \tfrac{\delta}{3})$ we find that $\varepsilon^{\delta - 2\alpha} \leq \varepsilon^{\alpha}$, so $C c_1' \varepsilon^{\delta} \leq C c_1' \varepsilon^{\delta-2\alpha} \leq \tfrac{1}{12} M$.
\end{enumerate}
Thus, for such a choice of $M = M(n, \eta, W, \delta_*, \mu)$, $\varepsilon \leq \varepsilon_0' = \varepsilon_0'(n, \eta, W, \delta_*, \mu, \alpha)$, we have
\[ \Phi \big( \mathcal{U}(\varepsilon; M) \times \mathcal{B}(\varepsilon; \mu) \times \operatorname{Met}_{\varepsilon,\eta}(\Omega) \big) \subset \mathcal{U}(\varepsilon; M). \]
We show that $\Phi(\cdot, \cdot, \cdot, \widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}, g)$ is a \emph{contraction} with respect to the norm
\begin{equation} \label{eq:dirichlet.data.interior.product.norm}
\Vert (v^\flat, v^\sharp, \zeta) \Vert_{\mathcal{U}} \triangleq \Vert v^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + \Vert v^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \varepsilon^{2\alpha} \Vert \zeta \Vert_{C^{2,\alpha}(\Sigma)},
\end{equation}
\emph{uniformly} with respect to $\widehat{v}^\flat$, $\widehat{v}^\sharp, \widehat{\zeta}, g$. Let's also define
\begin{equation} \label{eq:dirichlet.data.boundary.product.norm}
\Vert (\widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}) \Vert_{\mathcal{B}} \triangleq \Vert \widehat{v}^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\partial \Omega)} + \Vert \widehat{v}^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\partial \Sigma \times \mathbf{R})} + \Vert \widehat{\zeta} \Vert_{C^{2,\alpha}(\partial \Sigma)}.
\end{equation}
Let's set
\[ (V_1^\flat, V_1^\sharp, Z_1) \triangleq \Phi(v_1^\flat, v_1^\sharp, \zeta_1, \widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}, g), \]
\[ (V_2^\flat, V_2^\sharp, Z_2) \triangleq \Phi(v_2^\flat, v_2^\sharp, \zeta_2, \widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}, g). \]
By Lemma \ref{lemm:dirichlet.data.eq.3.26}, Lemma \ref{lemm:dirichlet.data.lemm.3.9}:
\begin{align} \label{eq:dirichlet.data.thm.contraction.vflat}
\Vert V_2^\flat - V_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} & \leq C \Vert \mathcal{L}_\varepsilon V_2^\flat - \mathcal{L}_\varepsilon V_1^\flat \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \nonumber \\
& = C \Vert N_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - N_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Omega)} \nonumber \\
& \leq C c_1' \varepsilon^{\delta} \Big( \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}(\Omega)} + \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)} \Big).
\end{align}
By Lemma \ref{lemm:dirichlet.data.prop.3.2}, Lemma \ref{lemm:dirichlet.data.lemm.3.9},
\begin{align} \label{eq:dirichlet.data.thm.contraction.vsharp}
\Vert V_2^\sharp - V_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} & \leq C \Vert L_\varepsilon V_2^\sharp - L_\varepsilon V_1^\sharp \Vert_{C^{0,\alpha}_{\varepsilon}(\Sigma \times \mathbf{R})} \nonumber \\
& = C \Vert \Pi_\varepsilon^\perp M_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - \Pi_\varepsilon^\perp M_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{C^{0,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \nonumber \\
& \leq C c_1' \varepsilon^\delta \Big( \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} + \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)} \Big).
\end{align}
Finally, by Lemma \ref{lemm:dirichlet.data.lemm.3.9}, \eqref{eq:dirichlet.data.sigma.nondegenerate}, and Schauder theory,
\begin{align} \label{eq:dirichlet.data.thm.contraction.zeta}
\Vert Z_2 - Z_1 \Vert_{C^{2,\alpha}(\Sigma)} \nonumber & \leq C \Vert J_\Sigma Z_2 - J_\Sigma Z_1 \Vert_{C^{0,\alpha}(\Sigma)} \nonumber \\
& = C \Vert \varepsilon^{-1} \Pi_\varepsilon M_\varepsilon(v_2^\flat, v_2^\sharp, \zeta_2) - \varepsilon^{-1} \Pi_\varepsilon M_\varepsilon(v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{C^{0,\alpha}(\Sigma)} \nonumber \\
& \leq C c_1' \Big[ \varepsilon^\delta \big( \Vert v_2^\flat - v_1^\flat \Vert_{\widetilde{C}^{2,\alpha}_\varepsilon(\Omega)} + \Vert \zeta_2 - \zeta_1 \Vert_{C^{2,\alpha}(\Sigma)} \big) + \varepsilon^{-\alpha} \Vert v_2^\sharp - v_1^\sharp \Vert_{C^{2,\alpha}_\varepsilon(\Sigma \times \mathbf{R})} \Big].
\end{align}
Adding \eqref{eq:dirichlet.data.thm.contraction.vflat}, \eqref{eq:dirichlet.data.thm.contraction.vsharp}, and $\varepsilon^{2\alpha}$ times \eqref{eq:dirichlet.data.thm.contraction.zeta}, using $\alpha < \tfrac{1}{3} \delta$, and the $\Vert \cdot \Vert_{\mathcal{U}}$ norm on $\mathcal{U}(\varepsilon; M)$:
\begin{equation} \label{eq:dirichlet.data.thm.contraction}
\Vert (V_2^\flat, V_2^\sharp, Z_2) - (V_1^\flat, V_1^\sharp, Z_1) \Vert_{\mathcal{U}}
\leq C c_1' \varepsilon^\alpha \Vert (v_2^\flat, v_2^\sharp, \zeta_2) - (v_1^\flat, v_1^\sharp, \zeta_1) \Vert_{\mathcal{U}}.
\end{equation}
This implies that $\Phi(\cdot, \cdot, \cdot, \widehat{v}^\flat, \widehat{v}^\sharp, \widehat{\zeta}, g)$ is uniformly Lipschitz, with Lipschitz constant $\leq C c_1' \varepsilon^\alpha$, and by \eqref{eq:dirichlet.data.thm.contraction.constant} we conclude that it's, in fact, a contraction mapping. This readily implies the existence of a fixed point $(v^\flat, v^\sharp, \zeta)$, which therefore satisfies \eqref{eq:dirichlet.data.pde}.
We finally move to prove the continuity of the solution map
\[ \mathcal{S} : \mathcal{B}(\varepsilon; \mu) \times \operatorname{Met}_{\varepsilon,\eta}(\Omega) \to \mathcal{U}(\varepsilon; M). \]
For $(\widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1)$, $(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2) \in \mathcal{B}(\varepsilon; \mu) \times \operatorname{Met}_{\varepsilon,\eta}(\Omega)$, we have, by the fixed point property:
\begin{align*}
& \mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2) - \mathcal{S}(\widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1) \\
& \qquad = \Big( \Phi(\mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2), \widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2) - \Phi(\mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2), \widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1) \Big) \\
& \qquad \qquad - \Big( \Phi(\mathcal{S}(\widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1), \widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1) - \Phi(\mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2), \widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1) \Big).
\end{align*}
The last parenthesis will be bounded using the contraction mapping property \eqref{eq:dirichlet.data.thm.contraction} on $(\widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1)$. The second-to-last parenthesis will be bounded by varying the four slots of $\Phi(\mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2), \cdot, \cdot, \cdot, \cdot)$ using the fundamental theorem of calculus. The $\widehat{v}^\flat$, $\widehat{v}^\sharp$, $\widehat{\zeta}$ derivatives of $\Phi(\mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2), \cdot, \cdot, \cdot, \cdot)$ can be controlled using Lemma \ref{lemm:dirichlet.data.eq.3.26}, Lemma \ref{lemm:dirichlet.data.prop.3.1}, and Schauder theory on $J_\Sigma$, respectively. Likewise, it is not hard to see that for $g \in \operatorname{Met}_{\varepsilon,\eta}(\Omega)$, the map
\begin{equation*}
g\mapsto \Phi(v^{\flat},v^{\sharp},\zeta,\widehat{v}^{\flat}, \widehat{v}^{\sharp}, \widehat{\zeta},g)
\end{equation*}
is uniformly Lipschitz with respect to $(v^{\flat},v^{\sharp},\zeta,\widehat{v}^{\flat}, \widehat{v}^{\sharp}, \widehat{\zeta}) \in \mathcal{U}(\varepsilon; M) \times \mathcal{B}(\varepsilon; \mu)$. Altogether, we have
\begin{align*}
& \Vert \mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2) - \mathcal{S}(\widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1)) \Vert_{\mathcal{U}} \\
& \qquad \leq c \Big( \Vert (\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2) - (\widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1) \Vert_{\mathcal{B}} + d(g_2, g_1) \Big) + C c_1' \varepsilon^\alpha \Vert \mathcal{S}(\widehat{v}_2^\flat, \widehat{v}_2^\sharp, \widehat{\zeta}_2, g_2) - \mathcal{S}(\widehat{v}_1^\flat, \widehat{v}_1^\sharp, \widehat{\zeta}_1, g_1)) \Vert_{\mathcal{U}},
\end{align*}
and the result follows by rearranging.
\end{proof}
\subsection{Approximation by superimposed heteroclinics} \label{subsec:jacobi.toda.setup} \label{subsec:approximate.solutions}
In this section we follow Wang-Wei's \cite{WangWei} investigation of local properties of solutions to the Allen-Cahn equation,
\[ \varepsilon^2 \Delta_g u = W'(u), \]
whose nodal set $\{ u = 0 \}$ can be (locally) decomposed as a union of graphs over a fixed hypersurface (to be denoted $\Sigma$), whose height functions (to be denoted $f_1, \ldots, f_Q$) are bounded in $C^2$ and small in $C^1$. The ultimate goal is to deduce, in a quantitative sense, that the height functions approximately satisfy a Jacobi-Toda system.
The reason we rework the setup is twofold:
\begin{enumerate}
\item First, most of the analysis in \cite{WangWei} was performed in $\mathbf{R}^n$, while here we include the details necessary to handle the Riemannian setting (cf.\ \cite[Section 16]{WangWei}).
\item Secondly (and more fundamentally), we combine the argument from \cite{WangWei} with a further bootstrap argument based on improved error estimates. This allows us to prove much sharper separation estimates than were obtained in \cite{WangWei}. Indeed, we will show that the behavior of the transition layers is dominated by mean curvature, rather than interaction between the layers. This will be crucial for our subsequent applications in Section \ref{sec:bounded.index}.
\end{enumerate}
Let's set things up. Suppose that $D^{n-1}$ is an $(n-1)$-dimensional disk, over which we take a topological cylinder $\Omega \triangleq D \times [-1,1]$, whose coordinates we label $X = (y, z) \in D \times [-1,1]$. Consider a smooth metric $g$ on $\Omega$, which we assume to be in Fermi coordinate form with respect to $\Sigma$; in $(y, z)$ coordinates:
\[ g = g_z + dz^2. \]
For convenience, we denote $\Sigma \triangleq D \times \{0\} \subset \Omega$.
Let us require that
\begin{equation} \label{eq:sheets.sff.bound}
\sum_{\ell = 0}^3 |\nabla_\Sigma^\ell \sff_\Sigma| \leq \eta.
\end{equation}
We additionally assume that $\Sigma$ is covered by $C^{4}$-coordinate charts so that the induced metric on $\Sigma$, $g_{0}$ is $C^{3}$-close to the Euclidean metric in the charts, i.e.,
\begin{equation}\label{eq:sheets.metric.bound}
\sum_{\ell=0}^{3} |\partial^{(\ell)}_{y} ((g_{0})_{ij} - \delta_{ij})| \leq \eta.
\end{equation}
We make no assumptions on the mean curvature of $\Sigma$ beyond what follows automatically from \eqref{eq:sheets.sff.bound}. Notice that, as a consequence of \eqref{eq:sheets.sff.bound}-\eqref{eq:sheets.metric.bound}, Fermi coordinates with respect to $\Sigma$ are a $C^4$ diffeomorphism.
In all that follows, we denote for $y_0 \in \Sigma \setminus \partial \Sigma$ and $0 < r < \dist_{g_0}(y_0, \partial \Sigma)$,
\[ B^{n-1}_r(y_0) \triangleq \{ y \in \Sigma : \dist_{g_0}(y, y_0) < r \}, \]
where $\dist_{g_0}$ is the intrinsic distance on $\Sigma$. We assume, without loss of generality, that $\Sigma = \overline{B}^{n-1}_2(0)$.
\begin{rema}\label{rema:scale-vs-WW}
We have chosen to work at the original scale, rather than rescaling by $\varepsilon$ as in \cite{WangWei}. This does not affect our subsequent analysis, but certain expressions will change by appropriate multiples of $\varepsilon$.
\end{rema}
Let $u : \Omega \to (-1,1)$ be a critical point of $E_\varepsilon \restr \Omega$, with
\begin{align}
\varepsilon & \leq \varepsilon_0, \label{eq:sheets.eps.bound} \\
(E_{\varepsilon} \restr \Omega)[u_i] & \leq E_0, \label{eq:sheets.energy.bound} \\
\varepsilon |\nabla u| & \geq c_0^{-1} > 0 \text{ on } \Omega \cap \{ |u| \leq 1-\beta \}, \label{eq:sheets.lower.density.bound} \\
|\mathcal{A}| & \leq c_0 \text{ on } \Omega \cap \{ |u| \leq 1-\beta \}; \label{eq:sheets.enhanced.sff.bound} \\
\intertext{By \eqref{eq:sheets.lower.density.bound}, \eqref{eq:sheets.enhanced.sff.bound}, and elliptic regularity, we automatically also get}
\varepsilon |\nabla \mathcal{A}| + \varepsilon^2 |\nabla^2 \mathcal{A}| & \leq c_0 \text{ on } \Omega \cap \{ |u| \leq 1-\beta \} \label{eq:sheets.enhanced.sff.grad.bound}
\end{align}
for a possibly larger $c_0 > 0$. See \cite[Lemma 8.1]{WangWei}. With regard to the nodal set of $u$, we require
\begin{align}
\{ u = 0 \} \cap \Omega & = \bigcup_{\ell=1}^Q \Gamma_\ell, \label{eq:sheets.graph.decomposition} \\
\intertext{where $\Gamma_\ell = \graph_\Sigma f_\ell$ denote normal graphs over $\Sigma$ ordered so that $f_{1} < f_{2}< \dots < f_{Q}$, and the graphing functions $f_\ell : \Sigma \to \mathbf{R}$ are assumed to satisfy}
|f_\ell| + |\nabla_\Sigma f_\ell| & \leq \eta, \label{eq:sheets.graph.apriori.C1.bounds} \\
\intertext{and (this alternatively follows automatically from \eqref{eq:sheets.sff.bound} and \eqref{eq:sheets.enhanced.sff.bound})}
|\nabla_\Sigma^2 f_\ell| & \leq c_0. \label{eq:sheets.graph.apriori.C2.bounds}
\end{align}
Finally, after possibly sending $z\mapsto -z$, we can assume that for $z \approx -1$, $u(y,z) \approx -1$. The constants that appear above are to be considered independent of $\varepsilon \leq \varepsilon_{0}$ and fixed so that
\begin{equation} \label{eq:sheets.constants}
c_0 \gg 1, \; 0 < \varepsilon_0, \beta, \eta \ll 1, \; Q \in \{1, 2, \ldots\}.
\end{equation}
Denote, for $\ell \in \{1, \ldots, Q\}$, $y_0 \in \Sigma$, $r > 0$:
\begin{enumerate}
\item $\Pi : \Omega \to \Sigma$ to be the closest point projection onto $\Sigma$ with respect to $g$.
\item $C_r(y_0) \triangleq \{ X \in \Omega : \Pi(X) \in B_r^{n-1}(y_0) \}$.
\item $\Gamma_\ell(r) \triangleq \Gamma_\ell \cap C_r(0)$.
\item $Z_{\Gamma_\ell} : \Gamma_\ell(3/2) \times [-1,1] \to \Omega$ to be the normal exponential map with respect to $\Gamma_\ell$.
\item $\Pi_\ell : \Omega \to \Gamma_\ell$ to be the closest point projection onto $\Gamma_\ell$ with respect to $g$.
\item $d_\ell : \Omega \to \mathbf{R}$ to be the signed distance from $\Gamma_\ell$ (with respect to $g$), which is positive above it and negative below it.
\item $D_\ell \triangleq \min \{ |d_{\ell-1}|, |d_{\ell+1}| \}$.
\end{enumerate}
Let us agree once and for all regarding Sections \ref{sec:jacobi.toda.reduction}-\ref{sec:stable.solutions}, that each $\Gamma_\ell$ is endowed with the same coordinates $(y^1, \ldots, y^{n-1})$ as $\Sigma$ via the diffeomorphism $\Pi|_{\Gamma_\ell} : \Gamma_\ell \xrightarrow{\approx} \Sigma$.
Set $\Omega' \triangleq B_1^{n-1}(0) \times [-2\eta,2\eta] \subset \Omega$. Consider arbitrary $C^2$ functions
\[ h_\ell : \Gamma_\ell \cap C_1(0) \to (-\tfrac{\eta}{2}, \tfrac{\eta}{2}), \; \ell \in \{1, \ldots, Q \}. \]
Let $\ve{h} = (h_1, \ldots, h_n)$, From $\ve{h}$, we construct an approximate critical point $U(\ve{h})$ of $E_\varepsilon \restr \Omega'$,
\begin{equation} \label{eq:approximate.critical.point}
U[\ve{h}] \triangleq \frac{(-1)^{Q+1}-1}{2} + \sum_{\ell=1}^Q \overline{\mathbb{H}}_{\varepsilon,\ell}.
\end{equation}
Here, each $\overline{\mathbb{H}}_{\varepsilon,\ell}$ is given by
\begin{multline} \label{eq:approximate.critical.point.model}
((Z_{\Gamma_\ell})^* \overline{\mathbb{H}}_{\varepsilon,\ell})(y, z) \triangleq \overline{\mathbb{H}}{}^{3 |\log \varepsilon|} \big((-1)^{\ell-1} \varepsilon^{-1} (z-h_\ell(y))\big) \\
\iff \overline{\mathbb{H}}_{\varepsilon,\ell} = \overline{\mathbb{H}}^{3|\log \varepsilon|}((-1)^{\ell-1} \varepsilon^{-1}(d_\ell - h_\ell \circ \Pi_\ell)),
\end{multline}
with $\overline{\mathbb{H}}{}^\Lambda : \mathbf{R} \to [-1,1]$ (here, $\Lambda = 3 |\log \varepsilon|$) being
\begin{equation} \label{eq:HLambda-cutoff-def}\overline{\mathbb{H}}{}^\Lambda(t) \triangleq \chi(\Lambda^{-1} t) \mathbb{H}(t) \pm (1-\chi(\Lambda^{-1} t)), \end{equation}
($\pm$ depending on $t > 0$ or $t < 0$). Here, $\chi (t) = 1$ for $t \in (-1,1)$ and $\support \chi \subset (-2,2)$ is a fixed cutoff function. These functions, $\overline{\mathbb{H}}{}^{3|\log \varepsilon|}$, are truncations of $\mathbb{H}$ that coincide with it on $(-3 |\log \varepsilon|, 3 |\log \varepsilon|)$, with $\pm 1$ outside $(-6 |\log \varepsilon|, 6 |\log \varepsilon|)$, and such that
\begin{equation} \label{eq:approximate.heteroclinic.behavior}
|(\overline{\mathbb{H}}{}^{3|\log \varepsilon|})'' - W'(\overline{\mathbb{H}}{}^{3 |\log \varepsilon|})|_{C^2(\mathbf{R})} = O(\varepsilon^3).
\end{equation}
See \cite[Section 9.1]{WangWei} for more details.
\begin{rema}
The components of $\ve{h}$ represent the vertical offset of the heteroclinic solutions we're superimposing relative to the nodal set of $u$.
\end{rema}
One can show (see \cite[Subsection 9.1]{WangWei}) that there exists $\ve{h}$ such that for every $\ell \in \{1, \ldots, Q \}$, $y \in \Gamma_\ell$, we have the orthogonality relation:
\begin{equation}\label{eq:h.defn.orth}
\int_{-\eta}^{\eta} ((Z_{\Gamma_\ell})^* (u - U[\ve{h}]))(y, z) \partial_{z} ((Z_{\Gamma_\ell})^* \overline{\mathbb{H}}_{\varepsilon,\ell})(y,z) \, dz = 0.
\end{equation}
Moreover (see \cite[Remark 9.2]{WangWei}):
\[ \sum_{j=0}^3 \varepsilon^{j-1} \Vert \nabla^j \ve{h} \Vert_{C^0(B_1^{n-1}(0))} = o(1) \text{ as } \varepsilon \to 0. \]
It will prove useful to introduce the notation
\begin{equation} \label{eq:discrepancy.function}
\phi \triangleq u - U[\mathbf{h}],
\end{equation}
seeing as to how we can conveniently bound $\ve{h}$ in terms of $\phi$, as Lemma \ref{lemm:h.phi.comparison} below shows.
\begin{lemm}[{\cite[Lemma 9.6]{WangWei}}] \label{lemm:h.phi.comparison}
For $\ell \in \{1, \ldots, Q\}$, $y \in \Gamma_\ell(\tfrac{9}{10})$,
\begin{align*}
\varepsilon^{-1} |h_\ell(y)|
& \leq c \left( |\phi |_{\Gamma_\ell}(y)| + \exp( -\sqrt{2} \varepsilon^{-1} D_\ell(y)) \right), \\
|\nabla_{\Gamma_\ell} h_\ell(y)|
& \leq c \Big( \varepsilon |\nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell})(y)| + o(1) \exp (-\sqrt{2}\varepsilon^{-1} D_\ell(y)) \Big), \\
\varepsilon |\nabla^2_{\Gamma_\ell} h_\ell(y)|
& \leq c \Big( \varepsilon^2 |\nabla^2_{\Gamma_\ell} (\phi|_{\Gamma_\ell})(y)| + \varepsilon^2 |\nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell})(y)|^2 + o(1) \exp (-\sqrt{2}\varepsilon^{-1} D_\ell(y)) \Big), \\
\varepsilon^{1+\theta} [\nabla^2_{\Gamma_\ell} h_\ell]_{\theta}
& \leq c' \Big( \varepsilon^{2+\theta} [\nabla^2_{\Gamma_\ell} (\phi|_{\Gamma_\ell})]_{\theta} + \varepsilon^{2+\theta} \Vert \nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell}) \Vert_{C^0} [ \nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell}) ]_\theta + \Vert \exp (-\sqrt{2}\varepsilon^{-1} D_\ell ) \Vert_{C^0} \Big),
\end{align*}
where $c = c(n,c_0,E_0,\eta,\beta)$, $c' = c'(n,c_0,E_0,\eta,\beta,\theta)$, and $o(1)$ is taken as $\varepsilon \to 0$ with the remaining parameters held fixed. In the last inequality, the H\"older seminorms and the $C^k$ norms are taken over all $y' \in \Gamma_\ell \cap C_\varepsilon(\Pi(y))$.
\end{lemm}
Wang--Wei deduce (see \cite[(10.2)]{WangWei}) the following Jacobi-Toda-like system; for $y \in \Gamma_\ell(\tfrac{9}{10})$,
\begin{align}
& \varepsilon (\Delta_{\Gamma_\ell} h_\ell(y) - H_{\Gamma_\ell}(y)) \label{eq:jacobi.toda} \\
& \qquad = \frac{4(A_0)^{2}}{h_0} \left( \exp(-\sqrt{2}\varepsilon^{-1} |d_{\ell-1}(y)|) - \exp(-\sqrt{2}\varepsilon^{-1} |d_{\ell+1}(y)|) \right) \nonumber \\
& \qquad + O \Big( \varepsilon^{-1} |h_\ell(y)| + \varepsilon^{-1} \Vert (h_{\ell-1} \circ \Pi_{\ell-1} \circ Z_{\Gamma_\ell})(y, \cdot )\Vert_{C^0} + \varepsilon^{\tfrac{1}{3}} \Big) \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-1}(y)|) \nonumber \\
& \qquad + O \Big( \varepsilon^{-1} |h_\ell(y)| + \varepsilon^{-1} \Vert (h_{\ell+1} \circ \Pi_{\ell+1} \circ Z_{\Gamma_\ell})(y, \cdot )\Vert_{C^0} + \varepsilon^{\tfrac{1}{3}} \Big) \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+1}(y)|) \nonumber \\
& \qquad + O(\exp(-(\tfrac{3}{2}\sqrt{2}) \varepsilon^{-1} |d_{\ell-1}(y)|)) + O(\exp(-(\tfrac{3}{2}\sqrt{2}) \varepsilon^{-1} |d_{\ell+1}(y)|)) \nonumber \\
& \qquad + O(\exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-2}(y)|)) + O(\exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+2}(y)|)) \nonumber \\
& \qquad + \sum_{m\neq \ell} \varepsilon^{-1} |d_m(y)| \exp(-\sqrt{2}\varepsilon^{-1}|d_m(y)|) \Big[ \varepsilon \Vert \Delta_{\Gamma_m} h_m - H_{\Gamma_m} \Vert_{C^0} + \Vert \nabla_{\Gamma_m} h_m \Vert^2_{C^0} \Big] \nonumber \\
& \qquad + \sup_{|t| < 6 \varepsilon |\log \varepsilon|} \Big[ \varepsilon^4 |(\nabla^2_{\Gamma_{\ell,t}} (\phi|_{\Gamma_{\ell,t}}))(Z_{\Gamma_\ell}(y,t))|^2 + \varepsilon^2 |(\nabla_{\Gamma_{\ell,t}} (\phi|_{\Gamma_{\ell,t}}))(Z_{\Gamma_\ell}(y,t))|^2 + |\phi(Z_{\Gamma_\ell}(y, t))|^2 \Big] \nonumber \\
& \qquad + O(\varepsilon^2). \nonumber
\end{align}
The $C^0$ norms appearing in the second and third term of the right hand side is taken over $|t| < 6\varepsilon|\log \varepsilon|$, and the $C^0$ norms appearing in the third term from the end is taken over $\Gamma_m \cap C_{\varepsilon^{4/3}}(\Pi(y))$.
\begin{rema}
$\Gamma_{\ell,t}$ denote $t$-level sets in Fermi coordinates $(y,t)$ relative to $\Gamma_\ell$, i.e., $\Gamma_{\ell,t} = \{ d_\ell = t \}$.
\end{rema}
\begin{rema}
Notice the sign difference in the mean curvature terms between \eqref{eq:jacobi.toda} and \cite[(10.2)]{WangWei}. For us, the mean curvature is the divergence of the upper pointing unit normal. For instance, the ambient Laplace-Beltrami operator expands as
\[ \Delta_g = \Delta_{\Gamma_{\ell,z}} + \partial_z^2 + H_{\Gamma_{\ell,z}} \partial_z. \]
For this reason, all instances of the mean curvature in this work have to have the opposite sign relative to \cite{WangWei}.
\end{rema}
It will also be convenient to introduce the notation
\begin{equation} \label{eq:sup.exp.distance}
A_\ell(r) \triangleq \sup \Big\{ \exp(-\sqrt{2} \varepsilon^{-1} D_\ell(y)) :
y \in \Gamma_\ell(r) \Big\}.
\end{equation}
We record \cite[(12.4)]{WangWei}, which will help estimate terms involving $h$, $\phi$, and the mean curvature:
\begin{equation} \label{eq:phi.c2a.estimate.full}
\Vert \phi \Vert_{C^{2,\theta}_\varepsilon(\mathcal{M}_\ell(r))} + \varepsilon \Vert \Delta_{\Gamma_\ell} h_\ell - H_{\Gamma_\ell} \Vert_{C^{0,\theta}_\varepsilon(\Gamma_\ell(r))} \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r+K\varepsilon |\log \varepsilon|),
\end{equation}
where we're using the weighted H\"older space notation from \eqref{eq:dirichlet.data.ckalpha.eps} (see Section \ref{sec:dirichlet.data}), and
\begin{equation*}
\mathcal{M}_\ell(r) \triangleq \{ X \in C_r(0) : |d_\ell(X)| < 1,
- d_{\ell-1}(X) < d_\ell(X) < - d_{\ell+1}(X) \}.
\end{equation*}
Likewise, we record \cite[(13.6)]{WangWei}:
\begin{equation} \label{eq:phi.improved.c2a.estimate.full}
\varepsilon \Vert ((Z_{\Gamma_\ell})_* \partial_{y_i}) \phi \Vert_{C^{1,\theta}_\varepsilon(\mathcal{M}_\ell(r))} \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa} + c' \varepsilon^\kappa \sum_{m=1}^Q A_m(r+2K \varepsilon |\log \varepsilon|),
\end{equation}
with $\kappa > 0$.
The expressions above, \eqref{eq:phi.c2a.estimate.full}-\eqref{eq:phi.improved.c2a.estimate.full}, are true for all $\ell \in \{1, \ldots, Q\}$, $r \leq 8/10$, $\theta \in (0,1)$, $\varepsilon \leq \varepsilon'$, where $c'$, $\varepsilon'$, $K$, $\kappa$, depend on $n$, $c_0$, $E_0$, $\eta$, $\beta$, $\theta$.
\begin{rema} \label{rema:major.goal}
In the remainder of Sections \ref{sec:jacobi.toda.reduction}-\ref{sec:stable.solutions}, we'll be actively interested in estimating the vertical distances $D_\ell$ from below. This is because Lemma \ref{lemm:h.phi.comparison}, \eqref{eq:sup.exp.distance}, \eqref{eq:phi.c2a.estimate.full}, and interior Schauder estimates together imply that, with $r$, $\theta$ as above:
\begin{equation} \label{eq:rema.major.goal}
\min_{\ell \in \{ 1, \ldots, Q \}} \inf_{\Gamma_\ell(r)} D_\ell \geq \tfrac{1+\theta}{2} \sqrt{2} \varepsilon |\log \varepsilon| \implies \Gamma_{\ell}(r') \text{ is uniformly } C^{2,\theta}
\end{equation}
for all $\ell \in \{ 1, \ldots, Q \}$, $r' \leq \sigma r$, $\sigma \in (0, 1)$, $\varepsilon \leq \varepsilon' = \varepsilon'(n, c_0, E_0, \eta, \beta, \theta, \sigma)$.
\end{rema}
\subsection{Bootstrapping regularity via sheet distance lower bounds} \label{subsec:bootstrapping.sheet.distance.bounds}
We recall the following lemma from \cite{WangWei}. (See \cite[Appendix C]{Mantoulidis} for necessary modifications for the Riemannian setting.)
\begin{lemm}[{\cite[Section 14]{WangWei}}] \label{lemm:stationary.estimates}
If $\ell \in \{1, \ldots, Q\}$, $y \in \Gamma_\ell(\tfrac{8.5}{10})$, and $\varepsilon \leq \varepsilon_1$, then
\[ D_\ell(y) \geq \tfrac{1}{2} \sqrt{2} \varepsilon |\log \varepsilon| - c_1 \varepsilon, \]
where $\varepsilon_1 = \varepsilon_1(n, c_0, E_0, \eta, \beta)$, $c_1 = c_1(n, c_0, E_0, \eta, \beta)$.
\end{lemm}
As a corollary of Lemma \ref{lemm:stationary.estimates}, we can bootstrap the proof of Lemma \ref{lemm:h.phi.comparison} and obtain the following \emph{improved} estimates:
\begin{lemm} \label{lemm:h.phi.comparison.improved}
For $\ell \in \{1, \ldots, Q\}$, $y \in \Gamma_\ell(\tfrac{8}{10})$,
\begin{align*}
\varepsilon^{-1} |h_\ell(y)|
& \leq c \left( |\phi|_{\Gamma_\ell}(y)| + \exp( -\sqrt{2} \varepsilon^{-1} D_\ell(y)) \right), \\
|\nabla_{\Gamma_\ell} h_\ell(y)|
& \leq c \left( \varepsilon |\nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell})(y)| + \varepsilon^\kappa \exp (-\sqrt{2}\varepsilon^{-1} D_\ell(y)) \right), \\
\varepsilon |\nabla^2_{\Gamma_\ell} h_\ell(y)|
& \leq c \Big( \varepsilon^2 |\nabla^2_{\Gamma_\ell} (\phi|_{\Gamma_\ell})(y)| + \varepsilon^2 |\nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell})(y)|^2 + \varepsilon^\kappa \exp (-\sqrt{2}\varepsilon^{-1} D_\ell(y)) \Big), \\
\varepsilon^{1+\theta} [\nabla^2_{\Gamma_\ell} h_\ell]_{\theta}
& \leq c' \Big( \varepsilon^{2+\theta} [\nabla^2_{\Gamma_\ell} (\phi|_{\Gamma_\ell})]_{\theta} + \varepsilon^{2+\theta} \Vert \nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell}) \Vert [ \nabla_{\Gamma_\ell} (\phi|_{\Gamma_\ell}) ]_\theta + \varepsilon^{\kappa'} \Vert \exp (-\sqrt{2}\varepsilon^{-1} D_\ell ) \Vert_{C^0} \Big),
\end{align*}
where $c = c(n,c_0,E_0,\eta,\beta)$, $c' = c'(n,c_0,E_0,\eta,\beta,\theta)$, $\kappa = \kappa(n,c_0,E_0,\eta,\beta)$, $\kappa' = \kappa'(n,c_0,E_0,\eta,\beta,\theta)$. The norms and seminorms in the last inequality are taken over all $y' \in \Gamma_\ell$ with $\Pi(y') \in B_\varepsilon^{n-1}(\Pi(y))$.
\end{lemm}
\begin{proof}
See Appendix \ref{app:proof.lem.comp.improved}.
\end{proof}
We now indicate how the enhanced second fundamental form tensor is affected by these estimates.
Fix $\ell \in \{1, \ldots, Q\}$. We see from \eqref{eq:phi.improved.c2a.estimate.full} that
\begin{align}
\varepsilon \Vert \nabla \phi - \langle \nabla \phi, \nabla d_\ell \rangle \nabla d_\ell \Vert_{C^0(\mathcal{M}_\ell(r))} & \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa} \label{eq:grad.phi.estimate} \\
\intertext{for some $\kappa = \kappa(n, c_0, E_0, \eta, \beta) > 0$. Likewise, from \eqref{eq:approximate.critical.point}, \eqref{eq:approximate.critical.point.model}, Lemmas \ref{lemm:stationary.estimates}-\ref{lemm:h.phi.comparison.improved}, and \eqref{eq:phi.improved.c2a.estimate.full}:}
\varepsilon \Vert \nabla U[\mathbf{h}] - \langle \nabla U[\mathbf{h}], \nabla d_\ell \rangle \nabla d_\ell \Vert_{C^0(\mathcal{M}_\ell(r))} & \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa}. \label{eq:grad.gluedU.estimate} \\
\intertext{Combining \eqref{eq:discrepancy.function}, \eqref{eq:grad.phi.estimate}, and \eqref{eq:grad.gluedU.estimate}, we get:}
\varepsilon \Vert \nabla u - \langle \nabla u, \nabla d_\ell \rangle \nabla d_\ell \Vert_{C^0(\mathcal{M}_\ell(r))} & \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa}. \label{eq:grad.u.estimate} \\
\intertext{Combining \eqref{eq:sheets.lower.density.bound} and \eqref{eq:grad.u.estimate}, we get:}
\Vert \nu - (-1)^{\ell-1} \nabla d_\ell \Vert_{C^0(\mathcal{M}_\ell(r) \cap \{ |u| \leq 1-\beta \})} & \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa}, \label{eq:normal.direction.estimate}
\end{align}
where $\nu = |\nabla u|^{-1} \nabla u$ denotes the normal to the level set of $u$ through each point. (The level set is smooth on $\{ |u| \leq 1-\beta \}$ in view of \eqref{eq:sheets.lower.density.bound}.)
For the remainder of this section we choose to work in Fermi coordinates $(y, t)$ relative to $\Gamma_\ell$; note that $t = d_\ell$. It is not hard to see that the only nontrivial Christoffel symbols in this coordinate system are $\Gamma_{ij}^t$, $\Gamma_{jt}^i$, $\Gamma_{tj}^i$, and $\Gamma_{ij}^k$. Set
\begin{equation} \label{eq:christoffel.symbol.sup}
\widehat{\Gamma}_\ell(r) \triangleq \sup_{\mathcal{M}_\ell(r) \cap \{ |u| \leq 1-\beta \}} |\Gamma_{ij}^t| + |\Gamma_{jt}^i| + |\Gamma_{tj}^i| + |\Gamma_{ij}^k|.
\end{equation}
By arguing as above, and relying on \eqref{eq:phi.improved.c2a.estimate.full}, we find that:
\begin{align}
& \varepsilon^2 \Vert \nabla^2 u - \nabla^2 u(\partial_t, \partial_t) \, dt^2 \Vert_{C^0(\mathcal{M}_\ell(r) \cap \{|u| \leq 1-\beta\})} \label{eq:hessu.estimate.i} \\
& \qquad \leq \varepsilon^2 \sum_{i=1}^{n-1} \Vert \nabla (((Z_{\Gamma_\ell})_* \partial_{y_i}) u) \Vert_{C^0(\mathcal{M}_\ell(r) \cap \{ |u| \leq 1-\beta \})} + \varepsilon^2 \widehat{\Gamma}_\ell(r) \Vert \nabla u \Vert_{C^0(\mathcal{M}_\ell(r) \cap \{ |u| \leq 1-\beta \})} \nonumber \\
& \qquad \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa} + c' \varepsilon \widehat{\Gamma}_\ell(r). \nonumber \\
\intertext{Using \eqref{eq:normal.direction.estimate} (note that $\partial_t = \nabla d_\ell$),}
& \varepsilon^2 \Vert \nabla^2 u(\partial_t, \partial_t) \, dt \otimes (dt - \langle dt, \nu \rangle \nu^\flat) \Vert_{C^0(\mathcal{M}_\ell(r) \cap \{ |u| \leq 1-\beta \})} \label{eq:hessu.estimate.ii} \\
& \qquad \leq c' \Vert \nu - \partial_t \Vert_{C^0(\mathcal{M}_\ell(r) \cap \{|u| \leq 1-\beta \})} \leq c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa}, \nonumber \\
\intertext{where $\nu^\flat$ denotes $\nu$'s dual 1-form. Finally, \eqref{eq:sheets.lower.density.bound}, \eqref{eq:hessu.estimate.i}, and \eqref{eq:hessu.estimate.ii} give:}
& \Vert \mathcal{A} \Vert_{C^0(\mathcal{M}_\ell(r) \cap \{ |u| \leq 1-\beta \})} \leq c' \varepsilon + c' \varepsilon^{-1} \sum_{m=1}^Q A_m(r + 2K \varepsilon |\log \varepsilon|)^{1+\kappa} + c' \widehat{\Gamma}_\ell(r). \label{eq:enhanced.sff.estimate} \\
\intertext{Now, we turn to estimating $H_{\Gamma_{\ell}}$. From Lemma \ref{lemm:h.phi.comparison.improved} and \eqref{eq:phi.improved.c2a.estimate.full} we have, for $y \in \Gamma_\ell(\tfrac{8}{10})$,}
& \varepsilon |\Delta_{\Gamma_\ell} h_\ell(y)| \label{eq:bootstrapped.i} \\
& \qquad \leq \varepsilon^2 |\nabla^2_{\Gamma_\ell} (\phi|_{\Gamma_\ell})(y)| + \varepsilon^\kappa \exp(-\sqrt{2} \varepsilon^{-1} D_\ell(y)) \nonumber \\
& \qquad \leq c' \varepsilon^2 + c' \varepsilon^{\kappa} \sum_{m=1}^Q A_m(|y| + 2K \varepsilon|\log \varepsilon|) + c' \sum_{m=1}^Q A_m(|y| + 2K \varepsilon |\log \varepsilon|)^{1+\kappa}. \nonumber
\end{align}
We're going to estimate the terms in \eqref{eq:jacobi.toda} from above by a function of $\varepsilon$ and the quantities in \eqref{eq:sup.exp.distance}. Fix $\ell \in \{1, \ldots, Q\}$, $y \in \Gamma_\ell(\tfrac{7}{10})$.
From Lemma \ref{lemm:stationary.estimates}, Lemma \ref{lemm:h.phi.comparison.improved}, and \eqref{eq:phi.c2a.estimate.full}, we have
\begin{align}
& (\varepsilon^{-1} |h_\ell| + \varepsilon^{-1} |h_{\ell-1} \circ \Pi_{\ell-1} \circ Z_{\Gamma_\ell}| + \varepsilon^{\tfrac{1}{3}}) \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-1}(y)|) \nonumber \\
& + (\varepsilon^{-1} |h_\ell| + \varepsilon^{-1} |h_{\ell+1} \circ \Pi_{\ell+1} \circ Z_{\Gamma_\ell}| + \varepsilon^{\tfrac{1}{3}}) \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+1}(y)|) \nonumber \\
& + \exp(-(\tfrac{3}{2}\sqrt{2}) \varepsilon^{-1} |d_{\ell-1}(y)|) + \exp(-(\tfrac{3}{2}\sqrt{2}) \varepsilon^{-1} |d_{\ell+1}(y)|) \nonumber \\
& + \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-2}(y)|)) + \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+2}(y)|) \nonumber \\
& \qquad \leq c' \varepsilon^2 + c' \varepsilon^{\kappa} \sum_{m=1}^Q A_m(|y|+K\varepsilon |\log \varepsilon|) + c' \sum_{m=1}^Q A_m(|y|+K \varepsilon |\log \varepsilon|)^{1+\kappa}. \label{eq:bootstrapped.ii}
\end{align}
By Lemma \ref{lemm:h.phi.comparison}, \eqref{eq:phi.c2a.estimate.full}, \eqref{eq:phi.improved.c2a.estimate.full}, every $m \neq \ell$ satisfies
\begin{multline} \label{eq:bootstrapped.iii}
\varepsilon^{-1} |d_m(y)| \exp(-\sqrt{2}\varepsilon^{-1}|d_m(y)|) \Big[ \varepsilon \Vert \Delta_{\Gamma_m} h_m - H_{\Gamma_m} \Vert_{C^0} + \Vert \nabla_{\Gamma_m} h_m \Vert^2_{C^0} \Big] \\
\leq c' A_m(|y| + 2K \varepsilon |\log \varepsilon|)^{1-\rho}\sum_{m'\not = m} A_{m}(|y|+2K\varepsilon |\log \varepsilon|)
+ c' \varepsilon^2 A_m(|y| + 2K \varepsilon |\log \varepsilon|)^{1-\rho}
\end{multline}
for small $\rho > 0$, $\varepsilon \leq \varepsilon'$. The $C^0$ norms are taken over $\Gamma_m \cap C_{\varepsilon^{4/3}}(\Pi(y))$. By Lemma \ref{lemm:stationary.estimates} and \eqref{eq:phi.c2a.estimate.full},
\begin{multline} \label{eq:bootstrapped.iv}
\sup_{|t| < 6 \varepsilon |\log \varepsilon|} \Big[ \varepsilon^4 |(\nabla^2_{\Gamma_{m,t}} (\phi|_{\Gamma_{m,t}}))(Z_{\Gamma_m}(y,t))|^2 + \varepsilon^2 |(\nabla_{\Gamma_{m,t}} (\phi|_{\Gamma_{\ell,t}})(Z_{\Gamma_m}(y,z))|^2 + |\phi(Z_{\Gamma_m}(y, z))|^2 \Big] \\
\leq c' \varepsilon^2 + c' \sum_{m'=1}^Q A_{m'}(|y| + K \varepsilon |\log \varepsilon|)^2.
\end{multline}
Combined, \eqref{eq:jacobi.toda} and \eqref{eq:bootstrapped.i}-\eqref{eq:bootstrapped.iv} give
\begin{align}
-\varepsilon H_{\Gamma_\ell}(y)
& = \frac{4(A_0)^{2}}{h_0} \Big( \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-1}(y)|) - \exp( -\sqrt{2} \varepsilon^{-1} |d_{\ell+1}(y)|) \Big) + \mathcal{R}_{\ell} \label{eq:bootstrapped.v} \\
\intertext{for all $y \in \Gamma_\ell(\tfrac{7}{10})$, where}
|\mathcal{R}_\ell(y)| & \leq c' \varepsilon^2 + c' \varepsilon^\kappa \sum_{m=1}^Q A_m(|y| + 2K \varepsilon |\log \varepsilon|) + c' \sum_{m=1}^Q A_m(|y| + 2K \varepsilon |\log \varepsilon|)^{1+\kappa} \label{eq:bootstrapped.error.i}.
\end{align}
\begin{lemm} \label{lemm:mean.curvature.laplacian}
Let $f : B_1^{n-1}(0) \to \mathbf{R}$ be as in \eqref{eq:sheets.graph.apriori.C1.bounds}-\eqref{eq:sheets.graph.apriori.C2.bounds}. If $G[f]$ is the normal graph of $f$ over $\Gamma_{\ell}$, i.e., $G[f] = \{ Z_{\Gamma_\ell}(y, f(y)) : y \in B_1^{n-1}(0) \}$, then
\[ H_{G[f]} - H_{\Gamma_\ell} = - (\mathcal{L} + |\sff_{\Gamma_\ell}|^2 + \ricc_g(\nu_{\Gamma_\ell},\nu_{\Gamma_\ell})|_{\Gamma_\ell})f + \mathcal{Q}(f), \]
where $\mathcal{L}$ is the linear uniformly elliptic operator
\begin{equation} \label{eq:mean.curvature.laplacian.operator}
\mathcal{L} \varphi = \mathcal{L}_{\Gamma_\ell,G[f]} \varphi \triangleq a(y)^{-1} \divg_{\Gamma_\ell} \left( a(y) \langle (Z_{\Gamma_\ell})_* \nu_{\Gamma_\ell}, \nu_{G[f]} \rangle \nabla_{G[f]} \varphi \right),
\end{equation}
with
\begin{equation} \label{eq:mean.curvature.laplacian.operator.a}
a(y) = a_{\Gamma_\ell,G[f]}(y) \triangleq \frac{\sqrt{g_{\Gamma_\ell}}}{\sqrt{g_{f(y)}}}.
\end{equation}
Here $(Z_{\Gamma_\ell})_* \nu_{\Gamma_\ell}$, $\nu_{G[f]}$ are upward pointing unit normal in Fermi coordinates and the upward pointing unit normal to $G[f]$, both evaluated at $Z_{\Gamma_\ell}(y, f(y))$. Note that the elliptic symbol coefficients are uniformly bounded away from $0$ and $\infty$ depending on \eqref{eq:sheets.graph.apriori.C1.bounds}. The (nonlinear) error term $\mathcal{Q}(f)$ satisfies
\begin{equation*}
|\mathcal{Q}(f)| \leq c' (|f|^2 + |\nabla_{\Gamma_\ell} f|^2).
\end{equation*}
\end{lemm}
\begin{proof}
This is a restatement of Lemma \ref{lemm:mean.curv.graphical.quad.error.new} from Appendix \ref{app:mean.curvature.graphs}.
\end{proof}
Notice that, by \eqref{eq:sheets.graph.apriori.C1.bounds}-\eqref{eq:sheets.graph.apriori.C2.bounds}, $\Gamma_{\ell+1}$ can be viewed as a normal graph of some function $f_{\ell,\ell+1}$ over $\Gamma_{\ell}$ that satisfies the conditions of Lemma \ref{lemm:mean.curvature.laplacian}. Let
\[ y' \triangleq Z_{\Gamma_\ell}(y,f_{\ell,\ell+1}(y)) \in \Gamma_{\ell+1}. \]
Applying \eqref{eq:bootstrapped.v} to $y$ at $\Gamma_\ell$ and to $y'$ at $\Gamma_{\ell+1}$, subtracting, and invoking Lemma \ref{lemm:mean.curvature.laplacian}, we see that:
\begin{align}
& \varepsilon (\mathcal{L} + |\sff_{\Gamma_\ell}|^2 + \ricc_g(\nu,\nu)|_{\Gamma_\ell} + \mathcal{Q}) f_{\ell,\ell+1}(y) \label{eq:bootstrapped.vi} \\
& \qquad = \varepsilon(H_{\Gamma_\ell}(y) - H_{\Gamma_{\ell+1}}(y')) \nonumber \\
& \qquad = \frac{4(A_0)^{2}}{h_0} \Big( \exp(-\sqrt{2} \varepsilon^{-1} f_{\ell,\ell+1}(y)) - \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+2}(y')|) \nonumber \\
& \qquad \qquad \qquad - \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-1}(y)|) + \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+1}(y)|) \Big) \nonumber \\
& \qquad \qquad - \mathcal{R}_{\ell}(y) + \mathcal{R}_{\ell+1}(y'). \nonumber
\end{align}
Here, $\mathcal{L}$ is the second order linear operator defined in \eqref{eq:mean.curvature.laplacian.operator}, and which depends on $\Gamma_\ell$, $\Gamma_{\ell+1}$. Note that (see Lemma \ref{lemm:WW.tilting.comparison.d}):
\begin{equation} \label{eq:distance.exchange.error}
\exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+1}(y)|) = \exp(-\sqrt{2} \varepsilon^{-1} f_{\ell,\ell+1}(y))
+ O(\varepsilon^{\tfrac{1}{3}}) \exp(-\sqrt{2} \varepsilon^{-1} D_{\ell}(y)).
\end{equation}
Absorbing the last term above into $\mathcal{R}_\ell$ in view of \eqref{eq:bootstrapped.error.i}, we conclude:
\begin{align}
& \varepsilon (\mathcal{L} + |\sff_{\Gamma_\ell}|^2 + \ricc_g(\nu,\nu)|_{\Gamma_\ell} + \mathcal{Q}) f_{\ell,\ell+1}(y) \label{eq:bootstrapped.vii} \\
& = \frac{4(A_0)^{2}}{h_0} \Big( 2 \exp(-\sqrt{2} \varepsilon^{-1} f_{\ell,\ell+1}(y)) - \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+2}(y')|) - \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-1}(y)|) \Big) \nonumber \\
& \qquad - \mathcal{R}_{\ell}(y) + \mathcal{R}_{\ell+1}(y'). \nonumber
\end{align}
Finally, dropping the negative terms gives:
\begin{equation} \label{eq:bootstrapped.viii}
\varepsilon (\mathcal{L} + |\sff_{\Gamma_\ell}|^2 + \ricc_g(\nu, \nu)|_{\Gamma_\ell} + \mathcal{Q}) f_{\ell,\ell+1}(y)
\leq \frac{8(A_0)^{2}}{h_0} \exp(-\sqrt{2} \varepsilon^{-1} f_{\ell,\ell+1}(y)) + c' |\mathcal{R}_\ell(y)| + |\mathcal{R}_{\ell+1}(y')|;
\end{equation}
the error terms $\mathcal{R}_\ell$, $\mathcal{R}_{\ell+1}$, are still as in \eqref{eq:bootstrapped.error.i}.
\section{Introduction}
Minimal surfaces---critical points of the area functional with respect to local deformations---are fundamental objects in Riemannian geometry due to their intrinsic interest and richness, as well as deep and surprising applications to the study of other geometric problems. Because many manifolds do not contain \emph{any} area-minimizing hypersurfaces, one is quickly led to the study of surfaces that are only critical points of the area functional. Such surfaces are naturally constructed by min-max (i.e., mountain-pass) type methods. To this end, Almgren and Pitts \cite{Pitts} have developed a far-reaching theory of existence and regularity (cf.\ \cite{SchoenSimon}) of min-max (unstable) minimal hypersurfaces. In particular, their work implies that any closed Riemannian manifold $(M^{n},g)$ contains at least one minimal hypersurface $\Sigma^{n-1}$ (in sufficiently high dimensions, $\Sigma$ may have a thin singular set). This result motivates a well-known question of Yau: ``do all $3$-manifolds contain infinitely many immersed minimal surfaces?'' \cite{Yau:problems}.
Recently, there have been several amazing applications of Almgren--Pitts theory to geometric problems, including the proof of the Willmore conjecture by Marques--Neves \cite{MarquesNeves:Willmore} and the resolution of Yau's conjecture for generic metrics in dimensions 3 through 7 by Irie--Marques--Neves \cite{IrieMarquesNeves}. In spite of this, certain basic questions concerning the Almgren--Pitts construction remain unresolved: including whether or not the limiting minimal surfaces can arise with multiplicity (for a generic metric) as well as whether or not one-sided minimal surfaces can arise as limits of an ``oriented'' min-max sequence (see, however, \cite{KMN:catenoid,MarquesNeves:multiplicity}).
\footnote{
Added in proof: There has been dramatic progress in Almgren--Pitts theory since we first posted this article. In particular, we note that A. Song \cite{Song:full-yau} has proved the full Yau conjecture in dimensions 3 through 7, and X. Zhou \cite{Zhou:multiplicity-one} proved the multiplicity one conjecture in the Almgren--Pitts setting, also in dimensions 3 through 7.
}
Guaraco \cite{Guaraco} has proposed an alternative to Almgren--Pitts theory, later extended by Gaspar--Guaraco \cite{GasparGuaraco}, which is based on study of a semilinear PDE known as the Allen--Cahn equation
\begin{equation} \label{eq:ac.pde}
\varepsilon^{2} \Delta_{g} u = W'(u)
\end{equation}
and its singular limit as $\varepsilon\searrow 0$. There is a well known expectation that, in $\varepsilon\searrow 0$ limit, solutions to \eqref{eq:ac.pde} produce minimal surfaces whose regularity reflects the solutions' variational properties. In particular:
\begin{enumerate}
\item It is known that the Allen--Cahn functional $\Gamma$-converges to the perimeter functional \cite{Modica,Sternberg}, so minimizing solutions to \eqref{eq:ac.pde} converge as $\varepsilon \searrow 0$ to minimizing hypersurfaces (and are thus regular away from a codimension $7$ singular set).
\item Under weaker assumptions on the sequence of solutions, one obtains different results. In general, solutions to \eqref{eq:ac.pde} on a Riemannian manifold $(M^n, g)$ have a naturally associated $(n-1)$-varifold obtained by ``smearing out'' their level sets of $u$, weighted by the gradient,
\[ V[u](\varphi) \triangleq h_0^{-1} \int \varphi(x, T_x \{ u = u(x) \}) \, \varepsilon |\nabla u(x)|^2 \, d\mu_g(x), \; \varphi \in C^0_c(\operatorname{Gr}_{n-1}(M)). \]
Here, $h_{0} > 0$ is a constant that is canonically associated with $W$ (see Section \ref{subsec:heteroclinic.solution}). A deep result of Hutchinson--Tonegawa \cite[Theorem 1]{HutchinsonTonegawa00} ensures that $V$ limits to a varifold with a.e.\ integer density as $\varepsilon\searrow0$. If, in addition, one assumes that the solutions are stable, Tonegawa--Wickramasekera \cite{TonegawaWickramasekera12} have shown that the limiting varifold is stable and satisfies the conditions of Wickramasekera's deep regularity theory \cite{Wickramasekera14}; thus the limiting varifold is a smooth stable minimal hypersurface (outside of a codimension $7$ singular set). In two dimensions, this was shown by Tonegawa \cite{Tonegawa05}.
\end{enumerate}
Guaraco's approach has certain advantages when compared with Almgren--Pitts theory:
\begin{enumerate}
\item A key difficulty in the work of Almgren--Pitts is a lack of a Palais--Smale condition, which is usually fundamental in mountain pass constructions. On the other hand, the Allen--Cahn equation does satisfy the usual Palais--Smale condition for each $\varepsilon>0$ (see \cite[Proposition 4.4]{Guaraco}), so this aspect of the theory is much simpler.
We note, however, that the bulk of the regularity theory in Guaraco's work is applied \emph{after} taking the limit $\varepsilon\searrow 0$ and thus relies on the deep works of Wickaramsekera \cite{Wickramasekera14} and Tonegawa--Wickramasekera \cite{TonegawaWickramasekera12}. This places a more serious burden on regularity theory than Almgren--Pitts.
\item In Almgren--Pitts theory, there is no ``canonical'' approximation of the limiting min-max surface by nearby elements of a sweepout. On the other hand, Allen--Cahn provides a canonical approximation built out of the function $u$ (which satisfies a PDE). It is thus natural to suspect that this might be useful when studying the geometric properties of the limiting surface.
For example, Hiesmayr \cite{Hiesmayr} and Gaspar \cite{Gaspar} have shown that index upper bounds for Allen--Cahn solutions directly pass to the limiting surface (we note that the Almgren--Pitts version of this result has been proven by Marques--Neves \cite{MarquesNeves:multiplicity}). Moreover, the second-named author has recently shown \cite{Mantoulidis} that $1$-parameter Allen--Cahn min-max on a surface produces a smooth immersed curve with at most one point of self-intersection; in general, Almgren--Pitts on a surface will only produce a geodesic net (cf.\ \cite{Aiex:ellipsoids}).
\end{enumerate}
Our main contributions in this work are as follows:
\begin{enumerate}
\item We show (see Theorem \ref{theo:curv.est} below) that the individual level sets of stable solutions to the Allen--Cahn equation on a $3$-manifold with energy bounds satisfy a priori curvature estimates (similar to stable minimal surfaces). Using this, we are can avoid the regularity theory of Wickramasekera and Tonegawa--Wickramasekera entirely, making the whole theory considerably more self-contained.
\item More fundamentally, our curvature estimates (and strong sheet separation estimates, which we will discuss below) allow us to study geometric properties of the limiting minimal surface using the ``canonical'' PDE approximations that exist \emph{prior} to taking the $\varepsilon \searrow 0$ limit. In particular, we will prove the multiplicity one conjecture of Marques--Neves \cite{MarquesNeves:multiplicity} in the Allen--Cahn setting (see Theorem \ref{theo:mult.intro-version} below) for min-max sequences on $3$-manifolds. In fact, we prove a strengthened version of the conjecture by ruling out (generically) stable components and one-sided surfaces.
\end{enumerate}
As an application of our multiplicity one results we are able to give a new proof of Yau's conjecture on infinitely many minimal surfaces in a $3$-manifold, when the metric is bumpy (see Corollary \ref{coro:yau-intro} below). This has been recently proven using Almgren--Pitts theory\footnote{We note that after the first version of this work was posted, Gaspar--Guaraco \cite{GasparGuaraco:weyl} gave a new proof of Yau's conjecture for generic metrics (in the spirit of Irie--Marques--Neves \cite{IrieMarquesNeves}) by proving a Weyl law for their Allen--Cahn $p$-widths.} by Irie--Marques--Neves \cite{IrieMarquesNeves}, for a slightly different class of metrics; their proof works in $(M^{n},g)$ for $3\leq n\leq 7$ and proves, in addition, that the minimal surfaces are dense. Our proof establishes several new geometric properties of the surfaces; in particular, we show that they are two-sided and that their area and Morse index behaves as one would expect, based on the theory of $p$-widths \cite{Gromov:waist,Guth:minimax,MarquesNeves:posRic,GasparGuaraco}.
We wish to emphasize two things:
\begin{enumerate}
\item Our results work at the level of sequences of critical points of the Allen--Cahn energy functional with uniform energy and Morse index bounds. At no point do we use any min-max characterization of the limiting surface; min-max is merely used as a tool to construct nontrivial sequences of critical points with energy and index bounds.
\item Our results highlight the philosophy that the solutions to Allen--Cahn provide a ``canonical'' approximation of the min-max surfaces.
\end{enumerate}
\subsection{Notation}
In all that follows, $(M^n, g)$ is a smooth Riemannian manifold.
\begin{defi}
A function $W \in C^{\infty}(\mathbf{R})$ is a \emph{double-well potential} if:
\begin{enumerate}
\item $W$ is non-negative and vanishes precisely at $\pm 1$;
\item $W$ satisfies $W'(0) = 0$, $t W'(t) < 0$ for $|t| \in (0,1)$, and $W''(0) \not = 0$;
\item $W''(\pm 1) =2 $;
\item $W(t) = W(-t)$.
\end{enumerate}
\end{defi}
The standard double-well potential is $W(t) = \frac 1 4 (1-t^{2})^{2}$, in which case \eqref{eq:ac.pde} becomes $\varepsilon^{2} \Delta_{g} u = u^{3}-u$.
The Allen--Cahn equation, \eqref{eq:ac.pde}, is the Euler--Lagrange equation for the energy functional
\[
E_{\varepsilon}[u] = \int_{M} \left( \frac \varepsilon 2 |\nabla u|^{2} + \frac{W(u)}{\varepsilon} \right) \, d\mu_{g}.
\]
Depending on what we wish to emphasize, we will go back and forth between saying that a function $u$ is a solution of \eqref{eq:ac.pde} on $M$ (or in a domain $U \subset M$) or a critical point of $E_\varepsilon$ (resp. of $E_\varepsilon \restr U$). The second variation of $E_{\varepsilon}$ is easily computed (for $\zeta,\psi \in C^{\infty}_{c}(M)$) to be
\begin{equation}\label{eq:second.var.AC}
\delta^{2}E_{\varepsilon}[u]\{\zeta,\psi\} =\int_{M} \left( \varepsilon \langle\nabla \zeta,\nabla \psi \rangle + \frac{W''(u)}{\varepsilon} \zeta\psi\right) \, d\mu_{g}.
\end{equation}
We are thus led to the notion of stability and Morse index (with respect to Dirichlet eigenvalues).
\begin{defi} \label{def:ac.stable} \label{def:ac.morse.index}
For $(M^{n},g)$ a complete Riemannian manifold and $U \subset M \setminus \partial M$ open, we say that a critical point of $E_{\varepsilon} \restr U$ is \emph{stable} on $U$ if $\delta^{2}E_{\varepsilon}[u]\{\zeta,\zeta\} \geq 0$ for all $\zeta \in C^{\infty}_{c}(U)$. More generally, we say $u$ has Morse index $k$, denoted $\ind(u) = k$, if
\[
\max \{ \dim V : \delta^{2}E_{\varepsilon}[u]\{\zeta,\zeta\} < 0 \text{ for all } \zeta \in V\setminus\{0\}\} = k,
\]
where the maximum is taken over all subspaces $V \subset C^{\infty}_{c}(U)$. Sometimes we will write $\ind(u;U)=k$ to emphasize the underlying set. Note that $\ind(u; U) = 0$ if and only if $u$ is stable on $U$.
\end{defi}
When $u$ is a solution of \eqref{eq:ac.pde} and $\nabla u(x) \neq 0$, we will write:
\begin{enumerate}
\item $\nu(x) = \tfrac{\nabla u(x)}{|\nabla u(x)|}$ for the unit normal of the level set of $u$ through $x$;
\item $\sff(x)$ for the second fundamental form of the level set of $u$ through $x$;
\item $\mathcal{A}(x)$ for the ``Allen--Cahn'' or ``enhanced'' second fundamental form of the level set:
\[ \mathcal{A} = \frac{\nabla^2 u - \nabla^2 u(\cdot, \nu) \otimes \nu^\flat}{|\nabla u|} \left( = \nabla \left( \frac{\nabla u}{|\nabla u |} \right)(x) \right). \]
\end{enumerate}
One may check that
\[
|\mathcal{A}(x)|^{2} = |\sff(x)|^{2} + |\nabla_{T} \log|\nabla u(x)||^{2},
\]
where $\nabla_{T}$ represents the gradient in the directions orthogonal to $\nabla u$; in other words, $|\mathcal{A}|$ strictly dominates the second fundamental form of the level sets.
Finally, we will often use Fermi coordinates centered on a hypersurface. To avoid confusion about which hypersurface the coordinates are associated to, we will define a function
\[ Z_\Sigma(y,z) \triangleq \exp_y(z \nu_\Sigma(y)), \; y \in \Sigma, \; z \in \mathbf{R}, \]
where $\nu_\Sigma$ will denote a distinguished normal vector to $\Sigma$. In this paper, $\nu_\Sigma$ is generally taken to be the upward pointing unit normal. Note that the pullback of the metric $g$ along $Z_{\Sigma}$ has the form $g_{z} + dz^{2}$, which is the setting that most of our analysis will take place below.
\subsection{Main results}
\subsubsection{Curvature estimates for stable solutions of \eqref{eq:ac.pde} on $3$-manifolds} We start this section by discussing the concept of stability applied to minimal surfaces, since that guides some aspects of our work in the Allen--Cahn setting.
We recall that a two-sided minimal surface $\Sigma^{2} \subset (M^{3},g)$ with normal vector $\nu$ is said to be \emph{stable} if it satisfies
\begin{equation}\label{eq:stable.min.surf}
\int_{\Sigma} \left( |\nabla_\Sigma \zeta|^{2} - (|\sff_\Sigma|^{2} + \ricc_g(\nu,\nu))\zeta^{2}\right) d\mu_g \geq 0
\end{equation}
for $\zeta \in C^{\infty}_{c}(\Sigma)$. Here, we briefly recall the well-known curvature estimates of Schoen \cite{Sch83} for stable minimal surfaces. If $\Sigma^{2}\subset (M^{3},g)$ is a complete, two-sided stable minimal surface, then the second fundamental form of $\Sigma$, $\sff_{\Sigma}$, satisfies
\begin{equation}\label{eq:curv.est.Schoen}
|\sff_{\Sigma}|(x) d(x,\partial\Sigma) \leq C = C(M,g).
\end{equation}
Observe that \eqref{eq:curv.est.Schoen} readily implies a stable Bernstein theorem: ``a complete two-sided stable minimal surfaces $\Sigma$ in $\mathbf{R}^{3}$ without boundary must be a flat plane.'' On the other hand, the stable Bernstein theorem (proven in \cite{Fischer-Colbrie-Schoen,doCarmoPeng,Pogorelov}) implies \eqref{eq:curv.est.Schoen} by a well known blow-up argument: if \eqref{eq:curv.est.Schoen} failed for a sequence of stable minimal surfaces $\Sigma_{j}$, then by choosing a point of (nearly) maximal curvature and rescaling appropriately (cf.\ \cite{White:PCMI}), we can produce $\tilde\Sigma_{j}$ a sequence of minimal surfaces in manifolds $(M_{j}^{3},g_{j})$ that are converging on compact sets to $\mathbf{R}^{3}$ with the flat metric, and so that $d_{g_{j}}(0,\partial\Sigma_{j}) \to \infty$, $|\sff_{\Sigma_{j}}|$ uniformly bounded on compact sets, and $|\sff_{\Sigma_{j}}|(0) = 1$. The second fundamental form bounds yield local $C^{2}$ bounds for the surfaces $\Sigma_{j}$, which may then be upgraded to $C^{k}$ bounds for all $k$. Thus, passing to a subsequence, the surfaces $\Sigma_{j}$ converge smoothly to a complete stable minimal surface $\Sigma_{\infty}$ without boundary in $\mathbf{R}^{3}$. Because the convergence occurs in $C^{2}$, the we see that $|\sff_{\Sigma_{\infty}}|(0) = 1$, so $\Sigma_{\infty}$ is non-flat. This contradicts the stable Bernstein theorem.
As such, before discussing curvature estimates for stable solution to Allen--Cahn, we must discuss the stable Bernstein theorem for complete solutions on $\mathbf{R}^{3}$. In general, it is not known if there are stable solutions to Allen--Cahn $\Delta u = W'(u)$ on $\mathbf{R}^{3}$ with non-flat level sets. However, under the additional assumption of quadratic energy growth, i.e.,
\[
(E_{1} \restr B_R(0))[u] \leq \Lambda R^{2},
\]
then it follows from the work of Ambrosio--Cabre \cite{AmbrosioCabre00} (see also \cite{FarinaMariValdinoci13}) that $u$ has flat level sets. We note that the corresponding stable Bernstein theorem on $\mathbf{R}^{2}$ is known to hold without any energy growth assumption; see the works of Ghoussoub--Gui \cite{GhoussoubGui98} and Ambrosio--Cabre \cite{AmbrosioCabre00}.
As such, one may expect that the blow-up argument described above may be used to prove curvature estimates. However, there is a fundamental difficulty present in the Allen--Cahn setting: if $u_{i}$ are stable solutions of \eqref{eq:ac.pde} on $(M^{3},g)$, then if their curvature (we will make this precise below) is diverging, then if we rescale by a factor $\lambda_{i}\to\infty$ in a blow-up argument this changes $\varepsilon_{i}$ to $\lambda_{i}\varepsilon_{i}$. If $\lambda_{i}\varepsilon_{i}$ converges to a non-zero constant, then standard elliptic regularity implies the rescaled functions limit smoothly to an entire stable solution of Allen--Cahn on $\mathbf{R}^{3}$. The smooth convergence guarantees that this solution will have non-flat level sets. If the original functions $u_{i}$ had uniformly bounded energy, we can show that the limit has quadratic area growth, which contradicts the aforementioned Bernstein theorem. However, if $\lambda_{i}\varepsilon_{i}$ still converges to zero, we must argue differently. In this case, we have a sequence of solutions to Allen--Cahn whose level sets are uniformly bounded in a $C^{2}$-sense. This can be used to show that the level sets converge to a plane (possibly with multiplicity) in the $C^{1,\alpha}$-sense. If the level sets behaved precisely like minimal surfaces, we could upgrade this $C^{1,\alpha}$-convergence using elliptic regularity, to conclude that the limit was not flat. However, in this situation, the level sets themselves do not satisfy a good PDE, so this becomes a significant obstacle.
Recently, a fundamental step in understanding this issue has been undertaken by Wang--Wei \cite{WangWei}. They have developed a technique for gaining geometric control of solutions to Allen--Cahn whose level sets are converging with Lipschitz bounds. Using this (and the $2$-dimensional stable Bernstein theorem) they have proven curvature estimates for individual level sets of stable solutions on two-dimensional surfaces. Moreover, they have shown that if one cannot upgrade $C^{2}$ bounds to $C^{2,\alpha}$ convergence, then by appropriately rescaling the height functions of the nodal sets, one obtains a nontrivial solution to the a system of PDE's known as the Toda system (see \cite[Remark 14.1]{WangWei}). Finally, their proof of curvature estimates in $2$-dimensions points to the crucial observation that it is necessary to use stability to upgrade the regularity of the convergence of the level sets.
This brings us to our first main result here, which is an extension of the Wang--Wei curvature estimates to $3$ dimensions. Our $3$-dimensional curvature estimates can be roughly stated as follows (see Theorem \ref{theo:curvature.estimate} for a slightly more refined statement and the proof)
\begin{theo}\label{theo:curv.est}
For a complete Riemannian metric on $\overline{B_{2}}(0) \subset \mathbf{R}^{3}$ and a stable solution $u$ to \eqref{eq:ac.pde} with $E_{\varepsilon}(u) \leq E_{0}$, the enhanced second fundamental form of $u$ satisfies
\[
\sup_{B_{1}(0) \cap \{|u| < 1-\beta\}} |\mathcal{A}|(x)\leq C = C(g,E_{0},W,\beta)
\]
as long as $\varepsilon >0$ is sufficiently small.
\end{theo}
We emphasize that Wang--Wei's $2$-dimensional estimates \cite[Theorem 3.7]{WangWei} do not require the energy bound (see also \cite[Theorem 4.13]{Mantoulidis} for the Riemannian modifications of this result). Note that we cannot expect to prove estimates with a constant that tends to $0$ as $\varepsilon\searrow 0$ (which was the case in \cite{WangWei}) since---unlike geodesics---minimal surfaces do not necessarily have vanishing second fundamental form.
We note that due to our curvature estimates, it is not hard to see that stable (and more generally, uniformly bounded index) solutions to the Allen--Cahn equation (with uniformly bounded energy) in a $3$-manifold limit to a $C^{1,\alpha}$ surface that has vanishing (weak) mean curvature. Standard arguments thus show that the surface is smooth. Thus, our estimates show that it is possible to completely avoid the regularity results of Wickramasekera and Wickramasekera--Tonegawa \cite{Wickramasekera14,TonegawaWickramasekera12} in the setting of Allen--Cahn min-max on a $3$-manifold (cf.\ \cite{Guaraco}).
\begin{rema}
We briefly remark on the possibility of extending curvature estimates to higher dimensions:
\begin{enumerate}
\item For $n \geq 8$, curvature estimates fail for stable (and even minimizing) solutions to the Allen--Cahn equation. See: \cite{PacardWei:stable,LiuWangWei}.
\item For $4\leq n \leq 7$, the Allen--Cahn stable Bernstein result is not known (even with an energy growth condition).
\end{enumerate}
Even if the stable Bernstein theorem were to be established in dimensions $4\leq n \leq 7$, we note that our proof currently uses the dimension restriction $n=3$ in one other place: we use a logarithmic cutoff function in the proof of our sheet separation estimates (Propositions \ref{prop:bootstrapped.stable.estimates} and \ref{prop:ultimate.stable.estimates}). \footnote{Added in proof: Wang--Wei have recently found \cite{WangWei2} the appropriate higher dimensional replacement for the log-cutoff argument used here. We note that the stable Bernstein problem for Allen--Cahn remains open in dimensions $4\leq n\leq 7$.}
On the other hand, we remark that the curvature estimate for minimizing solutions can be proven using the ``multiplicity one'' nature of minimizers \cite[Theorem 2]{HutchinsonTonegawa00}, together with \cite[Section 15]{WangWei} (or Remark \ref{rema:major.goal}).
We note that the case of complete minimizers is closely related to the well known ``De Giorgi conjecture.'' See \cite{GhoussoubGui98,AmbrosioCabre00,Savin:DGconj,delPinoKowalczykWei:DG-counterexample,Wang:Allard}.
\end{rema}
\subsubsection{Strong sheet separation estimates for stable solutions} A key ingredient in the proof of our curvature estimates is showing that distinct sheets of the nodal set of a stable solution to the Allen--Cahn equation remain sufficiently far apart. This aspect was already present in the work of Wang--Wei. For our applications to the case of uniformly bounded Morse index (and thus min-max theory), we must go beyond the sheet separation estimates proven in \cite{WangWei}. We prove in Proposition \ref{prop:ultimate.stable.estimates} that distinct sheets of nodal sets of a stable solution to the Allen--Cahn equation must be separated by a sufficiently large distance so that the location of the nodal sets becomes ``mean curvature dominated.''
In particular, as a consequence of these estimates, we show in Theorem \ref{theo:bounded.index} that if a sequence of stable solutions to the Allen--Cahn equation converge with multiplicity to a closed two-sided minimal surface $\Sigma$, then there is a positive Jacobi field along $\Sigma$ (which implies that $\Sigma$ is stable). It is interesting to compare this to the examples constructed by del Pino--Kowalczyk--Wei--Yang of minimal surfaces in $3$-manifolds with positive Ricci curvature that are the limit with multiplicity of solutions to the Allen--Cahn equation \cite{delPinoKowalczykWeiYang:interface}. Note that such a minimal surface cannot admit a positive Jacobi field, so the point here is that the Allen--Cahn solutions are not stable. (In fact, our Theorem \ref{theo:bounded.index} implies that they have diverging Morse index.) Note that the separation $D$ between the sheets of the examples constructed in \cite{delPinoKowalczykWeiYang:interface} satisfy, as $\varepsilon \searrow 0$,
\[
D \sim \sqrt{2} \varepsilon |\log\varepsilon| - \frac{1}{\sqrt{2}}\varepsilon \log | \log \varepsilon| ,
\]
while we prove in Proposition \ref{prop:ultimate.stable.estimates} that stability implies that the separation satisfies
\[
D - \left( \sqrt{2} \varepsilon |\log \varepsilon| - \frac{1}{\sqrt{2}} \varepsilon \log|\log \varepsilon| \right) \to -\infty.
\]
We emphasize that the improved separation estimates here are not contained in the work of Wang--Wei \cite{WangWei} and are fundamental for the subsequent applications of our results.
\subsubsection{The multiplicity one-conjecture for limits of the Allen--Cahn equation in $3$-manifolds}
In their recent work \cite{MarquesNeves:multiplicity}, Marques--Neves make the following conjecture:
\begin{conj}[Multiplicity one conjecture]
For generic metrics on $(M^{n},g)$, $3\leq n\leq 7$, two-sided unstable components of closed minimal hypersurfaces obtained by min-max methods must have multiplicity one.
\end{conj}
In \cite{MarquesNeves:multiplicity}, Marques--Neves confirm this in the case of a one parameter Almgren--Pitts sweepout. The one parameter case had been previously considered for metrics of positive Ricci curvature by Marques--Neves \cite{MarquesNeves:rigidity.min.max} and subsequently by Zhou \cite{Zhou:posRic}. See also \cite[Corollary E]{Guaraco} and \cite[Theorem 1]{GasparGuaraco} for results comparing the Allen--Cahn setting to Almgren--Pitts setting which establish multiplicity one for hypersurfaces obtained by a one parameter Allen--Cahn min-max method in certain settings. We also note that Ketover--Liokumovich--Song \cite{Song,KetoverLiokumovich,KetoverLiokumovichSong} have proven multiplicity (and index) estimates for one parameter families in the Simon--Smith \cite{SimonSmith} variant of Almgren--Pitts in $3$-manifolds.\footnote{Added in proof: As noted before, the full multiplicity one conjecture for Almgren--Pitts (in dimensions $3$ through $7$) has now been proven by X. Zhou \cite{Zhou:multiplicity-one}.}
We recall the following standard definition:
\begin{defi} \label{def:bumpy.metric}
We say that a metric $g$ on a Riemannian manifold $M^{n}$ is \emph{bumpy} if there is no immersed closed minimal hypersurface $\Sigma^{n-1}$ with a non-trivial Jacobi field.
\end{defi}
By work of White \cite{White:bumpy.old,White:bumpy.new}, bumpy metrics are generic in the sense of Baire category. Here, ``generic'' will always mean in the Baire category sense.
We are able to prove a strong version of the multiplicity one conjecture (when $n=3$) for minimal surfaces obtained by Allen--Cahn min-max methods with an \emph{arbitrary} number of parameters. Such a method was set up by Gaspar--Guaraco \cite{GasparGuaraco}.
Indeed, we prove that for \emph{any} metric $g$ on a closed $3$-manifold, the unstable components of such a surface are multiplicity one. Moreover, for a generic metric, we show that \emph{each} component of the surface occurs with multiplicity one (not just the unstable components). Finally, we are able to show for generic metrics on a $n$-manifold, $3\leq n\leq 7$, the minimal surfaces constructed by Allen--Cahn min-max methods are two-sided. For a one-parameter Almgren--Pitts sweepoints in a $n$-manifold $3\leq n\leq 7$ with positive Ricci curvature, this was proven by Ketover--Marques--Neves \cite{KMN:catenoid}. More precisely, our main results here are as follows (see Theorem \ref{theo:bounded.index} and Corollary \ref{coro:mult.one.conj} for the full statements).
\begin{theo}[Multiplicity and two-sidedness of minimal surfaces constructed via Allen--Cahn min-max]\label{theo:mult.intro-version}
Let $\Sigma^{2}\subset (M^{3},g)$ denote a smooth embedded minimal surface constructed as the $\varepsilon\searrow 0$ limit of solutions to the Allen--Cahn equation on a $3$-manifold with uniformly bounded index and energy. If $\Sigma$ occurs with multiplicity or is one-sided, then it carries a positive Jacobi field (on its two-sided double cover, in the second case).
Note that positive Jacobi fields do not occur when $g$ is bumpy or when $g$ has positive Ricci curvature. Thus, in either of these cases, each component of $\Sigma$ is two-sided and occurs with multiplicity one.
\end{theo}
\begin{rema}
We re-emphasize that our theorem applies generally to sequences of Allen--Cahn solutions with uniformly bounded energy and Morse index. Thus, unlike the proofs in the Almgren--Pitts setting, we do not need to make use of any min-max characterization of the limiting surface to rule out multiplicity.
\end{rema}
Our proof here is modeled on the study of bounded index minimal hypersurfaces in a Riemannian manifold. Indeed, Sharp has shown that minimal hypersurfaces in $(M^{n},g)$ for $3\leq n\leq 7$ with uniformly bounded area and index are smoothly compact away from finitely many points where the index can concentrate \cite{Sharp} (see also White's proof \cite{White:curvature} of the Choi--Schoen compactness theorem \cite{ChoiSchoen}). A crucial point there is to prove that higher multiplicity of the limiting surface produces a positive Jacobi field (even across the points of index concentration (where the convergence of the hypersurfaces need not occur smoothly). This can be handled via an elegant argument of White, based on the construction of a local foliation by minimal surfaces to use as a barrier for the limiting surfaces (cf.\ \cite{White:compactness.new}).
In the minimal surface setting, the existence of the foliation is a simple consequence of the implicit function theorem. However, in the Allen--Cahn setting, the singular limit $\varepsilon\searrow 0$ limit complicates this argument. Instead, we construct barriers by a more involved fixed point method in Theorem \ref{theo:dirichlet.data.construction}. Once that theorem is proven, we show how the barriers can be used to bound the Jacobi fields along the points of index concentration in the process of the proof of Theorem \ref{theo:bounded.index} by carrying out a new sliding plane type argument for the Allen--Cahn equation on Riemannian manifolds. Our proof of Theorem \ref{theo:dirichlet.data.construction} is modeled on the work of Pacard \cite{Pacard12} (with appropriate extension to the case of Dirichlet boundary conditions), but there is a significant technical obstruction here: we do not know that the level sets of the solution Allen--Cahn converge smoothly, but only in $C^{2,\alpha}$. To apply the fixed point argument, we need some control on higher derivatives. By an observation of Wang--Wei \cite[Lemma 8.1]{WangWei}, we control one higher derivative of the level sets, but only by a constant that is $O(\varepsilon^{-1})$ (see \eqref{eq:dirichlet.data.sigma.c3alpha}). This complicates the proof of Theorem \ref{theo:dirichlet.data.construction}.
\subsubsection{Index lower bounds}
Lower semicontinuity of the Morse index along the singular limit $\varepsilon\searrow 0$ of a sequence of solutions to the Allen--Cahn equation is proven by Hiesmayr \cite{Hiesmayr} (for two-sided surfaces) and Gaspar \cite{Gaspar} without assuming two-sidedness (see also \cite{Le:2ndvar}). On the other hand, upper semicontinuity of the index does not hold in general (cf.\ Example \ref{exam:upper.semi.fails.index}). Here, we establish upper semicontinuity of the index, in all dimensions, under the a priori assumption that the limiting surface is multiplicity one.\footnote{We note that Marques--Neves had previously announced the analogous index uppper-semicontinuity result for multiplicity one Almgren--Pitts limits and that their proof \cite{MarquesNeves:uper-semi-index} appeared shortly after the first version of this paper.} In particular we prove (see Theorem \ref{theo:index.lower.bounds} for the full statement)
\begin{theo}[Upper semicontinuity of the index in the multiplicity one case] \label{theo:index.semicontinuity}
Suppose that a smooth embedded minimal hypersurface $\Sigma^{n-1}\subset (M^{n},g)$ is the multiplicity one limit as $\varepsilon\searrow 0$ of a sequence of solutions $u$ to the Allen--Cahn equation. Then for $\varepsilon>0$ sufficiently small,
\[
\nul(\Sigma) + \ind(\Sigma) \geq \nul(u) + \ind(u).
\]
\end{theo}
To prove this upper semicontinuity, we need to delve deeper into the equation that controls the level sets of $u$ and obtain a more accurate approximation. What was done for Theorem \ref{theo:curv.est}---while well suited to understanding the phenomenon of multiplicity---does not suffice for Theorem \ref{theo:index.semicontinuity}.
\subsubsection{Applications related to Yau's conjecture on infinitely many minimal surfaces}
A well known conjecture of Yau posits that any closed $3$-manifold admits infinitely many immersed minimal surfaces \cite{Yau:problems}. By considering the $p$-widths introduced by Gromov \cite{Gromov:waist} (see also \cite{Guth:minimax}), Marques--Neves proved \cite{MarquesNeves:posRic} that a closed Riemannian manifold $(M^{n},g)$ (for $3\leq n\leq 7$) with positive Ricci curvature admits infinitely many minimal surfaces. Moreover, by an ingenious application of the Weyl law for the $p$-widths proven by Liokumovich--Marques--Neves \cite{LMN:Weyl}, Irie--Marques--Neves \cite{IrieMarquesNeves} (see also the recent work of Gaspar--Guaraco \cite{GasparGuaraco:weyl} that appeared after the first version of this paper was posted) have recently shown that the set of metrics on a closed Riemannian manifold $(M^{n},g)$ (with $3\leq n \leq 7$) with the property that the set of minimal surfaces is dense in the manifold is generic (see also \cite{MarquesNevesSong}).
We note that the arguments in each of \cite{MarquesNeves:posRic,IrieMarquesNeves,GasparGuaraco:weyl} to prove the existence of infinitely many minimal surfaces are \emph{necessarily} \emph{indirect}, as they do not rule out the $p$-widths being achieved with higher multiplicity. Having overcome this obstacle, we may give a ``direct'' proof (for $n=3$) of Yau's conjecture for bumpy metrics\footnote{We note that \cite{IrieMarquesNeves,GasparGuaraco:weyl} prove Yau's conjecture for a different (also generic) set of metrics.} with some new geometric conclusions (see Corollaries \ref{coro:mult.one.conj}, \ref{coro:Yau.conj} for proofs).
\begin{coro}[Yau's conjecture for bumpy metrics and geometric properties of the minimal surfaces]
\label{coro:yau-intro}
Let $(M^{3},g)$ denote a closed $3$-manifold with a bumpy metric. Then, there is $C=C(M,g,W)>0$ and a smooth embedded minimal surfaces $\Sigma_{p}$ for each positive integer $p>0$ so that
\begin{itemize}
\item each component of $\Sigma_{p}$ is two-sided,
\item the area of $\Sigma_{p}$ satisfies $C^{-1} p^{\frac 1 3}\leq \area_{g}(\Sigma_{p}) \leq C p^{\frac 1 3}$,
\item the index of $\Sigma_{p}$ is satisfies $\ind(\Sigma_{p}) = p$,
and
\item the genus of $\Sigma_{p}$ satisfies $\genus(\Sigma_p) \geq \frac p 6 - C p^{\frac 1 3}$.
\end{itemize}
In particular, thanks to the index estimate, all of the $\Sigma_{p}$ are geometrically distinct.
\end{coro}
We emphasize that each of the bullet points in the preceding corollary do not follow from the work of Irie--Marques--Neves \cite{IrieMarquesNeves}. Some of these properties were conjectured by Marques and Neves in \cite[p.\ 24]{marques:ICM}, \cite[p.\ 17]{neves:ICM}, \cite[Conjecture 6.2]{MarquesNeves:spaceOfCycles}. In particular, they conjectured that a generic Riemannian manifold contains an embedded two-sided minimal surface of each positive Morse index.
\begin{rema}[Yau's conjecture for $3$-manifolds with positive Ricci curvature]
We note that because the multiplicity-one property also holds even for non-bumpy metrics of positive Ricci curvature, we may also give a ``direct'' proof of Yau's conjecture for a $3$-manifold with positive Ricci curvature (this was proven by Marques--Neves \cite{MarquesNeves:posRic} in dimensions $3\leq n\leq 7$ using Almgren--Pitts theory). We obtain, exactly as in Corollary \ref{coro:Yau.conj}, the new conclusions that the surfaces $\Sigma_{p}$ are two-sided, have $\area(\Sigma_{p})\sim p^{\frac 13}$, $\ind(\Sigma_{p}) \leq p$ and $\nul(\Sigma_{p})+\ind(\Sigma_{p})\geq p$.
Moreover, approximating the metric by a sequence of bumpy metrics and passing to the limit (the limit occurs smoothly and with multiplicity one due to the positivity of the Ricci curvature, cf.\ \cite{Sharp}), we find that there is a sequence $\Sigma_{p}'$ (we do not know if this is the same sequence as $\Sigma_{p}$) with these properties and additionally satisfies the genus bound (note that $\Sigma_{p}$ is connected by Frankel's theorem) for possibly a larger constant $C$
\[
\genus(\Sigma_{p}') \geq \frac{p}{6} - Cp^{\frac 13}.
\]
It is interesting to observe that when $(M^{3},g)$ is the round $3$-sphere, combining our bound $\ind(\Sigma_{p}') \leq p$ with work of Savo \cite{Savo} implies that
\[
\genus(\Sigma_{p}') \leq 2 p - 8
\]
as long as $p$ is sufficiently large to guarantee that $\genus(\Sigma_{p}') \geq 1$. Similar conclusions can be derived in certain other $3$-manifolds embedded in Euclidean spaces by \cite{AmbrozioCarlottoSharp:index.genus}.
There has been significant activity concerning the index of the minimal surfaces constructed in \cite{MarquesNeves:posRic}, but before the present work, all that was known was that: for a bumpy metric of positive Ricci curvature, there are closed embedded minimal surfaces of arbitrarily large Morse index \cite{LiZhou,CKM,Carlotto:arb-large}, albeit without information on their area.
\end{rema}
\begin{rema}[Connected components in Corollary {\ref{coro:yau-intro}}]
Unless $(M, g)$ has the Frankel property (e.g., when it has positive Ricci curvature), the minimal surfaces $\Sigma_p$ obtained in Corollary \ref{coro:yau-intro} may be disconnected. In this case, every connected component $\Sigma_p'$ of $\Sigma_p$ must satisfy:
\begin{itemize}
\item $\Sigma_p'$ is two-sided and has $\area_g(\Sigma_p')\leq C p^{\frac 1 3}$,
\end{itemize}
and, by a counting argument, there will exist at least one component $\Sigma_p'$ of $\Sigma_p$ such that
\begin{itemize}
\item $\genus(\Sigma_p') \geq C^{-1} \ind(\Sigma_p') \geq C^{-1} p^{\frac 2 3}$.
\end{itemize}
See Corollary \ref{coro:Yau.conj.components}.
It is not clear that the component $\Sigma_{p}'$ will have unbounded area. In a follow up paper \cite{ChodoshMantoulidis:unbounded-area} we prove the following dichotomy; either
\begin{enumerate}
\item $(M,g)$ contains a sequence of connected closed embedded stable minimal surfaces with unbounded area, or
\item some connected component $\Sigma_{p}''$ of the surfaces $\Sigma_{p}$ obtained in Corollary \ref{coro:yau-intro} has $\area_{g}(\Sigma_{p}'')\geq Cp^{\frac 13}$.
\end{enumerate}
We note that by \cite{CKM,Carlotto:arb-large}, when $(M^{3},g)$ is a bumpy metric with positive scalar curvature the prior condition cannot hold, so the latter alternative holds and, moreover, $\ind(\Sigma_{p}'')\to\infty$. It would be interesting to determine if one can find a connected component $\Sigma_{p}''$ with arbitrarily large area and $\ind(\Sigma_{p}'')\geq c p$ for some $c\in (0,1)$.
\end{rema}
\subsection{One-dimensional heteroclinic solution, $\mathbb{H}$} \label{subsec:heteroclinic.solution}
Recall that the one-dimensional Allen-Cahn equation with $\varepsilon=1$ is $u'' = W'(u)$, for a function $u = u(t)$ of one variable. It's not hard to see that this ODE admits a unique bounded solution with the properties
\[ u(0) = 0, \; \lim_{t \to -\infty} u(t) = -1, \; \lim_{t \to \infty} u(t) = 1. \]
We call this the one-dimensional heteroclinic solution, and denote it as $\mathbb{H} : \mathbf{R} \to (-1, 1)$. It's also standard to see that the heteroclinic solution satisfies:
\begin{align}
\mathbb{H}(\pm t) & = \pm 1 \mp A_0 \exp(-\sqrt{2} t) + O(\exp(-2\sqrt{2} t)), \label{eq:heteroclinic.expansion.i} \\
\mathbb{H}'(\pm t) & = \sqrt{2} A_0 \exp(-\sqrt{2} t) + O(\exp(-2\sqrt{2} t)), \label{eq:heteroclinic.expansion.ii} \\
\mathbb{H}''(\pm t) & = - 2 A_0 \exp(-\sqrt{2} t) + O(\exp(-2\sqrt{2} t)), \label{eq:heteroclinic.expansion.iii}
\end{align}
as $t \to \infty$, for some fixed $A_0 > 0$ that depends on $W$. Moreover,
\[ \int_{-\infty}^\infty (\mathbb{H}'(t))^2 \, dt = h_0, \]
where $h_0 > 0$ also depends on $W$; it is explicitly given by
\[ h_0 = \int_{-1}^1 \sqrt{2W(t)} \, dt. \]
Finally, we also define
\begin{equation} \label{eq:heteroclinic.eps}
\mathbb{H}_\varepsilon(t) \triangleq \mathbb{H}(\varepsilon^{-1} t), \; t \in \mathbf{R},
\end{equation}
which is clearly a solution of $\varepsilon^2 \mathbb{H}_\varepsilon'' = W'(\mathbb{H}_\varepsilon)$.
\subsection{Organization of the paper}
In Section \ref{sec:jacobi.toda.reduction} we make precise \emph{the dependence of the regularity} of the nodal set $\{ u = 0 \}$ of bounded energy and bounded curvature solutions of \eqref{eq:ac.pde} \emph{on the distance} between its different sheets. The dependence is essentially modeled by a Toda system; see, e.g., \eqref{eq:jacobi.toda} and Remark \ref{rema:major.goal}. Restricting to $n=3$ dimensions, in Section \ref{sec:stable.solutions} we use the stability of Allen--Cahn solutions to bootstrap the distance estimates from Section \ref{sec:jacobi.toda.reduction} until they become sharp. In Section \ref{sec:bounded.index} we study solutions of \eqref{eq:ac.pde} with bounded energy and Morse index in $n=3$ dimensions. We use our strong sheet separation estimates from Section \ref{sec:stable.solutions} to construct, in the presence of multiplicity, positive Jacobi fields on the limiting minimal surface away from finitely many points. Then, a ``sliding plane'' argument (modulo a barrier construction deferred to Section \ref{sec:dirichlet.data}) allows us to extend the Jacobi field to the entire limiting surface.
In Section \ref{sec:multiplicity.one} we return to the arbitrary dimensional setting and prove the Morse index is lower semicontinuous for smooth multiplicity one limits.
In Section \ref{sec:applications} we apply all our tools to prove a strong form of Marques' and Neves' multiplicity one conjecture, and Yau's conjecture for generic metrics.
In Section \ref{sec:dirichlet.data} we construct curved sliding plane barriers for \eqref{eq:ac.pde} that resemble multiplicity-one heteroclinic solutions with prescribed Dirichlet data centered on nondegenerate minimal submanifolds-with-boundary $\Sigma^{n-1} \subset (M^n, g)$, $n \geq 3$.
In Appendix \ref{app:mean.curvature.graphs}, we recall several expressions related to the mean curvature and second fundamental form of graphical hypersurfaces in a Riemannian manifold. In Appendix \ref{app:WW-results} we recall several auxiliary results from \cite{WangWei}. In Appendix \ref{app:proof.lem.comp.improved}, we prove Lemma \ref{lemm:h.phi.comparison.improved} relating regularity of the ``centering'' functions $h_{\ell}$ to that of the function $\phi$ with improved error estimates. In Appendix \ref{app:proof.stab.inproved}, we derive the Toda-system stability inequality with improved error estimates \eqref{eq:toda.stability.estimate.sharper}. In Appendix \ref{app:interpolation.lemma} we recall an interpolation inequality for H\"older norms.
\subsection{Acknowledgments}
O.C. was supported in part by the Oswald Veblen fund and NSF Grant no.\ 1638352. He would like to thank Simon Brendle and Michael Eichmair for their continued support and encouragement, as well as Costante Bellettini, Guido De Philippis, Daniel Ketover, and Neshan Wickramasekera for their interest and for enjoyable discussions. C.M. would like to thank Rick Schoen, Rafe Mazzeo, and Yevgeniy Liokumovich for helpful conversations on topics addressed by this paper. Both authors would like to thank Fernando Cod\'a Marques and Andr\'e Neves very much for their interest and encouragement. They are also grateful to Davi Maximo for pointing out a mistake in the original version of Corollary \ref{coro:yau-intro}. This work originated during the authors' visit to the Erwin Schr\"odinger International Institute for Mathematics and Physics (ESI) during the ``Advances in General Relativity Workshop'' during the summer of 2017, which they would like to acknowledge for its support. Finally, the authors would like to thank the referee for their careful reading of the manuscript and many helpful suggestions.
\section{From phase transitions to Jacobi-Toda systems}
\input{jacobi-toda-reduction}
\section{Stable phase transitions ($n=3$)}
\input{stable}
\section{Phase transitions with bounded Morse index ($n=3$)}
\input{bounded-index}
\section{Phase transitions with multiplicity one}
\input{multiplicity-one}
\section{Geometric applications}
\input{applications}
\section{Barriers with Dirichlet data}
\input{dirichlet-data}
\subsection{Improved convergence} \label{subsec:multiplicity.one.convg}
Note that by scaling $M$, we can arrange that \eqref{eq:sheets.sff.bound}-\eqref{eq:sheets.metric.bound} hold; we will do so without further remark in the sequel. Note that then, due Lemma \ref{lemm:multiplicity.one.lower.gradient.bound} below, \eqref{eq:sheets.eps.bound}-\eqref{eq:sheets.enhanced.sff.grad.bound} hold as well. Thus, Section \ref{sec:jacobi.toda.reduction} applies (as does \cite[Section 15]{WangWei} in the flat setting).
\begin{lemm} \label{lemm:multiplicity.one.lower.gradient.bound}
Let $U \subset \subset M \setminus \partial M$ be a neighborhood of $\Sigma$, and $\beta \in (0, 1)$. Then, for sufficiently large $i$, $\varepsilon_i |\nabla u_i| \geq c > 0$ on $U \cap \{ |u_i| \leq 1-\beta \}$.
\end{lemm}
\begin{proof}
We argue by contradiction. If the result were false, we'd be able to pick a subsequence (labeled the same) along which there would exist $x_i \in U \cap \{ |u_i| \leq 1-\beta \}$ with $\varepsilon_i |\nabla u_i(x_i)| \to 0$. After rescaling by $\varepsilon_i^{-1}$ around $x_i$, the rescaled critical points $\widetilde{u}_i$ converge to a nontrivial critical point of $E_1$ on $\mathbf{R}^n$ with $|\widetilde{u}(0)| \leq 1-\beta$, $\nabla \widetilde{u}(0) = 0$. By the monotonicity formula (see \cite[Section 3]{HutchinsonTonegawa00} and \cite[Appendix B]{Guaraco}) and multiplicity one convergence at the original scale, we see that the tangent cone at infinity of $\widetilde{u}$ is a multiplicity one plane. Hence, by \cite[Theorem 11.1]{Wang:Allard} (cf.\ \cite[Theorem 3.6]{Mantoulidis}), $\widetilde{u}$ has flat level sets. This contradicts $|\widetilde{u}(0)| \leq 1-\beta$, $\nabla \widetilde{u}(0) = 0$.
\end{proof}
Combined with the multiplicity one analysis in \cite[Section 15]{WangWei} (cf. Section \ref{sec:jacobi.toda.reduction} and Remark \ref{rema:major.goal} above), we may argue as in the proof of Theorem \ref{theo:bounded.index} to conclude that $\Sigma=\supp V$ is a smooth two-sided embedded minimal hypersurface and the convergence of the level sets of $u_{i}$ to $\Sigma$ occurs in $C^{2,\theta}$. (Of course, convergence in the Hausdorff sense follows immediately from \cite[Theorem 1]{HutchinsonTonegawa00}.)
\begin{lemm} \label{lemm:multiplicity.one.convergence}
If $U \subset \subset M \setminus \partial M$ is a neighborhood of $\Sigma$, and $\theta,\beta \in (0, 1)$, then $U \cap \{ u_i = t \}$ converges uniformly in $C^{2,\theta}$ to $\Sigma$, for every $t \in (-1+\beta, 1-\beta)$.
\end{lemm}
\begin{proof}
By Section \ref{sec:jacobi.toda.reduction}, it suffices to check that the level sets are bounded in $C^2$. One uses a blow-up argument again, as in the proof of Theorem \ref{theo:curvature.estimate}. Suppose that the enhanced second fundamental form weren't bounded. Pick $x_i \in U \cap \{ |u_i| \leq 1-\beta \}$ such that $\lambda_i \triangleq |\mathcal{A}_i(x_i)|$ are within a factor of $\tfrac{1}{2}$ from $\sup_{U \cap \{|u_i| \leq 1-\beta\}} |\mathcal{A}_i|$; thus, $\lambda_i \to \infty$. Note that $\limsup_i \lambda_i \varepsilon_i < \infty$ by elliptic regularity. Moreover, we in fact have that $\limsup_i \lambda_i \varepsilon_i = 0$ because (by \cite[Theorem 11.1]{Wang:Allard} and monotonicity) there are no nontrivial (i.e., nonconstant and nonheteroclinic) entire critical points of $E_1$ in $\mathbf{R}^n$ with a planar tangent cone at infinity. In particular, rescaling by $\lambda_i^{-1}$ around $x_i$, we get a sequence $(\widetilde{u}_i, \widetilde{\varepsilon}_i)$ with $\widetilde{\varepsilon}_i \to 0$ and uniformly bounded enhanced second fundamental form, $|\widetilde{\mathcal{A}}_i(0)| = 1$, and which therefore converges to a $C^{1,1}$ minimal surface in $\mathbf{R}^n$. However, by monotonicity, this minimal surface is a plane; this contradicts $|\widetilde{\mathcal{A}}_i(0)| = 1$ by Remark \ref{rema:major.goal}.
\end{proof}
Let's return to the notation and conventions used in Section \ref{sec:jacobi.toda.reduction}. Also, we drop the subscript $i$.
Because of the multiplicity one assumption, we have reasonably strong estimates on $\phi,h,$ and $H_{\Gamma}$; see \eqref{eq:discrepancy.function}. We will write $h$ for $\ve{h}$, $U$ for $U[\ve{h}]$, $\Gamma$ for $\Gamma_1$, and $d$ for $d_1$, since $Q=1$. We record the specialization of \eqref{eq:phi.c2a.estimate.full} and Lemma \ref{lemm:h.phi.comparison} here (cf.\ \cite[Section 15]{WangWei}, and \cite[Theorem 3.6]{Mantoulidis}):
\begin{equation}\label{eq:mult.one.initial.bds}
\Vert \phi \Vert_{C^{2,\theta}_{\varepsilon}(\mathcal{M})} + \varepsilon \Vert \Delta _{\Gamma} h - H_{\Gamma}\Vert_{C^{0,\theta}_{\varepsilon}(\Gamma)} + \varepsilon^{-1} \Vert h \Vert_{C^{2,\theta}_{\varepsilon}(\Gamma)}\leq c' \varepsilon^{2},
\end{equation}
where $\mathcal{M} \triangleq \{X \in M : |d(X)| < 1\}$.
As we have already indicated, we must upgrade our estimates for $\Delta_{\Gamma} h - H_{\Gamma}$ in \eqref{eq:mult.one.initial.bds} as well as determine the $O(\varepsilon^{2})$ behavior of $\phi$.
Let us work in Fermi coordinates around $\Gamma$ so as not to write the diffeomorphism $Z_{\Gamma}$ explicitly below. We will also denote $\Gamma_z \triangleq \{ X \in \mathcal{M} : d(X) = z \}$, and will write $\overline{\mathbb{H}}$ for $\overline{\mathbb{H}}{}^{3 |\log \varepsilon|}$.
We can compute the equation for $\phi$ as follows. Using \eqref{eq:mean.curv.ddt.sff}, \eqref{eq:mean.curv.ddt.h}, \eqref{eq:mean.curv.ddt.laplace}, as well as \eqref{eq:sheets.eps.bound}-\eqref{eq:sheets.enhanced.sff.grad.bound}, \eqref{eq:approximate.heteroclinic.behavior}, and \eqref{eq:mult.one.initial.bds}, one computes (see \cite[(9.4)]{WangWei}) in $\mathcal{M}$
\begin{align} \label{eq:mult.one.phi.eqn.prelim}
\varepsilon^2 \Delta_g \phi
& = \varepsilon^{2} \Delta_{\Gamma_z} \phi + \varepsilon^{2} H_{\Gamma_z}\partial_{z}\phi + \varepsilon^2 \partial^{2}_{z}\phi \nonumber\\
& = W'(u) - \varepsilon^2 \Delta_{\Gamma_z} U - \varepsilon^{2} H_{\Gamma_z} \partial_{z} U - \varepsilon^2 \partial^{2}_{z} U \nonumber \\
& = W'(U + \phi) - (W'(U) + O(\varepsilon^3)) + \varepsilon (\Delta_{\Gamma_z} h - H_{\Gamma_z}) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \nonumber \\
& \qquad - |\nabla_{\Gamma_z} h|^{2} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \nonumber \\
& = W''(U)\phi + \varepsilon ((\Delta_{\Gamma} h - H_{\Gamma}) \circ \Pi_\Gamma) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \nonumber \\
& \qquad + \varepsilon ((|\sff_{\Gamma}|^{2} +\ricc_g(\partial_{z},\partial_{z}))\circ \Pi_\Gamma) \cdot z \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \nonumber \\
& \qquad + \varepsilon^2 O(|z|) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) + O( \varepsilon^3).
\end{align}
By using \eqref{eq:mult.one.initial.bds}, \eqref{eq:mult.one.phi.eqn.prelim}, and the multiplicity one assumption, one may revisit \cite[Appendix B]{WangWei} and establish the following bounds.
\begin{lemm}\label{lemm:mult.one.imp.hHeqn}
We can improve the estimate in \eqref{eq:mult.one.initial.bds} to $\varepsilon \Vert \Delta_{\Gamma} h - H_{\Gamma}\Vert_{C^{0}(\Gamma)} \leq c' \varepsilon^{3}$.
\end{lemm}
\begin{proof}
Multiply \eqref{eq:mult.one.phi.eqn.prelim} by $\overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))$ and integrate over $z \in [-\eta,\eta]$. We find (at $y \in \Sigma$ fixed)
\begin{align*}
& \int_{-\eta}^{\eta} (\varepsilon^2 (\Delta_{\Gamma_z} \phi + H_{\Gamma_z}\partial_{z}\phi + \partial^{2}_{z}\phi )- W''(U)\phi) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))) \, dz\\
& = \varepsilon^2 (h_0-o(1)) (\Delta_{\Gamma} h - H_{\Gamma}) + \varepsilon (|\sff_{\Gamma}|^{2} + \ricc_g(\partial_{z},\partial_{z})) \int_{-\eta}^{\eta} z \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2} \, dz\\
& \qquad + \int_{-\eta}^{\eta} \varepsilon^2 O(|z|) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2} \, dz + O(\varepsilon^4)\\
& = \varepsilon^2 (h_0 - o(1)) (\Delta_{\Gamma} h - H_{\Gamma}) + O( \varepsilon^{4}).
\end{align*}
We have used \eqref{eq:heteroclinic.expansion.ii} together with $\int_{-\infty}^{\infty} t \mathbb{H}'(t)^{2} dt = 0$ (which holds by parity).
Twice differentiating the orthogonality relation \eqref{eq:h.defn.orth} used to define $h$ (see Section \ref{subsec:approximate.solutions} and \cite[Appendix B]{WangWei}) and using \eqref{eq:mult.one.initial.bds}, we have
\[
\int_{-\eta}^{\eta} \varepsilon^2 (\Delta_{\Gamma_z} \phi) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, dz = O( \varepsilon^{4}).
\]
From \eqref{eq:mult.one.initial.bds}, we have
\[
\int_{-\eta}^{\eta} \varepsilon^2 H_{\Gamma_z} \partial_{z} \phi \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, dz = O( \varepsilon^{5}).
\]
Finally, an integration by parts shows that
\begin{align*}
& \int_{-\eta}^{\eta} \left(\varepsilon^2 \partial^{2}_{z} \phi \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) - W''(u) \phi \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \right) \, dz \\
& = \int_{-\eta}^{\eta} \left( \overline{\mathbb{H}}'''(\varepsilon^{-1}(z-h(y))) - W''(u) \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \right) \phi \, dz.
\end{align*}
Using \eqref{eq:approximate.heteroclinic.behavior} here, combined with the previous expressions, we conclude the proof.
\end{proof}
Thus, returning to \eqref{eq:mult.one.phi.eqn.prelim} we find that in $\mathcal{M}$, we have:
\begin{equation}\label{eq:mult.one.phi.upgrade}
\varepsilon^2 \Delta_g \phi - W''(U)\phi = \varepsilon((|\sff_{\Gamma}|^{2} +\ricc_g(\partial_{z},\partial_{z}))\circ \Pi_\Gamma) \cdot z \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) + O(\varepsilon^3).
\end{equation}
We have used the fact that $z \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) = O(\varepsilon)$.
Observe that the right hand side of \eqref{eq:mult.one.phi.upgrade} is only bounded in $O(\varepsilon^2)$. Thus, we expect this to represent the leading term of $\phi$, after inverting $ \varepsilon^2 \Delta_g - W''(U)$. To make this precise, we first define (cf.\ \cite[Section 3.2]{delPinoKowalczykWei}) a function $\mathbb{J}(t)$ to be the unique bounded solution of the ODE
\begin{equation}\label{eq:ODE.for.J}
\mathbb{J}''(t) = W''(\mathbb{H}(t)) \mathbb{J}(t) + t \mathbb{H}'(t), \text{ with } \mathbb{J}(0) = 0.
\end{equation}
Indeed, we even have the explicit expression (cf.\ \cite[p.\ 82]{delPinoKowalczykWei})
\[
\mathbb{J}(t) = \mathbb{H}'(t) \int_{0}^{t} \int_{-\infty}^{s} \tau \mathbb{H}'(s)^{-2} \, \mathbb{H}'(\tau)^{2} d\tau ds
\]
which shows that $\mathbb{J}$ is well defined and decays exponentially as $t\to\pm\infty$. It will be important in the sequel to observe that $\mathbb{J}(-t) = -\mathbb{J}(t)$, which follows from the parity of $\mathbb{H}(t)$ and either the uniqueness of solutions to the ODE, or the explicit integral expression.
Observe that $|\sff_{\Gamma}|^{2} +\ricc_g(\partial_{z},\partial_{z})$ converges to $|\sff_{\Sigma}|^{2} +\ricc_g(\nu,\nu)$ in $C^{0,\theta}$ because $\Gamma$ converges to $\Sigma$ in $C^{2,\theta}$ by Lemma \ref{lemm:multiplicity.one.convergence}. We fix functions $V :\Gamma \to\mathbf{R}$ with the property that $V$ still converges to $|\sff_{\Sigma}|^{2} +\ricc_g(\nu,\nu)$ in $C^{0}$ and $\Vert V \Vert_{C^{2}(\Gamma)} \leq C$. For definiteness we choose $V(y) = (|\sff_{\Sigma}|^{2} +\ricc_g(\nu,\nu))\circ \Pi_{\Sigma}(y)$, where $\Pi_{\Sigma}$ is the nearest point projection to $\Sigma$.
We claim that $\varepsilon^{2}V(y) \mathbb{J}(\varepsilon^{-1}(z-h(y)))$ represents the leading order term in $\phi$. To this end, in $\mathcal{M}$, we define a refined discrepancy function
\[ \widetilde{\phi}(y,z) \triangleq \phi(y,z) - \varepsilon^{2}(V \circ \Pi_\Gamma)(y, z) \cdot \mathbb{J}(\varepsilon^{-1}(z-h(y))). \]
We compute (using the $C^{2}$ bounds for $V$, as well as \eqref{eq:mult.one.initial.bds} and Lemma \ref{lemm:mult.one.imp.hHeqn}) that on $\mathcal{M}$, we have
\begin{align*}
& \varepsilon^{2} \Delta_g \widetilde{\phi} - W''(U) \widetilde{\phi} \\
& = \varepsilon ((|\sff_{\Gamma}|^{2} +\ricc_g(\partial_{z},\partial_{z}))\circ \Pi_\Gamma) \cdot z \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \\
& \qquad - \varepsilon^{2} (V \circ \Pi_\Gamma) \big[ \mathbb{J}''(\varepsilon^{-1}(z-h(y))) - W''(U) \cdot \mathbb{J}(\varepsilon^{-1}(z-h(y))) \big] + O(\varepsilon^3)\\
& = \varepsilon \big[ (|\sff_{\Gamma}|^{2} +\ricc_g(\partial_{z},\partial_{z}))\circ \Pi_\Gamma - V\circ \Pi_\Gamma \big] \cdot z \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \\
& \qquad - \varepsilon^{2} \big[ W''(\mathbb{H}(\varepsilon^{-1}(z-h(y)))) - W''(U) \big] (V\circ \Pi_\Gamma) \cdot \mathbb{J}(\varepsilon^{-1}(z-h(y))) + O(\varepsilon^3)\\
& = o(\varepsilon^{2}).
\end{align*}
We again used that $z \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) = O(\varepsilon)$ as well as the definition of $V$. We now use the defining property of $h$ to invert $\varepsilon^2 \Delta_g - W''(U)$.
\begin{prop}\label{prop:mult.one.improved.phi.behav}
We have that $\widetilde{\phi} = o(\varepsilon^{2})$ on $\mathcal{M}$.
\end{prop}
\begin{proof}
For contradiction, suppose that $\lambda \triangleq \sup_{\mathcal{M}} |\widetilde{\phi}| \geq \gamma \varepsilon^{2}$ for some $\gamma > 0$. Note that $\widetilde{\phi}$ is exponentially small at points that are uniformly bounded away from $\Gamma$, so it is clear that this supremum is achieved at some $X^* \in \mathcal{M}$ with $d(X^*) \to 0$. We can assume that $\widetilde{\phi}(X^{*}) = \lambda$. Write $X^* = (y^*,z^*)$ in Fermi coordinates over $\Gamma$. We split the argument into two cases: (i) $\varepsilon^{-1} |z^*|$ is uniformly bounded or (ii) $\varepsilon^{-1}|z^*| \to \infty$.
First we consider case (i). We can assume that $\varepsilon^{-1} z^* \to z_{\infty}$. Define $\widehat{\phi}(\widehat{X}) = \lambda^{-1} \widetilde{\phi}(X^* + \varepsilon \widehat{X})$, which, in blown up Fermi coordinates $\widehat{X} = (\widehat{y}, \widehat{z})$, satisfies:
\[
\Delta_{\widehat{g}} \widehat{\phi}(\widehat{y}, \widehat{z}) - W''(\overline{\mathbb{H}}(\varepsilon^{-1}z^* + \widehat{z} - \varepsilon^{-1}h(y^*+\varepsilon \widehat{y}))) \widehat{\phi}(\widehat{y}, \widehat{z}) = o(1),
\]
for $\widehat{z} \in (-\varepsilon^{-1} \eta, \varepsilon^{-1} \eta)$ and $\widehat{y} \in \Sigma$, and where $\widehat{g}$ is converging smoothly to the Euclidean metric. Moreover, $\widehat{\phi}(0) = 1$ and $|\widehat{\phi}|$ is uniformly bounded on compact sets. Interior Schauder estimates yield uniform bounds for $\widehat{\phi}$ in $C^{1,\theta}_{\loc}$. Thus, $\widehat{\phi}$ converges in $C^{1}$ to a weak (and thus strong, by elliptic regularity) solution of
\[ \Delta \widehat{\phi}(\widehat{y}, \widehat{z}) - W''({\mathbb{H}}(z_{\infty} + \widehat{z})) \widehat{\phi}(\widehat{y}, \widehat{z}) = 0 \]
on $\mathbf{R}^{n-1}\times \mathbf{R}$. By \cite[Lemma 3.7]{Pacard12} (see also \cite{PacardRitore03}), we have that
\[ \widehat{\phi}(\widehat{y}, \widehat{z}) = \rho {\mathbb{H}}'(z_{\infty} + \widehat{z}) \text{ for some } \rho \in\mathbf{R}, \]
because $\widehat{\phi} \in L^{\infty}(\mathbf{R}^{n-1}\times \mathbf{R})$. In fact, $\widehat{\phi}(0) = 1$ implies that $\rho = \mathbb{H}'(z_\infty)^{-1}$. At the original scale, write $X = (y, z)$ in Fermi coordinates over $\Gamma$. Then, for $K$ fixed sufficiently large, if $|z| \leq K\varepsilon$, we have:
\[ \widetilde{\phi}(y, z) = \lambda \big[ \mathbb{H}'(z_\infty)^{-1} {\mathbb{H}}'(z_{\infty} + \varepsilon^{-1} (z-z^*)) + o(1) \big]. \]
Therefore,
\begin{equation} \label{eq:mult.one.improved.phi.i}
\phi(y, z) = \lambda \big[ \mathbb{H}'(z_\infty)^{-1} {\mathbb{H}}'(z_{\infty} + \varepsilon^{-1} (z-z^*)) + o(1) \big] + \varepsilon^2 V(y) \mathbb{J}(\varepsilon^{-1}(z-h(y))).
\end{equation}
By estimating the exponential tail using \eqref{eq:heteroclinic.expansion.ii}, and then using the definition of $\phi$ and $h$, and also \eqref{eq:mult.one.initial.bds}, we have:
\begin{equation} \label{eq:mult.one.improved.phi.ii}
\int_{-K\varepsilon}^{K\varepsilon} \phi(y, z) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, dz = O(\varepsilon^3 e^{-\sqrt{2} K}).
\end{equation}
By parity ($\mathbb{H}'$ is even, $\mathbb{J}$ is odd) and similarly estimating an exponential tail we also have
\begin{equation} \label{eq:mult.one.improved.phi.iii}
\int_{-K\varepsilon}^{K\varepsilon} \mathbb{J}(\varepsilon^{-1}(z - h(y))) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, dz = O(\varepsilon e^{-\sqrt{2} K}).
\end{equation}
Finally:
\begin{equation} \label{eq:mult.one.improved.phi.iv}
\int_{-K\varepsilon}^{K\varepsilon} \mathbb{H}'(z_\infty+ \varepsilon^{-1}(z - z^*)) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) \, dz \geq (h_0 - O(e^{-\sqrt{2}K})) \varepsilon.
\end{equation}
Altogether, \eqref{eq:mult.one.improved.phi.i}-\eqref{eq:mult.one.improved.phi.iv} imply $\lambda = h_0^{-1} O(\varepsilon^2 e^{-\sqrt{2} K})$, which (for large $K$) contradicts our assumption that $\lambda \geq \gamma \varepsilon^2$ for a fixed $\gamma > 0$. This is a contradiction, completing the proof of case (i).
We now turn to case (ii). The proof here is analogous (and simpler). By rescaling as above, we find a non-zero smooth function $\widehat{\phi} \in L^{\infty}(\mathbf{R}^{n-1}\times \mathbf{R})$ solving $\Delta \widehat{\phi} - W''(\pm 1) \widehat{\phi} = 0$. An integration by parts, using $W''(\pm 1) > 0$, shows that $\widehat{\phi} = 0$. This is a contradiction, completing the proof of case (ii).
\end{proof}
\subsection{Relating the second variations and index upper semicontinuity}
We now can give the fundamental computation linking the index of $u$ as a critical point of $E_{\varepsilon}$ with the index of $\Sigma$ as a critical point of area. Our argument is closely related to the proof of \cite[Lemma 9.2]{delPinoKowalczykWei}. Recall from \eqref{eq:second.var.AC} that the second variation of $E_{\varepsilon}$ is given by
\begin{equation*}
\mathcal{Q}_{u}(\zeta,\psi) \triangleq \delta^{2}E_{\varepsilon}[u]\{\zeta,\xi\} = \int_{M} \left( \varepsilon \langle\nabla \zeta,\nabla \xi \rangle + \frac{W''(u)}{\varepsilon} \zeta \xi\right) \, d\mu_{g}.
\end{equation*}
Similarly, we recall that the second variation of area at $\Sigma$ is given by
\[
\mathcal{Q}_{\Sigma}(\zeta,\xi) \triangleq \delta^{2} \operatorname{Area}[\Sigma]\{\zeta,\xi\} = \int_{\Sigma} \left( \langle\nabla\zeta,\nabla \xi \rangle - (|\sff_{\Sigma}|^{2} + \ricc_g(\nu,\nu))\zeta\xi \right) \, d\mu_\Sigma
\]
\begin{lemm}\label{lemm:index.Hprime.comp}
For $f \in C^{2}(\Sigma)$, setting
\[ \psi(y, z) = f(y) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \]
for $(y,z)$ Fermi coordinates with respect to $\Gamma$ (the nodal set of $u$), and $\psi = 0$ far from $\Gamma$, we have that
\begin{align*}
\mathcal{Q}_{u}(\psi,\psi)
& = \varepsilon^{2}(h_0-o(1)) \int_{\Gamma} \left( |\nabla_{\Gamma} f|^{2} - \left( (|\sff_{\Gamma}|^{2} +\ricc_{g}(\partial_{z},\partial_{z}))\circ\Pi_\Gamma\right) f^{2} \right) d\mu_\Gamma \\
& \qquad + o(\varepsilon^{2}) \int_{\Gamma} \left( |\nabla_{\Gamma} f|^{2} + f^{2}\right) d\mu_\Gamma.
\end{align*}
Here, $\Pi_\Gamma$ denotes the nearest point projection onto $\Gamma$.
\end{lemm}
\begin{proof}
We compute, using \eqref{eq:approximate.heteroclinic.behavior}:
\begin{align*}
& \mathcal{Q}_{u}(\psi,\psi) \\
& = \int_{M} \left( -\varepsilon \psi \Delta_{g} \psi + \varepsilon^{-1} W''(u) \psi^{2} \right)\, d\mu_{g}\\
& = \int_{-\eta}^{\eta} \int_{\Gamma}\left( -\varepsilon \psi \Delta_{\Gamma_{z}} \psi - \varepsilon H_{\Gamma_{z}} \psi \partial_{z} \psi - \varepsilon \psi \partial^{2}_{z} \psi + \varepsilon^{-1} W''(u) \psi^{2} \right)\, d\mu_{g_{z}} dz\\
& = \int_{-\eta}^{\eta} \int_{\Gamma}\left( \varepsilon |\nabla_{\Gamma_{z}} \psi|^{2} - \varepsilon H_{\Gamma_{z}} \psi \partial_{z} \psi - \varepsilon \psi \partial^{2}_{z} \psi + \varepsilon^{-1} W''(u) \psi^{2} \right)\, d\mu_{g_{z}} dz\\
& = \int_{-\eta}^{\eta} \int_{\Gamma}\Big( \varepsilon |(\nabla_{\Gamma_{z}}f) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) - \varepsilon^{-1}f(y)(\nabla_{\Gamma_{z}}h) \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) |^{2}\\
& \qquad - H_{\Gamma_{z}} f(y)^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y)))\\
& \qquad - \varepsilon^{-1} f(y)^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \overline{\mathbb{H}}'''(\varepsilon^{-1}(z-h(y))) \\
& \qquad + \varepsilon^{-1} W''(u) f(y)^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2} \Big)\, d\mu_{g_{z}} dz.
\end{align*}
Using, additionally, \eqref{eq:mult.one.initial.bds}, our $C^2$ bounds on $\Gamma$, \eqref{eq:mean.curv.ddt.metric}, \eqref{eq:mean.curv.ddt.sff}, and \eqref{eq:mean.curv.ddt.h}:
\begin{align*}
& \mathcal{Q}_u(\psi, \psi) \\
& = \int_{-\eta}^{\eta} \int_{\Gamma}\Big( \varepsilon |(\nabla_{\Gamma_{z}}f) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) - \varepsilon^{-1}f(y)(\nabla_{\Gamma_{z}}h) \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) |^{2}\\
& \qquad - H_{\Gamma} f(y)^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y)))\\
& \qquad + \left( (|\sff_{\Gamma}|^{2} +\ricc_{g}(\partial_{z},\partial_{z})) \circ \Pi_\Gamma \right) f(y)^{2} \cdot z \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y)))\\
& \qquad + \varepsilon^{-1} (W''(U+\phi)-W''(U)) f(y)^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2} \Big)\, d\mu_{g_{z}} dz\\
& \qquad + O(\varepsilon^{3}) \int_{\Gamma} f(y)^{2} d\mu_\Gamma \\
& = \int_{-\eta}^{\eta} \int_{\Gamma}\Big( \varepsilon |\nabla_{\Gamma_{z}}f|^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2}\\
& \qquad + \left( (|\sff_{\Gamma}|^{2} +\ricc_{g}(\partial_{z},\partial_{z})) \circ \Pi_\Gamma \right) f(y)^{2} \cdot z \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y)))\\
& \qquad + \varepsilon^{-1} W'''(U) \phi f(y)^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2} \Big)\, d\mu_{g_{z}} dz\\
& \qquad + O(\varepsilon^{3}) \int_{\Gamma} (|\nabla_{\Gamma}f|^{2} + f^{2}) \, d\mu_\Gamma \\
& = \varepsilon^{2}(h_0-o(1)) \int_{\Gamma} \left( |\nabla_{\Gamma} f|^{2} - \left( (|\sff_{\Gamma}|^{2} +\ricc_{g}(\partial_{z},\partial_{z})) \circ \Pi_\Gamma \right) f(y)^{2} \right) d\mu_\Gamma \\
& \qquad + o(\varepsilon^{2}) \int_{\Gamma} \left( |\nabla_{\Gamma} f|^{2} + f^{2}\right) d\mu_\Gamma.
\end{align*}
In the final equality, we have used
\[
\int_{-\infty}^{\infty} t \mathbb{H}'(t) \mathbb{H}''(t) dt = \tfrac 12 \int_{-\infty}^{\infty} t \tfrac{d}{dt} \mathbb{H}'(t)^{2} dt = - \tfrac 12 h_0
\]
on the second term. We have also used $\phi = \varepsilon^{2} V(y) \mathbb{J}(\varepsilon^{-1}(z-h(y))) +o(\varepsilon^{2})$, $V(y) = (|\sff_{\Gamma}|^{2} +\ricc_{g}(\partial_{z},\partial_{z}))\circ\Pi_\Gamma + o(1)$, and the following identity that follows by differentiating \eqref{eq:ODE.for.J} once and integrating by parts:
\begin{align*}
\int_{-\infty}^{\infty} W'''(\mathbb{H}(t)) \mathbb{J}(t) \mathbb{H}'(t)^{2} dt & = \int_{-\infty}^{\infty} \left( \mathbb{J}'''(t)\mathbb{H}'(t) - W''(\mathbb{H}(t)) \mathbb{J}'(t)\mathbb{H}'(t) - \mathbb{H}'(t)^{2} - t\mathbb{H}'(t)\mathbb{H}''(t)
\right) dt\\
& = \int_{-\infty}^{\infty} \left( \mathbb{J}'(t)\mathbb{H}'''(t) - W''(\mathbb{H}(t)) \mathbb{J}'(t)\mathbb{H}'(t) - \mathbb{H}'(t)^{2} - t\mathbb{H}'(t)\mathbb{H}''(t)
\right) dt\\
& = \int_{-\infty}^{\infty} \left( - \mathbb{H}'(t)^{2} - t\mathbb{H}'(t)\mathbb{H}''(t)
\right) dt = -\tfrac 12 h_0.
\end{align*}
This completes the proof.
\end{proof}
Let $\Omega$ denote the $\eta$-tubular neighborhood of $\Gamma$ and consider the restriction $\mathcal{Q}_u^\Omega$ of $\mathcal{Q}_{u}$ to $\Omega$:
\[
\mathcal{Q}_{u}^{\Omega}(\zeta,\xi) \triangleq \delta^2 (E_\varepsilon \restr \Omega)[u]\{ \zeta, \xi \} = \int_{\Omega} \left( \varepsilon \langle\nabla \zeta,\nabla \xi \rangle + \frac{W''(u)}{\varepsilon} \zeta\xi\right) \, d\mu_{g}, \; \zeta, \xi \in C^{\infty}(\Omega).
\]
Consider $w \in C^{\infty}(\Omega)$.
We decompose $w$ as
\begin{equation} \label{eq:index.w.decomposition}
w(y,z) = f(y) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) + w^{\perp}(y,z)
\end{equation}
where
\begin{equation}\label{eq:index.w.perp.defn}
\int_{-\eta}^{\eta} w^{\perp}(y,z) \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, dz = 0.
\end{equation}
It is useful to write
\begin{equation} \label{eq:index.w.parallel}
\psi(y,z) = f(y) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))).
\end{equation}
Note that
\begin{align}\label{eq:index.decomp.L2.norm}
\int_{\Omega} w^{2} d\mu_{g}
& = \int_{-\eta}^{\eta} \int_{\Gamma} f^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2} \, d\mu_{g_{z}} \, dz + \int_{\Omega} (w^{\perp})^{2} \, d\mu_{g} \nonumber \\
& \qquad + 2 \int_{-\eta}^{\eta} \int_{\Gamma} f w^{\perp} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, d\mu_{g_{z}} \, dz \nonumber \\
& = (1+o(1))\int_{-\eta}^{\eta} \int_{\Gamma} f^{2} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))^{2} \, d\mu_\Gamma \, dz + (1+o(1))\int_{\Omega} (w^{\perp})^{2} \, d\mu_{g} \nonumber \\
& = \varepsilon (h_0-o(1)) \int_{\Gamma} f ^{2} \, d\mu_\Gamma + (1+o(1))\int_{\Omega} (w^{\perp})^{2} \, d\mu_{g}.
\end{align}
We now use this decomposition to estimate $\mathcal{Q}^{\Omega}_{u}(w,w)$.
\begin{lemm}\label{lemm:index.perp.comp}
For $w^{\perp}$ as in \eqref{eq:index.w.perp.defn}, there is $\gamma>0$ so that for $\varepsilon>0$ sufficiently small,
\[
\mathcal{Q}_{u}^{\Omega}(w^{\perp},w^{\perp}) \geq \gamma \int_{\Omega} \varepsilon |\nabla w^{\perp}|^{2} + \varepsilon^{-1} (w^{\perp})^{2} \, d\mu_{g}.
\]
\end{lemm}
\begin{proof}
Recall that there is some $\gamma = \gamma(W)>0$ so that if $f(t)$ satisfies $\int_{-\infty}^{\infty}f(t) \mathbb{H}'(t) dt =0$, then
\[
\int_{-\infty}^{\infty} f'(t)^{2} + W''(\mathbb{H}(t)) f(t)^{2} \, dt \geq 4\gamma \int_{-\infty}^{\infty} f'(t)^{2} + f(t)^{2} \, dt.
\]
(See, e.g.\ \cite[(9.28)]{delPinoKowalczykWei}.) A change of variables and a compactness argument imply that
\[
\int_{-\eta}^{\eta} \varepsilon (\partial_{z}w^{\perp}(y,z))^{2} + \varepsilon^{-1}W''(U) (w^{\perp}(y,z))^{2} \, dz \geq 3 \gamma \int_{-\eta}^{\eta} \varepsilon (\partial_{z}w^{\perp}(y,z))^{2} + \varepsilon^{-1}(w^{\perp}(y,z))^{2} \, dz
\]
as long as $\varepsilon>0$ is sufficiently small. From this, and \eqref{eq:mult.one.initial.bds}, we find
\begin{align*}
\mathcal{Q}_{u}^{\Omega}(w^{\perp},w^{\perp}) & = \int_{-\eta}^{\eta} \int_{\Gamma} \left( \varepsilon |\nabla_{\Gamma_{z}}w^{\perp}|^{2} + \varepsilon (\partial_{z}w^{\perp})^{2} + \varepsilon^{-1} W''(u) (w^{\perp})^{2} \right)d\mu_{g_{z}} dz\\
& = \int_{-\eta}^{\eta} \int_{\Gamma} \left( \varepsilon |\nabla_{\Gamma_{z}}w^{\perp}|^{2} + \varepsilon (\partial_{z}w^{\perp})^{2} + \varepsilon^{-1} W''(U) (w^{\perp})^{2} \right)d\mu_{g_{z}} dz\\
& \qquad + O(\varepsilon) \int_{\Omega} (w^{\perp})^{2} d\mu_{g} \\
& \geq 2\gamma \int_{-\eta}^{\eta} \int_{\Gamma} \left( \varepsilon (\partial_{z}w^{\perp})^{2} + \varepsilon^{-1} (w^{\perp})^{2} \right)d\mu_\Gamma dz\\
& \qquad + \int_{-\eta}^{\eta} \int_{\Gamma} \varepsilon |\nabla_{\Gamma_{z}}w^{\perp}|^{2} d\mu_{g_{z}} dz + O(\varepsilon) \int_{\Omega} (w^{\perp})^{2} d\mu_{g} \\
& \geq \gamma \int_{\Omega} \varepsilon |\nabla w^{\perp}|^{2} + \varepsilon^{-1}(w^{\perp})^{2} \, d\mu_{g}.
\end{align*}
This completes the proof.
\end{proof}
\begin{lemm}\label{lemm:index.mixed.terms}
For $\psi$, $f$, $w^{\perp}$ as in \eqref{eq:index.w.decomposition}-\eqref{eq:index.w.parallel}, we have
\begin{align*}
\mathcal{Q}_{u}^{\Omega}(\psi, w^{\perp}) & \geq - o(\varepsilon^{2}) \int_{\Gamma} |\nabla_\Gamma f|^{2} + f^{2}\, d\mu_\Gamma - o(1) \int_{\Omega} \varepsilon |\nabla w^{\perp}|^{2} + \varepsilon^{-1} (w^{\perp})^{2} \, d\mu_{g}.
\end{align*}
\end{lemm}
\begin{proof}
Repeatedly using \eqref{eq:heteroclinic.expansion.ii}, \eqref{eq:heteroclinic.expansion.iii}, \eqref{eq:mult.one.initial.bds}, Lemma \ref{lemm:mult.one.imp.hHeqn}, \eqref{eq:mean.curv.ddt.h}, \eqref{eq:mean.curv.ddt.grad}:
\begin{align*}
& \mathcal{Q}_{u}^{\Omega}(\psi,w^{\perp}) \\
& = \int_{\Omega} \left( - \varepsilon (\Delta_{g}\psi) w^{\perp} + \varepsilon^{-1} W''(u) \psi w^{\perp} \right) d\mu_{g}\\
& = \int_{-\eta}^{\eta} \int_{\Gamma} \Big( - \varepsilon (\Delta_{\Gamma_{z}} \psi) w^{\perp} - H_{\Gamma_{z}} f w^{\perp} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \\
& \qquad - \varepsilon^{-1} f w^{\perp} \cdot \overline{\mathbb{H}}'''(\varepsilon^{-1}(z-h(y))) + \varepsilon^{-1} W''(u) \psi w^{\perp} \Big) d\mu_{g_{z}}dz \\
& = \int_{-\eta}^{\eta} \int_{\Gamma} \Big( - \varepsilon (\Delta_{\Gamma_{z}}f) w^{\perp} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) + 2 \langle \nabla_{\Gamma_{z}}f,\nabla_{\Gamma_{z}} h\rangle w^{\perp} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \\
& \qquad + (\Delta_{\Gamma_{z}}h - H_{\Gamma_{z}}) f w^{\perp} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \\
& \qquad - \varepsilon^{-1} f w^\perp |\nabla_{\Gamma_z} h|^2 \cdot \overline{\mathbb{H}}'''(\varepsilon^{-1}(z - h(y))) \\
& \qquad - \varepsilon^{-1} f w^{\perp} \cdot \overline{\mathbb{H}}'''(\varepsilon^{-1}(z-h(y))) + \varepsilon^{-1} W''(u) \psi w^{\perp} \Big) d\mu_{g_z} dz \\
& = \int_{-\eta}^{\eta} \int_{\Gamma} \Big( - \varepsilon (\Delta_{\Gamma_{z}}f) w^{\perp} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) + 2 \langle \nabla_{\Gamma_{z}}f,\nabla_{\Gamma_{z}} h\rangle w^{\perp} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \\
& \qquad + (\Delta_{\Gamma_{z}}h - H_{\Gamma_{z}}) f w^{\perp} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \\
& \qquad + \varepsilon^{-1}(W''(U+\phi) - W''(U) + O(\varepsilon^3)) \psi w^\perp \Big) d\mu_{g_z} dz \\
& = \int_{-\eta}^{\eta} \int_{\Gamma} \Big( - \varepsilon (\Delta_{\Gamma_{z}}f) w^{\perp} \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) + 2 \langle \nabla_{\Gamma_{z}}f,\nabla_{\Gamma_{z}} h\rangle w^{\perp} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \\
& \qquad + (\Delta_{\Gamma_{z}}h - H_{\Gamma_{z}}) f w^{\perp} \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y)))) d\mu_{g_z} dz - O(\varepsilon) \int_\Omega |f w^\perp| \, d\mu_g. \\
& = \int_{-\eta}^\eta \int_\Gamma \Big( \varepsilon \langle \nabla_{\Gamma_z} f, \nabla_{\Gamma_z} w^\perp \rangle \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) + \langle \nabla_{\Gamma_z} f, \nabla_{\Gamma_z} h \rangle w^\perp \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \\
& \qquad + (\Delta_{\Gamma_z} h - H_{\Gamma_z}) f w^\perp \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \Big) d\mu_{g_z} dz - O(\varepsilon) \int_\Omega |f w^\perp| \, d\mu_g \\
& = \int_{-\eta}^\eta \int_\Gamma \Big( \varepsilon \langle \nabla_{\Gamma_z} f, \nabla_{\Gamma_z} w^\perp \rangle \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) + (\Delta_{\Gamma_z} h - H_{\Gamma_z}) f w^\perp \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \Big) d\mu_{g_z} dz \\
& \qquad - O(\varepsilon) \int_\Omega |f w^\perp| \, d\mu_g - O(\varepsilon^2) \int_\Omega |\nabla_{\Gamma_z} f| |w^\perp| \, d\mu_g \\
& = \int_{-\eta}^\eta \int_\Gamma \Big( \varepsilon \langle \nabla_{\Gamma_z} f, \nabla_{\Gamma_z} w^\perp \rangle\cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) + (\Delta_{\Gamma_z} h - H_{\Gamma_z}) f w^\perp \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \Big) d\mu_{g_z} dz \\
& \qquad - o(\varepsilon^3) \int_\Gamma |\nabla_{\Gamma} f|^2 + f^2 \, d\mu_\Gamma - o(1) \int_\Omega \varepsilon^{-1} (w^\perp)^2 \, d\mu_g.
\end{align*}
In the last inequality, we estimated, using Cauchy--Schwarz, $2 ab \leq \varepsilon^{1-\sigma} a^2 + \varepsilon^{-1+\sigma} b^2$, for $\sigma \in (0, 1)$, $(a, b) = (|f|, |w^\perp|)$, $(|\nabla_{\Gamma_z} f|, |w^\perp|)$. We can further estimate, using \eqref{eq:mult.one.initial.bds}, \eqref{eq:mean.curv.ddt.h}, \eqref{eq:mean.curv.ddt.grad}, and \eqref{eq:mean.curv.ddt.laplace}:
\[ \Delta_{\Gamma_z} h - H_{\Gamma_z} = \Delta_\Gamma h - H_\Gamma + O(|z|) = O(\varepsilon + |z|), \]
and
\[ \langle \nabla_{\Gamma_z} f, \nabla_{\Gamma_z} w^\perp \rangle = \langle \nabla_\Gamma f, \nabla_\Gamma w^\perp \rangle + O(\varepsilon + |z|) |\nabla_\Gamma f| |\nabla_\Gamma w^\perp|. \]
By the same Cauchy--Schwarz estimate applied to $(a, b) = (|f|, |w^\perp|)$, $(|\nabla_\Gamma f|, |\nabla_\Gamma w^\perp|)$ we get:
\begin{align*}
\mathcal{Q}_u^\Omega(\psi, w^\perp)
& = \int_{-\eta}^\eta \int_\Gamma \varepsilon \langle \nabla_\Gamma f, \nabla_\Gamma w^\perp \rangle\cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) \, d\mu_{g_z} dz \\
& \qquad - o(\varepsilon^2) \int_\Gamma |\nabla_\Gamma f|^2 + f^2 \, d\mu_\Gamma - o(1) \int_\Omega \varepsilon |\nabla w^\perp|^2 + \varepsilon^{-1} (w^\perp)^2 \, d\mu_g.
\end{align*}
Estimating $|d\mu_{g_z} - d\mu_\Gamma| = O(|z|) d\mu_\Gamma$ and using the same Cauchy--Schwarz inequality, we deduce:
\begin{align*}
\mathcal{Q}_u^\Omega(\psi, w^\perp)
& = \int_{-\eta}^\eta \int_\Gamma \varepsilon \langle \nabla_\Gamma f, \nabla_\Gamma w^\perp \rangle\cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) \, d\mu_\Gamma dz \\
& \qquad - o(\varepsilon^2) \int_\Gamma |\nabla_\Gamma f|^2 + f^2 \, d\mu_\Gamma - o(1) \int_\Omega \varepsilon |\nabla w^\perp|^2 + \varepsilon^{-1} (w^\perp)^2 \, d\mu_g \\
& = \int_\Gamma \int_{-\eta}^\eta \varepsilon \langle \nabla_\Gamma f, \nabla_\Gamma w^\perp \rangle\cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z - h(y))) \, dz \, d\mu_\Gamma\\
& \qquad - o(\varepsilon^2) \int_\Gamma |\nabla_\Gamma f|^2 + f^2 \, d\mu_\Gamma - o(1) \int_\Omega \varepsilon |\nabla w^\perp|^2 + \varepsilon^{-1} (w^\perp)^2 \, d\mu_g.
\end{align*}
Since $\langle \nabla_\Gamma f, \nabla_\Gamma w^\perp \rangle = g^{ij}_\Gamma \partial_{y_i} f \partial_{y_j} w^\perp$, whose first two factors are independent of $z$, we can use
\[
\int_{-\eta}^{\eta} \partial_{y_j} w^{\perp}\overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, dz = \varepsilon^{-1} \int_{-\eta}^{\eta} (\partial_{y_j} h) w^{\perp} \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \, dz,
\]
which follows from differentiating \eqref{eq:index.w.perp.defn} once horizontally. We thus have:
\begin{align*}
\mathcal{Q}_u^\Omega(\psi, w^\perp)
& = \int_{-\Gamma} \int_{-\eta}^\eta \langle \nabla_\Gamma f, \nabla_\Gamma h \rangle w^\perp \cdot \overline{\mathbb{H}}''(\varepsilon^{-1}(z-h(y))) \, dz \, d\mu_\Gamma\\
& \qquad - o(\varepsilon^2) \int_\Gamma |\nabla_\Gamma f|^2 + f^2 \, d\mu_\Gamma - o(1) \int_\Omega \varepsilon |\nabla w^\perp|^2 + \varepsilon^{-1} (w^\perp)^2 \, d\mu_g.
\end{align*}
This completes the proof, since we're already estimated terms of this form with the correct error term.
\end{proof}
\begin{lemm}\label{lemm:first.eig.stability.Q.low.bd}
There is $\sigma =\sigma(M,g,W,\Sigma)>0$ so that for $\varepsilon > 0$ sufficiently small and any $w \in C^{\infty}(\Omega)$, we have
\[
\mathcal{Q}^{\Omega}_{u}(w,w) \geq - \varepsilon \sigma \int_{\Omega} w^{2} d\mu_{g}.
\]
\end{lemm}
\begin{proof}
Because $\Gamma$ converges to $\Sigma$ in $C^{2,\theta}$, we find that for $\delta=\delta(M,g,\Sigma) \in (0, 1)$ and $\varepsilon>0$ sufficiently small, we have that
\[
\int_{\Gamma} |\nabla_{\Gamma} f|^{2} - \left( (|\sff_{\Gamma}|^{2} +\ricc_{g}(\partial_{z},\partial_{z}))\circ\Pi_\Gamma \right) f^{2} d\mu_\Gamma \geq \int_{\Gamma} \delta |\nabla_{\Gamma}f|^{2} -\delta^{-1}f^{2} \, d\mu_\Gamma.
\]
Thus, using \eqref{eq:index.w.decomposition}-\eqref{eq:index.w.parallel}, Lemmas \ref{lemm:index.Hprime.comp}, \ref{lemm:index.perp.comp}, and \ref{lemm:index.mixed.terms}, we find that
\begin{align*}
\mathcal{Q}^{\Omega}_{u}(w,w) & = \mathcal{Q}^{\Omega}_{u}(\psi,\psi) + \mathcal{Q}_u^{\Omega}(w^{\perp},w^{\perp}) + 2 \mathcal{Q}^{\Omega}_{u}(\psi,w^{\perp})\\
& \geq \varepsilon^{2} (h_0 - o(1)) \int_{\Gamma} \delta |\nabla_{\Gamma} f|^{2} - \delta^{-1} f^{2} \, d\mu_\Gamma + \gamma \int_{\Omega} \varepsilon |\nabla w^{\perp}|^{2} + \varepsilon^{-1} (w^{\perp})^{2} \, d\mu_{g}\\
& \qquad - o(\varepsilon^{2}) \int_{\Gamma} |\nabla f|^{2} + f^{2}\, d\mu_\Gamma - o(1) \int_{\Omega} \varepsilon |\nabla w^{\perp}|^{2} + \varepsilon^{-1} (w^{\perp})^{2} \, d\mu_{g}\\
& \geq - \varepsilon^{2} \delta^{-1}(h_0-o(1)) \int_{\Gamma} f^{2} d\mu_\Gamma \geq - \varepsilon \delta^{-1} (1+o(1)) \int_{\Omega} w^{2} d\mu_{g} .
\end{align*}
In the last inequality we used \eqref{eq:index.decomp.L2.norm}. This completes the proof.
\end{proof}
We are now able to prove the main theorem. In what follows:
\begin{itemize}
\item $\ind(u)$, $\nul(u)$ denote the index and nullity of the second variation of Allen--Cahn energy functional (see \eqref{eq:second.var.AC}), and
\item $\ind(\Sigma)$, $\nul(\Sigma)$ denote the index and nullity of the second variation of the area functional for the limiting multiplicity one smooth minimal surface (recall \eqref{eq:multiplicity.one.assumption}).
\end{itemize}
For simplicity, we will assume that $\partial M = \emptyset$, although we expect that the general strategy used here should extend to Dirichlet or Neumann boundary conditions with appropriate modifications.
\begin{theo} \label{theo:index.lower.bounds}
If $(M^n, g)$, $u$, $\Sigma$ are as above, and $\partial M = \emptyset$, then, for sufficiently small $\varepsilon > 0$,
\[ \ind(\Sigma) + \nul(\Sigma) \geq \ind(u) + \nul(u). \]
\end{theo}
\begin{proof}
For brevity, let's set $I_\Sigma \triangleq \ind(\Sigma) + \nul(\Sigma)$, $I_0 \triangleq \ind(u) + \nul(u)$. First, we show:
\begin{clai}
There are smooth functions $f_1, \ldots, f_{I_{\Sigma}} : \Gamma \to \mathbf{R}$ and a constant $\delta > 0$ so that if $f \in C^{1}(\Gamma)$ satisfies $\langle f, f_i \rangle_{L^2(\Gamma)} = 0$ for all $i = 1, \ldots, I_\Sigma$, then
\begin{equation}\label{eq:ind.gamma.pos.dir}
\mathcal{Q}_{\Gamma}(f,f) \geq \delta \int_{\Gamma} |\nabla_{\Gamma} f|^{2} + f^{2} \, d\mu_\Gamma.
\end{equation}
\end{clai}
\begin{proof}[Proof of claim]
Because the nodal set $\Gamma$ converges to $\Sigma$ in $C^{2,\theta}$ (by Lemma \ref{lemm:multiplicity.one.convergence}), it is not hard to see that is a uniform lower bound $\nu > 0$ for the first \emph{positive} eigenvalue of the second variation of area of $\Gamma$. Take $f_1, \ldots, f_{I_\Sigma}$ to be the first $I_\Sigma$ eigenfunctions of $\mathcal{Q}_\Gamma$. Then:
\[ \mathcal{Q}_\Gamma(f, f) = \int_\Gamma |\nabla_\Gamma f|^2 - (|\sff_\Gamma|^2 + \ricc_g(\partial_z, \partial_z)) f^2 \, d\mu_\Gamma \geq \nu \int_\Gamma f^2 \, d\mu_\Gamma, \]
for $f \in C^1(\Gamma)$, $\langle f, f_1 \rangle_{L^2(\Gamma)} = \ldots = \langle f, f_{I_\Sigma} \rangle_{L^2(\Gamma)} = 0$. If $|\sff_\Gamma|^2 + \ricc_g(\partial_z, \partial_z)\leq C$, then
\[ \tfrac{\nu}{2C} \mathcal{Q}_\Gamma(f, f) = \tfrac{\nu}{2C} \int_\Gamma |\nabla_\Gamma f|^2 - (|\sff_\Gamma|^2 + \ricc_g(\partial_z, \partial_z)) f^2 \, d\mu_\Gamma \geq \int_\Gamma \tfrac{\nu}{2C} |\nabla_\Gamma f|^2 - \tfrac{\nu}{2} f^2 \, d\mu_\Gamma. \]
The claim follows by adding these two inequalities.
\end{proof}
We define the linear functional $\Pi_\varepsilon : L^2(M) \to L^2(\Gamma)$:
\[ \Pi_\varepsilon(w)(y) \triangleq \varepsilon^{-1} \int_{-\eta}^\eta w(y, z) \cdot \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y))) \, dz, \]
and another linear functional $\mathcal{I}_\Gamma : C^1(\Gamma) \to \mathbf{R}^{I_\Sigma}$:
\[ \mathcal{I}_{\Gamma}(f) \triangleq \big( \langle f, f_1 \rangle_{L^2(\Gamma)}, \ldots, \langle f, f_{I_\Sigma} \rangle_{L^2(\Gamma)} \big), \]
so that $f \in \ker \mathcal{I}_\Gamma$ precisely implies \eqref{eq:ind.gamma.pos.dir}. We note one more property of elements of $\ker \mathcal{I}_\Gamma$:
\begin{clai}
Let $w \in C^{\infty}(\Omega)$ be such that $\Pi_\varepsilon(w) \in \ker \mathcal{I}_\Gamma$. Then,
\begin{equation} \label{eq:orth.ind.Q.big}
\mathcal{Q}^{\Omega}_{u} (w,w) \geq \varepsilon \sigma' \int_{\Omega} w^{2} \, d\mu_{g}
\end{equation}
for $\sigma'=\sigma'(M,g,W,\Sigma)>0$ and $\varepsilon>0$ sufficiently small.
\end{clai}
\begin{proof}[Proof of claim]
We proceed as in Lemma \ref{lemm:first.eig.stability.Q.low.bd} but we use the improved lower bound for $\mathcal{Q}_{\Gamma}(f,f)$, \eqref{eq:ind.gamma.pos.dir}, for $f = \Pi_\varepsilon(w)$. Write $\psi(y,z) = \Pi_\varepsilon(w) \overline{\mathbb{H}}'(\varepsilon^{-1}(z-h(y)))$. Then, using Lemmas \ref{lemm:index.Hprime.comp}, \ref{lemm:index.perp.comp}, and \ref{lemm:index.mixed.terms}:
\begin{align*}
\mathcal{Q}^{\Omega}_{u}(w,w)
& = \mathcal{Q}^{\Omega}_{u}(\psi,\psi) + \mathcal{Q}^{\Omega}(w^{\perp},w^{\perp}) + 2 \mathcal{Q}^{\Omega}_{u}(\psi,w^{\perp})\\
& \geq \varepsilon^{2} \delta (h_0 - o(1)) \int_{\Gamma} |\nabla_{\Gamma} f|^{2} + f^{2} \, d\mu_\Gamma + \gamma \int_{\Omega} \varepsilon |\nabla w^{\perp}|^{2} + \varepsilon^{-1} (w^{\perp})^{2} \, d\mu_{g}\\
& - o(\varepsilon^{2}) \int_{\Gamma} |\nabla f|^{2} + f^{2}\, d\mu_\Gamma - o(1) \int_{\Omega} \varepsilon |\nabla w^{\perp}|^{2} + \varepsilon^{-1} (w^{\perp})^{2} \, d\mu_{g}\\
& \geq \varepsilon \sigma' \int_{\Omega} w^{2} \, d\mu_{g}
\end{align*}
The claim follows.
\end{proof}
\begin{clai}
If $w \in C^{\infty}(M)$ satisfies $\mathcal{Q}_{u}(w,w) \leq 0$, then
\begin{equation} \label{eq:neg-dir-concentrate}
\int_{M\setminus \Omega} w^{2} \, d\mu_{g} \leq C \varepsilon^{2} \int_{\Omega} w^{2} \, d\mu_g,
\end{equation}
for $C=C(M,g,W,\Sigma,\eta)>0$ and $\varepsilon > 0$ sufficiently small.
\end{clai}
\begin{proof}[Proof of claim]
Using Lemma \ref{lemm:first.eig.stability.Q.low.bd} and that $W''(u) \geq \kappa > 0$ on $M\setminus\Omega$ for $\varepsilon>0$ small, we compute
\begin{equation*}
0 \geq \mathcal{Q}_{u}(w,w) \geq \mathcal{Q}_{u}^{\Omega}(w,w) + \varepsilon^{-1} \kappa \int_{M\setminus\Omega} w^{2} d\mu_{g} \geq -\varepsilon \sigma \int_{\Omega} w^{2} d\mu_{g} + \varepsilon^{-1} \kappa \int_{M\setminus\Omega} w^{2} d\mu_{g}.
\end{equation*}
Rearranging this completes the proof.
\end{proof}
Now, let $w_{1}, \ldots, w_{I_{0}} \in C^{\infty}(M)$ denote an $L^{2}(M)$-orthonormal set of eigenfunctions for $\mathcal{Q}_{u}$ with non-positive eigenvalue, and
\[ W_\Omega \triangleq \lspan\{ w_1|_\Omega, \ldots, w_{I_0}|_\Omega \} \subset C^\infty(\Omega), \; W_\Gamma \triangleq \left\{ \Pi_\varepsilon(w) : w \in \lspan\{ w_1, \ldots, w_{I_0} \} \right\}. \]
We emphasize that
\begin{equation} \label{eq:negative.eigenfunctions.cQ}
\mathcal{Q}_u(w, w) \leq 0 \text{ for all } w \in \lspan\{ w_1, \ldots, w_{I_0} \} \subset C^\infty(M).
\end{equation}
\begin{clai}
$\dim W_\Omega = \dim W_{\Gamma} = I_0$ for $\varepsilon > 0$ sufficiently small.
\end{clai}
\begin{proof}[Proof of claim]
To see $\dim W_\Omega = I_0$, it suffices to note that no nontrivial linear combination $w$ of $w_1, \ldots, w_{I_0}$ can vanish on $\Omega$ because of \eqref{eq:neg-dir-concentrate} and \eqref{eq:negative.eigenfunctions.cQ}.
Likewise, to see $\dim W_\Gamma = I_0$, it suffices to note that no nontrivial linear combination $w$ of $w_1, \ldots, w_{I_0}$ has $\Pi_\varepsilon(w) = 0$ because of \eqref{eq:orth.ind.Q.big}, \eqref{eq:neg-dir-concentrate}, and \eqref{eq:negative.eigenfunctions.cQ}.
\end{proof}
Finally, suppose, for the sake of contradiction, that $I_{\Sigma} < I_{0}$. Because $\dim W_\Gamma = I_{0} > I_{\Sigma}$, it must hold that there exists $w \in \lspan\{ w_1, \ldots, w_{I_0} \} \setminus \{ 0 \}$ such that $\mathcal{I}_\Gamma(\Pi_\varepsilon(w)) = 0$. For $\varepsilon>0$ sufficiently small so that $W''(u) \geq 0$ on $M\setminus\Omega$,
\[ 0 \geq \mathcal{Q}_{u}(w,w) = \mathcal{Q}_{u}^{\Omega}(w,w) + \int_{M\setminus \Omega} \varepsilon |\nabla w|^{2} + \varepsilon^{-1} W''(u) w^{2} \, d\mu_{g} \geq \mathcal{Q}_{u}^{\Omega}(w,w) \geq \varepsilon \sigma' \int_{\Omega} w^{2} d\mu_{g}. \]
We used \eqref{eq:orth.ind.Q.big} in the last step. Thus, $w \equiv 0$ on $\Omega$, so $w \equiv 0$ on $M$ by \eqref{eq:neg-dir-concentrate}, \eqref{eq:negative.eigenfunctions.cQ}, a contradiction.
\end{proof}
\subsection{Strong sheet distance lower bounds} \label{subsec:distance.estimates}
We continue to adopt the conventions and notation laid out in Section \ref{sec:jacobi.toda.reduction}. In particular, we emphasize that we continue to assume \eqref{eq:sheets.sff.bound}-\eqref{eq:sheets.enhanced.sff.bound} as well as assuming that $u$ is a stable critical point of $E_{\varepsilon}\restr\Omega$ (cf.\ Definition \ref{def:ac.morse.index}).
In \cite[(19.7)]{WangWei}, Wang--Wei derive the following stability inequality (in a slightly different setting) from the usual Allen--Cahn stability inequality.
\begin{equation} \label{eq:toda.stability.estimate}
\int_{\Gamma_\ell(7/10)} \zeta^2 \Big[ \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-1}|) + \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+1}|) \Big] \leq c' \int_{\Gamma_\ell(7/10)} \varepsilon^2 |\nabla_{\Gamma_\ell} \zeta|^2 + c' \varepsilon^{1+\kappa}\int_{\Gamma_\ell(7/10)} \zeta^2
\end{equation}
for all $\ell \in \{1, \ldots, Q\}$, $\zeta \in C^\infty_c(\Gamma_\ell(\tfrac{7}{10}))$, $\varepsilon \leq \varepsilon'$, where $\varepsilon'$, $c'$, $\kappa$ depend on $c_0$, $E_0$, $\eta$, $\beta$. In fact, by a careful inspection of Wang--Wei's derivation of \eqref{eq:toda.stability.estimate} from \cite[Section 19]{WangWei}, we see that the following stronger inequality is true here:
\begin{equation} \label{eq:toda.stability.estimate.sharper}
\int_{\Gamma_\ell(7/10)} \zeta^2 \Big[ \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell-1}|) + \exp(-\sqrt{2} \varepsilon^{-1} |d_{\ell+1}|) \Big] \leq c' \int_{\Gamma_\ell(7/10)} \varepsilon^2 |\nabla_{\Gamma_\ell} \zeta|^2 + |\mathcal{E}_{\zeta}| \int_{\Gamma_\ell(7/10)} \zeta^2
\end{equation}
with
\begin{equation} \label{eq:toda.stability.estimate.sharper.coefficient}
|\mathcal{E}_{\zeta}| \leq c' \varepsilon^2 + c' \sum_{m=1}^Q \sup\left\{\exp(-\sqrt{2} (1+\kappa) \varepsilon^{-1}D_{m}(y')) : y' \in \Gamma_m \cap \Pi_\ell^{-1}(B_{2K\varepsilon |\log \varepsilon|}^2(\support h)) \right\};
\end{equation}
here, $c', \kappa$ are independent of $\zeta$. We prove \eqref{eq:toda.stability.estimate.sharper} in Appendix \ref{app:proof.stab.inproved} in a general $n$-dimensional setting, $n \geq 3$. (Below, we use it for $n=3$.) Note that, by Lemma \ref{lemm:stationary.estimates}, this recovers \eqref{eq:toda.stability.estimate}.
Our first main result is the following sheet-distance estimate. (cf. Remark \ref{rema:major.goal}.)
\begin{prop}[Stable sheet distances, I] \label{prop:bootstrapped.stable.estimates}
If $u$ is a stable critical point of $E_\varepsilon \restr \Omega$, $\varepsilon \leq \varepsilon_3$, and $\nu \in (0, \tfrac{1}{2})$, then
\[ D_\ell \geq (1-\nu) \sqrt{2} \varepsilon |\log \varepsilon| \text{ on } \Gamma_\ell(\tfrac{1}{3}) \]
for all $\ell \in \{ 1, \ldots, Q \}$, where $\varepsilon_3 = \varepsilon_3(c_0,E_0,\eta,\beta,\nu)$.
\end{prop}
\begin{proof}
Take $\nu \in (0,\tfrac{1}{2})$ and assume, for contradiction, that
\begin{equation} \label{eq:bootstrapped.contradiction.assumption}
A_{\ell_0}(r) \geq A_{\ell_0}(\tfrac{1}{3}) > \varepsilon^{2(1-\nu)} \text{ for all } r \in [\tfrac{1}{3}, \tfrac{1}{2}] \text{ and some } \ell_0 \in \{1, \ldots, Q\}.
\end{equation}
We will aim to prove
\begin{equation} \label{eq:bootstrapped.claim}
\max_{\ell \in \{1, \ldots, Q\}} A_{\ell}(r-K_\nu\varepsilon^\nu) < \tfrac{1}{2} \max_{\ell \in \{1, \ldots, Q\}} A_{\ell}(r) \text{ for all } r \in [\tfrac{1}{3}, \tfrac{1}{2}]
\end{equation}
where $K_\nu = K_\nu(c_0, E_0, \eta, \beta, \nu) > 0$; this will in turn prove our claim by a simple iteration. (We denote the dependence of $K_\nu$ on $\nu$ explicitly to disambiguate with the previous constant $K$. Let's assume $K_\nu > 2K$.)
Let $r \in [\tfrac{1}{3}, \tfrac{1}{2}]$, $\alpha \triangleq \max \{ A_{\ell}(r) : \ell \in \{1, \ldots, Q\} \}$. Since
\begin{equation} \label{eq:bootstrapped.claim.i}
\alpha > \varepsilon^{2(1-\nu)}
\end{equation}
by \eqref{eq:bootstrapped.contradiction.assumption}, it follows that to prove \eqref{eq:bootstrapped.claim} it will suffice to prove
\begin{equation} \label{eq:bootstrapped.claim.ii}
A_{\ell}(r-\varepsilon K_\nu\alpha^{-\tfrac{1}{2}}) < \tfrac{1}{2} \alpha \text{ for all } \ell \in \{1, \ldots, Q\}.
\end{equation}
Suppose, by way of contradiction, that \eqref{eq:bootstrapped.claim.ii} is violated at some $\ell_0 \in \{1, \ldots, Q\}$ and $y \in \Gamma_{\ell_0}(r - \varepsilon K_\nu \alpha^{-\tfrac{1}{2}})$. From now on let's work in the coordinate chart induced on $\Gamma_{\ell_0}$ by $\Pi|_{\Gamma_{\ell_0}} \approx \Sigma$. For $\widetilde{y} \in B_{K_\nu/2}^2(0)$, define:
\begin{equation} \label{eq:bootstrapped.claim.ftilde}
\widetilde{f}(\widetilde{y}) \triangleq \varepsilon^{-1} f_{\ell_0,\ell_0+1}(y + \varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y}) - \frac{1}{\sqrt{2}} |\log \alpha|.
\end{equation}
If $\widetilde{\mathcal{L}}$ denotes the translation and rescaling of $\mathcal{L}$ that respects the stretched coordinate, $\widetilde{y}$, then from \eqref{eq:bootstrapped.viii} we find
\begin{align*}
\widetilde{\mathcal{L}} \widetilde{f}(\widetilde{y}) & = \varepsilon \alpha^{-1} \mathcal{L} f_{\ell_0,\ell_0+1}(y+\varepsilon\alpha^{-\tfrac{1}{2}}\widetilde{y}) \\
& \leq \frac{8(A_0)^{2}\alpha^{-1}}{h_0} \exp(-\sqrt{2}\varepsilon^{-1}f_{\ell_0,\ell_0+1}(y+\varepsilon \alpha^{-\tfrac{1}{2}}\widetilde{y})) \\
& \qquad + \alpha^{-1} c' |\mathcal{R}_{\ell_0}(y+\varepsilon \alpha^{-\tfrac{1}{2}}\widetilde{y})| + \alpha^{-1} |\mathcal{R}_{\ell_0+1}((y+\varepsilon \alpha^{-\tfrac{1}{2}}\widetilde{y})')| \\
& \qquad - \varepsilon\alpha^{-1} (|\sff_{\Gamma_{\ell_0}}|^2+\ricc_g(\nu,\nu)|_{\Gamma_{\ell_0}} + \mathcal{Q}) f_{\ell_0,\ell_0+1}(y+\varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y} ).
\end{align*}
Recalling \eqref{eq:bootstrapped.claim.ftilde}, the computation above readily implies that
\begin{align} \label{eq:bootstrapped.claim.iii}
\widetilde{\mathcal{L}} \widetilde{f}(\widetilde{y}) & \leq \frac{8(A_0)^{2}}{h_0} \exp(-\sqrt{2} \widetilde{f}(\widetilde{y})) \\
& \qquad + \alpha^{-1} c' |\mathcal{R}_{\ell_0}(y+\varepsilon \alpha^{-\tfrac{1}{2}}\widetilde{y})| + \alpha^{-1} |\mathcal{R}_{\ell_0+1}((y+\varepsilon \alpha^{-\tfrac{1}{2}}\widetilde{y})')| \nonumber \\
& \qquad - \varepsilon \alpha^{-1} (|\sff_{\Gamma_{\ell_0}}|^2+\ricc_g(\nu,\nu)|_{\Gamma_{\ell_0}} + \mathcal{Q}) f_{\ell_0,\ell_0+1}(y + \varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y}). \nonumber
\end{align}
From \eqref{eq:bootstrapped.error.i} and \eqref{eq:bootstrapped.claim.i}
we have:
\begin{equation} \label{eq:bootstrapped.claim.iv}
\alpha^{-1} c' |\mathcal{R}_{\ell_0}| + \alpha^{-1} |\mathcal{R}_{\ell_0+1}| \leq c' (\varepsilon^2 \alpha^{-1} + \varepsilon^\kappa + \alpha^\kappa)
\leq c'(\alpha^{\tfrac{\nu}{1-\nu}} + \alpha^{\tfrac{\kappa}{2(1-\nu)}}) \leq c'.
\end{equation}
Now define the auxiliary function $\psi \triangleq \exp(-\sqrt{2} \widetilde{f}) > 0$. From the chain rule, \eqref{eq:bootstrapped.claim.iii}, and \eqref{eq:bootstrapped.claim.iv}, we have
\begin{align} \label{eq:psi.differential.inequality}
\widetilde{\mathcal{L}} \psi & = - \sqrt{2} (\widetilde{\mathcal{L}} \widetilde{f}) \psi + 2 |\widetilde{\nabla} \widetilde{f}|^2 \psi \nonumber \\
& \geq - \frac{8\sqrt{2} (A_0)^{2}}{h_0} \psi^2 - c' \psi + \sqrt{2} \varepsilon \alpha^{-1} (|\sff_{\Gamma_{\ell_0}}|^2 + \ricc_g(\nu, \nu)|_{\Gamma_{\ell_0}}) f_{\ell_0,\ell_0+1} \psi \nonumber \\
& \qquad + \alpha^{-1} (\sqrt{2} \varepsilon \mathcal{Q} (f_{\ell_0,\ell_0+1}) + |\nabla_{\Gamma_{\ell_0}} f_{\ell_0,\ell_0+1}|^2) \psi \nonumber \\
& \geq - \frac{8\sqrt{2} (A_0)^{2}}{h_0} \psi^2 - c' \psi + \sqrt{2} \varepsilon \alpha^{-1} (|\sff_{\Gamma_{\ell_0}}|^2 + \ricc_g(\nu, \nu)|_{\Gamma_{\ell_0}}) f_{\ell_0,\ell_0+1} \psi \nonumber \\
& \qquad - \alpha^{-1} \left[ \sqrt{2} \varepsilon \mathcal{Q} (f_{\ell_0,\ell_0+1}) + |\nabla_{\Gamma_{\ell_0}} f_{\ell_0,\ell_0+1}|^2 \right]_- \psi
\end{align}
on $B_{K_\nu}^2(0)$. Here, $[\cdot]_-$ denotes the negative part of a real number (and is a nonnegative quantity). Using a logarithmic cutoff function in \eqref{eq:toda.stability.estimate}, which is $1$ on $B_{\varepsilon \alpha^{-1/2}\sqrt{K_\nu}}^2(0)$ and $0$ outside $B_{\varepsilon \alpha^{-1/2} K_\nu/2}^2(0)$, we get
\begin{equation} \label{eq:bootstrapped.claim.L1.bound}
\int_{B_{\sqrt{K_\nu}}^2(0)} \psi \leq c' (\log K_\nu)^{-1} + c' \alpha^{\tfrac{\kappa+2\nu-1}{2(1-\nu)}} K_\nu^2
\end{equation}
in the scale of $\psi$. By Moser's weak maximum principle on $B_1$ for \eqref{eq:psi.differential.inequality} (see, e.g.\ \cite[Theorem 4.1]{HanLin}), the $L^1$ bound in \eqref{eq:bootstrapped.claim.L1.bound} implies the $L^\infty$ bound
\begin{equation} \label{eq:bootstrapped.claim.sup.bound}
\psi(0) \leq C_\star \int_{B_1^2(0)} \psi \leq C_\star \left( (\log K_\nu)^{-1} + \alpha^{\tfrac{\kappa+2\nu-1}{2(1-\nu)}} K_\nu^2 \right),
\end{equation}
for a constant $C_\star$ that depends on the constants in \eqref{eq:sheets.constants} and the $L^\infty$ norm of the coefficients in the differential inequality \eqref{eq:psi.differential.inequality} on $B_1^2(0)$. We're assuming that \eqref{eq:bootstrapped.claim.ii} fails at $y$, so together with \eqref{eq:sheets.sff.bound}, \eqref{eq:sheets.graph.apriori.C1.bounds}, \eqref{eq:sheets.graph.apriori.C2.bounds}, and Lemma \ref{lemm:mean.curvature.laplacian}, we have
\begin{align}
& \sup_{\widetilde{y} \in B_1^2(0)} | \varepsilon \alpha^{-1} (|\sff_{\Gamma_{\ell_0}}|^2 + \ricc_g(\nu, \nu)|_{\Gamma_{\ell_0}}) f_{\ell_0,\ell_0+1}(y + \varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y})| \nonumber \\
& \qquad \leq c' \varepsilon \alpha^{-1} (|f_{\ell_0,\ell_0+1}(y)| + \operatorname{osc} \{ f_{\ell_0,\ell_0+1} : \Gamma_{\ell_0} \cap C_{\varepsilon \alpha^{-1/2}}(\Pi(y)) \}) \leq c' \varepsilon^2 \alpha^{-\tfrac{3}{2}} \leq c' \alpha^{\tfrac{3\nu-1}{2(1-\nu)}}. \label{eq:bootstrapped.claim.osc.bound}
\end{align}
Likewise, using Lemma \ref{lemm:mean.curvature.laplacian}, we can estimate
\[ \sqrt{2} \varepsilon |\mathcal{Q} (f_{\ell_0,\ell_0+1})| \leq c' \varepsilon (|f_{\ell_0,\ell_0+1}|^2 + |\nabla_{\Gamma_{\ell_0}} f_{\ell_0,\ell_0+1}|^2). \]
By absorbing the gradient term and estimating $f_{\ell_0,\ell_0+1}$ with the same argument as in \eqref{eq:bootstrapped.claim.osc.bound}, we also estimate
\begin{equation} \label{eq:bootstrapped.claim.quad.error.bound}
\alpha^{-1} \left[ 2 |\nabla_{\Gamma_{\ell_0}} f_{\ell_0,\ell_0+1}|^2 + \sqrt{2} \varepsilon \mathcal{Q} f_{\ell_0,\ell_0+1}(y + \varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y}) \right]_- \leq c' \varepsilon \alpha^{-1} f_{\ell_0,\ell_0+1}^2 \leq c' \varepsilon^3 \alpha^{-2} \leq c' \alpha^{\tfrac{4\nu-1}{2(1-\nu)}}.
\end{equation}
Thus, ignoring the unimportant dependencies on \eqref{eq:sheets.constants}, we have
\begin{equation} \label{eq:bootstrapped.claim.constant.dependence}
C_\star = C_\star(1 + \alpha^{\tfrac{\kappa+2\nu-1}{2(1-\nu)}} + \alpha^{\tfrac{3\nu-1}{2(1-\nu)}} + \alpha^{\tfrac{4\nu-1}{2(1-\nu)}}),
\end{equation}
which, as long as $\nu > \max \{ \tfrac 1 3, \tfrac{1-\kappa}{2} \}$, can be taken to be uniformly bounded independently of $\alpha$---though certainly depending on the constants in \eqref{eq:sheets.constants}---since $\alpha \leq 1$ by definition.
Since $C_\star$ is uniformly bounded per \eqref{eq:bootstrapped.claim.constant.dependence}, it follows from \eqref{eq:bootstrapped.claim.sup.bound} that by choosing suitably large $K_\nu = K_\nu(c_0, E_0, \eta, \beta, \nu) > 0$, $\psi(0)$ will become less than $\tfrac{1}{2}$ for small $\alpha$, contradicting our assumption that \eqref{eq:bootstrapped.claim.ii} is violated. Specifically, recalling \eqref{eq:bootstrapped.claim.sup.bound}, we may simply pick $K_\nu$ large enough that $C_\star (\log K_\nu)^{-1} < \tfrac14$, in which case $\psi(0) < \tfrac12$ as long as $\alpha$ is small enough that $C_\star \alpha^{\tfrac{\kappa+2\nu-1}{2(1-\nu)}} K_\nu^2 < \tfrac14$.
This concludes the proof of Proposition \ref{prop:bootstrapped.stable.estimates} for
\[ \nu \in (\nu_0, \tfrac{1}{2}), \text{ where } \nu_0 = \min \{ \tfrac{1}{3}, \tfrac{1-\kappa}{2} \}. \]
The next step is to show that $\nu_0$ can be taken to be arbitrarily small, at the expense of possibly having to rescale our domain a finite number of times.
Retracing the proof above, it's not hard to see that what one needs to improve are:
\begin{enumerate}
\item the exponent of $\alpha$ in \eqref{eq:bootstrapped.claim.L1.bound}, \eqref{eq:bootstrapped.claim.sup.bound}, and
\item the oscillation bounds in \eqref{eq:bootstrapped.claim.osc.bound}, \eqref{eq:bootstrapped.claim.quad.error.bound}.
\end{enumerate}
For the prior, we may use \eqref{eq:toda.stability.estimate.sharper}-\eqref{eq:toda.stability.estimate.sharper.coefficient} instead of \eqref{eq:toda.stability.estimate}; we get
\[ \psi(0) \leq C_\star \left( (\log K_\nu)^{-1} + (\alpha^{\tfrac{\nu}{1-\nu}} + \alpha^\kappa) K_\nu^2 \right), \]
a sufficient bound.
For the latter, we need to use a Harnack-type inequality on the elliptic \emph{equation} satisfied by $f_{\ell_0,\ell_0+1}$, \eqref{eq:bootstrapped.vii}. Recalling \eqref{eq:bootstrapped.error.i}, and using the fact that we now know Proposition \ref{prop:bootstrapped.stable.estimates} to hold for $\nu' \in (\nu_0, \tfrac{1}{2})$, we see that the right hand side of \eqref{eq:bootstrapped.vii} can be bounded in $L^\infty$ by
\[ c' \varepsilon^2 + c' \sum_{m=1}^Q A_m(|y| + 2K \varepsilon |\log \varepsilon|) \leq c' \varepsilon^{2(1-\nu')} \]
for some $\nu' \in (\nu_0, \tfrac{1}{2})$. Diving \eqref{eq:bootstrapped.vii} through by $\varepsilon$, we thus get a uniformly elliptic equation
\begin{equation} \label{eq:bootstrapped.ix}
(\mathcal{L} + |\sff_{\Gamma_\ell}|^2 + \ricc_g(\nu,\nu)|_{\Gamma_\ell} + \mathcal{Q}) f_{\ell,\ell+1}(y) = O(\varepsilon^{1-2\nu'}).
\end{equation}
Now we apply the inhomogeneous Harnack-type inequality found in \cite[Theorems 8.17, 8.18]{GilbargTrudinger01} to \eqref{eq:bootstrapped.vii}, multiplied through by the $a(y)$ in \eqref{eq:mean.curvature.laplacian.operator.a}, with some $q > 2$, $R = \varepsilon \alpha^{-\tfrac{1}{2}}$, and $g = O(\varepsilon^{1-2\nu'})$ (in the $L^\infty$ sense) and we get
\begin{equation*}
\sup \left\{ f_{\ell_0,\ell_0+1} : \Gamma_{\ell_0} \cap C_{\varepsilon \alpha^{-1/2}}(\Pi(y)) \right\} \leq c'\Big( f_{\ell_0,\ell_0+1}(y) + \varepsilon^{2} \alpha^{-1} \cdot \varepsilon^{1-2\nu'} \Big) = c' \Big( f_{\ell_0,\ell_0+1}(y) + \varepsilon^{3-2\nu'} \alpha^{-1} \Big).
\end{equation*}
Recall that we are assuming, by contradiction, that \eqref{eq:bootstrapped.claim.ii} is violated at our $y$, implying that $f_{\ell_0,\ell_0+1}(y)$ is an error term relative to the last term of the right hand side. Together with \eqref{eq:bootstrapped.claim.i}, this gives
\begin{align}
\sup \left\{ \varepsilon \alpha^{-1} f_{\ell_0,\ell_0+1} : \Gamma_{\ell_0} \cap C_{\varepsilon \alpha^{-1/2}}(\Pi(y)) \right\} & \leq c' \varepsilon^{4-2\nu'} \alpha^{-2} \leq c' \alpha^{\tfrac{2-\nu'}{1-\nu} - 2} = c' \alpha^{\tfrac{2\nu-\nu'}{1-\nu}}. \label{eq:bootstrapped.x}
\end{align}
This is $\leq c' \alpha^\delta$ for some $\delta > 0$ as long as $\nu > \nu_0' \triangleq \tfrac{1}{2} \nu'$. This gives the improved oscillation bound that we sought in place of \eqref{eq:bootstrapped.claim.osc.bound}, and Proposition \ref{prop:bootstrapped.stable.estimates} follows in full by iteratively pushing $\nu$, $\nu_0'$ down to zero.
\end{proof}
\begin{prop}[Stable sheet distances, II] \label{prop:ultimate.stable.estimates}
If $u$ is as in Proposition \ref{prop:bootstrapped.stable.estimates}, then
\[ \lim_{\varepsilon \to 0} \frac{\exp(-\sqrt{2} \varepsilon^{-1} D_\ell)}{\varepsilon^2 |\log \varepsilon|} = 0 \text{ on } \Gamma_\ell(\tfrac{1}{6}) \]
for all $\ell \in \{ 1, \ldots, Q \}$, uniformly in terms of $c_0$, $E_0$, $\eta$, $\beta$.
\end{prop}
\begin{proof}
The proof follows along the same lines as the bootstrap portion of the proof of Proposition \ref{prop:bootstrapped.stable.estimates}. However, the modifications are somewhat delicate so we give the argument here.
We first prove a weaker bound. We argue by contradiction, assuming that there exists $\ell \in \{1, \ldots, Q\}$ such that
\begin{equation} \label{eq:ultimate.i}
A_{\ell}(r) \geq A_{\ell}(1/5) > \varepsilon^2 |\log \varepsilon|^2 \text{ for all } r \in [\tfrac{1}{5}, \tfrac{1}{4}].
\end{equation}
Let $r \in [\tfrac{1}{5}, \tfrac{1}{4}]$, and then let $\alpha \triangleq \max \{ A_{\ell}(r) : \ell \in \{1, \ldots, Q\} \}$. Then
\begin{equation} \label{eq:ultimate.ii}
\alpha > \varepsilon^2 |\log \varepsilon|^2.
\end{equation}
We claim that
\begin{equation} \label{eq:ultimate.iii}
\max_{\ell \in \{1, \ldots, Q\}} A_{\ell}(r - \varepsilon K_0 \alpha^{-\tfrac{1}{2}}) < \tfrac{1}{2} \alpha
\end{equation}
for a constant $K_0 = K_0(c_0, E_0, \eta, \beta) > 0$.
Suppose that \eqref{eq:ultimate.iii} fails for $\ell_0 \in \{1, \ldots, Q\}$ and $y \in \Gamma_{\ell_0}(\varepsilon K_0 \alpha^{-\tfrac{1}{2}})$. Define
\begin{equation} \label{eq:ultimate.iv}
\widetilde{f}(\widetilde{y}) \triangleq \varepsilon^{-1} f_{\ell_0,\ell_0+1}(y + \varepsilon \alpha^{-\tfrac{1}{2}}\widetilde{y}) - \tfrac{1}{\sqrt{2}} |\log \alpha|,
\end{equation}
for $\widetilde{y} \in B_{K_0/2}^2(0)$. Proceeding as in \eqref{eq:bootstrapped.claim.iii}, we find that
\begin{align}
& \widetilde{\mathcal{L}} \widetilde{f}(\widetilde{y}) \label{eq:ultimate.v} \leq \frac{8A_0}{h_0} \exp(-\sqrt{2} \widetilde{f}(\widetilde{y})) + \alpha^{-1} c' |\mathcal{R}_{\ell_0}(y+\varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y})| + \alpha^{-1} |\mathcal{R}_{\ell_0+1}((y+\varepsilon \alpha^{-\tfrac{1}{2}}\widetilde{y})')| \nonumber \\
& \qquad \qquad - \varepsilon \alpha^{-1} (|\sff_{\Gamma_{\ell_0}}|^2 + \ricc_g(\nu, \nu)|_{\Gamma_{\ell_0}} + \mathcal{Q}) f_{\ell_0,\ell_0+1}(y + \varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y}). \nonumber
\end{align}
We also still have an estimate of the form
\begin{equation} \label{eq:ultimate.vi}
\alpha^{-1} c' |\mathcal{R}_{\ell_0}| + \alpha^{-1} |\mathcal{R}_{\ell_0+1}| \leq c',
\end{equation}
and the function $\psi \triangleq \exp(-\sqrt{2} \widetilde{f})$ still satisfies a differential inequality of the form
\begin{equation} \label{eq:ultimate.vii}
\widetilde{\mathcal{L}} \psi \geq - \frac{8\sqrt{2} (A_0)^{2}}{h_0} \psi^2 - c' \psi
+ \sqrt{2} \varepsilon \alpha^{-1}(|\sff_{\Gamma_{\ell_0}}|^2 + \ricc_g(\nu, \nu)|_{\Gamma_{\ell_0}} + \mathcal{Q}) f_{\ell_0,\ell_0+1}(y+\varepsilon \alpha^{-\tfrac{1}{2}} \widetilde{y}) \psi.
\end{equation}
Applying the same inhomogeneous Harnack-type inequality that led to \eqref{eq:bootstrapped.x} before, we obtain
\begin{align*}
\sup \left\{ f_{\ell_0,\ell_0+1} : \Gamma_{\ell_0} \cap C_{\varepsilon \alpha^{-1/2}}(\Pi(y)) \right\} & \leq c' \Big( f_{\ell_0,\ell_0+1}(y) + \varepsilon^{2} \alpha^{-1} \cdot \varepsilon^{-1}(\varepsilon^2 + \alpha) \Big)
\leq c' \Big( \varepsilon |\log \alpha| + \varepsilon^3 \alpha^{-1} + \varepsilon \Big).
\end{align*}
Thus, we have the following $L^\infty$ estimate on the coefficient in front of $\psi$ in the last term of \eqref{eq:ultimate.vii} on the domain $B_{\varepsilon \alpha^{-1/2}}^2(y)$:
\begin{equation} \label{eq:ultimate.viii}
\sup \left\{ \varepsilon \alpha^{-1} f_{\ell_0,\ell_0+1} : \Gamma_{\ell_0} \cap C_{\varepsilon \alpha^{-1/2}}(\Pi(y)) \right\}
\leq c' \Big( \varepsilon^2 \alpha^{-1} |\log \alpha| + \varepsilon^4 \alpha^{-2} + \varepsilon^{2} \alpha^{-1} \Big) \leq c',
\end{equation}
where we've used the simple fact that
\begin{equation} \label{eq:ultimate.aux.fact}
\eqref{eq:ultimate.ii} \iff \alpha > \varepsilon^2 |\log \varepsilon|^2 \implies \varepsilon^2 \alpha^{-1} |\log \alpha| = o(1).
\end{equation}
Thus, \eqref{eq:ultimate.vii} implies the uniformly elliptic partial differential inequality
\begin{equation} \label{eq:ultimate.ix}
\widetilde{\mathcal{L}} \psi \geq - \frac{8\sqrt{2} (A_0)^{2}}{h_0} \psi^2 - c' \psi.
\end{equation}
From Moser's weak maximum principle (see, e.g., \cite[Theorem 4.1]{HanLin}) applied to \eqref{eq:ultimate.ix} on $B_1^2(0)$, combined with \eqref{eq:toda.stability.estimate.sharper.coefficient}, we get the $L^\infty$ bound
\[ \psi(0) \leq c' \int_{B_1^2(0)} \psi \leq c'\Big( (\log K_0)^{-1} + (\varepsilon^2 \alpha^{-1} + \alpha^\kappa) K_0^2 \Big) \leq c' \Big( (\log K_0)^{-1} + (o(|\log \alpha|^{-1}) + \alpha^\kappa) K_0^2 \Big), \]
violating the assumption that \eqref{eq:ultimate.iii} fails, provided we take $K_0$ large and $\alpha$ small.
Thus, \eqref{eq:ultimate.iii} holds true with a fixed $K_0$. Notice, then, that
\[ \varepsilon K_0 \alpha^{-\tfrac{1}{2}} \leq K_0 |\log \varepsilon|^{-1}, \]
A backward iteration of \eqref{eq:ultimate.iii} from $r = \tfrac{1}{4}$ to $r = \tfrac{1}{5}$, followed by an application of Proposition \ref{prop:bootstrapped.stable.estimates} at radius $r = 1/4$ with $\nu < \tfrac{\log 2}{20K_0}$
yields
\begin{align*}
\log A_{\ell_0}(\tfrac{1}{5}) & \leq \log A_{\ell_0}(\tfrac{1}{4}) - \tfrac{\log 2}{20K_0} |\log \varepsilon| \leq 2(\nu-1) |\log \varepsilon| - \tfrac{\log 2}{20K_0} |\log \varepsilon| < -2 |\log \varepsilon| = \log \varepsilon^2,
\end{align*}
violating \eqref{eq:ultimate.i}.
We now prove the main claim. We argue by contradiction again assuming that there exists $\ell \in \{1, \ldots, Q\}$ such that
\begin{equation} \label{eq:ultimate.x}
A_{\ell}(r) \geq A_{\ell}(\tfrac{1}{5}) > \mu \varepsilon^2 |\log \varepsilon| \text{ for all } r \in [\tfrac{1}{6}, \tfrac{1}{5}],
\end{equation}
for some $\mu > 0$. Let $r \in [\tfrac{1}{6}, \tfrac{1}{5}]$, $\alpha \triangleq \max \{ A_{\ell}(r) : \ell \in \{ 1, \ldots, Q \}\}$. Then
\begin{equation} \label{eq:ultimate.xi}
\alpha > \mu \varepsilon^2 |\log \varepsilon|.
\end{equation}
We claim that
\begin{equation} \label{eq:ultimate.xii}
A_{\ell}(r - \varepsilon K_0' \alpha^{-\tfrac{1}{2}}) < \tfrac{1}{2} \alpha \text{ for every } \ell \in \{1, \ldots, Q\},
\end{equation}
for a constant $K_0' = K_0'(c_0, E_0, \eta, \beta) > 0$. This indeed follows from the same argument as above, modulo the fact that one needs to replace \eqref{eq:ultimate.aux.fact} with
\[ \eqref{eq:ultimate.xi} \iff \alpha > \mu \varepsilon^2 |\log \varepsilon| \implies \varepsilon^2 \alpha^{-1} |\log \alpha| \leq \mu^{-1} (2 + o(1)). \]
Notice, now, that
\[ \varepsilon K_0' \alpha^{-\tfrac{1}{2}} \leq \mu^{-\tfrac{1}{2}} K_0' |\log \varepsilon|^{-\tfrac{1}{2}}, \]
so that a backward iteration of \eqref{eq:ultimate.xii} from $r = 1/5$ to $r = 1/6$, together with the weaker assertion verified above, yields
\[ \log A_{\ell_0}(\tfrac{1}{6}) \leq \log A_{\ell_0}(\tfrac{1}{5}) - \mu^{\tfrac{1}{2}} \tfrac{\log 2}{20K_0'} |\log \varepsilon|^{\tfrac{1}{2}} \leq -2 |\log \varepsilon| + 2 \log |\log \varepsilon| - \mu^{\tfrac{1}{2}} \tfrac{\log 2}{20K_0'} |\log \varepsilon|^{\tfrac{1}{2}}. \]
However,
\[ \lim_{\varepsilon \to 0} \left( \log |\log \varepsilon| - \mu^{\tfrac{1}{2}} \tfrac{\log 2}{20 K_0'} |\log \varepsilon|^{\tfrac{1}{2}} \right) = -\infty, \]
so, for sufficiently small $\varepsilon$ (depending on $K_0$, $\mu$), this quantity is $< \log \mu$. Thus, for small $\varepsilon$,
\[ \log A_{\ell_0}(\tfrac{1}{6}) \leq \log \mu - 2 |\log \varepsilon| + \log |\log \varepsilon| = \log (\mu \varepsilon^2 |\log \varepsilon|), \]
which violates \eqref{eq:ultimate.x}. The result follows.
\end{proof}
In fact, Proposition \ref{prop:ultimate.stable.estimates} and \eqref{eq:bootstrapped.v}-\eqref{eq:bootstrapped.error.i} establish the following:
\begin{coro} \label{coro:ultimate.stable.estimates}
If $u$ is as in Proposition \ref{prop:ultimate.stable.estimates}, then
\[ \frac{H_{\Gamma_\ell}}{\varepsilon |\log \varepsilon|} \to 0 \text{ uniformly on } \Gamma_\ell(\tfrac{1}{6}) \]
as $\varepsilon \to 0$, for all $\ell \in \{1, \ldots, Q\}$.
\end{coro}
This estimate is key for our geometric applications, since it says that the mean curvature of the zero sets $u$ dominates the effect of interactions between the sheets. This will allow us to treat the sheets (essentially) like disjoint minimal surfaces.
\subsection{Curvature estimates} \label{subsec:stable.curvature.estimates}
In what follows, we let $B^n_r(0)$ be a smooth $n$-ball equipped with a Riemannian metric $g$ so that $B_{r}^{n}(0)$ is a geodesic $r$-ball centered at $0$ (with respect to $g$).
\begin{theo} \label{theo:curvature.estimate}
Suppose $\inj_g \geq 3$ and $|\riem_g| + |\nabla_g \riem_g| \leq 1$ on $B^3_1(0)$. If $\varepsilon \leq \varepsilon_1$, $u \in C^\infty(B^3_1(0); (-1,1))$ is a stable critical point of $E_\varepsilon \restr B^3_1(0)$, and $(E_\varepsilon \restr B^3_1(0))[u] \leq E_0$, then
\[ |\mathcal{A}(x)| \leq c_1 \text{ for all } x \in B^3_{1/2}(0) \cap \{ |u| \leq 1-\beta \}, \]
where $\varepsilon_1 = \varepsilon_1(n, E_0, \beta, W)$, $c_1 = c_1(n, E_0, \beta, W)$.
\end{theo}
\begin{rema}
We emphasize that, in one dimension lower, Wang--Wei \cite{WangWei} have proven that stable critical points of $E_{\varepsilon}$ satisfy curvature bounds even without the assumption of uniformly bounded energy.
\end{rema}
\begin{rema}
It's not immediately obvious that the enhanced second fundamental form $\mathcal{A}$ is well-defined on $B^3_{3/4}(0) \cap \{ |u| \leq 1-\beta \}$. This can be seen, for instance, by applying the following proposition with $n=3$. Its ``nonexistence'' condition, when $n=3$, is guaranteed in view of the work of Ambrosio--Cabr\'e \cite{AmbrosioCabre00} (see also the work of Farina--Mari--Valdinoci \cite{FarinaMariValdinoci13}).
\end{rema}
\begin{prop} \label{prop:lower.density.estimate}
Let $u : B^n_1(0) \to (-1,1)$ be a stable critical point of $E_\varepsilon \restr B^n_1(0)$ with $(E_\varepsilon \restr B^n_1(0))[u] \leq E_0$. If $\varepsilon \leq \varepsilon_0$ and $\mathbf{R}^n$ with the standard metric does not carry any nontrivial (i.e., heteroclinic or $\pm 1$) entire stable solutions with Euclidean energy growth then
\[ \varepsilon |\nabla u_i| \geq c_0^{-1} > 0 \text{ for all } x \in B^n_{3/4}(0) \cap \{ |u| \leq 1-\beta \}, \]
where $\varepsilon_0$, $c_0$ depend on $E_0$, $\beta$, $W$.
\end{prop}
\begin{proof}
We argue by contradiction. If the assertion were false, there would exist a sequence
\[ \{ (u_i, \varepsilon_i) \}_{i=1,2,\ldots} \subset C^\infty(B^n_1(0); (-1,1)) \times (0,\infty), \; \lim_i \varepsilon_i = 0, \]
where each $u_i : B^n_1(0) \to [-1,1]$ is a stable critical point for $E_{\varepsilon_i} \restr B^n_1(0)$, with $(E_{\varepsilon_i} \restr B^n_1(0))[u_i] \leq E_0$, and so that $\lim_i \varepsilon_i \nabla u_i(q_i) = 0$ along some $\{ q_i \}_{i=1,2,\ldots} \subset B^n_{3/4}(0)$. The rescaled functions
\[ v_i(x) \triangleq u_i(\varepsilon_i(x - q_i)) \]
are all stable critical points of $E_1 \restr B^n_{(1-|q_i|)/\varepsilon_i(0)}$ with Euclidean energy growth. Since the ellipticity constants are uniform at this scale, we may pass to a subsequence with $\lim_i v_i = v_\infty$ in $C^\infty_{\loc}(\mathbf{R}^n)$, where $v_\infty$ is a stable critical point of $E_1 \restr \mathbf{R}^n$ with Euclidean area growth, $|v_\infty(0)| \leq 1-\beta$, and $\nabla v_\infty(0) = 0$. No such $v_\infty$ exists: the only entire stable solutions on $\mathbf{R}^n$ with Euclidean energy growth are the constants $\pm 1$ and the one-dimensional heteroclinic solution.
\end{proof}
We are now in a position to prove Theorem \ref{theo:curvature.estimate}.
\begin{proof}[Proof of Theorem \ref{theo:curvature.estimate}]
If the assertion were false, there would exist a sequence
\[ \{ (u_i, \varepsilon_i) \}_{i=1,2,\ldots} \subset C^\infty(B_1^3(0); (-1,1)) \times (0,\infty), \; \lim_i \varepsilon_i = 0, \]
where each $u_i : B_1^3(0) \to (-1,1)$ is a stable critical point for $E_{\varepsilon_i} \restr B_1^3(0)$, with $(E_{\varepsilon_i} \restr B_1^3(0))[u_i] \leq E_0$, and so that the maximum value
\[ \max \left\{ \dist(x, \mathbf{R}^3 \setminus B_{3/4}^3(0)) |\mathcal{A}(x)| : x \in B^3_1(0) \cap \{ |u| \leq 1-\beta \} \right\} \]
is attained at some $q_i \in B_{3/4}^3(0)$ with
\[ \lim_i \dist(q_i, \partial B_{3/4}^3(0)) |\mathcal{A}_i(q_i)| = \infty. \]
Next, let $\lambda_i \triangleq |\mathcal{A}_i(q_i)|$, which we note also satisfies $\lim_i \lambda_i = \infty$.
\begin{clai}
$\liminf_i \varepsilon_i \lambda_i = 0$.
\end{clai}
\begin{proof}[Proof of claim]
Rescale to $v_i(x) \triangleq u_i(\varepsilon_i(x - q_i))$, a stable critical point of $E_1 \restr B^3_{(1-|q_i|)/\varepsilon_i}(0)$ with quadratic energy growth and $|v_i| \leq 1-\beta$. Since our ellipticity constants are uniform at this scale, we can pass to a subsequence such that $\lim_i v_i = v_\infty$ in $C^\infty_{\loc}(\mathbf{R}^3)$, where $v_\infty$ is a stable critical point of $E_1 \restr \mathbf{R}^3$ with $|v_\infty(0)| \leq 1-\beta$. The only such $v_\infty$ is the one-dimensional heteroclinic solution, for which $\mathcal{A}_\infty \equiv 0$, and, therefore $\liminf_i \varepsilon_i \lambda_i = |\mathcal{A}_\infty(0)| = 0$. This completes the proof of the claim.
\end{proof}
Pass to a subsequence for which $\liminf_i \varepsilon_i \lambda_i = 0$ is attained, and rescale to $\widetilde{u}_i(x) \triangleq u_i(\lambda_i^{-1}(x-q_i))$. This is a critical point of $E_{\varepsilon_i \lambda_i} \restr B^3_{(3/4 - |q_i|) \lambda_i}(0)$. We note that
\begin{equation} \label{eq:curvature.estimate.eps.lambda.limit}
\lim_i \varepsilon_i \lambda_i = 0, \; \lim_i (3/4 - |q_i|) \lambda_i = \infty.
\end{equation}
Moreover, for every $R \geq 1$,
\begin{equation} \label{eq:curvature.estimate.energy.growth}
(E_{\varepsilon_i \lambda_i} \restr B^3_{(3/4 - |q_i|) \lambda_i}(0))(B^3_R(0)) \leq cR^2
\end{equation}
for all sufficiently large $i$. Here, $c > 0$ is fixed. Combining \eqref{eq:curvature.estimate.eps.lambda.limit}, \eqref{eq:curvature.estimate.energy.growth}, together with the works of \cite[Theorem 1]{HutchinsonTonegawa00} and \cite[Appendix B]{Guaraco} for the Riemannian modifications, the $2$-varifolds
\begin{equation*}
V_{\varepsilon_i \lambda_i}[\widetilde{u}_i](\varphi) \triangleq \int \varphi(x, T_x \{ \widetilde{u}_i = \widetilde{u}_i(x) \}) \, \varepsilon_i \lambda_i |\nabla \widetilde{u}_i(x)|^2,
\text{ for } \varphi \in C^0_c(\operatorname{Gr}_2(B_{(3/4-|q_i|)\lambda_i}^3(0))),
\end{equation*}
converge weakly to a stationary integral varifold $V_\infty \in \mathbf{I}_2(\mathbf{R}^3)$. The enhanced second fundamental form estimates, moreover, imply that $\support \Vert V_\infty \Vert$ is $C^{1,1}$ and, therefore, a smooth minimal surface.
\begin{rema} \label{rema:HT.simplified}
We note that the most technical elements of \cite{HutchinsonTonegawa00}, such as proving that the limit varifold is integral, can be proven (in the setting at hand) in a much more direct manner given the curvature estimates we now know to be true.
\end{rema}
The stability of $\widetilde{u}_i$ is also known to imply stability of $\support \Vert V_\infty \Vert$. Indeed, one may plug $\zeta = \psi |\nabla \widetilde{u}_i|$, $\psi \in C^\infty_c(\mathbf{R}^3)$, into the second variation operator $\delta^2 E_{\varepsilon_i \lambda_i}|_{\widetilde{u}_i}$ and let $i \to \infty$, and recover the second variation operator for $\support \Vert V_\infty \Vert$ with $\psi|_{\support \Vert V_\infty \Vert}$ being the test function. See also \cite{Tonegawa05}.
Summarizing, $\support \Vert V_\infty \Vert$ is a smooth, stable, embedded minimal surface in $\mathbf{R}^3$. (In fact, with quadratic area growth.) Therefore, the limit is a a disjoint union of planes $P_1, \ldots, P_k \subset \mathbf{R}^3$ with integer multiplicities $m_1, \ldots, m_k \in \{1, 2, \ldots\}$. Without loss of generality, $P_j = \mathbf{R}^2 \times \{z_j\}$, with $0 = z_1 < z_2 < \ldots < z_k$.
We will only need to focus on one of these planes, e.g., $P_1$. Writing
\[ \{ \widetilde{u}_i = 0 \} \cap (B_1^2(0) \times [-z_2/2,z_2/2]) = \bigcup_{\ell=1}^{m_1} \graph f_{i,\ell}, \]
it follows from our rescaling that $f_{i,\ell} : B_1^2(0) \to \mathbf{R}$ all converge, in the $C^{1,\alpha}$ sense on $B_{1/2}^2(0)$, to the zero function as $i \to \infty$. In fact, by dilating as needed, we find ourselves in the setup of Sections \ref{subsec:jacobi.toda.setup}-\ref{subsec:distance.estimates}.
Therefore, by employing Proposition \ref{prop:ultimate.stable.estimates} (in fact, Proposition \ref{prop:bootstrapped.stable.estimates} suffices), it follows from \eqref{eq:enhanced.sff.estimate} that
\begin{equation} \label{eq:curvature.estimate.i}
\limsup_{i \to \infty} \Vert \widetilde{\mathcal{A}}_i \Vert_{C^0(\mathscr{M}_\ell(1/6) \cap \{ |\widetilde{u}_i| \leq 1-\beta\})} \leq c' \limsup_{i \to \infty} \widehat{\Gamma}_\ell(1/6),
\end{equation}
for all $\ell \in \{1, \ldots, Q\}$, where $\widehat{\Gamma}_\ell$ is as in \eqref{eq:christoffel.symbol.sup}.
\begin{clai}
The right hand side of \eqref{eq:curvature.estimate.i} is zero.
\end{clai}
Notice that this claim violates the fact that our dilations were such that $|\widetilde{\mathcal{A}}_i(0)| = 1$ for all $i = 1, 2, \ldots$, and Theorem \ref{theo:curvature.estimate} follows.
\begin{proof}[Proof of claim]
From the Riccati equation, \eqref{eq:mean.curv.ddt.sff}, it suffices to check that the second fundamental form of $\{ |\widetilde{u}_i| = 0 \}$ converges to zero. This follows from our H\"older estimate on the mean curvatures from \eqref{eq:phi.improved.c2a.estimate.full}, Lemma \ref{lemm:h.phi.comparison.improved}, and the fact that our graphing functions converge to zero in $C^1$.
\end{proof}
This concludes the proof of the curvature estimates.
\end{proof}
\begin{coro} \label{coro:curvature.estimates}
Let $(M, g)$, $u$, $\varepsilon$, $\varepsilon_1$ be as in Theorem \ref{theo:curvature.estimate}, and $\theta \in (0, 1)$. Then,
\[ [\sff_{\{u=t\}}]_{\theta, \{u=t\} \cap B^3_{1/3}(0)} \leq c_1' \text{ for all } |t| \leq 1-\beta, \]
where $c_1' = c_1'(n, E_0, \beta, \theta, W)$.
\end{coro}
\begin{proof}
\eqref{eq:bootstrapped.iii}, \eqref{eq:bootstrapped.iv}, Lemma \ref{lemm:h.phi.comparison.improved}, and Proposition \ref{prop:ultimate.stable.estimates} together give $C^\theta$ bounds on the mean curvatures of $\{ u = 0 \}$. The improvement to $C^{2,\theta}$ bounds on the level sets comes from (quasilinear) Schauder theory and Theorem \ref{theo:curvature.estimate}.
\end{proof}
|
{
"timestamp": "2019-08-30T02:15:35",
"yymm": "1803",
"arxiv_id": "1803.02716",
"language": "en",
"url": "https://arxiv.org/abs/1803.02716"
}
|
\section{Introduction}
\vspace{-1.5mm}
Active deployment of smart grid technologies in distribution systems has affected the way how these systems interact with the transmission systems. It is anticipated that distribution systems of the future will be equipped to actively engage in transmission system operations, \cite{TD1}. This will require a coordination mechanism to co-optimize generation resources available in both systems to achieve least-cost operations, while respecting objective functions and satisfying technical constraints of each system.
Coordination between the transmission and distribution systems has previously been investigated for economic dispatch and optimal power flow frameworks. References \cite{TD2, TD3} propose a decomposition approach for the coordinated economic dispatch of the transmission and distribution systems that can capture heterogeneous technical characteristics of these systems. In \cite{TD4}, the decomposition algorithm from \cite{TD2, TD3} is improved to handle AC power flow constraints for both the transmission and distribution systems. The interactions between the transmission and distribution system in the electricity market context is studied in \cite{Caramanis_2015}. The common caveat of \cite{TD2, TD3, TD4, Caramanis_2015} is that they do not endogenously model binary unit commitment (UC) decisions on conventional generators. To our knowledge, there is no approach to include binary UC decisions, while coordinating transmission and distribution operations.
Considering binary UC decisions invokes a number of challenges. First, it renders a mixed-integer linear program (MILP) that cannot always be solved efficiently with a standard branch-and-cut method. Second, traditional decomposition techniques, e.g. Lagrangian Relaxation (LR), are notorious for their unstable and often slow convergence due to the zigzagging effect of Lagrange multipliers. This paper deals with both challenges by using the Surrogate Lagrangian Relaxation (SLR) \cite{TD5}. The SLR enforces a ``surrogate optimality'' condition, which guarantees that ``surrogate'' subgradient directions form acute angles with directions toward the optimal multipliers. The ``surrogate optimality'' condition makes it unnecessary to solve all decomposed subproblems to optimality, thus speeding up the computations. Reference \cite{TD5} derives a stepsizing formula that guarantees the convergence and quantifiable solution accuracy without requiring any knowledge of the optimal dual Lagrangian function. Previously, the SLR was applied to large-scale transmission UC models \cite{TD6, TD9}, even with AC power flows \cite{TD10}.
This paper proposes a model to coordinate the transmission and distribution systems, which accounts for binary UC decisions and power flow physics. The model is solved using the SLR. Our case study describes the cost and computational performance of the proposed coordination and solution technique.
\vspace{-7pt}
\section{Model}
\vspace{-3pt}
This paper considers a power system layout typical to the US power sector, where multiple distribution systems are connected to the single transmission system. The transmission system is operated by the transmission system operator (TSO) using a wholesale electricity market. Each distribution system is operated by the distribution system operator (DSO) that dispatches its own generation and can also participate in the wholesale electricity market.
\vspace{-7pt}
\subsection{Preliminaries} \label{sec:prelim}
Let $\mathcal{B}^{[\cdot]}$, $\mathcal{I}^{[\cdot]}$ and $\mathcal{L}^{[\cdot]}$ be the sets of buses, generators and lines indexed by $b$, $i$, and $l$, where superscript ${[\cdot]}$ denotes the transmission ($\text{T}$) and distribution ($\text{D}$) system. Let $\mathcal{J}$ be the set of distribution systems indexed by $j$.
The transmission system and each distribution system are then given by graphs $\mathcal{G}^{\text{T}} = (\mathcal{B}^{\text{T}}, \mathcal{L}^{\text{T}})$ and $\mathcal{G}^{\text{D}}_j = (\mathcal{B}^{\text{D}}_j, \mathcal{L}^{\text{D}}_j)$. Graph $\mathcal{G}^{\text{T}}$ is chosen to be loopy (meshed) and $\mathcal{G}^{\text{D}}_j$ is chosen to be tree (radial) to represent common topologies of the transmission and distribution systems. Graph $\mathcal{G}^{\text{T}}$ and each graph $\mathcal{G}^{\text{D}}_j$ have strictly one connection point at the root bus of $\mathcal{G}^{\text{D}}_j$. The root bus of each distribution system is denoted as $b_{0,j}$. To denote the connection between the transmission and distribution systems, we use index $j(b)$, which is interpreted as distribution system $j$ is connected to transmission bus $b$. The set of transmission buses that have distribution systems is denoted as $\hat{B}^{\text{T}}$. Active and reactive power variables are distinguished by superscripts $\text{p}$ and $\text{q}$.
\subsection{DSO Model}\label{sec:dso}
The following model is formulated for each distribution system individually and therefore index $j$ is omitted for the sake of clarity. The DSO aims to maximize the social welfare in the distribution system by supplying its demand using available distribution and wholesale market resources:
\begin{flalign}
\max \!\! \big\{ o^{\text{D}} \big\} \!\! =\!\! \max \!\bigg\{\!\sum_{b \in \mathcal{B}}^{} \!L_b^{\text{p}} T\!\!-\!\!\sum_{i \in \mathcal{I}^{\text{D}}} C_i^{\text{g}} g_i^{\text{p}}\! \!+\! \lambda_{b_0} (p^{\text{$\uparrow$}}_{b_0} \!-\! p^{\text{$\downarrow$}}_{b_0})\!\! \bigg\}. \label{dso_obj}
\end{flalign}
The first term in \eqref{dso_obj} represents the payment collected by the DSO from consumers based on their active power consumption $L_b^{\text{p}}$ and flat-rate tariff $T$. The second term accounts for the production cost of conventional generators located in the distribution system and is computed based on their incremental generation cost $C_i^{\text{g}}$ and active power output $g_i^{\text{p}}$. The third term accounts for the cost of transactions performed by the DSO in the wholesale electricity market. Variables $p^{\text{$\downarrow$}}_{b_0}$ and $p^{\text{$\uparrow$}}_{b_0}$ represent the capacity bid/offered by the DSO in the wholesale market, while $\lambda_{b_0}$ denotes the locational marginal price (LMP) at the transmission bus, which is connected to the root bus of the distribution system. Thus, $p^{\text{$\uparrow$}}_{b_0} > 0$ indicates that the DSO offers to sell electricity in the wholesale market, while $p^{\text{$\downarrow$}}_{b_0} > 0$ signals that the DSO bids to purchase electricity
Note \eqref{dso_obj} neglects the fixed cost of conventional generators as it is normally negligible for distribution generators.
The output of distribution generators is constrained as:
\begin{flalign}
& \underline{G}^{\text{p}}_i \leq g^{\text{p}}_i \leq \overline{G}^{\text{p}}_i, \quad \forall i \in \mathcal{I}^{\text{D}}, \label{dso_eq1} \\
& \underline{G}^{\text{q}}_i \leq g^{\text{q}}_i \leq \overline{G}^{\text{q}}_i, \quad \forall i \in \mathcal{I}^{\text{D}},\label{dso_eq3}
\end{flalign}
where the minimum an maximum active power limits are $\underline{G}^{\text{p}}_i$ and $\overline{G}^{\text{p}}_i$, while the minimum and maximum reactive power limits are $\underline{G}^{\text{q}}_i$ and $\overline{G}^{\text{q}}_i$. sSince this paper considers a single-period optimization, the economic dispatch constraints do not include inter-temporal limits (e.g. ramp limits).
Since the distribution system is assumed to have a radial topology, AC power flows can be modeled using an exact second-order conic (SOC) relaxation; interested readers are referred to \cite{low_relaxation} for details of this relaxation given below:
\begin{flalign}
& \big[(f^{\text{p}}_l)^2 + (f^{\text{q}}_l)^2\big]\frac{1}{a_l} \leq v_{s(l)}, \quad \forall l \in \mathcal{L}^{\text{D}}, \label{dso_eq8} \\
& v_{r(l)} -
v_{s(l)} = 2 (R_l f^{\text{p}}_l + X_l f^{\text{q}}_l) - a_l (R_l^2 + X_l^2), \quad \forall l \in \mathcal{L}^{\text{D}}, \label{dso_eq7} \\
& (f^{\text{p}}_l)^2 + (f^{\text{q}}_l)^2 \leq \overline{S}_l^2, \quad \forall l \in \mathcal{L}^{\text{D}}, \label{dso_eq5} \\
& (f^{\text{p}}_l - a_l R_l)^2 + (f^{\text{q}}_l - a_l X_l)^2 \leq \overline{S}_l^2, \quad \forall l \in \mathcal{L}^{\text{D}} \label{dso_eq6} \\
& \underline{V}_b \leq v_b \leq \overline{V}_b, \quad \forall b \in \mathcal{B}^{\text{D}}. \label{dso_eq16}
\end{flalign}
Eq.~\eqref{dso_eq8} represents a relaxed expression for the current squared in branch $l$, denoted by auxiliary variable $a_l$, variables $f^{\text{p}}_l$ and $f^{\text{q}}_l$ denote active and reactive power flows across line $l$, and $v_{s(l)}$ is the voltage magnitude at the sending end of line $l$. The sending and receiving buses of branch $l$ are denoted as $s(l)$ and $r(l)$, respectively. Eq.~\eqref{dso_eq7} relates the sending and receiving bus voltages squared $v_{s(l)}$ and $v_{r(l)}$ via the voltage drop across branch $l$, where parameters $R_l$ and $X_l$ are the reactanace and impedance of branch $l$. Since the power flow at the sending and receiving buses of each branch $l$ differs due to losses incurred by transmission, the apparent power flow limit $\overline{S}_{l}$ is enforced for the sending and receiving buses separately in \eqref{dso_eq5} and \eqref{dso_eq6}. The bus voltages are constrained in \eqref{dso_eq16}, where $v_b$ denotes voltages squared limited by $\underline{V}_b$ and $\overline{V}_b$, see \cite{low_relaxation}.
With the exception of the root bus, which is discussed below, the nodal power balance is enforced as:
\begin{flalign}
& f_{l|s(l)=b}^{\text{p}} - \sum_{l|r(l)=b } (f_l^{\text{p}} - a_l R_l) - \sum_{i \in \mathcal{I}_b^{\text{U}}}g_i^{\text{p}} + L_b^{\text{p}} + v_b G_{l|s(l)=b} \nonumber \\ & \hspace{4.2cm} =0, \quad \forall b \in \mathcal{B}^{\text{D}} \backslash \big\{ b_0 \big\},\label{dso_eq9} \\
& f_{l|s(l)=b}^{\text{q}} -\!\! \sum_{l|r(l)=b } \!\!(f_l^{\text{q}} - a_l X_l) - \sum_{i \in \mathcal{I}_b^{\text{U}}}g_i^{\text{q}} + L_b^{\text{q}} - v_b B_{l|s(l)=b} \nonumber \\ & \hspace{4.2cm} =0, \quad\forall b \in \mathcal{B}^{\text{D}} \backslash \big\{ b_0 \big\} \label{dso_eq10},
\end{flalign}
where $L_b^{\text{p}}$ and $L_b^{\text{q}}$ denote the active and reactive power consumption at bus $b$ and $G_l$ is the conductance of branch $l$. In case of the root bus, \eqref{dso_eq9} and \eqref{dso_eq10} transform into:
\begin{flalign}
& \!\!- \!\! \sum_{l|r(l)=b_0 } \!\!\!\!(f_l^{\text{p}} - a_l R_l) \!- \!p^{\uparrow}_{b_0} + p^{\downarrow}_{b_0} + v_{b_0} G_{l|o(l)=b_0} =0, \label{dso_eq11} \\
& \!\!- \!\!\sum_{l|r(l)=b_0 } \!\!(f_l^{\text{q}} - a_l X_l) - v_{b_0} G_{l|o(l)=b_0} =0 \label{dso_eq12}.
\end{flalign}
Eq.~\eqref{dso_eq11} includes the power exchange with the transmission system based on the capacity bid ($p^{\text{$\uparrow$}}_{b_0}$) and offered ($p^{\text{$\downarrow$}}_{b_0}$) by the DSO in the electricity market. Since the DSO is assumed to meet its own reactive power needs, the reactive power balance for the root bus in \eqref{dso_eq12} has no reactive power exchange with the transmission system. Since the physical interface between the transmission and distribution systems is limited, $p^{\text{$\downarrow$}}_{b_0}$ and $p^{\text{$\uparrow$}}_{b_0}$ are limited as:
\begin{flalign}
& 0 \leq p^{\uparrow}_{b_0} \leq \overline{P}_{j(b)}, \label{dso_eq13}\\
& 0 \leq p^{\downarrow}_{b_0} \leq \overline{P}_{j(b)} \label{dso_eq14},
\end{flalign}
where $\overline{P}_{j(b)}$ and $\overline{P}_{j(b)}$ is the active power limit between distribution system $j$ and transmission bus $b$.
\subsection{TSO Model} \label{sec:tso}
As in \eqref{dso_obj}, the TSO aims to maximize the social welfare in the transmission system, which can be formalized as:
\begin{flalign}
& \max \big\{ o^{\text{T}} \big\} = \bigg\{ \sum_{b \in \mathcal{B}^{\text{T}}} C^{\text{b}}_b L^{\text{p}}_b- \sum_{i \in \mathcal{I}^{\text{T}}} C^{\text{o}}_i g^{\text{p}}_i \label{tso_obj} \\ & \hspace{3.2cm} +\sum_{ b \in \hat{\mathcal{B}^{\text{T}}} }\bigg( C_{j(b)}^{\text{$\downarrow$}} p^{\text{$\downarrow$}}_{j(b)} - C^{\text{$\uparrow$}}_{j(b)} p^{\text{$\uparrow$}}_{j(b)} \bigg) \bigg\} \nonumber .
\end{flalign}
The first term in \eqref{tso_obj} represents the payment collected from consumers connected directly to the transmission system based on their active power consumption $L_b^{\text{p}}$ and price bids $C^{\text{b}}_{b}$. The second term represents the cost of offers by conventional generation resources computed based on their offered price $C^{\text{o}}_i$ and power production $g^{\text{p}}_i $. The third term is the cost of active power exchange between the TSO and DSO, where $C^{\text{$\downarrow$}}_{j(b)}$ and $C^{\text{$\uparrow$}}_{j(b)}$ are the price bids and offers of the DSO $j$ located at transmission bus $b$.
The dispatch of conventional generators is constrained as:
\begin{flalign}
& \underline{G}^{\text{p}}_i \leq g^{\text{p}}_i \leq \overline{G}^{\text{p}}_i x_i, \quad \forall i \in \mathcal{I}^{\text{T}}, \label{tso_eq2}
\end{flalign}
where $x_i \in \big\{ 0, 1\big\}$ is a binary (on/off) decision on conventional generators. Since this paper considers a single-period case, inter-temporal ramp limits and minimum up an down times of conventional generators are omitted.
The network constraints are modeled using the DC power flow approximation to account for a meshed topology as customarily used in market clearing procedures:
\begin{flalign}
& f^{\text{p}}_l = \frac{1}{X_l} ( \theta_{o(l)} - \theta_{r(l)}) , \quad \forall l \in \mathcal{L}^{\text{T}}, \label{tso_eq6} \\
& -\overline{F}_l \leq f^{\text{p}}_l \leq \overline{F}_l , \quad \forall l \in \mathcal{L}^{\text{T}}, \label{tso_eq7}
\end{flalign}
where \eqref{tso_eq6} computes the active power flow in line $l$ and the active power flow limit $\overline{F}_l$ on each line $l$ is enforced in \eqref{tso_eq7}. The nodal active power balance is then modeled for transmission buses without and with interconnected distribution systems in \eqref{tso_eq1b} and \eqref{tso_eq1}:
\begin{flalign}
& \sum_{i \in I_b} g^{\text{p}}_i + \sum_{l|r(l)=b} f_l^{\text{p}} - \sum_{l|o(l)=b} f^{\text{p}}_l -
L^{\text{p}}_b =0, \nonumber \\
& \hspace{4.9cm} \quad \forall b \in \mathcal{B}^{\text{T}} \backslash \big\{ \hat{\mathcal{B}^{\text{T}}} \big\}, \label{tso_eq1b} \\
& \sum_{i \in \mathcal{I}_{b}} g^{\text{p}}_i + \sum_{l|r(l)=b} f_l^{\text{p}} - \sum_{l|o(l)=b} f^{\text{p}}_l + p^{\text{$\uparrow$}}_{j(b)}- p^{\downarrow}_{j(b)} -
L^{\text{p}}_{b} \nonumber \\ & \hspace{4.9cm} = 0, \forall b \in \hat{\mathcal{B}^{\text{T}}} : (\lambda_{b}) \label{tso_eq1},
\end{flalign}
where $\lambda_{b}$ is a Lagrangian multiplier of the power balance constraint, i.e. the wholesale LMP, at the transmission bus with an interconnected distribution system. Variables $p^{\text{$\uparrow$}}_{j(b)}$ and $p^{\text{$\downarrow$}}_{j(b)}$ in \eqref{tso_eq1} denote the power exchnage with distribution system as seen from the transmission side. Therefore, as in \eqref{dso_eq13}-\eqref{dso_eq14}, these flows are constrained:
\begin{flalign}
& 0 \leq p^{\downarrow}_{j(b)} \leq \overline{P}_{j(b)}, \quad \forall b \in \hat{\mathcal{B}^{\text{T}}}, \label{tso_eq4}\\
& 0 \leq p^{\uparrow}_{j(b)} \leq \overline{P}_{j(b)}, \quad \forall b \in \hat{\mathcal{B}^{\text{T}}}. \label{tso_eq5}
\end{flalign}
\subsection{Coordinated TSO-DSO Model} \label{sec:coordinated}
Operating decisions of the TSO and multiple DSOs can be coordinated by solving the following problem:
\begin{flalign}
& \text{Eq.~\eqref{dso_obj}-\eqref{dso_eq14}}, \quad \forall j \in \mathcal{J} , \label{coord_eq1}\\
& \text{Eq.~\eqref{tso_obj}-\eqref{tso_eq5}}, \label{coord_eq2} \\
& p^{\text{$\downarrow$}}_{b_{0,j(b)}} = p^{\text{$\downarrow$}}_{j(b)}, \quad \forall b \in \hat{\mathcal{B}^{\text{T}}} : (\psi_{b_{0,j}}^{\downarrow}), \label{tso_eq5aa} \\
& p^{\text{$\uparrow$}}_{b_{0,j(b)}} = p^{\text{$\uparrow$}}_{j(b)}, \quad \forall b \in \hat{\mathcal{B}^{\text{T}}} : (\psi_{b_{0,j}}^{\uparrow}). \label{tso_eq5bb}
\end{flalign}
Eq.~\eqref{coord_eq1} and \eqref{coord_eq2} list all DSO and TSO problems, while \eqref{tso_eq5aa} and \eqref{tso_eq5bb} enforce the power exchanges between the DSO and TSO problems. Note that $\psi_{b_{0,j}}^{\downarrow}$ and $\psi_{b_{0,j}}^{\uparrow}$ denote Lagrange multipliers of respective constraints. The problem in \eqref{coord_eq1}-\eqref{tso_eq5bb} cannot be solved efficiently for large-scale instances using off-the-shelf solution strategies. Furthermore, it is important to preserve the distributed nature of the coordination process between the TSO and DSOs. This motivates an iterative SLR-based solution technique described in Section~\ref{sec:solution_technique}.
\section{Solution Technique}\label{sec:solution_technique}
\begin{figure}[!b]
\centering
\begin{tikzpicture}[auto, node distance=2cm,auto, font=\normalsize]
\tikzstyle{startstop} = [rectangle, rounded corners, minimum width=3cm, minimum height=1cm,text centered, draw=black, fill=white!30]
\tikzstyle{process} = [rectangle, minimum width=3cm, minimum height=1cm, text centered, draw=black, fill=white!30]
\tikzstyle{decision} = [diamond, minimum width=2.5cm, minimum height=1cm, text centered, draw=black, fill=white!30]
\node (start) [startstop] {Initialize $\lambda^0_{b_{0,j}}, \psi^{\uparrow,0}_{b_{0,j}}, \psi^{\downarrow,0}_{b_{0,j}}, s^0, c^0, p^{0,\downarrow}_{b_{0,j}}, p^{0,\uparrow}_{b_{0,j}}, p^{0,\downarrow}_{j(b)}, p^{0,\uparrow}_{j(b)}$};
\node (DSO) [process, below of=start] {Solve the DSO problem, eq.~\eqref{alg_DSO_obj}-\eqref{alg_DSO_const}};
\node (TSO) [process, below of=DSO] {Solve the TSO problem, eq.~\eqref{relaxation_eq1}-\eqref{eq_opt}};
\node (Update) [process, below of=TSO] {Update $\lambda^k_{b_{0,j}}, \psi^{\uparrow,k}_{b_{0,j}}, \psi^{\downarrow,k}_{b_{0,j}}, s^k, c^k, \alpha^k$};
\node (Termination) [decision, below of=Update, node distance=2cm] {Stop?};
\node (end) [startstop, below of=Termination] {End};
\coordinate[right of=DSO, xshift=1.5cm] (c2);
\coordinate[right of=Termination, xshift=1.5cm] (c1);
\draw [->,thick, line width=0.25mm] (start) -- (DSO) node [midway] {};
\draw [->,thick, line width=0.25mm] (DSO) -- (TSO) node [midway] {$ p^{\uparrow,k}_{b_{0,j}}, p^{\downarrow,k}_{b_{0,j}}$};
\draw [->,thick, line width=0.25mm] (TSO) -- (Update) node [midway] {};
\draw [->,thick, line width=0.25mm] (TSO) -- (Update) node [midway] {$f^k_l, v_b^k, a_l^k, \theta_b^k$};
\draw [-,thick, line width=0.25mm] (Update) -- (TSO) node [midway] {$p^{\uparrow,k}_{b_{0,j}}, p^{\downarrow,k}_{b_{0,j}}, p^{\uparrow,k}_{j(b)}, p^{\downarrow,k}_{j(b)}$};
\draw [->,thick, line width=0.25mm] (Update) -- (Termination) node [midway] {};
\draw [->,thick, line width=0.25mm] (Termination) -- (end) node [midway] {};
\draw [-,thick, line width=0.25mm] (Termination) -- (c1) node [midway] {};
\draw [-,thick, line width=0.25mm] (c1) -- (c2) node [midway, right, xshift=0.25cm, yshift=-0.5cm, rotate=90] {$k=k+1$};
\draw [->,thick, line width=0.25mm] (c2) -- (DSO) node [midway] {};
\end{tikzpicture}
\caption{Flowchart of the proposed SLR-based solution technique. }
\label{fig_algortihm}
\end{figure}
The proposed SLR-based solution technique is illustrated in Fig.~\ref{fig_algortihm} and each step is detailed below:
\subsubsection{Initialization} Set the iteration counter $k=0$. Stepsize $s^{0}$ are initialized as in \cite{TD5} and penalty coefficient $c^{0}$ is chosen as in\cite{bertsekas}. Also, initialize $\lambda^{0}_{b_0,j}, \psi^{0,\downarrow}_{b_{0,j}}, \psi^{0,\uparrow}_{b_{0,j}}, p^{0,\downarrow}_{b_{0,j}}, p^{0,\uparrow}_{b_{0,j}}, p^{0,\downarrow}_{j(b)}, p^{0,\uparrow}_{j(b)}$.
\subsubsection{Solve the DSO problem} The following problem is solved for each DSO in a parallel manner ($\forall j \in \mathcal{J}$)
\begin{flalign}
& \max o^{\text{D}}_j (g_i, p_{b_{0,j}}^{\uparrow},p_{b_{0,j}}^{\downarrow}) + \psi^{k,\downarrow}_{b_{0,j}} (p^{\downarrow}_{b_{0,j}} - p^{\downarrow,k-1}_{j(b)}) \nonumber \\& +\frac{c^{k}}{2} |p^{\downarrow,k-1}_{b_{0,j}} - p^{\downarrow,k-1}_{j(b)}||p^{\downarrow}_{b_{0,j}} - p^{\downarrow,k-1}_{j(b)}|
+ \psi^{k,\uparrow}_{b_{0,j}} (p^{\uparrow}_{b_{0,j}} - p^{\uparrow,k-1}_{j(b)}) \nonumber \\& +\frac{c^{k}}{2} |p^{\uparrow,k-1}_{b_{0,j}} - p^{\uparrow,k-1}_{j(b)}||p^{\uparrow}_{b_{0,j}} - p^{\uparrow,k-1}_{j(b)}|, \label{alg_DSO_obj} \\
& \text{Eq.}~\eqref{dso_obj}-\eqref{dso_eq14} \label{alg_DSO_const}.
\end{flalign}
Since \eqref{tso_eq5aa}-\eqref{tso_eq5bb} are relaxed, the deviations from the TSO power flows at the previous iterations, $p^{\downarrow,k-1}_{j(b)}$ and $p^{\uparrow,k-1}_{j(b)}$, are penalized in \eqref{alg_DSO_obj}. As in \cite{TD9}, the absolute value penalties are used to avoid unnecessary linearization.
\subsubsection{Solve the TSO problem} Following the DSO problems, optimized values of $p^{\uparrow,k}_{b_{0,j}}, p^{\downarrow,k}_{b_{0,j}}$ are used in the TSO problem:
\begin{flalign}
& \max \big\{
L_{c^{k}}(\lambda^{k}_{b_{0,j}}, \psi^{\uparrow,k}_{b_{0,j}}, \psi^{\downarrow,k}_{b_{0,j}}; p^{\uparrow,k}_{b_{0,j}}, p^{\downarrow,k}_{b_{0,j}}; \\ & \hspace{5cm} f_l, g_l, \theta_b, p_{j(b)}^{\uparrow},p_{j(b)}^{\downarrow})
\big\},\label{relaxation_eq1} \\
& \text{Eq.}~\eqref{tso_eq2}-\eqref{tso_eq1b},~\eqref{tso_eq4}-\eqref{tso_eq5}, \label{relaxation_eq2} \\
& \tilde{L}_{c^{k}}(\lambda^{k}_{b_{0,j}}, \psi^{\uparrow,k}_{b_{0,j}}, \psi^{\downarrow,k}_{b_{0,j}}; p^{\uparrow,k}_{b_{0,j}}, p^{\downarrow,k}_{b_{0,j}}; f_l^{k}, g_l^{k}, \theta_b^{k}, p_{j(b)}^{\uparrow,k},
p_{j(b)}^{\downarrow,k}) > \nonumber \\ & \tilde{L}_{c^{k}}(\lambda^{k}_{b_{0,j}}, \psi^{\uparrow,k}_{b_{0,j}}, \psi^{\downarrow,k}_{b_{0,j}}; p^{\uparrow,k}_{b_{0,j}}, p^{\downarrow,k}_{b_{0,j}}; f_l^{k-1}, g_l^{k-1}, \theta_b^{k-1}, \nonumber \\ & \hspace{5cm} p_{j(b)}^{\uparrow,k-1},p_{j(b)}^{\downarrow,k-1}), \label{eq_opt}
\end{flalign}
where $L_{c^{k}}$ is the augmented Lagrangian function, and $\tilde{L}_{c^{k}}$ is the surrogate augmented dual value. The value of $\tilde{L}_{c^{k}}$ is defined as the value of (29) for its current feasible solution. Eq.~\eqref{eq_opt} represents the ``surrogate optimality'' condition from \cite{TD9}. As in Step 2, we relax and penalize constraints (20) and (25)-(26) within the augmented Lagrangian function $L_{c^{k}}$. The penalization is implemented as discussed in \cite{TD9}. Due to the pagination limit, we omit the procedure to derive the exact expressions for $L_{c^{k}}$ and $\tilde{L}_{c^{k}}$ and refer interested readers to \cite{TD9} for details.
\subsubsection{Update} Using the DSO and TSO solutions obtained at iteration $k$, the following parameters are updated:
\begin{flalign}
& c^{k+1} = c^k\beta, \beta > 1, \\
& \psi_{b_{0,j}}^{\downarrow,k+1} = \psi_{b_{0,j}}^{\downarrow,k} +s^k (p^{\downarrow,k}_{b_{0,j}} - p^{\downarrow,k}_{j(b)}), \\
& \psi_{b_{0,j}}^{\uparrow,k+1} = \psi_{b_{0,j}}^{\uparrow,k} +s^k (p^{\uparrow,k}_{b_{0,j}} - p^{\uparrow,k}_{j(b)}), \\
& \lambda_{b_{0,j}}^{k+1} = \lambda_{b_{0,j}}^{k} +s^k \tilde{h}_{b_0} (g_i^{\text{p},k}, f_{l}^{\text{p},k}, p_{b_{0,j}}^{\uparrow,k}, p_{b_{0,j}}^{\downarrow,k}), \\
& s^{k+1}= \alpha^k s^k \times \nonumber \\ &
\times \frac{||\tilde{H} (g_i^{\text{p},k}, f_{l}^{\text{p},k}, p_{b_{0,j}}^{\uparrow,k}, p_{b_{0,j}}^{\downarrow,k}, p_{b(j)}^{\uparrow,k}, p_{b(j)}^{\downarrow,k})||_2}{||\tilde{H} (g_i^{\text{p},k+1}, f_{l}^{\text{p},k+1}, p_{b_{0,j}}^{\uparrow,k+1}, p_{b_{0,j}}^{\downarrow,k+1}, p_{b(j)}^{\uparrow,k+1}, p_{b(j)}^{\downarrow,k+1})||_2},
\end{flalign}
where $\alpha^k$ is a step-sizing parameter
\begin{flalign}
& \alpha^k = 1-\frac{1}{M k^{1-1/k^r}}, M>1, r>0.
\end{flalign}
Value $\tilde{h}_{b_0} (g_i^{p,k}, f_{l}^{p,k}, p_{b_{0,j}}^{\uparrow,k}, p_{b_{0,j}}^{\downarrow,k})$ is defined as the level of constraint violation for a feasible solution of (29) defined for each distribution system as:
\begin{flalign}
& \tilde{h}_{b_0} (g_i^{p,k}, f_{l}^{p,k}, p_{b_{0,j}}^{\uparrow,k}, p_{b_{0,j}}^{\downarrow,k}) = \sum_{i \in \mathcal{I}_{b_0}} g^{p,k}_i + \sum_{l|r(l)=b_0} f_l^{p,k} \nonumber \\ & \hspace{1.0cm} - \sum_{l|o(l)=b_0} f^{p,k}_l + p^{\uparrow,k}_{b_{0,j}}- p^{\downarrow,k}_{b_{0,j}} -
L^{\text{p}}_{b_0}. \label{relaxation_eq3}
\end{flalign}
Accordingly, vector $\tilde{H} (g_i^{\text{p},k}, f_{l}^{\text{p},k}, p_{b_{0,j}}^{\uparrow,k}, p_{b_{0,j}}^{\downarrow,k}, p_{b(j)}^{\uparrow,k}, p_{b(j)}^{\downarrow,k}) $ is the surrogate subgradient direction. Each component of this vector represents the constraint violation of (20) and (25)-(26).
The procedure described in Step 1-4 repeats until the stopping criteria are satisfied such as CPU time, value of the surrogate subgradient norm, or the duality gap, \cite{TD5}.
\section{Case Study}
\subsection{Illustrative Example} \label{sec:study_illustration}
Fig.~\ref{fig_small_example} describes the illustrative test system. The transmission system includes one transmission line between nodes 1 and 2 with $\overline{F}_{1-2} = 100$ MW. The loads connected directly to the transmission system are $L^{\text{p}}_1=100$ MW and $L^{\text{p}}_2=200$ MW. The operating range of G1 and G2, i.e. the range between their minimum and maximum power outputs, is $\big[ 5, 75 \big]$ MW and $\big[ 5, 15 \big]$ MW, respectively, and their price offers are $C_1^{\text{o}}=\$16$/MW and $C_1^{\text{o}}=\$6$/MW. Each distribution system needs to supply $L^{\text{p}}_3=L^{\text{p}}_4=10$ MW. Generators G3 and G4 have the operating range $\big[ 10, 120 \big]$ MW each with the incremental costs of $C_3^{\text{o}}=\$6$/MW and $C_4^{\text{o}}=\$4$/MW. For clarity it is assumed that the distribution system has no reactive power loads, as well as power flow and voltage limits.
\begin{figure}[!b]
\centering
\begin{tikzpicture}[auto, node distance=1cm,auto, font=\normalsize]
\tikzstyle{block} = [circle, draw, fill=white!15, text width=0.4cm, text centered, minimum height=0.4cm]
\tikzstyle{block_rect} = [rectangle, draw, dotted, fill=white!10, text width=2.cm, text centered, minimum height=3.5cm]
\node[text width=3cm, xshift=-1.5cm] at (-4.25,2.0) {DSO-1};
\node[text width=3cm, xshift=-0.5cm] at (1.10,2.0) {DSO-2};
\node [block_rect,xshift=-.5cm, yshift=0.0cm)] (dso2) {} ;
\node [block_rect,xshift=-6.75cm, yshift=0.0cm)] (dso1) {} ;
\node [block, line width=0.25mm, yshift=1.25cm] (gen4) {G4} ;
\coordinate[below of=gen4] (c4);
\coordinate[left of=c4, xshift=-0.6cm] (c2);
\node[left of=c4, yshift=.25cm, xshift=-0.4cm] {2};
\node[right of=c4,yshift=.25cm, xshift=-0.7cm] {4};
\coordinate[below of=c4] (load4);
\coordinate[left of=c2, xshift=-3cm] (c1);
\coordinate[left of=c1, xshift=-0.6cm] (c3);
\node [block, line width=0.25mm, above of = c3] (gen3) {G3} ;
\coordinate[below of=c3] (load3);
\node[right of=c1, yshift=.25cm, xshift=-1.25cm] {1};
\node[left of=c1, yshift=.25cm, xshift=-0.9cm] {3};
\node [block, line width=0.25mm, right of = gen3, xshift=1.5cm] (gen1) {G1} ;
\node [block, line width=0.25mm, left of = gen4, xshift=-1.5cm] (gen2) {G2} ;
\coordinate[below of=gen1, yshift=-1cm] (load1);
\coordinate[below of=gen2, yshift=-1cm] (load2);
\draw [-,thick, line width=0.25mm] (gen1) -- (c1) node [midway] {};
\draw [-,thick, line width=0.25mm] (gen2) -- (c2) node [midway] {};
\draw [-,thick, line width=0.25mm] (gen4) -- (c4) node [midway] {};
\draw [->,thick, line width=0.25mm] (c4) -- (load4) node [near end, below, yshift=-0.5cm] {Load 4};
\draw [-,thick, line width=0.25mm] (c2) -- (c4) node [near end] {};
\draw [-,thick, line width=0.75mm] (c2) -- (c1) node [midway] {TSO};
\draw [-,thick, line width=0.25mm] (c1) -- (c3) node [near end] {};
\draw [-,thick, line width=0.25mm] (gen3) -- (c3) node [near end] {};
\draw [->,thick, line width=0.25mm] (c3) -- (load3) node [near end, below, yshift=-0.5cm] {Load 3};
\draw [->,thick, line width=0.25mm] (c1) -- (load1) node [near end, below, yshift=-0.5cm] {Load 1};
\draw [->,thick, line width=0.25mm] (c2) -- (load2) node [near end, below, yshift=-0.5cm] {Load 2};
\end{tikzpicture}
\caption{An illustrative example with two distribution systems (DSO-1 and DSO-2) connected to the transmission system (TSO-1). }
\label{fig_small_example}
\end{figure}
\begin{figure}[b!]
\centering
\includegraphics[width=0.8\linewidth, scale=0.5 ]{Doc4.eps}
\caption{Convergence of the proposed SLR-based approach compared to the convergence of the subgradinet method.}
\label{fig_convergence_gap}
\end{figure}
The optimal dispatch is $G_1=65$ MW, $G_2=15$ MW, $G_3=120$ MW, and $G_4=120$ MW and the LMPs are $\lambda_1=\lambda_2 = \$16$/MW. Note G1 is a price-maker as other generators are at their power output limit. The power flow in line between nodes 1 and 2 is 75 MW, and the power flows in distribution lines 1-3 and node 2-4 are 110 MW each. Fig.~\ref{fig_convergence_gap} compares the convergence of the proposed SLR-based approach observed at each iteration with the subgradient method, a common algorithmic benchmark. Relative to the benchmark, the proposed approach requires fewer iterations to achieve a higher accuracy of the optimal solution, e.g. after 400 iterations the accuracy gain is roughly 100x. Fig.~\ref{fig_convergence_lamdba} shows how $\lambda_1$ and $\lambda_2$ converge to their optimal values of \$16/MW. As shown in Fig.~\ref{fig_convergence_lamdba}, the SLR reduces zigzaging of Lagrangian multipliers relative to the standard subgradient method, which improves its convergence (Fig.~\ref{fig_convergence_gap}).
\begin{figure}[t!]
\centering
\includegraphics[width=0.8\linewidth]{Doc3.eps}
\caption{Convergence of the Lagrangian multipliers $\lambda_1$ and $\lambda_2$ (LMPs at node 1 and node 2) to their optimal value of \$16/MW.}
\vspace{-0.6cm}
\label{fig_convergence_lamdba}
\end{figure}
\vspace{-0.2cm}
\subsection{IEEE Benchmark} \label{sec:study_ieee}
\vspace{-0.5cm}
\begin{center}
\centering
\vspace{-0.1cm}
\begin{table}[!b]
\centering
\captionsetup{justification=centering, labelsep=period, font=footnotesize, textfont=sc}
\caption{Cost savings and computing times obtained with the proposed TSO-DSO coordination.}
\label{results_ieee_table}
\begin{tabular}{ c | c | c| c }
\hline
\multirow{ 2}{*}{\# of DSOs} & \multirow{ 2}{*}{\makecell{TSO cost \\ savings, \%}} &\multirow{ 2}{*}{\makecell{DSO* cost \\ savings, \%}} & \multirow{ 2}{*}{CPU time (s)} \\
& & & \\
\hline
1 & 0.82\% & 0.06\% & 2 \\
2 & 0.82\% & 0.05\% & 4 \\
4 & 0.82\% & 0.06\% & 5 \\
8 & 0.86\% & 0.07\% & 45 \\
16 & 1.61\% & 0.19\% & 112 \\
32 & 3.11\% & 0.18\% & 234 \\
64 & 4.29\% & 0.29\% & 422 \\
\hline
\end{tabular} \\
* Refers to the total cost of all DSOs cooridnated with the TSO.
\end{table}
\end{center}
This section uses the IEEE 118-bus data \cite{ieee_118} for the transmission system and each distribution system is modeled using the 34-bus IEEE distribution data \cite{ieee_34}. In the following simulations we increase the number of distribution systems connected to the transmission system and compare the results to the case when the TSO and DSO are operated without coordination. When added to the transmission system at a given bus, the distribution system is assumed to fully replace the transmission load at that bus. Each distribution system is assumed to have the same topology and the loads in each distribution system are scaled proportionally to match the total transmission load in the case when the transmission and distribution systems are not coordinated.
Table~\ref{results_ieee_table} summarizes the cost savings obtained with the proposed TSO-DSO coordination, as compared to the case without any coordination, and computing times obtained with the proposed solution technique. As the number of DSOs coordinated with the TSO increases, the relative TSO and DSO cost savings both increase. However, the TSO cost savings are roughly one order of magnitude grater than the DSO savings. This observation suggests that the TSO stands to benefit to a larger extent from the proposed coordination and therefore there is a need to design appropriate incentive mechanisms to engage DSOs in the proposed coordination. The computing times also increase with the number of DSOs engaged in the proposed coordination; however, the proposed solution technique is capable of solving all instances within a reasonable amount of time.
\vspace{-0.15cm}
\section{Conclusion \& Future Work}
\vspace{-0.15cm}
This paper presents an model to coordinate transmission and distribution system, while considering binary UC decisions. We solve the proposed model using the Surrogate Lagrangian Relaxation. Our case study demonstrates that both the transmission and distribution systems benefit from the proposed coordination. We also show that the proposed SLR solution technique outperforms existing methods.
The proposed model points to multiple directions for further investigation. First, it is important to extend the proposed model to a multi-period framework and include relevant inter-temporal constraints. Extending the model to multiple time periods will also require accounting for demand- and supply-side uncertainty in both the transmission and distribution system. It will also be important to refine the accuracy of AC and DC power flow models used in this work and avoid making restrictive assumptions on the system topology (meshed or radial). Finally, the proposed model and solution technique can be extended to a decentralized decision-making framework to respect privacy concerns of the DSO and TSO operators.
\vspace{-0.3cm}
|
{
"timestamp": "2018-03-08T02:09:07",
"yymm": "1803",
"arxiv_id": "1803.02681",
"language": "en",
"url": "https://arxiv.org/abs/1803.02681"
}
|
\section{Introduction}
The Laplace transform of a function $f(t)$ defined on $[0,\infty )$ is the
function $F(s)$, which is a unilateral transform defined b
\begin{equation*}
F(s)=\int_{0}^{\infty }f\left( t\right) e^{-st}dt,
\end{equation*
where $s$ is a complex number frequency parameter. The Laplace transform of
a function $f(t)$ is also denoted by $\mathcal{L}\left( f\right) $.
It is known that some special functions can be represented as corresponding
Laplace transforms, for example, Binet formula for gamma function
\begin{equation*}
\ln \Gamma \left( z\right) -\left( z-\frac{1}{2}\right) \ln z+z-\frac{1}{2
\ln \left( 2\pi \right) =\int_{0}^{\infty }\left( \frac{1}{e^{t}-1}-\frac{1}
t}+\frac{1}{2}\right) \frac{e^{-zt}}{t}dt\text{ \ }\mathbb{R}\left( z\right)
>0
\end{equation*
(see \cite[p. 21, Eq. (5)]{Erdelyi-HTF-I-1981}); the integral representation
of the modified Bessel functions of the second (see \cite[p. 181
{Watson-ATTBF-CUP-1922}
\begin{equation}
K_{v}\left( x\right) =\int_{0}^{\infty }e^{-x\cosh t}\cosh \left( vt\right)
dt, \label{Kv-Ir}
\end{equation
which, by replacing $\cosh t-1$ with $t$, can be expressed a
\begin{equation*}
K_{v}\left( x\right) =e^{-x}\int_{0}^{\infty }e^{-xt}\frac{\cosh \left(
\func{arccosh}\left( t+1\right) \right) }{\sqrt{t\left( t+2\right) }}dt;
\end{equation*
the Gaussian Q-function \cite{Simon-PDGRV-S-2006} defined b
\begin{equation*}
Q\left( x\right) =\frac{1}{\sqrt{2\pi }}\int_{x}^{\infty }e^{-t^{2}/2}d
\text{ for }x>0\text{,}
\end{equation*
which, by a change of variable $t=u+x$, is represented a
\begin{equation*}
Q\left( x\right) =\frac{1}{\sqrt{2\pi }}e^{-x^{2}/2}\int_{0}^{\infty
}e^{-u^{2}/2}e^{-ux}du.
\end{equation*
More examples can be found in \cite{Miller-ITSF-12-2001}, \cit
{Magnus-FTSFMPS-SV-1966}.
An important notion related to the Laplace transform is the completely
monotonic functions. A function $F$ is said to be completely monotonic on an
interval $I$, if $F$ has derivatives of all orders on $I$ and satisfie
\begin{equation}
(-1)^{n}F^{(n)}(x)\geq 0\text{ for all }x\in I\text{ and }n=0,1,2,....
\label{cm}
\end{equation
If the inequality (\ref{cm}) is strict, then $F$ is said to be strictly
completely monotonic on $I$. The classical Bernstein's theorem \cit
{Bernstein-AM-52-1929}, \cite{Widder-TAMS-33-1931} states that a function $F$
is completely monotonic (for short, CM) on $(0,\infty )$ if and only if it
is a Laplace transform of some nonnegative measure $\mu $, that is
\begin{equation*}
F\left( x\right) =\int_{0}^{\infty }e^{-xt}d\mu \left( t\right) ,
\end{equation*
where $\mu \left( t\right) $ is non-decreasing and the integral converges
for $0<x<\infty $.
Another important one is the Bernstein functions \cite{Schilling-BFTA-2010}.
A non-negative function $F$ is said to be a Bernstein function on an
interval $I$, if $F$ has derivatives of all orders on $I$ and satisfie
\begin{equation}
(-1)^{n}F^{(n)}(x)\leq 0\text{ for all }x\in I\text{ and }n=0,1,2,....
\label{bf}
\end{equation
Clearly, a function $F$ is a Bernstein function on $I$ if and only if
F^{\prime }$ is CM on $I$.
Very recently, Yang and Tian \cite{Yang-JIA-317-2017} established a
monotonicity rule for the ratio of two Laplace transforms as follows.
\begin{theorem}
\label{T-Rmr}Let the functions $f,g$ be defined on $\left( 0,\infty \right) $
such that their Laplace transforms $\mathcal{L}\left( f\right)
=\int_{0}^{\infty }f\left( t\right) e^{-xt}dt$ and $\mathcal{L}\left(
g\right) =\int_{0}^{\infty }g\left( t\right) e^{-xt}dt$ exist with $g\left(
t\right) \neq 0$ for all $t>0$. Then the ratio $\mathcal{L}\left( f\right)
\mathcal{L}\left( g\right) $ is decreasing (increasing) on $\left( 0,\infty
\right) $ if $f/g$ is increasing (decreasing) on $\left( 0,\infty \right) $.
\end{theorem}
By using this monotonicity rule, Yang and Tian proved that the functio
\begin{equation*}
x\mapsto \frac{1}{24x\left( \ln \Gamma \left( x+1/2\right) -x\ln x+x-\ln
\sqrt{2\pi }\right) +1}-\frac{120}{7}x^{2}
\end{equation*
is strictly increasing from $\left( 0,\infty \right) $ onto $\left(
1,1860/343\right) $. In another paper \cite{Yang-JMAA-455-2017}, this
monotonicity rule was applied to investigate the monotonicity of the functio
\begin{equation*}
x\mapsto \frac{\psi ^{\left( n+1\right) }\left( x\right) ^{2}}{\psi ^{\left(
n\right) }\left( x\right) \psi ^{\left( n+2\right) }\left( x\right) }
\end{equation*
on $\left( 0,\infty \right) $, where $\psi ^{\left( n\right) }$ for $n\in
\mathbb{N}$ is the polygamma functions, and obtained some new properties of
polygamma functions. These show that Theorem \ref{T-Rmr} is an efficient
tool of studying special functions.
Moreover, as shown in \cite[Remark 4]{Yang-JIA-317-2017}, if $g\left(
t\right) >0$ for all $t>0$, then by Theorem \ref{T-Rmr} and Bernstein's
theorem, both the function
\begin{equation*}
x\mapsto \mathcal{L}\left( f\right) -\beta \mathcal{L}\left( g\right) \text{
\ and \ }x\mapsto \alpha \mathcal{L}\left( g\right) -\mathcal{L}\left(
f\right)
\end{equation*
are CM on $\left( 0,\infty \right) $, where $\beta =\inf_{x>0}\left(
\mathcal{L}\left( f\right) /\mathcal{L}\left( g\right) \right) >-\infty $
and $\alpha =\sup_{x>0}\left( \mathcal{L}\left( f\right) /\mathcal{L}\left(
g\right) \right) <\infty $.
Inspired by the above comments, the aim of this paper is to further
establish the monotonicity rule of the ratio of two Laplace transforms
\mathcal{L}\left( f\right) /\mathcal{L}\left( g\right) $ under the condition
that there is a $t^{\ast }>0$ such that $f/g$ is increasing (decreasing) on
\left( 0,t^{\ast }\right) $ and decreasing (increasing) on $\left( t^{\ast
},\infty \right) $.
The rest of this paper is organized as follows. In Section 2, some lemmas
are given, which containing monotonicity rules for ratios of two power
series (polynomials). In Section 3, our main results (Theorems 1--3) are
proved by means of definition of integral and lemmas. As applications, two
monotonicity results involving psi function and modified Bessel functions of
the second kind are presented.
\section{Lemmas}
To state needed lemmas, we recall a useful auxiliary function $H_{f,g}$,
which was introduced in \cite{Yang-arxiv-1409.6408}. For $-\infty \leq
a<b\leq \infty $, let $f$ and $g$ be differentiable functions on $(a,b)$
with $g^{\prime }\neq 0$ on $(a,b)$. Then we defin
\begin{equation}
H_{f,g}:=\frac{f^{\prime }}{g^{\prime }}g-f. \label{H_f,g}
\end{equation
The auxiliary function $H_{f,g}$ has the following well properties \cite
Property 1]{Yang-arxiv-1409.6408}:
(i) $H_{f,g}$ is even with respect to $g$ and odd with respect to $f$, that
is
\begin{equation}
H_{f,g}\left( x\right) =H_{f,-g}\left( x\right) =-H_{-f,g}\left( x\right)
=-H_{-f,-g}\left( x\right) . \label{H-sr}
\end{equation}
(ii) If $g\neq 0$ on $\left( a,b\right) $, the
\begin{equation}
\left( \frac{f}{g}\right) ^{\prime }=\frac{g^{\prime }}{g^{2}}H_{f,g},
\label{df/g}
\end{equation
and therefore
\begin{equation}
\func{sgn}\left( \frac{f}{g}\right) ^{\prime }=\func{sgn}\left( g^{\prime
}\right) \func{sgn}\left( H_{f,g}\right) . \label{sgnd(f/g)}
\end{equation}
(iii) If $f$ and $g$ are twice differentiable on $(a,b)$, the
\begin{equation}
H_{f,g}^{\prime }=\left( \frac{f^{\prime }}{g^{\prime }}\right) ^{\prime }g.
\label{dH_f,g}
\end{equation}
The auxiliary function $H_{f,g}$ and its properties are very helpful to
investigate those monotonicity of ratios of two functions, see \cit
{Yang-arxiv-1409.6408}, \cite{Yang-JIA-2016-221}, \cite{Yang-JIA-2016-251},
\cite{Yang-JIA-2016-311}, \cite{Lv-JIA-2017-94}, \cite{Yang-MIA-20(3)-2017},
\cite{Yang-MIA-20(4)-2017}, \cite{Luo-RM-72-2017}, \cite{Yang-JMI-11-2017},
\cite{Yang-JMI-12-2018}. Recently, in \cite{Yang-JMAA-428-2015} they were
successfully applied to establish monotonicity rules for ratios of two power
series and of two polynomials under the condition that the ratio of
coefficients of two power series is increasing (decreasing) then decreasing
(increasing). The following monotonicity rule for ration of two polynomials
will be used in proof of our main results.
\begin{lemma}[{\protect\cite[Theorem 2.5]{Yang-JMAA-428-2015}}]
\label{L-PA/PB-pm}Let $A_{n}\left( t\right) =\sum_{k=0}^{n}a_{k}t^{k}$ and
B_{n}\left( t\right) =\sum_{k=0}^{n}b_{k}t^{k}$ be two real polynomials
defined on $\left( 0,r\right) $ ($r>0$) with $b_{k}>0$ for all $0\leq k\leq
n $. Suppose that for certain $m\in \mathbb{N}$ with $m<n$, the sequences
\{a_{k}/b_{k}\}_{0\leq k\leq m}$ and $\{a_{k}/b_{k}\}_{m\leq k\leq n}$ are
both non-constants, and are respectively increasing (decreasing) and
decreasing (increasing). Then the function $A_{n}/B_{n}$ is increasing
(decreasing) on $\left( 0,r\right) $ if and only if $H_{A_{n},B_{n}}\left(
r^{-}\right) \geq \left( \leq \right) 0$. While $H_{A_{n},B_{n}}\left(
r^{-}\right) <\left( >\right) 0$, there is a unique $t_{0}\in \left(
0,r\right) $ such that the function $A_{n}/B_{n}$ is increasing (decreasing)
on $\left( 0,t_{0}\right) $ and decreasing (increasing) on $\left(
t_{0},r\right) $.
\end{lemma}
The following monotonicity rule will be used in Proposition \ref{P-p1},
which first appeared in \cite[Lemma 6.4]{Belzunce-I:M-E-40-2007} without
giving the details of the proof. Two strict proofs were given in \cit
{Yang-JMAA-428-2015} and \cite{Xia-PJAM-7(2)-2016}.
\begin{lemma}[{\protect\cite[Corollary 2.6]{Yang-JMAA-428-2015}}]
\label{L-A/B-g}Let $A\left( t\right) =\sum_{k=0}^{\infty }a_{k}t^{k}$ and
B\left( t\right) =\sum_{k=0}^{\infty }b_{k}t^{k}$ be two real power series
converging on $\mathbb{R}$ with $b_{k}>0$ for all $k$. If for certain $m\in
\mathbb{N}$, the non-constant sequences $\{a_{k}/b_{k}\}_{0\leq k\leq m}$
and $\{a_{k}/b_{k}\}_{k\geq m}$ are respectively increasing (decreasing) and
decreasing (increasing), then there is a unique $t_{0}\in \left( 0,\infty
\right) $ such that the function $A/B$ is increasing (decreasing) on $\left(
0,t_{0}\right) $ and decreasing (increasing) on $\left( t_{0},\infty \right)
$.
\end{lemma}
The following lemma \cite[Lemma 2]{Yang-arxiv-1705-05704} offers a simple
criterion to determine the sign of a class of special series, which will be
used in proof of Proposition \ref{P-B1}.
\begin{lemma}[{\protect\cite[Lemma 2]{Yang-arxiv-1705-05704}}]
\label{L-sgnS}Let $\{a_{k}\}_{k=0}^{\infty }$ be a nonnegative real sequence
with $a_{m}>0$ and $\sum_{k=m+1}^{\infty }a_{k}>0$ and le
\begin{equation*}
S\left( t\right) =-\sum_{k=0}^{m}a_{k}t^{k}+\sum_{k=m+1}^{\infty }a_{k}t^{k}
\end{equation*
be a convergent power series on the interval $\left( 0,r\right) $ ($r>0$).
(i) If $S\left( r^{-}\right) \leq 0$, then $S\left( t\right) <0$ for all
t\in \left( 0,r\right) $. (ii) If $S\left( r^{-}\right) >0$, then there is a
unique $t_{0}\in \left( 0,r\right) $ such that $S\left( t\right) <0$ for
t\in \left( 0,t_{0}\right) $ and $S\left( t\right) >0$ for $t\in \left(
t_{0},r\right) $.
\end{lemma}
\begin{remark}
Clearly, when $r=\infty $ in Lemma \ref{L-sgnS}, there is a unique $t_{0}\in
\left( 0,\infty \right) $ such that $S\left( t\right) <0$ for $t\in \left(
0,t_{0}\right) $ and $S\left( t\right) >0$ for $t\in \left( t_{0},\infty
\right) $. This result appeared in \cite[Lemma 6.3]{Belzunce-I:M-E-40-2007}
without proof (see also \cite{Yang-JIA-2015-157}, \cite{Yang-JIA-2015-299}).
If $a_{k}=0$ for $k\geq n\geq m+1$, then Lemma \ref{L-sgnS} is reduced to a
polynomial version, which appeared in \cite{Yang-AAA-2014-702718} (see also
\cite{Yang-JMI-12-2018}).
\end{remark}
\section{Main results}
We are in a position to state and prove our main results.
\begin{theorem}
\label{MT-1}For $0<a<b<\infty $, let the functions $F$ and $G$ be defined on
$\left( 0,\infty \right) $ b
\begin{equation*}
F\left( x\right) =\int_{a}^{b}f\left( t\right) e^{-xt}dt\text{ \ and \
G\left( x\right) =\int_{a}^{b}g\left( t\right) e^{-xt}dt,
\end{equation*
where the functions $f,g$ are both continuous on $\left[ a,b\right] $ with
g\left( t\right) >0$ for $t\in \left[ a,b\right] $. If there is $t^{\ast
}\in \left( a,b\right) $ such that $f/g$ is strictly increasing (decreasing)
on $\left[ a,t^{\ast }\right] $ and strictly decreasing (increasing) on
\left[ t^{\ast },b\right] $, then the ratio $x\mapsto F\left( x\right)
/G\left( x\right) $ is decreasing (increasing) on $\left( 0,\infty \right) $
if and only i
\begin{equation*}
H_{F,G}\left( 0^{+}\right) =\lim_{x\rightarrow 0^{+}}\left( \frac{F^{\prime
}\left( x\right) }{G^{\prime }\left( x\right) }G\left( x\right) -F\left(
x\right) \right) \geq \left( \leq \right) 0,
\end{equation*
wit
\begin{equation}
\lim_{x\rightarrow 0^{+}}\frac{F\left( x\right) }{G\left( x\right) }=\frac
\int_{a}^{b}f\left( t\right) dt}{\int_{a}^{b}g\left( t\right) dt}\text{ \
and \ }\lim_{x\rightarrow \infty }\frac{F\left( x\right) }{G\left( x\right)
=\frac{f\left( a\right) }{g\left( a\right) }. \label{F/G-0,00}
\end{equation
While $H_{F,G}\left( 0^{+}\right) <\left( >\right) 0$, then there is at
least one $x^{\ast }>0$ such that $F/G$ is increasing (decreasing) on
\left( 0,x^{\ast }\right) $ and decreasing (increasing) on $\left( x^{\ast
},\infty \right) $.
\end{theorem}
\begin{proof}[Proof of Theorem \protect\ref{MT-1}]
We only prove this theorem under the condition that $f/g$ is strictly
increasing on $\left[ a,t^{\ast }\right] $ and strictly decreasing on $\left[
t^{\ast },b\right] $. If $f/g$ is strictly decreasing on $\left[ a,t^{\ast
\right] $ and strictly increasing on $\left[ t^{\ast },b\right] $, then
\left( -f\right) /g$ is strictly increasing on $\left[ a,t^{\ast }\right] $
and strictly decreasing on $\left[ t^{\ast },b\right] $, and then
corresponding conclusion of this theorem is also true, which suffices to
note that $H_{-F,G}\left( x\right) =-H_{F,G}\left( x\right) $ due to (\re
{H-sr}).
For $n\in \mathbb{N}$, given a partition of the interval $\left[ a,b\right]
:
\begin{equation*}
a=t_{0}<t_{1}<t_{2}<\cdot \cdot \cdot <t_{n}=b,
\end{equation*
with $\Delta t_{i}=t_{i}-t_{i-1}=\left( b-a\right) /n$, and so
t_{i}=a+\left( b-a\right) i/n$. Then we hav
\begin{eqnarray*}
\sum_{i=0}^{n-1}f\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}
&=&\sum_{i=0}^{n-1}f\left( t_{i}\right) e^{-ax}\left[ e^{-\left( b-a\right)
x/n}\right] ^{i}\frac{b-a}{n} \\
&=&\frac{b-a}{n}e^{-ax}\sum_{i=0}^{n-1}f\left( t_{i}\right) y^{i}=:\frac{b-
}{n}e^{-ax}F_{n}\left( y\right) ,
\end{eqnarray*
\begin{equation*}
\sum_{i=0}^{n-1}g\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}=\frac{b-a}{n
e^{-ax}\sum_{i=0}^{n-1}g\left( t_{i}\right) y^{i}:=\frac{b-a}{n
e^{-ax}G_{n}\left( y\right) ,
\end{equation*
where $y\equiv y_{n}\left( x\right) =e^{-\left( b-a\right) x/n}\in \left(
0,1\right) $. These imply tha
\begin{equation*}
\frac{\sum_{i=0}^{n-1}f\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}
\sum_{i=0}^{n-1}g\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}=\frac
\sum_{i=0}^{n-1}f\left( t_{i}\right) y^{i}}{\sum_{i=0}^{n-1}g\left(
t_{i}\right) y^{i}}=\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) },
\end{equation*
\begin{equation}
\frac{F\left( x\right) }{G\left( x\right) }=\frac{\int_{a}^{b}f\left(
t\right) e^{-xt}dt}{\int_{a}^{b}g\left( t\right) e^{-xt}dt}=\frac
\lim_{n\rightarrow \infty }\sum_{i=0}^{n-1}f\left( t_{i}\right)
e^{-xt_{i}}\Delta t_{i}}{\lim_{n\rightarrow \infty }\sum_{i=0}^{n-1}g\left(
t_{i}\right) e^{-xt_{i}}\Delta t_{i}}=\lim_{n\rightarrow \infty }\frac
F_{n}\left( y\right) }{G_{n}\left( y\right) }. \label{limFn/Gn}
\end{equation
Also, we hav
\begin{equation*}
\sum_{i=0}^{n-1}\left( t_{i}-a\right) f\left( t_{i}\right) e^{-xt_{i}}\Delta
t_{i}=\sum_{i=0}^{n-1}\frac{\left( b-a\right) i}{n}f\left( t_{i}\right)
e^{-ax}\left[ e^{-\left( b-a\right) x/n}\right] ^{i}\frac{b-a}{n}
\end{equation*
\begin{equation*}
=e^{-ax}\left( \frac{b-a}{n}\right) ^{2}y\sum_{i=0}^{n-1}if\left(
t_{i}\right) y^{i-1}=e^{-ax}\left( \frac{b-a}{n}\right) ^{2}yF_{n}^{\prime
}\left( y\right) ,
\end{equation*
\begin{equation*}
\sum_{i=0}^{n-1}\left( t_{i}-a\right) g\left( t_{i}\right) e^{-xt_{i}}\Delta
t_{i}=e^{-ax}\left( \frac{b-a}{n}\right) ^{2}yG_{n}^{\prime }\left( y\right)
.
\end{equation*
Therefore, we obtai
\begin{equation*}
\frac{\sum\limits_{i=0}^{n-1}\left( t_{i}-a\right) f\left( t_{i}\right)
e^{-xt_{i}}\Delta t_{i}}{\sum\limits_{i=0}^{n-1}\left( t_{i}-a\right)
g\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}=\frac{F_{n}^{\prime }\left(
y\right) }{G_{n}^{\prime }\left( y\right) },
\end{equation*
\begin{equation}
\frac{\int_{a}^{b}\left( t-a\right) f\left( t\right) e^{-xt}dt}
\int_{a}^{b}\left( t-a\right) g\left( t\right) e^{-xt}dt}=\frac
\lim_{n\rightarrow \infty }\sum\limits_{i=0}^{n-1}\left( t_{i}-a\right)
f\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}{\lim_{n\rightarrow \infty
}\sum\limits_{i=0}^{n-1}\left( t_{i}-a\right) g\left( t_{i}\right)
e^{-xt_{i}}\Delta t_{i}}=\lim_{n\rightarrow \infty }\frac{F_{n}^{\prime
}\left( y\right) }{G_{n}^{\prime }\left( y\right) }. \label{ldFn/dGn}
\end{equation
Sinc
\begin{eqnarray*}
\int_{a}^{b}\left( t-a\right) f\left( t\right) e^{-xt}dt
&=&\int_{a}^{b}tf\left( t\right) e^{-xt}dt-a\int_{a}^{b}f\left( t\right)
e^{-xt}dt=-F^{\prime }\left( x\right) -aF\left( x\right) , \\
\int_{a}^{b}\left( t-a\right) g\left( t\right) e^{-xt}dt
&=&\int_{a}^{b}tg\left( t\right) e^{-xt}dt-a\int_{a}^{b}g\left( t\right)
e^{-xt}dt=-G^{\prime }\left( x\right) -aG\left( x\right) ,
\end{eqnarray*
equation (\ref{ldFn/dGn}) also can be written a
\begin{equation}
\frac{F^{\prime }\left( x\right) +aF\left( x\right) }{G^{\prime }\left(
x\right) +aG\left( x\right) }=\lim_{n\rightarrow \infty }\frac{F_{n}^{\prime
}\left( y\right) }{G_{n}^{\prime }\left( y\right) }. \label{ldFn/dGn-a}
\end{equation
It then follows tha
\begin{equation*}
H_{F_{n},G_{n}}\left( y\right) =\frac{F_{n}^{\prime }\left( y\right) }
G_{n}^{\prime }\left( y\right) }G_{n}\left( y\right) -F_{n}\left( y\right)
\frac{\sum\limits_{i=0}^{n-1}\left( t_{i}-a\right) f\left( t_{i}\right)
e^{-xt_{i}}\Delta t_{i}}{\sum\limits_{i=0}^{n-1}\left( t_{i}-a\right)
g\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}\sum_{i=0}^{n-1}g\left(
t_{i}\right) y^{i}-\sum_{i=0}^{n-1}f\left( t_{i}\right) y^{i}
\end{equation*
\begin{eqnarray*}
&=&\frac{ne^{ax}}{b-a}\left( \frac{\sum_{i=0}^{n-1}\left( t_{i}-a\right)
f\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}{\sum_{i=0}^{n-1}\left(
t_{i}-a\right) g\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}\sum_{i=0}^{n-1
\frac{b-a}{n}e^{-ax}g\left( t_{i}\right) y^{i}-\sum_{i=0}^{n-1}\frac{b-a}{n
e^{-ax}f\left( t_{i}\right) y^{i}\right) \\
&=&\frac{ne^{ax}}{b-a}\left( \frac{\sum_{i=0}^{n-1}\left( t_{i}-a\right)
f\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}}{\sum_{i=0}^{n-1}\left(
t_{i}-a\right) g\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}
\sum_{i=0}^{n-1}g\left( t_{i}\right) e^{-xt_{i}}\Delta
t_{i}-\sum_{i=0}^{n-1}f\left( t_{i}\right) e^{-xt_{i}}\Delta t_{i}\right) \\
&:&=\frac{ne^{ax}}{b-a}C_{f,g}^{\left[ n\right] }\left( x\right) ,
\end{eqnarray*
and s
\begin{equation}
\func{sgn}\left( H_{F_{n},G_{n}}\left( y\right) \right) =\func{sgn}\left(
C_{f,g}^{\left[ n\right] }\left( x\right) \right) . \label{sgnHn-Cn}
\end{equation
Clearly, we hav
\begin{eqnarray*}
\lim_{n\rightarrow \infty }C_{f,g}^{\left[ n\right] }\left( x\right) &=
\frac{\int_{a}^{b}\left( t-a\right) f\left( t\right) e^{-xt}dt}
\int_{a}^{b}\left( t-a\right) g\left( t\right) e^{-xt}dt}\int_{a}^{b}g\left(
t\right) e^{-xt}dt-\int_{a}^{b}f\left( t\right) e^{-xt}dt \\
&=&\frac{F^{\prime }\left( x\right) +aF\left( x\right) }{G^{\prime }\left(
x\right) +aG\left( x\right) }G\left( x\right) -F\left( x\right) =\frac
G^{\prime }\left( x\right) }{G^{\prime }\left( x\right) +aG\left( x\right)
H_{F,G}\left( x\right) ,
\end{eqnarray*
which, in view of $G^{\prime }\left( x\right) <0$ an
\begin{equation}
G^{\prime }\left( x\right) +aG\left( x\right) =-\int_{a}^{b}\left(
t-a\right) g\left( t\right) e^{-xt}dt<0 \label{dG+aG<0}
\end{equation
for $x\in \left( 0,\infty \right) $, implies tha
\begin{equation}
\func{sgn}\left( \lim_{n\rightarrow \infty }C_{f,g}^{\left[ n\right] }\left(
x\right) \right) =\func{sgn}\left( H_{F,G}\left( x\right) \right)
\label{sgnC-H}
\end{equation
for $x\in \left( 0,\infty \right) $.
(i) The necessity follows fro
\begin{equation*}
\left( \frac{F\left( x\right) }{G\left( x\right) }\right) ^{\prime }=\frac
G^{\prime }\left( x\right) }{G\left( x\right) ^{2}}H_{F,G}\left( x\right)
\leq 0
\end{equation*
for $x>0$, which, due to $G^{\prime }\left( x\right) <0$ for all $x\in
\left( 0,\infty \right) $, implies that $H_{F,G}\left( 0^{+}\right) \geq 0$.
Conversely, if $H_{F,G}\left( 0^{+}\right) \geq 0$, then by the relation
\ref{sgnC-H}), there is a large $N\in \mathbb{N}$ such that $C_{f,g}^{\left[
n\right] }\left( 0^{+}\right) \geq 0$ for $n>N$, which in combination with
the relation (\ref{sgnHn-Cn}) gives that $H_{F_{n},G_{n}}\left( y\right)
\geq 0$ as $y\rightarrow 1^{-}$ for $n>N$. On the other hand, since $f/g$ is
increasing on $\left[ a,t^{\ast }\right] $ and decreasing on $\left[ t^{\ast
},b\right] $, we easily see that there is a $i_{0}\geq 1$ such that the
sequence $\{f\left( t_{i}\right) /g\left( t_{i}\right) \}$ is strictly
increasing for $0\leq i\leq i_{0}$ and decreasing for $i_{0}<i\leq n-1$. By
Lemma \ref{L-PA/PB-pm}, the ratio $F_{n}\left( y\right) /G_{n}\left(
y\right) $ is strictly increasing with respect to $y$ on $\left( 0,1\right)
, that is
\begin{equation*}
\frac{d}{dy}\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) }=\frac
G_{n}^{\prime }\left( y\right) }{G_{n}\left( y\right) }\left( \frac
F_{n}^{\prime }\left( y\right) }{G_{n}^{\prime }\left( y\right) }-\frac
F_{n}\left( y\right) }{G_{n}\left( y\right) }\right) >0\text{ for }n>N\text{
and }y\in \left( 0,1\right) ,
\end{equation*
which, due to $G_{n}\left( y\right) ,G_{n}^{\prime }\left( y\right) >0$,
yield
\begin{equation*}
\frac{F_{n}^{\prime }\left( y\right) }{G_{n}^{\prime }\left( y\right) }
\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) }>0\text{ for }n>N\text{
and }y\in \left( 0,1\right) .
\end{equation*
This together with (\ref{limFn/Gn}) and (\ref{ldFn/dGn-a}) give
\begin{equation*}
\frac{F^{\prime }\left( x\right) +aF\left( x\right) }{G^{\prime }\left(
x\right) +aG\left( x\right) }-\frac{F\left( x\right) }{G\left( x\right)
\geq 0\text{ for }x\in \left( 0,\infty \right) ,
\end{equation*
which indicates tha
\begin{equation*}
\left( \frac{F\left( x\right) }{G\left( x\right) }\right) ^{\prime }=\frac
G^{\prime }\left( x\right) +aG\left( x\right) }{G\left( x\right) }\left(
\frac{F^{\prime }\left( x\right) +aF\left( x\right) }{G^{\prime }\left(
x\right) +aG\left( x\right) }-\frac{F\left( x\right) }{G\left( x\right)
\right) \leq 0\text{ for }x\in \left( 0,\infty \right) ,
\end{equation*
where the inequality holds due to $G\left( x\right) >0$ and $G^{\prime
}\left( x\right) +aG\left( x\right) <0$ by (\ref{dG+aG<0}). This proves the
sufficiency.
The first limit of (\ref{F/G-0,00}) is clear. While the second one follows
from (\ref{limFn/Gn}), which implies tha
\begin{eqnarray*}
\lim_{x\rightarrow \infty }\frac{F\left( x\right) }{G\left( x\right) }
&=&\lim_{x\rightarrow \infty }\lim_{n\rightarrow \infty }\frac{F_{n}\left(
y\right) }{G_{n}\left( y\right) }=\lim_{n\rightarrow \infty
}\lim_{x\rightarrow \infty }\frac{F_{n}\left( y\right) }{G_{n}\left(
y\right) } \\
&=&\lim_{n\rightarrow \infty }\lim_{y\rightarrow 0}\frac{F_{n}\left(
y\right) }{G_{n}\left( y\right) }=\frac{f\left( t_{0}\right) }{g\left(
t_{0}\right) }=\frac{f\left( a\right) }{g\left( a\right) }.
\end{eqnarray*}
(ii) If $H_{F,G}\left( 0^{+}\right) <0$, by the relations (\ref{sgnC-H}) and
(\ref{sgnHn-Cn}), there is a large enough $N\in \mathbb{N}$ such that
H_{F_{n},G_{n}}\left( y\right) <0$ as $y\rightarrow 1^{-}$ for $n>N$. By
Lemma \ref{L-PA/PB-pm}, there is a unique $y_{0}^{\left[ n\right] }\in
\left( 0,1\right) $ for given $n>N$ such that the function $F_{n}\left(
y\right) /G_{n}\left( y\right) $ is increasing on $\left( 0,y_{0}^{\left[
\right] }\right) $ and decreasing on $\left( y_{0}^{\left[ n\right]
},1\right) $, that is
\begin{eqnarray*}
\frac{d}{dy}\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) } &=&\frac
G_{n}^{\prime }\left( y\right) }{G_{n}\left( y\right) }\left( \frac
F_{n}^{\prime }\left( y\right) }{G_{n}^{\prime }\left( y\right) }-\frac
F_{n}\left( y\right) }{G_{n}\left( y\right) }\right) >0\text{ for }n>N\text{
and }y\in \left( 0,y_{0}^{\left[ n\right] }\right) , \\
\frac{d}{dy}\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) } &=&\frac
G_{n}^{\prime }\left( y\right) }{G_{n}\left( y\right) }\left( \frac
F_{n}^{\prime }\left( y\right) }{G_{n}^{\prime }\left( y\right) }-\frac
F_{n}\left( y\right) }{G_{n}\left( y\right) }\right) <0\text{ for }n>N\text{
and }y\in \left( y_{0}^{\left[ n\right] },1\right) ,
\end{eqnarray*
where $y_{0}^{\left[ n\right] }$ is the unique solution of the equation
\left[ F_{n}\left( y\right) /G_{n}\left( y\right) \right] ^{\prime }=0$ on
\left( 0,1\right) $, namely
\begin{equation*}
\left[ \frac{d}{dy}\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) }\right]
_{y=y_{0}^{\left[ n\right] }}=\frac{G_{n}^{\prime }\left( y_{0}^{\left[
\right] }\right) }{G_{n}\left( y_{0}^{\left[ n\right] }\right) }\left( \frac
F_{n}^{\prime }\left( y_{0}^{\left[ n\right] }\right) }{G_{n}^{\prime
}\left( y_{0}^{\left[ n\right] }\right) }-\frac{F_{n}\left( y_{0}^{\left[
\right] }\right) }{G_{n}\left( y_{0}^{\left[ n\right] }\right) }\right) =
\text{ for }n>N.
\end{equation*}
In the same treatment as part (i) of the proof of this theorem, the above
three relations imply tha
\begin{eqnarray*}
\left( \frac{F\left( x\right) }{G\left( x\right) }\right) ^{\prime } &\leq &
\text{ for }x\in \left( x^{\ast },\infty \right) , \\
\left( \frac{F\left( x\right) }{G\left( x\right) }\right) ^{\prime } &\geq &
\text{ for }x\in \left( 0,x^{\ast }\right) ,
\end{eqnarray*
where $x^{\ast }=\lim_{n\rightarrow \infty }x_{0}^{\left[ n\right] }$,
x_{0}^{\left[ n\right] }=-n\left( \ln y_{0}^{\left[ n\right] }\right)
/\left( b-a\right) $, and satisfie
\begin{equation*}
\left[ \left( \frac{F\left( x\right) }{G\left( x\right) }\right) ^{\prime
\right] _{x=x^{\ast }}=0.
\end{equation*
Thus it remains to prove $x^{\ast }=\lim_{n\rightarrow \infty }x_{0}^{\left[
n\right] }\neq 0,\infty $. First, we claim that $x^{\ast
}=\lim_{n\rightarrow \infty }x_{0}^{\left[ n\right] }\neq 0$. If not, that
is, $\lim_{n\rightarrow \infty }x_{0}^{\left[ n\right] }=0$, then $F\left(
x\right) /G\left( x\right) $ is decreasing in $x$ on $\left( 0,\infty
\right) $. This, by part (i) of this theorem, implies that $H_{F,G}\left(
0^{+}\right) \geq 0$, which yields a contraction with the assumption that
H_{F,G}\left( 0^{+}\right) <0$.
Second, we also claim that $x^{\ast }=\lim_{n\rightarrow \infty }x_{0}^
\left[ n\right] }\neq \infty $. If not, that is, $\lim_{n\rightarrow \infty
}x_{0}^{\left[ n\right] }=\infty $, then $F\left( x\right) /G\left( x\right)
$ is increasing in $x$ on $\left( 0,\infty \right) $. It then follows that
for all $x>0$
\begin{equation*}
\frac{F\left( x\right) }{G\left( x\right) }<\lim_{x\rightarrow \infty }\frac
F\left( x\right) }{G\left( x\right) }=\frac{f\left( a\right) }{g\left(
a\right) }.
\end{equation*
Since $\lim_{n\rightarrow \infty }\left( F_{n}\left( y\right) /G_{n}\left(
y\right) \right) =F\left( x\right) /G\left( x\right) $, there exists a large
enough $N_{1}\in \mathbb{N}$ such that for $n>N_{1}$ the inequalit
\begin{equation}
\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) }<\frac{f\left( a\right) }
g\left( a\right) } \label{Fn/Gn<}
\end{equation
holds for all $y\in \left( 0,1\right) $. On the other hand, as shown just
now, the function $F_{n}\left( y\right) /G_{n}\left( y\right) $ is
increasing on $\left( 0,y_{0}^{\left[ n\right] }\right) $ and decreasing on
\left( y_{0}^{\left[ n\right] },1\right) $, which suggests that there exists
a small enough $\delta \in \left( 0,y_{0}^{\left[ n\right] }\right) $ such
that $F_{n}\left( y\right) /G_{n}\left( y\right) $ is increasing on $\left(
0,\delta \right) $. Therefore, for $y\in \left( 0,\delta \right) $ and $n>N$
\begin{equation*}
\frac{F_{n}\left( y\right) }{G_{n}\left( y\right) }\geq \frac{F_{n}\left(
0\right) }{G_{n}\left( 0\right) }=\frac{f\left( t_{0}\right) }{g\left(
t_{0}\right) }=\frac{f\left( a\right) }{g\left( a\right) }.
\end{equation*
This is in contradiction with the inequality (\ref{Fn/Gn<}) for all $y\in
\left( 0,1\right) $ and $n>N_{1}$.
Consequently, $x^{\ast }=\lim_{n\rightarrow \infty }x_{0}^{\left[ n\right]
}\neq 0,\infty $, which ends the proof.
\end{proof}
Letting $a\rightarrow 0^{+}$, $b\rightarrow \infty $ in Theorem \ref{MT-1}.
Then $F\left( x\right) $ and $G\left( x\right) $ are Laplace transforms of
the functions $f$ and $g$, respectively. By properties of uniformly
convergent improper integral with a parameter, we have the following
monotonicity rule, where the first limit of (\ref{LF/G-0,00}) follows from
the first one of (\ref{F/G-0,00}) and Cauchy mean value theorem, that is
\begin{eqnarray*}
\lim_{x\rightarrow 0^{+}}\frac{F\left( x\right) }{G\left( x\right) }
&=&\lim_{b\rightarrow \infty }\frac{\int_{a}^{b}f\left( t\right) dt}
\int_{0}^{b}g\left( t\right) dt}=\lim_{b\rightarrow \infty }\frac{\left[
\int_{a}^{s}f\left( t\right) dt\right] _{s=b}-\left[ \int_{a}^{s}f\left(
t\right) dt\right] _{s=a}}{\left[ \int_{a}^{s}g\left( t\right) dt\right]
_{s=b}-\left[ \int_{0}^{s}g\left( t\right) dt\right] _{s=a}} \\
&=&\lim_{b\rightarrow \infty }\frac{f\left( a+\theta \left( b-a\right)
\right) }{g\left( a+\theta \left( b-a\right) \right) }=\lim_{t\rightarrow
\infty }\frac{f\left( t\right) }{g\left( t\right) },
\end{eqnarray*
here $\theta \in \left( 0,1\right) $.
\begin{theorem}
\label{MT-2}Let $f$ and $g$ be both continuous functions on $\left( 0,\infty
\right) $ with $g\left( t\right) >0$ for $t\in \left( 0,\infty \right) $ and
let $F\left( x\right) =\mathcal{L}\left( f\right) $ and $G\left( x\right)
\mathcal{L}\left( g\right) $ converge for $x>0$. If there is a $t^{\ast }\in
\left( 0,\infty \right) $ such that $f/g$ is strictly increasing
(decreasing) on $\left( 0,t^{\ast }\right) $ and strictly decreasing
(increasing) on $\left( t^{\ast },\infty \right) $, then the function $F/G$
is decreasing (increasing) on $\left( 0,\infty \right) $ if and only i
\begin{equation*}
H_{F,G}\left( 0^{+}\right) =\lim_{x\rightarrow 0^{+}}\left( \frac{F^{\prime
}\left( x\right) }{G^{\prime }\left( x\right) }G\left( x\right) -F\left(
x\right) \right) \geq \left( \leq \right) 0,
\end{equation*
wit
\begin{equation}
\lim_{x\rightarrow 0^{+}}\frac{F\left( x\right) }{G\left( x\right)
=\lim_{t\rightarrow \infty }\frac{f\left( t\right) }{g\left( t\right) }\text{
\ and \ }\lim_{x\rightarrow \infty }\frac{F\left( x\right) }{G\left(
x\right) }=\lim_{t\rightarrow 0^{+}}\frac{f\left( t\right) }{g\left(
t\right) } \label{LF/G-0,00}
\end{equation
provide the indicated limits exist. While $H_{F,G}\left( 0^{+}\right)
<\left( >\right) 0$, there is at leas one $x^{\ast }>0$ such that $F/G$ is
increasing (decreasing) on $\left( 0,x^{\ast }\right) $ and decreasing
(increasing) on $\left( x^{\ast },\infty \right) $.
\end{theorem}
\begin{theorem}
\label{MT-3}Suppose that (i) both the functions $f$ and $g$ are continuous
on $\left( 0,\infty \right) $ with $g\left( t\right) >0$ for $t\in \left(
0,\infty \right) $; (ii) the function $\mu $ is positive, differentiable and
increasing from $\left( 0,\infty \right) $ onto $\left( \mu \left(
0^{+}\right) ,\infty \right) $; (iii) both the function
\begin{equation*}
F\left( x\right) =\int_{0}^{\infty }f\left( t\right) e^{-x\mu \left(
t\right) }dt\text{ \ and \ }G\left( x\right) =\int_{0}^{\infty }g\left(
t\right) e^{-x\mu \left( t\right) }dt
\end{equation*
converge for all $x>0$. Then the following statements are valid:
(i) If the ratio $f/g$ is increasing (decreasing) on $\left( 0,\infty
\right) $, then $F/G$ is decreasing (increasing) on $\left( 0,\infty \right)
$ wit
\begin{equation*}
\lim_{x\rightarrow 0^{+}}\frac{F\left( x\right) }{G\left( x\right)
=\lim_{t\rightarrow \infty }\frac{f\left( t\right) }{g\left( t\right) }\text{
\ and \ }\lim_{x\rightarrow \infty }\frac{F\left( x\right) }{G\left(
x\right) }=\lim_{t\rightarrow 0^{+}}\frac{f\left( t\right) }{g\left(
t\right) }.
\end{equation*}
(ii) If there is a $t^{\ast }\in \left( 0,\infty \right) $ such that $f/g$
is strictly increasing (decreasing) on $\left( 0,t^{\ast }\right) $ and
strictly decreasing (increasing) on $\left( t^{\ast },\infty \right) $, then
the ratio $F/G$ is decreasing (increasing) on $\left( 0,\infty \right) $ if
and only i
\begin{equation*}
H_{F,G}\left( 0^{+}\right) =\lim_{x\rightarrow 0^{+}}\left( \frac{F^{\prime
}\left( x\right) }{G^{\prime }\left( x\right) }G\left( x\right) -F\left(
x\right) \right) \geq \left( \leq \right) 0.
\end{equation*
While $H_{F,G}\left( 0^{+}\right) <\left( >\right) 0$, there is at least one
$x^{\ast }>0$ such that $F/G$ is increasing (decreasing) on $\left(
0,x^{\ast }\right) $ and decreasing (increasing) on $\left( x^{\ast },\infty
\right) $.
\end{theorem}
\begin{proof}
Let $\mu \left( t\right) -a=s$, where $a=\mu \left( 0^{+}\right) $. Then
F\left( x\right) $ and $G\left( x\right) $ are expressed a
\begin{equation*}
F\left( x\right) =e^{-xa}\int_{0}^{\infty }\frac{f\left( t\left( s\right)
\right) }{\mu ^{\prime }\left( t\left( s\right) \right) }e^{-xs}ds\text{ \
and \ }G\left( x\right) =e^{-xa}\int_{0}^{\infty }\frac{g\left( t\left(
s\right) \right) }{\mu ^{\prime }\left( t\left( s\right) \right) }e^{-xs}ds,
\end{equation*
where $t\left( s\right) =\mu ^{-1}\left( s+a\right) $, and $F\left( x\right)
/G\left( x\right) $ can be represented in the form of ratio of two Laplace
transforms
\begin{equation*}
\frac{F\left( x\right) }{G\left( x\right) }=\frac{e^{xa}F\left( x\right) }
e^{xa}G\left( x\right) }=\frac{\int_{0}^{\infty }\left[ f\left( t\left(
s\right) \right) /\mu ^{\prime }\left( t\left( s\right) \right) \right]
e^{-xs}ds}{\int_{0}^{\infty }\left[ g\left( t\left( s\right) \right) /\mu
^{\prime }\left( t\left( s\right) \right) \right] e^{-xs}ds}:=\frac
\int_{0}^{\infty }f^{\ast }\left( s\right) e^{-xs}ds}{\int_{0}^{\infty
}g^{\ast }\left( s\right) e^{-xs}ds}.
\end{equation*
It is easy to verify tha
\begin{equation}
\frac{f^{\ast }\left( s\right) }{g^{\ast }\left( s\right) }=\frac{f\left(
t\left( s\right) \right) }{g\left( t\left( s\right) \right) }\text{, \ \ \
\left( \frac{f^{\ast }\left( s\right) }{g^{\ast }\left( s\right) }\right)
^{\prime }=\left( \frac{f\left( t\right) }{g\left( t\right) }\right)
^{\prime }\times \frac{dt}{ds}=\frac{1}{\mu ^{\prime }\left( t\right)
\left( \frac{f\left( t\right) }{g\left( t\right) }\right) ^{\prime },
\label{f*/g*-f/g}
\end{equation
where $\mu ^{\prime }\left( t\right) >0$ for all $t>0$.
(i) If the ratio $f\left( t\right) /g\left( t\right) $ is increasing
(decreasing) on $\left( 0,\infty \right) $, then so is $f^{\ast }\left(
s\right) /g^{\ast }\left( s\right) $. By Theorem \ref{T-Rmr}, we easily find
that $\left( e^{xa}F\right) /\left( e^{xa}G\right) =F/G$ is decreasing
(increasing) on $\left( 0,\infty \right) $.
By the limit relations (\ref{LF/G-0,00}) we easily ge
\begin{eqnarray*}
\lim_{x\rightarrow 0^{+}}\frac{F\left( x\right) }{G\left( x\right) }
&=&\lim_{s\rightarrow \infty }\frac{f\left( t\left( s\right) \right) /\mu
^{\prime }\left( t\left( s\right) \right) }{g\left( t\left( s\right) \right)
/\mu ^{\prime }\left( t\left( s\right) \right) }=\lim_{t\rightarrow \infty
\frac{f\left( t\right) }{g\left( t\right) }, \\
\lim_{x\rightarrow \infty }\frac{F\left( x\right) }{G\left( x\right) }
&=&\lim_{s\rightarrow 0^{+}}\frac{f\left( t\left( s\right) \right) /\mu
^{\prime }\left( t\left( s\right) \right) }{g\left( t\left( s\right) \right)
/\mu ^{\prime }\left( t\left( s\right) \right) }=\lim_{t\rightarrow 0^{+}
\frac{f\left( t\right) }{g\left( t\right) }.
\end{eqnarray*}
(ii) If there is a $t^{\ast }\in \left( 0,\infty \right) $ such that
f\left( t\right) /g\left( t\right) $ is strictly increasing (decreasing) on
\left( 0,t^{\ast }\right) $ and strictly decreasing (increasing) on $\left(
t^{\ast },\infty \right) $, then by the second relation of (\ref{f*/g*-f/g
), there is a $s^{\ast }\in \left( 0,\infty \right) $ such that $f^{\ast
}\left( s\right) /g^{\ast }\left( s\right) $ is strictly increasing
(decreasing) on $\left( 0,s^{\ast }\right) $ and strictly decreasing
(increasing) on $\left( s^{\ast },\infty \right) $, where $s^{\ast }=\mu
\left( t^{\ast }\right) -a$. By Theorem \ref{MT-2}, the ratio
e^{xa}F/\left( e^{xa}G\right) =F/G$ is decreasing (increasing) on $\left(
0,\infty \right) $ if and only i
\begin{equation}
\lim_{x\rightarrow 0^{+}}H_{e^{xa}F,e^{xa}G}\left( x\right)
=\lim_{x\rightarrow 0^{+}}\left( \frac{\left( e^{xa}F\left( x\right) \right)
^{\prime }}{\left( e^{xa}G\left( x\right) \right) ^{\prime }}e^{xa}G\left(
x\right) -e^{xa}F\left( x\right) \right) \geq \left( \leq \right) 0.
\label{HF*,G*}
\end{equation
We claim that the limit relation is equivalent to $\lim_{x\rightarrow
0^{+}}H_{F,G}\left( x\right) \geq \left( \leq \right) 0$. In fact, we easily
check tha
\begin{eqnarray*}
H_{e^{xa}F,e^{xa}G}\left( x\right) &=&\frac{e^{xa}\left( F^{\prime }\left(
x\right) +aF\left( x\right) \right) }{e^{xa}\left( G^{\prime }\left(
x\right) +aG\left( x\right) \right) }e^{xa}G\left( x\right) -e^{xa}F\left(
x\right) \\
&=&e^{xa}\frac{F^{\prime }\left( x\right) G\left( x\right) +aF\left(
x\right) G\left( x\right) -F\left( x\right) G^{\prime }\left( x\right)
-aF\left( x\right) G\left( x\right) }{G^{\prime }\left( x\right) +aG\left(
x\right) } \\
&=&\frac{e^{xa}G^{\prime }\left( x\right) }{G^{\prime }\left( x\right)
+aG\left( x\right) }\left( \frac{F^{\prime }\left( x\right) }{G^{\prime
}\left( x\right) }G\left( x\right) -F\left( x\right) \right) =\frac
e^{xa}G^{\prime }\left( x\right) }{G^{\prime }\left( x\right) +aG\left(
x\right) }H_{F,G}\left( x\right) .
\end{eqnarray*
This together wit
\begin{eqnarray*}
G^{\prime }\left( x\right) &=&-\int_{0}^{\infty }\mu \left( t\right) g\left(
t\right) e^{-x\mu \left( t\right) }dt<0, \\
G^{\prime }\left( x\right) +aG\left( x\right) &=&-\int_{0}^{\infty }\left[
\mu \left( t\right) -\mu \left( 0^{+}\right) \right] g\left( t\right)
e^{-x\mu \left( t\right) }dt<0
\end{eqnarray*
for $x>0$ indicates tha
\begin{equation*}
\func{sgn}\left( H_{e^{xa}F,e^{xa}G}\left( x\right) \right) =\func{sgn
\left( H_{F,G}\left( x\right) \right) ,
\end{equation*
which proves the claim just now.
(iii) If $\lim_{x\rightarrow 0^{+}}H_{F,G}\left( x\right) <\left( >\right) 0
, then $\lim_{x\rightarrow 0^{+}}H_{e^{xa}F,e^{xa}G}\left( x\right) <\left(
>\right) 0$. By Theorem \ref{MT-2}, there is at least one $x^{\ast }>0$ such
that $e^{xa}F\left( x\right) /\left( e^{xa}G\left( x\right) \right) =F\left(
x\right) /G\left( x\right) $ is increasing (decreasing) on $\left( 0,x^{\ast
}\right) $ and decreasing (increasing) on $\left( x^{\ast },\infty \right) $.
Thus we complete the proof.
\end{proof}
\section{A unified treatment for certain bounds of harmonic number}
The Euler-Mascheroni constant is defined b
\begin{equation*}
\gamma =\lim_{n\rightarrow \infty }\left( H_{n}-\ln n\right) =0.577215664...,
\end{equation*
where $H_{n}=\sum_{k=1}^{n}k^{-1}$ is the $n$'th harmonic number. There is a
close connection between $H_{n}$ and the psi (or digamma) function. Indeed,
we have $H_{n}=\psi \left( n+1\right) +\gamma $. Several bounds for $H_{n}$
or $\psi \left( n+1\right) $ can see \cite{Tims-MG-55-1971}, \cit
{Young-MG-75-1991}, \cite{DeTemple-AMM-100-1993}, \cite{Negoi-GM-15-1997},
\cite{Alzer-AMSUH-68-1998}, \cite{Villarino-arXiv-0510585}, \cit
{Villarino-JIPAM-9(3)-2008}, \cite{Mortici-CMA-59-2010}, \cit
{Chen-AML-23-2010}, \cite{Qi-AMC-218-2011}, \cite{Chen-CMA-64-2013}, \cit
{Lu-JMAA-419-2014}, \cite{Yang-AMC-268-2015} \cite{Zhao-JIA-193-2015}.
In particular, Alzer \cite{Alzer-AMSUH-68-1998} obtained the double
inequality,
\begin{equation*}
\frac{1}{2\left( n+a\right) }\leq H_{n}-\ln n-\gamma <\frac{1}{2\left(
n+b\right) }
\end{equation*
holds for $n\in \mathbb{N}$ with the best constant
\begin{equation*}
a=\frac{1}{2\left( 1-\gamma \right) }-1\text{ \ and \ }b=\frac{1}{6}
\end{equation*
by proving the sequenc
\begin{equation}
A\left( n\right) =\frac{1}{2}\frac{1}{\psi \left( n+1\right) -\ln n}-n
\label{A(n)}
\end{equation
is strictly decreasing for $n\geq 1$.
Villarino \cite{Villarino-arXiv-0510585} showed tha
\begin{eqnarray}
H_{n} &=&\ln \sqrt{n\left( n+1\right) }+\gamma +\frac{1}{6n\left( n+1\right)
+L\left( n\right) }, \label{L(n)} \\
&=&\ln \left( n+\frac{1}{2}\right) +\gamma +\frac{1}{24\left( n+1/2\right)
^{2}+D\left( n\right) }, \label{D(n)}
\end{eqnarray
where both the sequences $L\left( n\right) $ and $D\left( n\right) $ are
increasing for $n\in \mathbb{N}$. Qi \cite{Qi-AMC-218-2011} showed that the
sequenc
\begin{equation}
Q\left( n\right) =\frac{1}{2}\frac{1}{\ln n+1/\left( 2n\right) -\psi \left(
n+1\right) }-12n^{2} \label{Q(n)}
\end{equation
is strictly increasing for $n\in N$. These monotonicity of sequences
L\left( n\right) $, $D\left( n\right) $ and $Q\left( n\right) $ similarly
yield corresponding sharp bounds for $H_{n}$ or $\psi \left( n+1\right) $.
We remark that it is difficult to deal with the monotonicity of the function
$A\left( x\right) $, $L\left( x\right) $, $D\left( x\right) $ and $Q\left(
x\right) $ on $\left( 0,\infty \right) $ by usual approach. However, if we
write them as ratios of two Laplace transforms, then we easily prove their
monotonicity on $\left( 0,\infty \right) $ by Theorems \ref{T-Rmr}\ and \re
{MT-2}. Here we chose $\Phi \left( x\right) =D\left( x-1/2\right) $ defined
by (\ref{D(n)}) and prove its monotonicity on $\left( 0,\infty \right) $. As
far as $A\left( x\right) $, $L\left( x\right) $ and $Q\left( x\right) $, we
only list their expressions in the form of ratios of Laplace transforms. In
fact, By means of the formula
\begin{eqnarray*}
\psi (x) &=&\int_{0}^{\infty }\left( \frac{e^{-t}}{t}-\frac{e^{-xt}}{1-e^{-t
}\right) dt, \\
\ln x &=&\int_{0}^{\infty }\frac{e^{-t}-e^{-xt}}{t}dt\text{, } \\
\frac{1}{x^{n}} &=&\frac{1}{\left( n-1\right) !}\int_{0}^{\infty
}t^{n-1}e^{-xt}dt,
\end{eqnarray*
we hav
\begin{equation*}
A\left( x\right) =\frac{1}{2}\frac{1}{\psi \left( x+1\right) -\ln x}-x=\frac
\int_{0}^{\infty }\left[ -p_{1}^{\prime }\left( t\right) \right] e^{-xt}dt}
\int_{0}^{\infty }p\left( t\right) e^{-xt}dt},
\end{equation*
wher
\begin{equation*}
p_{1}\left( t\right) =2\left( \frac{1}{t}-\frac{1}{e^{t}-1}\right) ;
\end{equation*
\begin{equation*}
L\left( x\right) =\frac{2}{2\psi \left( x+1\right) -\ln \left( x\left(
x+1\right) \right) }-6x\left( x+1\right) =\frac{\int_{0}^{\infty }\left[
-6\left( p_{2}^{\prime \prime }\left( t\right) +p_{2}^{\prime }\left(
t\right) \right) \right] e^{-xt}dt}{\int_{0}^{\infty }p_{2}\left( t\right)
e^{-xt}dt},
\end{equation*
wher
\begin{equation*}
p_{2}\left( t\right) =\frac{e^{2t}-2te^{t}-1}{t\left( e^{t}-1\right) e^{t}};
\end{equation*
\begin{equation*}
Q\left( x\right) =\frac{1}{\ln x+1/\left( 2x\right) -\psi \left( x+1\right)
-12x^{2}=\frac{\int_{0}^{\infty }\left[ -12p_{3}^{\prime \prime }\left(
t\right) \right] e^{-xt}dt}{\int_{0}^{\infty }p_{3}\left( t\right) e^{-xt}dt
,
\end{equation*
wher
\begin{equation*}
p_{3}\left( t\right) =\frac{1}{2}\left( \coth \frac{t}{2}-\frac{2}{t}\right)
.
\end{equation*
Next we prove the monotonicity of $\Phi \left( x\right) =D\left(
x-1/2\right) $ on $\left( 0,\infty \right) $ by Theorem \ref{MT-2}.
\begin{proposition}
\label{P-p1}The functio
\begin{equation*}
\Phi \left( x\right) =\frac{1}{\psi \left( x+1/2\right) -\ln x}-24x^{2}
\end{equation*
is strictly increasing from $\left( 0,\infty \right) $ onto $\left(
0,21/5\right) $.
\end{proposition}
\begin{proof}
We write $\Phi \left( x\right) =F\left( x\right) /G\left( x\right) $, wher
\begin{eqnarray*}
F\left( x\right) &=&1-24x^{2}\left( \psi \left( x+1/2\right) -\ln x\right) ,
\\
G\left( x\right) &=&\psi \left( x+1/2\right) -\ln x.
\end{eqnarray*
It has been shown in \cite{Yang-JIA-157-2015} tha
\begin{eqnarray*}
G\left( x\right) &=&\int_{0}^{\infty }q\left( t\right) e^{-xt}dt, \\
x^{2}G\left( x\right) &=&\frac{1}{24}+\int_{0}^{\infty }e^{-xt}q^{\prime
\prime }\left( t\right) dt,
\end{eqnarray*
wher
\begin{equation}
q\left( t\right) =\frac{1}{t}-\frac{1}{2\sinh \left( t/2\right) }. \label{q}
\end{equation
Then $F\left( x\right) $ can be written as
\begin{equation*}
F\left( x\right) =1-24x^{2}G\left( x\right) =\int_{0}^{\infty }\left(
-24q^{\prime \prime }\left( t\right) \right) e^{-xt}dt.
\end{equation*
Thus $\Phi \left( x\right) $ is expressed a
\begin{equation*}
\Phi \left( x\right) =\frac{F\left( x\right) }{G\left( x\right) }=\frac
\int_{0}^{\infty }\left[ -24q^{\prime \prime }\left( t\right) \right]
e^{-xt}dt}{\int_{0}^{\infty }q\left( t\right) e^{-xt}dt}.
\end{equation*
We first show that there is a $t^{\ast }>0$ such that the function
-24q^{\prime \prime }/q$ is decreasing on $\left( 0,t^{\ast }\right) $ and
increasing on $\left( t^{\ast },\infty \right) $. Direct computations giv
\begin{eqnarray*}
\frac{q^{\prime \prime }\left( t\right) }{q\left( t\right) } &=&\frac{1}{4
\frac{2\cosh ^{2}s\sinh s-s^{3}\cosh ^{2}s-2\sinh s-s^{3}}{s^{2}\left(
s-\sinh s\right) \sinh ^{2}s} \\
&=&\frac{1}{2}\frac{\sinh 3s-s^{3}\cosh 2s-3\sinh s-3s^{3}}{s^{2}\left(
\sinh 3s-2s\cosh 2s-3\sinh s+2s\right) },
\end{eqnarray*
where $s=t/2$. Expanding in power series yield
\begin{equation*}
\frac{q^{\prime \prime }\left( t\right) }{q\left( t\right) }=\frac{1}{2
\frac{\sum_{n=3}^{\infty }\frac{3^{2n-1}-\left( 2n-1\right) \left(
2n-2\right) \left( 2n-3\right) 2^{2n-4}-3}{\left( 2n-1\right) !}s^{2n-1}}
\sum_{n=2}^{\infty }\frac{3^{2n-3}-\left( 2n-3\right) 2^{2n-3}-3}{\left(
2n-3\right) !}s^{2n-1}}:=\frac{1}{2}\frac{\sum_{n=4}^{\infty }a_{n}\left(
s^{2}\right) ^{n-4}}{\sum_{n=4}^{\infty }b_{n}\left( s^{2}\right) ^{n-4}}.
\end{equation*
Sinc
\begin{equation*}
\left( 2n-1\right) !b_{n+1}-9\left( 2n-3\right) !b_{n}=\left( 10n-23\right)
2^{2n-3}+24>0\text{ for }n\geq 3
\end{equation*
and $b_{3}=0$, we see that $b_{n}>0$ for $n\geq 4$.\ Thus, to prove the
function $-24q^{\prime \prime }/q$ is decreasing on $\left( 0,t^{\ast
}\right) $ and increasing on $\left( t^{\ast },\infty \right) $, by Lemma
\ref{L-A/B-g} it suffices to prove that there is an integer $n_{0}>4$ such
that the sequence $\{a_{n}/b_{n}\}_{n\geq 4}$ is increasing for $4\leq n\leq
n_{0}$ and decreasing for $n>n_{0}$. For this end, we have to prove $d_{n}
\left[ \left( 2n-1\right) !\right] ^{2}\left(
a_{n}b_{n+1}-a_{n+1}b_{n}\right) <0$ for $4\leq n\leq 9$ and $d_{n}>0$ for
n\geq 10$.
Some elementary computations give
\begin{eqnarray*}
d_{n} &=&18\left( 4n-1\right) +\frac{2}{9}\left( 4n-1\right) 3^{4n}-\frac{1}
108}\left( 2n-1\right) \left( 20n^{4}-56n^{3}-77n^{2}+464n-243\right) 6^{2n}
\\
&&+\frac{4}{9}\left( 64n^{2}-132n+41\right) 3^{2n}-\frac{3}{4}\left(
2n-1\right) \left( 2n-3\right) \left( 6n^{3}+5n^{2}+2n-1\right) 2^{2n}.
\end{eqnarray*
We find tha
\begin{equation*}
\begin{array}{lll}
d_{4}=-66\,802\,176, & d_{5}=-13\,774\,616\,064, & d_{6}=-1570\,251\,36
\,536, \\
d_{7}=-127\,269\,822\,161\,664, & d_{8}=-7526\,731\,991\,528\,448, &
d_{9}=-240\,861\,038\,835\,686\,400
\end{array
\end{equation*
To prove $d_{n}>0$ for $n\geq 10$, we write $d_{n}$ a
\begin{equation*}
d_{n}=18\left( 4n-1\right) +\left( 4n-1\right) 6^{2n}\times a_{n}^{\ast
}+\left( 64n^{2}-132n+41\right) 2^{2n}\times b_{n}^{\ast },
\end{equation*
wher
\begin{eqnarray*}
a_{n}^{\ast } &=&\frac{2}{9}\left( \frac{3}{2}\right) ^{2n}-\frac{1}{108
\frac{\left( 2n-1\right) \left( 20n^{4}-56n^{3}-77n^{2}+464n-243\right) }
4n-1}, \\
b_{n}^{\ast } &=&\frac{4}{9}\left( \frac{3}{2}\right) ^{2n}-\frac{3}{4}\frac
\left( 2n-1\right) \left( 2n-3\right) \left( 6n^{3}+5n^{2}+2n-1\right) }
64n^{2}-132n+41}.
\end{eqnarray*
It is easy to verify tha
\begin{equation*}
a_{10}^{\ast }=\frac{710\,697\,141}{6815\,744}>0\text{, \ \ \ }b_{10}^{\ast
}=\frac{174\,443\,916\,097}{149\,159\,936}>0,
\end{equation*
and for $n\geq 10$
\begin{equation*}
a_{n+1}^{\ast }-\frac{9}{4}a_{n}^{\ast }=\frac{1}{432}\frac{\left(
2n^{2}+n-9\right) \left( 400n^{4}-2500n^{3}+1448n^{2}+1789n-777\right) }
\left( 4n-1\right) \left( 4n+3\right) }>0,
\end{equation*
\begin{equation*}
\begin{array}{l}
b_{n+1}^{\ast }-\dfrac{9}{4}b_{n}^{\ast }=\dfrac{3}{16}\dfrac{2n-1}{\left(
64n^{2}-4n-27\right) \left( 64n^{2}-132n+41\right) }\bigskip \\
\times \left( 4n^{4}\left( 960n^{2}-3004n-1181\right)
+19\,204n^{3}+16\,517n^{2}-684n-2697\right) >0
\end{array
\end{equation*
which yield $d_{n}>0$ for $n\geq 10$.
Second, it is easy to see tha
\begin{equation*}
\lim_{x\rightarrow 0^{+}}F\left( x\right) =1\text{, \ \ \
\lim_{x\rightarrow 0^{+}}F^{\prime }\left( x\right) =0,
\end{equation*
an
\begin{equation*}
\lim_{x\rightarrow 0^{+}}\frac{G\left( x\right) }{G^{\prime }\left( x\right)
}=\lim_{x\rightarrow 0^{+}}\frac{\psi \left( x+1/2\right) -\ln x}{\psi
^{\prime }\left( x+1/2\right) -1/x}=0,
\end{equation*
which yiel
\begin{equation*}
\lim_{x\rightarrow 0^{+}}H_{F,G}\left( x\right) =\lim_{x\rightarrow
0^{+}}\left( \frac{F^{\prime }\left( x\right) }{G^{\prime }\left( x\right)
G\left( x\right) -F\left( x\right) \right) =-1<0.
\end{equation*
It then follows by Theorem \ref{MT-2} that the function $\Phi =F/G$ is
strictly increasing on $\left( 0,\infty \right) $.
An easy computation give
\begin{equation*}
\lim_{x\rightarrow 0^{+}}\Phi \left( x\right) =\lim_{t\rightarrow \infty
\frac{-24q^{\prime \prime }\left( t\right) }{q\left( t\right) }=0\text{ \
and \ }\lim_{x\rightarrow \infty }\Phi \left( x\right) =\lim_{t\rightarrow 0
\frac{-24q^{\prime \prime }\left( t\right) }{q\left( t\right) }=\frac{21}{5},
\end{equation*
which completes the proof.
\end{proof}
\section{An application to Bessel functions}
The modified Bessel functions of the second kind $K_{v}$ is defined as \cite
p. 78]{Watson-ATTBF-CUP-1922
\begin{equation}
K_{v}\left( x\right) =\frac{\pi }{2}\frac{I_{-v}\left( x\right) -I_{v}\left(
x\right) }{\sin \left( v\pi \right) }, \label{Kv-Iv}
\end{equation
where $I_{v}\left( x\right) $ is the modified Bessel functions of the first
kind which can be represented by the infinite series as
\begin{equation}
I_{v}\left( x\right) =\sum_{n=0}^{\infty }\frac{\left( x/2\right) ^{2n+v}}
n!\Gamma \left( v+n+1\right) }\text{, \ }x\in \mathbb{R}\text{, \ }v\in
\mathbb{R}\backslash \{-1,-2,...\}, \label{I_v-is}
\end{equation}
and the right-hand side of (\ref{Kv-Iv}) is replaced by its limiting value
if $v$ is an integer or zero.
We easily see that $K_{v}\left( x\right) =K_{-v}\left( x\right) $ for all
v\in \mathbb{R}$ and $x>0$ by (\ref{Kv-Iv}), so we assume that $v\geq 0$ in
this section, unless otherwise specified.
As showed in proof of Theorem 3.1 in \cite{Baricz-BAMS-82-2010}, the identit
\begin{equation}
1-\frac{K_{v-1}\left( x\right) K_{v+1}\left( x\right) }{K_{v}\left( x\right)
^{2}}=\frac{1}{x}\left( \frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left(
x\right) }\right) ^{\prime } \label{T-xdK/K}
\end{equation
holds for $v\in \mathbb{R}$ and $x>0$, and the functio
\begin{equation*}
x\mapsto \frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left( x\right) }
\end{equation*
is strictly decreasing on $\left( 0,\infty \right) $ for all $v\in \mathbb{R}
$ (see also \cite{Yang-PAMS-145-2017}). As another application, in this
section we will determine the monotonicity of the function $x\mapsto
x+xK_{v}^{\prime }\left( x\right) /K_{v}\left( x\right) $ on $\left(
0,\infty \right) $ by Theorem \ref{MT-3}. More precisely, we have
\begin{proposition}
\label{P-B1}For $v\geq 0$, let $K_{v}\left( x\right) $ be the modified
Bessel functions of the second kind.
(i) If $v\in \left( 1/2,\infty \right) $, then the functio
\begin{equation*}
x\mapsto \Lambda \left( x\right) =x+\frac{xK_{v}^{\prime }\left( x\right) }
K_{v}\left( x\right) }
\end{equation*
is strictly increasing from $\left( 0,\infty \right) $ onto $\left(
-v,-1/2\right) $.
(ii) If $v\in \lbrack 0,1/2)$, then the function $x\mapsto \Lambda \left(
x\right) $ is strictly decreasing from $\left( 0,\infty \right) $ onto
\left( -1/2,-v\right) $.
(iii) If $v\neq 1/2$, then the double inequalit
\begin{equation}
-x-\max \left( v,\frac{1}{2}\right) <\frac{xK_{v}^{\prime }\left( x\right) }
K_{v}\left( x\right) }<-x-\min \left( v,\frac{1}{2}\right) \label{xdK/K<>}
\end{equation
holds for $x>0$ with the best constants $\min \left( v,1/2\right) $ and
\max \left( v,1/2\right) $.
\end{proposition}
Before proving Proposition \ref{P-B1}, we give the following lemmas.
\begin{lemma}
\label{L-hv-m}For $v>0$ with $v\neq 1/2$, let the function $h_{v}$ be
defined on $\left( 0,\infty \right) $ b
\begin{equation}
h_{v}\left( t\right) =\frac{\cosh \left( tv\right) +v\sinh \left( tv\right)
\sinh t}{\left( \cosh t+1\right) \cosh \left( tv\right) }. \label{hv}
\end{equation
(i) If $v\in \lbrack 1,\infty )$ then $h_{v}$ is increasing from $\left(
0,\infty \right) $ onto $\left( 1/2,v\right) $.
(ii) If $v\in \left( 1/2,1\right) $, then there is a $t^{\ast }\in \left(
0,\infty \right) $ such that $h_{v}$ is increasing on $\left( 0,t^{\ast
}\right) $ and decreasing on $\left( t^{\ast },\infty \right) $.
Consequently, the inequalitie
\begin{equation*}
\frac{1}{2}=\min \left( \frac{1}{2},v\right) <h_{v}\left( t\right) \leq
\theta _{v}
\end{equation*
hold for $x>0$, where $\theta _{v}=h_{v}\left( t^{\ast }\right) $, here
t^{\ast }$ is the unique solution of the equation $h_{v}^{\prime }\left(
t\right) =0$ on $\left( 0,\infty \right) $.
(iii) If $v\in \left( 0,1/2\right) $, then there is a $t^{\ast }\in \left(
0,\infty \right) $ such that $h_{v}$ is decreasing on $\left( 0,t^{\ast
}\right) $ and increasing on $\left( t^{\ast },\infty \right) $. Therefore,
it holds that for $x>0$
\begin{equation*}
\theta _{v}\leq h_{v}\left( t\right) <\max \left( \frac{1}{2},v\right)
\frac{1}{2},
\end{equation*
where $\theta _{v}$ is as in (ii).
\end{lemma}
\begin{proof}
Differentiating and simplifying yiel
\begin{equation*}
h_{v}^{\prime }\left( t\right) =\frac{r_{v}\left( t\right) }{\left( \cosh
t+1\right) ^{2}\cosh ^{2}\left( tv\right) },
\end{equation*
wher
\begin{equation*}
r_{v}\left( t\right) =v^{2}\left( 1+\cosh t\right) \sinh t+v\left( 1+\cosh
t\right) \cosh \left( tv\right) \sinh \left( tv\right) -\cosh ^{2}\left(
tv\right) \sinh t.
\end{equation*
Using "product into sum" formulas and expanding in power series give
\begin{eqnarray*}
4r_{v}\left( t\right) &=&\left( v+1\right) \sinh \left( 2vt-t\right) +\left(
v-1\right) \sinh \left( 2tv+t\right) \\
&&+2v\sinh \left( 2tv\right) +2v^{2}\sinh \left( 2t\right) +2\left(
2v^{2}-1\right) \sinh t \\
&:&=\sum_{n=1}^{\infty }\frac{a_{n}}{\left( 2n-1\right) !}t^{2n-1},
\end{eqnarray*
wher
\begin{equation*}
a_{n}=\left( v+1\right) \left( 2v-1\right) ^{2n-1}+\left( v-1\right) \left(
2v+1\right) ^{2n-1}+\left( 2v\right) ^{2n}+v^{2}2^{2n}+2\left(
2v^{2}-1\right) .
\end{equation*}
To confirm the monotonicity of $h_{v}$, we need to deal with the sign of
a_{n}$. For this end, we first give two recurrence relations. It is easy to
check tha
\begin{equation}
\begin{array}{c}
\dfrac{a_{n+1}-a_{n}}{2^{2n}}=2v\left( v^{2}-1\right) \left( v-\frac{1}{2
\right) ^{2n-1}+2v\left( v^{2}-1\right) \left( v+\frac{1}{2}\right)
^{2n-1}\bigskip \\
+\left( 4v^{2}-1\right) v^{2n}+3v^{2}:=b_{n}
\end{array}
\label{Dan}
\end{equation
\begin{equation}
\frac{b_{n+1}-b_{n}}{v\left( v^{2}-1\right) \left( v^{2}-1/4\right) }=\left(
2v-3\right) \left( v-\frac{1}{2}\right) ^{2n-2}+\left( 2v+3\right) \left( v
\frac{1}{2}\right) ^{2n-2}+4v^{2n-1}>0 \label{Dbn}
\end{equation
for $v\neq 1,1/2$. We now distinguish three cases to prove the desired
monotonicity.
\textbf{Case 1}: $v\geq 1$. It is clear that $a_{n}>0$ for $n\geq 1$, which
implies that $h_{v}^{\prime }\left( t\right) >0$ for $t\in \left( 0,\infty
\right) $.
\textbf{Case 2}: $v\in \left( 1/2,1\right) $. From the recurrence relation
\ref{Dbn}) we see that the sequence $\{b_{n}\}_{n\geq 1}$ is decreasing,
which together wit
\begin{eqnarray*}
b_{1} &=&2v^{2}\left( 2v-1\right) \left( 2v+1\right) >0, \\
\lim_{n\rightarrow \infty }b_{n} &=&\func{sgn}\left( v^{2}-1\right) \infty <0
\end{eqnarray*
yields that there is an integer $n_{1}>1$ such that $b_{n}\geq 0$ for $1\leq
n\leq n_{1}$ and $b_{n}\leq 0$ for $n\geq n_{1}$. This, by the recurrence
relation (\ref{Dan}), in turn implies that the sequence $\{a_{n}\}_{n\geq 1}$
is increasing for $1\leq n\leq n_{1}$ and decreasing for $n\geq n_{1}$.
Therefore, we obtai
\begin{equation*}
a_{n}\geq a_{1}=4\left( 2v-1\right) \left( 2v+1\right) >0\text{ for }1\leq
n\leq n_{1}.
\end{equation*
On the other hand, it is seen that
\begin{equation*}
\lim_{n\rightarrow \infty }\frac{a_{n}}{2^{2n}}=\func{sgn}\left( v-1\right)
\infty <0.
\end{equation*
It then follows that there is an integer $n_{0}>n_{1}$ such that $a_{n}\geq
0 $ for $1\leq n\leq n_{0}$ and $a_{n}\leq 0$ for $n\geq n_{0}$. By Lemma
\ref{L-sgnS}, there is a $t^{\ast }>0$ such that $h_{v}^{\prime }\left(
t\right) >0$ for $t\in \left( 0,t^{\ast }\right) $ and $h_{v}^{\prime
}\left( t\right) <0$ for $t\in \left( t^{\ast },\infty \right) $.
\textbf{Case 3}: $v\in \left( 0,1/2\right) $. In this case, we see that the
sequence $\{b_{n}\}_{n\geq 1}$ is increasing, which in combination wit
\begin{eqnarray*}
b_{1} &=&2v^{2}\left( 2v-1\right) \left( 2v+1\right) <0, \\
\lim_{n\rightarrow \infty }b_{n} &=&3v^{2}>0
\end{eqnarray*
indicates that there is an integer $n_{1}>1$ such that $b_{n}\leq 0$ for
1\leq n\leq n_{1}$ and $b_{n}\geq 0$ for $n\geq n_{1}$. This, by the
recurrence relation (\ref{Dan}), means that the sequence $\{a_{n}\}_{n\geq
1} $ is decreasing for $1\leq n\leq n_{1}$ and increasing for $n\geq n_{1}$.
Hence, we deduce tha
\begin{equation*}
a_{n}\leq a_{1}=4\left( 2v-1\right) \left( 2v+1\right) <0\text{ for }1\leq
n\leq n_{1}.
\end{equation*
Moreover, it is seen that
\begin{equation*}
\lim_{n\rightarrow \infty }\frac{a_{n}}{\left( 2v+1\right) ^{2n}}=\func{sgn
\left( v^{2}\right) \infty >0.
\end{equation*
It then follows that there is an integer $n_{0}>n_{1}$ such that $a_{n}\leq
0 $ for $1\leq n\leq n_{0}$ and $a_{n}\geq 0$ for $n\geq n_{0}$. By Lemma
\ref{L-sgnS}, there is a $t^{\ast }>0$ such that $h_{v}^{\prime }\left(
t\right) <0$ for $t\in \left( 0,t^{\ast }\right) $ and $h_{v}^{\prime
}\left( t\right) >0$ for $t\in \left( t^{\ast },\infty \right) $.
This completes the proof.
\end{proof}
\begin{lemma}
\label{L-HF,G}Let $K_{v}\left( x\right) $ be the modified Bessel functions
of the second kind and le
\begin{equation*}
F\left( x\right) =x\left[ K_{v}\left( x\right) +K_{v}^{\prime }\left(
x\right) \right] \text{ \ and \ }G\left( x\right) =K_{v}\left( x\right) .
\end{equation*
Then for $v\in \left( 0,1\right) $, we hav
\begin{equation*}
\lim_{x\rightarrow 0^{+}}H_{F,G}\left( x\right) =\lim_{x\rightarrow
0^{+}}\left( \frac{F^{\prime }\left( x\right) }{G^{\prime }\left( x\right)
G\left( x\right) -F\left( x\right) \right) =0.
\end{equation*}
\end{lemma}
\begin{proof}
By the asymptotic formulas \cite[p. 375, (9.6.9)]{Abramowitz-HMFFGMT-1972
\begin{equation}
K_{v}\left( x\right) \thicksim \frac{1}{2}\Gamma \left( v\right) \left(
\frac{x}{2}\right) ^{-v}\text{ for\ }v>0\text{ as }x\rightarrow 0\text{,}
\label{K_v-->0}
\end{equation
we have that$\quad $as $x\rightarrow 0$,
\begin{eqnarray*}
K_{v}^{\prime }\left( x\right) &\thicksim &-\frac{1}{4}\Gamma \left(
v+1\right) \left( \frac{x}{2}\right) ^{-v-1}\text{ for\ }v>0, \\
K_{v}^{\prime \prime }\left( x\right) &\thicksim &\frac{1}{8}\Gamma \left(
v+2\right) \left( \frac{x}{2}\right) ^{-v-2}\text{ for\ }v>0.
\end{eqnarray*
Then, for $v>0$, as $x\rightarrow 0^{+}$
\begin{eqnarray*}
F\left( x\right) &=&x\left[ K_{v}\left( x\right) +K_{v}^{\prime }\left(
x\right) \right] \thicksim x\left[ \frac{1}{2}\Gamma \left( v\right) \left(
\frac{x}{2}\right) ^{-v}-\frac{1}{4}\Gamma \left( v+1\right) \left( \frac{x}
2}\right) ^{-v-1}\right] \\
&=&\Gamma \left( v\right) \frac{x-v}{2}\left( \frac{x}{2}\right) ^{-v},
\end{eqnarray*
\begin{eqnarray*}
F^{\prime }\left( x\right) &=&K_{v}\left( x\right) +\left( x+1\right)
K_{v}^{\prime }\left( x\right) +xK_{v}^{\prime \prime }\left( x\right) \\
&\thicksim &\frac{1}{2}\Gamma \left( v\right) \left( \frac{x}{2}\right)
^{-v}-\frac{x+1}{4}\Gamma \left( v+1\right) \left( \frac{x}{2}\right)
^{-v-1}+x\frac{1}{8}\Gamma \left( v+2\right) \left( \frac{x}{2}\right)
^{-v-2} \\
&=&\frac{1}{2}\Gamma \left( v\right) \frac{\left( 1-v\right) x+v^{2}}{x
\left( \frac{x}{2}\right) ^{-v},
\end{eqnarray*
which yiel
\begin{eqnarray*}
H_{F,G}\left( x\right) &=&\frac{F^{\prime }\left( x\right) }{G^{\prime
}\left( x\right) }G\left( x\right) -F\left( x\right) \\
&\thicksim &\frac{\frac{1}{2}\Gamma \left( v\right) \frac{\left( 1-v\right)
x+v^{2}}{x}\left( \frac{x}{2}\right) ^{-v}}{-\frac{1}{4}\Gamma \left(
v+1\right) \left( \frac{x}{2}\right) ^{-v-1}}\frac{1}{2}\Gamma \left(
v\right) \left( \frac{x}{2}\right) ^{-v}-\Gamma \left( v\right) \frac{x-v}{2
\left( \frac{x}{2}\right) ^{-v} \\
&=&-\frac{\Gamma \left( v\right) }{v}\left( \frac{x}{2}\right)
^{1-v}\rightarrow 0\text{ as }x\rightarrow 0^{+}\text{ if }v\in \left(
0,1\right) .
\end{eqnarray*
This completes the proof.
\end{proof}
We now are in a position to prove Proposition \ref{P-B1}.
\begin{proof}[Proof of Proposition \protect\ref{P-B1}]
We have
\begin{equation*}
\Lambda \left( x\right) =\frac{x\left[ K_{v}\left( x\right) +K_{v}^{\prime
}\left( x\right) \right] }{K_{v}\left( x\right) }=\frac{F\left( x\right) }
G\left( x\right) }.
\end{equation*
By the integral representation (\ref{Kv-Ir}) we get tha
\begin{equation}
G\left( x\right) =K_{v}\left( x\right) =\int_{0}^{\infty }\cosh \left(
vt\right) e^{-x\cosh t}dt; \label{G-ir}
\end{equation
\begin{eqnarray*}
K_{v}\left( x\right) +K_{v}^{\prime }\left( x\right) &=&\int_{0}^{\infty
}\cosh \left( vt\right) e^{-x\cosh t}dt-\int_{0}^{\infty }\cosh \left(
vt\right) \left( \cosh t\right) e^{-x\cosh t}dt \\
&=&-\int_{0}^{\infty }\cosh \left( vt\right) \left( \cosh t-1\right)
e^{-x\cosh t}dt,
\end{eqnarray*
then integration by parts yield
\begin{eqnarray}
F\left( x\right) &=&x\left( K_{v}\left( x\right) +K_{v}^{\prime }\left(
x\right) \right) =\int_{0}^{\infty }\cosh \left( vt\right) \frac{\cosh t-1}
\sinh t}de^{-x\cosh t} \notag \\
&=&-\int_{0}^{\infty }\frac{\cosh \left( vt\right) +v\sinh t\sinh \left(
vt\right) }{\cosh t+1}e^{-x\cosh t}dt. \label{F-ir}
\end{eqnarray
Thus $\Lambda \left( x\right) $ can be written a
\begin{equation*}
\Lambda \left( x\right) =\frac{F\left( x\right) }{G\left( x\right) }=\frac
\int_{0}^{\infty }\left( -\frac{\cosh \left( vt\right) +v\sinh t\sinh \left(
vt\right) }{\cosh t+1}\right) e^{-x\cosh t}dt}{\int_{0}^{\infty }\cosh
\left( vt\right) e^{-x\cosh t}dt}:=\frac{\int_{0}^{\infty }f\left( t\right)
e^{-x\mu \left( t\right) }dt}{\int_{0}^{\infty }g\left( t\right) e^{-x\mu
\left( t\right) }dt},
\end{equation*
where $\mu \left( t\right) =\cosh t$ an
\begin{equation*}
f\left( t\right) =-\frac{\cosh \left( vt\right) +v\sinh t\sinh \left(
vt\right) }{\cosh t+1}\text{, \ \ \ }g\left( t\right) =\cosh \left(
vt\right) .
\end{equation*
Clearly, $\mu \left( t\right) $ is positive, differentiable and increasing
on $\left( 0,\infty \right) $, while $f\left( t\right) $ and $g\left(
t\right) $ are differentiable on $\left( 0,\infty \right) $ with $g\left(
t\right) >0$ for $t>0$. Also, we hav
\begin{equation*}
\frac{f\left( t\right) }{g\left( t\right) }=\frac{-\left[ \cosh \left(
vt\right) +v\sinh t\sinh \left( vt\right) \right] /\left( \cosh t+1\right) }
\cosh \left( vt\right) }=-h_{v}\left( t\right) .
\end{equation*}
(i) If $v\geq 1$, then by the first assertion of Lemma \ref{L-hv-m} we see
that $f/g$ is strictly decreasing on $\left( 0,\infty \right) $. It follows
by part (i) of Theorem \ref{MT-3} that $\Lambda =F/G$ is strictly increasing
on $\left( 0,\infty \right) $.
(ii) If $v\in \left( 1/2,1\right) $, then by the second assertion of Lemma
\ref{L-hv-m} we see that there is a $t^{\ast }\in \left( 0,\infty \right) $
such that $f/g$ is decreasing on $\left( 0,t^{\ast }\right) $ and increasing
on $\left( t^{\ast },\infty \right) $. Also, $\lim_{x\rightarrow
0^{+}}H_{F,G}\left( x\right) =0$ due to Lemma \ref{L-HF,G}. These, by part
(ii) of Theorem \ref{MT-3}, yield that $\Lambda =F/G$ is strictly increasing
on $\left( 0,\infty \right) $.
(iii) If $v\in \left( 0,1/2\right) $, then the third assertion of Lemma \re
{L-hv-m} it is seen that there is a $t^{\ast }\in \left( 0,\infty \right) $
such that $f/g$ is increasing on $\left( 0,t^{\ast }\right) $ and decreasing
on $\left( t^{\ast },\infty \right) $. And, $\lim_{x\rightarrow
0^{+}}H_{F,G}\left( x\right) =0$ due to Lemma \ref{L-HF,G}. It then follows
from part (ii) of Theorem \ref{MT-3} that $\Lambda =F/G$ is strictly
decreasing on $\left( 0,\infty \right) $.
(iv) If $v=0$, then $f\left( t\right) /g\left( t\right) =-1/\left( \cosh
t+1\right) $ is clearly increasing on $\left( 0,\infty \right) $. It follows
from part (i) of Theorem \ref{MT-3} that $\Lambda =F/G$ is strictly
decreasing on $\left( 0,\infty \right) $.
The limit values ar
\begin{eqnarray*}
\Lambda \left( 0^{+}\right) &=&\lim_{x\rightarrow 0^{+}}\frac{F\left(
x\right) }{G\left( x\right) }=-\lim_{t\rightarrow \infty }\frac{\cosh \left(
vt\right) +v\sinh t\sinh \left( vt\right) }{\left( \cosh t+1\right) \cosh
\left( vt\right) }=-v, \\
\Lambda \left( \infty \right) &=&\lim_{x\rightarrow \infty }\frac{F\left(
x\right) }{G\left( x\right) }=-\lim_{t\rightarrow 0^{+}}\frac{\cosh \left(
vt\right) +v\sinh t\sinh \left( vt\right) }{\left( \cosh t+1\right) \cosh
\left( vt\right) }=-\frac{1}{2}.
\end{eqnarray*
The double inequality (\ref{xdK/K<>}) follows from the monotonicity of $F/G$
on $\left( 0,\infty \right) $, which ends the proof.
\end{proof}
As a consequence of \cite[Theorem 5]{Miller-ITSF-12(4)-2001}, Miller and
Samko showed that the function $x\mapsto \sqrt{x}e^{x}K_{v}\left( x\right) $
is strictly decreasing on $\left( 0,\infty \right) $ for $\left\vert
v\right\vert >1/2$. Yang and Zheng in \cite[Corollary 3.2
{Yang-PAMS-145-2017} reproved this assertion and further proved this
function is strictly increasing on $\left( 0,\infty \right) $ for
\left\vert v\right\vert <1/2$. Now we have a more general result by
Proposition \ref{P-B1}.
\begin{corollary}
\label{C-B1}For $v\geq 0$, let $K_{v}\left( x\right) $ be the modified
Bessel functions of the second kind. Then the functio
\begin{equation*}
x\mapsto x^{r}e^{x}K_{v}\left( x\right)
\end{equation*
is strictly increasing on $\left( 0,\infty \right) $ if and only if $r\geq
\max \left( v,1/2\right) $, and decreasing if and only if $r\leq \min \left(
v,1/2\right) $. While $v>\left( <\right) 1/2$ and $1/2<r<v$ ($v<r<1/2$),
there is an $x_{0}>0$ such that this function is decreasing (increasing) on
\left( 0,x_{0}\right) $ and increasing (decreasing) on $\left( x_{0},\infty
\right) $.
\end{corollary}
\begin{proof}
Differentiation yield
\begin{eqnarray*}
\left[ x^{r}e^{x}K_{v}\left( x\right) \right] ^{\prime }
&=&x^{r}e^{x}K_{v}\left( x\right) +rx^{r-1}e^{x}K_{v}\left( x\right)
+x^{r}e^{x}K_{v}^{\prime }\left( x\right) \\
&=&x^{r-1}e^{x}K_{v}\left( x\right) \left( r+x+\frac{xK_{v}^{\prime }\left(
x\right) }{K_{v}\left( x\right) }\right) :=x^{r-1}e^{x}K_{v}\left( x\right)
\times \phi \left( x\right) .
\end{eqnarray*
By Proposition \ref{P-B1}, $\left[ x^{r}e^{x}K_{v}\left( x\right) \right]
^{\prime }\geq \left( \leq \right) 0$ for all $x>0$ if and only i
\begin{eqnarray*}
r &\geq &-\inf_{x>0}\left( x+\frac{xK_{v}^{\prime }\left( x\right) }
K_{v}\left( x\right) }\right) =\left\{
\begin{array}{cc}
v & \text{if }v\in \left( 1/2,\infty \right) \\
\frac{1}{2} & \text{if }v\in \left( 0,1/2\right
\end{array
\right. =\max \left( v,\frac{1}{2}\right) , \\
r &\leq &-\sup_{x>0}\left( x+\frac{xK_{v}^{\prime }\left( x\right) }
K_{v}\left( x\right) }\right) =\left\{
\begin{array}{cc}
\frac{1}{2} & \text{if }v\in \left( 1/2,\infty \right) \\
v & \text{if }v\in \left( 0,1/2\right
\end{array
\right. =\min \left( v,\frac{1}{2}\right) .
\end{eqnarray*
When $v>1/2$ and $1/2<r<v$, by part (i) of Proposition \ref{P-B1}, we see
that $x\mapsto r+x+xK_{v}^{\prime }\left( x\right) /K_{v}\left( x\right)
=\phi \left( x\right) $ is increasing on $\left( 0,\infty \right) $, which
in combination with $\phi \left( 0^{+}\right) =r-v<0$ and $\phi \left(
\infty \right) =r-1/2>0$ yields that there is an $x_{0}>0$ such that $\phi
\left( x\right) <0$ for $x\in \left( 0,x_{0}\right) $ and $\phi \left(
x\right) >0$ for $x\in \left( x_{0},\infty \right) $. That is to say, the
function $x\mapsto x^{r}e^{x}K_{v}\left( x\right) $ is decreasing on $\left(
0,x_{0}\right) $ and increasing on $\left( x_{0},\infty \right) $.
Similarly, by part (ii) of Proposition \ref{P-B1} we can prove that for
v\in \lbrack 0,1/2)$ and $v<r<1/2$, there is an $x_{0}>0$ such that the
function $x\mapsto x^{r}e^{x}K_{v}\left( x\right) $ is increasing on $\left(
0,x_{0}\right) $ and decreasing on $\left( x_{0},\infty \right) $. This
completes the proof.
\end{proof}
\begin{remark}
Corollary \ref{C-B1} implies that for all $0<x<y$, the double inequalit
\begin{equation}
e^{y-x}\left( \frac{y}{x}\right) ^{r_{1}}<\frac{K_{v}\left( x\right) }
K_{v}\left( y\right) }<e^{y-x}\left( \frac{y}{x}\right) ^{r_{2}}
\label{Kx/Ky<>}
\end{equation
if and only if $r_{1}\leq \min \left( \left\vert v\right\vert ,1/2\right) $
and $r_{2}\geq \max \left( \left\vert v\right\vert ,1/2\right) $. Obviously,
our inequalities (\ref{Kx/Ky<>}) are superior to those earlier results
appeared in \cite{Bordelon-SIAMREV-15-1973}, \cite{Ross-SIAMREV-15-1973},
\cite{Paris-SIAM-JMA-15(1)-1984}, \cite{Joshi-JAMS-50-1991}, \cit
{Laforgia-JVAM-34(3)-1991}. Detailed comments can see \cite[Section 3
{Baricz-PEMS-53-2010}.
\end{remark}
By Lemma \ref{L-hv-m} and Bernstein's theorem, we give a class of completely
monotonic function.
\begin{corollary}
\label{C-B2}For $v\geq 0$, the functio
\begin{equation*}
P_{\lambda }\left( x\right) =\left( x+\lambda \right) e^{x}K_{v}\left(
x\right) +xe^{x}K_{v}^{\prime }\left( x\right)
\end{equation*
is CM on $\left( 0,\infty \right) $ if and only i
\begin{equation*}
\lambda \geq \left\{
\begin{array}{cc}
v & \text{if }v\in \lbrack 1,\infty ), \\
\theta _{v} & \text{if }v\in \left( \frac{1}{2},1\right) , \\
\frac{1}{2} & \text{if }v\in \left( 0,\frac{1}{2}\right)
\end{array
\right.
\end{equation*
and so is $-P_{\lambda }\left( x\right) $ if and only if
\begin{equation*}
\lambda \leq \left\{
\begin{array}{cc}
\frac{1}{2} & \text{if }v\in \left( \frac{1}{2},\infty \right) , \\
\theta _{v} & \text{if }v\in \left( 0,\frac{1}{2}\right)
\end{array
\right.
\end{equation*
where $\theta _{v}$ is as in Lemma \ref{L-hv-m}.
\end{corollary}
\begin{proof}
From integral representations (\ref{F-ir}) and (\ref{G-ir}), we hav
\begin{eqnarray*}
P_{\lambda }\left( x\right) &=&e^{x}F\left( x\right) +\lambda e^{x}G\left(
x\right) \\
&=&\int_{0}^{\infty }\left( -\frac{\cosh \left( vt\right) +v\sinh t\sinh
\left( vt\right) }{\cosh t+1}\right) e^{-x\left( \cosh t-1\right)
}dt+\lambda \int_{0}^{\infty }\cosh \left( vt\right) e^{-x\left( \cosh
t-1\right) }dt
\end{eqnarray*
\begin{equation*}
=\int_{0}^{\infty }\left( \lambda -h_{v}\left( t\right) \right) \cosh \left(
vt\right) e^{-x\left( \cosh t-1\right) }dt=\int_{0}^{\infty }\left( \lambda
-h_{v}\left( t\left( s\right) \right) \right) \frac{\cosh \left( vt\left(
s\right) \right) }{\sinh \left( t\left( s\right) \right) }e^{-xs}ds,
\end{equation*
where $h_{v}\left( t\right) $ is defined by (\ref{hv}) and $t\left( s\right)
=\cosh ^{-1}\left( s+1\right) $. By Bernstein's theorem and Lemma \re
{L-hv-m} $P_{\lambda }\left( x\right) $ is CM on $\left( 0,\infty \right) $
if and only i
\begin{equation*}
\lambda \geq \sup_{t>0}\left( h_{v}\left( t\right) \right) =\left\{
\begin{array}{cc}
v & \text{if }v\in \lbrack 1,\infty ), \\
\theta _{v} & \text{if }v\in \left( \frac{1}{2},1\right) , \\
\frac{1}{2} & \text{if }v\in \left( 0,\frac{1}{2}\right)
\end{array
\right.
\end{equation*
and so is $-P_{\lambda }\left( x\right) $ if and only i
\begin{equation*}
\lambda \leq \inf_{t>0}\left( h_{v}\left( t\right) \right) =\left\{
\begin{array}{cc}
\frac{1}{2} & \text{if }v\in \lbrack 1,\infty ), \\
\frac{1}{2} & \text{if }v\in \left( \frac{1}{2},1\right) , \\
\theta _{v} & \text{if }v\in \left( 0,\frac{1}{2}\right)
\end{array
\right.
\end{equation*
which completes the proof.
\end{proof}
\begin{remark}
It was proved in \cite[Theorem 5]{Miller-ITSF-12(4)-2001} that the function
x\mapsto \sqrt{x}e^{x}K_{v}\left( x\right) $ is CM on $\left( 0,\infty
\right) $ if $\left\vert v\right\vert >1/2$. Now we present a new proof by
Corollary \ref{C-B2}. Differentiation yield
\begin{equation*}
-\left[ \sqrt{x}e^{x}K_{v}\left( x\right) \right] ^{\prime }=-\frac{1}{\sqrt
x}}\left[ \left( x+\frac{1}{2}\right) e^{x}K_{v}\left( x\right)
+xe^{x}K_{v}^{\prime }\left( x\right) \right] =\frac{1}{\sqrt{x}}\left[
-P_{1/2}\left( x\right) \right] .
\end{equation*
Since $x\mapsto 1/\sqrt{x}$ and $-P_{1/2}\left( x\right) $ are CM on $\left(
0,\infty \right) $ by Corollary \ref{C-B2}, we find that so is $x^{-1/2
\left[ -P_{1/2}\left( x\right) \right] $, and so is $\sqrt{x
e^{x}K_{v}\left( x\right) $ on $\left( 0,\infty \right) $.
\end{remark}
\begin{remark}
Baricz \cite{Baricz-PEMS-53-2010} conjectured that $x\mapsto \sqrt{x
e^{x}K_{v}\left( x\right) $ for all $|v|<1/2$ is a Bernstein function, which
was proved in \cite[Remark 3.3]{Yang-PAMS-145-2017}. By Corollary \ref{C-B2
, we can give a simple proof. Indeed, it suffices to prove $\left[ \sqrt{x
e^{x}K_{v}\left( x\right) \right] ^{\prime }$ for $v\in \left( 0,1/2\right) $
is CM on $\left( 0,\infty \right) $. Differentiation yield
\begin{equation*}
\left[ \sqrt{x}e^{x}K_{v}\left( x\right) \right] ^{\prime }=\frac{1}{\sqrt{x
}\left[ \left( x+\frac{1}{2}\right) e^{x}K_{v}\left( x\right)
+xe^{x}K_{v}^{\prime }\left( x\right) \right] =\frac{1}{\sqrt{x}
P_{1/2}\left( x\right) .
\end{equation*
Since the function $x\mapsto 1/\sqrt{x}$ is CM on $\left( 0,\infty \right)
, while $P_{1/2}\left( x\right) $ is CM on $\left( 0,\infty \right) $ in
view of Corollary \ref{C-B2}, it then follows from \cite[Theorem 1
{Miller-ITSF-12(4)-2001} that so is $x^{-1/2}P_{1/2}\left( x\right) $ on
\left( 0,\infty \right) $.
\end{remark}
We note that (\ref{T-xdK/K}) can be written a
\begin{equation*}
1+\frac{1}{x}-\frac{K_{v-1}\left( x\right) K_{v+1}\left( x\right) }
K_{v}\left( x\right) ^{2}}=\frac{1}{x}\left( x+\frac{xK_{v}^{\prime }\left(
x\right) }{K_{v}\left( x\right) }\right) ^{\prime }=\frac{1}{x}\Lambda
^{\prime }\left( x\right) ,
\end{equation*
which, by $\Lambda ^{\prime }\left( x\right) >\left( <0\right) $ if
\left\vert v\right\vert >\left( <\right) 1/2$ given in Proposition \ref{P-B1
, we derive the following corollary.
\begin{corollary}
Let $v\in \mathbb{R}$ with $\left\vert v\right\vert \neq 1/2$. Then the
following inequalit
\begin{equation*}
\frac{K_{v-1}\left( x\right) K_{v+1}\left( x\right) }{K_{v}\left( x\right)
^{2}}<\left( >\right) 1+\frac{1}{x},
\end{equation*
or equivalently
\begin{equation}
K_{v}\left( x\right) ^{2}-K_{v-1}\left( x\right) K_{v+1}\left( x\right)
>\left( <\right) -\frac{1}{x}K_{v}\left( x\right) ^{2} \label{T-I}
\end{equation
holds for $x>0$ if $\left\vert v\right\vert >\left( <\right) 1/2$.
\end{corollary}
\begin{remark}
The inequality (\ref{T-I}) is the Tur\'{a}n type inequality for modified
Bessel functions of the second kind, which first appeared in \cit
{Baricz-EM-33-2015}. More such inequalities can be found in \cit
{Ismail-SIAM-JMA-9(4)-1978}, \cite{Haeringen-JMP-19-1978}, \cit
{Laforgia-JIA-2010-253035}, \cite{Baricz-BAMS-82-2010}, \cit
{Segura-JMAA-374-2011}, \cite{Baricz-PAMS-141(2)-2013}, \cit
{Baricz-FM-26(1)-2014} \cite{Baricz-EM-33-2015}, \cite{Yang-PAMS-145-2017}.
\end{remark}
Finally, we give an improvement of the double inequality (\ref{xdK/K<>}).
\begin{corollary}
Let $v\in \mathbb{R}$ with $\left\vert v\right\vert \neq 1/2$. Then the
following inequalit
\begin{equation}
-\sqrt{x^{2}+x+\max \left( \left\vert v\right\vert ,\frac{1}{2}\right) ^{2}}
\frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left( x\right) }<-\sqrt
x^{2}+x+\min \left( \left\vert v\right\vert ,\frac{1}{2}\right) ^{2}}
\label{xdK/K<>-a}
\end{equation
holds for $x>0$.
\end{corollary}
\begin{proof}
The desired inequalities are equivalent t
\begin{eqnarray*}
-\sqrt{x^{2}+x+v^{2}} &<&\frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left(
x\right) }<-x-\frac{1}{2}\text{ if }\left\vert v\right\vert >\frac{1}{2}, \\
-x-\frac{1}{2} &<&\frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left(
x\right) }<-\sqrt{x^{2}+x+v^{2}}\text{ if }\left\vert v\right\vert <\frac{1}
2}.
\end{eqnarray*
By the double inequality (\ref{xdK/K<>}), it suffices to prov
\begin{equation}
\frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left( x\right) }>\left(
<\right) -\sqrt{x^{2}+x+v^{2}}\text{ if }\left\vert v\right\vert >\left(
<\right) 1/2. \label{xdK/K><}
\end{equation
To this end, we use the recurrence relations (see \cite[p. 79
{Watson-ATTBF-CUP-1922}
\begin{eqnarray}
\frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left( x\right) }+v &=&-\frac
xK_{v-1}\left( x\right) }{K_{v}\left( x\right) }, \\
\frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left( x\right) }-v &=&-\frac
xK_{v+1}\left( x\right) }{K_{v}\left( x\right) }
\end{eqnarray
to get the identit
\begin{equation*}
\left( \frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left( x\right) }\right)
^{2}-v^{2}=x^{2}\frac{K_{v-1}\left( x\right) K_{v+1}\left( x\right) }
K_{v}\left( x\right) ^{2}}.
\end{equation*
This in combination with the identity (\ref{T-xdK/K}) yield
\begin{equation*}
x^{2}+x+v^{2}-\left( \frac{xK_{v}^{\prime }\left( x\right) }{K_{v}\left(
x\right) }\right) ^{2}=x\left( x+\frac{xK_{v}^{\prime }\left( x\right) }
K_{v}\left( x\right) }\right) ^{\prime }=x\Lambda ^{\prime }\left( x\right) .
\end{equation*
Using the inequality $\Lambda ^{\prime }\left( x\right) >\left( <\right) 0$
if $\left\vert v\right\vert >\left( <\right) 1/2$ due to Proposition \re
{P-B1} and noting $K_{v}\left( x\right) >0$ and $K_{v}^{\prime }\left(
x\right) <0$, the inequality (\ref{xdK/K><}) follows. This completes the
proof.
\end{proof}
\begin{remark}
The bound $-\sqrt{x^{2}+x+v^{2}}$ for $xK_{v}^{\prime }\left( x\right)
/K_{v}\left( x\right) $ given in inequality \ref{xdK/K><} is better than
some known ones, which refer to \cite[p. 242--243]{Baricz-EM-33-2015}. While
the another one $-x-1/2$ seems to be a new and simple one.
\end{remark}
\section{Conclusions}
In this paper, by the monotonicity rule for the ratio of two polynomials
with the same highest degree (Lemma \ref{L-PA/PB-pm}) and definition of
integral, we find that the decreasing (increasing) property of three ratio
\begin{equation*}
\frac{F\left( x\right) }{G\left( x\right) }:=\frac{\int_{a}^{b}f\left(
t\right) e^{-xt}dt}{\int_{a}^{b}g\left( t\right) e^{-xt}dt}\text{, \ \ \
\frac{\int_{0}^{\infty }f\left( t\right) e^{-xt}dt}{\int_{0}^{\infty
}g\left( t\right) e^{-xt}dt}\text{, \ \ \ }\frac{\int_{0}^{\infty }f\left(
t\right) e^{-x\mu \left( t\right) }dt}{\int_{0}^{\infty }f\left( t\right)
e^{-x\mu \left( t\right) }dt},
\end{equation*
under the condition that $f/g$ has the monotonicity pattern that $\nearrow
\searrow $ ($\searrow \nearrow $), according as $H_{F,G}\left( 0^{+}\right)
\geq \left( \leq \right) 0$. Otherwise, the three ratios $F/G$ are unimodal.
Since many special functions have integral representations in the form of
\int_{a}^{b}\left( \cdot \right) e^{-xt}dt$, our three theorems in this
paper are efficient tools of researching certain special functions.
|
{
"timestamp": "2018-03-08T02:03:58",
"yymm": "1803",
"arxiv_id": "1803.02513",
"language": "en",
"url": "https://arxiv.org/abs/1803.02513"
}
|
\section*{Acknowledgments}
We are pleased to thank Rob Myers and Robb Mann for useful conversations and comments.
The work of PAC is funded by Fundaci\'on la Caixa through a ``la Caixa - Severo Ochoa" international pre-doctoral grant. The work of PAC was also supported by the MINECO/FEDER, UE grant FPA2015-66793-P and by the Spanish Research Agency (Agencia Estatal de Investigaci\'on) through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016- 0597. PAC also thanks Perimeter Institute ``Visiting Graduate Fellows" program. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Research, Innovation and Science.
\end{acknowledgments}
|
{
"timestamp": "2018-03-30T02:02:57",
"yymm": "1803",
"arxiv_id": "1803.02795",
"language": "en",
"url": "https://arxiv.org/abs/1803.02795"
}
|
\section{Introduction} \label{sec:intro}
Young massive OB stars produce strong ultraviolet radiation that significantly influences the structure, chemistry, thermal balance, and dynamical evolution of the nearby interstellar medium \citep{Hollenbach97}. Their extreme-ultraviolet (EUV) photons ionize the surrounding gas and form H{\scriptsize II} regions. Photon-dominated or photo-dissociated regions (PDRs) start at the H{\scriptsize II} region/neutral cloud boundary where the EUV radiation vanishes and the far-ultraviolet (FUV) radiation becomes dominant, dissociating H$_2$ and ionizing heavier elements. The gas inside the PDR transits from atomic to molecular as the FUV flux decreases due to dust extinction and H$_2$ absorption \citep[e.g.,][]{Goicoechea16}. PDRs exist in many astrophysical environments and span a board range of spatial scales, from the nuclei of starburst galaxies \citep[e.g.,][]{Fuente08} to the illuminated surfaces of protoplanetary disks \citep[e.g.,][]{Agundez08}. The study of PDRs could also help to understand the impact from massive young stars on subsequent star formation in nearby molecular clouds.
The Orion Bar is probably the best studied PDR in our Galaxy. It is located between the Orion Molecular Cloud 1 and the H{\scriptsize II} region excited by the Trapezium cluster, and is exposed to a FUV field a few $10^4$ times the mean interstellar radiation field. Owing to its proximity \citep[417 pc,][]{Menten07} and nearly edge-on orientation, the Bar provides an ideal laboratory for testing PDR models \citep[e.g.,][]{Jansen95,Gorti02,Andree17} and a primary target for observational studies of physical and chemical structures of PDRs \citep[e.g.,][]{Tielens93,Walmsley00,van09,Arab12,Peng12,Goicoechea16,Nagy17}. Observations of various molecular spectral lines have shown that the emissions could be better interpreted with an inhomogeneous density structure containing an extended and relatively low density ($n_{\rm H}\sim10^4$ -- $10^5$ cm$^{-3}$) medium and a compact and high density ($n_{\rm H}\sim10^5$ -- $10^6$ cm$^{-3}$) component \citep[e.g.,][]{Hogerheijde95,Young00,Leurini06,Leurini10,Goicoechea16}. However, due to the scarce of high resolution observations capable of spatially resolving the density structure, the nature of the high density clumps or condensations is still not well understood. \citet{Lis03} mapped the Bar in H$^{13}$CN (1--0) with the Plateau de Bure Interferometer (PdBI) at an angular resolution of about $5''$, and detected 10 dense clumps. They proposed that the H$^{13}$CN clumps are in virial equilibrium and may be collapsing to form stars. \citet{Goicoechea16} performed Atacama Large Millimeter/submillimeter Array (ALMA) HCO$^+$ (4--3) observations of the Bar and detected over-dense substructures close to the cloud edge, and found that the substructures have masses much lower than the mass needed to make them gravitationally unstable. These two interferometric observations both target molecular spectral lines. A high resolution map of the dust continuum emission of the Bar, which is highly desirable in constraining the mass and density of the dense condensations, is still lacking. Here we report our Submillimeter Array (SMA) observations of the dust continuum and molecular spectral line observations of the Bar.
\section{Observations} \label{sec:obs}
The SMA\footnote{The SMA is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia
Sinica.} observations were carried out in 2009 and 2012. The observations in 2009 were taken on January 14th, January 30th, and February 3rd, with 5, 7, and 7 antennas, respectively, in the Sub-compact configuration. The weather conditions were good, with the zenith atmospheric opacity at 225 GHz, $\tau_{\rm 225GHz}$, in the range of 0.05 to 0.15. We observed two fields, one in the northeast (NE) centered at (R.A., decl.)$_{\rm J2000}$ = ($5^{\rm h}35^{\rm m}25.\!^{\rm s}2$, $-5{\arcdeg}24{\arcmin}34.\!{\arcsec}6$) and the other in the southwest (SW) centered at (R.A., decl.)$_{\rm J2000}$ = ($5^{\rm h}35^{\rm m}22.\!^{\rm s}1$, $-5{\arcdeg}25{\arcmin}13.\!{\arcsec}4$), and used Titan for flux calibration, Uranus and 3C454.3 for bandpass calibration, and J0423-013, J0607-085 for time dependent gain calibration. The 230~GHz receivers were tuned to cover rest frequencies of 234.76 -- 236.76 GHz in the lower sideband (LSB) and 244.76 -- 246.76 GHz in the upper sideband (USB). Signals from each sideband were processed by correlators consisting of 24 chunks with each chunk having a bandwidth of 104~MHz divided into 256~channels, resulting in a uniform spectral resolution of 406.25 kHz ($\sim$0.5~km\,s$^{-1}$). With this setup we could simultaneously observe the 1.2~mm continuum and molecular spectral lines of CS~(5--4) and H$_2$CS (7$_{1,6}$--6$_{1,6}$). Motived by the detection of dust continuum and spectral line emissions with the 2009 observations, we performed additional observations in 2012 to further constrain the physical conditions of the dense gas in the Bar. The observations were performed on 2012 January 1st with 7 antennas in the Compact configuration. The weather was good, with $\tau_{\rm 225GHz}$ varying from 0.07 to 0.18. We observed Callisto for flux calibration, 3C279 for bandpass calibration, and J0423-013, J0607-085 for time dependent gain calibration. By the time of our 2012 observations, the SMA correlators had been upgraded to be able to process signals across a bandwidth of 4~GHz divided into 48 chunks in each sideband. The 230~GHz receivers were then tuned to cover 192.26 -- 196.26 GHz in the LSB and 204.26 -- 208.26 GHz in the USB, and the correlators were configured to provide a spectral resolution of 812.5 kHz ($\sim$1.2~km\,s$^{-1}$) across all the chunks expect chunk 42, which covered C$^{34}$S (4--3) and was set to provide a high spectral resolution of 203.125 kHz ($\sim$0.3 km\,s$^{-1}$). This setup also covered spectral lines of CS~(4--3), C$^{33}$S (4--3), and H$_2$CS (6$_{0,6}$--5$_{0,5}$), (6$_{2,4}$--5$_{2,3}$).
The raw data were calibrated using the IDL MIR package\footnote{\url{https://www.cfa.harvard.edu/~cqi/mircook.html}} and the calibrated visibilities were exported to MIRIAD for further processing. The visibilities were separated into continuum and spectral line data before imaging. Given that the frequency setups between the 2009 and 2012 observations differed by about 40.5~GHz and that the 2009 observations had a better $(u,v)$ coverage, we made the 1.2 mm continuum map with the 2009 data and obtained a synthesized beam with a full-width-half-maximum (FWHM) size of $3.\!''6{\times}3.\!''0$ and a position angle (PA) of $9^{\circ}$. The root mean square (RMS) noise level of the continuum map is 3.0~mJy\,beam$^{-1}$. Maps of the CS~(5--4) and H$_2$CS (7$_{1,6}$--6$_{1,6}$) lines, covered by the 2009 observations, have synthesized beams approximately the same as that of the 1.2~mm continuum map. The RMS noise level of the CS~(5--4) map is about 40~mJy\,beam$^{-1}$ per 0.5~km\,s$^{-1}$, while the H$_2$SC line is close to an atmospherical absorption feature and thus has a higher RMS noise level of 60~mJy\,beam$^{-1}$ per 0.5~km\,s$^{-1}$. Maps of the CS, C$^{33}$S, and C$^{34}$S (4--3), SO~(5$_4$--4$_3$), and H$_2$CS (6$_{0,6}$--5$_{0,5}$), (6$_{2,4}$--5$_{2,3}$) were made from the 2012 data. The CS, C$^{33}$S, C$^{34}$S, and SO maps have synthesized beams with a FWHM size about $4.\!''0{\times}2.\!''7$ and a PA of $-50^{\circ}$. The RMS noise levels are about 60~mJy\,beam$^{-1}$ per 1.2~km\,s$^{-1}$ for the CS, C$^{33}$S, and SO maps, and 120~mJy\,beam$^{-1}$ per 0.3~km\,s$^{-1}$ for the C$^{34}$S map. To optimize the signal-to-noise (S/N) ratios, a Gaussian taper of $4''\times4''$ were applied to the H$_2$CS lines during the imaging process, resulting in a synthesized beam with a FWHM size of $5.\!''0{\times}4.\!''5$ and a PA of $-71^{\circ}$, and an RMS noise level of about 50~mJy\,beam$^{-1}$ per 1.2~km\,s$^{-1}$.
The \emph{Spitzer} IRAC data were obtained from the Spitzer archive (PID: 8334669). We adopted Post Basic Calibrated Data provided by the Spitzer Science Center.
\section{Results} \label{sec:results}
\subsection{Dust Continuum Emission} \label{subsec:cont}
Figure~\ref{fig:spitzer}(a) shows a \emph{Spitzer} IRAC color-composite image of a $\sim4'\times4'$ region covering the Kleinmann-Low (KL) nebula, Trapezium cluster, and Orion Bar from northwest to southeast. The Bar stands out by virtue of its bright 5.8~$\mu$m emission, which is most likely FUV excited polycyclic aromatic hydrocarbon (PAH) emission and delineates an atomic layer \citep{Goicoechea16}. In Figure ~\ref{fig:spitzer}(b), the SMA 1.2~mm continuum observations reveal dust emission immediately behind the PAH ridge of the Bar, providing unambiguous evidence for the existence of dense and clumpy molecular gas shielded from FUV emission.
We identify the dust condensations and measure their fluxes, positions, and sizes from the dust continuum map with a two-step process. First we decompose the continuum map into individual sources with a ``dendrogram'' analysis \citep[e.g.,][]{Goodman09} and a 2D CLUMPFIND algorithm \citep{Williams94}; we find that the results from the two analyses are in general consistent with each other. Second, we perform a multi-component 2D Gaussian fitting on the continuum map using the MIRIAD task IMFIT, with the sources identified by both the dendrogram and CLUMPFIND analyses as the initial estimates. Figure~\ref{fig:cont} shows a comparison between the observed continuum map, the Gaussian fitting results, and the residual map obtained by subtracting the fitted Gaussian components from the observed map. We measure an RMS noise level of 3.2~mJy\,beam$^{-1}$ from the residual map, which is fairly close to the RMS noise level of the observed map. We identify a total of 9 condensations with peak fluxes at least 7 times the RMS noise level of the observations. These condensations are denoted as SMA1 to SMA9 in a decreasing order of the peak flux in Figure~\ref{fig:cont}(a). SMA2 is a foreground pre-main sequence star with a protoplanetary disk \citep{Mann10}, and will be excluded from the following analyses.
With the measured flux of each condensation and assuming that the dust emission at 1.2~mm is optically thin, we estimate the dust mass, $M_{\rm dust}$, according to $$M_{\rm dust} = \frac{F_{\nu}D^2}{B_{\nu}(T_{\rm dust})\kappa_{\nu}},$$
where $F_\nu$ is the dust emission flux, $D$ is the distance (417 pc), $B_{v}(T_{\rm dust}$) is the Planck function at dust temperature $T_{\rm dust}$, which is adopted to be 73 K (Section \ref{subsec:lines}) by assuming thermal equilibrium between gas and dust and that all the condensations have the same temperature, and $\kappa_{\nu}$ is the dust opacity adopted to be 1.0~cm$^2$ g$^{-1}$ following \citet{Ossenkopf94} for dust grains with ice mantles in regions of gas densities of order $10^6$ -- $10^8$ cm$^{-3}$. The dust mass is converted to gas mass, $M_{\rm g}$, with a gas-to-dust mass ratio of 100, and the volume density of H$_2$ is computed with the gas mass and the measured size. We list the measured and computed parameters of each condensation in Table~\ref{tab:prop}.
\subsection{Molecular Spectral Line Emissions} \label{subsec:lines}
With the two frequency setups we detect spectral line emissions in CS~(5--4), (4--3), C$^{34}$S~(4--3), C$^{33}$S~(4--3), H$_2$CS~(7$_{1,6}$--6$_{1,6}$), (6$_{0,6}$--5$_{0,5}$), (6$_{2,4}$--5$_{2,3}$), and SO~(5$_4$--4$_3$). Among these lines, maps of CS~(5--4), (4--3), H$_2$CS~(7$_{1,6}$--6$_{1,6}$), and SO~(5$_4$--4$_3$) reveal the distribution of dense molecular gas and complement the dust continuum map. Optically thinner lines of C$^{34}$S and C$^{33}$S~(4--3) are used to estimate the velocity dispersion of the compact structures. The H$_2$CS~(6$_{0,6}$--5$_{0,5}$) and (6$_{2,4}$--5$_{2,3}$) lines help to constrain the dense gas temperature.
\subsubsection{CS~(5--4), H$_2$CS~(7$_{1,7}$--6$_{1,6}$), and SO~(5$_4$--4$_3$) emissions} \label{subsubsec:distribution}
Figure~\ref{fig:linesm0} shows velocity integrated emissions in CS~(5--4), H$_2$CS~(7$_{1,6}$--6$_{1,6}$), and SO~(5$_4$--4$_3$), and Figures~\ref{fig:cschan}--\ref{fig:sochan} show velocity channel maps of the lines. The CS~(5--4) map in Figure~\ref{fig:linesm0}(a) reveals dense molecular gas around dust condensations SMA1, SMA3, SMA5, and SMA7 in the NE field and SMA6 and SMA8 in the SW field. The CS~(5--4) emission also probes additional structures not detected or very faint in the dust continuum map; such structures are further away from the dissociation front of the Bar, and are seen as a clump to the southeast of SMA9 in Figure~\ref{fig:linesm0}(a) and as clumpy and elongated structures in velocity channels of 8.0--9.0~km\,s$^{-1}$ in Figure \ref{fig:cschan}. The CS~(4--3) map shows gas structures similar to those seen in the CS~(5--4) map, but has a lower sensitivity and image quality, and thus is not shown here. In Figure \ref{fig:linesm0}(b), prominent H$_2$CS~(7$_{1,6}$--6$_{1,6}$) emission is only detected in the NE field, and traces dense gas around SMA1, SMA3, SMA5, and SMA7. A velocity gradient in an orientation perpendicular to the axis of the Bar is seen in the velocity channel map of this line (Figure~\ref{fig:h2cschan}). In general, the brightest emissions in CS~(5--4) and H$_2$CS~(7$_{1,6}$--6$_{1,6}$) are roughly consistent with the dust continuum emission. However, both Figure~\ref{fig:linesm0}(c) and Figure~\ref{fig:sochan} show that the SO~(5$_4$--4$_3$) emission traces dense gas structures very different from those seen in the CS and H$_2$CS emissions. Except a compact structure associated with SMA1 and faint emission associated with SMA8, the SO emission reveals structures not seen in the dust continuum; the emission is dominated by clumpy and elongated structures lying behind the dust condensations in the SW field.
\subsubsection{H$_2$CS~(6$_{0,6}$--5$_{0,5}$) and (6$_{2,4}$--5$_{2,3}$) emissions} \label{subsubsec:temp}
H$_2$CS is a slightly asymmetric rotor, a heavier analogue to H$_2$CO \citep{Chandra10}. So one can measure the gas temperature by comparing the intensities of two $K$-components from the same ${\Delta}J=1$ transition of the same symmetry species \citep{Mangum93}. With this motivation we observed H$_2$CS~(6$_{0,6}$--5$_{0,5}$) and (6$_{2,4}$--5$_{2,3}$) lines. We detect H$_2$CS~(6$_{0,6}$--5$_{0,5}$) or (6$_{2,4}$--5$_{2,3}$) emission only toward the NE field. Figure \ref{fig:h2cs} shows velocity integrated emissions of the two lines. Weak emissions are seen around SMA1 and SMA3: the 6$_{0,6}$--5$_{0,5}$ map shows a peak S/N ratio of 6 and the 6$_{2,4}$--5$_{2,3}$ map has a peak S/N ratio of 4. H$_2$CS (6$_{0,6}$--5$_{0,5}$) and (6$_{2,4}$--5$_{2,3}$) have upper energy levels above the ground of 35~K and 87~K, respectively, thus their flux ratio is sensitive to gas temperatures from a few 10 to more than 100~K, suitable for a temperature diagnostic of dense molecular gas in the Bar. We estimate the gas kinematic temperature from the flux ratio toward the peak position of the 6$_{0,6}$--5$_{0,5}$ emission, assuming local thermodynamical equilibrium (LTE) and optically thin emissions for both lines, and obtain a temperature of $73\pm58$~K. This represents an average temperature for gas around SMA1 and SMA3. The large uncertainty of the temperature determination is due to relatively low S/N ratios of the detections.
\subsubsection{C$^{34}$S and C$^{33}$S~(4--3) emissions} \label{subsubsec:dispersion}
We observed C$^{34}$S~(4--3) with a higher spectral resolution (0.3~km\,s$^{-1}$) to measure the line width toward the dust condensations. The simultaneously observed C$^{33}$S~(4--3) can be used to constrain the optical depth of the C$^{34}$S emission and improve the accuracy of the line width measurement. Emissions around dust condensations in these two lines are only detected in the NE field. Figure \ref{fig:csm0} shows velocity integrated emissions. Both lines are detected and reveal clumpy structures around SMA1, SMA3, and SMA7. We measure the line widths by performing Gaussian fittings to the C$^{34}$S spectra. Figure~\ref{fig:csfit} shows the C$^{34}$S spectra extracted from the positions of SMA1, SMA3, and SMA7, overlaid with Gaussian fittings. A spectral line could be broadened by the optical depth. Assuming that C$^{34}$S and C$^{33}$S (4--3) have the same excitation temperature, the line ratio is a function of the optical depth and is given by
$$\frac{F_\nu({\rm C^{34}S})}{F_\nu({\rm C^{33}S})}=\frac{1-e^{-\tau_{\rm C^{34}S}}}{1-e^{-\tau_{\rm C^{34}S}/\chi}},$$
where $F_\nu$ is the peak flux, $\tau_{\rm C^{34}S}$ is the C$^{34}$S optical depth at the line center, and $\chi$ is the abundance ratio of C$^{34}$S to C$^{33}$S and is about 4.9 according to the measured abundance ratio of $^{34}$S to $^{33}$S in the Orion KL region \citep{Persson07}. After smoothing the C$^{34}$S spectra to a spectral resolution of 1.2~km\,s$^{-1}$, the same as the C$^{33}$S spectra, we derive optical depths of $2.29\pm0.24$, $2.86\pm0.27$, and $0.70\pm0.29$ toward SMA1, SMA3, and SMA7, respectively, for the C$^{34}$S lines. The intrinsic line width, ${\Delta}V_{\rm int}$, is then deduced following \citet{Beltran05}: $$\frac{{\Delta}V_{\rm obs}}{{\Delta}V_{\rm int}} = \frac{1}{\sqrt{ln2}}\sqrt{-ln\left[-\frac{1}{\tau_{\rm C^{34}S}}ln\left(\frac{1+e^{-\tau_{\rm C^{34}S}}}{2}\right)\right]},$$ where ${\Delta}V_{\rm obs}$ is the line width derived from Gaussian fittings (see Table \ref{tab:prop}). Consequently, the non-thermal velocity dispersion, $\sigma_{\rm nth}$, can be calculated from $\sigma_{\rm nth}=\sqrt{{\Delta}V_{\rm int}^2/8ln2-\sigma_{\rm th}^2}$, where $\sigma_{\rm th}=0.12$~km\,s$^{-1}$ is the thermal velocity dispersion at 73~K. We find that $\sigma_{\rm nth}$ ranges from 0.24 to 0.50~km\,s$^{-1}$ (Table \ref{tab:prop}), and is smaller than or at most comparable with the speed of sound, $c_{\rm s}$, which is 0.51~km\,s$^{-1}$ at 73~K.
\section{Discussion} \label{sec:discuss}
\subsection{Dense molecular gas revealed by the dust continuum and molecular spectral line emissions} \label{subsec:densegas}
We detect clumpy and dense gas structures in both dust continuum and molecular spectral line observations of the Orion Bar. By comparing with the IRAC image, we find that the dust condensations lying from immediately behind the dissociation front to an extension of $\sim15''$ (0.03~pc) toward the interior of the molecular cloud behind the Bar. Among the detected spectral lines, C$^{34}$S, C$^{33}$S~(4--3) and H$_2$CS~(6$_{0,6}$--5$_{0,5}$), (6$_{2,4}$--5$_{2,3}$) show unresolved and faint emissions toward the brightest dust condensations. The other spectral lines reveal a larger extent of dense gas and could complement the dust continuum map. Most CS~(5--4) clumps and all the H$_2$CS~(7$_{1,7}$--6$_{1,6}$) clumps appear to be associated with the dust emission. The CS emission also reveals gas structures further away from the dissociation front. We then compare previous PdBI H$^{13}$CN~(1--0) observations with our spectral line maps for a more comprehensive overview of the distribution of dense molecular gas. In Figure~\ref{fig:linesm0}(a), all the H$^{13}$CN clumps seem to be associated with the CS emission. Such an association is much clearer in Figure~\ref{fig:cschan}, where a CS counterpart of every H$^{13}$CN clump can be identified in some certain velocity channels. The SO~(5$_4$--4$_3$) map, however, has a distinctive behavior; the emission in this line is dominated by structures devoid of dust emission and undetected in any other line.
Therefore, while the dust emission reveals a majority of the dense gas structures within a distance of $\sim$0.03~pc from the dissociation front, some molecular spectral lines such as CS, H$^{13}$CN, and SO lines could trace additional structures deeper into the cold molecular cloud. In addition to the variation of physical conditions as a function of distance from the dissociation front, chemistry should also play a role in producing the difference in the distribution of various spectral line emissions. We refrain from discussing chemical stratification in the distribution of molecular gas in the Bar, since it is out of the scope of this work. Here we focus on the dust condensations, which are characteristic of dense gas structures immediately behind the dissociation front, and their masses could be estimated based on the dust emission.
\subsection{On the nature of the detected dust condensations} \label{subsec:diffmass}
We identify a total of 9 condensations from the 1.2~mm continuum map. Except one condensation (SMA2) associated with a protoplanetary disk and regarded as a foreground source, all the other condensations are dense gas structures in the Bar. The masses of the condensations are estimated to be $\sim$0.03 -- 0.3 $M_{\odot}$, a few times lower than the virial masses of the H$^{13}$CN clumps \citep{Lis03} and one to two orders of magnitude greater than that of the HCO$^+$ substructures \citep{Goicoechea16}. Are these condensations gravitationally bound? Would they collapse to form stars? How were they formed?
To address these questions, we first calculate the Jeans mass, $M_{\rm J}$, of the condensations following $$M_{\rm J} = \frac{4\pi}{3}(\frac{\lambda_{\rm J}}{2})^3\rho,$$ where $\lambda_{\rm J} = \sqrt{{\pi}c_{\rm s}^2/G\rho}$ is the Jeans length, and $\rho$ is the mass density. By comparing $M_{\rm J}$ with $M_{\rm g}$ listed in Table \ref{tab:prop}, it is immediately clear that all the detected condensations have masses more than an order of magnitude lower than $M_{\rm J}$. For three condensations, SMA1, SMA3, and SMA7, with the measured line widths we can calculate the virial mass, $M_{\rm vir}$, following $$M_{\rm vir} = 1165(\frac{R}{\rm 1~pc})(\frac{\sigma_{\rm tot}}{\rm 1~km~s^{-1}})^{2},$$
where $R$ is taken as half of the geometric mean of the deconvolved major and minor axes of the condensation (Table \ref{tab:prop}), and $\sigma_{\rm tot} = \sqrt{\sigma_{\rm nth}^2 + c_{\rm s}^2}$ is the total velocity dispersion. From Table \ref{tab:prop}, the calculated virial mass is close to the Jean mass, and significantly greater than the gas mass of the condensations. The condensations in the Bar are believed to be surrounded by a less dense medium and the external pressure may play a role in the dynamical evolution of the condensations. We then estimate the Bonnor-Ebert mass, $M_{\rm BE}$, given by $1.182c_{\rm s}^3/\sqrt{G^3P_0}$, where $P_0$ is the external pressure \citep{McKee07}. The density of the surrounding medium is suggested to be $10^4$ -- $10^5$~cm$^{-3}$ \citep{Hogerheijde95,Young00,Goicoechea16}, resulting in $M_{\rm BE}\sim22$ -- 7.1~$M_{\odot}$, which is again orders of magnitude greater than $M_{\rm g}$. All these calculations indicate that the detected dust condensations have masses significantly lower than that is required for making them gravitationally bound or for self-gravity to play a crucial role in their dynamical evolution. Thus it seems unlikely that the dust condensations could collapse to form stars in the future.
The dust condensations could not arise from a thermal Jeans fragmentation process. If that is the case, with a density of $10^4$ -- $10^5$~cm$^{-3}$ for the surrounding medium, one may expect the mass of the condensations on the order of 20--50~$M_{\odot}$ (the Jeans mass at $10^5$ -- $10^4$~cm$^{-3}$) and the nearest separations between the condensations around 0.2--0.5~pc (the Jeans length at $10^5$ -- $10^4$~cm$^{-3}$); both are clearly inconsistent with the observations. Alternatively, small dense structures can be temporary density fluctuations frequently created and destroyed by supersonic turbulence \citep[e.g.,][]{Elmegreen99,Biskamp03,Falgarone03}. However, $\sigma_{\rm nth}$ is found to be subsonic or at most transonic. \citet{Goicoechea16} also found that there is only a gentle level of turbulence in the Bar. So the turbulence dose not seem to be strong enough to produce the condensations. Another force that could potentially compress the cloud and produce high density structures is a high pressure wave from the expansion of the H{\scriptsize II} region. \citet{Goicoechea16} detected a fragmented ridge of high density substructures at the molecular cloud surface and three periodic emission maxima in HCO$^+$ (4--3) from the cloud edge to the interior of the Bar, providing evidence that a high pressure wave has compressed the cloud surface and moved into the cloud to a distance of $\sim$15$''$ from the dissociation front. The dust condensations are also located within a distance of $15''$ from the dissociation front, and thus are very likely over-dense structures created as the compressive wave passed by. The complex clumpy appearance of the condensations is probably related to the front instability of the compressive wave \citep{Goicoechea16}, or an instability developed across different layers of the Bar \citep[e.g., the thin-shell instability,][]{Garcia96}. The velocity structure of dense gas around the dust condensations may provide insights into the feasibly of this scenario. Figure~\ref{fig:h2csm1} shows the intensity-weighted velocity map of the H$_2$CS~(7$_{1,7}$--6$_{1,6}$) emission. A velocity gradient in a northwest-southeast direction (i.e., along the direction of the propagation of the compressive wave) is seen in the map, and is consistent with the scenario that the gas is being compressed by a high pressure wave.
\subsection{Uncertainty of the mass estimate} \label{subsec:uncert}
We estimate the masses of the condensations based on the dust emission. The uncertainty of this estimate mainly depends on the accuracy in determining the dust opacity and temperature. We adopted a dust opacity of 1 cm$^2$ g$^{-1}$ (at 249~GHz), which is equivalent to adopting a dust opacity index, $\beta$, of 1.5 according to $\kappa_{\nu}=10(\nu/1.2{\rm THz})^{\beta}$ \citep{Hildebrand83}. It agrees well with the value measured by \citet{Arab12}, who obtained $\beta=1.62\pm0.13$ toward the dust emission peak inside the Bar based on Herschel 70 to 500 $\mu$m observations. This renders support for the appropriateness of our adopted dust opacity. Hence we expect that the largest uncertainty in the mass estimate is ascribed to the dust temperature determination.
With the newly detected H$_2$CS lines, we measured a gas temperature of $73\pm58$ K. There have been a number of molecular spectral line observations providing constraints on the temperature of dense gas in the Bar \citep[e.g.,][]{Hogerheijde95,Batrla03,Goicoechea11,Goicoechea16,Nagy17}. These studies suggest a gas temperature of $\gtrsim$100~K; it appears that our derived value of 73~K lies at the lower end of the range. However, the dust temperature could be lower than the gas temperature. \citet{Goicoechea16} illustrate that for the gas behind the dissociation front in a PDR such as the Orion Bar, the gas temperature decreases from 500 K to 100 K while the dust temperature varies from 75 to 50 K. Indeed, \citet{Arab12} measured a dust temperature of 49~K toward the dust emission peak inside the Bar based on Herschel observations. Considering that the dust condensations detected in our SMA map are located close to the dissociation front, a value of 50~K may represent a lower limit for the dust temperature of the condensations. For $T_{\rm dust}\sim50$~K, $M_{\rm g}$ in Table \ref{tab:prop} would be increased by about 50\% and $M_{\rm J}$ would be decreased by about 50\%. But $M_{\rm J}$ and $M_{\rm vir}$ would be still clearly greater than $M_{\rm g}$, indicating that the condensations are anyway gravitationally unbound. In the less likely case that $T_{\rm dust}>73$~K, $M_{\rm J}$ becomes greater and $M_{\rm g}$ gets lower, and all the discussions remain unchanged.
\section{Summary} \label{sec:sum}
We have performed SMA dust continuum and molecular spectral line observations toward the Orion Bar. The 1.2~mm continuum map reveals, for the first time, dust condensations in the Bar at arc~second resolutions. These condensations are distributed immediately behind the dissociation front. They have an average temperature of about 73~K, and exhibit subsonic to transonic turbulence. The masses of the condensations span a range of $\sim$0.03--0.3~$M_{\odot}$, which are all too low to enable self-gravity to play a crucial role. Thus the condensations do not seem to be able to collapse and form stars. The formation of these condensations is mostly likely induced by a compressive wave originating from the expansion of the H{\scriptsize II} region.
\acknowledgments K.Q. acknowledges the support from National Natural Science Foundation of China (NSFC) through grants 11473011 and 11590781. Q. Z. acknowledges the support from NSFC through grant 11629302.
|
{
"timestamp": "2018-03-08T02:11:43",
"yymm": "1803",
"arxiv_id": "1803.02785",
"language": "en",
"url": "https://arxiv.org/abs/1803.02785"
}
|
\section{Introduction}
In \cite{Simon} Simon proved a decomposition of a non-negative form defined on a dense subspace of a Hilbert space into the sum of two non-negative forms such that one is the greatest non-negative form which is smaller than the form and {\it closable}. The second form is referred as the {\it singular} part.
However, the definition of singular non-negative form, in terms of sequences, goes back to Koshmanenko \cite{Kos_sing} (see also his book \cite{Kos} dedicated to singular forms).
Simon, again in \cite{Simon}, stated the correspondent decomposition of a non-negative form $\t$ into the sum of a closable (or, with another terminology, {\it absolutely continuous}) form $\t_a$ and a singular form $\t_s$ with respect to a second non-negative form $\w$ (see also \cite{Kos}). In this setting, $\t$ and $\w$ are defined on a common complex vector space.
The study of this last so-called {\it Lebesgue decomposition} was continued by Hassi, Sebestyén, De Snoo in \cite{HSdeS}.
Their framework involves the notion of parallel sum of forms, which is inspired by the one for non-negative operators used by Ando \cite{Ando}. A proof with a different approach was developed by Sebestyén, Tarcsay and Titkos \cite{STT}.
The Lebesgue decomposition of non-negative forms, as the name suggests, is inspired to the classical Lebesgue decomposition of non-negative measures (or, in more generality, additive set functions). Moreover, these notions are related. Indeed, a non-negative measure induces a non-negative form and the absolutely continuous parts are in correspondence, as well as the singular parts (see \cite[Theorem 5.5]{HSdeS} and also \cite[Theorems 3.2 and 3.4]{STT}).
Recently, Di Bella and Trapani \cite{Tp_DB} have given a notion of regularity and singularity for a (non-necessarily non-negative) sesquilinear form with respect to a non-negative one and then they proved a correspondent Lebesgue decomposition theorem.
More precisely, let $\w,\t$ be forms on $\D$, $\w$ being non-negative. We denote by $M(\t)$ the set of non-negative sesquilinear forms $\s$ satisfying the inequality $|\t(\xi,\eta)|\leq \s[\xi]^\mez \s[\eta]^\mez$ for all $\xi,\eta \in \D$. Then a sesquilinear form $\t$ is $\w$-{\it regular} if there exists $\s\in M(\t)$ such that $\s$ is $\w$-absolutely continuous. On the other hand, $\t$ is {\it $\w$-singular} if for every $\phi\in \D$ there exists a sequence $(\phi_n)\subset\D$ with
\begin{equation*}
\label{def:sing}
\lim_{n\to +\infty} \w[\phi_n]=0 \;\;\text{ and }\; \lim_{n\to +\infty} \t[\phi-\phi_n] =0.
\end{equation*}
Furthermore, Theorem 4.3 of \cite{Tp_DB} states that if $M(\t)\neq \varnothing$, then $\t=\t_r+\t_s$ where $\t_r$ is $\w$-regular and $\t_s$ is $\w$-singular.
In this paper Di Bella and Trapani's theorem is reconsidered. First of all, in analogy to the notion of $\w$-regularity, one can give a notion of singularity of a form $\t$ (coherent to the classical one in the non-negative case) as follows
\begin{equation}
\tag{ss}\label{def_intr_ss}
\text{$\exists\s\in M(\t)$ such that $\s$ is $\w$-singular.}
\end{equation}
This idea is supported by the following fact from the Theory of Measure.
If $\mu,\nu$ are (complex) measure on the same $\sigma$-algebra and $\nu$ is non-negative, then $\mu$ is $\nu$-absolutely continuous (resp. $\nu$-singular) if and only if it is dominated by an $\nu$-absolutely continuous (resp. $\nu$-singular) non-negative measure. \\
Nevertheless, condition (\ref{def_intr_ss}) does not always hold for the singular part of a form in \cite{Tp_DB} (see Remark \ref{cexm}), but actually it is a stronger notion. For this reason, we give to a form $\t$ satisfying (\ref{def_intr_ss}) the name of $\w$-{\it strongly singular form}. \\
However, it turns out (Theorem \ref{Leb_th}) that every sesquilinear form $\t$ such that $M(\t)\neq \varnothing$ can be decomposed as $\t=\t_{r}+\t_m+\t_{ss}$, where $\t_{r}$ is the $\w$-regular part, $\t_{ss}$ the $\w$-strongly singular part and $\t_m$ is a form (called $\w$-{\it mixed}) which is dominated by the product of a non-negative $\w$-absolutely continuous form and a non-negative $\w$-singular form. This is the version of the Lebesgue decomposition that we states in the present article.
The organization of this paper is as follows. In Sections \ref{sec_pre} we establish some properties and characterizations of the forms considered above, as well as some examples. Under simple conditions on the values of a $\t$ (Proposition \ref{pro_N_t}) one can see cases where a $\w$-mixed form is identically zero (for example assuming the condition of non-negativity) or that the notions of $\w$-singularity and $\w$-strongly singularity are equivalent. Section \ref{sec_Leb} contains the Lebesgue decomposition of forms as stated above, and shows also that it is not the same if one chooses a different non-negative dominant form $\s\in M(\t)$. Finally, relations between measures and forms are investigated in Section \ref{sec_meas}. In particular, the Lebesgue decomposition of a complex measure with respect to a non-negative one is proved through sesquilinear forms.
\section{Preliminaries}
\label{sec_pre}
To make the topic on sesquilinear forms as self-contained as possible we begin recalling basic notions and properties.
A {\it sesquilinear form} $\t$ on a complex vector space $\D$ (called the {\it domain} of $\t$) is a map $\D\times \D\to \C$ which is linear in the first component and anti-linear in the second one. The map $\D\to \C$ defined by $\phi \mapsto \t[\phi]:=\t(\phi,\phi)$ is the {\it quadratic form} associated to $\t$. The {\it polarization identity}
$$
\t(\phi,\psi)=\frac{1}{4} \sum_{k=0}^{3} i^k\t[\phi+i^k\psi], \qquad \forall \phi,\psi \in \D
$$
connects quadratic and sesquilinear forms. The {\it scalar multiple} $\alpha \t$, with $\alpha \in \C$, is defined as
\begin{align*}
(\alpha \t)(\phi,\psi)&:=\alpha\t(\phi,\psi), \qquad \phi,\psi \in \D.
\end{align*}
Given two sesquilinear forms $\t_1,\t_2$ on $\D_1$ and $\D_2$, respectively, the {\it sum} $\t_1+\t_2$ is the sesquilinear form
\begin{align*}
(\t_1+\t_2)(\phi,\psi)&:=\t_1(\phi,\psi)+\t_2(\phi,\psi),\qquad \phi,\psi \in \D_1\cap \D_2.
\end{align*}
\no Classic forms associated to a sesquilinear form $\t$ on $\D$ are:
\begin{itemize}
\item the {\it adjoint} $\t^*$ of $\t$, defined as
$$
\t^*(\phi,\psi)=\ol{\t(\psi,\phi)}, \qquad \phi, \psi\in \D;
$$
\item the {\it real part} $\Re \t$ of $\t$, defined as $\Re \t := \mez(\t+\t^*)$;
\item the {\it imaginary part} $\Im \t$ of $\t$, defined as $\Im \t := \frac{1}{2i}(\t-\t^*)$.
\end{itemize}
A sesquilinear form $\t$ on $\D$ is called {\it symmetric} if $\t=\t^*$ and, in particular, {\it non-negative} (in symbol $\t\geq 0$) if $\t[\phi]\geq 0$ for all $\phi \in \D$. In this latter case the {\it Cauchy-Schwarz} and {\it triangle inequalities} hold; i.e.,
\begin{align*}
|\t(\phi,\psi)|&\leq \t[\phi]^\mez\t[\psi]^\mez,\\
\t[\phi+\psi]^\mez &\leq \t[\phi]^\mez + \t[\psi]^\mez, \qquad \forall \phi,\psi\in \D.
\end{align*}
If $\s_1$ and $\s_2$ are non-negative sesquilinear forms on $\D$, we write $\s_1\leq\s_2$ when $\s_1[\phi]\leq \s_2[\phi]$ for all $\phi \in \D$.
If $\D$ is a subspace of a Hilbert space $\H$ with inner product $\pint$ and corresponding norm $\nor$, a sesquilinear form $\t$ on $\D$ satisfying for some $C\geq 0$,
$|\t(\phi,\psi)|\leq C \n{\phi}\n{\psi}$ for all $\phi,\psi \in \D,$
is called ({\it normed}) {\it bounded} on $\H$. For this form there exists a bounded operator $T$ on $\H$ such that $\t(\phi,\psi)=\pin{T\phi}{\psi}$, for all $\phi,\psi \in \D$. Moreover, if $\D$ is dense in $\H$, then $T$ is unique (with norm not greater than $C$) and $\t$ can be extended to a bounded form defined on the whole of $\H$, called the {\it closure} of $\t$. \\
For reader's convenience we also summarize the definitions presented and motivated in the Introduction, part of which are taken from \cite{Tp_DB}.
Let $\D$ be a complex vector space and $\t,\w$ sesquilinear forms on $\D$. Throughout the paper $\w$ will be non-negative.\\
We write $M(\t)$ for the set of non-negative sesquilinear forms $\s$ on $\D$ satisfying
\begin{equation*}
\label{def_M_t}
|\t(\phi,\psi)|\leq \s[\phi]^\mez\s[\psi]^\mez, \qquad \forall \phi,\psi\in \D.
\end{equation*}
\no The set $M(\t)$ is not empty if and only if there exists a form $\s\geq 0$ on $\D$ such that $|\t[\phi]|\leq \s[\phi]$ for all $\phi \in \D$ (it follows by an argument like in the proof of Lemma 11.1 in \cite{Schm}).
\no The following definitions will also be needed in the sequel:
\begin{itemize}
\item if $\t$ is non-negative, $\t$ is {\it $\w$-absolutely continuous} (in symbols $\t\ll\w$) if for every sequence $(\phi_n)\subset \D$ such that $\w[\phi_n]\to 0$ and $\t[\phi_n-\phi_m]\to 0$ one has $\t[\phi_n]\to 0$;
\item $\t$ is {\it $\w$-singular} (in symbols $\t\perp\w$)
if for every $\phi\in \D$ there exists a sequence $(\phi_n)\subset\D$ verifying
\begin{equation*}
\lim_{n\to +\infty} \w[\phi_n]=0 \;\;\text{ and }\; \lim_{n\to +\infty} \t[\phi-\phi_n] =0,
\end{equation*}
or, equivalently, if for every $\psi\in \D$ there exists a sequence $(\psi_n)\subset\D$ verifying
\begin{equation*}
\lim_{n\to +\infty} \w[\psi-\psi_n]=0 \;\;\text{ and }\; \lim_{n\to +\infty} \t[\psi_n] =0,
\end{equation*}
(if $\t$ is non-negative, then it is $\w$-singular if and only if for every non-negative form $\mathfrak{p}$ with $\mathfrak{p} \leq \w$ and $\mathfrak{p} \leq \t$ one has $\mathfrak{p}=0$);
\item $\t$ is {\it $\w$-regular} if there exists $\s\in M(\t)$ such that $\s\ll \w$;
\item $\t$ is {\it $\w$-strongly singular} if there exists $\s\in M(\t)$ such that $\s\perp \w$.\\
\end{itemize}
The fundamental result in the theory of absolutely continuous and singular forms is the following decomposition (for the proof see \cite[Theorem 2.11]{HSdeS}, \cite[Theorem 2.3]{STT} or \cite[Corollary 4.5]{Tp_DB}).
\begin{theo}[Lebesgue decomposition of non-negative forms]
\label{Leb_pos}
Let $\s,\w$ be non-negative sesquilinear forms on $\D$. Then
$$
\s=\s_a+\s_s,
$$
where $\s_a$ and $\s_s$ are non-negative, $\w$-absolutely continuous and $\w$-singular forms, respectively.
Moreover, if $0\leq \mathfrak{u} \leq \s$ and $\mathfrak{u}$ is $\w$-absolutely continuous, then $\mathfrak{u}\leq \s_a$.
\end{theo}
\begin{rem}
\label{rem_stong->sing}
\begin{enumerate}
\item[(i)] A simple class of $\w$-regular forms is the class of {\it $\w$-bounded} forms $\t$, verifying for some $C\geq 0$ the inequality $|\t(\phi,\psi)|\leq C \w[\phi]^\mez\w[\psi]^\mez$, for all $\phi,\psi\in \D$; i.e, $C\w\in M(\t)$.
\item[(ii)] A non-negative $\w$-absolutely continuous form is $\w$-regular.
\item[(iii)] A $\w$-strongly singular form $\t$ is $\w$-singular. Moreover, the converse holds if $\t$ is non-negative.
\end{enumerate}
\end{rem}
The $\w$-regularity in the non-negative case is weaker than the $\w$-absolute continuity as the next two examples show, in contrast with what stated in \cite[Proposition 4.8]{Tp_DB}.
\begin{exm}
Let $\H$ be a Hilbert space with inner product $\pint$ and let $H$ be an unbounded positive self-adjoint operator with domain $D(H)$. Take $\kappa\notin D(H)$ and consider the projector $P\xi=\pin{\xi}{\kappa}\kappa$, $\xi\in \H$.
We indicate by $\w$, $\t$ and $\s$ the non-negative sesquilinear forms
$$
\w(\phi,\psi)=\pin{\phi}{\psi}, \qquad \t(\phi,\psi)=\pin{PH\phi}{H\psi}, \qquad \s(\phi,\psi)=\pin{H\phi}{H\psi},
$$
for $\phi,\psi\in D(H)$, respectively. We have that $\s\ll \w$ and $\s\in M(\t)$, then $\t$ is $\w$-regular. Nevertheless, $\t$ is not $\w$-absolutely continuous. Indeed, were it so, then from
$$
\t[\phi]=\n{PH\phi}^2, \qquad \forall \phi \in D(H),
$$
$PH$ would be a closable operator in $\H$. But its adjoint $HP$ is not densely defined.
\end{exm}
As known, a non-negative form which is both $\w$-absolutely continuous and $\w$-singular is identically zero. The situation in our context is very different even in the non-negative case.
\begin{exm}
Basing on \cite[Theorem 4.4]{HSdeS}, if $\s$ is a non-negative $\w$-absolutely continuous form but not $\w$-bounded, then there exists a non-negative $\w$-singular form $\t\neq 0$ such that $\t\leq \s$. This shows that there exist non-trivial (non-negative) forms $\t$ which are both $\w$-regular and $\w$-singular ($\w$-strongly singular). However, a particular case is given by the next proposition.
\end{exm}
\begin{pro}
The only sesquilinear form which is $\w$-bounded and $\w$-singular is the null form.
\end{pro}
\begin{proof}
Let $\t$ be a $\w$-bounded and $\w$-singular sesquilinear form on $\D$. For every $\phi\in \D$ there exists a sequence $(\phi_n)\subset\D$ with the property that
\begin{equation*}
\lim_{n\to +\infty} \w[\phi_n]=0 \;\;\text{ and }\; \lim_{n\to +\infty} \t[\phi-\phi_n] =0.
\end{equation*}
Note that, by the triangle inequality, $\{\w[\phi-\phi_n]\}$ is a bounded sequence. Therefore, for some $C\geq0$,
\begin{align*}
|\t[\phi]|&\leq |\t(\phi_n,\phi)|+|\t(\phi-\phi_n,\phi_n)|+|\t[\phi-\phi_n]|\\
&\leq C\w[\phi_n]^\mez\w[\phi]^\mez+C\w[\phi-\phi_n]^\mez\w[\phi_n]^\mez+|\t[\phi-\phi_n]|\to 0;
\end{align*}
i.e., $\t=0$.
\end{proof}
\no Two subsets of $\D$ related to a sesquilinear form $\t$ on $\D$ are
\begin{align*}
K(\t)&=\{\phi\in \D:\t[\phi]=0\},\\
\ker(\t)&=\{\phi\in \D: \t(\phi,\psi)=0, \forall \psi\in \D\}.
\end{align*}
In particular, the second one is a subspace of $\D$. Clearly, $\ker(\t)\subseteq K(\t)$ and the equality holds if $\t$ is non-negative by Cauchy-Schwarz inequality. Note that if $\t$ is not symmetric then $\ker(t)$ and $\ker(\t^*)$ may be different; however, we have also $\ker(t^*)\subseteq K(t)$ and $K(\t)=K(\t^*)$.
There is a classical way to define a Hilbert space associated to a non-negative form $\w$ on $\D$. More precisely, the quotient $\D/\ker(\w)$ can be endowed with the inner product $\pin{\pi_\w(\phi)}{\pi_\w(\psi)}_\w:=\w(\phi,\psi)$, for all $\phi,\psi \in \D$, where $\pi_\w:\D\to \D/\ker(\w)$ is the canonical projection. The completion of $(\D/\ker(\w),\pint_\w)$ is denoted by $\H_\w$.
\begin{rem}
\begin{enumerate}
\item[(i)] If $\t$ is a $\w$-regular form, then $\ker(\w)\subseteq\ker(\t)$.
\item[(ii)] Suppose that $\D$ has finite dimension. A form $\t$ is $\w$-regular if and only if $\t$ is $\w$-bounded if and only if $\ker(\w)\subseteq\ker(\t)$. By Remark \ref{rem_stong->sing} and the previous point, we have to prove only one implication. Namely, if $\ker(\w)\subseteq\ker(\t)$, then the form
\begin{equation*}
\overset{\sim}{\t}(\pi_{\w}(\phi),\pi_{\w}(\psi)):=\t(\phi,\psi), \qquad \pi_{\w}(\phi),\pi_{\w}(\psi)\in \D/\ker(\w),
\end{equation*}
is well-defined and therefore bounded by the norm of $\H_\w$; i.e., $\t$ is $\w$-bounded.
\end{enumerate}
\end{rem}
It is worth mentioning a characterization of non-negative singular forms involving the Hilbert spaces associated to them.
\begin{lem}[{\cite[Theorem 6.1]{Kos}}]
\label{H_s_w_sing}
A non-negative sesquilinear form $\s$ is $\w$-singular if and only if $\H_{\s+\w}$ is isomorphic to the cartesian product of $\H_\s$ and $\H_\w$
($\H_{\s+\w}\simeq\H_\s \times \H_\w$).
\end{lem}
We also recall that Theorem 3.6 of \cite{Tp_DB} gives a characterization of the $\w$-regular forms in terms of a representation in the space $\H_\w$. This expression is studied in another (but affine) context in \cite{Second} when $\w$ is the inner product of a Hilbert space.
\begin{exm}[{\cite[Remark 5.3]{Kos}}]
\label{exm_ker_dens}
Let $\t$ be a sesquilinear form on $\D$. If $\pi_\w(K(\t))$ is dense in $\H_\w$ (in particular, if $\pi_\w(\ker(\t))$ or $\pi_\w(\ker(\t^*))$ is dense in $\H_\w$), then $\t$ is trivially $\w$-singular.
\end{exm}
\begin{rem}
\label{cexm}
One might ask if, in analogy to Theorem \ref{Leb_pos}, a sesquilinear form can be decomposed as a sum of a $\w$-regular form and a $\w$-strongly singular one. Here we prove that this is not allowed.
Indeed, consider $\D=\C^2$ and the sesquilinear forms given by
\begin{align*}
\t(\underline{x},\underline{y})&= x_1\ol{y_1}-x_2\ol{y_2}\\
\w(\underline{x},\underline{y})&= x_1\ol{y_1}+x_1\ol{y_2}+x_2\ol{y_1}+x_2\ol{y_2}
\end{align*}
for all $\underline{x}:=(x_1,x_2),\underline{y}:=(y_1,y_2)\in \C^2$. Assume that
\begin{equation}
\label{counter}
\t=\t_{r}+\t_{ss}
\end{equation}
where $\t_{r}$ is a $\w$-regular form and $\t_{ss}$ is $\w$-strongly singular form. Then there exist two non-negative forms $\s_a$ and $\s_s$ such that $\s_a \ll \w$, $\s_s \perp \w$, $\s_a\in M(\t_r)$ and $\s_s\in M(\t_{ss})$.
Since $\w[\underline{p}]=0$, where $\underline{p}=(1,-1)$, $\s_a[\underline{p}]=0$ and $\t_{r}[\underline{p}]=0$.
One has that $\t_{ss}=0$. In fact, it is clear if $\s_s=0$; on the other hand, if $\s_s\neq 0$
by Lemma \ref{H_s_w_sing} there exists $\underline{q}\in \C^2$ for which $\C^2=\l\underline{p},\underline{q}\r$ and $\s_s[\underline{q}]=0$. This implies that $\t_{ss}(\underline{x},\underline{q})=0$ for all $\underline{x}\in \C^2$. Moreover, $0=\t[\underline{p}]=\t_r[\underline{p}]+\t_{ss}[\underline{p}]=\t_{ss}[\underline{p}]$. Therefore, $\t_{ss}=0$.
Hence, $|\t(\underline{x},\underline{p})|\leq \s_a[\underline{x}]^\mez \s_a[\underline{p}]^\mez=0$ for all $\underline{x}\in \C^2$. But this leads to a contradiction since $\t((1,1),\underline{p})\neq 0$. We conclude that (\ref{counter}) does not hold.
\end{rem}
In Theorem \ref{Leb_th} we will give a decomposition inspired to Theorem \ref{Leb_pos} involving one more type of form which is introduced by the next lemma.
\begin{lem}
\label{lem_def_mixed}
Let $\t$ be a sesquilinear form on $\D$. The following statements are equivalent.
\begin{enumerate}
\item There exist non-negative forms $\af,\bb$ such that $\af \ll \w$, $\bb \perp \w$, $\af \perp \bb$ and
\begin{equation}
\label{def_car_mix}
|\t[\phi]|\leq \af[\phi]^\mez\bb[\phi]^\mez, \qquad\forall \phi\in \D.
\end{equation}
\item There exist non-negative forms $\af,\bb$ such that $\af \ll \w$, $\bb \perp \w$, $\af \perp \bb$, $\af+\bb \in M(\t)$ and $\t[\phi]=0$ if $\af[\phi]=0$ or $\bb[\phi]=0$.
\item There exist non-negative forms $\af,\bb$ such that $\af \ll \w$, $\bb \perp \w$, $\af \perp \bb$ and
$$
|\t(\phi,\psi)|\leq \af[\phi]^\mez \bb[\psi]^\mez+\af[\psi]^\mez \bb[\phi]^\mez, \qquad \forall \phi,\psi\in \D.
$$
\item There exist forms $\t_1,\t_2,\af,\bb$ on $\D$ such that
$\t=\t_1+\t_2$, $\af,\bb$ are non-negative forms, $\af\ll\w$, $\bb\perp \w$, $\af \perp \bb$ and
$$
|\t_1(\phi,\psi)|\leq \af[\phi]^\mez \bb[\psi]^\mez, \qquad |\t_2(\phi,\psi)|\leq \af[\psi]^\mez \bb[\phi]^\mez, \qquad \forall \phi,\psi \in \D.
$$
\end{enumerate}
\end{lem}
\begin{proof}
(i) $\Rightarrow$ (ii) It is immediate.\\
(ii) $\Rightarrow$ (iii) Let us consider the bounded sesquilinear form
\begin{equation}
\label{eq_t_tilde_a_b}
\overset{\sim}{\t}(\pi_{\af+\bb}(\phi),\pi_{\af+\bb}(\psi)):=\t(\phi,\psi), \qquad \pi_{\af+\bb}(\phi),\pi_{\af+\bb}(\psi)\in \D/\ker(\af+\bb),
\end{equation}
and its closure $\overline{\t}$ on $\H_{\af+\bb}$.
With similar meanings, we consider also the forms $\overline{\af}$ and $\overline{\bb}$. By Lemma \ref{H_s_w_sing}, $\H_{\af+\bb}$ can be decomposed as orthogonal sum of two subspaces, $\H_{\af+\bb}=M_1\oplus M_2$, where $\overline{\af}$ is zero on $M_1$ and $\overline{\bb}$ is zero on $M_2$.
Consequently, if $P$ is the orthogonal projector on $M_2$, the forms $\af$ and $\bb$ have the following expressions
$$
\af[\phi]=\n{P\pi_{\af+\bb}(\phi)}_{\af+\bb}^2, \qquad \bb[\phi]=\n{(I-P)\pi_{\af+\bb}(\phi)}_{\af+\bb}^2, \qquad \forall \phi \in \D.
$$
Since $\overline{\af}[P\pi_{\af+\bb}(\phi)]=0$ for all $\phi\in \D$, one has $\overline{\t}[P\pi_{\af+\bb}(\phi)]=0$ for all $\phi\in \D$ and, by the polarization identity, $\overline{\t}(P\pi_{\af+\bb}(\phi),P\pi_{\af+\bb}(\psi))=0$ for all $\phi,\psi\in \D$. In the same way, $\overline{\t}((I-P)\pi_{\af+\bb}(\phi),(I-P)\pi_{\af+\bb}(\psi))=0$ for all $\phi,\psi\in \D$. Hence,
\begin{align*}
|\t(\phi,\psi)| &= |\overline{\t}(P\pi_{\af+\bb}(\phi),(I-P)\pi_{\af+\bb}(\psi))| \\
&+ |\overline{\t}((I-P)\pi_{\af+\bb}(\phi),P\pi_{\af+\bb}(\psi))|\\
&\leq \n{P\pi_{\af+\bb}(\phi)}_{\af+\bb}\n{(I-P)\pi_{\af+\bb}(\psi)}_{\af+\bb}\\
&+\n{P\pi_{\af+\bb}(\psi)}_{\af+\bb}\n{(I-P)\pi_{\af+\bb}(\phi)}_{\af+\bb}\\
&=\af[\phi]^\mez \bb[\psi]^\mez+\af[\psi]^\mez \bb[\phi]^\mez, \qquad\qquad \forall \phi,\psi \in \D.
\end{align*}
(iii) $\Rightarrow$ (iv) Clearly, $2(\af+\bb)\in M(\t)$. Following the proof of the previous part, the sesquilinear forms on $\D$ defined by
\begin{align*}
\t_1(\phi,\psi)&=\overline{\t}(P\pi_{\af+\bb}(\phi),(I-P)\pi_{\af+\bb}(\psi)), \\ \t_2(\phi,\psi)&=\overline{\t}((I-P)\pi_{\af+\bb}(\phi),P\pi_{\af+\bb}(\psi)),
\end{align*}
satisfy the statement, up to rename $2\af$ and $2\bb$ with $\af$ and $\bb$, respectively.\\
(iv) $\Rightarrow$ (i) One obtains (\ref{def_car_mix}) replacing $\af$ with $2\af$ and $\bb$ with $2\bb$, which are still $\w$-absolutely continuous and $\w$-singular, respectively, and singular with respect to each other.
\end{proof}
\begin{defin}
A sesquilinear form is said {\it $\w$-mixed} if it satisfies one of the statements in Lemma \ref{lem_def_mixed}.
\end{defin}
We now conclude this section by giving some examples.
\begin{exm}
It is easy to see, using Lemma \ref{lem_def_mixed}(ii), that the form $\t$ of Remark \ref{cexm} is $\w$-mixed, taking $\af=\w$ and $\bb$ defined by $\bb(\underline{x},\underline{y})= x_1\ol{y_1}-x_1\ol{y_2}-x_2\ol{y_1}+x_2\ol{y_2}$, for all $\underline{x},\underline{y}\in \C^2$. However, $\t$ is also $\w$-singular. Indeed, for $\phi=(x_1,x_2)$ the constant sequence $\phi_n:=\mez (x_1+x_2,x_1+x_2)$ satisfy $\t[\phi_n]=0$ and $\w[\phi-\phi_n]=0$. This fact and Remark \ref{cexm} show that there exist $\w$-singular forms which are not $\w$-strongly singular.
\end{exm}
\begin{exm}
Let $H$ be a self-adjoint operator with domain $D(H)$ on a Hilbert space $(\H,\pint)$. Define two sesquilinear form on $\D:=D(H)\times D(H)$ as
\begin{align*}
\w(\underline{\xi}, \underline{\eta})=\pin{\xi_1}{\eta_1}, \qquad
\t(\underline{\xi}, \underline{\eta})=\pin{H\xi_1}{\eta_2}+\pin{H\xi_2}{\eta_1},
\end{align*}
for $\underline{\xi}=(\xi_1,\xi_2),\underline{\eta}=(\eta_1,\eta_2)\in \D$. It is easy to check that $\t$ satisfies (\ref{def_car_mix}) with
$$
\af(\underline{\xi}, \underline{\eta})=\pin{H\xi_1}{H\eta_1},
\qquad
\bb(\underline{\xi}, \underline{\eta})=\pin{\xi_2}{\eta_2}, \qquad \underline{\xi},\underline{\eta}\in \D.
$$
\end{exm}
\begin{exm}
Let $\D:=C(0,1)$ stand for the vector space of continuous functions on the interval $[0,1]$. It is well-known that the non-negative forms
$$
\w(f,g)=\int_0^1 f(x)\ol{g(x)}dx, \qquad \bb(f,g)=f(0)\ol{g(0)}, \qquad f,g\in \D,
$$
are singular with respect to each other (in particular, $\bb$ is a form of the type of Example \ref{exm_ker_dens}). Consequently, the sesquilinear form
$$
\t(f,g)=f(0)\int_0^1 \ol{g(x)}dx, \qquad f,g\in \D,
$$
is $\w$-mixed.
\end{exm}
\begin{exm}
Let $\H_- \supset \H \supset \H_+$ be a rigged Hilbert space with duality $\pint$ between $\H_-$ and $\H_+$. Given $\omega,\varrho\in \H_-$ we define the sesquilinear form
$$
\t(\xi,\eta)=\pin{\omega}{\xi}\ol{\pin{\varrho}{\eta}}, \qquad \xi,\eta \in \H_+,
$$
and let $\w(\xi,\eta)=\pin{\xi}{\eta}$ for $\xi,\eta \in \H_+$. Taking into account \cite[Examples 1.15, 5.5, 5.9]{Kos}, we can state that
\begin{itemize}
\item if $\omega,\varrho\in \H$, then $\t$ is $\w$-bounded;
\item if $\omega\in\H_-\backslash\H,\varrho\in \H$ or $\varrho\in \H_-\backslash\H,\omega\in \H$, then $\t$ is $\w$-mixed;
\item if $\omega$ or $\rho$ is in $\H_-\backslash\H$, then $\ker(\t)$ is dense in $\H$ and therefore $\t$ is $\w$-singular;
\item if $V\cap \H=\{0\}$, where $V$ is the subspace of $\H_-$ generated by $\omega$ and $\varrho$, then $\t$ is $\w$-strongly singular.
\end{itemize}
\end{exm}
In the rest of this section we analyze the definitions given at the beginning in some special cases. We recall that in our approach a form is not in general non-negative; however forms with a restricted set of values can have a interest (see Proposition \ref{pro_N_t} below). We start with the following relations between a form, its adjoint, the real and the imaginary parts, which are easy to prove.
\begin{pro}
\label{pro_re_im}
Let $\t$ be a sesquilinear form on $\D$.
\begin{enumerate}
\item The sets $M(\t)$ and $M(\t^*)$ are equal. Furthermore,
$$
M(\Re\t)+M(\Im\t)\subseteq M(\t)\subseteq M(\Re\t)\cap M(\Im\t),
$$
where $M(\Re\t)+M(\Im\t):=\{\s_1+\s_2:\s_1\in M(\Re\t),\s_2\in M(\Im\t)\}$.
\item If $\t$ is $\w$-regular ($\w$-singular, $\w$-strongly singular or $\w$-mixed), then the same holds for $\t^*$, $\Re \t$ and $\Im \t$.
\end{enumerate}
\end{pro}
\no We denote by $N(\t)$ the {\it positively homogeneous} subset of $\C$
$$
N(\t) :=\{\t[\phi]:\phi\in \D\}.
$$
Positively homogeneous means that $\alpha N(\t)=N(\t)$ for all $\alpha >0$. By definition, $\t$ is non-negative if and only if $N(\t) = [0,+\infty)$. Moreover, $\t$ is symmetric if and only if $N(\t)\subseteq \R$.
\begin{rem}
If $\D$ is a subspace of a Hilbert space with norm $\nor$, then a more important (convex) set is the so-called {\it numerical range} (see \cite[Chapter VI]{Kato} and \cite{Halmos_nr,Schm} for the operator case) defined by
$$
\mathfrak{N}(\t):=\{\t[\phi]:\phi\in \D, \n{\phi}=1\}.
$$
Clearly, $\mathfrak{N}(\t) \subseteq N(\t)$ and $N(t)$ is contained in one of the following subsets of $\C$
\begin{align*}
[0,+\infty),\qquad\qquad\;\;\; \R,\qquad\qquad\;\; \mathcal{Q}:=\{\lambda \in \C:\Re \lambda \geq 0, \Im \lambda \geq 0\},\\
\Pi:=\{\lambda \in \C:\Re \lambda \geq 0\}, \qquad
\mathcal{S}_c:=\{\lambda \in \C:|\Im \lambda|\leq c\Re \lambda\} \;\;(c\geq 0),
\end{align*}
if and only if $\mathfrak{N}(\t)$ is contained in the same one. We mention that the last subset (a {\it sector} of $\C$) above plays a special role in the theory of representation by a linear operator of a sesquilinear form (see \cite[Chapter VI]{Kato} and \cite{RC_CT,Second} for generalizations).
\end{rem}
For forms $\t$ with special set $N(\t)$ the notions introduced in the previous section are simplified.
\begin{pro}
\label{pro_N_t}
Let $\t$ be a sesquilinear form on $\D$. The following statements hold.
\begin{enumerate}
\item If $\t$ is non-negative and $\w$-mixed, then $\t=0$.
\item Assume that $N(\t)\subseteq \mathcal{Q}$. Then
\begin{enumerate}
\item[\emph{(a)}] $2(\Re\t+\Im\t)\in M(\t)$;
\item[\emph{(b)}] $\t$ is $\w$-singular if and only if $\t$ is $\w$-strongly singular if and only if $\Re\t+\Im\t$ is $\w$-singular;
\item[\emph{(c)}] if $\t$ is $\w$-mixed, then $\t=0$.
\end{enumerate}
\item Assume that $N(\t)\subseteq \mathcal{S}_c$, with $c\geq0$. Then
\begin{enumerate}
\item[\emph{(a)}] $(1+c)\Re\t\in M(\t)$;
\item[\emph{(b)}] $\t$ is $\w$-singular if and only if $\t$ is $\w$-strongly singular if and only if $\Re\t$ is $\w$-singular;
\item[\emph{(c)}] if $\t$ is $\w$-mixed, then $\t=0$.
\end{enumerate}
\item If $N(\t)\subseteq\Pi$ and $\t$ is $\w$-mixed, then $\Re\t=0$.
\end{enumerate}
\end{pro}
\begin{proof}
\begin{enumerate}
\item[(i)] Assume that (\ref{def_car_mix}) holds and adopt the notation of the proof of Lemma \ref{lem_def_mixed}. The space $\H_{\af+\bb}$ is the orthogonal sum of two subspaces $M_1$ and $M_2$ where $\overline{\af}$ is zero on $M_1$ and $\overline{\bb}$ is zero on $M_2$. Moreover let $\overline{\t}$ be closure of the form in (\ref{eq_t_tilde_a_b}). By (\ref{def_car_mix}) $\overline{\t}$ vanishes on $M_1$ and on $M_2$; hence $\overline{\t}=0$ on $\H_{\af+\bb}$, because of the Cauchy-Schwarz inequality.
\item[(ii)] For (a) we have
\begin{align*}
\label{eq_|Q|}
|\t(\phi,\psi)|&\leq |\Re\t(\phi,\psi)|+|\Im\t(\phi,\psi)| \\
&\leq \Re\t[\phi]^\mez\Re\t[\psi]^\mez+\Im\t[\phi]^\mez\Im\t[\psi]^\mez \nonumber \\
&\leq 2(\Re\t+\Im \t)[\phi]^\mez (\Re\t+\Im \t)[\psi]^\mez, \nonumber \qquad \forall \phi,\psi\in \D.
\end{align*}
To prove (b) we notice that if $\t$ is $\w$-singular, then so $\Re \t+ \Im \t$ is, because $|\t[\phi]|^2=\Re \t[\phi]^2+\Im\t[\phi]^2$. The singularity of $\Re \t+ \Im \t$ implies that $\t$ is $\w$-strongly singular. \\
For proving (c) assume that $\t$ is $\w$-mixed. Proposition \ref{pro_re_im} implies that $\Re \t, \Im \t$ are $\w$-mixed. Since $\Re \t, \Im \t \geq 0$, by the previous case, $\t=0$. The last implication we need is given by Remark \ref{rem_stong->sing}.
\item[(iii)] Similar considerations as above apply to this statement.
\item[(iv)] In this case $\Re\t\geq 0$ and $\w$-mixed. Therefore, $\Re\t=0$ by point (i). \qedhere
\end{enumerate}
\end{proof}
\section{Lebesgue decomposition theorem}
\label{sec_Leb}
Now, we prove the main theorem of this paper, whose proof is based on Theorem 4.3 of \cite{Tp_DB}. To do this we will use the following construction of \cite{STT} of the $\w$-absolutely continuous $\s_a$ and $\w$-singular $\s_s$ parts of a non-negative form $\s$.\\
Let $J$ be the embedding operator $\pi_{\s+\w}(\phi)\to \pi_w (\phi)$, from $\D/\ker{(\s+\w)}\sub \H_{\s+\w}$ into $\H_\w$. In particular, $J$ is a densely defined contraction and $J^{**}$ is the closure of $J$.
If $P$ is the orthogonal projection of $\H_{s+w}$ onto $\{\ker J^{**}\}^\perp$, then for all $\phi,\psi \in \D$,
\begin{align*}
(\s_a+\w)(\phi,\psi)&=\pin{P\pi_{\s+\w}(\phi)}{\pi_{\s+\w}(\psi)}_{\s+\w} \\
\s_s(\phi,\psi)&=\pin{(I-P)\pi_{\s+\w}(\phi)}{\pi_{\s+\w}(\psi)}_{\s+\w} \nonumber.
\end{align*}
\no We stress that $\s_a+\w$ and $\s_s$ are also singular with respect to each other.
\begin{theo}
\label{Leb_th}
Let $\t,\w$ be sesquilinear forms on $\D$, with $\w$ non-negative and $M(\t)\neq \varnothing$. Then, for any $\s\in M(\t)$,
$$
\t=\t_r+\t_m+\t_{ss},
$$
where $\t_r$ is a $\w$-regular form, $\t_m$ is $\w$-mixed form and $\t_{ss}$ is a $\w$-strongly singular form on $\D$.
\end{theo}
\begin{proof}
Take $\s\in M(\t)$.
A well-defined bounded sesquilinear form on $\D/\ker(\s+\w)$ can be defined as
\begin{equation*}
\label{t_tilde}
\overset{\sim}{\t}(\pi_{\s+\w}(\phi),\pi_{\s+\w}(\psi)):=\t(\phi,\psi), \qquad \forall \pi_{\s+\w}(\phi),\pi_{\s+\w}(\psi)\in \D/\ker(\s+\w).
\end{equation*}
There exists a unique bounded operator $T$ on $\H_{s+w}$, whose norm is not greater than $1$, such that
\begin{equation*}
\label{eq_t_T}
\t(\phi,\psi)=\pin{T\pi_{\s+\w}(\phi)}{\pi_{\s+\w}(\psi)}_{\s+\w}, \qquad \forall \phi,\psi \in \D.
\end{equation*}
Set
\begin{align}
\label{t_r} \t_r(\phi,\psi)&:=\pin{TP\pi_{\s+\w}(\phi)}{P\pi_{\s+\w}(\psi)}_{\s+\w}\\
\t_m(\phi,\psi)&:=\pin{TP\pi_{\s+\w}(\phi)}{(I-P)\pi_{\s+\w}(\psi)}_{\s+\w} \nonumber \\
&+ \pin{T(I-P)\pi_{\s+\w}(\phi)}{P\pi_{\s+\w}(\psi)}_{\s+\w} \nonumber\\
\t_{ss}(\phi,\psi)&:=\pin{T(I-P)\pi_{\s+\w}(\phi)}{(I-P)\pi_{\s+\w}(\psi)}_{\s+\w} \nonumber
\end{align}
for all $\phi,\psi \in \D$. We have $\t=\t_r+\t_m+\t_{ss}$. In addition, $\t_r$ is $\w$-regular, $\t_m$ is $\w$-mixed and $\t_{ss}$ is $\w$-strongly singular. In fact, for all $\phi,\psi \in \D$,
\begin{align*}
|\t_r(\phi,\psi)|&\leq \nosw{T}\n{P\pi_{\s+\w}(\phi)}_{\s+\w} \n{P\pi_{\s+\w}(\psi)}_{\s+\w} \\
&= (\s_a+\w)[\phi]^\mez(\s_a+\w)[\psi]^\mez;\\
|\t_m(\phi,\psi)|&\leq \nosw{T}\n{P\pi_{\s+\w}(\phi)}_{\s+\w} \n{(I-P)\pi_{\s+\w}(\psi)}_{\s+\w}\\
&+
\nosw{T}\n{(I-P)\pi_{\s+\w}(\phi)}_{\s+\w} \n{P\pi_{\s+\w}(\psi)}_{\s+\w} \\
&\leq (\s_a+\w)[\phi]^\mez\s_s[\psi]^\mez + (\s_a+\w)[\psi]^\mez \s_s[\phi]^\mez;\\
|\t_{ss}(\phi,\psi)|&\leq \nosw{T}\n{(I-P)\pi_{\s+\w}(\phi)}_{\s+\w} \n{(I-P)\pi_{\s+\w}(\psi)}_{\s+\w}\\
&\leq \s_s[\phi]^\mez\s_s[\psi]^\mez. \qedhere
\end{align*}
\end{proof}
\begin{rem}
The sesquilinear form $\t_s:=\t_m+\t_{ss}$ is the $\w$-singular part of $\t$ according to \cite[Theorem 4.3]{Tp_DB}. To prove that $\t_s$ is actually $\w$-singular, let $\phi \in \D$ and $(\phi_n)\subset \D$ such that $\pi_{\s+\w}(\phi_n)\to (I-P)\pi_{\s+\w}(\phi)$. Therefore, $\w[\phi_n]\leq (\w+\s_a)[\phi_n]\to 0$ and $\t_s[\phi-\phi_n]\to 0$.
\end{rem}
\begin{rem}
The decomposition in Theorem \ref{Leb_pos} is a special case of Theorem \ref{Leb_th} taking $\s=\t$. In particular, with the notations of these theorems, $\t_{r}=\t_a$, $\t_m=0$ and $\t_{ss}=\t_s$.
\end{rem}
\begin{rem}
The decomposition of a form $\t$ into a sum of $\w$-regular, $\w$-mixed and $\w$-strongly singular parts is not unique, even if $\t$ is non-negative, as it is well-known (see \cite[Theorem 4.6]{HSdeS}). In addition, the particular decomposition given by Theorem \ref{Leb_th} depends also on the choice of $\s\in M(\t)$ as we show here (we will follow the construction of the proof above).
Set $\D=\C^3$. We indicate by $e_1,e_2,e_3$ the vectors $(1,0,0),(0,1,0),(0,0,1)$, respectively. Here, for convenience, we represent all sesquilinear forms by their associated matrices with respect to the basis $\{e_1,e_2,e_3\}$. Consider the sesquilinear forms $\t,\s,\w$ on $\C^3$ which are represented by the following matrices
$$
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}
,\qquad
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}
,\qquad
\begin{pmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
$$
Clearly $\s\in M(\t)$ and $J^{**}=J$ is defined as
$J^{**}:\H_{\s+\w} \to \C^3/ \ker(\w)$, $J^{**}:\phi \mapsto \phi +\text{span} \{e_1\},$
where $\H_{\s+\w}$ is the space $\C^3$ with the norm $\nor_{\s+\w}$. Moreover, $\ker J^{**}=\text{span} \{e_1\}$, $\{\ker J^{**}\}^\perp=\text{span}\{ e_2,e_3 \}$ and the projector $P$ is defined as $P(\phi_1,\phi_2,\phi_3) = (0,\phi_2,\phi_3)$.
The Lebesgue decomposition $\s=\s_a+\s_s$ of $\s$ with respect to $\w$ is then
$$
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix} =\begin{pmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}+
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}.
$$
Note that $\t(\phi,\psi)=\pin{T\phi}{\psi}_{\s+\w},$ for all $\phi,\psi \in \C^3$, where $T(\phi_1,\phi_2,\phi_3)=(-\phi_1,\mez\phi_2,0)$. With this we recover that the Lebesgue decomposition $\t=\t_r+\t_m+\t_{ss}$ of $\t$ with respect to $\w$ and taking $\s\in M(\t)$ is
$$
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}=
\begin{pmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}+
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}+
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}.
$$
Now, let $\mathfrak{u}$ the non-negative sesquilinear form which corresponds to the matrix
$$
\begin{pmatrix}
\frac{5}{3} & -\frac{4}{3} & 0 \\
-\frac{4}{3} & \frac{5}{3} & 0 \\
0 & 0 & 0
\end{pmatrix}.
$$
We have that $\mathfrak{u}-\t$ and $\mathfrak{u}+\t$ are non-negative forms, then $\mathfrak{u}\in M(\t)$.
Therefore, $\H_{\mathfrak{u}+\w}$ is $\C^3$ with the norm $\nor_{\mathfrak{u}+\w}$, the new operator $J^{**}$ is defined as before and $\ker J^{**}=\text{span} \{e_1\}$. But now $\{\ker J^{**}\}^\perp=\text{span} \{(4,5,0),(0,0,1)\}$ and the projection $P_\mathfrak{u}$ on $\{\ker J^{**}\}^\perp$ is $P_\mathfrak{u}(\phi_1,\phi_2,\phi_3) = (\frac{4}{5}\phi_2,\phi_2,\phi_3)$.
The Lebesgue decomposition $\mathfrak{u}=\mathfrak{u}_a+\mathfrak{u}_s$ of $\s$ with respect to $\w$ is
$$
\begin{pmatrix}
\frac{5}{3} & -\frac{4}{3} & 0 \\
-\frac{4}{3} & \frac{5}{3} & 0 \\
0 & 0 & 0
\end{pmatrix}=
\begin{pmatrix}
0 & 0 & 0 \\
0 & \frac{3}{5} & 0 \\
0 & 0 & 0
\end{pmatrix}+
\begin{pmatrix}
\frac{5}{3} & -\frac{4}{3} & 0 \\
-\frac{4}{3} & \frac{16}{15} & 0 \\
0 & 0 & 0
\end{pmatrix}
$$
Moreover,
$\t(\phi,\psi)=\pin{T_{\mathfrak{u}}\phi}{\psi}_{\mathfrak{u}+\w},$ for all $\phi,\psi \in \C^3$, where $T_\mathfrak{u}(\phi_1,\phi_2,\phi_3)=(-\phi_1-\frac{1}{2}\phi_2,\frac{1}{2}\phi_1+\frac{5}{8}\phi_2,0)$ and, finally, the Lebesgue decomposition $\t=\t_r'+\t_m'+\t_{ss}'$ of $\t$ with respect $\w$ and taking $\mathfrak{u}\in M(\t)$
$$
\begin{pmatrix}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}=
\begin{pmatrix}
0 & 0 & 0 \\
0 & \frac{9}{25} & 0 \\
0 & 0 & 0
\end{pmatrix}+
\begin{pmatrix}
0 & -\frac{4}{5} & 0 \\
-\frac{4}{5} & \frac{32}{25} & 0 \\
0 & 0 & 0
\end{pmatrix}+
\begin{pmatrix}
-1 & \frac{4}{5} & 0 \\
\frac{4}{5} & -\frac{16}{25} & 0 \\
0 & 0 & 0
\end{pmatrix}.
$$
In the rest of the paper, we refer to Theorem \ref{Leb_th} as the {\it Lebesgue decomposition} of a form $\t$ with respect to $\w$ and $\s\in M(\t)$.
\end{rem}
\begin{pro}
Let $\t=\t_r+\t_m+\t_{ss}$ be the Lebesgue decomposition of a sesquilinear form $\t$ with respect to $\w$ and $\s\in M(\t)$.
\begin{enumerate}
\item The Lebesgue decomposition with respect to $\w$ and $\s$ of $\t^{*}, \Re \t$ and $\Im \t$ are
\begin{align*}
\t&=(\t_r)^*+(\t_m)^*+(\t_{ss})^*,\\
\Re\t&=\Re(\t_r)+\Re(\t_m)+\Re(\t_{ss}),\\
\Im\t&=\Im(\t_r)+\Im(\t_m)+\Im(\t_{ss}),
\end{align*}
respectively. In particular, if $\t$ is symmetric, then $\t_r,\t_m$ and $\t_{ss}$ are symmetric.
\item The sets $N({\t_{r}}),N({\t_{ss}})$ are contained in $N(\t)$. In particular, if $\t$ is non-negative, then $\t_r$ and $\t_{ss}$ are non-negative.
\end{enumerate}
\end{pro}
The $\w$-mixed part is not in general non-negative (and consequently the null form by Proposition \ref{pro_N_t}) if $\t$ is non-negative. For instance, one can take $\w$-mixed part of $\t$ with respect to $\w$ and $\s\in M(\t)$, where $\t,\s,\w$ are represented by the matrices
$$
\begin{pmatrix}
2 & 1 & 0 \\
1 & 2 & 0 \\
0 & 0 & 0
\end{pmatrix}
,\qquad
\begin{pmatrix}
3 & 0 & 0 \\
0 & 3 & 0 \\
0 & 0 & 0
\end{pmatrix}
,\qquad
\begin{pmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix},
$$
respectively.
\section{Measures and sesquilinear forms}
\label{sec_meas}
In this section we show that one can prove the Lebesgue decomposition of (complex) measures with the help of Theorem \ref{Leb_th}. We refer to \cite{Rudin} for the notions and results of the Measure Theory (see also \cite{Halmos_m,Rao}). All the measures that we will consider are finite.
Let $\RR$ stand for a $\sigma$-algebra on a non-empty set $\mathcal{A}$. We write $\D:=S(\mathcal{A},\RR)$ for the complex vector space of simple functions on $(\mathcal{A},\RR)$. Let $\mu$ be a (complex) measure on $(\mathcal{A},\RR)$. We said that $\mu$ is
\begin{itemize}
\item {\it signed} if $\mu(A)\in \R$ for all $A\in \RR$;
\item {\it non-negative} if $\mu(A)\geq 0$ for all $A\in \RR$.
\end{itemize}
The {\it total variation} of a measure $\mu$ is the non-negative measure $|\mu|$ on $(\mathcal{A},\RR)$ defined on $A\in \RR$ as
$$
|\mu|(A):=\sup \sum_{k=1}^{\infty} |\mu(A_k)|,
$$
where the supremum is taken over all sequences $\{A_k\}$ of disjoint subsets in $\RR$ such that $\bigcup_k A_k=A$. The importance of $|\mu|$ is that it is the smaller non-negative measure $\kappa$ that {\it bounds} $\mu$; i.e., $|\mu(A)|\leq \kappa(A)$ for all $A\in \RR$.\\
The {\it characteristic function} of a subset $A\in \RR$ will be indicated by $\chi_A$.
Given two measures $\mu,\nu$ on $(\mathcal{A},\RR)$ with $\nu$ non-negative, $\mu$ is {\it $\nu$-absolutely continuous} (in symbol $\mu \ll \nu$) if the following equivalent conditions are satisfied:
\begin{enumerate}
\item[(a1)] if $\nu(A)=0$ implies $\mu(A)=0$;
\item[(a2)] for every $\epsilon>0$ there exists $\delta>0$ such that $|\mu(A)|<\epsilon$ for all $A\in \RR$ with $\nu(A)<\delta$, or equivalently in a different notation, $\displaystyle \lim_{\nu(A)\to 0} \mu(A) =0$.
\end{enumerate}
On the other hand, $\mu$ is {\it $\nu$-singular} (in symbol $\mu \perp \nu$) if one of the following equivalent conditions is satisfied (see \cite[Theorem 6.1.17]{Rao})
\begin{enumerate}
\item[(s1)] there exists $E\in \RR$ such that $\nu(A)=\nu(A\cap E)$ and $\mu(A)=\mu(A\cap E^c)$;
\item[(s2)] $\forall \epsilon>0$ there exists $E_\epsilon\in \RR\text{ such that } \mu_s(E_\epsilon)<\epsilon \text{ and }\nu(\mathcal{A}\backslash E_\epsilon)<\epsilon$.
\end{enumerate}
Furthermore, $\mu$ is $\nu$-absolutely continuous (resp. $\nu$-singular) if and only if $|\mu|$ is $\nu$-absolutely continuous (resp. $\nu$-singular) if and only if there exists an $\nu$-absolutely continuous (resp. $\nu$-singular) non-negative measure $\tau$ on $(\mathcal{A},\RR)$ bounding $\mu$.
A sesquilinear form $\t$ on $\D=S(\mathcal{A},\RR)$ is said to be {\it induced} by the measure $\mu$ on $(\mathcal{A},\RR)$ if
$$
\t(\phi,\psi)=\int_\mathcal{A} \phi \ol{\psi}d\mu, \qquad \forall \phi,\psi\in \D.
$$
Let $\mu,\nu$ be two measures on $(\mathcal{A},\RR)$ with $\nu$ non-negative. Consider the sesquilinear forms induced by $\mu,|\mu|$ and $\nu$; i.e.,
\begin{equation}
\label{form_meas}
\t(\phi,\psi)=\int_\mathcal{A} \phi \ol{\psi}d\mu, \qquad \s(\phi,\psi)=\int_\mathcal{A} \phi \ol{\psi}d|\mu|, \qquad
\w(\phi,\psi)=\int_\mathcal{A} \phi \ol{\psi}d\nu,
\end{equation}
for all $\phi,\psi\in \D=S(\mathcal{A},\RR)$, respectively. Obviously, $\s\in M(\t)$ and $\t$ is non-negative (resp. symmetric) if and only if $\mu$ is non-negative (resp. signed).
\begin{lem}
\label{lem_reg_for_meas}
The following statements hold.
\begin{enumerate}
\item The form $\t$ is $\w$-regular if and only if $\mu$ is $\nu$-absolutely continuous.
\item If $\mu$ is $\nu$-singular, then $\mu$ is $\w$-strongly singular. The converse is true if $\t$ is non-negative.
\item If $\s$ is $\w$-singular, then $\mu$ is $\nu$-singular.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}
\item[(i)] Assume $\t$ is $\w$-regular. By definition, there exists $\mathfrak{u}\in M(\t)$ and $\mathfrak{u} \ll \w$. If $A\in \RR$ and $\nu(A)=0$ then $\chi_A\in \ker \w \sub \ker \mathfrak{u} \sub \ker \t$. Therefore, $\mu(A)=0$. Conversely, if $\mu$ is $\nu$-absolutely continuous, then so $|\mu|$ is and $\s\ll\w$ by \cite[Theorem 3.2]{STT}. Since $\s\in M(\t)$, $\t$ is $\w$-regular.
\item[(ii)] In \cite[Theorem 3.2]{STT} it was proved that if $\mu$ is non-negative, then $\t$ is $\w$-singular if and only if $\mu$ is $\nu$-singular. In the general case, assume that $\mu$ is $\nu$-singular. This means that $|\mu|$ is $\nu$-singular and, consequently, $\s\perp \w$. Finally, $\s\in M(\t)$ implies that $\t$ is $\w$-strongly singular.
\item[(iii)] If $\s$ is $\w$-singular, then $|\mu|$ is $\nu$-singular. Hence, $\mu$ is $\nu$-singular. \qedhere
\end{enumerate}
\end{proof}
Now we can give the announced proof of the Lebesgue decomposition theorem of finite measures based on the ideas developed in this paper. We state it for reader's convenience.
\begin{theo}
Let $\RR$ be a $\sigma$-algebra on a non-empty set $\mathcal{A}$. Let $\nu,\mu$ be measures on $(\mathcal{A},\RR)$, $\nu$ being non-negative.
There exist unique measures $\mu_a,\mu_s$ on $(\mathcal{A},\RR)$ such that
\begin{enumerate}
\item $\mu=\mu_a+\mu_s$;
\item $\mu_a$ is $\nu$-absolutely continuous and $\mu_s$ is $\nu$-singular.
\end{enumerate}
\end{theo}
\begin{proof}
The uniqueness follows easily by the following argument. Indeed, assume that $\mu=\mu_a + \mu_s=\mu_a' + \mu_s'$, where $\mu_a,\mu_a'\ll \nu$ and $\mu_s,\mu_s '\perp \nu$. Then $\mu_a-\mu_a'=\mu_s'-\mu_s$; i.e., $\mu_a-\mu_a'$ is both absolutely continuous and singular with respect to $\nu$.
Thus, clearly, $\mu_a=\mu_a'$ and $\mu_s=\mu_s'$.
To prove the existences, let us define the forms $\t,\w,\s$ as in (\ref{form_meas}).
First of all, assume that $\mu$ is non-negative; i.e., $\t=\s\geq 0$.
Consider the Lebesgue decomposition $\s=\s_a+\s_s$ of $\s$ with respect to $\w$ as in Theorem \ref{Leb_pos}.
Moreover, with the notations introduced before Theorem \ref{Leb_th}, for all $\phi,\psi\in \D$,
\begin{align*}
(\s_a+\w)(\phi,\psi)&=\pin{P\pi_{\s+\w}(\phi)}{P\pi_{\s+\w}(\psi)}_{\s+\w},\\
\s_s(\phi,\psi)&=\pin{(I-P)\pi_{\s+\w}(\phi)}{(I-P)\pi_{\s+\w}(\psi)}_{\s+\w}.
\end{align*}
We know (see \cite[Theorem 3.4]{STT
) that there exist additive set functions $\mu_a$ and $\mu_s$ on $(\mathcal{A},\RR)$ satisfying $\mu=\mu_a+\mu_s$ and
\begin{align*}
(\s_a+\w)(\phi,\psi)&=\int_\mathcal{A} \phi \ol{\psi}d(\mu_a+\nu), \\
\s_s(\phi,\psi)&=\int_\mathcal{A} \phi \ol{\psi}d\mu_s, \qquad \forall \phi,\psi\in \D.
\end{align*}
In addition, $\mu_a$ is a $\nu$-absolutely continuous and $\mu_s$ is a $\nu$-singular in the sense of \cite[Section 3]{STT}; i.e, $\displaystyle \lim_{\nu(A)\to 0} \mu_a(A)=0$
and
$$
\forall\epsilon>0\; \exists E_\epsilon\in \RR\text{ such that } \mu_s(E_\epsilon)<\epsilon \text{ and }\nu(\mathcal{A}\backslash E_\epsilon)<\epsilon.
$$
We prove that $\mu_a,\mu_s$ are continuous from below; then they must be measures (see \cite[Theorem 5.F]{Halmos_m}). Take $A\in \RR$ and $(A_n)\subset \RR$ an increasing sequence with $\bigcup_{n} A_n=A$. Therefore,
$$
(\s+\w)[\chi_A-\chi_{A_n}]=(\s+\w)[\chi_{A\backslash{A_n}}]=(\mu+\nu)(A\backslash{A_n})\to 0,
$$
because $\mu+\nu$ is a measure. This means that $\pi_{\s+\w}(\chi_{A_n})\to\pi_{\s+\w}(\chi_A)$ in $\H_{\s+\w}$ and, by continuity,
\begin{align*}
\mu_a(A)&=\n{P\pi_{\s+\w}(\chi_A)}_{\s+\w}^2=\lim_{n\to +\infty} \n{P\pi_{\s+\w}(\chi_{A_n})}_{\s+\w}^2=\lim_{n\to +\infty} \mu_a(A_n).
\end{align*}
In the same way, $\displaystyle\mu_s(A)=\lim_{n\to +\infty} \mu_s(A_n)$. Consequently, $\mu_a$ is a $\nu$-absolutely continuous measure and $\mu_s$ is a $\nu$-singular measure.
Before we move on the general case without any condition on the sign of $\mu$, we give an expression to the projector $P$. There exists $E\in \RR$ such that $\mu_s(E)=0$ and $\nu(E^c)=0$
and, consequently, for all $A\in \RR$,
\begin{align}
\label{def_sing_meas3}
(\mu_a+\nu)(A\cap E)=(\mu_a+\nu)(A), \qquad \mu_s(A\cap E)=0.
\end{align}
Let $\phi,\psi\in \D$. Thus, $\phi\ol{\psi}=\sum_{k=1}^{n}a_k \chi_{A_k}$, for some $n\geq1$ and $A_k\in \RR$, disjoint subsets. Applying (\ref{def_sing_meas3}) we obtain that
\begin{align*}
(\s_a+\w)(\phi,\psi)&=\int_\mathcal{A} \phi \ol{\psi}d(\mu_a+\nu) =\sum_{k=1}^n a_i (\mu_a+\nu)(A_k) \\
&=\sum_{k=1}^n a_i (\mu_a+\nu)(A_k \cap E) +\sum_{k=1}^n a_i \mu_s(A_k \cap E)\\
&=\int_\mathcal{A} \chi_E \phi \ol{\psi}d(\mu_a+\nu) +\int_\mathcal{A} \chi_E \phi \ol{\psi}d\mu_s \\
&=\int_\mathcal{A} \chi_E \phi \ol{\psi}d(\mu+\nu).
\end{align*}
Clearly, $\chi_E \phi\in \D$. Therefore, we can write
$$
\pin{P\pi_{\s+\w}(\phi)}{\pi_{\s+\w}(\psi)}_{\s+\w}=(\s_a+\w)(\phi,\psi)=\pin{\pi_{\s+\w}(\chi_E\phi)}{\pi_{\s+\w}(\psi)}_{\s+\w}.
$$
Since $\D\backslash \ker(\s+\w)$ is dense in $\H_{\s+\w}$, we have
\begin{equation}
\label{expr_P}
P\pi_{\s+\w}(\phi)=\pi_{\s+\w}(\chi_E\phi), \qquad \forall \phi \in \D.
\end{equation}
Now, let $\mu$ be a (complex) measure on $(\mathcal{A}, \RR)$.
Let $\t=\t_r+\t_m+\t_{ss}$ be the Lebesgue decomposition of $\t$ with respect to $\s\in M(\t)$ and $\w$.\\
We can repeat the arguments above for $\s$ which is non-negative. Thus, $P$ act as in (\ref{expr_P}) with some $E\in \RR$. Taking into account (\ref{t_r}),
\begin{align*}
\t_r(\phi,\psi)&=\pin{TP\pi_{\s+\w}(\phi)}{P\pi_{\s+\w}(\psi)}_{\s+\w}\\
&=\pin{T\pi_{\s+\w}(\chi_E\phi)}{\pi_{\s+\w}(\chi_E\psi)}_{\s+\w}\\
&=\t(\chi_E\phi,\chi_E\psi)\\
&=\int_\mathcal{A} \chi_E\phi\ol{\psi}d\mu, \qquad\qquad \forall \phi,\psi\in \D.
\end{align*}
Hence, $\mu_a(A):=\t_r[\chi_A]=\mu(A\cap E)$, for $A\in \RR$, is a measure on $(\mathcal{A},\RR)$. We can conclude that $\mu_a$ is $\nu$-absolutely continuous applying Lemma \ref{lem_reg_for_meas}. Also $\mu_s:=\mu-\mu_a$ is a measure on $(\mathcal{A},\RR)$ and, in particular, $\mu_s(A)=\t_s[\chi_A]=\mu(A\cap E^c)$. This shows that $\nu \perp \mu_s$.
\end{proof}
This proof does not involve the Jordan decomposition of a signed measure and, in the general case, it works also taking for $\s$ the sesquilinear form induced by any non-negative measure which bounds $\mu$.
\section*{Acknowledgments}
The author gratefully acknowledges the hospitality of the Alfréd Rényi Institute of Mathematics - Hungarian Academy of Sciences - and the Eötvös Loránd University of Budapest. He also wishes to thank Dr. T. Titkos and Dr. Zs. Tarcsay for many conversations. \\
This work was supported by the ''National Group for Mathematical Analysis, Probability and their Applications'' (GNAMPA – INdAM, project ''Problemi spettrali e di rappresentazione in quasi *-algebre di operatori'' 2017).
|
{
"timestamp": "2018-03-08T02:11:25",
"yymm": "1803",
"arxiv_id": "1803.02772",
"language": "en",
"url": "https://arxiv.org/abs/1803.02772"
}
|
\section{Introduction}
In this paper we continue our investigation in \cite{Ng17a} of the Dirichlet problem for the complex Monge-Amp\`ere equation in a bounded strictly pseudoconvex domain $\Omega \subset \mathbb{C}^n$, provided the existence of a H\"older continuous subsolution. We refer the reader to \cite{BT76}, \cite{GKZ08}, and \cite{ko98} for a more detailed historical account on the subject (see also \cite{DDGKPZ14}, \cite{DGZ16}, \cite{DN16}, \cite{DMN17} \cite{ko08}, \cite{hiep} and \cite{viet16} for geometric motivations and applications).
Let $\varphi \in PSH(\Omega) \cap C^{0,\alpha} (\bar\Omega)$ for some $0<\alpha \leq 1$. Assume also that
\[\notag
\varphi = 0 \quad \mbox{on } \d\Omega.
\]
We consider the following set
\[\notag
\mathcal{M} (\varphi, \Omega):= \left\{
\mu \mbox{ is positive Borel measure: } \mu \leq (dd^c\varphi)^n
\mbox{ in } \Omega \right\}.\]
We also say that $\varphi$ is a H\"older continuous subsolution to measures in $\mathcal{M}(\varphi,\Omega)$. Given $\psi$ a H\"older continuous function on the boundary $\d\Omega$ and a measure $\mu$ in $\mathcal{M}(\varphi, \Omega)$ we look for a real-valued function $u$ satisfying
\[\label{eq:dirichlet-prob}
\begin{aligned}
& u\in PSH\cap L^\infty(\Omega), \\
& (dd^c u)^n = \mu \quad \mbox{in } \Omega, \\
& \lim_{z\to x} u(z) = \psi(x) \quad \mbox{for } x\in\d\Omega,
\end{aligned}\]
and
\[\label{eq:holder}
u \in C^{0,\alpha'}(\bar\Omega) \quad \mbox{for some } 0< \alpha' \leq 1. \\
\]
The Dirichlet problem~\eqref{eq:dirichlet-prob} was solved by Ko\l odziej \cite{ko95} provided that there exists a bounded plurisubharmonic subsolution. In our setting, the H\"older continuity of $\psi$ on $\d\Omega$ and of $\varphi$ on $\bar\Omega$ are necessary in order to solve the Dirichlet problem \eqref{eq:dirichlet-prob} $\&$ \eqref{eq:holder}. In \cite{Ng17a} this problem is solved under the extra assumptions:
\[\notag
\psi \equiv 0 \quad \mbox{ and } \quad \int_\Omega (dd^c \varphi)^n <+\infty.
\]
This gave also an affirmative answer to a question of A. Zeriahi \cite[Question 17]{DGZ16} when the subsolution $\varphi$ has finite Monge-Amp\`ere total mass. Our goal now is to remove these extra assumptions. The first main result of this paper is as follows.
\medskip
\noindent{\bf Theorem~A. }{\em
Let $\psi \in C^0(\d\Omega)$ and $\mu \in \mathcal{M}(\varphi,\Omega)$. Then, there exists a unique solution $u\in C^0(\bar\Omega)$ to the Dirichlet problem \eqref{eq:dirichlet-prob}.}
\medskip
This theorem is closely related to a question of S.\,Ko\l odziej \cite[Question 14]{DGZ16} where he asked if one could prove Theorem~A when the subsolution $\varphi$ is only \em continuous? \rm The question is still open in general.
The next result gives a necessary and sufficient condition under which a positive Borel measure admitting a H\"older continuous plurisubharmonic potential. In particular, the answer to the above question of A.\,Zeriahi is affirmative.
\medskip
\noindent{\bf Theorem B. } {\em Assume that $\psi$ is H\"older continuous and $\mu \in \mathcal{M}(\varphi,\Omega)$. Then, the Dirichlet problem \eqref{eq:dirichlet-prob} $\&$ \eqref{eq:holder} is solvable.}
\medskip
Thanks to this we obtain easily the convexity of the set of Monge-Amp\`ere measures of H\"older continuous plurisubharmonic functions in $\Omega$. Another important consequence is the $L^p$ property. Thus, results in \cite{BKPZ16}, \cite{Cha15a, Cha15b} are special cases of ours.
\medskip
\noindent{\bf Corollary C.} {\em Let $\mu \in \mathcal{M}(\varphi,\Omega)$ and $f\in L^p(\Omega, d\mu)$, $p>1$, a nonnegative function. Suppose that $\varphi$ is H\"older continuous plurisubharmonic function on a neighborhood of $\bar\Omega$. Then, $f\mu \in \mathcal{M}(\tilde\varphi,\Omega)$ for a H\"older continuous plurisubharmonic function $\tilde\varphi$ in $\Omega.$
}
\bigskip
\noindent{\em Acknowledgement. } I am very grateful to S\l awomir Ko\l odziej for many useful discussions. I would like to thank Kang-Tae Kim for his generous support and encouragement.
The author is supported by the NRF Grant 2011-0030044 (SRC-GAIA) of The Republic of Korea.
\section{Preliminaries}
In this section we will recall results that are needed in the proofs of Theorems A and B, Corollary~C. If there is no other indication, then the notations in this section will be used for the rest of the paper.
Let $\Omega$ be a bounded strictly pseudoconvex domain in $\mathbb{C}^n$. Let $\rho\in C^2(\bar\Omega)$ be a strictly plurisubharmonic defining function for $\Omega$. Namely,
\[\label{eq:defining-fct}
\Omega = \{\rho < 0\} \quad \mbox{and} \quad d\rho \neq 0 \mbox{ on } \d\Omega.
\]
Let us denote by $\beta = dd^c |z|^2$ the standard K\"ahler form in $\mathbb{C}^n$. Without loss of generality we may assume that
\[
dd^c\rho \geq \beta \quad \mbox{on } \bar\Omega.
\]
Throughout the paper the H\"older continuous subsolution $\varphi$ and the associated set of measures $\mathcal{M}(\varphi, \Omega)$ are defined as in the introduction.
The following estimate will be very useful for us. For simplicity we write
\[
\|\cdot\|_\infty := \sup_{\Omega} |\cdot |.
\]
\begin{lem}[B\l ocki \cite{Bl93}] \label{lem:blocki}
Let $v_1,...,v_n, v, h \in PSH \cap L^\infty(\Omega)$ be such that $v_i \leq 0$ for $i =1, ...,n$, and $v\leq h$. Assume that $\lim_{z\to \d\Omega} [h(z) - v(z)] =0$. Then, for an integer $1\leq k \leq n$,
\[\begin{aligned}
& \int_{\Omega} (h-v)^k dd^c v_1 \wedge \cdots \wedge dd^c v_n \\
&\leq k! \|v_1\|_\infty \cdots \|v_k\|_\infty \int_{\Omega} (dd^cv)^k \wedge dd^cv_{k+1} \wedge \cdots \wedge dd^c v_n.
\end{aligned}\]
\end{lem}
Consider also the following Cegrell class:
\[ \mathcal{E}_0 = \left\{
v \in PSH \cap L^\infty(\Omega) \biggm|
\begin{aligned}& \lim_{x\to z} v(x) =0 \quad \forall z\in \d\Omega, \\
&\mbox{ and } \int_\Omega (dd^c v)^n < +\infty \end{aligned}\right\}.
\]
The Cegrell inequaliy in this class reads:
\begin{lem}[Cegrell \cite{Ce04}] \label{lem:cegrell}
Let $v_1,..., v_n \in \mathcal{E}_0$. Then,
\[
\int_\Omega dd^c v_1 \wedge \cdots \wedge dd^c v_n \leq
\left(\int_\Omega (dd^cv_1)^n\right)^\frac{1}{n} \cdots
\left(\int_\Omega (dd^cv_n)^n\right)^\frac{1}{n}.
\]
\end{lem}
We need also to work with a subclass of the Cegrell class:
\[
\mathcal{E}_0' := \left\{v\in \mathcal{E}_0: \int_\Omega (dd^cv)^n \leq 1\right\}.
\]
The decay of the volume of sublevel sets of functions in the class $\mathcal{E}_0'$ is equivalent to the volume-capacity inequality. This inequality plays a crucial role in the capacity method due to Ko\l odziej to obtain the \em a priori and stability estimates \rm for weak solutions of complex Monge-Amp\`ere equation. Here the capacity is the Bedford-Taylor capacity and it is defined as follows. For a Borel set $E\subset \Omega$
\[
cap(E,\Omega):= \sup\left\{\int_E (dd^cw)^n: w\in PSH(\Omega), \; 0\leq w\leq 1\right\}.
\]
In what follows we shall write $cap(E)$ instead of $cap(E,\Omega)$ for simplicity as the domain $\Omega$ is already fixed.
\section{Proof of Theorem A}
In this section we shall prove the following result.
\begin{prop}\label{prop:vol-cap} Assume that $\mu \in \mathcal{M}(\varphi, \Omega)$. Then, there exist uniform constants $\alpha_0,C>0$ depending only on $\varphi, \Omega$ such that for every compact set $K\subset\Omega$,
\[
\mu(K) \leq C cap(K) \exp \left(\frac{-\alpha_0}{[cap(K)]^\frac{1}{n}}\right).
\]
\end{prop}
Notice that under the assumption
$
\int_\Omega (dd^c\varphi)^n < +\infty
$
a similar inequality, without the factor $cap(K)$ on the right hand side, was proven in \cite{Ng17a}.
\begin{remark} Theorem~A will follows immediately from the proposition and a result of Ko\l odziej \cite[Theorem~5.9]{ko05} as $\mu$ belongs to the class $\mathcal{F}(A, h)$ with $h = e^{\alpha_0 x}$ and a uniform $A>0$.
\end{remark}
We will need the following two lemmas. The first one tells us how fast the Monge-Amp\`ere mass of $(dd^c\varphi)^n$ on large sublevel sets goes to infinity.
\begin{lem}\label{lem:cegrell-sublevel}
Let $v\in \mathcal{E}_0'$. Then, there exists a uniform constant $C$ such that for $s>0$,
\[
\int_{\{v<-s\}} (dd^c \varphi)^n \leq \frac{C\|\varphi\|_\infty^n}{s^n}.
\]
\end{lem}
\begin{proof} Set $v_s := \max\{v, -s\}$. Then, $v_s = v$ on a neighborhood of $\d\Omega$. Moreover,
\[
v_{s/2} - v \geq \frac{s}{2} \quad \mbox{on } \{v< -s\} \subset\subset \Omega.
\]
Therefore,
\[\label{eq:lcs}\begin{aligned}
\int_{\{v<-s\}} (dd^c\varphi)^n
&\leq \left(\frac{2}{s}\right)^n \int_{\Omega} (v_{\frac{s}{2}} -v)^n (dd^c \varphi)^n \\
&\leq \frac{2^n n!}{s^n} \|\varphi\|_\infty^n \int_\Omega (dd^cv)^n,
\end{aligned}\]
where the second inequality follows from Lemma~\ref{lem:blocki}.
\end{proof}
On the other hand the volume, with respect to the measure $(dd^c\varphi)^n$, of small sublevel sets of functions in $\mathcal{E}_0'$ decays exponentially fast to zero. The H\"older continuity of $\varphi$ is crucially important to prove such an estimate.
\begin{lem} \label{lem:exp-decay}
There exist uniform constants $\tau>0$ and $C>0$ such that for $v\in \mathcal{E}_0'$ and $s\geq 2$,
\[
\int_{\{v<-s\}} (dd^c\varphi)^n \leq C e^{- \tau s}.
\]
\end{lem}
\begin{proof}
With the same notations as in the proof of Lemma~\ref{lem:cegrell-sublevel} we have for $s\geq 2$
\[ \label{eq:exp-decay-A}
\begin{aligned}
\int_{\{v<-s\}} (dd^c\varphi)^n
&\leq \frac{2}{s} \int_{\Omega} (v_{\frac{s}{2}} -v) (dd^c \varphi)^n\\
&\leq \int_{\Omega} (v_{\frac{s}{2}} -v) (dd^c \varphi)^n.
\end{aligned}
\]
Let us denote
\[\notag
S_k := (dd^c\varphi)^k \wedge \beta^{n-k},
\]
where $\beta= dd^c |z|^2$ and $0 \leq k\leq n$ is integer. Our first goal is to show that there exist $\alpha_k>0$ and $C>0$ (independent of $v$ and $s$) such that for $v\in \mathcal{E}_0'$ and $s\geq 1$,
\[\label{eq:holder-cegrell-class}
\int_\Omega (v_s -v) S_k \leq C \left(\int_\Omega (v_s -v) dV_{2n}\right)^{\alpha_k},
\]
where $v_s = \max\{v, -s\}$. Indeed, without loss of generality we may assume that
\[\label{eq:small-ass}
0< \|v_s -v\|_1 < 1/100.
\]
Otherwise, if $\|v_s -v\|_1 =0$, then the inequality trivially holds. If $\|v_s -v\|_1 \geq 1/100$, then we have, using $s\geq 1$, $v\leq 0$ and Lemma~\ref{lem:blocki}, that
\[\begin{aligned}
\int_\Omega (v_s -v) S_k
&= \int_{\{v<-s\}} (-s-v) S_k \\
&\leq \int_{\Omega} (-v)^k S_k\\
&\leq C \|\varphi\|_\infty^k.
\end{aligned}\]
This implies the inequality.
Next, under the assumption \eqref{eq:small-ass} we prove the inequality by induction in $k$. The case $k=0$ is obvious. Assume that
for every integer $m\leq k$ we have
\[
\int_\Omega (v_s -v) S_m \leq C \left(\int_\Omega (v_s -v) dV_{2n}\right)^{\alpha_m}.
\]
Then, we need to show that there exists $0< \alpha_{k+1} \leq 1$ such that
\[
\int_\Omega (v_s -v) S_{k+1} \leq C \left(\int_\Omega (v_s -v) dV_{2n}\right)^{\alpha_{k+1}}.
\]
For simplicity we write
\[
S:= (dd^c \varphi)^{k} \wedge \beta^{n-k-1}.
\]
Let us still write $\varphi$ to be the H\"older continuous extension of $\varphi$ onto a neighbourhood $U$ of $\bar \Omega$. Consider the convolution of $\varphi$ with the standard smooth kernel $\chi$, i.e.,
$\chi \in C^\infty_c(\mathbb{C}^n)$ such that $\chi(z) \geq 0$, $\chi(z)= \chi(|z|)$, $\mbox{supp } \chi \subset\subset B(0,1) $ and $\int_{\mathbb{C}^n}\chi (z) dV_{2n} =1$.
Namely, for $z\in U$ and $\delta>0$ small,
\[\begin{aligned}
\varphi * \chi_t (z)
&= \int_{B(0,1)} \varphi(z-t z') \chi (z') dV_{2n}(z') \\
&= \frac{1}{t^{2n}} \int_{B(z, t)} \varphi (z') \chi \left(\frac{z -z'}{t} \right) dV_{2n} (z').
\end{aligned}\]
Observe that
\[\label{eq:holder-convol}
\begin{aligned}
\varphi * \chi_t (z) - \varphi(z)
&= \int_{B(0,1)} [\varphi(z-t z') -\varphi(z)] \chi (z') dV_{2n}(z') \\
& \leq Ct^\alpha,
\end{aligned}\]
and
\[\label{eq:sec-der-convol}
\left|\frac{\d^2 \varphi * \chi_t}{\d z_j \d \bar z_k} (z)\right| \leq \frac{C \|\varphi\|_\infty}{t^2}.
\]
We first have
\[ \label{eq:i1+i2}
\begin{aligned}
\int_\Omega (v_s-v) dd^c \varphi \wedge S
&\leq \left| \int_\Omega (v_s -v) dd^c \varphi*\chi_t \wedge S \right| \\
&\quad + \left| \int_\Omega (v_s -v) dd^c (\varphi*\chi_t -\varphi) \wedge S\right| \\
&=: I_1+ I_2.
\end{aligned}\]
It follows from \eqref{eq:sec-der-convol} that
\[
I_1 \leq \frac{C \|\varphi\|_\infty}{t^2} \int_\Omega (v_s -v) S \wedge \beta =
\frac{C \|\varphi\|_\infty}{t^2} \int_\Omega (v_s -v) S_{k} .
\]
Hence,
\[
I_1 \leq \frac{C \|\varphi\|_\infty}{t^2} \|v_s -v\|_1^{\alpha_k}.
\]
We turn to the estimate of the second integral $I_2$. By integration by parts
\[\label{eq:i2-int}\begin{aligned}
\int_\Omega (v_s-v) dd^c (\varphi*\chi_t -\varphi) \wedge S
&= \int_{\Omega} (\varphi*\chi_t -\varphi) dd^c (v_s -v) \wedge S \\
&= \int_{\{v < -\frac{s}{2}\}} (\varphi*\chi_t -\varphi) dd^c (v_s -v) \wedge S
\end{aligned}\]
as $v_s = v$ on $ \{v \geq -s\}$. Hence,
\[\begin{aligned}
I_2
&\leq \int_{\{v<-\frac{s}{2}\}} |\varphi*\chi_t -\varphi| (dd^c v + dd^c v_s) \wedge S\\
&\leq C t^\alpha \int_{\{v<-\frac{s}{2}\}} (dd^c v + dd^c v_s) \wedge S.
\end{aligned}\]
For the first term of the integral on the right hand side we have
\[\label{eq:ind-i2-a1}\begin{aligned}
\int_{\{v<-\frac{s}{2}\}} dd^c v \wedge S
&\leq \left(\frac{4}{s}\right)^k \int_{\{v<-\frac{s}{4}\}} (v_{\frac{s}{4}} -v)^k dd^c v \wedge S \\
&\leq \frac{C}{s^k} \int_\Omega (v_{\frac{s}{4}} -v)^k dd^c v \wedge (dd^c \varphi)^k \wedge \beta^{n-k-1}.
\end{aligned}\]
Applying Lemma~\ref{lem:blocki} we conclude that
\[\label{eq:ind-i2-a2}
\int_{\Omega} (v_{s/4} -v)^k dd^cv \wedge (dd^c\varphi)^k \wedge \beta^{n-k-1} \leq C \|\varphi\|_\infty^k \int_\Omega (dd^cv)^{k+1} \wedge \beta^{n-k-1}.
\]
Using $dd^c\rho \geq \beta$ (see Preliminaries) and Cegrell's inequality we get that
\[\label{eq:ind-i2-a3}\begin{aligned}
\int_\Omega (dd^cv)^{k+1} \wedge \beta^{n-k-1}
&\leq \int_\Omega (dd^cv)^{k+1} \wedge (dd^c\rho)^{n-k-1} \\
&\leq \left(\int_\Omega (dd^cv)^n\right)^\frac{k+1}{n}
\left(\int_\Omega (dd^c\rho)^n\right)^\frac{n-k-1}{n}.
\end{aligned}\]
Combining \eqref{eq:ind-i2-a1}, \eqref{eq:ind-i2-a2} and \eqref{eq:ind-i2-a3} we have for $s\geq 1$,
\[
\int_{\{v<-s/2\}} dd^c v \wedge S \leq C \|\varphi\|_\infty^k.
\]
Notice that $v_s \in \mathcal{E}_0'$. The same arguments as above imply that for $s\geq 1$,
\[
\int_{\{v<-s/2\}} dd^c v_s \wedge S \leq C \|\varphi\|_\infty^k.
\]
Thus, altogether we have
\[
I_1+ I_2 \leq \frac{C \|\varphi\|_\infty}{t^2} \|v_s-v\|_1^{\alpha_k} + C\|\varphi\|_\infty^k t^{\alpha}.
\]
If we choose
\[
t = \|v_s -v\|_1^\frac{\alpha_k}{3}, \quad
\alpha_{k+1} = \frac{\alpha \alpha_k}{3},
\]
then the proof of \eqref{eq:holder-cegrell-class} is completed.
We now conclude the proof of the lemma. It follows from \cite[Eq. (2.26)]{Ng17a} and \cite[Lemma~4.1]{ko05} that
$$
\int_\Omega (v_s -v) dV_{2n} \leq C e^{-\tau_0 s},
$$
where $\tau_0>0$ and $C>0$ are uniform constants independent of $v$ and $s$. Combining this with \eqref{eq:exp-decay-A} and the inequality \eqref{eq:holder-cegrell-class} for $k=n$ the lemma follows.
\end{proof}
We are ready to prove the main result of this section.
\begin{proof}[Proof of Proposition~\ref{prop:vol-cap}]
Let us denote $\nu := (dd^c\varphi)^n$. First, we show that for $v\in \mathcal{E}_0'$
there exist uniform constants $\alpha_1, C>0$ such that
\[\label{eq:sharp-vol-cap}
\nu (v<-s) \leq \frac{Ce^{-\alpha_1s}}{s^n} \quad \forall s>0.
\]
Indeed, there are two possibilities either $s\geq 2$ or $s<2$. If $s \geq 2$, then the inequality follows from Lemma~\ref{lem:exp-decay} as
$$s^n e^{-\tau s/2} \leq \left(\frac{2n}{\tau}\right)^n e^{-n}.$$
(We can take $\alpha_1 = \tau/2$). Otherwise, if $0< s <2$, then we have $e^{-\alpha_1 s} \geq C.$ Then, the desired inequality follows from Lemma~\ref{lem:cegrell-sublevel}.
To complete the proof of the proposition we use an argument which is inspired by the proofs in \cite{ACKPZ09}. Let $K\subset \Omega$ be compact. Since $\nu$ is dominated by a Monge-Amp\`ere measure of a bounded plurisubharmonic function, it vanishes on pluripolar sets. Hence, we may assume that $K$ is non-pluripolar. Let $h_K^*$ be the relative extremal function of $K$ with respect to $\Omega$.
Since $K \subset \Omega$ is compact, it is well-known that
\[\notag
\lim_{\zeta \to \d \Omega} h_{K}^*(\zeta)=0.
\]
By \cite[Proposition~5.3]{BT82} we have
\[\notag
\tau^n:= cap(K, \Omega) = \int_{\Omega} (dd^ch_K^*)^n >0.
\]
Let $0<x<1.$ Since the function $w:= \frac{h_K^*}{\tau}$ satisfies assumptions of the inequality \eqref{eq:sharp-vol-cap}, we have
\[\notag
\nu (h_K^* < -1 +x) = \nu \left(w < \frac{-1+x}{\tau} \right) \leq C \frac{\tau^n}{\alpha_1^n (1-x)^n}\exp\left({-\frac{\alpha_1(1-x)}{\tau}}\right).
\]
Let $x\to 0^+$, we obtain
\[\label{eq:cap-ineq1}
\nu(h_K^* \leq -1) \leq \frac{C}{\alpha_1^n} cap(K,\Omega)\exp\left({\frac{-\alpha_1}{\left[cap(K,\Omega)\right]^\frac{1}{n}}}\right).
\]
Since $h_K = h_K^*$ outside a pluripolar set, we have
\[\label{eq:cap-ineq2}
\nu(K) \leq \nu(h_K =-1) = \nu(h_K^* =-1) \leq \nu(h_K^* \leq -1).
\]
We combine \eqref{eq:cap-ineq1} and \eqref{eq:cap-ineq2} to finish the proof.
\end{proof}
\section{Proof of Theorem B}
In this section we will prove the H\"older continuity of the solution obtained in Theorem A provided furthermore that the boundary data $\psi$ is H\"older continuous. Notice that the zero boundary values of the subsolution $\varphi$ is not essential. We can modify it by adding an appropriate envelope, similar to \eqref{eq:envelope}, because no condition has been imposed on the total mass of the subsolution.
By Theorem A there exists a unique continuous solution to the Dirichlet problem \eqref{eq:dirichlet-prob}, namely, $u \in PSH(\Omega)\cap C^0(\bar\Omega)$ solving
\[\label{eq:sol-B}
(dd^c u)^n = \mu, \quad u(z) = \psi (z) \quad \forall z\in \d\Omega.
\]
We are going to show that $u \in C^{0,\alpha'}(\bar\Omega)$ for some exponent $0<\alpha'\leq 1.$
\medskip
{\em Outline of the proof.\rm} Let us sketch the proof of Theorem~B.
Overall we follow the steps in the proof of \cite{Ng17a} which in turns followed \cite{GKZ08}. Though, we need to consider the problem on an increasing exhaustive sequence of relatively compact domains in $\Omega$. Denote for $\delta>0$ small
\[
\Omega_\delta:= \{z\in \Omega: dist(z,\d\Omega) >\delta\};
\]
and for $z\in \Omega_\delta$ we define
\[\begin{aligned}
& u_\delta(z) := \sup_{|\zeta| \leq \delta} u(z+\zeta), \\
& \hat u_\delta(z):= \frac{1}{\sigma_{2n}\delta^{2n}} \int_{|\zeta| \leq \delta} u(z+\zeta) dV_{2n}(\zeta),
\end{aligned}\]
where $\sigma_{2n}$ the volume of the unit ball.
Then, we wish to show that
$$\sup_{\Omega_\delta} (\hat u_\delta - u) \lesssim \delta^{\varpi}
$$
for some $0<\varpi \leq 1$.
Thanks to the H\"older continuity of the boundary data we can extend $\hat u_\delta$ to $\tilde u$
by a gluing process such that the new function is plurisubharmonic on $\Omega$ and equal to $u$ outside $\Omega_\varepsilon$ for some (small) $\varepsilon > \delta$. Moreover, we shall still have
$$
\sup_{\Omega_\delta} (\hat u_\delta - u)
\leq \sup_{\Omega} (\tilde u -u) + C \varepsilon^\alpha,
$$
where $\alpha$ is the H\"older exponent of the boundary data $\psi$.
Next, we shall show that
\[\notag
\int_{\Omega_\varepsilon} (dd^c\varphi)^n \lesssim \frac{1}{\varepsilon^n}.
\]
This estimate enables us to invoke the results of \cite{Ng17a}. It gives a precise quantitative estimate $\sup_{\Omega} (\tilde u -u)$ in terms of $\delta$ and $\varepsilon$.
Finally, we can choose $\varepsilon = \delta^{\varpi'}$ with $\varpi'>0$ so small that our desired inequality holds.
\medskip
We now proceed to give details of the argument.
For the rest of the arguments we fix a small $\delta_0>0$ and consider two parameters $\delta, \varepsilon$ such that
\[\label{eq:eps-del}
0 < \delta \leq \varepsilon < \delta_0.
\]
We may assume that $\psi\in C^{0,2\alpha}(\d\Omega)$, where $0<\alpha\leq 1/2$ (decreasing $\alpha$ if necessary) is the H\"older exponent of the subsolution $\varphi$. Then, we define
\[\label{eq:envelope}
h(z) = \sup\{v(z) \in PSH(\Omega)\cap C^0(\bar\Omega): h_{|_{\d\Omega}} \leq \psi\}.
\]
It is well-known \cite[Theorem~6.2]{BT76} that $h \in PSH(\Omega)\cap C^{0,\alpha}(\bar\Omega)$ and $h = \psi$ on $\d\Omega$, which is also the solution of the homogeneous Monge-Amp\`ere equation in $\Omega$. Hence, we may assume that
\[
\psi \in PSH(\Omega) \cap C^{0,\alpha}(\bar\Omega)\quad \mbox{and}\quad (dd^c\psi)^n \equiv 0.
\]
Thanks to the comparison principle \cite{BT82} we get that
\[\label{eq:boundary-hol}
\psi + \varphi \leq u \leq \psi \quad \mbox{ on } \bar\Omega.
\]
\begin{lem} \label{lem:boundary-holder}
We have for $z\in \bar\Omega_{\delta}\setminus\Omega_{\varepsilon}$,
\[ u_\delta(z) \leq u(z) + C \varepsilon^\alpha.
\]
In particular,
\[
\sup_{\Omega_\delta} (\hat u_\delta - u) \leq \sup_{\Omega_\varepsilon} (\hat u_\delta - u) + C \varepsilon^\alpha.
\]
\end{lem}
\begin{remark} It is important to keep in mind that the uniform constants $C>0$ appeared in the lemma, and many times below are independent of $\delta$ and $\varepsilon$.
\end{remark}
\begin{proof} Fix a point $z\in \bar\Omega_\delta\setminus\Omega_\varepsilon$. Since $u$ is continuous, there is $\zeta_1 \ \in \mathbb{C}^n$ with $|\zeta_1| \leq \delta$ such that
\[\label{eq:point1}
u_\delta(z) = u(z + \zeta_1).
\]
Moreover, there exists $\zeta_2\in \mathbb{C}^n$ with $|\zeta_2| \leq \varepsilon$ such that $z+\zeta_2 \in \d\Omega$. Using this and \eqref{eq:boundary-hol} we get that
\[\begin{aligned}
u_\delta(z) - u(z)
&\leq \psi(z+ \zeta_1) - [\psi(z) +\varphi(z)] \\
&= [\psi (z+ \zeta_1) - \psi(z)] + [\varphi(z) - \varphi(z+\zeta_2)] \\
&\leq C_1 |\zeta_1|^{\alpha} + C_2|\zeta_2|^{\alpha},
\end{aligned}\]
where $C_1 = \|\psi\|_{C^{0,\alpha}}, C_2= \|\varphi\|_{C^{0,\alpha}}$. Since $\delta \leq \varepsilon$ we conclude the proof of the first part.
To prove the second part, we observe that $u \leq \hat u_\delta \leq u_\delta$. Therefore,
\[
\sup_{\Omega_\delta} (\hat u_\delta - u) \leq \sup_{\Omega_\varepsilon} (\hat u_\delta - u) + \sup_{\Omega_\delta \setminus \Omega_\varepsilon} (u_\delta -u).
\]
Combining this with the first part we get the the second part.
\end{proof}
The lemma above tells us that to obtain the H\"older continuity of the solution $u$ it is enough to get the estimate on the domain $\Omega_\varepsilon$ for $\varepsilon$ being of a small constant compared to $\delta$. To achive our goal we will work on the domain $\Omega_\varepsilon$ and keep track of the (negative) exponent of $\varepsilon$.
Recall that
\[\label{eq:omega-epsilon}
\Omega_\varepsilon= \{z\in \Omega: dist(z,\d\Omega) >\varepsilon\}.
\]
We define
\[\label{eq:d-domain} D_{\varepsilon}:= \{\rho(z) < - \varepsilon\},
\]
where $\rho$ is the defining function of $\Omega$ as in \eqref{eq:defining-fct}.
The following lemma is very similar to Lemma~\ref{lem:cegrell-sublevel}. The main observation is that the domains $D_\varepsilon$ and $\Omega_\varepsilon$ are comparable.
\begin{lem}\label{lem:mass-est}
Let $1\leq k \leq n$ be an integer. Let $v\in PSH\cap L^\infty(\Omega)$. Then,
\[
\int_{\Omega_\varepsilon} (dd^cv)^k \wedge \beta^{n-k} \leq \frac{C \|v\|_\infty^k}{\varepsilon^k},
\]
where $C$ is independent of $\varepsilon.$
\end{lem}
\begin{proof} Observe that, from Hopf's lemma,
\[ \label{eq:dist-bound}
|\rho(z)| \geq c_0 dist(z, \d\Omega)
\]
for a uniform constant $0 < c_0 \leq 1$. Therefore,
\[\label{eq:inclusion}
\Omega_\varepsilon \subset \{\rho(z) < - c_0 \varepsilon\}.
\]
Since $\max\{\rho, -\varepsilon'/2\} - \rho \geq \varepsilon'/2$ with $\varepsilon'= c_0\varepsilon$ on the latter set, it follows that
\[\begin{aligned}
& \int_{\Omega_\varepsilon} (dd^cv)^k \wedge \beta^{n-k} \\
&\leq \left(\frac{2}{\varepsilon'}\right)^k\int_{\Omega} \left(\max\{\rho, -\varepsilon'/2\} - \rho\right)^k (dd^cv)^k \wedge \beta^{n-k} \\
&\leq \frac{C\|v\|_\infty^k}{\varepsilon^k} \int_{\Omega} (dd^c\rho)^k\wedge \beta^{n-k},
\end{aligned}\]
where we used Lemma~\ref{lem:blocki} for the second inequality. The last integral is bounded by the $C^2-$smoothness of $\rho$ on $\bar\Omega$.
\end{proof}
We will now approximate the subsolution $\varphi$. Let us denote
\[
\varphi_\varepsilon := \max\{\varphi-\varepsilon, A \rho/\varepsilon\},
\]
where $A:= 1+ \|\varphi\|_\infty$.
\begin{lem} \label{lem:sub-sol-mass-est}
We have \[
\int_\Omega (dd^c\varphi_\varepsilon)^n \leq \frac{C A^n}{\varepsilon^{n}}.
\]
Moreover,
\[ {\bf 1}_{D_\varepsilon} \cdot \mu \leq (dd^c \varphi_\varepsilon)^n\] as two measures, where $D_\varepsilon$ is defined in \eqref{eq:d-domain}.
\end{lem}
\begin{proof} To estimate the Monge-Amp\`ere mass of $\varphi_\varepsilon$ we use a result of Bedford and Taylor \cite[Corollary~4.3]{BT82} which is a consequence of the comparison principle.
Since $\frac{A \rho}{\varepsilon} \leq \varphi_\varepsilon \leq 0$ and the functions have the zero values on the boundary,
\[
\int_{\Omega} (dd^c\varphi_\varepsilon)^n
\leq \frac{A^n}{\varepsilon^n}\int_{\Omega} (dd^c \rho)^n.
\]
The last integral is finite as $\rho$ is $C^2$ on a neighborhood of the closure of $\Omega$. Furthermore, since $\varphi_\varepsilon(z) = \varphi(z)-\varepsilon$ on $D_\varepsilon = \{\rho<-\varepsilon\}$ as $\varepsilon>0$ small, it is clear that $${\bf 1}_{D_\varepsilon} \cdot \mu \leq (dd^c \varphi_\varepsilon)^n.$$
This completes the proof of the lemma.
\end{proof}
\begin{remark} \label{rmk:mass-control}
Using the same argument we also get that for an integer $1\leq k \leq n$
\[
\int_\Omega (dd^c\varphi_\varepsilon)^k \wedge \beta^{n-k} \leq \frac{CA^k}{\varepsilon^k}.
\]
\end{remark}
We obtain now the volume-capacity inequality for the approximation sequence.
\begin{cor} \label{cor:vol-cap}
There exists uniform constants $\alpha_1>0$ and $C>0$ which are independent of $\varepsilon$ such that for every compact set $K\subset \Omega$,
\[
\int_K (dd^c\varphi_\varepsilon)^n \leq \frac{C}{\varepsilon^{n}} \cdot cap(K) \cdot \exp\left(\frac{-\alpha_1}{[cap(K)]^\frac{1}{n}}\right).
\]
In particular, for a fixed $\tau>0$, there is a constant $C(\tau)>0$ such that
for every compact set $K\subset \Omega$,
\[
\int_K (dd^c\varphi_\varepsilon)^n \leq \frac{C(\tau)}{\varepsilon^{n}} \left[cap(K)\right]^{1+\tau}.
\]
\end{cor}
\begin{proof} This is the analogue of Proposition~\ref{prop:vol-cap} with $\varphi$ is replaced by $\varphi_\varepsilon.$ The proof is a repetition of the one of this proposition. Here we need to take into account three facts:
\[
\|\varphi_\varepsilon\|_{\infty} \leq \frac{C}{\varepsilon} \quad\mbox{and}\quad \|\varphi_\varepsilon\|_{C^{0,\alpha}(\bar\Omega)} \leq \frac{C}{\varepsilon},
\]
and for an integer $1\leq k \leq n$ (Remark~\ref{rmk:mass-control}),
\[
\int_\Omega (dd^c\varphi_\varepsilon)^k \wedge \beta^{n-k} \leq \frac{C}{\varepsilon^{k}}.
\]
This explains why we need an extra factor $C/\varepsilon^{n}$ on the right hand side of the inequality.
\end{proof}
Next, we have the following stability estimate for the Monge-Amp\`ere equation similar to \cite[Theorem~1.1]{GKZ08}. However, it also takes into account the possibility of infinite total mass of the measure on the right hand side.
\begin{prop} \label{prop:stability} Let $u$ be the solution of the equation \eqref{eq:sol-B} and $\Omega_\varepsilon$ is defined by \eqref{eq:omega-epsilon}.
Let $v\in PSH\cap L^\infty(\Omega)$ be such that $v= u$ on $\Omega\setminus \Omega_\varepsilon$. Then, there is $0<\alpha_2\leq 1$ such that
\[
\sup_{\Omega} (v - u) \leq \frac{C}{\varepsilon^{n}} \left(\int_{\Omega} \max\{v -u, 0\} d\mu\right)^{\alpha_2}.
\]
\end{prop}
\begin{proof} Without loss of generality we may assume that $ \sup_{\Omega} (v-u)>0$. Set \[s_0 := \inf_{\Omega} (u-v).\] We know that
for $0<s < |s_0|$,
\[
U(s):=\{u< v + s_0 +s\} \subset \subset \Omega_\varepsilon.
\]
\begin{lem}\label{lem:capacity-growth}
Fix $\tau>0$. For every $0< s, t < \frac{|s_0|}{2}$. Then,
\[
t^n cap(U(s)) \leq \frac{C(\tau)}{\varepsilon^{n}} \left[ cap(U(s+t)) \right]^{1+\tau}.
\]
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:capacity-growth}]
Let $0 \leq w \leq 1$ be a plurisubharmonic function in $\Omega$. We have the following chain of inequalites.
\[\begin{aligned}
t^n \int_{U(s)} (dd^cw)^n
&= \int_{\{u< v+s_0+s\}} [dd^c (tw)]^n \\
&\leq \int_{\{u<v+s_0 +s+tw\}} [dd^c (v + tw)]^n \\
&\leq \int_{\{u<v+s_0 +s+tw\}} (dd^c u)^n, \\
\end{aligned}\]
where we used the comparison principle \cite[Theorem~4.1]{BT82} for the last inequality. Since $\{u<v+s_0 +s+tw\} \subset U(s+t)$ and $w$ is arbitrary, we get that
\[
t^n cap(U(s)) \leq \int_{U(s+t)} d\mu.
\]
If we denote $\varepsilon' := c_0\varepsilon$, where $c_0$ is the constant in \eqref{eq:dist-bound}, then
$${\bf 1}_{D_{\varepsilon'}} \cdot d\mu \leq (dd^c\varphi_{\varepsilon'})^n$$
as two measures.
Since $U(s+t) \subset \Omega_\varepsilon \subset D_{\varepsilon'}$, it follows that
\[\begin{aligned}
\int_{U(s+t)} d\mu
&\leq \int_{U(s+t)} (dd^c\varphi_{\varepsilon'})^n \\
&\leq \frac{C(\tau)}{(c_0\varepsilon)^{n}} \left[ cap(U(s+t)) \right]^{1+\tau},
\end{aligned}\]
where the last inequality followed from Corollary~\ref{cor:vol-cap}. The proof of the lemma is complete.
\end{proof}
Now together with Lemma~\ref{lem:capacity-growth}, the rest of the proof of the proposition is the same as in \cite[Theorem~1.1]{GKZ08} (see also \cite[Theorem~3.11]{KN3}).
\end{proof}
The following result is a variation of Lemma~2.7 in \cite{Ng17a} where we considered the H\"older continuity of a measure $\nu$ on $\mathcal{E}_0'$. Though, the situation now is differrent as $\nu(\Omega)$ is no longer finite.
\begin{thm} \label{thm:l1-l1}
Let $u$ be the solution of the equation \eqref{eq:sol-B} and $\Omega_\varepsilon$ is defined by \eqref{eq:omega-epsilon}.
Let $v \in PSH \cap L^\infty(\Omega)$ be such that that $v=u$ on $\Omega\setminus\Omega_\varepsilon$. Then, there exists $0<\alpha_3\leq 1$ such that
\[
\int_\Omega |v-u| d\mu \leq \frac{C}{\varepsilon^{n+1}} \left(\int_\Omega |v-u| dV_{2n} \right)^{\alpha_3}.
\]
\end{thm}
\begin{proof} This is a variation of the inequality \eqref{eq:holder-cegrell-class} with
\[
S_{k,\varepsilon} := (dd^c\varphi_\varepsilon)^k \wedge \beta^{n-k},
\]
where $\varphi_\varepsilon= \max\{\varphi - \varepsilon, A\rho/\varepsilon\}$ and $0\leq k \leq n$ is integer. Since $\mu \leq S_{n,\varepsilon}$ on $\Omega_\varepsilon$, it is enough to show that there is $0<\tau \leq 1$ satisfying
\[\label{eq:thm-l1-l1-2}
\int_\Omega (v-u) S_{n,\varepsilon} \leq \frac{C}{\varepsilon^{n+1}} \|v-u\|_1^{\tau}.
\]
for $v\geq u$ on $\Omega$. (In the general case we use the identity
\[\notag
|v-u| = (\max\{v,u\} -u) + (\max\{v,u\} -v)
\]
and apply twice the inequality \eqref{eq:thm-l1-l1-2} to get the theorem.)
Now we can repeat the inductive arguments of the proof of \eqref{eq:holder-cegrell-class} with $v, u$ and $\varphi_\varepsilon$ in the places of $v_s, v$ and $\varphi$, respectively. However, there are differences as follows. First, $v, u$ are no longer in $\mathcal{E}_0'$. Second, let us extend $\varphi$ as in proof of Lemma~\ref{lem:exp-decay}, then $\varphi_\varepsilon= \max\{\varphi - \varepsilon, A\rho/\varepsilon\}$ is also defined on the neighborhood $U$ of $\bar\Omega$, and
$$
\|\varphi_\varepsilon\|_{C^{0,\alpha}(U)} \leq \frac{C}{\varepsilon}.
$$
Taking into account above differences, to pass from the $k$-th step to the step number $(k+1)$ we need the following inequality,
corresponding to \eqref{eq:i1+i2}, (with notation $S_\varepsilon:= (dd^c \varphi_\varepsilon)^{k} \wedge \beta^{n-k-1}$)
\[\label{eq:ind-est}\begin{aligned}
\int_\Omega (v-u) dd^c \varphi_\varepsilon \wedge S_\varepsilon
&\leq \left| \int_\Omega (v-u) dd^c \varphi_\varepsilon * \chi_t \wedge S_\varepsilon \right| \\
&\quad +\left| \int_\Omega (v-u) dd^c (\varphi_\varepsilon * \chi_t -\varphi_\varepsilon) \wedge S_\varepsilon\right| \\
&=: I_{1,\varepsilon}+ I_{2,\varepsilon}.
\end{aligned}\]
Since \[\label{eq:observe-holder-b}
\left|\frac{\d^2 \varphi_\varepsilon * \chi_t}{\d z_j \d \bar z_k} (z)\right| \leq \frac{C \|\varphi\|_\infty}{\varepsilon t^2},
\]
and the induction hypothesis at the step $k$-th, there exists $0<\tau_k \leq 1$ such that
$$
\int_\Omega (v-u) S_\varepsilon \wedge \beta \leq \frac{C}{\varepsilon^{k+1}} \|v-u\|_1^{\tau_k},
$$
we conclude that
\[\label{eq:ind-est-a}
\begin{aligned} I_{1,\varepsilon}
&\leq \frac{C\|\varphi\|_\infty}{\varepsilon\, t^2} \int_\Omega (v-u) S_\varepsilon \wedge \beta \\
&\leq \frac{C\|\varphi\|_\infty}{\varepsilon^{k+2}\, t^2} \|v-u\|_1^{\tau_k}.
\end{aligned}\]
Similar to \eqref{eq:i2-int}, by integration by parts, $u=v$ on $\Omega\setminus \Omega_\varepsilon$, and
\[\notag \label{eq:observe-holder-a}
\begin{aligned}
\left|\varphi_\varepsilon * \chi_t (z) - \varphi_\varepsilon(z) \right|
& \leq \frac{Ct^{\alpha}}{\varepsilon},
\end{aligned}\]
it follows that
\[\label{eq:i2e-a}\begin{aligned}
I_{2,\varepsilon}
&\leq \frac{C t^\alpha}{\varepsilon} \int_{\Omega_\varepsilon} (dd^c v+ dd^cu) \wedge S_\varepsilon. \\
\end{aligned}\]
At this point as $u, v$ do not belong to $\mathcal{E}_0'$ we need to use a different argument to bound $I_{2,\varepsilon}$. Namely, similarly to Lemma~\ref{lem:mass-est}, we have
\[\label{eq:i2e-b}
\int_{\Omega_\varepsilon} (dd^c u + dd^c v) \wedge S_\varepsilon \leq \frac{C\|u+v\|_\infty (1+\|\varphi\|_\infty)^k}{\varepsilon^{k+1}} .
\]
Indeed, we first have
\[\notag\begin{aligned}
&\int_{\Omega_\varepsilon} dd^c (u+v) \wedge (dd^c \varphi_\varepsilon)^k\wedge \beta^{n-k-1} \\
&\leq \frac{2}{\varepsilon'}\int_{\Omega} \left(\max\{\rho, - \varepsilon'/2\} - \rho\right) \wedge dd^c (u+v) \wedge (dd^c \varphi_\varepsilon)^k\wedge \beta^{n-k-1} \\
&\leq \frac{C}{\varepsilon} \|u+v\|_\infty \int_\Omega (dd^c\rho)\wedge (dd^c \varphi_\varepsilon)^k\wedge \beta^{n-k-1},
\end{aligned}\]
where $\varepsilon' = c_0\varepsilon$ with $c_0$ defined by \eqref{eq:dist-bound}.
The desired inequality \eqref{eq:i2e-b} follows from Remark~\ref{rmk:mass-control}.
Now, combining \eqref{eq:i2e-a} and \eqref{eq:i2e-b} we get that
\[\label{eq:ind-est-b}
I_{2,\varepsilon} \leq \frac{C t^\alpha}{\varepsilon^{k+2}}.
\]
Next, it is easy to see (from Lemma~\ref{lem:sub-sol-mass-est}) that
\[\notag
\int_{\Omega} (v-u) S_n \leq \frac{C \|u\|_\infty (1+ \|\varphi\|_\infty)^n}{\varepsilon^{n}}.
\]
Therefore, we can assume that $0< \|v-u\|_1 < 0.01$.
Thanks to \eqref{eq:ind-est-a} and \eqref{eq:ind-est-b} we have
\[\notag
\int_\Omega (v-u) dd^c \varphi_\varepsilon\wedge S_\varepsilon \leq \frac{C}{\varepsilon^{k+2} \, t^2} \|v-u\|_1^{\tau_k} + \frac{C t^\alpha}{\varepsilon^{k+2}}.
\]
If we choose
$
t = \|v-u\|_1^\frac{\tau_k}{3},
\quad \tau_{k+1} = \frac{\alpha\tau_k}{3},
$
then
\[\notag
\int_\Omega (v-u) S_\varepsilon \wedge dd^c \varphi_\varepsilon \leq \frac{C}{\varepsilon^{k+2}} \|v-u\|_1^{\tau_{k+1}}.
\]
Thus, the induction argument is completed, and the theorem follows.
\end{proof}
The last ingredient to prove Theorem~B was proved first in \cite{BKPZ16} (see also \cite[Lemma~2.12]{Ng17a}). Here, the estimate is sharper and the proof is simpler too.
\begin{lem} \label{lem:lap-est}
For $\delta>0$ small we have
\[
\int_{\Omega_\delta} |\hat u_\delta -u | dV_{2n} \leq C \delta.
\]
\end{lem}
\begin{proof} First, we know from the classical Jensen formula (see e.g. \cite[Lemma 4.3]{GKZ08}) that
\[
\int_{\Omega_{2\delta}} |\hat u_\delta -u| \leq C \delta^2 \int_{\Omega_\delta} \Delta u(z).
\]
Again, it follows from Lemma~\ref{lem:mass-est} applied for $k =1$ and $\delta = \varepsilon$, that
\[
\int_{\Omega_\delta} \Delta u(z) \leq \frac{C}{\delta}.
\]
Therefore,
\[
\int_{\Omega_\delta} |\hat u_\delta -u| dV_{2n} \leq \int_{\Omega_{2\delta}} |\hat u_\delta -u| dV_{2n} + \|u\|_\infty \int_{\Omega_\delta\setminus \Omega_{2\delta}} dV_{2n} \leq C\delta.
\]
This is the required inequality.
\end{proof}
We are ready to prove the H\"older continuity of the solution.
\begin{proof}[End of Proof of Theorem~B] Let us fix $\delta$ such that $0< \delta < \delta_0$ small and let $\varepsilon$ be such that $ \delta \leq \varepsilon < \delta_{0}$ which is to be determined later.
Thanks to Lemma~\ref{lem:boundary-holder} and $\hat u_\delta \leq u_\delta$ we have $\hat u_\delta - C\varepsilon^\alpha \leq u$ on $\d\Omega_\varepsilon$. Therefore, the function
\[ \label{eq:extend-solution}
\tilde u :=
\begin{cases}
\max\{\hat u_\delta - C \varepsilon^\alpha, u\} \quad
&\mbox{ on } \Omega_{\varepsilon},\\
u \quad &\mbox{ on } \Omega\setminus \Omega_{\varepsilon},
\end{cases}
\]
belongs to $PSH(\Omega)\cap C^0(\bar\Omega)$. Notice that $\tilde u \geq u$ in $\Omega$, and
\[
\tilde u = u \quad\mbox{ on } \Omega\setminus \Omega_\varepsilon.
\]
Again, by the second part of Lemma~\ref{lem:boundary-holder} we have that
\[\label{eq:holder-eq1}
\begin{aligned}
\sup_{\Omega_\delta}(\hat u_\delta -u)
&\leq \sup_{\Omega_\varepsilon} (\hat u_\delta -u) + C \varepsilon^\alpha \\
&\leq \sup_{\Omega} (\tilde u - u) + C \varepsilon^\alpha + C\varepsilon^\alpha.
\end{aligned}\]
By the stability estimate (Proposition~\ref{prop:stability}) there exists $0<\alpha_2 \leq 1$ such that
\[\label{eq:holder-eq2}\begin{aligned}
\sup_{\Omega} (\tilde u - u)
&\leq \frac{C}{\varepsilon^{n}} \left(\int_{\Omega} \max\{\tilde u - u,0\} d\mu\right)^{\alpha_2} \\
&\leq \frac{C}{\varepsilon^{n}} \left(\int_{\Omega} |\tilde u - u| d\mu\right)^{\alpha_2},
\end{aligned}\]
where we used the fact that $\tilde u = u$ outside $\Omega_\varepsilon$.
Using Theorem~\ref{thm:l1-l1}, there is $0<\alpha_3\leq 1$ such that
\[\label{eq:holder-eq3}
\begin{aligned}
\sup_{\Omega} (\tilde u -u)
&\leq \frac{C}{\varepsilon^{n+(n+1)\alpha_2}} \left(\int_{\Omega} |\tilde u - u| dV_{2n}\right)^{\alpha_2\alpha_3} \\
&\leq \frac{C}{\varepsilon^{2n+1}} \left(\int_{\Omega_\delta} |\hat u_\delta - u| dV_{2n}\right)^{\alpha_2\alpha_3},
\end{aligned}\]
where we used $0\leq \tilde u - u \leq {\bf 1}_{\Omega_\varepsilon} \cdot (\hat u_\delta -u)$ and $\Omega_\varepsilon \subset \Omega_{\delta}$ for the second inequality. It follows from \eqref{eq:holder-eq1}, \eqref{eq:holder-eq3} and Lemma~\ref{lem:lap-est} that
\[
\sup_{\Omega_\delta} (\hat u_\delta - u) \leq C \varepsilon^\alpha + \frac{C \delta^{\alpha_2\alpha_3}}{\varepsilon^{2n+1}}.
\]
Now, we choose $\alpha_4 = \alpha\alpha_2\alpha_3/(2n+1+\alpha)$ and
$$ \varepsilon = \delta^{\frac{\alpha_2\alpha_3 }{2n+1 + \alpha}}.
$$
Then,
$
\sup_{\Omega_\delta} (\hat u_\delta - u) \leq C \delta^{\alpha_4}.
$
Finally, thanks to \cite[Lemma~4.2]{GKZ08} we infer that
$
\sup_{\Omega_\delta} (u_\delta - u) \leq C \delta^{\alpha_4}.
$
The proof of the theorem is finished.
\end{proof}
\section{proof of Corollary~C}
Let $\mu\in \mathcal{M}(\varphi,\Omega)$ and $0\leq f\in L^p(\Omega, d\mu)$ with $p>1$. We wish to show that there exists $\tilde\varphi \in PSH(\Omega) \cap C^{0,\tilde\alpha}(\bar\Omega)$ with $0<\tilde\alpha \leq 1$ such that
\[
f d\mu \in \mathcal{M}(\tilde\varphi,\Omega).
\]
The proof of the corollary is similar to the one of Theorem~B with the aid of following two lemmas.
\begin{lem} Fix a constant $\tau>0$. Then, there exists a uniform constant $C(\tau)$ such that for every compact set $K\subset \Omega$,
\[
\int_K fd\mu \leq C(\tau) \left[cap(K)\right]^{1+\tau}.
\]
\end{lem}
\begin{proof} H\"older's inequality and Proposition~\ref{prop:vol-cap} give us
\[\begin{aligned}
\int_K fd\mu
&\leq \|f\|_{L^p(\Omega, d\mu)} \left[\mu(K)\right]^\frac{p-1}{p} \\
&\leq C \left[cap(K) \cdot \exp\left(\frac{-\alpha_0}{[cap(K)]^\frac{1}{n}}\right)\right]^\frac{p-1}{p}.
\end{aligned}
\]
Let $0< a,b,c <1$ be fixed. The following elementary inequality holds for $x>0$,
$$x^a \exp\left(\frac{-c}{x^b}\right) \leq C(\tau) x^{1+\tau},$$
where $C(\tau) = C(\tau,a,b,c)$ depends only on $\tau, a, b, c$. Thus, the desired inequality follows.
\end{proof}
Thanks to the lemma and \cite[Theorem~5.9]{ko05} we can solve the Monge-Amp\`ere equation
\[
u\in PSH(\Omega) \cap C^0(\bar\Omega), \quad
(dd^cu)^n = fd\mu, \quad u_{|_{\d\Omega}} =0.
\]
Moreover, the above lemma will enable us to have the stability estimate (Proposition~\ref{prop:stability}).
The next lemma is also a consequence of the generalized H\"older inequality which was proved in \cite[Corollary~2.14]{Ng17a}.
\begin{lem} \label{lem:lp-property}
Let $v \in PSH(\Omega) \cap C^0(\bar\Omega)$ be such that $v\geq u$ in $\Omega$ and $v= u$ near $\d\Omega$. Then, there exist uniform constants $C>0$ and $0< \tilde\alpha_3 <1$ such that
\[
\int_\Omega (v-u) f d\mu \leq C \|v-u\|_{L^1(d\mu)}^{\tilde\alpha_3}.
\]
\end{lem}
Next, we use the extendability assumption of $\varphi$ to get the one similar to Lemma~\ref{lem:boundary-holder} in the current setting. Namely, let $\tilde\Omega$ be a striclty pseudoconvex neighborhood of $\bar\Omega$ such that $\varphi \in PSH(\tilde\Omega)$ and H\"older continuous on the closure of $\tilde\Omega$. Thanks to the results in \cite{Ng17a} there exists $v \in PSH(\tilde\Omega)$ and H\"older continuous in $\tilde\Omega$ satisfying
\[\notag
(dd^c v)^n = {\bf 1}_{\Omega} f d \mu \quad \mbox{in } \tilde\Omega, \quad
v = 0 \quad \mbox{on } \d\tilde\Omega.
\]
Consider $h$ to be the maximal pluriharmonic extension into $\Omega$ of $(- v)_{|_{\d\Omega}}$ which is H\"older continuous on $\d\Omega$. So is $h$ on $\bar\Omega$. Then, by the comparison principle,
\[\notag
v + h \leq u \leq 0 \quad \mbox{ on } \bar\Omega.
\]
From this we easily deduce the desired estimate near boundary for $u$.
Now the rest of the proof goes exactly as in the proof of Theorem~B. Namely, the inequality \eqref{eq:holder-eq2} holds for the measure $fd\mu$, next use Lemma~\ref{lem:lp-property} and Theorem~\ref{thm:l1-l1} to get the inequality \eqref{eq:holder-eq3}. Then we get the H\"older continuity of $u$. Notice that the H\"older exponent is worse by a factor $\tilde\alpha_3$. Thus, $fd\mu\in \mathcal{M}(u,\Omega)$.
|
{
"timestamp": "2018-03-08T02:03:54",
"yymm": "1803",
"arxiv_id": "1803.02510",
"language": "en",
"url": "https://arxiv.org/abs/1803.02510"
}
|
\section{Introduction}{\label{intro}}
In recent years, dynamics of quantum-many body systems is one of the
most actively studied subjects in physics.
Process in which a system approach to an equilibrium is of fundamental
interests, and also evolution of system under a quench has attracted many
physicists.
Nowadays, ultra-cold atomic gas systems play a very important role
for the study on these subjects because of their versatility, controllability
and observability~\cite{ob}.
Theoretical ideas proposed to understand transient phenomena are to be
tested by experiments on ultra-cold atomic systems.
This is one of examples of so-called quantum simulations
\cite{Nori,Cirac,coldatom1,coldatom2}.
For the second-order thermal phase transition,
time-evolution of systems under a change in temperature
has been studied extensively so far.
From the view point of cosmology, Kibble \cite{kibble1,kibble2} claimed that
the phase transitions lead to disparate local choices of the broken
symmetry state and as a result, topological defects called cosmic strings are generated.
Later, Zurek \cite{zurek1,zurek2,zurek3} pointed out that a similar phenomenon
is realized in laboratory experiments on
the condensed matter systems like the superfluid (SF) of $^4$He.
After the above seminal works, many theoretical and experimental studies on
the Kibble-Zurek (KZ) mechanism have appeared~\cite{IJMPA}.
Concerning to experiments on Bose-condensed ultra-cold atomic gases,
the correlation length of the SF and the rate of topological defect formation
were measured and the KZ scaling hypothesis was examined~\cite{navon,Chomaz}.
To study dynamics of quantum many-body systems,
the parameters in the Hamiltonian are varied through a quantum phase transition (QPT),
i.e., the quantum
quench~\cite{Chomaz,dziarmaga,pol,Zoller,Fischer1,Fischer2,Be1,Be2,
sondhi,Zu1,Sonner,francuz,Zu2,SKHI},
and the system evolution is observed.
Experiments on this problem have been already done
using the various ultra-cold atomic gases~\cite{Chen,Braun,Anquez,clark,cui,bernien}.
Among them,
works in Refs.~\cite{Chen,Braun} questioned the applicability of the KZ scaling
theory to the QPT, whereas Refs.~\cite{Anquez,clark} concluded that
the observed results were in good agreement with the KZ scaling law.
In this paper, we focus on the two-dimensional (2D) Bose-Hubbard model
(BHM)~\cite{BHM1,BHM2},
which is a canonical model of the bosonic ultra-cold atomic gas systems
in an optical lattice.
In particular, we add nearest-neighbor (NN) repulsions between atoms.
Then, the resultant system
is described by an extended Bose-Hubbard model (EBHM).
As a result, a parameter region corresponding to the density wave (DW) appears
in the ground-state phase diagram, in addition to the Mott insulator and SF.
Near the half-filling, there exists a first-order phase transition between
the SF and DW \cite{firstO}.
We shall study the quench dynamics of the EBHM on passing across the SF and DW
phase boundary.
There are only a few works for the dynamical properties of quantum systems
at first-order phase transitions under a quench \cite{firstPT1,firstPT2,firstPT3},
and therefore
detailed study on that problem is desired.
This paper is organized as follows.
In Sec.~2, we introduce the EBHM and explain the Gutzwiller (GW) methods,
which are used in the present work.
In Sec.~3, quench dynamics of the first-order phase transition from
the DW to SF is studied.
Behavior of SF and DW orders are investigated by solving the Schr\"{o}dinger
equation by means of time-dependent GW (tGW) methods.
We focus on the order parameters, correlation length, vortex number, etc,
in particular, scaling laws of these quantities with respect to the quench time
$\tau_{\rm Q}$.
Contrary to the common expectation, we find that scaling laws hold
for the correlation length and vortex density.
In Sec.~4, we give a possible explanation of the observed results
from viewpoint of the SF bubble-nucleation process.
We employ a time-dependent Ginzburg-Landau theory and show that
scaling laws with small deviations from the KZ scaling hold in the vicinity
of a triple point in the phase diagram.
Applicability of the GW methods is also discussed there.
In Sec.~5, we study the time evolution of the system from the SF to DW
crossing the first-order phase transition.
We find that even for very slow quench, a genuine DW does not form if
we start the time evolution with the ground-state obtained by the static
GW methods.
Numerical result shows that a coexisting state of the SF and DW appears instead.
On the other hand, if SF states with small coherent phase fluctuations are
employed as an initial state, the system acquires a DW domain structure of large
size with thin domain walls.
Section 6 is devoted for conclusion.
In appendix, we show the results obtained for the hard-core Bose-Hubbard model,
in which the first-order phase transition between the DW and SF exists as in the
soft-core system of the present work.
We discuss the behavior of the correlation length and vortex density compared to
the soft-core case.
\section{Extended Bose-Hubbard model and slow quench}{\label{BHM}}
We consider the EBHM
whose Hamiltonian is given by \cite{Dutta},
\begin{eqnarray}
H_{\rm BH}&=&-J\sum_{\langle i,j \rangle}(a^\dagger_i a_j+\mbox{H.c.})
+{U \over 2}\sum_in_i(n_i-1) \nonumber\\
&&+V\sum_{\langle i,j \rangle}n_in_j-\mu\sum_in_i,
\label{HEBH}
\end{eqnarray}
where $\langle i, j\rangle$ denotes NN sites of a square lattice,
$a_i^\dagger \ (a_i)$ is the creation (annihilation) operator of boson at site $i$,
$n_i=a^\dagger_i a_i$, and $\mu$ is the chemical potential.
$J(>0)$ and $U(>0)$ are the hopping amplitude and the on-site repulsion, respectively.
We also add the NN repulsion with the coefficient $V$, which plays an important role
in the present work.
\begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=8cm]{fig1}
\end{center}
\vspace{-0.5cm}
\caption{Ground-state phase diagram of the extended Bose-Hubbard model
for $V=0.05$ obtained by the static GW methods.
There exist three phases, the density wave (DW), superfluid (SF) and
supersolid (SS).
Mean particle density $\rho\approx 1/2$.
}
\label{groundstate}
\end{figure}
\begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=7cm]{fig2}
\end{center}
\caption{Physical quantities in the DW and SF critical region in various system sizes;
the hopping $J$-term energy, amplitude of SF order ($|\Psi|$),
and mean density ($\rho$).
The obtained results show that the phase transition is of first order as dictated
by Landau-Ginzburg-Wilson paradigm.
Critical point is estimated as $J_c/U\approx 0.022$.}
\label{static}
\end{figure}
In this study, we are interested in cases near the half filling, i.e.,
$\rho\equiv {1 \over N_s}\sum_i\langle n_i\rangle\approx 1/2$,
where $N_s$ is the total number of the lattice sites, and
we take $N_s=64\times 64$ or $100\times 100$ for the practical calculation.
We set $U=1$ as the energy unit, and time $t$ is measured in the unit $\hbar/U$.
We investigated the system in Eq.(\ref{HEBH}) by using
the static GW approximation and show obtained ground-state phase diagram
in Fig.~\ref{groundstate} for $V/U=0.05$.
There exist three phases, i.e., the DW, SF and supersolid (SS) although the area of
the SS in the phase diagram is small for $V/U=0.05$.
We also show the system energy, particle density and amplitude of the SF order
parameter,
$|\Psi|\equiv {1 \over N_s}\sum_i|\Psi_i|$, where $\Psi_i\equiv \langle a_i\rangle$,
in Fig.~\ref{static} for $\mu/U=0.1$.
From the results in Fig.~\ref{static}, it is obvious that the system exists near the
half filling $\rho\approx 1/2$, and a first-order phase transition between the DW and SF
takes place at $J_c/U\simeq 0.022$ as a finite jump in $|\Psi|$ indicates.
The existence of the first-order phase transition is quite plausible as the DW and SF
have both the own long-range order.
In recent paper~\cite{KZIII}, we studied the EBHM for $V/U=0.375$ and near
the unit filling $\rho\approx 1$.
There exists a substantially finite region of the SS in addition to the DW and SF.
These three phases are separated by two second-order phase transitions.
This result is in agreement with the quantum Monte-Carlo study~\cite{QMC}.
In the following, we shall study dynamics of the system under ``slow quenchs".
To this end, we employ the tGW
methods~\cite{tGW1,tGW2,tGW3,tGW4,tGW5,tGW6,aoki}.
In the tGW approximation, the Hamiltonian of the EBHM in Eq.(\ref{HEBH})
is approximated by a single-site Hamiltonian $H_i$, which is derived
by introducing the expectation value $\Psi_i=\langle a_i \rangle$,
\begin{eqnarray}
&&H_{\rm GW}=\sum_i H_i, \nonumber\\
&&H_i=-J\sum_{j\in i{\rm NN}}(a^\dagger_i\Psi_j+\mbox{H.c.})
+{U \over 2}n_i(n_i-1) \nonumber\\
&&\hspace{1cm}+V\sum_{j\in i{\rm NN}}n_i\langle n_j\rangle-\mu n_i,
\label{HGW}
\end{eqnarray}
where $i{\rm NN}$ denotes the NN sites of site $i$, and Hartree-Fock type
approximation has been used for the hopping and NN repulsion.
To solve the quantum system $H_{\rm GW}$ in Eq.(\ref{HGW}),
we introduce GW wave function,
\begin{eqnarray}
|\Phi_{\rm GW}\rangle=\prod^{N_s}_i\Big(\sum^{n_c}_{n=0}
f^i_n(t)|n\rangle_i\Big),
\;\; \hat{n}_i|n\rangle_i=n|n\rangle_i,
\label{GW}
\end{eqnarray}
where $n_c$ is the maximum number of particle at each site, and we mostly take
$n_c=6$ in the present work.
Some quantities are calculated with $n_c=10$ to verify that $n_c=6$ is large enough
for the study of the half filling case.
See Fig.~\ref{SForder} and Fig.~\ref{correlationtime}.
In terms of $\{f^i_n(t)\}$, the order parameter of the SF is given as,
\begin{eqnarray}
\Psi_i=\langle a_i \rangle=\sum^{n_c}_{n=1}\sqrt{n}f^{i\ast}_{n-1}f^i_n,
\label{BEC}
\end{eqnarray}
and $\{f^i_n(t)\}$ are determined by solving the following Schr\"{o}dinger equation
for various initial states,
\begin{eqnarray} i\hbar \partial_t |\Phi_{\rm GW}\rangle
=H_{\rm GW}(t)|\Phi_{\rm GW}\rangle.
\label{SEq}
\end{eqnarray}
The time dependence of $H_{\rm GW}(t)$ in Eq.(\ref{SEq}) comes from the quench
$J\to J(t)$ with fixed $U$ and $V$ as explained in the following section.
Practically, the time evolution above is calculated by the
fourth-order Runge-Kutta method.
\begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=9cm]{fig3}
\includegraphics[width=7cm]{fig3a}
\end{center}
\caption{(Upper panel) Phase of the SF order parameter $\Psi_i$
for $\tau_{\rm Q}=300$ as a function of time.
(Middle panel) Amplitude of the SF order parameter $\Psi_i$
for $\tau_{\rm Q}=300$ as a function of time.
Relevant times $\hat{t}$ and $t_{\rm eq}$ are $\hat{t}\approx 70$ and
$t_{\rm eq}\approx 120$, respectively.
On the other hand, $t_{\rm ex}\approx 400$, at which the oscillation of
$|\Psi|$ terminates.
From $t_{\rm eq}$ to $t_{\rm ex}$, coarsening process of the phase of $\Psi_i$
takes place in large scales \cite{SKHI}.
(Lower panel) Calculation of $|\Psi|$ in the $n_c=10$ case is also shown.
It is in good agreement with that of $n_c=6$.
}
\label{SForder}
\end{figure}
\section{Dynamics of phase transition from density wave to superfluid}\label{DWtoSF}
We first study the dynamics from the DW to SF.
In this section, the hopping amplitude is varied as
\begin{eqnarray}
{J(t)-J_c \over J_c}\equiv \epsilon(t)={t \over \tau_{\rm Q}},
\label{protocol}
\end{eqnarray}
where $\tau_{\rm Q}$ is the quench time, which is a controllable parameter in
experiments.
We employed 10 samples as the initial state at $t=-\tau_{\rm Q}$
(i.e., $J(-\tQ)=0$), which have
the DW order with small local density fluctuations from the perfect DW.
Then, we solve Eq.(\ref{SEq}) to obtain $|\Phi_{\rm GW}\rangle$.
Physical quantities for which scaling lows are examined are obtained by
averaging over samples.
The linear quench in Eq.(\ref{protocol}) is terminated at $t=t_{\rm f}$ with
$J(t_{\rm f})=0.044(>J_c)$ in the numerical study.
Subsequent behavior of the system is also observed to see how the system
approaches to an equilibrium.
\begin{figure}[h]
\centering
\begin{center}
\includegraphics[width=7cm]{fig4}
\end{center}
\caption{$\Delta_{\rm DW}$, $\Delta^{\rm c}_{\rm DW}$
and $\Delta_{\rm SF}$ as a function of time
for $\tau_{\rm Q}=300$.
After passing the equilibrium critical point $J_c/U\simeq 0.022$, the both quantities
start to evolve with oscillations.
}
\label{Diff}
\end{figure}
We show the typical behavior of $|\Psi|$ as a function of $t$ in Fig.~\ref{SForder}
for $\tau_{\rm Q}=300$.
At $t=0$, the system crosses the critical point at $J_c/U\simeq 0.022$.
After crossing the critical point, $|\Psi|$ remains vanishingly small for some period,
and then it develops very rapidly.
After the rapid increase, $|\Psi|$ starts to fluctuate and coarsening of
the phase of the SF order parameter takes place there~\cite{SKHI}.
$\hat{t}$ in Fig.~\ref{SForder} is defined as $|\Psi(\hat{t})|=2|\Psi(0)|$,
and $t_{\rm eq}$ is the time at which the oscillation of $|\Psi|$ starts.
Similarly, $t_{\rm ex}$ is the time at which that oscillation terminates.
Similar behavior to the above was observed in the Mott to SF quench dynamics and
examined carefully~\cite{SKHI}.
Compared with the Mott to SF dynamics, the SF amplitude $|\Psi|$ is smaller,
e.g., for $t>t_{\rm eq}$, $|\Psi|\sim (0.8-0.9)$ in the Mott to SF transition, whereas
$|\Psi|\sim 0.5$ in the present case.
This difference simply comes from the difference of the mean particle density, i.e.,
$\rho \sim 1$ in the Mott to SF transition case.
The DW order parameters
$\Delta_{\rm DW}\equiv {1 \over N_s}\sum_i(-)^i\langle n_i\rangle$,
$\Delta^{\rm C}_{\rm DW}\equiv {1 \over 2N_s}\sum_{\langle i,j\rangle}
|\langle (n_i-n_j)\rangle|$,
and the even-odd deference of the SF order parameter defined as
$\Delta_{\rm SF}\equiv {1 \over 2N_s}\sum_{\langle i,j\rangle}
||\Psi_i|-|\Psi_j||$
are shown in Fig.~\ref{Diff}.
These quantities exhibit fluctuations as a function of time until $J\approx 0.045$.
These fluctuations are getting smaller, i.e., the system is approaching to a
homogeneous SF.
The system with other vales of $\tau_{\rm Q}$ exhibits a similar behavior, although
the reaction of the system starts at larger value of $J/U$ for
smaller value of the quench time $\tau_{\rm Q}$.
It is interesting to study the correlation length $\xi$ of the SF order parameter
and the vortex density $N_{\rm v}$ as a function of the quench time $\tau_{\rm Q}$.
These quantities are defined as follows;
\begin{eqnarray}
&&\langle \Psi_i^\ast \Psi_j\rangle \propto \exp (-|i-j|/\xi), \nonumber \\
&&N_{\rm v}=\sum_i|\Omega_i|, \nonumber \\
&&\Omega_i={1 \over 4}\Big[\sin (\theta_{i+\hat{x}}-\theta_i)
+\sin (\theta_{i+\hat{x}+\hat{y}}-\theta_{i+\hat{x}})
\nonumber \\
&&\hspace{1cm} -\sin (\theta_{i+\hat{x}+\hat{y}}-\theta_{i+\hat{y}})
-\sin (\theta_{i+\hat{y}}-\theta_{i})\Big],
\end{eqnarray}
where $\theta_i$ is the phase of $\Psi_i$ ($\Psi_i=|\Psi_i|e^{i\theta_i}$)
and $\hat{x} \ (\hat{y})$ is the unit vector in the $x \ (y)$ direction.
For continuous second-order phase transitions, the KZ hypothesis
predicts a scaling law such as $\xi\propto \tau_{\rm Q}^b$ and
$N_{\rm v}\propto \tau_{\rm Q}^{-d}$.
Recently, applicability of the above KZ scaling law for
{\em second-order quantum
phase transition} has been discussed for several quantum systems.
On the other hand for first-order phase transitions, it is commonly expected that
such a scaling law does not hold as the relaxation time
cannot be defined properly.
For a classical statistical model, another type of scaling law was proposed
for first-order phase transitions~\cite{firstPT1}.
It should be also noted that off-equilibrium dynamics of a quantum Ising ring
was investigated recently and finite-size scaling laws for first-order phase
transitions were proposed~\cite{firstPT11}.
There, off-equilibrium scaling variables were given in terms of an energy gap
and quench time, and physical quantities were obtained as a function of time.
\begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=7.5cm]{fig5}
\end{center}
\caption{Scaling laws observed for the correlation length $\xi$, vortex number
$N_{\rm v}$ at $t=\hat{t}$, and $\hat{t}$
with respect to $\tau_{\rm Q}$.
}
\label{scalinghatt}
\end{figure}
\begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=7.5cm]{fig6}
\end{center}
\caption{Scaling lows observed for the correlation length, vortex number at
$t=\teq$, and $\teq$ with respect to $\tQ$.
}
\label{scalingteq}
\end{figure}
To see if scaling law exists or not, we measured $\xi$ and $N_{\rm v}$ at
$t=\hat{t}$ and $t=\teq$.
In the original KZ hypothesis for continuous phase transitions~\cite{IJMPA},
$\hat{t}$ is the time at which
the system re-enters an equilibrium after the freezing (or impulse) period.
On the other hand, $\teq$ is the time at which a coarsening process
of the SF phase coherence starts~\cite{SKHI}.
We show the obtained results in Figs.~\ref{scalinghatt} and \ref{scalingteq}.
The results show that
at $t=\hat{t}$, both $\xi$ and $N_{\rm v}$ satisfy the scaling law with exponents
$b=0.25$ and $d=0.26$, respectively, and also $\hat{t}\propto \tQ^{0.45}$.
On the other hand at $t=\teq$, data at each $\tQ$ exhibits slightly large fluctuations
but scaling laws for the correlation length, $N_{\rm v}$ and $\teq$ seem to exist
for $\tQ>20$.
The above results indicate that besides the KZ mechanism, there exists
another mechanism to generate the scaling laws.
Possible explanation is given in Sec.~4.
It should be noted that
after passing the critical point, $\Delta_{\rm DW}$ and $\Delta_{\rm SF}$
have even-odd site fluctuations, and therefore, the system is not homogeneous.
We think that because of this inhomogeneity, the critical exponents of $\xi$ and
$N_{\rm v}$ at $t=\hat{t}$ do not satisfy the expected relation such as $b=d/2$.
On the other hand at $t=\teq$, the system is rather homogeneous, and therefore
$b\sim d/2$.
In appendix, we consider the hard-core version of the EBHM and show
the calculations of the scaling laws with respect to $\tQ$ in Fig.~\ref{HCtQ}.
There, $\xi(\hat{t})$ and $N_{\rm v}(\hat{t})$ fluctuate rather strongly.
This behavior comes from the fact that fluctuations of the particle number
at each site is smaller compared with the soft-core case, and as a result,
the stability of the phase degrees of freedom of the SF order parameter
is weakened.
\begin{figure}[h]
\centering
\begin{center}
\includegraphics[width=6.5cm]{fig7}
\includegraphics[width=6.5cm]{fig7b} \vspace{0.5cm} \\
\includegraphics[width=5cm]{fig7c}
\includegraphics[width=5cm]{fig7d} \\
\includegraphics[width=5cm]{fig7e}
\includegraphics[width=5cm]{fig7f}
\end{center}
\caption{(Upper left panel) For a typical initial state at $t=-\tQ$,
the correlation length is calculated as a function of time.
After passing $t=\teq$, increase of the correlation length becomes weak.
(Upper right panel) We also show the results for the $n_c=10$ case.
(Lower panels) The correlation functions
$G(r)={1 \over 2N_s}\sum_i\langle a_i^\dagger a_{i+r}\rangle$ exhibit
very close behavior in the $n_c=6$ and $n_c=10$ cases.
}
\label{correlationtime}
\end{figure}
We terminate the linear quench at $t_{\rm f}=\tQ=300$.
After $t_{\rm f}$, the system approaches to an equilibrium as the results
in Figs.~\ref{SForder} and \ref{Diff} indicate.
It is interesting to see how the correlation length of the SF develops.
As the results in Fig.~\ref{correlationtime} show,
the correlation length increases after passing the critical point as it is expected.
However, its increase gets weak at $t\sim\teq$, and it
saturates at $t\sim 500$ and keeps a finite value.
To study the resultant phase, we measured $N_{\rm v}$ and found
that there exist no vortices at $t>500$.
One may expect that the system settles in a {\em finite-temperature} ($T$) SF phase
for sufficiently large $t$ with an effective $T$, $T_{\rm eff}$.
The finite-$T$ SF in 2D has a quasi-long range order and the correlation length
diverges, i.e., the Kosterlitz-Thouless (KT) phase.
The above result seems to indicate that some other state is realized in the final
stage of the present process.
However, the system behavior may strongly depend on the average particle
density $\rho$.
Further study is needed to clarify this interesting problem.
In fact, we studied this problem in the case of the mean particle density
$\rho \approx 1$ and $V/U=0.375$~\cite{KZIII}.
In the quench process such as the DW $\to$ SS $\to$ SF, the correlation length
continues to increase even for large $t$.
This result seems to indicate that a KT phase of the SF is realized there.
\section{Consideration by the Ginzburg-Landau theory}
In the previous section, we showed that the results obtained by the GW methods
indicate the scaling laws of $\hat{t}$, $\teq$ and the correlation length with
respect to the quench time $\tQ$.
It is interesting and also important to study the origin of these observations
from more universal and intuitive point of view.
To this end, the Ginzburg-Landau (GL) theory is quite useful.
In fact very recently, it was pointed out that the GL theory can drive the
scaling laws for the second-order phase transition by analytical transformation of
the associated equations of motion~\cite{Niko}.
In this section, we first review the above derivation of the scaling laws for the ordinary
second-order phase transition, and then give an intuitive picture of the scaling laws
by using a classical solution representing decay of the false vacuum.
Then, we extend the methods to the present case involving the SF and DW
order parameters.
This consideration also gives an insight about the physical meaning and limitation of the
GW methods.
\subsection{Second-order phase transition}
Let us start with the stochastic GL equation for a complex order parameter
(condensate) $\phi(\vec{r}, t)$,
\be
{\partial \phi \over \partial t}=\nabla^2_r \phi-{\epsilon(t) \over 2}\phi
-{1\over 2}|\phi|^2\phi+\Theta(\vec{r},t),
\label{GLeq}
\ee
where $\Theta(\vec{r},t)$ represents the uncorrelated white-noise variables with
$\langle \Theta(\vec{r},t)\Theta(\vec{r}',t')\rangle=T \delta(\vec{r}-\vec{r}')
\delta(t-t')$ and $T$ is the temperature of particles ensemble not participating
the Bose-Einstein condensate.
As in Ref.~\cite{Niko}, we consider the critical parameter $\epsilon(t)$ such as
\be
\epsilon(t)=-\Big|{t\over \tQ}\Big|^\lambda \mbox{sgn} (t),
\label{epsilont}
\ee
where $\lambda$ is a parameter for the quench protocol.
Then, let us change variables as follows,
\be
\eta=\alpha t, \;\; \vec{\ell}=(\alpha)^{1/2}\vec{r}, \;\;
\tilde{\phi}=\phi/(\alpha)^{1/2},
\label{change}
\ee
where $\alpha=\tQ^{-\l/(\l+1)}$.
In terms of the new variables, the equation of motion (\ref{GLeq}) leads to
\be
{\partial \tilde{\phi} \over \partial \eta}=\nabla^2_\ell \tilde{\phi}
-{1 \over 2}|\eta|^\l \mbox{sgn}(\eta)\tilde{\phi}
-{1 \over 2}|\tilde{\phi}|^2\tilde{\phi}+{1 \over \alpha}\Theta(\vec{\ell}, \eta).
\label{GLeq1}
\ee
In Eq.(\ref{GLeq1}), the $\tQ$-dependence in Eq.(\ref{epsilont}) disappears except
the last white-noise term.
From the above fact, it is concluded in Ref.~\cite{Niko} that
the $\tQ$-dependence of $\hat{t}$ and $\xi(\hat{t})$ are expected to follow
the transformation in Eq.(\ref{change}), and they are given as follows
for sufficiently low $T$,
\be
\hat{t} \propto \alpha^{-1}=\tQ^{\l/(\l+1)}, \;\;\;
\xi(\hat{t}) \propto \alpha^{-1/2}=\tQ^{\l/2(\l+1)}.
\label{scalGL}
\ee
For the linear quench $\l=1$, $\hat{t}\propto \tQ^{1/2}$ and
$\xi(\hat{t})\propto \tQ^{1/4}$.
The above estimations agree with those of the KZ scaling with the mean-field
exponents such as $\nu=1/2$ and $z=2$.
As we show, the above scaling transformation gives an intuitive picture that derives
the KZ scaling law.
To this end, we put $\Theta(\vec{r},t)=0$ in Eq.(\ref{GLeq}) and consider
a static potential such as $\epsilon(t)=-\epsilon_0<0$.
In this case, the static ground state is given as $\phi=\sqrt{\epsilon_0}$.
To study the {\em sudden quench dynamics}, we consider the decay of the false vacuum
$\phi=0$ to the true ground state $\phi=\sqrt{\epsilon_0}$.
In 1D case, a classical solution representing the decay is obtained as
follows~\cite{firstPT3},
\be
\phi(t,x)=\sqrt{\epsilon_0}
\Big[1+\exp \Big({\sqrt{\epsilon_0} \over 2}(x-v_0t)\Big)\Big]^{-1},
\label{solution1}
\ee
where $v_0={3\sqrt{\epsilon_0} \over 2}$, and $\phi(t,-\infty)=\sqrt{\epsilon_0}$
and $\phi(t,\infty)=0$.
The solution Eq.(\ref{solution1}) obviously represents the situation in which the true
vacuum $\phi=\sqrt{\epsilon_0}$ born in the false vacuum expands with the speed $v_0$.
Let us consider the ``slow" quench dynamics and study bubble nucleation-evolution
process in the SF formation.
We expect that this process corresponds to the numerical studies in the previous sections.
We have to find the solution to Eq.(\ref{GLeq}) that describes a single SF-bubble
evolution in the false vacuum $\phi=0$, but we cannot find an exact solution.
However,
the above solution in Eq.(\ref{solution1}) suggests that a spherically-symmetric
solution in higher dimensions and also for the time-dependent $\epsilon(t)$
has the following form for $\epsilon(t)<0$,\footnote{Solution in Eq.(\ref{solution2})
might be regarded a solution in the slow quench limit, in which the time-derivative
of $\epsilon(t)$ is small. However, it also satisfies the scaling transformation with
$\epsilon(t)$ in Eq.(\ref{epsilont}). See the discussion below.}
\be
\phi_s(\vec{r},t)=\sqrt{|\epsilon(t)|}F\Big(\sqrt{|\epsilon(t)}|(r-v_tt)\Big),
\;\;\; r>0,
\label{solution2}
\ee
where $v_t=C_0\sqrt{|\epsilon(t)|}$ with a certain constant $C_0$, and
$F(x)$ is a decreasing function such as $F(-\infty)=1$ and $F(\infty)=0$.
In fact, we can show that the function $\phi_s(\vec{r},t)$ in
Eq.(\ref{solution2}) satisfies the scaling transformation in Eq.(\ref{change})
for the time-dependent $\epsilon(t)$ in Eq.(\ref{epsilont}), i.e.,
\be
\tilde{\phi}_s(\eta,\vec{\ell})=\phi_s/(\alpha)^{1/2}
=\sqrt{\eta^\l}F\Big(\sqrt{\eta^\l}(\ell-v(\eta)\eta)\Big),
\;\; v(\eta)=C_0\sqrt{\eta^\l},
\label{solution3}
\ee
does not depend on $\tQ$.
As far as the above picture holds in the time evolution of the system,
Eq.(\ref{solution3}) implies that typical events and phenomena are
observed similarly in systems with various $\tQ$'s, and corresponding times have
$\tQ$-dependence such as $\tQ^{\l/(\l+1)}$.
For example, we numerically obtained $\hat{t}$ and $\teq$ for various $\tQ$'s in
Sec.~3 by starting with qualitatively the same initial states.
These values are related to $\tQ$-independent $\hat{\eta}$
and $\eta_{\rm eq}$ that are obtained by the rescaled picture from
Eq.(\ref{solution3}), i.e.,
$\hat{t}$ and $\teq$ in the $\tQ$-system are given by
$\hat{t}=\tQ^{\l/(\l+1)}\hat{\eta}$ and
$\teq=\tQ^{\l/(\l+1)}\eta_{\rm eq}$.\footnote{Rough estimation of $\hat{\eta}$
and $\eta_{\rm eq}$ are the followings. As $\hat{t}$ is determined by the condition
such as $|\Psi(\hat{t})|=2|\Psi(0)|$,
$\sqrt{\hat{\eta}^\l}(v(\hat{\eta})\hat{\eta})^2=$constant for the 2D case.
On the other hand, as $\teq$ is the time at which the overlap of SF bubbles starts
\cite{SKHI}, $v(\eta_{\rm eq})\eta_{\rm eq}=$constant.
Simulation for various $\l$'s is a future work.}
Furthermore, a typical linear size of the bubble at $t$, i.e., the correlation length at $t$,
$\xi(t)$, is given as
\be
\xi(t)=\int_0^{t} v_t dt\propto {1 \over \tQ^{\l/2}} \ t^{\l/2+1},
\ee
and therefore, $\xi(\hat{t})\propto (\tQ)^{\l/2(\l+1)}\hat{\eta}^{(\l+2)/2}$ and
$\xi(\teq)\propto (\tQ)^{\l/2(\l+1)}\eta_{\rm eq}^{(\l+2)/2}$.
{\em After $\teq$, the merging and coarsening process of SF bubbles takes
place \cite{SKHI}, and therefore the above picture and also the resultant
scaling laws do not hold anymore.}
\subsection{GL theory, GW methods and quantum Monte-Carlo simulation}
Here, it is suitable to comment on the GW approximation.
The GL theory and also the Gross-Pitaevskii (GP) equation consider only
the mean field and totally ignore fluctuations around it.
On the other hand in the GW approximation, we focus on a wave function
of site factorization, and wave function at each site is obtained by
solving the site-factorized Hamiltonian in which
the NN operators are replaced with their expectation values \cite{SKHI}.
The uncertainty relation between the particle number and phase at each site is faithfully
taken into account although an equation of motion similar to the GL (GP) equation
is derived by the GW methods.
This is an advantage of the GW approximation over the GL and GP theories.
As more reliable methods, let us consider the quantum Monte-Carlo (MC) simulations
of the coherent-state path integral in the imaginary-time formalism.
In this MC simulations, quantum operators are reduced into classical variables
and the quantum superpositions are treated by the fluctuations in the imaginary-time
direction.
Large number of configurations are generated by the MC updates and
physical quantities are calculated by averaging them over generated
configurations.
In the Metropolis MC algorithm, the local updates are applied to variables at each site
by calculation a local energy around that site.
In the vicinity of a phase transition point, a large number of configurations contribute
equally, and calculations by large CPU times are required in order to take into
account all relevant configurations.
On the other hand away from the critical point, the number of important
configurations is not so large.
From the viewpoint of the MC simulation with the local update, we can get
an interesting insight into the GW approximation.
That is, let us imagine that we perform a GW calculation for a system with size
$10^4\times 10^4$.
When we calculate expectation values, we divide the $10^4\times 10^4$ system
into $10^4$ number of $10^2\times 10^2$ subsystems.
We obtain the expectation values by averaging values calculated
in each subsystem.
Compared with the path-integral MC simulation, this method is more reliable
as the uncertainty relation is faithfully respected.
[In the path-integral MC simulation, this relates to the problem
how accurately effects of the Berry phase are taken into account.
See for example, Ref.~\cite{KSI}.]
However in the vicinity of the phase transition, $10^4$ configurations are not
sufficient to obtain physical quantities closely related to the singularities of
the phase transition.
The above consideration suggests that the GW methods are a fairly good
approximation for calculating physical quantities that are finite even for
the critical regime, e.g., finite order parameters.
In other words, the estimation of the critical exponents by the GW methods
is not reliable even for using very large systems.
The above consideration may over estimate the reliability and applicability
of the GW methods, but it explains why the GW methods often succeed
in obtaining correct results such as the phase diagrams, etc.
We expect that the GW methods also works for the correlation functions
as far as the correlation length is finite as the quantum MC simulations do,
although at present there are no ways to verify it in the quench dynamics.
\subsection{First-order phase transition in vicinity of triple point}
As the phase diagram in Fig.~\ref{groundstate} shows, the present first-order
phase transition is located in the vicinity of the triple point of the DW,
SF and SS.
The GL theory for the quench dynamics in Sec.~4.1 can be applied to this case
with some modification.
Besides the SF order parameter, we introduce a coarse-grained real DW order
parameter, $D(\vec{r},t) [\sim (-)^in_i]$.
GL equations are given as
\be
&&{\partial \phi \over \partial t}=\nabla^2_r \phi-\epsilon(t)\phi -g_1|\phi|^2\phi
-g_3D^2\phi, \label{GLeq2} \\
&&{\partial D \over \partial t}=\nabla_r^2D+m(t)D-g_2D^3-2g_3D|\phi|^2,
\label{GLeq3}
\ee
where the positive parameters $g_1, g_2$ and $g_3$ are phenomenological ones,
which are to be determined by the parameters $U$ and $V$.
The positivity of $g_3$ comes from the fact that the SF and DW are competing
orders in the original EBHM.
On the other hand, $\epsilon(t)$ and $m(t)$ are parameters that are determined
by $J(t)$, $U$ and $V$.
In the quench from the DW to SF, both $\epsilon(t)$ and $m(t)$ are decreasing
functions of $t$.
Let us consider a slow quench, and denote the phase transition time from
the DW to SF by $t_c$.
At $t=t_c-\delta \ (\delta \to +0)$, the system is in the DW and then,
$\epsilon(t_c)+g_3D^2(t_c)=\epsilon(t_c)+{g_3 \over g_2}m(t_c)>0$, $\phi=0$
and $D^2={m(t_c) \over g_2}$.
On the other hand at $t=t_c+\delta \ (\delta \to +0)$, the system is in the SF,
and $m(t_c)-2g_3|\phi|^2=m(t_c)+2{g_3 \over g_1}\epsilon(t_c)<0$,
$|\phi|^2=-{\epsilon(t_c) \over g_1}$ and $D=0$.
From the above equations, we obtain the constraint for the occurrence of
the direct DW to SF
transition such as $2g^2_3>g_1g_2$, and $\epsilon(t_c)<0, \ m(t_c)>0$.
The critical time, $t_c$, is determined by the condition that the potential
energy ${\cal V}=\epsilon(t)|\phi|^2+{g_1 \over 2}|\phi|^4+g_3D^2|\phi|^2
-m(t){D^2 \over 2}+{g_2 \over 4}D^4$ has the same value in the DW and
SF states at $t=t_c$.
This condition gives $\epsilon^2(t_c)={g_1 \over 2g_2}m^2(t_c)$.
On the other hand,
the triple point is realized by $\epsilon(t_c)=m(t_c)=0$ or $2g^2_3=g_1g_2$.
Let us focus on the SF for $t\geq t_c$.
In this case, $D=0$ and we only consider the GL equation in Eq.(\ref{GLeq2})
with $D=0$.
We assume the same protocol with Eq.(\ref{epsilont}) and
then, the transformation in Eq.(\ref{scalGL}) can be applied as in the case of
the second-order phase transition.
Correlation length at time $t$ is estimated as
\be
\xi(t)=\int_{t_c}^t v_t dt={1 \over \tQ^{\l/2}}(t^{\l/2+1}-t_c^{\l/2+1}).
\label{Lt2}
\ee
The second term on the RHS in Eq.(\ref{Lt2}) comes from the finite jump
of $\phi$ at the critical point and indicates the deviation from the genuine
second-order phase transition.
However for sufficiently small $t_c$ such as $t_c \ll \hat{t}, \ \teq$,
the correlation length satisfies almost the same scaling law with the KZ one.
\section{Dynamics of phase transition from superfluid to density wave}\label{SFtoDW}
This section considers the temporal evolution of the system under a quench
from the SF to DW.
We found that behaviors of the system strongly depend on the initial state.
We shall show the results in the following two subsections.
\subsection{Evolution from the GW ground-state of SF}
\begin{figure}[h]
\centering
\begin{center}
\includegraphics[width=9cm]{fig8}
\end{center}
\caption{Transition from SF to DW with $J(t_{\rm f})=0$, Case A.
The system passes through the critical point $J_c$ at $t=0$.
Even for $t>0$, both the SF amplitude and DW order parameter do not
exhibit the typical behaviors of the DW.
}
\label{SFtoDW1}
\end{figure}
\begin{figure}[h]
\centering
\begin{center}
\includegraphics[width=8.5cm]{fig9}
\end{center}
\caption{Snapshots of SF local density (amplitude), particle density,
SF phase degrees of freedom, and vortex density at $t=\tQ$ ($J/U=0$).
Global coherence of $\Psi_i$ does not exist, and finite-size domains of the DW
partially form as indicated by the red circles.
}
\label{SnapSFtoDW2}
\end{figure}
Let us consider the dynamics of the phase transition from the SF to DW.
The hopping amplitude is varied as follows in the linear quench,
\be
{J_c-J(t) \over J_c}\equiv -\epsilon(t)={t \over \tau_{\rm Q}}.
\label{protocol2}
\ee
In order to clarify the quench dynamics, we shall consider three cases
in this subsection.
In the first case, Case A, we start with configurations at $J(t=-\tQ)=2J_c=0.044$
and terminate the quench at $t=\tQ$ with $J(\tQ)=0$.
We employ the tGW methods to study the system.
In Case A, as well as Cases B and C in the later study in this subsection,
{\em the initial state is the lowest-energy state obtained by
the static GW methods}.
The obtained
results of $|\Psi|$, $\Delta_{\rm DW}$ and $\Delta_{\rm SF}$ are shown in
Fig.~\ref{SFtoDW1} for $\tQ=300$.
$|\Psi|$ exhibits fluctuations in the SF for $t<0$,
whereas it becomes stable in the region $J<J_c$ (i.e., $t>0)$.
This behavior comes from the fact that $\Psi_i$ has a phase coherence in the SF,
which induces amplitude fluctuations, as the amplitude and phase of the SF order
parameter are quantum conjugate variables with each other.
On the other hand in the would-be DW region for $t>0$, the phase coherence is lost,
and then the SF amplitude is stable.
The DW order parameter $\Delta_{\rm DW}$ does not have a stable finite value even
after passing through the critical point at $t=0$.
These results indicate that some kind of domain structure forms there, i.e.,
small DW domains may coexist with local SF regions.
Calculations of the amplitude of $\Psi_i$ and the particle density at $t=\tQ$
are shown in Fig.~\ref{SnapSFtoDW2}.
As expected above, DW domains and regions with finite SF amplitude coexist
without overlapping with each other.
In Case A, the quench stops with $J(\tQ)=0$, and therefore no movement
of particles occurs after the quench, and the {\em particle-density} snapshot
in Fig.~\ref{SnapSFtoDW2} continues to describe the states for $t>\tQ$.
Similarly, we expect that the coherence of the phase of $\Psi_i$ is
destroyed at $t=\tQ$ because $J(\tQ)=0$ and also $\tQ=300$ is a slow quench.
See Fig.~\ref{SnapSFtoDW2}.
In order to verify the expected behavior of $\Psi_i$, we measured the vortex
density as a function of time.
At $t=\tQ$, $N_{\rm v}\sim 300$ is sufficiently large.
In summary,
in Case A with $\tQ=300$, an inhomogeneous state with local DW and SF domains
forms after quench.
SF order parameter gradually loses its phase coherence during the slow quench.
On the other hand for cases of smaller $\tQ=100$ and 50, the SF order parameter
$\Psi_i$ is finite even at $t=\tQ$, and it varies after $t=\tQ$.
The {\em phase of $\Psi_i$} gradually loses its long-range coherence by the existence
of the repulsive interactions for $t>\tQ$.
\begin{figure}[t]
\centering
\begin{center}
\includegraphics[width=8cm]{fig10a}
\includegraphics[width=6cm]{fig10b}
\end{center}
\caption{Transition from SF to DW with $J(t_{\rm f})=0.01$, Case B.
Genuine global DW order does not form.
After passing $J_c$ at $t=0$, $N_{\rm v}$ keeps a small value for a while, and
the SF order survives there.
After passing $t_{\rm f}=0.55\tQ=27.5$, the total energy of the system keeps
a constant value
as the system is and isolated one.
}
\label{SFtoDW2}
\end{figure}
\begin{figure}[h]
\centering
\begin{center}
\includegraphics[width=8.5cm]{fig11}
\end{center}
\caption{Transition from SF to DW with $J(t_{\rm f})=0.02$, Case C.
Increase of $N_{\rm v}$ is slow compared to the cases $J(t_{\rm f})=0$
and $J(t_{\rm f})=0.01$.
SF amplitude $|\Psi|$ also keeps a finite value even for $t\to$large.
However, $N_{\rm v}$ increases smoothly, and therefore, the supercooled
state formed in the quench is not a meta-stable state.
}
\label{SFtoDW3}
\end{figure}
\begin{figure*}[t]
\centering
\begin{center}
\includegraphics[width=6cm]{fig12a} \hspace{0.5cm}
\includegraphics[width=5cm]{fig12b} \vspace {1cm} \\
\includegraphics[width=5cm]{fig12c} \hspace{0.6cm}
\includegraphics[width=5cm]{fig12d}
\end{center}
\caption{(Upper-left) Vortex number as a function of time.
Each point denotes the following time;
(a) $t=-50$, (b) $t=0$, (c) $t=150$, and (d) $t=450$.
(Upper-right) Particle density snapshot in Case C.
At t=0, a typical DW domain appears as indicated in the red circle.
(Lower-left) SF density snapshot in Case C.
(Lower-right) Snapshot of phase degrees of freedom of SF order parameter in Case C.
}
\label{CaseCsnap}
\end{figure*}
As Case B, we consider a quench such as $J(-\tQ)=0.044$ and $J(0)=J_c=0.022$
as before but it terminates at $t=t_{\rm f}$ with $J(t_{\rm f})=0.01$, i.e.,
$t_{\rm f}=0.55\tQ$ (see Fig.~\ref{SFtoDW2}).
We also study how the system evolves after $t_{\rm f}$.
Observed quantities are shown in Fig.~\ref{SFtoDW2} for $\tQ=50$.
The DW order parameter $\Delta_{\rm DW}$ develops but its value fluctuates
in rather long period after passing $J_c$ as in Case A.
The total energy slightly decreases until $t_{\rm f}$, and the kinetic and
on-site energies exhibit fluctuating behavior for $t<t_{\rm f}$ although
the NN interaction energy is rather stable.
This behavior mostly originates from the local density fluctuations,
and the stability of the NN interaction comes from the cancellation
mechanism between NN sites $j\in i{\rm NN}$.
After passing the critical point at $t=0$, the $\Psi_i$ keeps a coherent
SF order for some period as the calculation of the vortex number $N_{\rm v}$ indicates.
At $t\approx 100$, it starts to lose the coherence and the SF is destroyed
as the increase in $N_{\rm v}$ indicates.
The state at $t\sim t_{\rm f}$ is a {\em supercooled state}, and
a coexisting phase of local domains of the DW and SF is realized there.
The observed phenomenon after $t>t_{\rm f}$, therefore, has very similar nature to
the {\em glass transition}, in which
the phase coherence and superfluidity are getting lost as the supercooled state
evolves after the quench.
We call it {\em quantum glass transition (QGT)} as the hopping amplitude $J$, instead of
temperature, is the controlled physical quantity and the relevant transition
is quantum mechanical one instead of thermal one.
We have verified that similar phenomenon is observed for other values of $\tQ$,
e.g., $\tQ=20$ and $200$.
In both Case A and Case B, the above mentioned QGT is observed {\em dynamically
as a nonequilibrium phenomenon}, i.e., the QGT point is passed through as the system
evolves.
Therefore as the next problem,
it is interesting to see whether there exits a genuine glass transition point,
$J_{\rm g}(<J_c)$.
Below $J_{\rm g}$, the supercooled state is meta-stable or at least has a long life time,
and the SF survives without losing its phase coherence.
For Cases A and B, $J<J_{\rm g}$.
Then as Case C, we studied the quench whose finial point is
$J(t_{\rm f})=0.02$, i.e., very close to the equilibrium critical point.
Obtained order parameter $|\Psi|$ and vortex number $N_{\rm v}$ are shown
in Fig.~\ref{SFtoDW3} for $\tQ=50$, and time evolution of the particle density,
amplitude and phase of $\Psi_i$ are shown in Fig.~\ref{CaseCsnap}.
After passing the critical point $J=J_c$ at $t=0$, the domain formation of
the DW starts as shown by the particle-density snapshot in Fig.~\ref{CaseCsnap},
whereas the long-range coherence of the SF order parameter
$\Psi_i$ exists there.
Compared with the cases of $J(t_{\rm f})=0$ and $J(t_{\rm f})=0.01$,
the destruction of SF and formation of the DW region are slow, but
after $t>450$, the quantum glass state forms.
Local DW domains develop but also empty regions (voids) form.
SF order loses a long-range coherence.
This result indicates that $J_{\rm g}$ cannot be observed.
Similar results are obtained for the case of $\tQ=20$ and $\tQ=200$.
\subsection{Evolution from SF state with small phase fluctuations}
\begin{figure*}[t]
\centering
\begin{center}
\includegraphics[width=6cm]{fig13a}
\includegraphics[width=12cm]{fig13b}
\end{center}
\caption{(First) SF order parameter as a function of time.
Each point denotes the following time;
(a) $t=-300$, (b) $t=115$, and (c) $t=300$.
(Second) Particle density snapshot in Case D.
(Third) SF density snapshot in Case D.
(Lowest) Snapshot of phase of SF order parameter in Case D.
At $t=300$, a large scale DW domain structure with thin domain walls forms.
Coherence of SF phase is lost there.
}
\label{CaseDsnap}
\end{figure*}
In Sec.~5.1, we studied dynamical evolution of the system from the SF to DW.
In that study, the initial state is set to the ground-state obtained by
the equilibrium GW methods.
It is interesting to see how the dynamical phenomena depend on the initial state
as we are considering the first-order phase transition.
In order to study this problem, we consider a SF state that is uniform and
has almost perfect phase coherence with very small random fluctuations.
For the practical calculation, we employ an initial state GW wave function in Eq.(\ref{GW})
corresponding to $\Psi_j=\sqrt{\rho}e^{i\delta\theta_j}$
with random numbers $\{\delta\theta_j\}$ from a uniform distribution
$[-0.005,0.005]\times\pi$.
The other condition is the same with the Case A, (please refer to the left panel
in Fig.~\ref{SFtoDW1}).
We call the present study Case D.
We investigated the time evolution of the system by the tGW methods, and
obtained results are shown in Fig.~\ref{CaseDsnap}.
Interestingly enough, the system behavior after passing across the critical point
$J_c$ is substantially different from that in Cases A.
The SF order parameter $|\Psi|$ decreases a finite amount at $t\sim 100$,
and the density difference at even-odd sublattice increases there.
On the other hand, the vortex number starts to increase rapidly at $t\sim 150$.
Snapshots of the particle density, SF amplitude and SF phase are shown in
Fig.~\ref{CaseDsnap}.
Contrary to Case A, the DW pattern starts to form at $t\sim 115$ and
it develops to the whole system at $t\sim 300$, even though there
exist domain walls.
It should be noticed that
a similar behavior was observed for the classical first-order phase
transition in Ref.~\cite{firstPT1}.
On the other hand, the SF phase coherence exists at $t<115$, whereas
it is destroyed at $t\sim 300$.
The initial state of Case D has higher energy than that of Case A.
The above numerical result indicates that there exists an energy barrier
between the supercooled SF state and the genuine DW, and some amount
of energy is need to overcome the barrier.
Furthermore, the above result also indicates that the existence of the SF
phase coherence in large spatial regions prevents the formation of large size DW domains.
In other words, local fluctuations of the superfluidity coherence substantially develops
under a quench even if they are initially tiny, and the DW is preferred as a result.
We expect that the above interesting phenomenon is observed by experiments on
ultra-cold atomic gases in the near future.
\section{Conclusion}\label{conclusion}
In this work, we studied dynamical behavior of the EBHM in 2D by using
the tGW methods.
In the ground-state phase diagram, there are three phases, the SF, DW, and SS.
In particular, we are interested in the first-order phase transition between
the SF and DW under a slow quench of the hopping amplitude.
First, we investigated the dynamics of the EBHM in the transition from the DW to SF.
In the practical calculation, we fix the strength of the one-site and NN repulsions,
and vary the hopping parameter $J$.
After passing through the equilibrium critical point $J_c$, the amplitude of
the SF order parameter, $|\Psi|$, remains vanishingly small until $t=\hat{t}$.
After $\hat{t}$, it develops quite rapidly.
Therefore, $\hat{t}$ has the meaning of the reentry time to the adiabatic region
passing from the frozen regime although the present phase transition is of first order.
At $\teq(>\hat{t})$, $|\Psi|$ stars to oscillate until $t=t_{\rm ex}$.
This behavior is quite similar to that in the second-order phase
transition from the Mott insulator to SF, which we observed in the previous
work~\cite{SKHI}.
Then we are interested in whether some kind of scaling laws between
the correlation length/vortex number and the quench time $\tQ$ exist.
Our numerical study shows that the scaling laws such as $\xi\propto \tQ^b$
and $N_{\rm v}\propto \tQ^{-d}$ in fact hold.
This result is against to the simple expectation that such scaling laws do not exist
in the first-order phase transitions because the simple relaxation-time picture
and the concept of the (dynamical) critical exponents are not applicable.
From this result, we think that there exists another mechanism, besides the KZ
mechanism, to generate the scaling laws.
As a possible explanation, we studied the present system by using the GL-type theory
suggested by Ref.~\cite{Niko}.
This consideration indicates that the observed scaling laws come from the fact
that the present phase transition point is located in the vicinity of the triple point.
In the second half, we studied the dynamics of the EBHM in the quench of
the opposite direction, i.e., from the SF to DW.
We focused on how the final value of the hopping amplitude of the quench,
$J(t_{\rm f})$, influences the dynamics of the system during and after the quench.
Our numerical study showed very interesting phenomena.
First, in the case for the GW ground-state as the initial state, the
genuine DW state does not form even for very slow quench $\tQ=300$.
Instead, the coexisting state composed of DW and SF domains appears and
spatially inhomogeneous structure of that state is stable after the quench.
In cases with $J(t_{\rm f})>0$, the SF order parameter has a phase coherence
at $t=t_{\rm f}$, and after the quench, the SF order is getting weak by the
generation of vortices.
Obviously, the quench produces a {\em supercooled state} in which the domain
structure of the DW and SF local (i.e., short-range) coherent state forms.
These two domains have an off-set structure with each other.
Then, after termination of the quench, the SF is destroyed.
This phenomenon is a reminiscent of the glass transition in classical
polymers etc, and we call the observed phenomenon quantum glass transition.
On the other hand, if we start with the uniform SF state with tiny fluctuations in the
phase of the SF order, the system evolves into the DW with thin domain walls.
In the phase diagram of the EBHM near the half-filling shown in
Fig.~\ref{groundstate}, there is the SS phase, and the SS has two
phase boundaries with the DW and SF.
In the case of the mean particle density $\rho=1$ and strong NN repulsion,
the region of the SS is large and two second-order phase transitions are
observed clearly from the SS to the DW and SF, respectively.
It is interesting to study the dynamics in that region, that is, how the system
develops crossing through two second-order phase boundaries.
Some related problem was recently studied in classical systems, and
a modified KZ scaling law was proposed~\cite{hybrid}.
We studied the above problem in the EBHM by using tGW methods,
and results are published in Ref.~\cite{KZIII}.
|
{
"timestamp": "2018-08-09T02:04:55",
"yymm": "1803",
"arxiv_id": "1803.02548",
"language": "en",
"url": "https://arxiv.org/abs/1803.02548"
}
|
\section{Introduction}
\label{intro}
The ideal qubit consists of a pair of orthogonal quantum states. However most systems used for quantum computing (QC) are multilevel systems and these additional levels allow for leakage out of the qubit subspace. Leakage errors result in the quantum system leaving the computational space and are suffered by trapped ions \cite{duan2001geometric, haffner2008quantum, cirac1995quantum, plenio1997decoherence}, quantum dots \cite{byrd2005universal, fong2011universal, mehl2015fault}, superconducting qubits \cite{zhou2005rapid, motzoi2009simple, ferron2010intrinsic, herrera2013tradeoff, ghosh2013understanding} and anyons \cite{xu2008constructing, ainsworth2011topological}.
Because leakage faults occur outside the computational space, traditional methods for correcting Pauli type errors are ineffective on them. Instead, the issue of leakage requires a separate set of techniques for reducing the faults. At the physical level, leakage errors can be mitigated through the use of different pulse techniques \cite{motzoi2009simple, ferron2010intrinsic, mcconkey2017mitigating, chen2016measuring}. Leakage errors can also be detected and converted to Pauli or erasure errors by constructing suitable leakage reducing units (LRUs) \cite{byrd2005universal, byrd2004overview, wu2002efficient, suchara,fowler, fowler2012surface, ghosh2013understanding, ghosh2015leakage}. It is also possible to construct a system that does not suffer from leakage \cite{lucas2004isotope}. Thus when designing the architecture of a quantum computer is it worthwhile to examine the resources needed to deal with leakage.
Ion trapped computers are a leading candidate for QC \cite{haffner2008quantum}. Quantum information is encoded in the internal states of the ion, often a pair of levels in the $S_{1/2}$ ground state. The two states are connected by a magnetic dipole transition with a small frequency difference, typically a radio-frequency for Zeeman qubits and a microwave frequency for hyperfine qubits, resulting in a practically infinite lifetime of the excited level due to spontaneous decay \cite{wineland, toolbox, ozeri}. In ions with $I=0$, the only $S_{1/2}$ levels available are that of the two Zeeman states. Zeeman qubits do not suffer from leakage in the ground manifold states, but have a first order dependence on magnetic fields \cite{lucas2004isotope, keselman2011high, poschinger2009coherent, ratschbacher2013decoherence}. In ions with $I \neq 0$, the qubit can be encoded into any pair of hyperfine states. However, the existence of other hyperfine states means there is a potential for leakage. Hyperfine qubits based on clock-states, have a second order dependance on magnetic fields but spontaneous scattering during stimulated Raman processes can lead to leakage errors \cite{brown2011single, ballance2016high, blinov2004quantum, olmschenk2007manipulation}.
Two-photon Raman transitions are often used to manipulate qubits in ion traps \cite{wineland, toolbox, ozeri, haffner2008quantum, lucas2004isotope, ratschbacher2013decoherence, poschinger2009coherent, brown2011single}. Quantum gates rely on coupling to excited states through electric dipole transitions. Since laser light is used to drive these transitions, spontaneous scattering of photons is inevitable. While detuning the laser frequency away from allowed optical transitions can suppress this scattering, it is impossible to completely eliminate. Both Raman and Rayleigh scattering can lead to decoherence but each manifest differently depending on qubit choice \cite{ozeri, uys, ozeri2005hyperfine}. We note that scattering errors can be avoided by using only microwave gates \cite{HartyPRL2016, WeidtPRL2016, KhromovaPRL2012, OspelkausPRL2008}, but leakage due to background gas collisions or imperfections in operations could still occur.
This work seeks to quantify these errors in the context of quantum error correction (QEC). First we describe the characteristics associated with each type of qubit as well as their magnetic field dependence. Next we discuss the calculation of the different errors associated with spontaneous scattering from driven Raman transitions. Finally we compare the ions in the context of the surface code. Our results show leakage is more prominent than expected, and given a stable enough magnetic field, Zeeman qubits require a smaller distance surface code to produce the same logical error rate as a logical qubit composed from a physical hyperfine qubit.
\section{Yb$^+$ Model and Associated Errors}
Yb$^+$ has many naturally occurring isotopes but we examine, $^{174}$Yb$^+$ ($I=0$) and $^{171}$Yb$^+$ ($I=1/2$), whose nuclear spin yield a Zeeman and hyperfine qubit, respectively. This makes Yb$^+$ the perfect candidate to study the associated error rates between these two types of qubits. The atomic structures and associated possible errors resulting from spontaneous scattering for both isotopes are illustrated in Fig. \ref{YbModel}. While there are other sources of noise that could be considered, we choose to focus on two types of noise that are the most relevant to the comparison of the two types of qubits: magnetic field fluctuations that lead to dephasing in Zeeman qubits and spontaneous scattering that lead to leakage errors in hyperfine qubits.
\begin{figure}[h]
\includegraphics[width=7cm, height=5cm]{Fig1.pdf}
\caption{Atomic structure of Yb$^+$ isotopes and errors associated with different scattering events from the $^{2}$P states assuming the ion starts in the lower qubit state. Spontaneous Raman scattering can cause bit flip noise or leakage errors. Spontaneous Rayleigh scattering can lead to dephasing errors.}
\label{YbModel}
\end{figure}
\subsection{Unstable Magnetic Field}
For the Zeeman qubit, $^{174}$Yb$^+$, the qubit is encoded into the electron spin states $\ket{S=1/2,m_s=-1/2}$ and $\ket{S=1/2, m_s=1/2}$. While there is no possibility for leakage (in this discussion we assume higher-level leakage states in the D and F manifolds are quickly repumped to the ground state), because the qubit itself is encoded in Zeeman energy splitting, it will be highly susceptible to magnetic field fluctuations. The applied magnetic field required for the ion trap causes the well known Zeeman energy splitting and the first order effects grow linearly with the magnetic field. Any deviations in the magnetic field yield a first order frequency shift given by
\begin{equation}
\Delta \nu = \frac{g_s \mu_B }{\hbar} \Delta B
\end{equation}
where $g_s$ is the Land\'{e} \textit{g}-factor, $\mu_B$ is the Bohr magneton, $\hbar$ is Planck's constant and $\Delta B$ is the difference from between the actual magnetic field and the ideal magnetic field \cite{PhysRev.49.324}. Such magnetic field noise can cause dephasing and is the main disadvantage of using a Zeeman qubit.
For $^{171}$Yb$^+$, the qubit is encoded into the clock states $\ket{F=1,m_F=0}$ and $\ket{F=0,m_F=0}$. These states are magnetic field insensitive transitions that do not suffer from first order effects. The second order magnetic field dependence can be derived from the Briet-Rabi formula with the frequency shift due to uncertainties in the magnetic field given by
\begin{equation}
\Delta \nu = \frac{(g_J - g_I)^2 \mu_B^2 }{2 \hbar^2 \omega} (2B_0\Delta B +(\Delta B)^2)
\end{equation}
where $g_J$ and $g_I$ are the Land\'{e} \textit{g}-factors for the electron and the nucleus, $\omega$ is the angular frequency of the hyperfine splitting, $B_0$ is the ideal magnetic field strength, and $\Delta B$ is the deviation from the ideal magnetic field~\cite{PhysRev.49.324, zeeman}. Because the second order effect is so small, clock states are negligibly affected by magnetic field noise, a clear advantage when using hyperfine qubits. At typical values of applied magnetic fields for hyperfine qubits, the effective frequency fluctation is 10$^{-3}$ to 10$^{-4}$ smaller than for the Zeeman qubit. However, the existence of the other hyperfine states $\ket{1,+1}$ and $\ket{1, -1}$ in $^{171}$Yb$^{+}$ can lead to leakage events.
Using equations $(1)$ and $(2)$, we assumed a Gaussian distribution and calculated the probability of error based on gate time and magnetic field stability. For low errors, the error from the first order Zeeman effect grows quadratically with increasing field fluctuations. For fields with high fluctuations, this probability is well above the threshold error value of the surface code of ~1$\%$ \cite{dennis2002topological, raussendorf2007fault}.
The probability of error resulting from the second order effects grows quartically with field fluctuations in the limit of zero average magnetic field. Even at fields with low stability, this error remains below threshold. Table \ref{table1} lists these probabilities with varying magnetic field stabilities for both single and two-qubit $\hat{I}$ gates. The more stable the field, the less error. The errors vary drastically for Zeeman qubits and are almost negligible for hyperfine qubits.
\begin{table}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|}
\hline
&
\multicolumn{2}{|c|}{Single Qubit Gate}&
\multicolumn{2}{|c|}{Two-Qubit Gate}\\
&
\multicolumn{2}{|c|}{$\tau_{gate}=1$ $\mu$s}&
\multicolumn{2}{|c|}{$\tau_{gate}=200$ $\mu$s}\\
\hline
Probability & $^{171}$Yb$^{+}$ & $^{174}$Yb$^{+}$ & $^{171}$Yb$^{+}$ &$ ^{174}$Yb$^{+}$ \\ \hline
${P_{\sigma=10^{-2}}}$ &$1.90\times10^{-14}$&$1.93\times 10^{-3}$ & $7.62\times10^{-10}$ & $0.50$ \\
\hline
${P_{\sigma=10^{-3}}}$ &$1.90\times10^{-18}$&$1.93\times 10^{-5}$ & $7.62\times10^{-14}$ & $0.39$ \\
\hline
${P_{\sigma=10^{-4}}}$ &$1.90\times10^{-22}$&$1.93 \times 10^{-7}$ & $7.62\times10^{-18}$ & $7.69 \times 10^{-3}$ \\
\hline
${P_{\sigma=10^{-5}}}$ &$1.90\times10^{-26}$&$1.93\times10^{-9}$ & $7.62\times10^{-22}$ & $7.75\times 10^{-5}$ \\
\hline
${P_{\sigma=10^{-6}}}$ &$1.90\times10^{-30}$&$1.93\times 10^{-11}$ & $7.62\times10^{-26}$ & $7.75\times 10^{-7}$ \\
\hline
\end{tabular}
}\caption{A list of error probabilities caused by the first order Zeeman effect ($^{174}$Yb$^+$) and the second order Zeeman effect ($^{171}$Yb$^+$). The gate times for one and two-qubits gates were 1 $\mu$s and 200 $\mu$s, respectively. $\sigma$ is the standard deviation of the magnetic field strength in G. The table shows $^{171}$Yb$^+$ error for zero average magnetic field. For typical magnetic fields yielding 1 MHz Zeeman splittings, the error for $^{171}$Yb$^+$ for a given $\sigma$ is comparable to the error for $^{174}$Yb$^+$ with $\sigma^\prime = 10^{-4} \sigma $. }
\label{table1}
\end{table}
\subsection{Spontaneous Scattering}
Additional errors arise from the scattering of photons during gates. Two-photon Raman coupling is among the most popular choices for gate implementation \cite{haffner2008quantum, lucas2004isotope, poschinger2009coherent, brown2011single, ballance2016high, blinov2004quantum, olmschenk2007manipulation, wineland, ozeri, uys}. Lasers detuned off-resonance drive qubit transitions through interactions with excited states. This use of stimulated transition to perform a qubit rotation lends itself to spontaneous emission errors. Raman scattering is usually thought of as the biggest contributor to these errors as all qubit types suffer from it \cite{ozeri}. Spontaneous Raman scattering can lead to leakage errors, or change the qubit in the computational basis ($\hat{X}$/$\hat{Y}$ error). Unlike leakage errors, Pauli type errors can be corrected using standard quantum error correction codes (QECC). Rayleigh scattering is typically less of a contributor to errors as it does not necessarily cause decoherence in all qubit types and in certain cases can be ignored \cite{ozeri, uys, ozeri2005hyperfine}. Rayleigh scattering leads to dephasing errors ($\hat{Z}$), similar to the magnetic field fluctuations. This decoherence rate is dependent on the scattering amplitudes of the qubit levels and thus varies from isotope to isotope.
To calculate the different error rates for the two ions, we followed the procedure outlined in \cite{uys}. The rate at which the ion in state $\ket{i}$ scatters a photon and ends in state $\ket{j}$ is given by the Kramers-Heisenberg formula
\begin{equation}
\Gamma_{ij} =(\frac{\mu E_0 }{2\hbar})^2 \gamma \sum_{\lambda}(\sum_{J}A^{i \rightarrow j}_{J,\lambda})^2
\end{equation}
where $\mu$ is the largest element of the dipole matrix, $E_0$ is the magnitude of a nonresonant light field of the lasers, $\gamma$ is the spontaneous decay rate of the excited states and $A^{i \rightarrow j}_{J,\lambda}$ are the scattering amplitudes \cite{uys, ozeri}.
The total Raman and effective Rayleigh scattering rates are given by
\begin{equation}
\Gamma_{Ram} = \Gamma_{ij}+\Gamma_{ji}
\end{equation}
\begin{equation}
\Gamma_{el} = (\frac{\mu E_0}{2\hbar})^2 \gamma \sum_{\lambda}(\sum_J A^{j \rightarrow j}_{J,\lambda} - \sum_{J^\prime}A_{J^\prime,\lambda}^{i \rightarrow i})^2
\end{equation}
respectively, where $i$ and $j$ represent the two qubit levels. When Rayleigh scattering rates from the two ion qubit states are different, the scattered photons will measure the qubit state causing decoherence \cite{uys}. Thus the effective Rayleigh scattering that will cause dephasing is given by this difference.
We calculated fidelity for both single ($\tau = 1$ $\mu$s) and two-qubit ($\tau = 200$ $\mu$s) gates of a $\pi$ rotation about the x-axis on the Bloch sphere. These gates were assumed to be driven by co-propogating linearly polarized Raman beams, blue detuned from the $P_{1/2}$ level with laser frequency of $355$ nm and a beam waist $w_0 = 20$ $\mu$m. The choice of these parameters was motivated by desired gate times, the minimization of spontaneous scattering and by recent experiments performed using a $355$ nm laser \cite{linke2017fault, leung2018robust, debnath2016demonstration, fallek2016transport}.
Table \ref{table2} shows the different scattering errors for both the $^{174}$Yb$^+$ Zeeman and $^{171}$Yb$^+$ hyperfine qubit. When the Rayleigh scattering amplitudes of the two qubit levels are approximately equal, their contributions can add up destructively. The decoherence rate due to Rayleigh scattering will be small and decoherence will be dominated by Raman scattering \cite{uys}. This is precisely what we see for $^{171}$Yb$^+$. However, even when amplitudes are approximately equal, they can have opposite sign and their different contributions can add up constructively leading to large Rayleigh scattering decoherence \cite{uys}, as in the case for $^{174}$Yb$^+$. For $^{174}$Yb$^+$, Rayleigh scattering was approximately equal to the Raman scattering. In this sense, $^{174}$Yb$^+$ can be modeled anisotropically, with double the amount of Pauli $\hat{Z}$ type errors for every single Pauli $\hat{X}$ or $\hat{Y}$ type error. For $^{171}$Yb$^+$, Raman scattering that resulted in leakage was equal to the scattering which caused Pauli type errors.
When looking at overall error rates, it is clear that a single $^{171}$Yb$^+$ is prone to less physical error. However, this hides the fact that leakage errors can be damaging to QECC. A majority of the errors that occur via spontaneous scattering in $^{171}$Yb$^+$ (leakage errors) requires extra overhead to correct relative to pure Pauli errors. To gain a better understanding of this, we must look at how each type of qubit performs with a QECC.
\begin{table}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|}
\hline
&
\multicolumn{2}{|c|}{Single Qubit Gate}&
\multicolumn{2}{|c|}{Two-Qubit Gate}\\
&
\multicolumn{2}{|c|}{$\tau_{gate}=1$ $\mu$s}&
\multicolumn{2}{|c|}{$\tau_{gate}=200$ $\mu$s}\\
\hline
Probability & $^{171}$Yb$^{+}$ & $^{174}$Yb$^{+}$ & $^{171}$Yb$^{+}$ &$ ^{174}$Yb$^{+}$ \\ \hline
$P_{Raman}$ &$2.42\times10^{-6}$ & $4.8\times10^{-6}$ & $6.37\times 10^{-5}$ & $12.6 \times 10^{-5}$ \\
\hline
$P_{Leakage}$ &$2.42\times10^{-6}$ & N/A &$6.37\times 10^{-5}$ & N/A \\
\hline
$P_{Rayleigh}$ &$1.60\times10^{-13}$ &$4.88\times10^{-6}$ & $4.21\times10^{-12}$ & $12.6 \times 10^{-5}$ \\
\hline
\end{tabular}
}
\caption{A list of error probabilities caused by spontaneous scattering from stimulated Raman transitions. The gate times for one and two-qubits gates were 1 $\mu$s and 200 $\mu$s. The gates were assumed to by driven by co-propogating linearly polarized Raman beams with $f = 355$ nm and a beam waist of $w_0 = 20$ $\mu$m. For $^{174}$Yb$^+$, Rayleigh scattering was just as dominant as Raman scattering. For $^{171}$Yb$^+$, Raman scattering which resulted in leakage was equal to that of bit flip noise. }
\label{table2}
\end{table}
\section{Surface Code Model and LRC}
The toric code was the first example of a topological code and is well studied \cite{kitaev2002classical, kitaev1997quantum, PhysRevA.90.032326}. The toric code is a two dimensional surface code with periodic boundary conditions and thus has a natural mapping onto the surface of a torus. Qubits are positioned in an array and either function as data qubits or ancilla/measurement qubits. Data qubits are used to encode the information while ancilla qubits are used to measure stabilizers, which in turn help infer where errors occurred.
A six step cycle is implemented in order to perform one round of error correction. First, all ancilla qubits are initialized in their respective eigen basis (either $\ket{0}$ for $\hat{Z}$ or $\ket{+}$ for $\hat{X}$). Next, four CNOT gates are performed between each ancilla and data qubit. Finally, each ancilla is measured in it's respective basis. This is precisely the circuit outlined in Fig. \ref{surface}. The problem of inferring the most probable error given the observed syndrome is mapped to a minimum weight perfect matching problem that can be solved with Edmond's algorithm \cite{suchara}. Such error correcting schemes have been studied both with and without leakage \cite{ghosh2013understanding, ainsworth2011topological, fowler2012surface, fowler2012topological, suchara, fowler, ghosh2015leakage, PhysRevA.90.032326}.
\begin{figure}[h]
\includegraphics[trim=250 450 250 125, width=2cm, height=3.5cm]{Fig2.pdf}
\caption{Standard circuits to measure surface code check operators. The open white circles represent data qubits while the closed dark circles represent measure/ancilla qubits. The blue and green diamonds represent $\hat{Z}$ and $\hat{X}$ stabilizers respectively. }
\label{surface}
\end{figure}
\begin{figure}
\includegraphics[trim=110 425 20 115, clip, height=4cm, width=7cm]{Fig3.pdf}
\caption{The QUICK LRC required to perform error detection in the presence of leakage. After each cycle, the physical qubits get swaped. Data qubits become ancilla and ancilla qubits become data qubits. The information is transferred and leaked qubits get measured and reset every other cycle \cite{suchara}.}
\label{LRC}
\end{figure}
This six step error correction cycle is all that is needed to correct Pauli type errors. Handling leakage errors requires the use of LRU's. The idea of incorporating LRU's was first used to show an accuracy threshold exists even in the presence of leakage errors \cite{aliferis2005fault}. The most common type of LRU implements gate teleportation in some fashion \cite{aliferis2005fault, byrd2005universal, byrd2004overview, wu2002efficient}. The additional circuitry required to perform the teleportation is referred to as a leakage reducing circuit or LRC. Different strategies for implementing LRCs into surface codes have been studied \cite{aliferis2005fault, ghosh2013understanding, suchara, fowler, ghosh2015leakage}, in order to grasp the tradeoff between circuit complexity and effectiveness of leakage reduction.
In our work, we chose to implement the Quick LRC \cite{suchara}, as depicted in Fig. \ref{LRC}. The Quick LRC adds a SWAP gate after the last CNOT of the standard circuit. At the end of each cycle, the physical qubits trade roles. Data qubits become ancilla qubits and ancilla qubits become data qubits. The cycle starts again reinitializing ancilla qubits. Leaked data qubits now get measured and reinitialized as ancilla qubits, and thus leaked qubits do not live for more than two cycles with this LRC implemented. Through the use of gate identities and gate cancellation, the implementation of this LRC requires only one additional CNOT. The Quick LRC is the simplest of all current LRCs and was shown to produced comparable results to that of more complicated LRCs \cite{suchara}. Other LRCs require more SWAP gates per cycle but did not show significant improvement compared to the QUICK LRC. In short, the Quick LRC effectively reduces leakage using the smallest overhead.
\section{Results and Discussion}
Using the error probabilities calculated from the magnetic field fluctuations and the spontaneous scattering rates, we analyzed the performance of the Zeeman and hyperfine qubits on the toric code. The Zeeman qubit was demonstrated on the standard circuit (Fig. \ref{surface}) while the hyperfine qubit was demonstrated on the Quick LRC (Fig. \ref{LRC}).
In our model, after every gate magnetic field noise was introduce with probabilities corresponding to the magnetic field susceptibility of the qubits (Table \ref{table1}). Additionally, spontaneous scattering errors occurred after every gate with the ratios of the probability for a particular error corresponding to the calculated spontaneous scattering rate of the qubits (Table \ref{table2}), e.g. leakage was twice as probable as a Pauli $\hat{X}$/$\hat{Y}$, with the total probability of an scattering event equal to p. Spontaneous scattering also allows leaked qubits to return to the qubit subspace. The two qubits involved in a CNOT gate have independent probabilities to leak after the gate. Once the qubit leaked, it would remain leaked until a spontaneous scattering event returns it to the qubit subspace or the qubit is reset by the Quick LRC. While this means a leaked qubit was corrected at maximum every other error correction cycle, long lived leaked qubits had the potential of corrupting other qubits.
When a CNOT is performed between a leaked qubit and a qubit in the computational basis, the latter suffers a random single-qubit Pauli error (including the trivial error $\hat{I}$), with equal probability. When a CNOT is performed between two qubits in the computational basis, the standard error propagation rules are applied. Magnetic field noise and spontaneous scattering errors are only applied after the gates to model environmental noise. Finally when a leakage qubit is measured, it yields a $\ket{+1}$ eigenvalue. This is physically motivated by the atomic structure of $^{171}$Yb$^+$ because any leaked state will be in the $F=1$ manifold and will be detected as such (see Fig. \ref{YbModel}).
\begin{figure}[h]
\includegraphics[width=\columnwidth]{Fig4.pdf}
\caption{Comparison of various magnetic field stabilities for a distance 5 code per 2 qubit gate. The hyperfine qubits (black) have the LRC implemented (Fig. \ref{LRC}) while the Zeeman qubits have only the standard circuit implemented (Fig. \ref{surface}). The LRC swaps data and ancilla qubits, effectively reinitizating leaked qubits back into the computational subspace. If the magnetic field is stabilized to below $\approx 30$ $\mu$G, the logical error of the Zeeman qubit is better than that of the hyperfine for the scattering rates considered.}
\label{mag}
\end{figure}
\begin{figure}[h]
\includegraphics[width=\columnwidth]{Fig5.pdf}
\caption{Comparison of various distances for hyperfine qubits with the LRC (black) and Zeeman qubits with a magnetic field fluctuations (red) with a standard deviation of 10 $\mu$G, per 2 qubit gate. The Zeeman qubit yields lower logical error for codes of the same distance.}
\label{dis}
\end{figure}
As expected we found that the success of the Zeeman qubit depended heavily on the stability of the magnetic field. A comparison of the Zeeman and hyperfine qubits at varying magnetic field stabilities is shown in Fig. \ref{mag}. It is clear from this graph that if the magnetic field is not stable enough, the error rate is above threshold and QECC will not help. There is also a stability where the performance of the Zeeman qubit and hyperfine qubit are about equal ($\sigma = 31.62$ $\mu$G), but when the probability of a spontaneous scattering event is low enough, ($\approx10^{-4}$), then the main source of error for the Zeeman qubit is from the magnetic field fluctuation. This base error results in a plateau on the graph were the logical error rate cannot be improved by reducing the scattering. Finally, if the magnetic field can be stabilized to $10$ $\mu$G or less, corresponding to a qubit dephasing error probability per gate of $7.75 \times 10^{-7}$, then the Zeeman qubit produced a lower logical error rate than hyperfine qubit. There did not appear to be a significant improvement of the logical error rate past $10$ $\mu$G for the scattering rates studied. When the field reaches a certain magnitude of stability, the main source of error comes from the spontaneous scattering, which is independent of the magnetic field. Thus the behavior at higher stabilities is more or less the same.
Using a stability of $10$ $\mu$G, we looked at the behavior of different distance toric codes. Fig. \ref{dis} compares the performance of the two qubits using $d = 3, 5, 7$ codes. It is clear from this that, given the $10$ $\mu$G stability, the Zeeman qubit produces the smaller logical error. With the addition of the LRC, the hyperfine qubits performance was suppressed to that of a lower distance code. The LRC data for $d = 5$ is nearly identical to the standard circuit data for $d = 3$. Similarly, the LRC data for $d = 7$ is comparable to that of the standard circuit data for $d = 5$. A similar behavior was also found by Fowler \cite{fowler}. This behavior suggested a single leakage error may act like two Pauli errors. This is evidence that not all errors are equally damaging. Some errors (such as leakage) can be more harmful to QECC compared to others. Not only do these error require more resources to correct, they suppress the effectiveness of QECC.
In this sense it is clear that the Zeeman qubit outperforms the hyperfine qubit as it does not require additional circuitry that suppress its performance. However this of course comes with the caveat that the applied magnetic field be stabilized to $\leq 10$ $\mu$G. The existence of a Zeeman qubit in a field of stabilized to $10$ nG has already been physically realized \cite{ruster2016long}.
\section{Conclusions}
Zeeman qubits are prone to more overall physical errors resulting from both magnetic field fluctuations and spontaneous scattering. When the stability of the applied magnetic field is above $30$ $\mu$G, the Zeeman qubit's logical error rate is higher than that of the hyperfine qubit. However, when the magnetic field is stabilized to $\leq 10$ $\mu$G, the logical error rate is suppressed and is less than that of the hyperfine qubit.
For hyperfine qubits, leakage due to spontaneous scattering is a prominent source of error. These errors are problematic for two reasons: 1) when entangled with other qubits via the CNOT gates, they corrupt the other qubit state and 2) these errors cannot be corrected using standard QEC schemes and require the use of LRCs to correct. For standard QEC schemes, a single physical leakage error has the ability to produce a logical error. This limits the effectiveness of a QECC.
We have not considered additional physical differences between the hyperfine and Zeeman qubits involving state preparation and measurement. We have also not considered physical methods of leakage reduction. For example, perfect polarized $\pi$ light tuned resonant with the S$_{1/2}$, $F=1$ to P$_{1/2}$, $F=1$ transition will remove population from the leaked states for the hyperfine qubit. The qubit $\ket{1}$ states will have a small probability ($\approx 10^{-4}$) to leak or suffer a bit flip error due to off-resonant $\Delta F = -1$ transitions. In practice, leakage errors during this procedure will be larger due to imperfect polarization.
In our study, we also examined the toric code which may be less practical than the planar surface code depending on the layout. Modular architectures could implement the toric code directly \cite{NickersonNatComm2013}, while architectures based on local geometry are better suited to the surface code \cite{LekitscheSciAdv2017}. For small devices implementing the code in a single ion chain \cite{trout2017simulating}, either the torus or plane would work. To implement the leakage reduction circuit in the plane, additional circuits on the boundary are necessary to enable the swap.
We have shown that the ideal qubit for near term experiments may not be the ideal qubit for large scale fault-tolerant quantum computation. Our simulation has centered on trapped ions, but we expect that the design of small quantum systems and error corrected quantum systems will yield different requirements on the qubits. In particular, for solid-state qubits where the qubits are constructed from multiple components, we expect there will be many interesting tradeoffs between the fidelities of small systems and the overhead required to reach a target logical error.
\section{Acknowledgments}
We thank Kenton Brown and Wes Campbell for useful discussions on Zeeman and hyperfine qubits. The toric surface code simulator was provided by Andrew Cross and Martin Suchara, with permission from IBM. This work was supported by the Office of the Director of National Intelligence - Intelligence Advanced Research Projects Activity through Army Research Office (ARO) contract W911NF-10-1-0231, ARO MURI on Modular Quantum Systems W911NF-16-1-0349, and the National Science Foundation grant PHY-1415461.
|
{
"timestamp": "2018-04-18T02:03:19",
"yymm": "1803",
"arxiv_id": "1803.02545",
"language": "en",
"url": "https://arxiv.org/abs/1803.02545"
}
|
\section{Introduction}
The Internet has revolutionized the world by transforming it into a global entity.
Widespread advantages of Internet have spawned new industries and services.
However, this connectivity comes at the cost of privacy.
Every Internet client has a unique identity in the form of an Internet protocol (IP) address which can be translated to its location by the local Internet service provider (ISP).
This lack of privacy has serious implications, particularly for journalists, freedom fighters and ordinary citizens.
Lack of privacy has lead to the use of anonymous communication networks (ACN).
ACNs hide client IP addresses through various techniques.
There are a number of ACNs including Tor, Java anonymous proxy (JAP), Hotspot Shield and UltraSurf etc.
Among various ACNs, Tor is the one of the most popular network, owing to its distributed nature which makes it difficult to connect the two end points of a session.
Recently, Tor has been used for bomb hoax at Harvard \cite{Brandom2013}. Similarly, it has been used by the Russians to bypass online censorship \cite{Khrennikov2016}.
A number of attempts are being made by FBI and other organizations to breach Tor network \cite{Hern2016}\cite{Graham-Smith2016}\cite{Neal2016}.
In this paper, we survey various studies conducted on the Tor network covering the scope of these studies.
We quantify the studies into three broad but distinct groups, including (1) deanonymization, (2) path selection, (3) analysis and performance improvements, and several sub-categories.
To the authors' best knowledge, this is the most comprehensive attempt at analyzing Tor network research with a focus over its anonymity mechanism.
Table \ref{tab: Comparison of other Surveys with this survey} presents a comparison of this survey with previous surveys covering the scope of researches and implementation (experiments), verification (simulations) and analysis of various research works. Categorization of first column is made by listing all Tor areas considered in our study. AlSabah and Goldberg \cite{alsabah2016performance} presented the most comprehensive study covering complete Tor network and our paper is complementary to their survey paper. However, our paper pays more focus to the anonymity and breaching aspects of Tor than their paper. Their research paper presents only twenty references related to anonymity while we present more than $120$\footnote{This paper has 146 references, some of the references are to tools rather than research works; in all, we are considering a research corpus of 120 references.} references.
\begin{table*}[t]
\caption{Comparison of other surveys with this survey.}\label{tab: Comparison of other Surveys with this survey}
\centering
\small
\begin{tabular}{|
@{}>{\centering}p{2.3cm}@{}|
@{}>{\centering\arraybackslash}p{2.5cm}@{\hspace{0.015in}}|
@{}>{\centering\arraybackslash}p{2.2cm}@{\hspace{0.015in}}|
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@{}>{\centering\arraybackslash}p{1.4cm}@{\hspace{0.015in}}|
@{}>{\centering\arraybackslash}p{0.9cm}@{\hspace{0.015in}}|
@{}>{\centering\arraybackslash}p{1.5cm}@{\hspace{0.015in}}|
@{}>{\centering\arraybackslash}p{1.2cm}@{\hspace{0.015in}}|
@{}>{\centering\arraybackslash}p{1.4cm}@{\hspace{0.015in}}|
@{}>{\centering\arraybackslash}p{1.3cm}@{\hspace{0.015in}}|
@{}>{\centering\arraybackslash}p{1.15cm}@{\hspace{0.015in}}|
}
\hline
\multirow{14}{*}{Scope} & \multicolumn{2}{|c|}{[Areas]$\downarrow$ / [Research, Year]$\rightarrow$} & $<$This Paper$>$ & AlSabah and Goldberg \cite{alsabah2016performance} & Koch \emph{et al.} \cite{koch2016anonymous} & AlSabah and Goldberg \cite{alsabah2015performance} & Conrad and Shirazi \cite{conrad2014survey} & Jagerman \emph{et al.} \cite{jagerman2014fifteen} & Ren and Wu \cite{ren2010survey} & Johnson and Kapadia \cite{johnson2007chaum} \\
\cline{2-11}
& \multicolumn{2}{|c|}{Year$\rightarrow$} & 2017 & 2016 & 2016 & 2015 & 2014 & 2014 & 2010 & 2007 \\
\cline{2-11}
& \multicolumn{2}{|c|}{Coverage (Studies)$\rightarrow$} & 120 & 120 & 10 & 99 & 40 & 37 & 109 & 32 \\
\cline{2-11}
& \multirow{6}{*}{Deanonymization} & Hidden Services & \checkmark & \checkmark & & \checkmark & & & \checkmark & \checkmark \\
& & Finger printing & \checkmark & \checkmark & & \checkmark & & & & \\
& & Attacks & \checkmark & \checkmark & & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\
& & Traffic Analysis & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark & & \checkmark \\
& & Improvements & \checkmark & \checkmark & & \checkmark & & & & \\
& & Bypassing Tor & \checkmark &\checkmark & & & \checkmark & \checkmark & \checkmark & \checkmark \\
\cline{2-11}
& \multirow{2}{*}{Path Selection} & Algorithm design & \checkmark & \checkmark & & \checkmark & & & & \\
& & Analysis & \checkmark & \checkmark & & \checkmark & & & & \\
\cline{2-11}
& \multirow{4}{*}{Analysis} & General & \checkmark & & \checkmark & \checkmark & \checkmark & & \checkmark & \checkmark \\
& & Modelling & \checkmark & & & & & & & \\
& & Analysis & \checkmark & & \checkmark & & & & & \\
& & Improvement & \checkmark & \checkmark & & & & & & \\
& & Mobile Tor & \checkmark & & & & & & & \\
\hline
\multirow{5}{*}{Implementation} & \multirow{5}{*}{Experiments} & Private Setup & \checkmark & & & & & & & \\
& & PlanetLab & \checkmark & & & & & & & \\
& & Cloud Services & \checkmark & & & & & & & \\
& & OpenFlow & \checkmark & & & & & & & \\
& & UC Framework & \checkmark & & & & & & & \\
\cline{1-11}
\multirow{4}{*}{Verification} & \multirow{5}{*}{Simulations} & Cus. Simulator & \checkmark & & & & & & & \\
& & ExperimenTor & \checkmark & & & & & & & \\
& & Shadow Simulator & \checkmark & & & & & & & \\
& & ModelNet & \checkmark & & & & & & & \\
\hline
Analysis & Parameters & & \checkmark & & & & & & & \checkmark \\
\hline
\end{tabular}
\end{table*}
Analysis of keywords used in various studies shows that anonymity, security and privacy have been used the most.
Our study shows that majority of the research works have been made in the field of ``deanonymization'' track, followed by ``performance analysis and architectural improvements''.
In the deanonymization track, a major chunk of research is devoted to \emph{breaching attacks} followed by \emph{traffic analysis}.
In the path selection track, most research works focused on the development of new algorithms.
Relays, protocol messages and traffic interception have been the most frequently exploited factors in the Tor's deanonymization track.
In the path selection track, performance and anonymity have been the most commonly used factors.
Performance, relay selection and anonymity have been the most studied parameters in the performance analysis and improvement track.
Analysis over simulations and experiments shows that 60\% of studies used experiments and 86\% of those experiments were carried out on private testbed networks.
Among simulations, 75\% of the studies developed their own simulator to analyze Tor network.
Analysis of simulation parameters shows that there is no distinct pattern of parameters. However, majority of the studies used bandwidth and latency.
Table \ref{tab: Glossary of all abbreviations used in the text.} presents a glossary of the important abbreviations used in our survey paper. This paper is organized as follows: Section \ref{sec: Tor Architecture} introduces the architecture of Tor network and its comparison with other anonymity services. Section \ref{sec:Tor Research Areas} presents the studies covering deanonymization, path selection, and performance analysis and architectural improvements. Section \ref{sec:Platforms for Tor Research} presents the simulations and experiments conducted in previous studies. Section \ref{sec: Discussion} presents the Tor performance metrics, our findings and open research areas in Tor. Finally, section \ref{sec: Conclusion} concludes the paper.
\begin{table*}[t]
\centering
\caption{Glossary of the important abbreviations used in the text.}\label{tab: Glossary of all abbreviations used in the text.}
\small
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{4}{|c|}{Glossary}\\
\hline \hline
ACK & Acknowledgement &
MRA & Multi-Resolution Analysis \\
ACN & Anonymous Communication Network &
NAT & Network address translation \\
ADSL & Asymmetric digital subscriber line &
NTP & Network Time Protocol \\
AES & Advanced Encryption Standard &
OP & Onion Proxy \\
AS & Autonomous System &
OR & Onion Router \\
CGI & Computer-generated imagery &
PGP & Pretty Good Privacy \\
CSRF & Cross site request forgery &
POP3 & Post Office Protocol 3 \\
DHCP & Dynamic Host Configuration Protocol &
PPTP & Point-to-Point Tunneling Protocol \\
DNS & Domain Name System &
P2P & Peer-to-Peer \\
DoS & Denial of Service &
QoE & Quality of Experience \\
DS & Directory Server &
QoS & Quality of Service \\
DSL & Digital Subscriber Line &
ROC & Region of Convergence \\
EWMA & Exponentially weighted moving average &
RRD & Round Robin Database \\
FIFO & First In First Out &
RTT & Round Trip Time \\
FN & False Negative &
SMTP & Simple Mail Transfer Protocol \\
FP & False Positive &
SSH & Secure Shell \\
HTML & HyperText Markup Language&
SVM & Support Vector Machine \\
HTTP & Hypertext Transfer Protocol &
TAP & Tor Authentication Protocol \\
IMAP & Internet Message Access Protocol &
TCP & Transmission Control Protocol \\
ICMP & Internet Control Message Protocol &
TMT & Tunable mechanism of Tor \\
IP & Internet Protocol &
Tor & The Onion Router \\
ISP & Internet Service Provider &
TP & True Positive \\
I2P & Invisible Internet Project &
TTL & Time To Live \\
JAP & Java Anonymous Proxy &
URL & Uniform Resource Locator \\
JVM & Java virtual machine &
VDE & Virtual Distributed Ethernet \\
LAN & Local area network &
VM & Virtual Machine \\
L2TP & Layer 2 Tunneling Protocol &
VPN & Virtual Private Network \\
ML & Machine Learning &
VPS & Virtual Private Server \\
\hline
\end{tabular}
\end{table*}
\section{Anonymity tools}
\label{sec: Tor Architecture}
In this section, we present and discuss Tor and other anonymity tools.
In the first part, we present the architecture of Tor network before presenting details of the research in Tor.
In the second part, we present the comparison and working mechanism of other anonymity tools which compete with Tor.
\subsection{Architecture of Tor network}
Tor network is composed of a decentralized distributed network of relays operated by volunteer users \cite{goldschlag1999onion}.
In July $2016$, nearly $10,000$ users (per day) participated in the Tor network (as Tor relays and Tor bridges) to provide anonymity services to nearly half a million users daily \cite{torMetricPortal2016}.
History of Tor dates back to late 1990's when Goldschlag, Reed and Syverson presented the architecture and implemented onion routing in several papers \cite{reed1996proxies, goldschlag1996hiding, reed1998anonymous, syverson2001towards} which laid the foundation of Tor network by providing proxy servers which are resilient to eavesdropping and effectively hide client's IP address.
The Tor network consists of routers which cooperate with each other to provide low latency anonymity services to users.
Central servers help Tor establish and update links between Tor routers.
User participation as Tor relays (router) is optional, but it is recommended because it improves the chances of staying anonymous, because it increases the traffic to the user.
Tor's architecture has three types of components, namely onion proxy (OP), onion router (OR) and directory server (DS).
OPs are used by Tor users to obtain up-to-date information of operating relays from DS.
OPs also creates connections using the information from a DS.
Users may configure OPs to select specific routers.
ORs are Tor relay routers, operated by volunteer users, to act as entry (guard), middle and exit relays.
Information of all online relays is available at DS.
To counter attacks on Tor that block Tor relays, a secret group of Tor relays exists with the DS, called \emph{bridges}.
A set of three bridge relays is available through unique \emph{Gmail} addresses.
Once a connection is established, every OR knows only immediate predecessor and successor node.
Nine authorities acting as Tor DSs keep an up-to-date record of all available ORs and broadcast the bandwidth, IP, public key, exit policies etc. to OPs.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{tor_architecture.pdf}
\caption{Architecture of Tor network.}
\label{fig:tor_architecture}
\end{figure}
Figure \ref{fig:tor_architecture} shows a circuit established from a user to a server through three Tor relays.
The various steps involved in circuit establishment are listed below:
\begin{enumerate}
\item OP sends HTTP requests to DS for information about Tor relays.
\item OP selects three Tor routers (entry relay, intermediate relay and exit relay) using Tor's path selection algorithm considering maximum anonymity and performance.
\item OP sends a \emph{Create Cell} request (containing half of Diffie-Hellman handshake \cite{bresson2001provably}) to the entry node. The entry node replies with the hash of the negotiated key.
\item Next, OP sends a \emph{Extend Cell} request to the entry node containing the address of the intermediate relay and encryption key. Entry node forwards the cell to the intermediate node. Similar to previous case, the intermediate node replies with \emph{Created cell} response. Similar process continues till client negotiates the key with the exit relay.
\item OP constructs IP packet $P1$ with source and destination IP addresses of exit relay and destination server, respectively, and packet size of $512$ bytes.
\item OP encrypts the packet further with the key $E3$, negotiated between client and exit relay, and containing source and destination addresses of intermediate relay and exit relay.
\item Next, OP encrypts with the key $E2$, negotiated between client and intermediate relay, and containing source and destination IPs of entry relay and intermediate relay.
\item Finally, OP constructs packet $P2$ encrypted with key $E1$, negotiated between client and entry relay, and containing source and destination IPs of OP and entry relay.
\item Packet $P2$ is transmitted from the entry relay, which decrypts the packet and forwards it to intermediate relay. All relays decrypt the packet using their specialized decryption keys and forward it towards the destination.
\end{enumerate}
\emph{Cell} refers to the \emph{Tor Packet}, comprising of payload data and headers, with an aggregate size of 512 bytes \cite{pries2008new, barbera2013cellflood, wang2013improved}.
Padding is used to fill cells with less data.
Tor relays communicate with each other by pairwise TCP connections.
Traffic multiplexing is used to transfer data between any pair of relays.
Tor employs token buckets to rate limit connections.
Buckets are filled and removed with tokens based on configured bandwidth limits and data read, respectively.
TCP buffers are read using a round-robin scheduling mechanism.
For flow control, edges (client and exit node) keep track of data flow by maintaining an active window about the packets in flight.
Data packets are processed in a first-in-first-out (FIFO) manner from the queues of Tor relays.
Multiplexing of packets, from Tor relays to relay links, is performed using exponentially-weighted moving average (EWMA) scheduler.
\subsection{Comparison with other anonymization services}
In this section, we present the features and working mechanisms of other deanonymization services which compete with Tor network. Table \ref{tab: Comparison of Tor with other anonymization services} presents a comparison of Tor with other deanonymization services. Table \ref{tab: Comparison of Tor with other anonymization services} shows that Tor is the only anonymity service which provides various services (http, https, visible TCP port, remote DNS, hides IP and user-to-proxy encryption) under all circumstances. On the contrary, JonDo, I2P, CGI and socks5 provide some services in limited circumstances only. A summary of various anonymity services is presented in following subsections.
\begin{table*}[t]
\caption{Comparison of Tor with other anonymization services (`\checkmark $\star$' refers to `In limited circumstances').}\label{tab: Comparison of Tor with other anonymization services}
\centering
\begin{tabular}{|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|p{1.5cm}|}
\hline
Proxy/ Anon. Service & HTTP & HTTPS & Visible TCP Port & UDP & Remote DNS & Hides IP & user-to-proxy encryption \\
\hline
\hline
http & \checkmark & & & & \checkmark & \checkmark $\star$ & \\
\hline
https & \checkmark & \checkmark & & & \checkmark & \checkmark $\star$ & \\
\hline
socks4 & \checkmark & \checkmark & & & & \checkmark & \\
\hline
socks4a & \checkmark & \checkmark & & & \checkmark & \checkmark & \\
\hline
socks5 & \checkmark & \checkmark & & \checkmark & \checkmark & \checkmark & \\
\hline
CGI & \checkmark $\star$ & \checkmark $\star$ & & & \checkmark & \checkmark $\star$ & \checkmark $\star$ \\
\hline
I2P & \checkmark $\star$ & \checkmark $\star$ & & \checkmark & \checkmark & \checkmark & \checkmark \\
\hline
JonDo & \checkmark & \checkmark & & \checkmark $\star$ & \checkmark & \checkmark & \checkmark \\
\hline
\textbf{Tor} & \checkmark & \checkmark & \checkmark & & \checkmark & \checkmark & \checkmark \\
\hline
\end{tabular}
\end{table*}
\subsubsection{Cross Platform Anonymity Tools}
A number of cross platforms anonymity tools are used now-a-days. In below lines, we summarize the basic working mechanism of prominent anonymity tools.
\begin{itemize}
\item Java Anonymous Proxy (JAP or JonDonym) \cite{gordon2016official}: Users can select among several Mix Cascades, different from P2P.
\item PacketiX.NET \cite{softether2016}: Virtual LAN card and Virtual HUB by Ethernet and can provide layer $2$ VPN virtualization.
\item JanusVM \cite{janusvm2016}: Uses Virtual Private Network (VPN) connection - No update on project since $2010$.
\item proXPN \cite{proxpn2016}: A personal VPN that provides you safety and privacy while using the Internet.
\item USAIP \cite{USAIP2016}: A VPN service provider with servers in Switzerland, Luxembourg and Hungary etc.
\item VPNReactor \cite{VPNreactor2016}: Uses a VPN connection with time limits for free and pro service and user logs are kept for $5$ days.
\end{itemize}
\subsubsection{Windows Based Anonymity Tools}
A number of windows based anonymity tools are large competitors of Tor network. Basic mechanisms of prominent anonymity tools are summarized in below lines:
\begin{itemize}
\item xB Browser \cite{xbbrowser2009}: A browser designed to run over the Tor network and XeroBank anonymity network.
\item Hotspot Shield \cite{hotspotshield2016}: Uses VPN. Hosts web servers accessible through proxy and has a central server that can be compromised.
\item AdvTor \cite{AdvTor2016}: Acts as a portable client and server for the Tor network. Improvements include the UNICODE path, HTTP and HTTPS protocols, estimates AS paths etc.
\item SecurityKISS \cite{securitykiss2016}: A VPN service based on OpenVPN, PPTP and L2TP.
\item UltraSurf \cite{ultrsurf2016}: Uses HTTP proxy to bypass censorship and uses encryption protocols for privacy.
\item CyberGhost VPN \cite{cyberghost2016}: OpenVPN based proprietary client, Centralized server with VPN using 1024-SSL encryption.
\item Freegate \cite{freegate2016}: Uses range of proxy servers (called Dynaweb) along with encryption.
\end{itemize}
\subsubsection{Linux based solutions}
Linux, being the prominent platform, is used by various anonymity tools to guarentee anonymity to its users. Working mechanism of some tools is summarized in below lines:
\begin{itemize}
\item Tails (Amnesic Incognito Live System) \cite{tails2016}: Has a Debian Linux distribution using Tor network.
\item Privatix \cite{privatix2016}: Provides encryption with anonymous web browsing (using Tor, Torbutton and firefox).
\end{itemize}
\subsubsection{Anonymous Search Engines}
Several search engines also provided anonymity to their users by collecting no user information. Some of these anonymous search engines are as follows:
\begin{itemize}
\item Ixquick \cite{ixquick2016}: Opens all search results through a proxy for anonymity.
\item DuckDuckGo \cite{duckduckgo2016}: Provides searcher's privacy and avoids ``filter bubble''. Displays same information to all users for a given search.
\end{itemize}
\subsubsection{Anonymous Emails}
Anonymous emails are provided by many servers. In following lines, we have summarized the tools provided anonymous email capability to their users.
\begin{itemize}
\item Anonymous E-mail \cite{anonymousemail2016}: Sends the emails with anonymous senders.
\item Safe-mail \cite{safemail2016}: Provides a secure communication, storage, distribution and sharing system for the internet.
\item HushMail \cite{hushmail2016}: Provides a PGP encrypted email service.
\item 10 Minute Mail \cite{10minutemail2016}: Gives an email address which expires after 10 minutes to counter spam emails.
\item Yopmail \cite{yopmail2016}: Provides a disposable email address with no registration and password.
\end{itemize}
\section{Tor Research Areas}
\label{sec:Tor Research Areas}
A large number of networks have utilized Tor networks for various purposes.
To get an overview of the research areas dealt in various studies, we created a word cloud of the keywords of these studies.
Figure \ref{fig:Word cloud of keywords used in Tor researches} shows the word cloud of keywords (on log-scale) of all studies cited in this paper.
Data of keywords shows that anonymity, privacy and security are the most important terms dealt in various studies.
In our review, we observed that research works on Tor could be broadly classified into three tracks/categories which include (1) deanonymization, (2) path selection, and (3) Analysis and performance improvement. Figure \ref{fig: Taxonomy of Tor research} shows the classification of our survey paper along with a list of all research works present in various subcategories.
\begin{figure*}[t]
\centering
\includegraphics[width=1.5\columnwidth]{wordle_v1.png}\\
\caption{Word cloud of \emph{keywords} used in Tor research works.}\label{fig:Word cloud of keywords used in Tor researches}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=2.0\columnwidth]{Tor_classification_v5.pdf}\\
\caption{Taxonomy of Tor research. Tor literature can be broadly classified into three areas: deanonymization, path selection, and performance analysis and architectural improvements.}\label{fig: Taxonomy of Tor research}
\end{figure*}
In deanonymization track, research works were observed in six different categories covering (1) Hidden services which limit their scope to hidden servers identification, (2) Fingerprinting which are based on pinpointing Tor network, (3) Attacks which are focused over breaching Tor network, (4) Traffic analysis which analyze Tor traffic to pinpoint the weaknesses, (5) Studies studying improvements in Tor to avoid deanonymization, and (6) Anonymity without Tor which suggest alternate methods to provide anonymity by pinpointing weaknesses in Tor.
In the path-selection track, all research works are either based upon (1) Development of new algorithms providing better efficiency and anonymity, and (2) Analysis of Tor's algorithm to study its strong and weak points in circuit establishment mechanism.
Lastly, analysis and performance improvement track focuses on four sub-areas which include (1) Generalized studies over Tor providing usability and social implications, (2) Modelling studies which focus on the development of model for analysis of Tor, (3) Analysis studies which cover QoS, relays, servers, etc., (4) Performance improvement studies provide modification in relays and architecture to provide better QoS, and (5) Development of efficient mechanisms for Tor clients with mobility.
Figure \ref{fig: Classification of Tor research areas.} shows the classification of various research areas studied in the Tor network.
It is pertinent to mention that all numeric values used for all pie charts, figures and tables in this paper have been calculated by the authors.
Source of all numeric values is the `Reference' section at the end of the paper, which includes scholarly research articles. Moreover, references have been collected by the authors from Tor repository\footnote{https://www.freehaven.net/anonbib/} with a particular bias towards papers covering `Tor' network only. Span of collection varies from $2007$ to $2017$ in reputed international conferences and journals. We also included the important studies in this field before $2007$ which play helpful role in understanding of Tor network, such as \cite{reed1996proxies}\cite{reed1998anonymous}.
Many articles were also collected from `ACM digital library' and `IEEE Xplore digital library' with a particular focus towards anonymity and security in Tor.
\begin{figure}[b]
\centering
\includegraphics[width=0.9\columnwidth]{classification.pdf}\\
\caption{Classification of Tor research areas.}\label{fig: Classification of Tor research areas.}
\end{figure}
\subsection{Tor Deanonymization}
Breaching the Tor network is one of the most widely studied research problems.
In fact, the majority of the studies describe deanonymization attacks without identifying any counter-measures \cite{arp2014torben}.
Research works covering deanonymization can be subdivided into a number of sub-categories including (1) Hidden services identification, (2) Tor traffic identification, (3) Attacking Tor network, (3) Traffic fingerprinting, (4) Focusing over Tor improvements, and (6) Providing anonymity without Tor.
Classification of various research problems is shown in the pie chart in Figure \ref{fig: Classification of Tor's deanonymization approaches.}.
\begin{figure}[b]
\centering
\includegraphics[width=0.8\columnwidth]{deanonymization.pdf}\\
\caption{Classification of Tor's deanonymization approaches.}\label{fig: Classification of Tor's deanonymization approaches.}
\end{figure}
Table \ref{tab: Researches on Tor's Deanonymization.} presents a comparison of various research works in the Tor's deanonymization track.
Prominent patterns show that relay compromise and traffic interception are the most frequent factors in deanonymizing Tor.
This suggests that relays and traffic are more susceptible for exploitation than other factors.
Individual details of various research works in following subsections would explain this exploitation in much detail.
\begin{table*}
\caption{Research works focused on Tor's deanonymization. Table entries symbolize attacks (Att.), counter-attacks (Cou. Att.), Analysis (Ana.), relays (Rel.), Autonomous systems (AS), browser (brows.), server (serv.), decoy traffic (Dec. Traf.), protocol messages (Prot. Mess.), traffic interception (Traf. Interc.), Flag cheating (Flag Cheat.).}
\label{tab: Researches on Tor's Deanonymization.}
\footnotesize
\centering
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\hline
Research & \multicolumn{3}{|c|}{Focus} & \multicolumn{8}{|c|}{Exploited Tor's weakness} & Idea\\
\cline{2-12}
& Att. & Cou. & Ana. & \multicolumn{4}{|c|}{Compromised} & Dec. & Prot. & Traf. & Flag & \\
\cline{5-8}
& & Att. & & Rel. & AS & brows. & serv. & traf. & mess. & interc. & Cheat. &\\
\hline \hline
Overlier and Syverson \cite{overlier2006locating} & \checkmark & & \checkmark & & & & & & & \checkmark & &Timing signature analysis attack, service location attack, predecessor attack and distance attack \\
Elices \emph{et al.} \cite{elices2011fingerprinting} & \checkmark & & & & & & & & & \checkmark & &Fingerprint analysis attack \\
Zhang \emph{et al.} \cite{zhang2011application} & \checkmark & & & & & & & & & \checkmark & & Application level time correlation attack \\
Biryukov \emph{et al.} \cite{biryukov2013trawling} & \checkmark & \checkmark & & & & & & & & & \checkmark & Using corrupted relay node and cheating Tor's flag marking mechanism \\
\hline
\multicolumn{13}{|c|}{Tor's Traffic identification} \\
\hline
Bai \emph{et al.} \cite{bai2008traffic} & \checkmark & & & & & & & & & \checkmark & &Packet examination, context checking and matching \\
Barker \emph{et al.} \cite{barker2011using} & \checkmark & & & & & & & & & \checkmark & &Using unsupervised macine learning tachniques over packet sizes \\
AlSabah \emph{et al.} \cite{alsabah2012enhancing} & \checkmark & & & & & & & & & \checkmark & &Application level time correlation attack \\
Houmansadr \emph{et al.} \cite{houmansadr2013parrot} & \checkmark & & & & & & &\checkmark & & & &Passive and active attacs to bypass traffic imitation technique \\
Chakravarty \emph{et al.} \cite{chakravarty2008identifying} & \checkmark & \checkmark & & & & & & & & & & Observing bandwidth fluctuations through compromised node\\
Winter and Lindskog \cite{winter2012great} & \checkmark & & & & & & & & &\checkmark & & Using port tuples\\
\hline
\multicolumn{13}{|c|}{Tor attacks} \\
\hline
Sulaiman and Zhioua \cite{sulaiman2013attacking} & \checkmark & & & \checkmark & & & & & & & &Using unpopular ports over compromised relays \\
Chan-Tin \emph{et al.} \cite{chan2013revisiting} & \checkmark & & & \checkmark & & & \checkmark & & & & &Using malicious servers to observe traffic fluctuations over relays \\
Pries \emph{et al.} \cite{pries2008new} & \checkmark & & & \checkmark & & & & \checkmark & & & &Passing duplicate cells through compromised entry relay \\
Wang \emph{et al.} \cite{wang2009novel} & \checkmark & & & \checkmark & & & & & & & &Returning malicious page through compromised exit relay \\
Wagner \emph{et al.} \cite{wagner2012breaking} & \checkmark & & & \checkmark & & \checkmark & & & & & &Compromised exit node to send images which is used by semi-supervised learning algorithm \\
Benmeziane and Badache \cite{benmeziane2010tor} & \checkmark & & & & & & & & \checkmark & & &Using destroy and DSN requests for man-in-the-middle attack \\
Jansen \emph{et al.} \cite{jansen2014sniper} & \checkmark & & & & & & & & \checkmark & & &Using valid protocol messages over relays to perform denial of service attack \\
Abbot \emph{et al.} \cite{abbott2007browser} & \checkmark & & & \checkmark & & \checkmark & & & & & &Use compromised relay and user's browser for man-in-the-middle attack \\
Evans \emph{et al.} \cite{evans2009practical} & \checkmark & & & \checkmark & & & & & & & &Use exit relay to inject javascript for DoS attack \\
Bauer \emph{et al.} \cite{bauer2007low} & \checkmark & & & \checkmark & & & & & & & &Advertise low bandwidth to divert traffic towards malicous nodes \\
Edman and Syverson \cite{edman2009awareness} & \checkmark & & & & \checkmark & & & & & & &Using autonomous systems to breach Tor traffic \\
Barbera \emph{et al.} \cite{barbera2013cellflood} & \checkmark & & & & & & \checkmark & & & & &Perform DoS attack by placing large load on Tor routers \\
Le Blond \emph{et al.} \cite{blond2011one} & \checkmark & & & \checkmark & & & & & & & &Using peer-to-peer applications with compromised exit relays to deanonymize users \\
Geddes \emph{et al.} \cite{geddes2013low} & \checkmark & & & \checkmark & & & & & & & & Exploiting compromised exit node to advertise high stats to attract large traffic \\
Chakravarty \emph{et al.} \cite{chakravarty2011detecting} & & \checkmark & & & & & & &\checkmark & & & Using decoy traffic to detect traffic interception \\
\hline
\multicolumn{13}{|c|}{Tor traffic analysis attacks} \\
\hline
Johnson \emph{et al.} \cite{johnson2013users} & \checkmark & & & & & & & & &\checkmark & & Correlation based attacks using a single compromised relay \\
Murdoch and Danezis \cite{murdoch2005low} & \checkmark & & & \checkmark & & & & & & \checkmark & & Use timing signature attack by passing large traffic from corrupted node to Tor relays \\
Chakravarty \emph{et al.} \cite{chakravarty2014effectiveness} & \checkmark & & & & & & \checkmark & & & \checkmark & & Generate traffic between two servers and map relays through correlation \\
Zhang \emph{et al.} \cite{zhang2008novel} & & \checkmark & & & & & & & & \checkmark & & Suggested priority queue algorithm to bypass correlation between load and latency \\
Song \emph{et al.} \cite{song2013anonymize} & \checkmark & & & & & & & & &\checkmark & & Use time and stream size with k-means algorithm to deanonymize users \\
Panchenko \emph{et al.} \cite{panchenko2011website} & \checkmark & & & & & & & & &\checkmark & & Use volume, time and direction for classification \\
Wang and Goldberg \cite{wang2013improved} & \checkmark & & & & & & & &\checkmark & & & Use Tor cells for website fingerprinting \\
Jin and Wang \cite{jin2009effectiveness} & \checkmark & & & & & & & & & \checkmark & & use wavelet based decomposition to estimate timing distortion \\
Gilad and Herzberg \cite{gilad2012spying} & \checkmark & & & & & & & & \checkmark & & & Breach by off-path TCP connection or eavesdrop on clients \\
\hline
\multicolumn{13}{|c|}{Tor Improvements} \\
\hline
Gros \emph{et al.} \cite{gros2010protecting} & & \checkmark & \checkmark & \checkmark & & & & & & & & Proposed Honeywall to rank node's reliability \\
Winter and Lindskog \cite{winter2014spoiled} & & \checkmark & & \checkmark & & & & & & & & Proposed exit relay scanner to avoid misuse of exit node \\
Xin and Neng \cite{xin2009design} & & \checkmark & & \checkmark & & & & & & & & Proposed a tuning mechanism to keep a track of reliable nodes \\
Backes \emph{et al.} \cite{backes2012provably} & & \checkmark &\checkmark & & & & & & \checkmark & & & Identified flaws in current key exchange mechanisms \\
Marks \emph{et al.} \cite{marks2010unleashing} & & \checkmark & & & & & & & \checkmark & & & Suggested separate bi-directional TCP links to increase anonymity \\
Nowlan \emph{et al.} \cite{nowlan2013reducing} & & \checkmark &\checkmark & & & & & & \checkmark & & & Suggested use of uTCP and uTLS to avoid head of line blocking problem \\
Danner \emph{et al.} \cite{danner2012effectiveness} & & \checkmark &\checkmark & \checkmark & & & & & & & & Investigated DoS attack and proposed improvements to avoid it \\
\hline
\multicolumn{13}{|c|}{Anonymity without Tor} \\
\hline
Herzberg \emph{et al.} \cite{herzberg2011camouflaged} & &\checkmark &\checkmark & & & & & \checkmark & & & & Suggested camouflaged web server by mimicking GMAIL traffic \\
Mendonca \emph{et al.} \cite{mendonca2012flexible} & &\checkmark & & & \checkmark & & & & & & & Proposed concealed source identifier through network service provider \\
\hline
\end{tabular}
\end{table*}
\subsubsection{Tor Hidden Services}
An important feature of the Tor network is provisioning of Tor service through a hidden server.
A series of protocols used by hidden server and Tor users can make location of hidden server invisible to client \cite{murdoch2006hot}.
However, several studies listed below address the deanonymization of hidden servers.
\emph{Locating Hidden Server:} Overlier and Syverson \cite{overlier2006locating} presented new attack strategies to detect the location of hidden servers using only one Tor node. They proposed changes in route selection and relay selection to increase anonymity. The average duration of the attack varied from minutes to a few hours. The various attacks they considered included the timing signature analysis attack, service location attack, predecessor attack and distance attack. Their proposed solution included introducing middleman nodes to connect to rendezvous points, introducing dummy traffic, extending hidden server path to rendezvous point and using guard entry nodes.
\emph{Timing Signature Attack:} Elices \textit{et al.} \cite{elices2011fingerprinting} presented a fingerprint analysis attack for Tor's hidden services. They used timestamps from logs of machines hosting hidden services on the Tor network to generate detectable fingerprints. The authors studied delay properties of the Tor network and other users' log entries to make the fingerprint attack feasible.
\emph{Application Layer Correlation Attack:} Zhang \textit{et al.} \cite{zhang2011application} described an application level HTTP-based attack for Tor's hidden services. Time correlation was used to assess the resemblance between web accesses and the traffic generated in a compromised Tor router. This attack assumes that the compromised onion router can operate as an entry relay.
\emph{Detection, Measurement and Deanonymization of hidden services:} Biryukov \textit{et al.} \cite{biryukov2013trawling} analyzed weaknesses in hidden services which can be exploited by attackers to detect, measure and deanonymize hidden services running over the Tor network. Services of three different applications were analyzed, (1) Botnet for command and control, (2) Silk Road\footnote{Silk road was an online market place which provided anonymity to its customers by way of the Tor network. It was used in great part for the sale of drugs and illegal materials and was shutdown by the FBI. Defunct Website: \url{http://silkroad6ownowfk.onion}} and (3) DuckDuckGo\footnote{DuckDuckGo is a search engine that does not track its users and provides anonymity to users by giving same search results for any query to all users. Website: \url{https://duckduckgo.com}}. The study identified major flaws of Tor that included the inflation and cheating of bandwidth by a corrupted relay node, and cheating marking mechanism of flags in Tor network from attacker relay node.
\subsubsection{Tor Traffic Detection}
A number of studies focus their research on the identification of Tor traffic form other network traffic. These studies suggest that differentiation of traffic can ultimately be used to block Tor traffic, as done by China a number of times in the recent past \cite{winter2012great}. Various approaches for traffic identification are summarized as follows.
\emph{Tor Traffic Identification from Network Traffic:} Bai \textit{et al.} \cite{bai2008traffic} studied the traffic identification mechanisms of popular anonymity tools, i.e., Tor and Web-Mix. Authors used fingerprint identification (packet examination and packet context checking) followed by matching to identify the traffic. Key attributes used for traffic identification from other network traffic include specific strings, packet length and packet transmission frequency in the network.
\emph{Differentiate Tor Traffic from Encrypted Traffic:} Barker \textit{et al.} \cite{barker2011using} showed that traffic from the Tor network can be differentiated form encrypted traffic in the network. They captured regular HTTPS, Tor HTTPS and HTTP traffic routed through Tor and analyzed their packet sizes and developed an unsupervised machine learning (ML) classifier that operates only on packet size attribute with $97.54\%$ true positive (TP) and $1.06\%$ false positive (FP) rates.
\emph{Differentiating Tor Traffic:} AlSabah \textit{et al.} \cite{alsabah2012enhancing} developed an ML classifier to differentiate web traffic from bulk download traffic. AlSabah \textit{et al.} used the following four features to classify Tor traffic: (1) Circuit lifetime, (2) data transferred, (3) cell inter-arrival times, and (4) recently sent cells. They tested na\"ive Bayes, Bayesian networks and decision tree classifiers. Using the proposed classification method, they reported $75\%$ improvement in responsiveness and $86\%$ decrease in download rates.
\emph{Fingerprinting Tor traffic:} Houmansadr \textit{et al.} \cite{houmansadr2013parrot} aimed to differentiate the traffic of anonymous networks from other network traffic. They claimed that mimicking other traffic is an obsolete way for anonymity. They devised a number of passive and active attack strategies to breach anonymous networks. Their study suggested the use of partial imitation and use of new strategies by incorporating popular protocols like HTTPS email etc.
\emph{Tor Proxy Node Identification:} Chakravarty \textit{et al.} \cite{chakravarty2008identifying} described a novel attack that identifies all Tor relays participating in a given circuit. The attack modulates the bandwidth of an anonymous connection through a compromised server, router or an entry point and observes the resultant fluctuations in the Tor network using \textit{LinkWidth} \cite{chakravarty2008linkwidth}. LinkWidth sends a train of pulses comprising of alternate TCP-SYN and TCP-RST packets and capacity is computed at the receiver end by estimating packet dispersion. Authors reported a $59.46\%$ TP rate and $10\%$ true negatives rate for compromised Tor relays using the proposed strategy.
\emph{Identification of Tor Bridges:} Winter and Lindskog \cite{winter2012great} conducted an extensive investigation into the blocking of Tor relays and bridges by China. Their investigation showed that Tor bridges were blocked by port tuples, rather than IP addresses and that bridges were blocked only when they were active. Their investigation also showed that adversaries did not conduct traffic fingerprinting for domestic traffic and that packet fragmentation could be used to circumvent China's firewall.
\emph{Fingerprinting Keywords in Search Queries:} Oh \emph{et al.} \cite{oh2017fingerprinting} investigated the viability of keyword fingerprinting attacks in the Tor network. Study showed that effective feature selection can help any passive adversary in figuring out the identity of the user. Time and volume of traffic play the most crucial role in determining the identity of the user. Among other features keyword sets, incremental search and high security search are other features used for classification. Experimental results demonstrated recall, precision and accuracy of 80\%, 91\% and 48\%, respectively for one of 300 targeted keywords of Google.
\subsubsection{Tor Attacks}
Attacking the Tor network is an interesting research dimension which ultimately aims to block access to it. Several attempts by China and other countries have failed in the recent past because Tor is being improved continuously \cite{winter2012great}. In this subsection, we summarize various studies covering Tor attacks.
\emph{Unpopular Ports Attack:} Sulaiman and Zhioua \cite{sulaiman2013attacking} described an attack they developed which can compromise circuits in the Tor network. Their attack takes advantage of unpopular ports in the Tor network. Sulaiman and Zhioua added a small number of compromised entry /exit relays to the Tor network ($\thicksim30$ relays) which permit the use of unpopular ports. By doing so, $50\%$ of developed circuits can be compromised, which significantly decreases the anonymity of the Tor network.
\emph{Circuit Clogging Attack:} Chan-Tin \textit{et al.} \cite{chan2013revisiting} proposed an attack that can identify the Tor routers used in any circuit. For the proposed attack a client connects to a malicious server which sends data to the client in large bursts and in small amounts. During large bursts, Tor routers take long times to process the extra amount of data. Authors showed that continuous monitoring of all Tor relays can identify the Tor relays used in the particular circuit. A mechanism to detect the behavior of malicious routers by the client was also evaluated, which measured network latency of the client.
\emph{Replay Attack:} Pries \textit{et al.} \cite{pries2008new} suggested a replay attack to detect the exit routers in the Tor network. The replay attack assumes that the entry onion router is compromised. The replay attack duplicates packets coming from a sender. Tor uses counter-mode Advanced Encryption Standard (AES-CTR)\cite{daemen2013design}\footnote{Advanced Encryption Standard (AES) is an encryption standard which is based upon substitution-permutation technique. It has three members Rijndael family each with block size of 128 bits and key lengths of 128, 192 and 256 bits.\cite{standard2001announcing}} for encryption and decryption, any duplicate cells will give a cell recognition error at the exit routers. This behavior leaks exit router information to the entry router by simple correlation.
\emph{Flow Multiplication Attack:} Wang \textit{et al.} \cite{wang2009novel} designed a flow multiplication attack similar to a man-in-the-middle attack. The attack assumes that the exit router is compromised. Whenever a client sends a request to target server, the exit router returns a malicious page which triggers certain fetch requests in the client browser over the same circuit. An accomplice at the entry router can see the requests, and together with knowledge of the exit relay, identify the complete Tor circuit.
\emph{Attack Using Game Theory and Data Mining:} Wagner \textit{et al.} \cite{wagner2012breaking} proposed an attack which exploits the exit malicious exit node to cluster observed traffic flows using an active tag injection scheme. The proposed method has two steps, (1) image tags are injected into HTML replies from the exit node to the user, and (2) a semi-supervised learning algorithm based upon deep data mining is used to reconstruct the entire browsing session of the user. The authors model the Tor network in form of a game theoretical concept where all Tor users and rogue nodes play a game for identification of malicious node. Once a rogue node has been identified, it's game is over because no other user uses it due to presence of special flag in it. Authors main aim is to work over over the equilibrium between rogue nodes and Tor users.
\emph{Attack using Destroy and DNS Requests:} Benmeziane and Badache \cite{benmeziane2010tor} investigated possible breaches of Tor targeting its network requests. They exploited \textit{destroy requests} (Tor's circuit destruction requests) and DNS requests to break anonymity. Destroy requests are not encrypted, which poses a serious threat to Tor. Moreover, a local eavesdropper can use the man-in-the-middle attack strategy against DNS requests, which are unprotected.
\emph{The Sniper Attack:} Jansen \textit{et al.} \cite{jansen2014sniper} presented the Sniper attack, a low-resource denial-of-service (DoS) attack against the Tor network which can disable arbitrary relays. The adversary builds a Tor circuit through the target relay and starts obtaining a large file by continuously sending the SENDME cells (protocol messages for continuously receiving the file), which increases the congestion window size. By repeating over multiple circuits, memory of host of target relay would exhaust which can disrupt the functioning of Tor relay. Experiments showed that an adversary can consume $2,187$ KB/s memory of a victim relay at the cost of very little bandwidth and decrease Tor network bandwidth by as much as $35\%$.
\emph{Browser-based Attacks:} Abbot \textit{et al.} \cite{abbott2007browser} proposed a novel attack that tricks a user's web browser into sending a distinctive signal over the Tor network (by installing a Java or HTML script). An attacker that controls an exit relay can use it in a man-in-the-middle attack to mirror and forward duplicated traffic to a malicious server. By analyzing the data, the malicious server can deanonymize the Tor user. However, this study makes two significant assumptions: the ability to control the exit relay and the ability to configure / compromise a targeted user's web browser.
\emph{Congestion Attack using Long Paths:} Evans \textit{et al.} \cite{evans2009practical} proposed an extension to the congestion attack proposed by Murdoch and Danezis \cite{murdoch2005low} owing to the enormity of the current Tor network. Evans \textit{et al.} proposed the combination of Javascript injection and DoS attack. A Tor exit relay is used by the attacker to inject Javascript code into a user's browser, which makes the browser send a response every second. They suggested modifications such as disabling JavaScript, thwarting DoS attack by disabling ability to control latency of routers. In the modified design, routers keep a track of all paths with flags and disable any request for latency by using flags.
\emph{Exploiting Routing Algorithm:} Bauer \textit{et al.} \cite{bauer2007low} exploited Tor's routing algorithm to steer a disproportionate number of users towards selecting their entry and exit relays from a set of malicious Tor routers. Bauer \textit{at al.} suggested that low-latency constraints force Tor's routing algorithm to prefer nodes advertising high bandwidths. Instead of performing complex traffic analysis techniques, the authors suggested to collect detailed flow logs from malicious nodes (both entry and exit nodes) and use the information of node selection to deanonymize flows.
\emph{Analyzing Autonomous Systems for Tor Path Selection:} Edman and Syverson \cite{edman2009awareness} analyzed the effect of autonomous systems (AS) for path selection in Tor network. They studied the selection of AS residing in different countries and found it quite effective. Traffic analysis of the Tor network showed that majority of traffic passes through a few ASs because all established paths focus over latency and anonymity which occurs better in some ASs. Analysis shows that increase in relays has not increased the diversity to a large extent.
\emph{DoS Attack using Cell Flooding}: Barbera \textit{et al.} \cite{barbera2013cellflood} presented a novel attack which generates a few circuits requiring large computing and networking resources. Their study showed that this attack requires only $0.2\%$ resources for old routers and $1-16\%$ router resources for new attacks, which makes it an inexpensive attack to execute. Barbera \textit{et al.} proposed a mitigation scheme by placing an upper cap on the utilization of resources at routers.
\emph{Exploiting peer-to-peer application:} Le Blond \textit{et al.} \cite{blond2011one} suggested that peer-to-peer applications can be exploited to trace IP addresses of users running Tor. Moreover, scan of malicious Tor exit relays should be used to correlate various user streams for deanonymization. Experiments showed that their `bad apple' attack was able to identify $193\%$ more streams, including $27\%$ HTTP streams, and reveal IP addresses of $10,000$ Tor users. This constituted $9\%$ of all the flows passing through the exit relays under their control.
\emph{Induced Throttling Attacks:} Geddes \textit{et al.} \cite{geddes2013low} proposed a new attack which breaches the Tor network by exploiting its selection bias in favor of high capacity relay nodes. Authors showed that induced throttling at the corrupt exit node by exploiting congestion or traffic shaping algorithms can induce similar traffic patterns at other relays associated with the corrupted exit relay.
\emph{Using Decoy Traffic:} Chakravarty \textit{et al.} \cite{chakravarty2011detecting} used decoy traffic on anonymous networks to detect traffic interception. The proposed strategy is based on the idea of injecting traffic containing bait credentials for decoy services requiring user authentication. Chakravarty \textit{et al.} set up decoy IMAP and SMTP servers and identified ten instances of traffic interception over ten months.
\subsubsection{Tor Traffic Analysis Attacks}
A few studies have focused on the analysis of Tor traffic for breaching this network. Analysis shows that traffic analysis can provide an efficient mechanism for deanonymization. A few of these studies are summarized as follows.
\emph{Traffic Correlation Attacks:} Johnson \textit{et al.} \cite{johnson2013users} conducted a thorough analysis of the Tor network with a deep focus on the development of a threat model. They built the Tor path simulator (TorPS) to assess Tor's vulnerability to correlation based attacks. Their study suggested that a single Tor relay adversary can deanonymize $80\%$ of users within six months. This research showed that set of relays is dependent upon the user's application which reduces security of the Tor network.
\emph{Using Timing Signature:} Murdoch and Danezis \cite{murdoch2005low} presented a simple mechanism to evaluate the Tor nodes being used in a circuit. In the proposed scheme, a malicious node sends probe data to the Tor relays. All Tor relays used in the circuit will experience a delay and client-server communication will be modulated. Hence, correlation between delay and modulation gives insight about the relays being used in a circuit.
\emph{Traffic Analysis Attack:} Chakravarty \textit{et al.} \cite{chakravarty2014effectiveness} used NetFlow data to analyze the effectiveness of traffic analysis attacks against Tor network. Their proposed attack creates variations in traffic at the server end and observes the effects at a colluding server at the other end. They reported $81.4\%$ accuracy in real-world experiments with $6.4\%$ FP rates.
\emph{Queue Scheduling and Resource Allocation:} Zhang \textit{et al.} \cite{zhang2008novel} proposed a priority queue scheduling mechanism to reduce the correlation between high load and high latency which would ultimately increase the level of anonymity. However, increase in anonymity comes at the cost of latency which degrades quality of service at the user end. Extensive experiments using the proposed mechanism showed an increase in anonymity due to decrease in correlation between load and latency.
\emph{Correlation Using K-means Algorithm:} Song \textit{et al.} \cite{song2013anonymize} applied machine learning techniques to deanonymize Tor flows at the first hop and last hop in the network. They used the time / stream size tuple of attributes together with the \emph{k-means} algorithm to deanonymize by matching first hop traffic with last hop traffic. Their results showed that as little as $8$ packets are enough to deanonymize a Tor stream with greater than $99\%$ accuracy.
\emph{Website Fingerprinting Using Machine Learning:} Panchenko \emph{et al.} \cite{panchenko2011website} suggested the use of machine learning approaches requiring feature selection and classification for website fingerprinting. Authors used Support Vector Machine (SVM) classifier with various features including packet sizes (except 52 size packets because of excess use in acknowledgements), packet size markers to express direction of flow, HTML markers, total transmitted bytes, number markers, occurring packet sizes, percentage incoming packets and number of packets. Extensive research showed that volume, time and direction of the traffic were the most promising features and classification of close-world and open-world dataset gave $55\%$ detection rate. However, camouflaging the traffic decreased the detection rate to $3\%$.
\emph{Website Fingerprinting Using New Metrics:} Wang and Goldberg \cite{wang2013improved} proposed the use of Tor cells as a unit of data transfer rather TCP/IP packets for website fingerprinting. Authors collected data using realistic assumptions on adversaries from client to entry guard node. The study suggested the removal of SENDME cells as they do not play any significant role in improving performance. Proposed metrics use the observation that dynamic content is present in only incoming packets and it is present at the end of the packets. Upto $95\%$ recall rate and $0.2\%$ FP rate is observed using SVM classifier.
\emph{Wavelet Decomposition Attack:} Jin and Wang \cite{jin2009effectiveness} suggested a wavelet based decomposition mechanism to estimate the distortion in timing at the receiver end of Tor network. Authors showed that wavelet based multi-resolution analysis (MRA) captures the variability of the timing distortion at all levels, with better granularity than traditional estimation of timing distortion. Deanonymization rate of 96\% was obtained for Tor at a packet rate of 4 pkts/sec in 3 minutes without changing established paths (circuits) of Tor. Analysis showed that Tor circuit rotation could decrease the accuracy of deanonymization to 72\% after 5 Tor circuit rotations in $3$ minutes.
\emph{Exploiting Side-channels to Identify Clients:} Gilad and Herzberg \cite{gilad2012spying} exploited three kinds of side-channels including (1) globally incrementing IP identifiers, (2) packet processing delays, and (3) bogus-congestion events. Sequential port allocation is also used to identify the clients. Two scenarios for breaching have been presented including (1) fully off-path attack to detect TCP connections, and (2) detecting Tor connections by eavesdropping on clients.
\subsubsection{Tor Improvements}
In this section, we present some miscellaneous studies about improvement in Tor network by focusing on Tor relays, path selection mechanisms, transport layer protocols and application layer improvements.
\emph{Misusing Tor Exit Node:} Gros \emph{et al.} \cite{gros2010protecting} studied the abuse and misuse of Tor exit nodes to compromise the anonymity of the Tor network. Authors proposed a mechanism, called \emph{Honeywall}, to avoid misuse of any Tor exit node. According to Honeywall, whenever any exit node detects a malicious behavior, it lowers the reputation of the immediate predecessor router and also sends an alert to it. Similarly, the intermediate router lowers the reputation of its predecessor router. Through this strategy, all ``bad'' nodes eventually end up with lower reputations and all ``good'' nodes have higher reputation.
\emph{Exposing Exit Relays:} Winter and Lindskog \cite{winter2014spoiled} detected malicious exit Tor relays and profiled their behavior. An exit relay scanner was built to identify all outgoing Tor traffic and identify the malicious nodes and avoid man-in-the-middle attacks. Patches were built for the Tor browser bundle to collect certificates through multiple paths to check authenticity of the destination server.
\emph{Tuning mechanism for Tor:} Xin and Neng \cite{xin2009design} showed that Tor lacks the evaluation system for the node store. Authors presented and theoretically analyzed a tuning mechanism for Tor. The proposed tuning system included the establishment of an evaluation system and optimization of Tor node store and output mode. Through the evaluation system, all nodes are ranked based on their anonymity, uptime, bandwidth and latency. In the optimization stage authors suggested to use a fixed number of circuits such that traffic load has least effect on latency.
\emph{Increasing Security of Tor Network:} Backes \emph{et al.} \cite{backes2012provably} conducted research on the security of Tor network for anonymous browsing and presented a novel security protocol. Authors elaborated the concept of security in anonymity softwares. Their study showed that current key exchange algorithms are inefficient and a number of security enhancements were suggested including cryptographic requirements for secure browsing.
\emph{Transport Layer Improvements:} Marks \emph{et al.} \cite{marks2010unleashing} studied TCP based deficiencies in the Tor network. By studying the transmission mechanism of Tor, authors proposed to split bidirectional links into two separate TCP links. Experiments with separate TCP links showed 100\% increase in throughput with a decrease in throughput variance from 43000 KB/s to 10000 KB/s \cite{marks_unleashingtor_thesis}.
\emph{Switching to uTCP and uTLS:} Nowlan \emph{et al.} \cite{nowlan2013reducing} probed into the cross-stream head of line blocking problem of TCP in the Tor network. Their study suggested the use of unordered TCP (uTCP) and unordered TCL (uTLS) for reducing inter-dependence in inter-leaving streams, due to the requirement of low latency in the Tor network.
\emph{Feasibility of DOS attack over Tor:} Danner \emph{et al.} \cite{danner2012effectiveness} conducted a deep investigation on the feasibility of Denial-of-Service (DoS) attack (proposed in a previous study by Borisov \emph{et al.} \cite{borisov2007denial}) over Tor network. Authors showed through simulations and analytical evaluations that corrupted relay nodes can be used to exploit Tor network and perform DoS attack. Authors suggested the use of reliable guard nodes (entry and exit) which can decrease the probability of selection of a corrupt Tor relay.
\subsubsection{Anonymity without Tor}
To present a glimpse of studies providing anonymity without Tor, we present a few studies focusing on packet encapsulation and central server based anonymity mechanism.
\emph{Anonymity Using a Central Server:} Herzberg \emph{et al.} \cite{herzberg2011camouflaged} proposed a camouflaged browsing design using a camouflaged server. The basic idea is to communicate with the camouflaged server using a manner similar to popular web services. Encrypted communication, URLs of GMAIL with packet frequency and sizing similar to GMAIL can easily pass unnoticed through any adversary. Although this design provides better anonymity, it suffers from a single point of failure.
\emph{In-Network IP Anonymization Service:} Mendonca \emph{et al.} \cite{mendonca2012flexible} presented a novel idea of user anonymity by working with a network service provider. Proposed service \emph{AnonyFlow} used an in-network IP anonymization service. The fundamental idea was to conceal the source identifier from the other side of the network. An OpenFlow based implementation was used for performance evaluation. However, anonymity could be breached by compromising the network service provider.
\subsection{Tor Path Selection}
Tor selects three relays based upon its path selection algorithm which incorporates anonymity and reliability characteristics of relays and users \cite{dingledine2004tor}. By compromising the path selection mechanism, the complete anonymity mechanism of Tor can be breached. In this subsection, we present studies covering (1) new algorithms for path selection, and (2) analysis of path selection algorithms. An overview showing the classification of major studies is shown in Figure \ref{fig: Classification for Tor's Path Selection approaches.}.
\begin{table*}[t]
\caption{Research works on Tor's path selection. Table entries symbolize New algorithms (New Algo), Analysis (Anal.), Autonomous Systems (AS), Relay Locations (Relay loc.), Hops, Performance-Latency-Bandwidth (Perf., Lat, BW), Multi-path, Load, Relay Capacity (Rel. Cap.) and Anonymity (Anon).}
\label{tab: Researches on Tor's Path Selection.}
\centering
\small
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}
\hline
& \multicolumn{2}{|c|}{Focus} & \multicolumn{8}{|c|}{Path Selection Parameters} & Idea\\
\cline{2-11}
Research & New & Anal. & AS & Relay & Hops & Perf. & Multi- & Load & Rel. & Anon. & \\
& Algo & & & loc. & & Lat, BW & path & & Cap. & & \\
\hline
\multicolumn{12}{|c|}{New Path Selection Algorithms} \\
\hline
Akhoondi \emph{et al.} \cite{akhoondi2012lastor} & \checkmark & & \checkmark &\checkmark & & & & & & & Included relay locations and autonomous system reliability\\
Chen \emph{et al.} \cite{chen2010toward} & \checkmark & & &\checkmark &\checkmark &\checkmark & & & & \checkmark & Included hops and geographic distance in path selection\\
Karaoglu \emph{et al.} \cite{karaoglu2012multi} &\checkmark & & & & & \checkmark & \checkmark & & & & Studied multipath design\\
Panchenko \emph{et al.} \cite{panchenko2012improving} &\checkmark & & & & & \checkmark & & \checkmark & \checkmark & & Studied Load and Capacity at nodes\\
Li \emph{et al.} \cite{li2012tmt} &\checkmark & & & & & \checkmark & & & & \checkmark & Proposed tunable mechanism varying between anonymity and performance\\
Panchenko \emph{et al.} \cite{panchenko2008performance} &\checkmark &\checkmark & & \checkmark & & \checkmark & & &\checkmark & & Studied latency, link capacity and load at nodes\\
Liu and Wang \cite{liu2009random} &\checkmark & & & & &\checkmark & &\checkmark & &\checkmark & Proposed random walk based circuit building protocol\\
Liu and Wang \cite{liu2009improved} &\checkmark & & & & &\checkmark & & & &\checkmark & Proposed new relay selection mechanism with backup circuit algorithm\\
Snader and Borisov \cite{snader2011improving} &\checkmark & & & & &\checkmark &\checkmark & & &\checkmark & Studied malicious nodes, proposed balance between anonymity and performance\\
Li \emph{et al.} \cite{li2012relay} &\checkmark & & & & &\checkmark & & & &\checkmark & Proposed relay recommendation system\\
Tang and Goldberg \cite{tang2010improved} &\checkmark & & & & &\checkmark & & & & & Suggested the use of bursty circuits instead of busy paths\\
Wang \emph{et al.} \cite{wang2012congestion} &\checkmark & & & & &\checkmark & & & & & Included latency as a measure of congestion in path selection\\
Snader and Borisov \cite{snader2008tune} &\checkmark & & & & &\checkmark & & & &\checkmark & Suggested opportunistic bandwidth measurement with priority based traffic handling\\
Elahi \emph{et al.} \cite{elahi2012changing} &\checkmark &\checkmark & &\checkmark & & & & & & & Discouraged short term entry guard churn and time-based entry guard rotation\\
\hline
\multicolumn{12}{|c|}{Analysis of Path Selection} \\
\hline
Bauer \emph{et al.} \cite{bauer2009predicting} & &\checkmark & &\checkmark & & & & &\checkmark & & Suggested random or Snader-Borisov approach for router selection\\
Wacek \emph{et al.} \cite{wacek2013empirical} & &\checkmark & & & &\checkmark & & & &\checkmark & Suggested bandwidth weighted relay selection and avoidance of congested circuits\\
\hline
\end{tabular}
\end{table*}
\begin{figure}[b]
\centering
\includegraphics[width=0.8\columnwidth]{path_selection.pdf}\\
\caption{Focus of published research on Tor's Path Selection approaches.}\label{fig: Classification for Tor's Path Selection approaches.}
\end{figure}
Table \ref{tab: Researches on Tor's Path Selection.} presents a comparison of various research works in Tor's path selection track. Comparison shows that performance and anonymity were the most frequently studied parameters for path selection. However, majority studies neglected autonomous systems, relay locations, hop counts, multi-path mechanisms, load and relay capacity. Moreover, majority research works focus on the development of new algorithms while few studies analysed the current path selection algorithms.
\subsubsection{New Path Selection Algorithms}
\emph{LASTor - Low Latency with Better Anonymity Algorithm:} Akhoondi \emph{et al.} \cite{akhoondi2012lastor} proposed a new path selection algorithm namely \emph{LASTor}. LASTor incorporates the locations of relays before choosing paths and does not always select the shortest path as it reduces the entropy of path selection. Moreover, LASTor avoids paths passing through ASs which can compromise anonymity of the system by traffic correlation.
\emph{Optimizing Hops, Performance flags and Geographic Distance:} Chen and Pasquale \emph{et al.} \cite{chen2010toward} studied the path selection mechanism by varying the number of hops, performance ratings and changing the geographic distance between routers. Trade-offs between anonymity and other parameters (latency etc.) were extensively evaluated. The authors concluded that reduction in hops and geographic distance can increase throughput and decrease anonymity.
\emph{Using MultiPath Routing} Karaoglu \emph{et al.} \cite{karaoglu2012multi} evaluated the multipath design for Tor network to avoid congestion and overcome the limitations in Tor's circuit construction. Evaluations revealed a four-fold increase in throughput with better load balancing and traffic mixing. However, high buffer costs at the Tor proxies were the major limitations of multipath design.
\emph{Path Selection Using \emph{Load} and \emph{Capacity} of Nodes:} Panchenko \emph{et al.} \cite{panchenko2012improving} studied the delays in the Tor network and provided new measures in path selection to improve user experience. Two factors used for path selection design are ``load'' at the nodes and maximum ``capacity'' at the nodes. Authors showed that these factors can increase the performance by $70\%$. Their study concludes that nodes, not edges, are the deciding factors for performance.
\emph{Tunable Mechanism of Tor:} Li \emph{et al.} \cite{li2012tmt} emphasized the development of a tunable mechanism for Tor users depending on \emph{anonymity} and \emph{performance} required by users. Authors used ``path length'' as a metric to tune user requirements based upon anonymity and performance followed by client side modifications of Tor protocol. Results showed that browsing time deteriorates quickly from $14.4$ to $140.1$secs with a $37.3\%$ increase in failure rate by increasing the path length from $2$ to $6$. The proposed mechanism requires only client side modification.
\emph{Using Latency and Link Capacity:} Panchenko \emph{et al.} \cite{panchenko2008performance} evaluated the impact of different factors on the performance of the Tor network. Factors considered included overloaded nodes and links and geographical diversity of nodes. Authors presented a novel path selection algorithm based on latency experienced by the nodes and link capacity. Metrics used for evaluations included circuit setup duration, round trip time (RTT), stream throughput and influence of penetration.
\emph{Random Walk Based Circuit Building Protocol:} Liu and Wang \cite{liu2009random} presented a random walk based circuit building protocol (RWCBP) which is a two-step method: circuit construction, followed by application message transmission. Network latency, computational latency and transmission loads were used to analyze the performance of the proposed protocol. Using indexes of performance and anonymity, resilience of the proposed protocol was analyzed.
\emph{New Circuit Building Protocol:} Liu and Wang \cite{liu2009improved} studied the current protocol of the Tor network and proposed a novel circuit building design with two phases: selection of user selectable relay nodes and circuit construction. Authors presented enhancements in the selection of relay nodes, fast circuit construction and backup circuit algorithm. Better performance and user experience are obtained with the new protocol while achieving the same level of anonymity.
\emph{Tunable Path Selection for Better Security and Performance:} Snader and Borisov \cite{snader2011improving} addressed the issue of the selection of malicious nodes in the path selection due to self-advertised bandwidth. Authors proposed an algorithm which is based upon the anonymity and performance in the Tor network. Significant performance gains were observed using the proposed strategy with single and multipath route selection.
\emph{Relay Recommendation System:} Li \emph{et al.} \cite{li2012relay} proposed a relay recommendation system to provide reliable information about all relays for building circuits (paths). Its main goals include the mitigation of low-resource attacks, better performance and tradeoffs between anonymity and performance. Authors proposed path selection algorithms for increased anonymity. Significant performance gains with increase in anonymity were observed in the simulations of the proposed scheme.
\emph{Preferring Bursty Circuits over Busy Circuits:} Tang and Goldberg \cite{tang2010improved} proposed a new algorithm which suggests the use of bursty circuits instead of busy circuits. Authors suggest that bursty circuits (such as web browsing) can provide less latency than the busy circuits (used for bulk data transfer). Proposed circuit selection algorithm uses exponentially weighted moving average (EWMA) of cells sent on any path and uses the path with lowest EWMA (because new and bursty paths have high EWMA).
\emph{Incorporating Congestion in Path Selection:} Wang \emph{et al.} \cite{wang2012congestion} proposed a novel path selection algorithm which incorporates the latency of nodes as a measure for congestion. The proposed algorithm favors nodes which provide lower latency. Study suggests that node latency is greater than the link latency in majority of the cases. Authors conclude that the proposed algorithm can reduce latency by upto $40\%$.
\emph{Opportunistic Bandwidth Measurement Algorithm:} Snader and Borisov \cite{snader2008tune} addressed Tor's shortcoming of favoring high bandwidth nodes based on advertised bandwidth. Their study showed that an opportunistic measurement of bandwidth for all routers by other connected routers can reduce the vulnerability risk by any adversary in Tor. Moreover, priority based traffic handling, i.e., high performance or high anonymity can reduce the risk of partitioning attacks.
\emph{Analyzing and Improving Entry Guard Selection:} Elahi \emph{et al.} \cite{elahi2012changing} conducted an in depth investigation on the selection of entry guards in Tor network. The study showed that short-term entry guard churn and explicit time-based entry guard rotation result in an increased usage of entry guards in clients, which results in a greater number of profiling attacks.
\emph{Trust-Aware Path Selection Algorithm:} Johnson \emph{et al.} \cite{johnson2017avoiding} proposed a path selection algorithm which uses the probability based distribution to keep itself aware of the location of adversaries in the Tor network. In developing trust based model, authors take the relays uptime as the most trustworthy factor in determining the selection of the path. Bypassing the paths containing adversaries can mitigate the traffic analysis attacks conducted by the adversaries.
\subsubsection{Analysis of Path Selection}
\emph{Predicting Path Compromise:} Bauer \emph{et al.} \cite{bauer2009predicting} showed that the current mechanism of Tor is vulnerable to path compromise because Tor selects paths based on bandwidth capabilities of routers. Study shows that the application level protocol is a significant factor to predict path compromise. Research suggests that router selection should be random or through Snader-Borisov approach to avoid any bias in router selection. Study showed that most robust applications for path compromise are HTTP and HTTPs applications while the most vulnerable are peer-to-peer applications.
\emph{Empirical Evaluation of Relay Selection:} Wacek \emph{et al.} \cite{wacek2013empirical} evaluated the relay selection mechanism of Tor to estimate latency. Performance and anonymity were analyzed for a number of relay selection techniques under varying load conditions. The authors suggest that a combination of bandwidth-weighted relay selection and avoidance of congested circuits can provide better throughput and less latency.
\subsection{Tor Analysis and Performance Improvements}
In this section, we cover studies on Tor dealing with its analysis and performance improvement mechanisms. Classification of various studies is shown in Figure \ref{fig: Classification of approaches on Tor's Analysis.}.
\begin{figure}[b]
\centering
\includegraphics[width=0.8\columnwidth]{analysis.pdf}\\
\caption{Focus of various research works on the analysis of Tor.}\label{fig: Classification of approaches on Tor's Analysis.}
\end{figure}
Table \ref{tab: Researches on general study and modelling of Tor} presents a comparison of various generalized studies covering the modelling of Tor network. Comparison shows that majority studies focused over analysis of Tor network. Moreover, usability analysis and anonymity analysis were the most frequently studied topics followed by performance analysis. Very few research works focused over the sociability issues of Tor network.
\begin{table*}
\caption{Research works on general study and modelling of Tor. Table entries symbolize Discussion (Dis.), Analysis (Anal.), Propose (Propos.), Sociability Issues (Soci. Issu.), Usability Issues (Usab.), Performance Latency (Perf. Lat.), Performance - bandwidth (Perf. BW), Anonyimty (Anon.).}\label{tab: Researches on general study and modelling of Tor}
\centering
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\hline
& \multicolumn{3}{|c|}{Study focus} & \multicolumn{5}{|c|}{Research Parameters} & Idea\\
\cline{2-10}
Research & Dis. & Anal. & Propos. & Soci. & Usab. & Perf. & Perf. & Anon. & \\
& & & & issu. & [St./eas] & lat & BW & & \\
\hline
\multicolumn{10}{|c|}{General Study over Tor}\\
\hline
Dingledine \emph{et al.} \cite{dingledine2007deploying} &\checkmark & & & \checkmark & & & & & Discussed challenges and social issues, and studied Tor network\\
Abou-Tair \emph{et al.} \cite{abou2009usability} &\checkmark &\checkmark & &&\checkmark & &\checkmark &\checkmark & Studied usability, bandwidth and anonymity over anonymous networks\\
Clark \emph{et al.} \cite{clark2007usability} & &\checkmark & &&\checkmark & & &\checkmark & Performed usability analysis of Tor with other anonymity tools\\
Edmundson \emph{et al.} \cite{edmundson14security} & &\checkmark & && &\checkmark & &\checkmark & Compared anonymity and performance of \emph{Safeplug} with Tor\\
Barthe \emph{et al.} \cite{barthe2010robustness} & &\checkmark & && &\checkmark &\checkmark & & Studied robustness in Tor network\\
Mulazzani \emph{et al.} \cite{mulazzani2010anonymity} & &\checkmark & &&\checkmark & & &\checkmark & Analysed monitoring and anonymity issues in Tor\\
Huber \emph{et al.} \cite{huber2010tor} & &\checkmark & &&\checkmark & & &\checkmark & Studied anonymity using HTTP usage statistics\\
McCoy \emph{et al.} \cite{mccoy2008shining} & &\checkmark & &&\checkmark & & &\checkmark & Studied applications, usage statistic and misusage of Tor\\
Loesing \emph{et al.} \cite{loesing2010case} & &\checkmark & &&\checkmark & & &\checkmark & Studied country and port usage of Tor\\
Chen \emph{et al.} \cite{chen2009xpay} & & &\checkmark &\checkmark & & & &\checkmark & Proposed anonymous payments over anonymous network\\
\hline
\multicolumn{10}{|c|}{Modelling Tor network}\\
\hline
Jansen \emph{et al.} \cite{jansen2012methodically} & &\checkmark &\checkmark && &\checkmark &\checkmark & & Proposed graph based Tor topology\\
Jansen and Hopper \cite{jansen2011shadow} & & &\checkmark && &\checkmark &\checkmark & & Developed discrete event Tor simulatork\\
Bauer \emph{et al.} \cite{bauer2011experimentor} & & &\checkmark && &\checkmark &\checkmark & & Developed emulation toolkit \emph{ExperimenTor} for Tor\\
\hline
\end{tabular}
\end{table*}
\subsubsection{General Studies of Tor}
Several studies covering pros and cons of Tor and analyzing statistics of Tor referring to users' quality of experience are summarized in the paragraphs below.
\emph{Understanding Challenges and Social Factors:} In \cite{dingledine2007deploying}, Dingledine \emph{et al.} described the challenges in implementation of Tor and discussed social issues. Tor network design and its details were also discussed with reference to the previous state-of-the-art. Possible avenues for improvements in the Tor network and flaws in the current system were presented including abuse, security implications and perceived social value.
\emph{Who is More User Friendly ?} Abou-Tair \emph{et al.} \cite{abou2009usability} focused on the usability of different anonymizing solutions including Tor, I2P\footnote{I2P is an anonymous overlay network which supports both TCP and UDP traffic. Web: \url{https://geti2p.net/en/}}, JAP/JonDo (Java Anonymous Proxy)\footnote{Java Anon Proxy allows web browsing with pseudonymity using its proxy based system. Web: \url{https://anonymous-proxy-servers.net/}} and Mixmaster\footnote{Mixmaster is a Chaumian mix network which is an anonymous remailer providing security against traffic analysis and sender deanonymization. Web: \url{http://mixmaster.sourceforge.net/}}. The installation of all softwares was analyzed with regard to ease-of-use. They measured the bandwidth consumption of all softwares. The authors concluded that I2P and Mixmaster provide better anonymity but are more complex. On the contrary, Tor and JAP are easy to use but comprise somewhat on the degree of anonymity they provide.
\emph{Usability Analysis of Tor:} Clark \emph{et al.} \cite{clark2007usability} conducted usability analysis for deployment of Tor and software tools associated with Tor including Vidalia, Privoxy, Torbutton and FoxyProxy. Research showed that all implementations have associated pros and cons. The study presented guidelines for future implementations for maximum usability of anonymity tools. Research spanned over the installation, configuration, usage menu, verification and switch-off features of various anonymity tools.
\emph{Safeplug vs. Tor:} Edmundson \emph{et al.} \cite{edmundson14security} analyzed the security provided by Safeplug in comparison to the Tor network. Safeplug\footnote{\url{https://pogoplug.com/safeplug}} is a plug-and-play network device which is plugged into the router and it acts as an HTTP proxy by directing all web traffic through the Tor network. Safeplug was launched to provide ease in access for Tor users. On the contrary, Tor network can be accessed through Tor browser bundle provided by Tor. Study showed that Safeplug was vulnerable to first and third-party trackers, through which users can be deanonymized. Attacker can modify the settings of Safeplug externally through cross-site request forgery (CSRF). Safeplug provided more latency and less protection than Tor.
\emph{Robustness of Tor:} Barthe \emph{et al.} \cite{barthe2010robustness} argued that \emph{robustness} has always been neglected while \emph{privacy} is the issue that receives most attention. Authors defined general and flexible definitions for robustness and studied the Golle and Juels protocol. By identifying the weaknesses in the current protocol, novel enhancements were also proposed for robustness.
\emph{Anonymity and Monitoring on Tor:} Mulazzani \emph{et al.} \cite{mulazzani2010anonymity} addressed the \emph{Monitoring} and \emph{Anonymity} issues in the current Tor network. A dataset was collected over a period of six months. Analysis showed that a sinusoidal pattern in users is observed with half of servers located in Germany and United States. A proposed implementation has been added into \emph{TorStatus}\footnote{\url{https://torstatus.blutmagie.de}}, which is the project displaying Tor network status, available routers, bandwidths, hosts and availability history.
\emph{Tor Traffic Statistics:} Huber \emph{et al.} \cite{huber2010tor} analyzed the HTTP usage of the Tor network. Research showed that $78\%$ of Tor users do not use Tor using TorButton, which can be used for deanonymization. $1\%$ of Tor requests are vulnerable to piggybacking attacks. $7\%$ requests, pertaining to social networks, contain identifiable information. The authors suggested the use of HTTPS instead of HTTP for secure communication.
\emph{Tor Usage Statistics:} McCoy \emph{et al.} \cite{mccoy2008shining} studied the applications, user countries and usage of Tor network. Statistics collected from Tor showed that non-interactive protocols (BitTorrent traffic), comprising of a minority of connections, consumed majority of resources. Non-secure protocols like HTTP can be exploited by the exit router to log sensitive information. The study suggested a protocol for identification of all exit routers capturing POP3 traffic. Usage statistics revealed that USA, Germany and China are major users of Tor.
\emph{Statistical Data of Tor:} Loesing \emph{et al.} \cite{loesing2010case} collected the statistics from the live Tor network to measure two aspects of communication, i.e., (1) country wise usage, and (2) traffic port numbers for exiting traffic. Both these statistics can be used for future improvements in the Tor network for better anonymity services. The study also revealed that port $80$ receives most traffic.
\emph{Micropayments Using Tor:} Chen \emph{et al.} \cite{chen2009xpay} proposed a novel mechanism of anonymous payments for network services. The proposed mechanism allows users to make untraceable micro-payments to each other. Authors included features of offline verification, overspending prevention, aggregation and low overheads. Experiments showed only $4\%$ overhead for the proposed strategy.
\subsubsection{Modeling Tor Network}
In this section, we present the modeling techniques used for analyzing Tor.
\emph{Modeling Topology and Hosts of Tor:} Jansen \emph{et al.} \cite{jansen2012methodically} developed a model of Tor which closely resembled the Tor network. Authors developed a graph for Tor topology where vertexes related to downstream bandwidth, upstream bandwidth and packet loss, and edges related to latency, jitter and packet loss. All hosts including relays, authorities, clients and Internet servers were mapped to the developed graph based on characteristics obtained from Tor.
\emph{Shadow: Simulating Tor Network} Jansen and Hopper \cite{jansen2011shadow} developed an open source discrete event simulator for simulating the network layer of Tor on a single machine. Authors compared the performance of \emph{Shadow} with real-world simulation results from the PlanetLab testbed.
\emph{Emulation Toolkit for Tor Experimentation:} Bauer \emph{et al.} \cite{bauer2011experimentor} developed ExperimenTor, an emulation toolkit for Tor network. Their research was focused on the toolkit rather than the analysis of the Tor network.
\subsubsection{Analysis of Tor}
Analysis of Tor network has been a part of many studies covering delays, bandwidth, quality of service, relay selection and authentication protocols.
Several studies covering these areas are summarized in following sections.
Table \ref{tab: Researches on analysis and performance improvements of Tor} presents a comparison of various research works in analysis and performance improvement track. Comparison shows that relay selection and latency analysis are the most frequently studied topics followed by anonymity, bandwidth and quality of service analysis. Very few studies focused over queues, traffic shaping techniques and protocol messages.
\begin{table*}
\caption{Research studies on analysis and performance improvements of Tor. Table entries symbolize New algorithms (New Algo), Analysis (Anal.), Relay Selection (Relay Sel.), Performance Latency (Perf. Lat.), Performance Bandwidth (Perf. BW.), Quality of Service (QoS), Queues, Protocol Messages (Prot. Msgs.), Traffic Shaping (Traff. Shap.), Anonymity (Anon.).}\label{tab: Researches on analysis and performance improvements of Tor}
\centering
\small
\begin{tabular}{|
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}
\hline
\multirow{3}{*}{Research} & \multicolumn{2}{|c|}{Focus} & \multicolumn{8}{|c|}{Path Selection Parameters} & \multirow{3}{*}{Idea}\\
\cline{2-11}
& New & Anal. & Relay & Perf. & Perf. & QoS & Queues & Prot. & Traff. & Anon. & \\
& Algo & & Sel. & Lat. & BW. & & & Msgs. & Shap. & & \\
\hline
\multicolumn{12}{|c|}{Analysis of Tor network}\\
\hline
Dhungel \emph{et al.} \cite{dhungel2010waiting} & &\checkmark & \checkmark & \checkmark & & & & & & & Analysed latency for Tor relays\\
Loesing \emph{et al.} \cite{loesing2008performance} & &\checkmark & & \checkmark & & \checkmark & & & & & Analysed latency, QoS and performance of Tor\\
Ehlert \emph{et al.} \cite{ehlert2011i2p} & &\checkmark & &\checkmark &\checkmark & & & & & & Studied bandwidth and latency in Tor\\
Pries \emph{et al.} \cite{pries2008performance} & &\checkmark & & & \checkmark & \checkmark & & & & & Investigated bandwidth for various path selection algorithms\\
Liu and Wang \cite{liu2009anti} &\checkmark & &\checkmark &\checkmark & &\checkmark & & & &\checkmark & Proposed relay reliability mechanism considering performance anonymity and QoS\\
Wang \emph{et al.} \cite{wang2013empirical} & & \checkmark &\checkmark & & & & & & & & Performed an empirical analysis over family nodes\\
Tschorsch and Scheuermann \cite{tschorsch2011tor} & \checkmark & & &\checkmark & & & \checkmark & & & & Proposed fairness model for efficient and fair resource allocation\\
Chaabane \emph{et al.} \cite{chaabane2010digging} & & \checkmark &\checkmark & & && & & & & Studied misuse of Tor's exit nodes as proxies\\
Hopper \emph{et al.} \cite{hoppershort} & & \checkmark & & & && &\checkmark & & & Analysed Tor's performance considering key exchange mechanisms\\
Lenhard \emph{et al.} \cite{lenhard2009performance} & & \checkmark & &\checkmark & && & & & & Studied communication overhead in low bandwidth networks for Tor's hidden services\\
Goldberg \cite{goldberg2006security} & & \checkmark &\checkmark & & && &\checkmark & & & Analysed anonymity with Tor;s authentication protocol\\
\hline
\multicolumn{12}{|c|}{Tor performance improvement}\\
\hline
Jansen \emph{et al.} \cite{jansen2010recruiting} & \checkmark & &\checkmark & & && & & & & Proposed token based performance mechanisms for recruiting more relays\\
Dingledine \emph{et al.} \cite{dingledine2010building} & \checkmark & &\checkmark & & && & & & & Proposed priority based traffic handling for relays\\
Wang \emph{et al.} \cite{wang2013rbridge} & \checkmark & &\checkmark & & && & & &\checkmark & Proposed node reliability mechanism to avoid blockage of bridges\\
Smits \emph{et al.} \cite{smits2011bridgespa} & \checkmark & &\checkmark & & && &\checkmark & &\checkmark & Proposed packet authorization based mechanism to protect bridges from eavesdroppers\\
Moghaddam \emph{et al.} \cite{mohajeri2012skypemorph} & \checkmark & & & & && & &\checkmark & \checkmark & Proposed traffic morphing (using Skype traffic) to avoid censorship\\
Weinberg \emph{et al.} \cite{weinberg2012stegotorus} & \checkmark & & & & && & &\checkmark & \checkmark & Proposed traffic shaping (by assembling regular HTTP traffic) to avoid deanonymization\\
Gopal and Heninger \cite{gopal2012torchestra} & \checkmark & & &\checkmark & && & & & & Suggested latency reduction by separate TCP connections for interactive and bulk traffic\\
AlSabah \emph{et al.} \cite{alsabah2011defenestrator} & \checkmark & & &\checkmark & &&\checkmark & & & & Proposed traffic morphing (using Skype traffic) to avoid censorship\\
Jansen \emph{et al.} \cite{jansen2012throttling} & \checkmark & & &\checkmark & && & & &\checkmark & Proposed throttling mechanisms for reducing latency by avoiding bulk traffic\\
\hline
\end{tabular}
\end{table*}
\emph{Understanding Delays in Tor:} Dhungel \emph{et al.} \cite{dhungel2010waiting} analyzed the delays in the entire Tor network. Authors suggested that overlay network plays the most significant role in Tor. The study revealed that $11\%$ of Tor routers are overloaded with traffic which resulted in very high delays. In $7.5\%$ of circuits, overall latency introduced a $450$ms delays. \emph{Guard} routers incorporate more delay than \emph{non-guard} routers. There is high fluctuation in delay for all routers except for those having high bandwidths.
\emph{Measurement and Statistics:} Loesing \emph{et al.} \cite{loesing2008performance} studied the latencies inside the Tor network. A deep investigation was conducted to evaluate the individual delays and QoS properties. The authors showed that circuit building time (Introduction and Rendezvous) is the most crucial delay period in Tor. Fr\'{e}chet and exponential distributions were combined to analyze the response times.
\emph{Comparing Bandwidth with Latency:} Ehlert \cite{ehlert2011i2p} compared the bandwidth and latency performance of Tor network with the popular I2P network. Authors measured the core latency (HTTP GET requests durations), average latency (webpage download times including external threads and pictures) and bandwidth (download speeds). This research showed that I2P network provides lower core latency and Tor network excels in average latency and bandwidth, owing to the nodes distribution and penetration of the Tor network.
\emph{Tor QoS with Path Selection Strategy:} Pries \emph{et al.} \cite{pries2008performance} suggested that TCP suffers severe performance degradation from the random path selection of Tor. Slight QoS improvement is achieved with Tor's bandwidth weighted path selection algorithm. The main reason attributed for small improvements is low bandwidth of Tor routers.
\emph{Investigating Tor's Exit Policies:} Liu and Wang \cite{liu2009anti} studied the exit policies of the exit nodes and addressed the short-comings in the current Tor architecture. A new protocol was proposed which comprised of three parts: (1) reporting misbehavior protocol, (2) building global blacklist protocol, and (3) blocking misbehavior protocol for users. User experience, performance and anonymity were the key indexes used for evaluation.
\emph{Behavior of Family Nodes:} Wang \emph{et al.} \cite{wang2013empirical} presented an empirical analysis of Tor family nodes. A rich dataset of live Tor network comprising of three years was used to study the impact of family nodes. The study suggested that family nodes provide stable and better service than other nodes. Moreover, attacks on family nodes can disrupt the Tor network more severely than random Tor nodes.
\emph{Fairness in Tor:} Tschorsch and Scheuermann \cite{tschorsch2011tor} analyzed the fairness issues in the current Tor network. Large unfairness was observed in the current resource allocation mechanism of the Tor network. Authors proposed a max-min fairness based model for efficient and fair scheduling of resources. The proposed design was analyzed with Tor's $N23$ congestion feedback mechanism.
\emph{Misuse of Tor:} Chaabane \emph{et al.} \cite{chaabane2010digging} showed that Tor network was being used for transmitting P2P traffic (Bit torrent etc.) over the Tor network. HTTP and Bit torrent were analyzed on the Tor network. The study showed that Tor exit nodes are being used as one hop SOCKS proxies through tunneling. New techniques were devised to detect such abnormalities in exit nodes' behavior. Research showed that simple crawling over exit nodes can be used to collect as many bridge identities as needed.
\emph{Challenges for Hidden Services of Tor:} Hopper \cite{hoppershort} conducted research on the poor performance of Tor based on the fact that users of Tor increased from $1$ million to nearly $6$ million but no dramatic change was observed in the network. Hopper attributed the poor performance to the key exchange mechanism of Tor, which was later updated. Study showed the possible research dimensions of limiting request rates from botnets, throttling entry guard, reusing failed partial circuits and isolating hidden services circuits.
\emph{Tor Hidden Services in Low Bandwidth Access Networks:} Lenhard \emph{et al.} \cite{lenhard2009performance} conducted a measurement and statistical analysis for estimating the communication overhead of Tor hidden services in low bandwidth access networks. Research showed that boot strapping time, RTT and circuit building time were the major bottlenecks to performance. Due to numerous delays, an increase in timeout value was suggested to avoid repeated retransmissions.
\emph{Analysis of Tor Authentication Protocol:} Goldberg \cite{goldberg2006security} analyzed the security of Tor's authentication protocol (TAP). The authors argued that any security breach by a single malicious Tor relay can deanonymize users' sessions. Through empirical evaluations, research showed that TAP is secure in random oracle model.
\emph{Statistics Collection Mechanism of Tor:} Mani and Sherr \cite{mani2017historvarepsilon} analysed the data collection mechanism of Tor through `PrivEx'. They showed that statistics of PrivEx can be easily compromised by the present of adversary nodes in the Tor network. As a result of shortcomings of PrivEx, authors proposed `HisTor', a privacy preserving statistics collection mechanism of Tor which is much more diverse than PrivEx. HisTor uses the count of queries by exit nodes and relays in form of a histogram where individual nodes have little control over the aggregate statistics.
\subsubsection{Tor Performance Improvement}
Owing to the increasing demand for Tor, various studies have proposed performance improvements to cope with future demands. In this section, we present these studies covering Tor node selection, traffic distribution and latency management etc.
\emph{Node Recruitment for Tor:} Jansen \emph{et al.} \cite{jansen2010recruiting} focused their research on recruitment of new Tor relays, motivated by the fact that only $1.5\%$ nodes participate as relays. Authors proposed BRAIDS which is a token based mechanism providing high bandwidth to those users who employ BRAIDS. Proposed scheme characterizes traffic into high throughput, low latency and normal traffic. Based upon usage of BRAIDS and node networking stats, tickets are generated which can be used to increase bandwidth.
\emph{Encouraging Tor nodes for traffic relaying:} Dingledine \emph{et al.} \cite{dingledine2010building} proposed a mechanism to encourage Tor nodes for traffic relaying. Study suggested a priority based traffic handling, which gives more weight (in form of bandwidth and delays) to those nodes contributing resources to Tor. However, all Tor relays carry an additional load of priority based traffic handling. Directory authorities need to assign priority levels to all Tor users participating in Tor relays.
\emph{Improving Distribution Mechanism of Tor Bridges:} Wang \emph{et al.} \cite{wang2013rbridge} improved the distribution mechanism of Tor bridges by implementing node reliability statistics to avoid the blockage of bridges by corrupt nodes. The uptime of assigned bridges is used to give reputation points to users. In case of any blockage of a bridge, a new bridge address is given on payment of earned credit. To ensure anonymity, reputation information is stored on users' systems by using a privacy-preserving technique which cannot be circumvented by malicious users.
\emph{Packet Authorization for Tor Bridges:} Smits \emph{et al.} \cite{smits2011bridgespa} proposed BridgeSPA, a packet authorization based mechanism, to protect users of Tor hosting Bridges. All Tor user hosting bridges are susceptible to traffic analysis attacks. To counter this attack, the authors suggest the transmission of a bridge key by bridge distribution authorities which is valid for a limited time, as determined by the bridge. For any communication with the bridge, Tor users should use that key within the assigned time period.
\emph{SkypeMorph - Tor traffic Shaping:} Moghaddam \emph{et al.} \cite{mohajeri2012skypemorph} proposed a new mechanism namely SkypeMorph to avoid the censorship of Tor bridges. The fundamental idea was to hide Tor traffic as Skype video traffic (a widely used protocol). SkypeMorph, which runs side by side with Tor, makes it hard to distinguish Tor traffic from Skype traffic. Two schemes were suggested for traffic morphing, (1) using the target stream attributes, (2) incorporating both source and destination streams to incorporate packet timings. Both streams provided nearly identical performance, but the former had lower complexity.
\emph{StegoTorus - Steganographing Tor Traffic:} Weinberg \emph{et al.} \cite{weinberg2012stegotorus} proposed a novel technique to bypass censorship on Tor. Their scheme is based upon the idea of chopping Tor traffic into multiple streams, resembling HTTP traffic, before passing through the censor. StegoTorus acted as a proxy on Tor clients.
\emph{Torchestra - Separate connections for Interactive and Bulk Traffic:} Gopal and Heninger \cite{gopal2012torchestra} proposed the transmission of interactive and bulk traffic over two separate TCP connections among all nodes in the Tor network. Exponentially weighted moving average (EWMA) algorithm was used to distinguish between interactive and bulk traffic on all circuits. Upto $40\%$ reduction in delays was observed as compared to standard Tor for the proposed strategy.
\emph{Reducing Latency in Tor:} AlSabah \emph{et al.} \cite{alsabah2011defenestrator} proposed a mechanism for congestion control and flow control in order to reduce latency in the Tor network. The study suggested the use of small fixed size windows and small dynamic windows which can reduce the packets in flight. For flow control, the study proposed an N23 algorithm which caps the queue lengths of Tor routers and provided a $65\%$ increase in webpage responses and $32\%$ decrease in page loading time.
\emph{Throttling Tor bulk users:} Jansen \emph{et al.} \cite{jansen2012throttling} addressed the poor performance of Tor network using bulk data transfers. Three dynamic throttling algorithms were proposed for reducing network congestion and latency. The guard relay capped the bandwidth capacity of nodes, so, only local relay information was used. Simulations showed that throttling reduces the web page latency and increases the anonymity of Tor network.
\subsubsection{Tor Client Mobility}
In this section, we study the research works focused on the mobility of Tor network with a particular emphasis on anonymity.
Table \ref{tab: Researches on Tor's mobile devices} shows the research works in path selection track and shows that performance and anonymity have been the most frequently studied parameters. Details are presented in below paragraphs.
\begin{table*}
\caption{Research works on Tor's client mobility. Table entries symbolize New algorithms (New Algo), Analysis (Anal.), Autonomous Systems (AS), Relay Locations (Relay loc.), Hops, Performance-Latency-Bandwidth (Perf., Lat, BW), Multi-path, Load, Relay Capacity (Rel. Cap.) and Anonymity (Anon).}\label{tab: Researches on Tor's mobile devices}
\centering
\small
\begin{tabular}{|
@{}>{\centering}p{3cm}@{}|
@{}>{\centering\arraybackslash}p{0.6cm}@{\hspace{0.035in}}|
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}
\hline
\multirow{3}{*}{Research} & \multicolumn{2}{|c|}{Focus} & \multicolumn{8}{|c|}{Path Selection Parameters} & \multirow{3}{*}{Idea}\\
\cline{2-11}
& New & Anal. & AS & Relay & Hops & Perf. & Multi- & load & Rel. & Anon. & \\
& Algo & & & Loc. & & Lat, BW & path & & Cap. & & \\
\hline
Doswell \emph{et al.} \cite{doswell2013novel} & &\checkmark & & & &\checkmark & & & & & Suggested bridge relays to avoid bandwidth issues while roaming\\
Andersson \emph{et al.} \cite{andersson2007practical} &\checkmark & & & & & \checkmark & & & & \checkmark & Proposed trade-of between anonymity and performance\\
\hline
\end{tabular}
\end{table*}
\emph{Using Bridge Relays:} Doswell \emph{et al.} \cite{doswell2013novel} analyzed the performance of Tor for wireless devices roaming across multiple networks. Analysis showed that the \emph{speed} of mobile wireless devices significantly affects the circuit building time and Tor's performance. Authors studied the use of bridge relays to provide persistent Tor connections for mobile devices.
\emph{New Architectural Designs:} Andersson \emph{et al.} \cite{andersson2007practical} proposed several new architectural designs for a mobile Tor network. A trade-off between anonymity and performance was evaluated. Several criteria used in performance estimation included usability, availability, trust and practicality. The study concluded that the single Tor client option offers lowest degree of anonymity.
\begin{figure}[b]
\centering
\includegraphics[width=0.7\columnwidth]{implementation.pdf}\\
\caption{Classification of platforms for Tor's research.}\label{fig: Classification of platforms for Tor's research.}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=1.2\columnwidth]{Tor_implemtation_bd.pdf}\\
\caption{Taxonomy of platforms employed in Tor research.}\label{fig: Taxonomy of platform for Tor researches}
\end{figure*}
\section{Platforms for Tor Research}
\label{sec:Platforms for Tor Research}
In this section, we study the platforms used to study Tor network. Our observations spanning over decades of anonymity research shows that all research works have studied the Tor network using three different techniques, (1) Experiment, (2), Simulations, and (3) Analysis. Figure \ref{fig: Classification of platforms for Tor's research.} shows that 60\% of the studies used in this paper conducted experiments. Only 27\% of the studies conducted experiments. In the experiment section, majority studies developed their own testbed followed by experiments on cloud services and PlanetLab testbeds. In the simulations section, majority research works used extensive simulations to study Tor network. Finally, some studies analyze Tor network by collecting statistics and discussing the sociability and usability issues of Tor network. These three classification categories are elaborated in Figure \ref{fig: Taxonomy of platform for Tor researches} which shows the platforms used to study Tor network.
\subsection{Tor Experiments}
Studies covering Tor experiments have focused over several areas including (1) private setup establishment, (2) PlanetLab experiments, (3) cloud services (4) OpenFlow networks and (5) universal composability framework.
Table \ref{tab: Experimental setups used in different researches} presents the clients, relays, servers, Tor services and Tor implementation used by various research works. Comparison shows that majority studies deployed their private testbeds with 1-2 clients and 1-2 servers. Several studies deployed limited number of relays for experiments. Number of clients were increased drastically in the PlanetLab and cloud setup for Tor experiments. Moreover, traffic analysis was the most frequently studied topic. Majority research works used the default Tor setup without any modifications. Figure \ref{fig: Classification of experiments on Tor.} shows the classifications of experiments on Tor. Analysis of figure shows that majority of studies deployed their own private testbeds.
\begin{table*}
\centering
\caption{Experimental setups used in different research works.}
\label{tab: Experimental setups used in different researches}
\footnotesize
\begin{tabular}{| p{4cm} | p{1.3cm} | p{1.3cm}| p{2cm}| p{2.7cm}| p{3.3cm} |}
\hline
Research & Servers & Relays & Clients & Service & Tor implementation\\
\hline
\hline
\multicolumn{6}{|c|}{\emph{Private Setup}}\\
\hline
Overlier and Syverson \cite{overlier2006locating} & 2 & & 1 & Hidden service & default Tor\\
\hline
Andersson and Panchenko \cite{andersson2007practical} & 1 & & 1 & Mobile Tor & Onion Coffea\\
\hline
Panchenko \emph{et al.} \cite{panchenko2008performance} & & & 1 & Download Service & Def. Tor + Onion Coffea\\
\hline
Pries \emph{et al.} \cite{pries2008performance} & 1 & & 1 & Download Service & Privoxy\\
\hline
Wagner \emph{et al.} \cite{wagner2012breaking} & 2 & 1 & 1 & Log processing & WebProxy\\
\hline
Chan-Tin \emph{et al.} \cite{chan2013revisiting} & 1 & & 2 & Traffic Analysis & Def. Tor\\
\hline
Pries \emph{et al.} \cite{pries2008new} & 1 & 2 & 1 & TCP data coll. & Tor mod.\\
\hline
Herzberg \emph{et al.} \cite{herzberg2011camouflaged} & & & 1 & Web page download & Def. Tor\\
\hline
Bauer \emph{et al.} \cite{bauer2009predicting} & & 6+ & 1+ & Path compromise & Def. Tor\\
\hline
Song \emph{et al.} \cite{song2013anonymize} & & 6 & 1 & Traffic Analysis & Tor mod.\\
\hline
Dhungel and Steiner \cite{dhungel2010waiting} & 1 & 2 & 1 & Traffic Analysis & Def. Tor\\
\hline
Gros \emph{et al.} \cite{gros2010protecting} & & & 2+ & Traffic Analysis & Honeywall\\
\hline
Wang \emph{et al.} \cite{wang2009novel} & & 2 & & Traffic Analysis & Privoxy\\
\hline
Zhang \emph{et al.} \cite{zhang2011application} & 1 & 3 & 1 & Hidden Service & Polipo\\
\hline
Loesing \emph{et al.} \cite{loesing2008performance} & & 1 & 1+ & Access Attempt & Def. Tor\\
\hline
Chen and Pasquale \cite{chen2010toward} & 1 & & 10 & Download & Def. Tor\\
\hline
Panchenko \emph{et al.} \cite{panchenko2012improving} & $1+$ & & 1+ & Traffic Analysis & Def. Tor\\
\hline
Houmansadr \emph{et al.} \cite{houmansadr2013parrot} & $4$ & & $3$ & Traffic Analysis & $-$\\
\hline
Li \emph{et al.} \cite{li2012tmt} & 1 & & 1 & Download Analysis & Def. Tor\\
\hline
Chakravarti \emph{et al.} \cite{chakravarty2008identifying} & 1 & & 2 & Download Analysis & Def. Tor\\
\hline
Mulazzani \emph{et al.} \cite{mulazzani2010anonymity} & & & 1+ & Traffic Analysis & Tor Status\\
\hline
Chaabane \emph{et al.} \cite{chaabane2010digging} & $1+$ & 6 & $1+$ & Traffic Analysis & Def. Tor\\
\hline
Bai \emph{et al.} \cite{bai2008traffic} & 2 & & 6 & Traffic Analysis & Def. Tor\\
\hline
Barker \emph{et al.} \cite{barker2011using} & 3 & 15 & 1+ & Traffic Analysis & Def. Tor\\
\hline
Marks \emph{et al.} \cite{marks2010unleashing} & 3 & & 3 & Download Analysis & \\
\hline
Jin and Wang \cite{jin2009effectiveness} & 1 & & 1 & Traffic Analysis & Tor mod.\\
\hline
Tang and Goldberg \cite{tang2010improved} & 1 & 1 & 1 & Download Analysis & Def. Tor\\
\hline
Alsabah \emph{et al.} \cite{alsabah2012enhancing} & & & 1 (3 Apps)& Traffic Analysis & Def. Tor\\
\hline
Moghaddam \emph{et al.} \cite{mohajeri2012skypemorph} & & & 2+ & Traffic Analysis & SkypeMorph\\
\hline
Weinberg \emph{et al.} \cite{weinberg2012stegotorus} & 1 & & 1 & Download Analysis & StegoTorus\\
\hline
Evans \emph{et al.} \cite{evans2009practical} & & & 1 & Traffic Analysis & Def. Tor\\
\hline
Wang \emph{et al.} \cite{wang2012congestion} & & & 1 & Traffic Analysis & Def./Mod. Tor\\
\hline
Ehlert \cite{ehlert2011i2p} & & & 1 & Traffic Analysis & Def. Tor\\
\hline
Barbera \emph{at al.} \cite{barbera2013cellflood} & & 2 & 4 & Traffic Analysis & Def. Tor\\
\hline
Winter and Lindskog \cite{winter2012great} & & 2 & 2+ & Traffic Analysis & Tor mod.\\
\hline
Edmundson \emph{et al.} \cite{edmundson14security} & & & 1 & Download Analysis & Def. Tor\\
\hline
Huber \emph{et al.} \cite{huber2010tor} & & 1 & & Traffic Analysis & Def. Tor\\
\hline
Blond \emph{et al.} \cite{blond2011one} & & 6 & 1+ & Traffic Analysis & Def. Tor\\
\hline
Lenhard \emph{et al.} \cite{lenhard2009performance} & & & 1+ & Hidden Service & Def. Tor\\
\hline
McCoy \emph{et al.} \cite{mccoy2008shining} & & 3 & & Traffic Analysis & Tor mod.\\
\hline
Chakravarty \emph{et al.} \cite{chakravarty2011detecting} & & & 1 & Traffic Analysis & Def. Tor\\
\hline
Snader and Borisov \cite{snader2008tune} & 1 & & 1 & Traffic Analysis & Tunable Tor + Vanilla\\
\hline
Gilad and Herzberg \cite{gilad2012spying} & 1 & & 1 & Traffic Analysis & Def. Tor\\
\hline
Loesing \emph{et al.} \cite{loesing2010case} & & & 1+ & Traffic Analysis & Def. Tor\\
\hline
Chen \emph{et al.} \cite{chen2009xpay} & & 3+ (VMs) & 2+ (VMs)& Traffic Analysis & Def. Tor\\
\hline
Panchenko \emph{et al.} \cite{panchenko2011website} & & & 1+ & Traffic Analysis & Def. Tor\\
\hline
Wang and Goldberg \cite{wang2013improved} & & & 200 cores & Traffic Analysis & Def. Tor\\
\hline \hline
\multicolumn{6}{|c|}{\emph{PlanetLab Setup}}\\
\hline
Akhoondi \emph{et al.} \cite{akhoondi2012lastor} & & & 50 & Traffic Analysis & LASTor \\
\hline
Murdoch and Danezis \cite{murdoch2005low} & 1 & & 2 & Traffic Analysis & Tor Mod. \\
\hline
Bauer \emph{et al.} \cite{bauer2007low} & 6 & 2-6 & 40-90 & 40 node network & \\
\cline{2-6}
& 6 & 3-6 & 60-90 & 60 node network & \\
\hline \hline
\multicolumn{6}{|c|}{\emph{Cloud Setup (Amazon EC2)}}\\
\hline
Sulaiman and Zhioua \cite{sulaiman2013attacking} & 1 & & & Traffic Analysis & Def./Mod Tor \\
\hline
Karaoglu \emph{et al.} \cite{karaoglu2012multi} & 1 & & 4 & Traffic Analysis & Def. Tor \\
\hline
Biryukov \emph{et al.} \cite{biryukov2013trawling} & & & 50 & Hidden Services & Def. Tor \\
\hline
\end{tabular}
\end{table*}
\begin{figure}[b]
\centering
\includegraphics[width=0.8\columnwidth]{experiments.pdf}\\
\caption{Classification of platforms used in experimental Tor research.}\label{fig: Classification of experiments on Tor.}
\end{figure}
\subsubsection{Private Setup connected with Tor}
Overlier and Syverson \cite{overlier2006locating} performed experiments by setting up two nodes (one in Europe and other in US) running hidden services at two ends of the Tor network. Access to webpages and images was provided using these services. The client PC was setup both as a client and a middleman node, and all sampling takes place at this client node.
Andersson and Panchenko \cite{andersson2007practical} performed experiments to verify the performance of their proposed mobile protocol. Mobile Tor was setup on a laptop connected to the Tor network. The content server hosting the files was placed at Karlstadt University. Experiments used OnionCoffee, which is a Java project developed under the PRIME project.
Panchenko \emph{et al.} \cite{panchenko2008performance} performed experiments using a Pentium Dual Core $1$GHz CPU with $2$GB RAM as a client nodes. Two existing Tor implementations (default Tor and OnionCoffee) were used on the client nodes. The Internet connection had a $10$Gbps bandwidth while the local backbone was $100$Gbps. Actual Tor relays were used to analyze the performance.
Pries \emph{et al.} \cite{pries2008performance} performed experiments by downloading a $458$kB file from a school web server. Command line utility \emph{wget} was used as the downloading tool. \emph{wget's} http-proxy and ftp-proxy were configured to download all files through \emph{Privoxy} from the server. Tor release \texttt{0.1.1.26} was configured on the exit and entry nodes.
Wagner \emph{et al.} \cite{wagner2012breaking} implemented a novel architecture using Tor. Three machines were setup running Tor exit node, BIND (DNS server with tcpdump), and Apache webserver, respectively. All machines were synchronised by NTP. Connected to Tor network, WebProxy was implemented in Perl. \texttt{iptables} was used to re-route traffic from Tor exit node to Perl proxy server. All processing of web server logs and proxy logs was performed using Perl, sqlite and modified tcpick.
Chan-Tin \emph{et al.} \cite{chan2013revisiting} setup a limited network for probing Tor network using client, burst server and probe machines. Entry and middle routers were chosen randomly while exit node was forced by choice. Four Tor relays were probed for the experiment and data of probes was collected after every $5$secs. Five connections were setup by the client using multi-threading.
Pries \emph{et al.} \cite{pries2008new} setup client, server, entry malicious router and exit malicious router by setting up four devices. A TCP client application was built which sent and received TCP data. Test server used port $41$ and received and displayed data on the screen. The client used \texttt{tsocks} to transport its TCP stream through onion proxy. The Tor configuration file was configured to select designated Tor entry and exit routers.
Herzberg \emph{et al.} \cite{herzberg2011camouflaged} implemented their proposed camouflaged browsing design over a test machine with an ADSL connection to the Internet with $1,269$kBps downlink and $103$ kBps uplink bandwidth. Four different URLs were tested with $100$ measurements and access time for browsers was recorded. \texttt{wget} was used to download web pages.
Bauer \emph{et al.} \cite{bauer2009predicting} built an extensive experimental setup by establishing circuits for different kinds of applications with a number of malicious routers. The simulator generated $10,000$ circuits with $6$ to $106$ malicious routers. The path compromise rate for different applications was estimated by the selection of malicious routers.
Song \emph{et al.} \cite{song2013anonymize} used an \emph{Au3} script to capture Tor traffic. Six nodes located at distinct places (India, Romania, Luxemburg, New Zealand, Chile, and Russia) were deployed as exit nodes. Onion proxy running on a local PC was configured to use the deployed exit nodes. Traffic of all routers was captured to analysis.
Dhungel and Steiner \cite{dhungel2010waiting} measured delay of Tor network by setting up two relays instead of three. Client, exit router and destination server were fixed while the entry router was selected from the list of available routers. To cope with varying network characteristics, the experiment was repeated for eight months with each duration of $40$ minutes. All $1,426$ available routers were pinged five times for measurements.
Gros \emph{et al.} \cite{gros2010protecting} performed experiments by using the proposed Honeywall mechanism. All vulnerable clients using Tor were placed on one side of the Honeywall and Internet cloud was present on the other side of the Honeywall. All Tor clients had distinct private addresses while Honeywall had a single public IP address.
Wang \emph{et al.} \cite{wang2009novel} conducted experiments in a partially controlled environment. The OP code was modified to use the designated entry and exit Tor routers. Entry and exit Tor nodes were configured to record the data relayed through them. Internet Explorer was used at the Tor client through Privoxy. The middle Tor router was selected through Tor router selection algorithms.
Zhang \emph{et al.} \cite{zhang2011application} used Mozilla Firefox on Fedora $11$ to access the hidden service using \emph{Polipo}. The hidden server was configured to use the bridge whose traffic was being logged continuously. Clients and bridges were configured to record the circuit ID, command, stream ID and arrival time.
Loesing \emph{et al.} \cite{loesing2008performance} conducted experiments on Tor by configuring the Tor client to use fixed first / entry relay, which was being monitored continuously. Second and third Tor relays were chosen randomly by Tor's router selection algorithm. A single access attempt was performed by creating new Tor clients after every five minutes over a $72$ hour duration.
Chen and Pasquale \cite{chen2010toward} analyzed the throughput by downloading a $100$kB file through nearly $100$ unique paths with $10$ times repeated downloads over each path. $10$ Tor clients were configured over PlanetLab testbed distributed around the globe. A file server containing $100$kB file was hosted in the US using \texttt{thttpd}. \texttt{cURL} was used to conduct downloads. Python was used to write measurement scripts using the \texttt{TorCtl} library for Tor control port. Tor circuits were configured to be replaced after every $30$ minutes instead of $10$ minutes so that no middle replacement takes place.
Panchenko \emph{et al.} \cite{panchenko2012improving} performed experiments in the current Tor network with the estimation of delay and throughput. In first experiment, onion routers are fixed but links connecting the circuit are variable. $2,000$ sets were built with random onion routers. In the second experiment, ICMP Ping was used to measure the delay between sending a \emph{SYN} and receiving a \emph{SYN-ACK} packet. Each ping was iterated $20$ times to calculate the mean value.
Houmansadr \emph{et al.} \cite{houmansadr2013parrot} conducted a deep experimental investigation on the Tor network. Application layer softwares (Skype, CensorSpoofer) were executed in VirtualBox virtual machines (VMs) on a Funtoo Linux machine. Various VMs were connected through virtual distributed Ethernet (VDE). Authors built their own plugin for VDE which could drop packets at variable rates and also modify packet contents. Various VDE switches were connected to the central switch which provides DHCP connectivity to the Internet.
Li \emph{et al.} \cite{li2012tmt} tested their proposed tunable mechanism of Tor (TMT) over the real Tor network. Two virtual private servers (VPSes), acting as client and server, were configured on \emph{Linode}. The client was configured with the TMT enhancement while the server hosted a web page. Time to load the file, number of attempts and number of failure attempts were measured to estimate the performance of TMT.
Chakravarti \emph{et al.} \cite{chakravarty2008identifying} setup their own client, server and probing host machine at three distinct locations inside US. $100$MB file was placed at the web server which gave sufficient downloading time to the client. Linux traffic controller was used to shape the client-server bandwidth. $26$ distinct Tor circuits were created and probed through different locations and compromised links were detected.
Mulazzani \emph{et al.} \cite{mulazzani2010anonymity} collect data by using \emph{TorStatus} and updating its script \emph{tns-update.pl} and \emph{network-history.php}. \emph{RRDtool} was used to store the values in a round robin database (RRD). The collected dataset was used for basic network monitoring.
Chaabane \emph{et al.} \cite{chaabane2010digging} conduct a deep traffic analysis of Tor using HTTP and Bit Torrent protocols. The authors created and monitored six Tor relay nodes (placed in US, Germany, France, Japan, Taiwan) advertising $100$kB available bandwidth for $23$ days. On average $20$GB of data is provided by each server on every day. Data was collected at entry and exit relays.
Bai \emph{et al.} \cite{bai2008traffic} setup eight PCs with one PC running Tor and one PC running java anonymous proxy (JAP). Dummy traffic was generated from the other six PCs. Traffic was captured through \texttt{ethereal}. \emph{Winsock Packet Editor} was used to record packets generated by a specific application. Duration of the test was about $120$mins with five repetitions.
Barker \emph{et al.} \cite{barker2011using} collected Tor network traces by developing a complete Tor setup. Firefox running on Ubuntu was used on all machines. Using the Selenium browser testing framework, $170$ simulations were executed by accessing $30$ websites. Three directory servers with $15$ relays were configured to be used for experiments. Regular HTTPs traffic and HTTP and HTTPs traffic through private Tor network were collected.
Marks \emph{et al.} \cite{marks2010unleashing} conducted a simple experiment using three PCs running Linux kernel (\texttt{2.6.26} and CUBIC TCP). All three machines were connected via an Ethernet switch. All Ethernet interfaces were configured to be $10$Mbps full-duplex links. The first and last two devices setup TCP connections. First device sent data to the second for $250$secs while the second retransmitted after $50$secs delay for a duration of $250$secs.
Jin and Wang \cite{jin2009effectiveness} conducted extensive experiments by monitoring both anonymous traffic and Tor traffic during two experiments. In the first experiment, an Apache webserver on a Dell PC using Redhat Enterprise 4 Linux was configured. A watermark encoder was installed on the Apache proxy. A Dell Precision $390$ was configured as a NAT router to route traffic between client and the anonymous server. In the second experiment, an SSH server and watermark encoder were installed on one machine acting as server. SSH client and watermark decoder were installed on another machine. Three random characters were sent every second from one machine to the other through Tor. Entry and exit relays were fixed.
Tang and Goldberg \cite{tang2010improved} setup their own node (acting as the middle node). Entry and exit nodes were selected from the Tor nodes of the directory server. Authors avoided the use of PlanetLab testbed because majority nodes were providing only $100$KB/s. \emph{Webfetch} was used to download the target file ($87$KB) from author's web server. Connecting circuits and load was varied to verify the proposed path selection strategy.
Alsabah \emph{et al.} \cite{alsabah2012enhancing} performed real world experiments by collecting offline data of $200$ circuits from three distinct application traces. All three applications (BitTorrent client, web browsing client and stream client) were setup on the same machine which was configured to use a specified Tor node as the entry node. All $200$ circuits included browsing ($122$), BitTorrent ($49$) and streaming circuits ($28$). All applications collected $24$ hours of data over a $6$ week period with periodic intervals.
Moghaddam \emph{et al.} \cite{mohajeri2012skypemorph} implemented their proposed SkypeMorph technique on Linux using C and C++ with boost libraries. Authors collected traces of Skype data set for modeling using multiple machines. The proposed SkypeMorph scheme was tested by downloading multiple files with and without it.
Weinberg \emph{et al.} \cite{weinberg2012stegotorus} implemented the proposed scheme \emph{StegoTorus} by deploying an experimental setup. The client was a desktop PC in California with DSL link to the Internet (downstream $5.6$Mb/s, upstream $0.7$Mb/s) and the virtual host was situated in New Jersey inside a commercial data center. $1$MB files were downloaded over several trials to test the performance.
Evans \emph{et al.} \cite{evans2009practical} performed experiments on the real Tor network for their proposed congestion attack. The victim user (to be breached) was using Javascript on her browser. Entry node was fixed but the other two Tor relay nodes were selected at random (by Tor's router selection algorithm).
Wang \emph{et al.} \cite{wang2012congestion} measured the network delays during congestion by collecting delay readings of all Tor routers for $72$ hours in August $2011$. At the next stage, authors collected RTT measurements of the modified and unmodified Tor client to setup $255$ circuits. In all experiments, client machines were modified to incorporate the proposed algorithm and measure the delay.
Ehlert \cite{ehlert2011i2p} measured the performance of I2P and Tor network. For I2P network, experimental setup consisted of two machines, acting as dedicated outproxy and client. $500$ most visited websites were used for downloading webpages. For Tor, a client machine was connected to the Tor network and performance parameters were measured similar to I2P proxy.
Barbera \emph{at al.} \cite{barbera2013cellflood} conducted controlled experiments by setting up $100$Mb/s network connected to four hosts (possessing $2.66$GHz Core 2 Duo CPUs). For real time network experiments, the authors used their two OR nodes, acting as Tor relays. CellFlood attacks were performed on these routers and performance of attack and mitigation scheme was analyzed.
Winter and Lindskog \cite{winter2012great} deployed one relay in Russia and two bridges in Singapore and Sweden. Multiple clients were present in China for connection setup to Tor through designated bridges and relays. In Singapore, a Tor relay was hosted in an Amazon EC2 cloud. Bridge and relay in Sweden and Russia were hosted by an institution and data center, respectively. For vantage points in China, $32$ SOCKS proxies and a VPS running Linux was used.
Edmundson \emph{et al.} \cite{edmundson14security} analyzed the security of Safeplug and Tor by conducting separate experiments for both applications. Authors measured the latency of the system, and investigated the effect of cookies and third party trackers over both applications.
Huber \emph{et al.} \cite{huber2010tor} deployed a Tor exit node which logs the HTTP requests. Nine million HTTP requests were recorded in several weeks. All requests were analyzed for available patterns and statistics were presented in the research.
Blond \emph{et al.} \cite{blond2011one} conducted experiments by deploying Tor exit nodes. Authors instrumented and monitored six Tor nodes for a period of three weeks. One exit node was configured to accept TCP connection for Bit torrent, in order to perform the hijacking attack.
Lenhard \emph{et al.} \cite{lenhard2009performance} ran Tor processes on their devices connected to the Tor network. The hidden services were accessed through low bandwidth access network edge. A modem provided a data rate of $56$kb/s downstream and $44$kb/s upstream. For EDGE, data rate was around $230$kb/s. The broadband network provided $100$Mb/s.
McCoy \emph{et al.} \cite{mccoy2008shining} setup their router connected to $1$Gb/s network link with a rank of top $5\%$ Tor routers and flagged as \emph{Running}. At most $20$bytes were logged to avoid information breaching laws. Setup was configured for both experiments separately covering (1) exit router and (2) non-exit router. Entrance and middle router traffic was logged for $15$ days comprising of time stamp, previous hop's IP, TCP port, next hop's IP and circuit identifier. For exit traffic logging, \texttt{tcpdump} was used over the router which relayed $709$GB of traffic and only the first $150$bytes of packet were logged. \texttt{Ethereal} was used for protocol analysis.
Chakravarty \emph{et al.} \cite{chakravarty2011detecting} transmitted decoy traffic over a custom client supporting IMAP and SMTP protocols. The client was implemented using Perl and service protocol emulation was provided by \emph{Net::IMAPClient} and \emph{Net::SMTP}. The client hosted on Intel Xeon CPU running Ubuntu Server Linux \texttt{v8.04}.
Snader and Borisov \cite{snader2008tune} performed experiments on Tor by downloading $1$MB files over HTTP connections through various exit routers. All other entities including guard routers, client and web server remain fixed for the entire duration of the experiment. $20,000$ and $40,000$ trials were performed for tunable Tor and standard Tor respectively spanning a duration of two months.
Gilad and Herzberg \cite{gilad2012spying} conducted an empirical investigation for the performance of proposed attacks in the Tor network. Indirect rate reduction attack was evaluated by experiments in the live network. For experiments, a Linux machine ran an Apache web server. Data at the rate of $0.5$KBps was transmitted.
Loesing \emph{et al.} \cite{loesing2010case} collected Tor statistics by following the legal requirements, user privacy, ethical approvals, informed consent and community acceptance. Authors collected data from the Tor network and evaluated the port numbers and country of origin of the obtained IP addresses.
Chen \emph{et al.} \cite{chen2009xpay} developed ORPay which uses out-of-band communication for payment primitives and control messages. The ``bank'' was built using C language and OpenSSL for encryption. Authors performed controlled experiments consisting of a set of interconnected PCs running directory servers and Tor routers on VMs. Inter-client bandwidth was $500-600$KB/s with $1-2$ms average latency and $0.5$ms for inter-VMs on the same machine. One micropayment was made for every $20$ packets.
Panchenko \emph{et al.} \cite{panchenko2011website} using standard PCs for fetching websites using Firefox with disabled active components (Java, Flash etc.) and Chickenfoot used as the default plugin. The closed-world dataset was collected from previous studies, to obtain labeled ground truth dataset.
Wang and Goldberg \cite{wang2013improved} performed experiments on SHARCNET, a parallel computing cluster. Upto $200$ cores were used for computation of SVM kernel matrix. \texttt{torrc} was configured to close the circuits manually instead of fixed $10$mins duration and fixed entry guard selection was disabled. iMacros and Tor controller was used to automate site accesses. For closed world circuits, fingerprinting was performed on $100$ sites with $40$ instances each and using $10$-fold cross validation. For open-world experiments, Alexa's top $1,000$ sites list was used.
\subsubsection{PlanetLab Experiments}
Akhoondi \emph{et al.} \cite{akhoondi2012lastor} performed experiments in the real Tor network by modifying the Tor Client with their proposed \emph{LASTor} protocol. \texttt{LASTor} is a Java application controlling the Tor client through \emph{Control Port}. $50$ \emph{PlanetLab} nodes running \texttt{LASTor} were used as Tor clients to access top $200$ websites. Both latency and anonymization were tested by collecting the traces of data set at the client nodes.
Murdoch and Danezis \cite{murdoch2005low} performed experiments on the Tor network by setting up a probe PC. A modified version of Tor was used in the probe PC to choose routes of length one. A TCP client was also established at the node which connects to the SOCKS interface of Tor using \emph{socat}. Original Tor relays were used with a corrupt destination Tor server recording the traffic traces. The probe server ran at the University of Cambridge Computer Laboratory while victim and corrupt server were run on PlanetLab nodes. Data from $13$ probing Tor nodes was collected and analyzed in \emph{GNU R}.
Bauer \emph{et al.} \cite{bauer2007low} performed experiments over PlanetLab testbed by setting up two independent node networks comprising of $40$ and $60$ nodes, respectively. Two and six malicious nodes were added in the $40$ node network while three and six malicious nodes were added in the $60$ node network. Traffic was generated by six machines running $60$ and $90$ clients (requesting files of less than $10$MB size using HTTP protocol) in the $40$ and $60$ node network, respectively. To avoid flooding of network requests, clients sleep in the $0-60$sec interval for random periods after every random number of web requests.
\subsubsection{Cloud Services}
Sulaiman and Zhioua \cite{sulaiman2013attacking} performed extensive experiments using Amazon \emph{EC2} cloud services. An Apache web server was used to host a simple web page. \texttt{socket.io} with \texttt{node.js.Socket.io} was installed which supported WebSocket to help users' browsers in using OP and using unpopular ports. For path selection, simulations were also performed for entrance router selection algorithm and non-entrance router selection algorithm. Several experiments were conducted on a number of unpopular ports with $1,500$ circuits established per experiment and compromised links were detected.
Karaoglu \emph{et al.} \cite{karaoglu2012multi} implemented a unidirectional scenario of client uploading a file to a web server. A client established multiple socket connections for multipath transmissions. A $1.5$MB file was uploaded through clients. To incorporate geo-diversity, client softwares were installed in the US and at Amazon EC2 sites in Singapore, Ireland and North Virginia. A web server, placed at the Emulab Utah facility, listened on multiple ports.
Biryukov \emph{et al.} \cite{biryukov2013trawling} performed deanonymization by spending less than $100$ USD on Amazon EC2 cloud. $50$ Amazon EC2 instances were generated which captured $59,130$ publication requests. Data from $120$ running hidden services from the Tor network was collected. Collected data was used to identify the vulnerability of Tor hidden services.
\subsubsection{OpenFlow Enabled Network}
Mendonca \emph{et al.} \cite{mendonca2012flexible} used OpenFlow implementation for their proposed AnonyFlow scheme. An experimental testbed used Linux to connect the two subnetworks. Each subnetwork was connected to two OpenFlow enabled switches and two Net FPGA based switches. All these OpenFlow switches were governed by a NOX controller. \texttt{iperf} was used at the two client hosts with each running for nearly $25$secs.
\subsubsection{Universal Composability framework}
Backes \emph{et al.} \cite{backes2012provably} provided security enhancements to the currently used Tor network. New algorithms were provided and setup in the universal composability (UC) framework.
\subsection{Tor Simulations}
Tor simulations have been performed by (1) developing custom simulator (2) using ExperimenTor, (3) employing Shadow simulator, and (4) using ModelNet, as shown in Figure \ref{fig: Classification of Simulations on Tor.}.
Figure \ref{fig: Classification of Simulations on Tor.} shows that 75\% of the researches (comprising of simulations) used in this study developed their custom simulator. Only $13\%$ used ExperimenTor. $8\%$ and $4\%$ of the studies used Shadow simulator and Modelnet, respectively.
\begin{figure}[b]
\centering
\includegraphics[width=0.8\columnwidth]{simulations.pdf}\\
\caption{Classification of simulations on Tor.}\label{fig: Classification of Simulations on Tor.}
\end{figure}
\subsubsection{Custom Simulator}
Tschorsch and Scheuermann \cite{tschorsch2011tor} conducted simulations on \emph{ns-3} to implement Tor network with and without \emph{N23} modifications. To replicate the onion routers environment, all onion routers were connected to a central node. Access links of all onion routers had an $80$ms delay and $100$Mbps bandwidth. Sending hosts generate data at a rate of $400$kbps and Tor nodes had a maximum bandwidth limit of $600$kbps.
Doswell \emph{et al.} \cite{doswell2013novel} used the generic network simulator OMNET++ to simulate mobile Tor. Wireless access points were placed $75$m apart and results were estimated using linear mobility. Average throughput (kbps) was selected as the performance metric. A $300$kB webpage was downloaded after every $2$secs over the time-frame of $600$secs. An artificial latency was also introduced to incorporate congestion.
Edman and Syverson \cite{edman2009awareness} implemented the multi-thread path selection algorithm in C. Relationships between different ASs were borrowed from predecessor studies. RIBs collected by University of Oregon's RouteViews project were used.
Ngan \emph{et al.} \cite{dingledine2010building} built a discrete event simulator, in Java, for the Tor network. $64$-bit AMD Opteron $252$ dual core processors were used with $4$GB RAM and operating on Sun's JVM and RedHat Enterprise Linux. Tor network with $150$ relays was simulated and all cells from every client were simulated at every hop. Link latency was $100$ms and link capacity was $500$KB/s. All scenarios were tested comprising of Tor's original design, proposed design and a hybrid mechanism.
Benmeziane and Badache \cite{benmeziane2010tor} built their own simulator which incorporates public communications, DNS requests, and anonymous communications by Tor. The authors used $500$ senders using $100$ Tor relay nodes with $10$ executions per sender. Authors increased the number of recipients to $200$. Much of the simulator details were skipped.
Li \emph{et al.} \cite{li2012relay} developed their own discrete event simulator for Tor network. Key data structures and algorithms of Tor were used to simulate several thousand nodes. However, authors did not perform encryption, decryption and data transmission to avoid complexity. Moreover, simulations were driven by initialization and termination events. For a closer look, realistic values of bandwidth and uptime were obtained from the Tor metrics portal. Effective bandwidth of relays was set to $155$kBps with a $750$ standard deviation. $3000$ relays with millions of clients were used for simulations.
Snader and Borisov \cite{snader2011improving} developed a custom flow-level simulator for the Tor network. Using the Tor metric portal, bandwidth of actual Tor relays was used to simulate a $1,000$ node network. $10,000$ flows were simulated for each time unit of the simulator. Fair queueing was used for flow scheduling.
Jansen \emph{et al.} \cite{jansen2010recruiting} built a discrete event simulator for the Tor network comprising of $19,400$ web clients, $300$ Tor relays, $2,000$ servers and $600$ file sharing nodes. For file sharing, web traffic comprised of $12$Mbps downstream with $1.3$Mbps upstream bandwidth.
Johnson \emph{et al.} \cite{johnson2013users} built the \emph{TorPS} simulator for selection of Tor paths. Simulations were carried out for six months with an adversary model containing one guard relay and one exit relay having $83$MBps and $16.7$MBps. For analysis of client behavior, $50,000$ Monte Carlo simulations were carried out spanning a period of three months.
Nowlan \emph{et al.} \cite{nowlan2013reducing} developed a setup for a small virtual Tor network to estimate the performance of the proposed modification. Tor network comprises of three directory authorities, three relay servers and single onion proxy. The link delay had a mean of $50$ms with a $5\%$ path loss for the second onion router.
Jansen \emph{et al.} \cite{jansen2012methodically} performed extensive simulations for their proposed model on both small and large-scale networks. Loss rate and latency have been borrowed from Ookla and iPlane estimation services. For small scale network, $50$ relays and $500$ clients have been configured using $50$ HTTP file servers. For large scale networks, $100$ relays and $1,000$ clients are linked with $100$ HTTP servers. Files of $320$KB and $5$MB are downloaded for performance analysis.
Danner \emph{et al.} \cite{danner2012effectiveness} carried out extensive simulations of their proposed analytical model. However, authors do not focus on experiments or discrete event simulations.
Wang \emph{et al.} \cite{wang2013rbridge} analyzed the performance of their proposed bridge distribution mechanism on an event-based simulator. Aggressive blocking, conservative blocking and event-driven based blocking of bridges were tested. The authors also developed an analytical model for performance prediction.
Jansen and Hopper \cite{jansen2011shadow} developed a discrete event simulator to replicate the real-world Tor network in software running on a single machine. Performance was validated against $402$ node PlanetLab network. Through HTTP client and server plugins, data was transferred through Shadow for verifications of simulations.
Smits \emph{et al.} \cite{smits2011bridgespa} developed an open source implementation of the proposed mechanism. Implementation is based on Linux version \emph{2.6.4}. Bridge distribution authorities needed to be reconfigured for distribution of keys.
Elahi \emph{et al.} \cite{elahi2012changing} simulated Tor entry guard selection and rotation mechanism on multicore servers with each simulation run comprising of $80,000$ users. The entry guard data was collected from real Tor network spanning a duration of eight months.
Zhang \emph{et al.} \cite{zhang2008novel} developed a complete Tor setup containing client, server and three onion routers. A probe server and user nodes were deployed in different network segments. Tor code in the nodes was configured to use the designated three relay nodes. Data from the probe server was sent in bursts after every $0.2$secs while corrupt server sends data after every $10-15$secs.
\subsubsection{ExperimenTor}
Bauer \emph{et al.} \cite{bauer2011experimentor} built a toolkit for emulation of Tor network named by \emph{ExperimenTor}. The \emph{ModelNet} network emulation platform has been used as the baseline approach. Scalability is one of the issues in ExperimenTor, owing to high resource consumption for large number of nodes.
Wacek \emph{et al.} \cite{wacek2013empirical} performed network experiments over ExperimenTor for a variety of network topologies. Authors also performed simulations on their simulator which modeled a $1,524$ relay network.
Gopal and Heninger \cite{gopal2012torchestra} used ExperimenTor framework for simulations of their proposed Torchestra approach. ExperimenTor was setup on two physical machines working as edge node and emulator. For performance analysis, small and large files were downloaded starting from $300$KB. In the following stage, web and SSH traffic were simulated.
\subsubsection{Shadow Simulator}
Geddes \emph{et al.} \cite{geddes2013low} used the Shadow simulator with real Tor code on a simulated network. The simulated network consisted of $160$ exit relays, $240$ non-exit relays, $2375$ web clients, $125$ bulk clients, $150$ small and medium \texttt{Torperf} clients and $400$ HTTP servers. Experiments consisted of downloads of a $320$KB file from random servers after random delays ($1-60$secs). Bulk clients downloaded $5$MB file without any wait time. For \texttt{TorPerf} clients, $50$KB, $1$MB and $5$MB files were downloaded after every ten minutes.
Jansen \emph{et al.} \cite{jansen2011shadow} performed simulations over Shadow with a setup of $200$ HTTP servers, $950$ Tor web clients, $50$ Tor bulk clients and $50$ Tor relays. Bulk clients downloaded $5$MB file while web clients downloaded $5$KB page. Latency of network was borrowed from the latency of PlanetLab nodes. Performance was estimated by varying the load from $25$ (light) to $50$ (medium) to high ($100$) bulk users.
\subsubsection{ModelNet}
AlSabah \emph{et al.} \cite{alsabah2011defenestrator} used the \texttt{ModelNet} network emulation platform along with practical traffic models for performance evaluations. For small-scale experiments, $200$ downloads are made of the $300$KB and $5$MB files by two clients in two separate experiments. For large scale experiment, $20$ Tor routers are deployed with real Tor networks' bandwidth. Each link has $80$ms RTT delay. Ten clients download $1$--$5$MB file and $190$ clients download $100$--$500$KB file.
\subsection{Tor's Analysis}
A number of studies limited their research works to the analysis of current Tor network instead of simulations and experiments of Tor.
Analysis occurs in the subfields of usability of Tor, path selection mechanism, empirical analysis and development of theoretical model.
In below lines, we present the individual studies comprising of Tor's analysis.
\subsubsection{Analytical Model} Several studies develop analytical model for analysis of Tor network. These studies are presented in following lines.
\emph{Anti-misbehaviour Policy Analysis:} Liu and Wang \cite{liu2009anti} proposed anti-misbehavior policies and analyzed it with the original Tor architecture. No simulations or experiments were conducted.
\emph{Security Analysis:} Goldberg \cite{goldberg2006security} built an analytical model for analyzing the security of Tor's authentication protocol. Authors focused on analytical evaluations rather than simulations or experiments.
\emph{Botnet Abuse Analysis:} Hopper \cite{hoppershort} analyzed the various possibilities for avoiding botnet abuse in the Tor network. Majority schemes were discussed only, and a few schemes were tested to verify the performance. Several schemes were analyzed analytically.
\emph{Anonymity Model:} Xin \emph{et al.} \cite{xin2009design} developed a theoretical model to increase the anonymity of the Tor network. They aimed to implement the proposed system on PlanetLab testbed, in future.
\subsubsection{Empirical Analysis} A number of studies performed empirical analysis of Tor network without performing simulations or experiments. In following lines, we present the findings of these studies.
\emph{Statistical Analysis:} Wang \emph{et al.} \cite{wang2013empirical} performed empirical analysis and used data available from ``Tor Metric Portal'' for analysis. There were no simulations or experiments performed in the research. In another study, Elices \emph{et al.} \cite{elices2011fingerprinting} analyzed their attack on Tor using empirical analysis. Access logs from seven web servers were obtained to analyze user request pattern. Moreover, Abbott \emph{et al.} \cite{abbott2007browser} conducted a statistical evaluation by measuring the probabilities of breaching Tor using the proposed scheme.
\emph{Robustness Analysis:} Barthe \emph{et al.} \cite{barthe2010robustness} analyzed the robustness of the Tor network and proposed enhancements in the current network. Cryptographic enhancements were evaluated without any simulation or experimental validations.
\emph{Path Selection Protocol:} Liu and Wang \cite{liu2009improved} presented an improved circuit building protocol with no simulations or experiments. Proposed algorithm was analyzed considering various aspects. In another study, Liu and Wang \cite{liu2009random} presented random walk based algorithm for Tor circuit construction. Anonymity and performance were the key metrics evaluated in their study. However, the scope of this study did not cover simulations or experimental evaluations.
\subsubsection{Usability Analysis} Usability analysis of Tor has not been carried out by a lot of studies. However, some studies referring to usability analysis are summarized in below lines.
Clark \emph{et al.} \cite{clark2007usability} conducted a usability analysis by installing various components of Tor including Vidalia, Privoxy, Torbutton and Foxyproxy on a standard machine.
In another study, Abou-Tair \emph{et al.} \cite{abou2009usability} presented the usability analysis of the various anonymous service applications including Tor. Various anonymity tools were installed on a machine and usability, ease of installation and use was analyzed.
\section{Discussion}
\label{sec: Discussion}
In this section, we present the discussion and our findings of Tor network. In the first part, we present the performance metrics used to evaluate the Tor network in different research works. In the second part, we present our findings of Tor research works referred in this study. In the last part, we show our findings for open research areas in the field of Tor network which may be used for future research works.
\subsection{Tor Performance Metrics}
\label{sec:Tor Performance Metrics}
Analyzing the performance metrics is a crucial task for future research, analysis, simulations and experiments in the Tor network.
Table \ref{table: Performance Metrics used in various researches} presents the performance metrics of Tor used in various studies. No clear patterns were observed, so, authors described the metrics used in individual studies. A brief overview of the table shows that throughput (bandwidth) and latency are the most frequently used metrics. However, every research formalized its own performance metric based upon the requirement of the experiment.
\begin{table*}
\centering
\small
\caption{Performance metrics used in various research works.}\label{table: Performance Metrics used in various researches}
\begin{tabular}{|p{1.8cm}|p{7cm}|p{8cm}|}
\hline
Domain & Performance Metrics & Research Works \\
\hline \hline
Quality of Service of Tor & Throughput; Bandwidth; Packet rate; Bit rate; Goodput & Mendonca \emph{et al.} \cite{mendonca2012flexible}, Zhang \emph{et al.} \cite{zhang2011application}, Pries \emph{et al.} \cite{pries2008new}, Jin and Wang \cite{jin2009effectiveness}, Marks \emph{et al.} \cite{marks2010unleashing}, Panchenko \emph{et al.} \cite{panchenko2008performance}, Chen and Pasquale \cite{chen2010toward}, Karaoglu \emph{et al.} \cite{karaoglu2012multi}, Li \emph{et al.} \cite{li2012relay}, Panchenko \emph{et al.} \cite{panchenko2012improving}, Snader and Borisov \cite{snader2011improving}, Houmansadr \emph{et al.}, Pries \emph{et al.} \cite{pries2008performance}, Tschorsch and Scheuermann \cite{tschorsch2011tor}, Andersson and Panchenko \cite{andersson2007practical}, Doswell \emph{et al.} \cite{doswell2013novel}, Jansen \emph{et al.} \cite{jansen2010recruiting}, Moghaddam \emph{et al.}, Weinberg \emph{et al.} \cite{weinberg2012stegotorus}, Jansen \emph{et al.} \cite{jansen2012methodically}, Ehlert \cite{ehlert2011i2p}, Barbera \emph{et al.} \cite{barbera2013cellflood}, Hopper \cite{hoppershort}, Nowlan \emph{et al.} \cite{nowlan2013reducing}, Ngan \emph{et al.} \cite{dingledine2010building}, Panchenko \emph{et al.} \cite{panchenko2008performance}, Wang \emph{et al.} \cite{wang2013empirical}, Jansen \emph{et al.} \cite{jansen2010recruiting}, Tang and Goldberg \cite{tang2010improved}, AlSabah \emph{et al.}, Wack \emph{et al.} \cite{wacek2013empirical}, Geddes \emph{et al.} \cite{geddes2013low}, AlSabah \emph{et al.} \cite{alsabah2011defenestrator}, Jansen and Hopper \emph{et al.} \cite{jansen2011shadow}, Jansen \emph{et al.} \cite{jansen2012throttling} \\
%
\cline{2-3}
%
& Latency; Webpage loading time; Round trip time; Download Time; Router latency; Circuit setup duration; Boot strap duration; Time to first byte; Time to last byte; Ping reply delay; Per hop latency; SYN and SYN ACK difference; Delay per cell; Jitter; Inter packets delay distribution & Mendonca \emph{et al.} \cite{mendonca2012flexible}, Overlier and Syverson \cite{overlier2006locating}, Loesing \emph{et al.} \cite{loesing2008performance}, Chan-Tin \emph{et al.} \cite{chan2013revisiting}, Herzberg \emph{et al.} \cite{herzberg2011camouflaged}, Murdoch and Danezis \cite{murdoch2005low}, Zhang \emph{et al.} \cite{zhang2008novel}, Akhoondi \emph{et al.} \cite{akhoondi2012lastor}, Panchenko \emph{et al.} \cite{panchenko2008performance}, Li \emph{et al.} \cite{li2012tmt}, Dhungel and Steiner \cite{dhungel2010waiting}, Andersson and Panchenko \cite{andersson2007practical}, Doswell \emph{et al.} \cite{doswell2013novel}, AlSabah \emph{et al.}, Moghaddam \emph{et al.}, Weinberg \emph{et al.} \cite{weinberg2012stegotorus}, Evans \emph{et al.} \cite{evans2009practical}, Wang \emph{et al.} \cite{wang2012congestion}, Jansen \emph{et al.} \cite{jansen2012methodically}, Ehlert \cite{ehlert2011i2p}, Hopper \cite{hoppershort}, Winter and Lindskog \cite{winter2012great}, Edmundson \emph{et al.} \cite{edmundson14security}, Ngan \emph{et al.} \cite{dingledine2010building}, Panchenko \emph{et al.} \cite{panchenko2008performance}, Lenhard \emph{et al.} \cite{lenhard2009performance}, Wack \emph{et al.} \cite{wacek2013empirical}, Geddes \emph{et al.} \cite{geddes2013low}, AlSabah \emph{et al.} \cite{alsabah2011defenestrator}, Jansen and Hopper \emph{et al.} \cite{jansen2011shadow}, Snader and Borisov \cite{snader2008tune}, Jansen \emph{et al.} \cite{jansen2012throttling}, Chen \emph{et al.} \cite{chen2009xpay}, Smits \emph{et al.} \cite{smits2011bridgespa}, Gopal and Heninger \cite{gopal2012torchestra}, Panchenko \emph{et al.} \cite{panchenko2012improving}, Panchenko \emph{et al.} \cite{panchenko2008performance} \\
%
\hline
%
Performance of Tor's breaching attempts & True Positive; True Negative; False Positive; False Negative; Region of Convergence; Recognition rate; Mis-recognition rate; Accuracy; Recall; Precision; F-measure & Chakravarti \emph{et al.} \cite{chakravarty2008identifying}, Barker \emph{et al.} \cite{barker2011using}, Chan-Tin \emph{et al.} \cite{chan2013revisiting}, Akhoondi \emph{et al.} \cite{akhoondi2012lastor}, Danner \emph{et al.} \cite{danner2012effectiveness}, Gilad and Herzberg \cite{gilad2012spying}, Panchenko \emph{et al.} \cite{panchenko2011website}, Song \emph{et al.}, Wang and Goldberg \cite{wang2013improved}, Wang and Goldberg \cite{wang2013improved}, Elices \emph{et al.} \cite{elices2011fingerprinting}, Bai \emph{et al.} \cite{bai2008traffic}, AlSabah \emph{et al.}, Wagner \emph{et al.} \cite{wagner2012breaking}\\
%
\cline{2-3}
%
& Timing Attack correlation & Overlier and Syverson \cite{overlier2006locating}, Zhang \emph{et al.} \cite{zhang2011application}, Pries \emph{et al.} \cite{pries2008new}, Wang \emph{et al.} \cite{wang2009novel}, Houmansadr \emph{et al.}, Murdoch and Danezis \cite{murdoch2005low}, Song \emph{et al.}, Panchenko \emph{et al.} \cite{panchenko2008performance} \\
%
\cline{2-3}
%
& Compromised relays; Compromised circuits; Compromised Streams; Time for first compromised stream; Failure rate; Compromise time; Compromised links; Compromised Tunnels; Detection rate; Compromised Clients; Compromised router bandwidth; Compromise probability; Congestion attack time & Overlier and Syverson \cite{overlier2006locating}, Sulaiman and Zhioua \cite{sulaiman2013attacking}, Chen and Pasquale \cite{chen2010toward}, Li \emph{et al.} \cite{li2012relay}, Li \emph{et al.} \cite{li2012tmt}, Bauer \emph{et al.} \cite{bauer2007low}, Johnson \emph{et al.} \cite{johnson2013users}, Evans \emph{et al.} \cite{evans2009practical}, Danner \emph{et al.} \cite{danner2012effectiveness}, Chakravarty \emph{et al.} \cite{chakravarty2011detecting}, Snader and Borisov \cite{snader2008tune}, Panchenko \emph{et al.} \cite{panchenko2011website}, Elahi \emph{et al.} \cite{elahi2012changing}, Abbott \emph{et al.} \cite{abbott2007browser}, Bauer \emph{et al.} \cite{bauer2009predicting}, Wang \emph{et al.} \cite{wang2012congestion}\\
\hline
%
Analysis of Tor & Packet Sizes; Probability difference plots; Energy plots; Recipient probabilities; Queued messages length; Anonymity vs performance; Router bandwidth; HTTP content distribution; Tor servers; Tor traffic; Generated Paths; Client resource usuage; Tor load per circuit; Node Connection pattern; IP TTL difference; Service, browser, file format usuage; Tor location usuage; Boot strap time; Exit traffic stats; Tor bridges statistics; hidden service descriptor request rate; Botnet decay rate; Tor overhead; Router statistics
& Barker \emph{et al.} \cite{barker2011using}, Loesing \emph{et al.} \cite{loesing2008performance}, Benmeziane \emph{et al.}, Jin and Wang \cite{jin2009effectiveness}, Zhang \emph{et al.} \cite{zhang2008novel}, Liu and Wang \cite{liu2009random}, Liu and Wang \cite{liu2009anti}, Dhungel and Steiner \cite{dhungel2010waiting}, Chaabane \emph{et al.} \cite{chaabane2010digging}, Mulazzani \emph{et al.} \cite{mulazzani2010anonymity}, Moghaddam \emph{et al.}, Edman and Syverson \cite{edman2009awareness}, Barbera \emph{et al.} \cite{barbera2013cellflood}, Hopper \cite{hoppershort}, Winter and Lindskog \cite{winter2012great}, Huber \emph{et al.} \cite{huber2010tor}, Blond \emph{et al.} \cite{blond2011one}, Lenhard \emph{et al.} \cite{lenhard2009performance}, McCoy \emph{et al.} \cite{mccoy2008shining}, Wang \emph{et al.} \cite{wang2013rbridge}, Biryukov \emph{et al.} \cite{biryukov2013trawling}, Loesing \emph{et al.} \cite{loesing2010case}, Chen \emph{et al.} \cite{chen2009xpay}, Panchenko \emph{et al.} \cite{panchenko2011website}, Elahi \emph{et al.} \cite{elahi2012changing}, Marks \emph{et al.} \cite{marks2010unleashing}\\
%
\cline{2-3}
%
& Empirical Evaluations: Usability analysis, security model analysis, general discussion, proposed mechanism validation & Clark \emph{et al.} \cite{clark2007usability}, Abou-Tair \emph{et al.} \cite{abou2009usability}, Goldberg \cite{goldberg2006security}, Kuhn \emph{et al.}, Barthe \emph{et al.} \cite{barthe2010robustness}, Gros \emph{et al.} \cite{gros2010protecting}\\
\hline
\end{tabular}
\end{table*}
\begin{comment}
\onecolumn
\begin{longtable}{|p{4cm}|p{10cm}|}
\hline \hline
Research & Investigated Performance Metrics \\
\hline \hline
Mendonca \emph{et al.} \cite{mendonca2012flexible} & TCP Throughput, Webpage loading time \\
\hline
Overlier and Syverson \cite{overlier2006locating} & Round Trip Time, Time to first match, Matched Circuits \\
\hline
Zhang \emph{et al.} \cite{zhang2011application} & Bandwidth, Time Correlation \\
\hline
Elices \emph{et al.} \cite{elices2011fingerprinting} & Mis-detection and False Positive Probability \\
\hline
Loesing \emph{et al.} \cite{loesing2008performance} & Time periods (Min, Max, Mean, Var, Std dev etc.), Probability difference plots \\
\hline
Chakravarti \emph{et al.} \cite{chakravarty2008identifying} & TP, TN, FP, FN rate for compromised circuits \\
\hline
Bai \emph{et al.} \cite{bai2008traffic} & Recognition rate and mis-recognition rate \\
\hline
Barker \emph{et al.} \cite{barker2011using} & Tor Packet Sizes, TP, FP and ROC \\
\hline
Pries \emph{et al.} \cite{pries2008new} & Bandwidth, Correlation \\
\hline
Sulaiman and Zhioua \cite{sulaiman2013attacking} & Count of compromised entry and exit relays on $9$ relays \\
\hline
Chan-Tin \emph{et al.} \cite{chan2013revisiting} & Latency for entry, middle and exit relay, TP and FP rate \\
\hline
Wagner \emph{et al.} \cite{wagner2012breaking} & Sensitivity, Specificity \\
\hline
Wang \emph{et al.} \cite{wang2009novel} & Correlation (Webpage injection and Traffic pattern detection time) \\
\hline
Benmeziane \emph{et al.} & Recipient probabilities of the sender \\
\hline
Houmansadr \emph{et al.} & Time, Packet rate with time \\
\hline
Jin and Wang \cite{jin2009effectiveness} & Energy plots, Packet rate (pkts/sec) \\
\hline
Gros \emph{et al.} \cite{gros2010protecting} & Validation of proposed mechanism \\
\hline
Herzberg \emph{et al.} \cite{herzberg2011camouflaged} & Latency for webpage accessing \\
\hline
Marks \emph{et al.} \cite{marks2010unleashing} & Throughput, Sequence no. and ACK progression \\
\hline
Murdoch and Danezis \cite{murdoch2005low} & Latency, Correlation \\
\hline
Song \emph{et al.} & Accuracy, recall, Correlation \\
\hline
Zhang \emph{et al.} \cite{zhang2008novel} & Latency, length of queued messages \\
\hline
Akhoondi \emph{et al.} \cite{akhoondi2012lastor} & Latency, FN and FP rate \\
\hline
Panchenko \emph{et al.} \cite{panchenko2008performance} & Throughput, Circuit Setup Duration, RTT vs. hops, Correlation \\
\hline
Bauer \emph{et al.} \cite{bauer2009predicting} & Claimed bandwidth by malicious routers for various ports \\
\hline
Chen and Pasquale \cite{chen2010toward} & Throughput, Percent failures and average number of attempts \\
\hline
Karaoglu \emph{et al.} \cite{karaoglu2012multi} & Downloading Time (1.5 MB and 300 KB file) \\
\hline
Li \emph{et al.} \cite{li2012relay} & Bandwidth, Rate of Lying relays \\
\hline
Panchenko \emph{et al.} \cite{panchenko2012improving} & TCP Ping, RTT, Jitter, Throughput \\
\hline
Snader and Borisov \cite{snader2011improving} & Throughput \\
\hline
Liu and Wang \cite{liu2009random} & Anonymity levels and performance tested empirically \\
\hline
Li \emph{et al.} \cite{li2012tmt} & Browsing time and failure rate \\
\hline
Pries \emph{et al.} \cite{pries2008performance} & Throughput\\
\hline
Liu and Wang \cite{liu2009anti} & Anonymity, performance and QoE evaluation empirically \\
\hline
Dhungel and Steiner \cite{dhungel2010waiting} & Router Delay, Advertised and Consensus bandwidth\\
\hline
Wang \emph{et al.} \cite{wang2013empirical} & Router bandwidth, Online time, aggregate bandwidth \\
\hline
Tschorsch and Scheuermann \cite{tschorsch2011tor} & Throughput\\
\hline
Chaabane \emph{et al.} \cite{chaabane2010digging} & Geopolitical relay distribution, HTTP content distribution, Tor Peers \\
\hline
Mulazzani \emph{et al.} \cite{mulazzani2010anonymity} & No. of Tor servers, daily pattern, exit servers, guard servers \\
\hline
Abou-Tair \emph{et al.} \cite{abou2009usability} & Download, Installation, Configuration, Verification, Deactivation, Bandwidth consumption \\
\hline
Andersson and Panchenko \cite{andersson2007practical} & Throughput, Latency (msec) \\
\hline
Doswell \emph{et al.} \cite{doswell2013novel} & Bit rate, Circuit build time \\
\hline
Abbott \emph{et al.} \cite{abbott2007browser} & Breaching probability from Tor statistics\\
\hline
Bauer \emph{et al.} \cite{bauer2007low} & Compromised Ciruits (\%)\\
\hline
Jansen \emph{et al.} \cite{jansen2010recruiting} & Circuit throughput, download time, bandwidth utilization\\
\hline
Tang and Goldberg \cite{tang2010improved} & Downloading time\\
\hline
AlSabah \emph{et al.} & Precision, Recall, F-measure, accuracy, data downloaded per application, downloading time, time to first byte\\
\hline
Moghaddam \emph{et al.} & Goodput, network bandwith used, packet size distribution, inter-packet delays distribution\\
\hline
Weinberg \emph{et al.} \cite{weinberg2012stegotorus} & Goodput, RTT\\
\hline
Johnson \emph{et al.} \cite{johnson2013users} & Time to first compromise, Streams compromised\\
\hline
Evans \emph{et al.} \cite{evans2009practical} & Latency, Detection time, Number of runs\\
\hline
Wang \emph{et al.} \cite{wang2012congestion} & Latency (RTT), Congestion time\\
\hline
Jansen \emph{et al.} \cite{jansen2012methodically} & Throughput, Times for first and last bytes\\
\hline
Danner \emph{et al.} \cite{danner2012effectiveness} & FP rate, FN rate, Compromised links\\
\hline
Edman and Syverson \cite{edman2009awareness} & Circuits generated by proposed path selection\\
\hline
Ehlert \cite{ehlert2011i2p} & Latency, Bandwidth\\
\hline
Barbera \emph{et al.} \cite{barbera2013cellflood} & Client resource usage, Bandwidth\\
\hline
Hopper \cite{hoppershort} & Download time, circuit building time, onion-skin load per circuit \\
\hline
Winter and Lindskog \cite{winter2012great} & Delay of ping reply, IP TTL difference, Scan of connection pattern\\
\hline
Nowlan \emph{et al.} \cite{nowlan2013reducing} & Download completion time\\
\hline
Edmundson \emph{et al.} \cite{edmundson14security} & Latency\\
\hline
Huber \emph{et al.} \cite{huber2010tor} & Services usage, browser usage, file format usage \\
\hline
Ngan \emph{et al.} \cite{dingledine2010building} & Download time, ping time \\
\hline
Panchenko \emph{et al.} \cite{panchenko2008performance} & Bandwidth, RTT and Jitter\\
\hline
Blond \emph{et al.} \cite{blond2011one} & Bit Torrent Contents and websites usage, country wise Tor usage\\
\hline
Lenhard \emph{et al.} \cite{lenhard2009performance} & Circuit completion time, boot strapping time \\
\hline
McCoy \emph{et al.} \cite{mccoy2008shining} & Exit traffic protocol, geo-political distribution (client and router), Tor router bandwidth distribution\\
\hline
Wang \emph{et al.} \cite{wang2013rbridge} & Probability of alive bridges, time of bridges usage, $\%$ of thirsty hours \\
\hline
Wack \emph{et al.} \cite{wacek2013empirical} & Throughput, time to first byte and average ping RTT\\
\hline
Biryukov \emph{et al.} \cite{biryukov2013trawling} & hidden service descriptor request rate, relays, decay rate of botnet, connections \\
\hline
Geddes \emph{et al.} \cite{geddes2013low} & Throughput, latency \\
\hline
AlSabah \emph{et al.} \cite{alsabah2011defenestrator} & Time to first byte, download time\\
\hline
Chakravarty \emph{et al.} \cite{chakravarty2011detecting} & Time for exposure of decoy credentials and first connect back attempt\\
\hline
Jansen and Hopper \emph{et al.} \cite{jansen2011shadow} & Bandwidth, latency, time to first byte, download time\\
\hline
Snader and Borisov \cite{snader2008tune} & Transfer times for 1 MB file, fraction of tunnels compromised by attacker\\
\hline
Gilad and Herzberg \cite{gilad2012spying} & FP, FN and success vs. attack duration\\
\hline
Jansen \emph{et al.} \cite{jansen2012throttling} & Time to first byte, download time for web and bulk load\\
\hline
Goldberg \cite{goldberg2006security} & Empirical model of security \\
\hline
Clark \emph{et al.} \cite{clark2007usability} & Ease of Usability including installation, configuration and verification\\
\hline
Loesing \emph{et al.} \cite{loesing2010case} & Country of origin and port number of exiting traffic \\
\hline
Chen \emph{et al.} \cite{chen2009xpay} & Per-hop latency, transactions per second, amortized time cost\\
\hline
Smits \emph{et al.} \cite{smits2011bridgespa} & SYN/SYN ACK difference + std. dev\\
\hline
Panchenko \emph{et al.} \cite{panchenko2011website} & Detection rate, overhead, TP rate, FP rate \\
\hline
Elahi \emph{et al.} \cite{elahi2012changing} & Number of guards, client compromise rates, router up and down times \\
\hline
Gopal and Heninger \cite{gopal2012torchestra} & Time to download first byte, download completion time, delay per cell \\
\hline
Wang and Goldberg \cite{wang2013improved} & ROC Area, Accuracy, FP rate \\
\hline
\caption{Performance metrics used in various researches}\label{table: Performance Metrics used in various researches}
\end{longtable}
\twocolumn
\end{comment}
\subsection{Survey Findings}
In this section, we summarize our findings for the onion router by comparing all studies with a deep focus over the key concepts and ideas used in different research works.
We divide our research evaluations in three subcategories considering (1) research areas, (2) research platforms, and (3) performance metrics.
\subsubsection{Research Areas}
The majority of Tor research (nearly 55\%) covering anonymity is focused over deanonymization of Tor network. Around 20\% studies are related to the path selection mechanism. Only 25\% research studies are on performance analysis and improvement mechanism of Tor network. According to Dingledine (the co-founder of Tor project), majority research works focus their attention on the breaching Tor.
\emph{Deanonymization:}
In the deanonymization track, 35\% of the studies design deanonymization attacks for Tor while 21\% deanonymize Tor using traffic analysis. 16\% focus on improvements to bypass deanonymization while 14\% study fingerprinting mechanisms to identify Tor traffic on the Internet. Only 9\% identify hidden services while 2\% focus on anonymity mechanisms without using Tor.
All deanonymization related Tor studies have exploited its inherent weaknesses. Compromised relays are the most exploited weaknesses followed by traffic interception and protocol messages. Very few studies focus on the compromised autonomous systems, browsers, servers, decoy traffic, and flag cheating.
\emph{Path Selection:}
In the path selection track, 87\% of the studies focus on the design of new path selection algorithms and 13\% research works analyze currently developed algorithms.
Our analysis shows that anonymity and performance (bandwidth and latency) are the most important parameters used in the design and analysis of path selection algorithms. Relays have been incorporated in the design of path selection algorithms covering both location and capacity of relays. Other parameters include autonomous systems, hops, multi-path mechanism and load.
\emph{Performance Analysis and Architectural Improvements:}
In the performance analysis and improvement track, 32\% of the research works cover analysis and 27\% of studies focus on performance improvement mechanisms. 29\% of the studies provide general analysis of Tor covering usability and sociability issues. 10\% of the research works focus on modeling of the Tor network while 2\% address client mobility.
Analysis of various research studies show that performance (latency and bandwidth), relay selection, and anonymity are the most used parameters. Other studies also pay attention to queues, QoS, protocol messages and traffic shaping.
\subsubsection{Research Platforms}
An interesting feature revealed in analysis is the fact that 60\% of the research works were conducted by performing real-world experiments on the Tor network. Although special measures were taken to protect the identity of users but majority research works failed to analyze legal or ethical requirements of capturing user data and performing experiments by developing attacks in real network. Only 27\% of studies developed their own simulator and 13\% conducted analysis without experiments or simulations.
\emph{Experiments:}\\
Our survey shows that 86\% of the research works developed their own testbed for experiments. The majority of studies deployed 1-2 clients with 1-3 servers for experiments. Research works covering relays used 1-3 relays. However, some research works increased the number of relays by using virtual machines and PlanetLab. A limited number of studies used cloud-based setups.
\emph{Simulations:}\\
Interestingly, 75\% of the research works developed their own simulator without any common parameters used for Tor network. ns-3, OMNET, C and Java were used for the development of custom simulators. 13\% of the research works used ExperimenTor. Our research shows that ExperimenTor is the most common toolkit used by majority of the research. 8\% and 3\% of the studies used Shadow simulator and ModelNet, respectively.
\subsubsection{Performance Metrics}
We considered the performance metrics used in various works. Analysis shows that no hard and fast rule exists for use of performance metrics. Every study developed its own metrics to measure performance, anonymity and QoS. Moreover, no baseline techniques exist for the comparison of results.
\subsection{Open Research Areas}
Our survey shows that majority of the research works are concentrated in a few domains. However, a number of major challenges exist owing to the peer-to-peer nature of Tor.
A number of key areas have also been identified by the Tor team. We identified the following areas which require further research.
\begin{enumerate}
\item \emph{Data Estimation:}
Estimation of key network statistics is the most critical task in the Tor network because it is a peer-to-peer network. No one can see the entire traffic so it is not possible to estimate the size of Tor network. Some of the statistics requiring attention are as follows:
\begin{itemize}
\item Number of clients in the network: Peer-to-peer networks make it impossible to estimate the total traffic statistics because no user can see the complete traffic.
\item Capabilities of relays: There is limited information available about the relays which are the most crucial parameters in path selection. Incorporation of relay capabilities into anonymity of Tor and performance model is a key research area as done in a number of studies.
\item Performance of the network: Estimation of network performance at any given time is a crucial task. Owing to the P2P nature, only health of relays is known to the Tor administration. \emph{How is the network performing at any given instant?} is still a crucial task.
\item Number of clients connecting via bridges: Tor authorities provide secret relay addresses to clients who can't access Tor due to blockage of relays in their location. However, very little is known about the quantity of clients connecting through bridges and their traffic statistics.
\item Exit network traffic: Significant research is required about the exit network traffic. All clients pass their data through relays and very little is known about the statistics of traffic exiting exit relays.
\end{itemize}
\item \emph{Analysis:}
Deep analysis of the current Tor network is required. Analysis may be based upon an extension of previous research into path length, anonymity, latency, etc. Analysis of the optimal performance parameters is required.
\item \emph{Measurement and Attack tools:}
Development of novel attack methodologies to identify the shortcomings of the current Tor network. Tor has no automatic mechanism to identify anomalies and assess the health of the network. Attack tools should be developed which should prevent attacks occurring from compromised relays and servers. Comprised relays are vulnerable to botnet based attacks comprising of DDOS attacks, fingerprinting attacks etc. Despite large amount of research in botnet attacks, it is still open to research which would make Tor a more stable and secure network.
\item \emph{Defenses against Attacks:}
Develop novel defense methodologies to counter attacks on the Tor network. Although majority research works have focused on the development of novel attack methodologies, very little is known about viable counter-measures. Our survey shows that relays are mostly vulnerable because they can be deployed by any eavesdropper. Counter-measures against congestion attacks, latency measuring attacks, throughput measuring attacks, etc. can help in the improvement of Tor.
\end{enumerate}
\section{Conclusion}
\label{sec: Conclusion}
This paper deals with the survey, classification, quantification and comparative analysis of various research works covering Tor network.
To the author's best knowledge, no other survey/research has performed such a deep and thorough analysis of Tor studies.
Our study shows that Tor research areas can be broadly classified into (1) deanonymization, (2) path selection, (3) analysis and performance improvements.
More than half studies carried out address `deanonymization' with major subdivisions into deanonymization `attacks' and `traffic analysis' attacks.
In the `path selection' area, more than $85\%$ of the studies have focused on the development of new algorithms.
In the `analysis and performance improvement' area, the majority of studies are a mixed bag, followed by analysis, followed by performance improvement studies.
Our analysis of Tor platforms shows that $60\%$ of studies performed experiments while $27\%$ performed simulations.
Among experiments, $86\%$ of the studies deployed private testbeds.
Among simulations, $75\%$ developed their own simulators.
Analysis of parameters (used in various studies) shows that their is no little consistency across various studies.
However, a majority of the studies used variations of throughput and latency for performance analysis.
\section{Bibliography}
\bibliographystyle{unsrt}
|
{
"timestamp": "2018-03-08T02:12:36",
"yymm": "1803",
"arxiv_id": "1803.02816",
"language": "en",
"url": "https://arxiv.org/abs/1803.02816"
}
|
\section{Introduction}
The state of a finite dimensional quantum system is described by
the density operator $\varrho \in \mathcal{B}(\mathcal{H}_d)$
acting on the Hilbert space $\mathcal{H}_d$, $d = {\rm
dim}\mathcal{H}_d$. The density operator is Hermitian, positive
semidefinite, and has unit trace.
In the theory of open quantum systems, the most general form of
the density operator transformation due to its own evolution and
interaction with some environment (initially uncorrelated from the
system) is given by a quantum channel
$\Phi:\mathcal{B}(\mathcal{H}_{d_1}) \mapsto
\mathcal{B}(\mathcal{H}_{d_2})$, which is a completely positive
and trace preserving linear map (Ref.~\cite{holevo-2012}, section
6.3). In what follows, we consider the case when the system
dimension does not change, i.e., $d_1 = d_2 = d$.
There is a special class of unital quantum channels, which
preserve the maximally mixed state $\frac{1}{d} I$, where $I$ is
the identity operator, i.e. $\Phi[\frac{1}{d} I] = \frac{1}{d} I$.
The seminal result of Landau and Streater~\cite{Landau-Streater}
is that all unital channels $\Phi:\mathcal{B}(\mathcal{H}_2)
\mapsto \mathcal{B}(\mathcal{H}_2)$ are random unitary, i.e.
$\Phi[\varrho] = \sum_i p_i U_i \varrho U_i^{\dag}$ for some
probability distribution $\{p_i\}$ and unitary operators $U_i$
acting on $\mathcal{H}_2$; whereas for larger dimensions this is
not the case~\cite{Landau-Streater}. Moreover, Landau and
Streater~\cite{Landau-Streater} provided an example of quantum
channel $\Phi:\mathcal{B}(\mathcal{H}_d) \mapsto
\mathcal{B}(\mathcal{H}_d)$, which is unital but not random
unitary for all $d \geqslant 3$.
The main goal of this paper is to explore quantum informational
properties of the Landau--Streater map
\begin{equation} \label{LS-map}
\Phi[\rho] = \frac{1}{j(j+1)} \left( J_x \rho J_x + J_y \rho J_y +
J_z \rho J_z \right)
\end{equation}
\noindent defined through the $SU(2)$ generators $J_x$, $J_y$,
$J_z$ acting on a $(2j+1)$-dimensional Hilbert space
$\mathcal{H}_{2j+1}$. Physically, this space corresponds to the
state space of a spin-$j$ particle. Hermitian operators $J_x$,
$J_y$, $J_z$ are spin projection operators onto axes $x$, $y$,
$z$, respectively. Hereafter, we will use indices $x,y,z$ and
$1,2,3$ interchangeably. The map \eqref{LS-map} is completely
positive as it has diagonal sum (Kraus) representation
(Ref.~\cite{holevo-2012}, Corollary 6.13), and is trace preserving
as $\sum_{k=1}^3 J_k^2 = j(j+1) I$ (Ref.~\cite{Varshalovich},
section 6.1.2, formula (5)). The latter formula is also
responsible for unitality of the map~\eqref{LS-map}.
If $j=\frac{1}{2}$, then $J_k = \frac{1}{2} \sigma_k$, where
$\{\sigma_k\}_{k=1}^3$ is the conventional set of Pauli operators.
In this case $d=2$ and $\Phi$ is random
unitary~\cite{Landau-Streater}. Such a channel $\Phi$ is also
referred to as the best physical approximation of the universal
NOT gate for qubits~\cite{buzek-1999}.
If $j \geqslant 1$, then the map \eqref{LS-map} is an extremal
channel in the set of all channels, and therefore it cannot be
random unitary~\cite{Landau-Streater}. Also, in contrast to the
case $j = \frac{1}{2}$, the channel~\eqref{LS-map} differs from
the spin polarization-scaling channels~\cite{fm-2018} if $j
\geqslant 1$.
Since the Landau--Streater channel \eqref{LS-map} is essentially
unexplored, the goal of this paper is to fill the gap in both the
fundamental properties (such as covariance and spectrum of the
output states) and more specific quantum informational properties
(such as classical and quantum capacities, entanglement dynamics).
The paper is organized as follows.
In section~\ref{section-covariance}, we study covariance
properties of the Landau--Streater channel. In
section~\ref{section-spectrum-map}, we explicitly find the
spectrum of $\Phi$ for all $j$. In
section~\ref{section-spectrum-output}, we analyze the spectrum of
the output operator $\Phi[X]$ and reveal its peculiarities in the
case $j=1$. Such peculiarities are attributed to the fact that the
Landau--Streater channel for $j=1$ reduces to the Werner--Holevo
channel~\cite{wh-2002}. In section~\ref{section-multiplicativity},
we explicitly find the maximal $p$-norm and the minimal output
entropy of the general Landau--Streater channel. In
section~\ref{section-complementary}, physical realization of the
Landau--Streater channel and its complementary version is
discussed. In section~\ref{section-capacities}, different
capacities of the Landau--Streater channel are evaluated. In
section~\ref{section-entanglement}, we examine the entanglement
annihilation property of the channel $\Phi \otimes \Phi$. In
section~\ref{section-conclusions}, brief conclusions and outlook
are presented.
\section{Covariance} \label{section-covariance}
Following Refs.~\cite{holevo-1993}, consider a group $G$ and a
unitary representation $g \rightarrow U_g$, $g \in G$, in
$\mathcal{H}_d$. The channel $\Phi:\mathcal{B}(\mathcal{H}_d)
\mapsto \mathcal{B}(\mathcal{H}_d)$ is called covariant with
respect to representation $U_g$ if there exists a unitary
representation $g \rightarrow V_g$, $g \in G$, in $\mathcal{H}_d$
such that
\begin{equation}
\Phi[U_g X U_g^{\dag}] = V_g \Phi[X] V_g^{\dag}
\end{equation}
\noindent for all $g \in G$ and $X \in
\mathcal{B}(\mathcal{H}_d)$. Covariance of a channel has many
implications, e.g. the strong converse property of the
entanglement-assisted classical capacity~\cite{datta-2016}. Many
other properties and the structure of irreducibly covariant
quantum channels are reviewed in Ref.~\cite{mozrzymas-2017}.
We start with the simple observation that the Landau--Streater
channel \eqref{LS-map}, by construction, is endowed with the
$SU(2)$ covariance.
\begin{proposition} \label{prop-SU2-cov}
The Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \rightarrow
\mathcal{B}(\mathcal{H}_{2j+1})$ is covariant with respect to the
unitary representation of $SU(2)$ for all spins $j$, namely,
$\Phi[U_g X U_g^{\dag}] = U_g \Phi[X] U_g^\dag$ for all $g \in
SU(2)$ and $X \in \mathcal{B}(\mathcal{H}_{2j+1})$.
\end{proposition}
\begin{proof}
Up to an irrelevant phase, any $U_g$, $g \in SU(2)$ can be
expressed through the $SU(2)$ generators
$\{J_{\alpha}\}_{\alpha\in\{x,y,z\}}$ as follows:
\begin{equation}
U_g = \exp \left( - i \theta \sum_{\alpha} n_{\alpha} J_{\alpha}
\right),
\end{equation}
\noindent where ${\bf n} = (n_x, n_y, n_z)$ is a unit vector in
$\mathbb{R}^3$, which defines the rotation axis, and $\theta \in
\mathbb{R}$ is the rotation angle. The operators $J_x,J_y,J_z$
satisfy the commutation relation $[J_{\alpha},J_{\beta}] = i
\sum_{\gamma=1}^3 e_{\alpha\beta\gamma} J_{\gamma}$, where
$e_{\alpha\beta\gamma}$ is the Levi-Civita symbol
(Ref.~\cite{Varshalovich}, section 2.1.2, formula (7)). Using such
a commutation relation, it is not hard to see that
(Ref.~\cite{Varshalovich}, section 3.1.3, formula (11); section
3.1.6, item (a); section 1.4.5, formula (33))
\begin{equation} \label{rotation}
U_g^{\dag} J_{\alpha} U_g = \sum_{\beta} Q_{\alpha\beta}
J_{\beta},
\end{equation}
\noindent where $(Q_{\alpha\beta})$ is the orthogonal matrix
describing the rotation in $\mathbb{R}^3$ around the axis ${\bf
n}$ by angle $\theta$ (Ref.~\cite{Varshalovich}, section 1.4.2).
Finally, we get
\begin{eqnarray}
\Phi[U_g X U_g^{\dag}] &=& \frac{1}{j(j+1)} \sum_{\alpha}
J_{\alpha} U_g X U_g^{\dag} J_{\alpha} \nonumber\\
&=& \frac{1}{j(j+1)} \sum_{\alpha} U_g (U_g^{\dag} J_{\alpha} U_g)
X (U_g^{\dag} J_{\alpha} U_g) U_g^{\dag} \nonumber\\
&=& \frac{1}{j(j+1)} \sum_{\alpha,\beta,\gamma} Q_{\alpha\beta}
Q_{\alpha\gamma} U_g J_{\beta}
X J_{\gamma} U_g^{\dag} \nonumber\\
&=& \frac{1}{j(j+1)} \sum_{\beta\gamma} \delta_{\beta\gamma} U_g
J_{\beta} X J_{\gamma} U_g^{\dag} \nonumber\\
&=& U_g \Phi[X] U_g^{\dag},
\end{eqnarray}
\noindent where we have taken into account that $\sum_{\alpha}
Q_{\alpha\beta} Q_{\alpha\gamma} = \delta_{\beta\gamma}$, the
Kronecker delta.
\end{proof}
Since the Landau--Streater channel is extreme in the set of all
channels~\cite{Landau-Streater} it follows that it is also an
extreme point of $SU(2)$ irreducibly covariant channels
(abbreviated as an EPOSIC channel). The general properties of an
EPOSIC channel such as the Kraus representation, the Choi matrix,
and the dual channel are reviewed in Ref.~\cite{nuwairan-2014}.
It turns out that in the case $j=1$ the Landau--Streater channel
is not only $SU(2)$ covariant, but also globally unitarily
covariant. It means that in dimension $d=3$ the
channel~\eqref{LS-map} possesses $U(3)$ covariance.
\begin{proposition} \label{prop-U3-cov}
In the case $j=1$, the Landau--Streater channel is globally
unitarily covariant, namely, for all $X \in
\mathcal{B}(\mathcal{H}_3)$ and unitary operators $U$ on
$\mathcal{H}_3$ the following holds
\begin{equation} \label{U3-covariance}
\Phi[U X U^\dag] = V \Phi[X] V^\dag,
\end{equation}
\noindent where the unitary operator $V$ is expressed through $U$
in the orthonormal basis of eigenvectors of $J_z$ via formula
\begin{equation} \label{UV-relation-U3}
U =
\begin{pmatrix}
u_{11} & u_{12} & u_{13} \\
u_{21} & u_{22} & u_{23} \\
u_{31} & u_{32} & u_{33}
\end{pmatrix},
\qquad
V =
\begin{pmatrix}
\overline{u_{33}} & -\overline{u_{32}} & \overline{u_{31}}
\\
- \overline{u_{23}} & \overline{u_{22}} & -\overline{u_{21}}
\\
\overline{u_{13}} & -\overline{u_{12}} & \overline{u_{11}}
\end{pmatrix}.
\end{equation}
\end{proposition}
\begin{proof}
Note that $V$ is unitary if $U$ is unitary. In the basis of
eigenvectors of $J_z$, the operators
$\{J_{\alpha}\}_{\alpha=x,y,z}$ have the following form if $j=1$:
\begin{equation}
J_x=
\begin{pmatrix} 0&\frac{1}{\sqrt{2}}&0\\
\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\\0&\frac{1}{\sqrt{2}}&0\end{pmatrix},
\quad J_y=
\begin{pmatrix}0&-\frac{ i }{\sqrt{2}}&0\\ \frac{i}{\sqrt{2}}&0&-\frac{i}{\sqrt{2}}\\0&\frac{i}{\sqrt{2}}&0\end{pmatrix},
\quad J_z=
\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}.
\end{equation}
\noindent Substituting these operators in equation~\eqref{LS-map},
the direct calculation justifies the validity of formulas
\eqref{U3-covariance}, \eqref{UV-relation-U3}.
\end{proof}
The global unitary covariance is known to be a peculiar property
of the tracing, transposition, and identity maps. This feature
allowed one to find specific results for the Werner--Holevo
channel~\cite{datta-2004} and transpose-depolarizing
channels~\cite{datta-2006} as well as to prove additivity of
classical capacity for depolarizing quantum
channels~\cite{king-depol-2003}. We have just found out that the
Landau--Streater map for $j=1$ is globally unitarily covariant
too. However, as we show below, in the case $j>1$ the
Landau--Streater channel loses the property of global unitary
covariance.
\begin{proposition}
The Landau--Streater channel is not globally unitarily covariant
if $j>1$.
\end{proposition}
\begin{proof}
We prove the statement by constructing a counterexample. Suppose
the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$, $j>1$, is covariant with respect
to representation of $U(2j+1)$. Then for any unitary operator $U$
acting on $\mathcal{H}_{2j+1}$, there exists a unitary operator
$V$ such that
\begin{equation} \label{UV-assumption}
\Phi[U \rho U^\dag] = V \Phi[\rho] V^\dag
\end{equation}
\noindent holds true for all density operators $\rho$. This
implies that the output density operators $\Phi[U \rho U^\dag]$
and $\Phi[\rho]$ have identical spectra.
Consider eigenvectors of the spin projection onto $z$-axis
(Ref.~\cite{Varshalovich}, section 6.1.2, formula(5)):
\begin{equation}
J_z |j,m\rangle = m |j,m\rangle, \quad m = j, j-1, \ldots, -j.
\end{equation}
Let $\rho = |j,j\rangle\langle j,j|$ and
\begin{equation}
U = |j,j-1\rangle\langle j,j| + |j,j\rangle\langle j,j-1| +
\sum_{k=-j}^{j-2} |j,k\rangle\langle j,k|,
\end{equation}
\noindent then $\Phi[|j,j-1\rangle\langle j,j-1|]$ and
$\Phi[|j,j\rangle\langle j,j|]$ must have the same spectra.
On the other hand, action of the Landau--Streater map on the
states $|j,m\rangle \langle j,m|$ with definite spin projection
$m$ onto $z$-axis can be expressed explicitly by introducing
auxiliary operators $J_{\pm} = J_x \pm i J_y$ satisfying
(Ref.~\cite{Varshalovich}, section 2.3.3, formula (7); section
6.1.2, formula (13))
\begin{equation} \label{J-pm}
J_{\pm} | j,m \rangle = \sqrt{( j \mp m)(j \pm m + 1)} |j, m \pm 1
\rangle.
\end{equation}
\noindent Since $\Phi[X] = [j(j+1)]^{-1} \left( \frac{1}{2}J_- X
J_+ + \frac{1}{2} J_+ X J_- + J_z X J_z \right)$, we get
\begin{eqnarray} \label{Phi-on-jm}
\Phi[|j,m\rangle\langle j,m|] &=& \frac{1}{j(j+1)} \bigg[
\frac{1}{2}\left( j(j+1)-m(m-1) \right) |j,m-1\rangle\langle
j,m-1| \nonumber\\
&& + \frac{1}{2}\left( j(j+1)-m(m+1) \right) |j,m+1\rangle\langle
j,m+1| + m^2|j,m\rangle\langle j,m| \bigg],
\end{eqnarray}
\noindent from which we make conclusion about the spectrum of the
output state $\Phi[|j,m\rangle\langle j,m|]$:
\begin{equation} \label{spec-phi-jm}
{\rm Spec}\left( \Phi[|j,m\rangle\langle j,m|] \right) = \left\{
\frac{j(j+1)-m(m+1)}{2j(j+1)}, \frac{j(j+1)-m(m-1)}{2j(j+1)},
\frac{m^2}{j(j+1)}, 0,0,\ldots \right\}.
\end{equation}
\noindent If $j>1$, then ${\rm Spec}\left( \Phi[|j,j\rangle\langle
j,j|] \right) \neq {\rm Spec}\left( \Phi[|j,j-1\rangle\langle
j,j-1|] \right)$. This contradiction concludes the proof.
\end{proof}
\section{Spectral properties}
\subsection{Spectrum of the map} \label{section-spectrum-map}
The Landau--Streater channel $\Phi$ is Hermitian as it coincides
with its dual $\Phi^{\dag}$, therefore its spectrum
$\{\lambda_k\}_{k=0}^{(2j+1)^2-1}$ is real. Hermitian
eigenoperators $X_k$ satisfy $\Phi[X_k] = \lambda_k X_k$. Due to
unitality of $\Phi$, the identity operator $I$ is the
eigenoperator, so we can fix the corresponding eigenvalue
$\lambda_0 = 1$ for all $j$. By determinant ${\rm det}\Phi$ of the
channel $\Phi$ we will understand the product of its eigenvalues
$\prod_k \lambda_k$.
If $j=\frac{1}{2}$, then $J_x$, $J_y$, $J_z$ are eigenoperators of
$\Phi$ and $\lambda_1 = \lambda_2 = \lambda_3 = - \frac{1}{3}$. In
this case, ${\rm det}\Phi = - \frac{1}{27} < 0$, so the channel
$\Phi$ is not infinitesimally divisible~\cite{wolf-2008} and
cannot be obtained as a result of Markovian evolution, although it
can be realized physically, e.g., via collision
models~\cite{fpmz-2017}.
If $j=1$, then $J_x$, $J_y$, $J_z$ are eigenoperators of $\Phi$
with corresponding eigenvalues $\lambda_1 = \lambda_2 = \lambda_3
= \frac{1}{2}$. Five more eigenoperators have the form $3 \left(
\sum_{\alpha} n_{\alpha}^{(k)}J_{\alpha} \right)^2 - 2I$, ${\bf
n}^{(k)} \in \mathbb{R}^3$, $k=1,\ldots,5$ (Ref.~\cite{fm-2010},
formula (8) and text after formula (36)) and correspond to
eigenvalues $\lambda_4 = \ldots = \lambda_8 = - \frac{1}{2}$.
Similarly, ${\rm det}\Phi < 0$, so such a channel cannot be a
result of Markovian evolution.
In what follows, we find spectrum of the Landau--Streater map
$\Phi: \mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ for an arbitrary integer or
half-integer $j$. As we show, the eigenoperators of $\Phi$ are
particularly related with the irreducible tensor operator
$T_{LM}^{(j)}$ for the $SU(2)$ group, which is also known as the
polarization operator (Ref.~\cite{Varshalovich}, section 2.4.2,
formula (6); section 8.4.3, formula (10)):
\begin{equation}
T_{LM}^{(j)} = \sqrt{\frac{2L+1}{2j+1}} \, \sum_{m_1,m_2=-j}^j
C_{j m_1 L M}^{j m_2} | j m_2 \rangle \langle j m_1 | =
\sum_{m_1,m_2 = -j}^{j} (-1)^{j-m_1} C_{j m_2 j -m_1}^{L M} | j
m_2 \rangle \langle j m_1 |,
\end{equation}
\noindent where $C_{j_1 m_1 j_2 m_2}^{J M}$ is the conventional
Clebsch--Gordan coefficient.
\begin{proposition} \label{prop-spectrum-LS}
The spectrum of the Landau--Streater map $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ comprises $(2L+1)$-fold
degenerate eigenvalues
\begin{equation} \label{lambda-L}
\lambda_L = 1 - \frac{L(L+1)}{2j(j+1)}, \qquad L = 0, 1, \ldots,
2j.
\end{equation}
\noindent The corresponding eigenoperators are linearly
independent operators of the form $U_g T_{L0}^{(j)} U_g^{\dag}$,
where the operators $U_g$ belong to the unitary representation of
the $SU(2)$ group.
\end{proposition}
\begin{proof}
We start with the observation that $T_{L0}^{(j)}$ is the
eigenoperator of $\Phi$. To prove this fact we rewrite the
Landau--Streater channel in the form $\Phi[X] = [j(j+1)]^{-1}
(\frac{1}{2} J_+ X J_- + \frac{1}{2} J_- X J_+ + J_z X J_z)$ and
use the commutation relations $[J_{\pm},T_{L0}^{(j)}] =
\sqrt{L(L+1)} T_{L \pm 1}^{(j)}$ and $[J_z,T_{L0}^{(j)}] = 0$ (see
Ref.~\cite{Varshalovich}, section 2.4.1, formula (1); section
2.3.3, formula (7); for the Clebsch-Gordan coefficients $C_{L0 1
\pm 1}^{L \pm 1} = \mp \frac{1}{\sqrt{2}}$ and $C_{L0 10}^{L0} =
0$ see section 8.5.1, formula (8)). We get
\begin{equation} \label{Phi-on-TL0}
\Phi \left[ T_{L0}^{(j)} \right] = \frac{1}{j(j+1)} \left[ \left(
\frac{1}{2} J_+ J_- + \frac{1}{2} J_- J_+ + J_z J_z \right)
T_{L0}^{(j)} - \frac{\sqrt{L(L+1)}}{2} \left( J_+ T_{L-1}^{(j)} +
J_- T_{L1}^{(j)} \right) \right].
\end{equation}
\noindent The first expression in parentheses $(\cdot)$ is
$j(j+1)I$, whereas the second expression in parentheses $(\cdot)$
can be simplified because $J_{\pm} = \mp
\sqrt{\frac{2j(j+1)(2j+1)}{3}} T_{1 \pm 1}^{(j)}$
(Ref.~\cite{Varshalovich}, section 2.4.2, formula (10); section
2.3.3, formula (7)) and the product $T_{1 \pm 1}^{(j)} T_{L
\mp 1}^{(j)}$ is known (Ref.~\cite{Varshalovich}, section 2.4.4,
formula (16)):
\begin{eqnarray}
&& J_+ T_{L-1}^{(j)} + J_- T_{L1}^{(j)} = -
\sqrt{\frac{2j(j+1)(2j+1)}{3}} \left( T_{1 1}^{(j)} T_{L
-1}^{(j)} - T_{1 -1}^{(j)} T_{L
1}^{(j)} \right) \nonumber\\
&& = - \sqrt{\frac{2j(j+1)(2j+1)}{3}} \sum_{L'} (-1)^{2j+L'}
\sqrt{3(2L+1)} \left\{
\begin{array}{ccc}
1 & L & L' \\
j & j & j \\
\end{array} \right\} \left( C_{11L-1}^{L'0} - C_{1-1L1}^{L'0}
\right) T_{L'0}^{(j)}. \nonumber\\
&& \label{difference-T}
\end{eqnarray}
\noindent The Clebsch-Gordan coefficients $C_{11L-1}^{L'0}$ and
$C_{1-1L1}^{L'0}$ coincide if $L'=L \pm 1$
(Ref.~\cite{Varshalovich}, section 8.4.3, formula (11)) and vanish
if $L' < L-1$ or $L' > L+1$ (Ref.~\cite{Varshalovich}, section
8.1.1, formula (1)), so the only contribution
to~\eqref{difference-T} makes $L'=L$, when $C_{11L-1}^{L0} -
C_{1-1L1}^{L0} = \sqrt{2}$ (Ref.~\cite{Varshalovich}, section
8.5.1, formula (8)). The Wigner 6$j$-symbol $\left\{
\begin{array}{ccc}
1 & L & L \\
j & j & j \\
\end{array} \right\} = \frac{1}{2} (-1)^{2j+L+1} \sqrt{\frac{L(L+1)}{j(j+1)(2j+1)(2L+1)}}$ (Ref.~\cite{Varshalovich}, section 9.5.4,
formula (21)). Finally, $J_+ T_{L-1}^{(j)} + J_- T_{L1}^{(j)} =
\sqrt{L(L+1)} T_{L0}^{(j)}$. Substituting this result in
formula~\eqref{Phi-on-TL0}, we conclude that $T_{L0}^{(j)}$ is the
eigenoperator of $\Phi$ and corresponds to the eigenvalue
$\lambda_L$ given by formula~\eqref{lambda-L}.
Due to the $SU(2)$-covariance of $\Phi$
(proposition~\ref{prop-SU2-cov}), the operators $U_g T_{L0}^{(j)}
U_g^{\dag}$ are eigenoperators of $\Phi$ too and correspond to the
eigenvalue $\lambda_L$. It is known that there are exactly $2L+1$
linear independent operators $U_g T_{L0}^{(j)} U_g^{\dag}$ if
$U_g$ is a representation of the $SU(2)$ group (see
Ref.~\cite{fm-2009}, formula (11), where $S_L^{(j)}$ is
proportional to the operator $T_{L0}^{(j)}$, and
Ref.~\cite{fm-2010}, text after formula (36)). Therefore,
eigenvalues $\lambda_L$ are $(2L+1)$-fold degenerate. Since
$\sum_{L=0}^{2j} (2L+1) = (2j+1)^2$, the given eigenvalues are the
only ones and constitute the spectrum of $\Phi$.
\end{proof}
In the latter proposition, for the case $L=1$ the generators
$J_x$, $J_y$, $J_z$ are exactly the three linear independent
eigenoperators of $\Phi: \mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ corresponding to the eigenvalue
$1 - \frac{1}{j(j+1)}$.
It is not hard to see that if $L=2j$, then $\lambda_L < 0$.
Negativity of the eigenvalue implies that the Landau--Streater
channel cannot be obtained via positive divisible Hermitian
evolution~\cite{chruscinski-macchiavello-maniscalco-2017} for any
$j$.
\subsection{Spectrum of the output} \label{section-spectrum-output}
Let us consider spectral properties of the output operator, i.e.,
the spectrum of $\Phi[X]$, where $X$ is a Hermitian input
operator.
It is not hard to see that in the case $j=\frac{1}{2}$ the
spectrum of $\Phi[X]$ is $\{\frac{1}{3}(x_1+2x_2),\frac{1}{3}(2x_1
+ x_2)\}$ provided the spectrum of the input density operator
$\rho$ is $\{x_1,x_2\}$.
\begin{corollary} \label{corollary-qubit-entropy}
The output purity and entropy of the Landau--Streater channel
$\Phi: \mathcal{B}(\mathcal{H}_2) \mapsto
\mathcal{B}(\mathcal{H}_2)$ for all pure spin-$\frac{1}{2}$ input
states $| \psi \rangle \langle \psi |$ are equal to $\frac{5}{9}$
and $\log 3 - \frac{2}{3}$, respectively.
\end{corollary}
The case $j=1$ is more involved, but in this case the spectrum of
the output also depends only on the spectrum of the input, as we
show in the following proposition.
\begin{proposition} \label{prop-spectrum-output}
Suppose a Hermitian operator $X \in \mathcal{B}(\mathcal{H}_3)$
with spectrum $\{x_1,x_2,x_3\}$. The output operator $\Phi[X]$ of
the Landau--Streater channel $\Phi: \mathcal{B}(\mathcal{H}_3)
\mapsto \mathcal{B}(\mathcal{H}_3)$ has spectrum $\{
\frac{1}{2}(x_1+x_2),\frac{1}{2}(x_1+x_3),\frac{1}{2}(x_2+x_3)
\}$.
\end{proposition}
\begin{proof}
In the basis of eigenvectors of $J_z$, the action of the
Landau--Streater channel reads
\begin{equation}
\Phi \left[ \begin{pmatrix}
X_{11} & X_{12} & X_{13} \\
X_{21} & X_{22} & X_{23} \\
X_{31} & X_{32} & X_{33}
\end{pmatrix} \right] =
\frac{1}{2}
\begin{pmatrix}
X_{11} + X_{22} & X_{23} & -X_{13} \\
X_{32} & X_{11} + X_{33} & X_{12} \\
-X_{31} & X_{21} & X_{22} + X_{33}
\end{pmatrix}.
\end{equation}
Since the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_3) \mapsto \mathcal{B}(\mathcal{H}_3)$ is
globally unitarily covariant by proposition~\ref{prop-U3-cov}, the
spectrum of the output operator $\Phi[X]$ depends only on the
spectrum of the input operator $X$. To find the explicit relation
between the spectra we consider the unitary operator $U$ realizing
the transition from the basis of eigenvectors of $X$ to the basis
of eigenvectors of $J_z$. Then in the basis of eigenvectors of
$J_z$ we have $U X U^{\dag} = {\rm diag}(x_1,x_2,x_3)$ and $\Phi[U
X U^{\dag}] = {\rm diag} \left( \frac{1}{2} (x_1+x_2),\frac{1}{2}
(x_1+x_3),\frac{1}{2} (x_2+x_3) \right)$. Thanks to the global
unitary covariance, the latter diagonal matrix is exactly the
spectrum of $\Phi[X]$.
\end{proof}
The spectral property of the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_3) \mapsto \mathcal{B}(\mathcal{H}_3)$
resembles that of the depolarizing channel, but the
Landau--Streater channel is not depolarizing in the case $j = 1$.
This peculiarity is ascribed to the close relation between the
Landau--Streater channel and the Werner--Holevo channel $\Phi_{\rm
WH}: \mathcal{B}(\mathcal{H}_d) \mapsto
\mathcal{B}(\mathcal{H}_d)$ defined through transposition $\top$
in some orthonormal basis via formula $\Phi_{\rm WH}[X] =
\frac{1}{d-1} \left( {\rm tr}[X] I - X^{\top} \right)$,
Ref.~\cite{wh-2002}. It turns out that if $d=3$ and transposition
$\top$ is performed in the basis of eigenstates of $J_z$, then the
Landau--Streater channel $\Phi: \mathcal{B}(\mathcal{H}_3) \mapsto
\mathcal{B}(\mathcal{H}_3)$ is merely the Werner--Holevo channel
concatenated with a unitary channel:
\begin{equation} \label{LS-WH}
\Phi[X] = \Phi_{\rm WH}[W X W^{\dag}] = \frac{1}{2} \left( {\rm
tr}[X] I - W X^{\top} W^{\dag} \right),
\end{equation}
\noindent where $W = | 1,1 \rangle \langle 1,-1 | - | 1,0 \rangle
\langle 1,0 | + | 1,-1 \rangle \langle 1,1 |$ is a unitary
operator.
Since the spectrum of pure states consists of $1$ and zeros, we
can make conclusions about the output purity $\mu_{\rm out} = {\rm
tr}\left[ (\Phi[\rho])^2 \right]$ and the output entropy $S_{\rm
out} = - {\rm tr} \left[ \Phi[\rho] \log \Phi[\rho] \right]$ for
pure input states in cases $j=\frac{1}{2}$ and $j=1$. Hereafter,
$\log$ is understood as $\log_2$ if one measures the entropy and
capacity in bits.
\begin{corollary} \label{corollary-qutrit-entropy}
The output purity and entropy of the Landau--Streater channel
$\Phi: \mathcal{B}(\mathcal{H}_3) \mapsto
\mathcal{B}(\mathcal{H}_3)$ for all pure spin-$1$ input states $|
\psi \rangle \langle \psi |$ are equal to $\frac{1}{2}$ and $1$,
respectively.
\end{corollary}
In the case $j > 1$, the spectrum of $\Phi[X]$ depends not only on
the spectrum of $X$ but also on the particular form of the
operator $X$. For instance, in the case $j=\frac{3}{2}$, one can
consider two different pure input states
$|\frac{3}{2},\frac{3}{2}\rangle\langle \frac{3}{2},\frac{3}{2}|$
and $|\frac{3}{2},\frac{1}{2}\rangle\langle
\frac{3}{2},\frac{1}{2}|$ with identical spectra $\{1,0,0,0\}$. By
formula~\eqref{spec-phi-jm} ${\rm Spec} \left( \Phi \left[
|\frac{3}{2},\frac{3}{2}\rangle\langle \frac{3}{2},\frac{3}{2}|
\right] \right) = \{\frac{3}{5},\frac{2}{5},0,0\}$ and ${\rm Spec}
\left( \Phi \left[ |\frac{3}{2},\frac{1}{2}\rangle\langle
\frac{3}{2},\frac{1}{2}| \right] \right) =
\{\frac{2}{5},\frac{1}{4},\frac{8}{15},0\}$, so spectra of output
states may not coincide. Similarly to the case $j=\frac{3}{2}$,
for any $j>1$ one can always take pure input states
$|j,j\rangle\langle j,j|$ and $|j,j-1\rangle\langle j,j-1|$ and
make sure that ${\rm Spec} \left( \Phi \left[ |j,j\rangle\langle
j,j| \right] \right) \ne {\rm Spec} \left( \Phi \left[
|j,j-1\rangle\langle j,j-1| \right] \right)$.
\subsection{The maximal $p$-norm and the minimal output entropy} \label{section-multiplicativity}
The maximal $p$-norm of a channel $\Phi$ is defined by the formula
\begin{equation}
\nu_p(\Phi) = \sup_{\rho} \{ \| \Phi[\rho] \|_p \},
\end{equation}
\noindent where $\| \Phi[\rho] \|_p = \left\{ {\rm tr}
(\Phi[\rho])^p \right\}^{1/p}$ is the Schatten $p$-norm of
$\Phi[\rho]$. The maximal 2-norm is merely the square root of the
maximal output purity. Before we proceed to the analysis of
maximal $p$-norm and minimal output entropy of the
Landau--Streater channel, we prove auxiliary results following
from the theory of angular momentum.
\begin{lemma} \label{lemma-1}
Let ${\bf k}\in\mathbb{R}^3$ be a unit vector, $|{\bf k}| =
\sqrt{k_1^2+k_2^2+k_3^2} = 1$. The spectrum of operator
$\sum_{\alpha=1}^3 k_{\alpha} J_{\alpha}$ is $\{ m \}_{m=-j}^j$.
\end{lemma}
\begin{proof}
Physically, the operator $\sum_{\alpha=1}^3 k_{\alpha} J_{\alpha}$
is the spin projection operator onto axis ${\bf k}$ and therefore
has the same spectrum as any of operators $J_x$, $J_y$, $J_z$.
Mathematically, there exists a unitary operator $U_g: {\cal
B}({\cal H}_{2j+1}) \mapsto {\cal B}({\cal H}_{2j+1})$, $g\in
SU(2)$, such that $U_g^{\dag} J_z U_g = \sum_{\beta=1}^3 k_{\beta}
J_{\beta}$, cf. formula~\eqref{rotation} with $k_{\beta} =
Q_{3\beta}$, where $Q$ is orthogonal. Hence, ${\rm Spec}\left(
\sum_{\alpha=1}^3 k_{\alpha} J_{\alpha} \right) = {\rm Spec}(J_z)
= \{m\}_{m=-j}^j$.
\end{proof}
The eigenvector $| \psi_{\bf k} \rangle$ of operator
$\sum_{\alpha=1}^3 k_{\alpha} J_{\alpha}$ corresponding to the
maximal eigenvalue $j$ will be referred to as a vector with the
maximal spin polarization. Clearly, $| \psi_{\bf k} \rangle =
U_g^{\dag} | j,j \rangle$, where $U_g$ is the unitary operator
used in the proof of Lemma~\ref{lemma-1}.
\begin{lemma} \label{lemma-2}
Let ${\bf k}\in\mathbb{R}^3$, $|{\bf k}|=1$. The maximum of
expression $ \left\| \sum_{\alpha=1}^3 k_{\alpha} J_{\alpha} |
\psi \rangle \right\|^2$ with respect to normalized vectors $|
\psi \rangle \in {\cal H}_{2j+1}$ equals $j^2$ and is attained at
the state $| \psi_{\bf k} \rangle$ with the maximal spin
polarization.
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma-1}, the spectrum of $\left( \sum_{\alpha=1}^3
k_{\alpha} J_{\alpha} \right)^2$ reads $\{m^2\}_{m=-j}^j$.
Therefore, $\| \left( \sum_{\alpha=1}^3 k_{\alpha} J_{\alpha}
\right) | \psi \rangle \|^2 = \langle \psi | \left(
\sum_{\alpha=1}^3 k_{\alpha} J_{\alpha} \right)^2 | \psi \rangle
\leq j^2 \langle \psi | \psi \rangle = j^2$ and \newline $\langle
\psi_{\bf k} | \left( \sum_{\alpha=1}^3 k_{\alpha} J_{\alpha}
\right)^2 | \psi_{\bf k} \rangle = j^2$.
\end{proof}
\begin{lemma} \label{lemma-3}
If $j \geq 1$, then $\langle \psi | J_z | \psi \rangle^2 \leq 9
j^2 \frac{j^2 - \langle \psi | J_x^2 | \psi \rangle}{2j-1}$ for
all normalized vectors $| \psi \rangle \in {\cal H}_{2j+1}$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lemma-1}, the spectral decomposition of $J_x^2$
reads $J_x^2 = j^2 (P_{j} + P_{-j}) + \sum_{m=-j+1}^{j-1} m^2
P_m$, where $P_m = |j,m\rangle_x\langle j,m|$ and $J_x
|j,m\rangle_x = m |j,m\rangle_x$. The average value $\langle J_x^2
\rangle = j^2 - \epsilon \leq p j^2 + (1-p)(j-1)^2$, where $p =
\langle (P_{j} + P_{-j}) \rangle$. Therefore, $1-p \leq
\frac{\epsilon}{2j-1} = \frac{j^2 - \langle J_x^2 \rangle}{2j-1}$.
Let $|\psi\rangle = c_j|j,j\rangle_x + c_{-j}|j,-j\rangle_x +
\sum_{m=-j+1}^{j-1} c_m |j,m\rangle_x$. Note that ${}_x\langle
j,j| J_z |j,j\rangle_x = 0$, ${}_x\langle j,-j| J_z |j,-j\rangle_x
= 0$, $ {}_x\langle j,j| J_z |j,-j\rangle_x = 0$ if $j \geq 1$, $p
= |c_j|^2+|c_{-j}|^2$, and $1-p = \sum_{m=-j+1}^{j-1} |c_m|^2$. We
have
\begin{eqnarray}
\langle \psi | J_z | \psi \rangle & = & 2 {\rm Re} \left(
\sum_{m=-j+1}^{j-1} \overline{c_m} \, {}_x\langle j,m| \right) J_z
\left( c_j|j,j\rangle_x + c_{-j}|j,-j\rangle_x \right)
\nonumber\\
&& + \left(\sum_{m=-j+1}^{j-1} \overline{c_m} \, {}_x\langle j,m|
\right) J_z \left(\sum_{m'=-j+1}^{j-1} c_{m'} | j,m' \rangle_x
\right) \nonumber \\
& \leq & 2\sqrt{p}\sqrt{1-p}\, j + (1-p) j \leq 3 \sqrt{1-p} \, j.
\end{eqnarray}
Noticing that $-J_z$ has the same spectrum as $J_z$ and arguing as
above, we see that $\langle \psi | J_z | \psi \rangle \geq
2\sqrt{p}\sqrt{1-p}\, (-j) + (1-p) (-j) \geq - 3 \sqrt{1-p} \, j$.
Thus, $|\langle \psi | J_z | \psi \rangle| \leq 3 \sqrt{1-p} \,
j$. Squaring both sides of this inequality and recalling $1-p \leq
\frac{j^2 - \langle J_x^2 \rangle}{2j-1}$, we get the statement of
Proposition~ concludes the proof.
\end{proof}
Lemma~\ref{lemma-3} shows that the average value $\langle J_z
\rangle$ cannot be large when $\langle J_x^2 \rangle$ is close to
its maximal value $j^2$. Lemma~\ref{lemma-3} obviously remains
valid if one replaces $J_x$ by $J_y$.
\begin{lemma} \label{lemma-4}
Let the vectors ${\bf k},{\bf l} \in \mathbb{R}^3$ satisfy $|{\bf
k}|^2 + |{\bf l}|^2 = 1$, then $\left\| \sum_{\alpha=1}^3
(k_{\alpha} + i l_{\alpha}) J_{\alpha} | \psi \rangle \right\|^2
\leq \max(j,j^2)$ for all normalized vectors $| \psi \rangle \in
{\cal H}_{2j+1}$.
\end{lemma}
\begin{proof}
Suppose ${\bf k}$ and ${\bf l}$ are linearly dependent, i.e.,
${\bf k} = |{\bf k}| {\bf n}$ and ${\bf l} = |{\bf l}| {\bf n}$
for some unit vector ${\bf n} \in \mathbb{R}^3$. Then $\left\|
\sum_{\alpha=1}^3 (k_{\alpha} + i l_{\alpha}) J_{\alpha} | \psi
\rangle \right\|^2 = (|{\bf k}|^2 + |{\bf l}|^2) \left\|
\sum_{\alpha=1}^3 n_{\alpha} J_{\alpha} | \psi \rangle \right\|^2
\leq j^2$ by Lemma~\ref{lemma-2}.
Suppose ${\bf k}$ and ${\bf l}$ are linearly independent. Note
that $\left\| \sum_{\alpha=1}^3 (k_{\alpha} + i l_{\alpha})
J_{\alpha} | \psi \rangle \right\|^2 = \langle \psi | F | \psi
\rangle$, where
\begin{equation} \label{F-operator}
F:=\sum_{\alpha,\beta}(k_{\alpha} - i l_{\alpha})(k_{\beta} + i
l_{\beta}) J_{\alpha} J_{\beta} = \left( \sum_{\alpha} k_{\alpha}
J_{\alpha} \right)^2 + \left( \sum_{\alpha} l_{\alpha} J_{\alpha}
\right)^2 + \sum_{\gamma} [{\bf l} \times {\bf k}]_{\gamma}
J_{\gamma}.
\end{equation}
\noindent Here, we have used the commutation relation
$[J_{\alpha},J_{\beta}] = i \sum_{\gamma} e_{\alpha\beta\gamma}
J_{\gamma}$ and the notation $[{\bf l} \times {\bf k}]$ for the
conventional cross product of vectors ${\bf l}$ and ${\bf k}$. Let
the angle between vectors ${\bf k}$ and ${\bf l}$ be $\vartheta$.
Consider a rotation $Q$ in $\mathbb{R}^3$ such that $Q\left(
\frac{\bf k}{|{\bf k}|} + \frac{\bf l}{|{\bf l}|} \right)$ is
aligned with the positive direction of axis $x$ and $Q[{\bf l}
\times {\bf k}]$ is aligned with the positive direction of axis
$z$. In other words, the vectors $Q{\bf k}$ and $Q{\bf l}$ belong
to the $xy$-plane, and the axis $x$ is a bisector of the angle
between vectors $Q{\bf k}$ and $Q{\bf l}$. The vector $Q[{\bf l}
\times {\bf k}]$ is perpendicular to both $Q{\bf k}$ and $Q{\bf
l}$ and has length $|[{\bf l} \times {\bf k}]| = |{\bf k}| \,
|{\bf l}| \sin\vartheta$. Therefore, the vector $Q{\bf k}$ has
coordinates $(|{\bf k}|\cos\frac{\vartheta}{2},|{\bf
k}|\sin\frac{\vartheta}{2},0)$, the vector $Q{\bf l}$ has
coordinates $(|{\bf l}|\cos\frac{\vartheta}{2},-|{\bf
l}|\sin\frac{\vartheta}{2},0)$, and the vector $Q[{\bf l} \times
{\bf k}]$ has coordinates $(0,0,|{\bf k}| \, |{\bf l}|
\sin\vartheta)$. The corresponding unitary rotation $U_Q \in
\{U_g\}_{g\in SU(2)}$ transforms the spin operators in accordance
with formula~\eqref{rotation} as follows:
\begin{eqnarray}
\label{Jx-rotation} U_Q \left( \sum_{\alpha} k_{\alpha} J_{\alpha}
\right) U_Q^{\dag} &=& \sum_{\beta} (Q{\bf k})_{\beta} J_{\beta} =
|{\bf k}| \left( \cos\frac{\vartheta}{2} J_x +
\sin\frac{\vartheta}{2} J_y \right), \\
U_Q \left( \sum_{\alpha} l_{\alpha} J_{\alpha} \right) U_Q^{\dag}
&=& \sum_{\beta} (Q{\bf l})_{\beta} J_{\beta} = |{\bf l}| \left(
\cos\frac{\vartheta}{2} J_x -
\sin\frac{\vartheta}{2} J_y \right), \\
U_Q \left( \sum_{\gamma} [{\bf l} \times {\bf k}]_{\gamma}
J_{\gamma} \right) U_Q^{\dag} &=& \sum_{\beta} (Q[{\bf l} \times
{\bf k}])_{\beta} J_{\beta} = |{\bf k}| \, |{\bf l}| \sin\vartheta
J_z. \label{Jz-rotation}
\end{eqnarray}
\noindent Substituting \eqref{Jx-rotation}--\eqref{Jz-rotation} in
\eqref{F-operator} and taking into account that $|{\bf k}|^2+|{\bf
l}|^2 = 1$, we get
\begin{eqnarray}
U_Q F U_Q^{\dag} &=& \cos^2\frac{\vartheta}{2} J_x^2 +
\sin^2\frac{\vartheta}{2} J_y^2 + (|{\bf k}|^2 - |{\bf l}|^2)
\sin\frac{\vartheta}{2} \cos\frac{\vartheta}{2} \left( J_x J_y +
J_y J_x \right) + |{\bf
k}| \, |{\bf l}| \sin\vartheta J_z \nonumber\\
&=& \frac{1}{4} \cos\vartheta (J_+^2 + J_-^2) + \sin\vartheta
\left( \frac{1}{4i} (|{\bf k}|^2 - |{\bf l}|^2) (J_+^2 - J_-^2) +
|{\bf k}| \, |{\bf l}| J_z \right) + \frac{1}{2} (J_x^2 +
J_y^2).\nonumber\\
\end{eqnarray}
\noindent Quantities $|{\bf k}|^2 - |{\bf l}|^2$ and $2|{\bf k}|
\, |{\bf l}|$ can be treated as $\cos\eta$ and $\sin\eta$ for some
real $\eta$, respectively, because $(|{\bf k}|^2 - |{\bf l}|^2)^2
+ 4 |{\bf k}|^2 |{\bf l}|^2 = (|{\bf k}|^2 + |{\bf l}|^2)^2 = 1$.
Hence,
\begin{equation}
U_Q F U_Q^{\dag} = \frac{1}{4} \cos\vartheta (J_+^2 + J_-^2) +
\sin\vartheta \left( \frac{\cos\eta}{4i} (J_+^2 - J_-^2) +
\frac{\sin\eta}{2} J_z \right) + \frac{1}{2} (J_x^2 + J_y^2).
\end{equation}
If $j=1/2$, then $J_+^2 = J_-^2 = 0$, $J_x^2=J_y^2=\frac{1}{4}I$,
and $U_Q F U_Q^{\dag} = \frac{1}{2}\sin\vartheta\sin\eta J_z +
\frac{1}{4}I$. Clearly, $\langle \psi | U_Q F U_Q^{\dag} | \psi
\rangle \leq \frac{1}{2} = j$.
If $j=1$, then the matrix of operator $U_Q F U_Q^{\dag}$ in the
basis $\{|j,m\rangle\}_{m=-j}^j$ has a rather simple form. Its
eigenvalues do not depend on $\vartheta$ and $\eta$ and read
$1,1,0$.
For the cases $j=3/2$ and $j=2$ one can find eigenvalues of $U_Q F
U_Q^{\dag}$ and maximize them with respect to $\vartheta$ and
$\eta$ to get upper bounds $9/4$ and $4$, respectively.
For $j>2$ we develop the following technique.
Note that $A \cos\eta + B \sin\eta \leq \sqrt{A^2 + B^2}$ for
$A,B,\eta\in\mathbb{R}$, so the average value
\begin{equation}
\left\langle \left( \frac{1}{4i} (|{\bf k}|^2 - |{\bf l}|^2)
(J_+^2 - J_-^2) + |{\bf k}| \, |{\bf l}| J_z \right) \right\rangle
\leq \frac{1}{4} \sqrt{ \langle i(J_+^2 - J_-^2) \rangle^2 + 4
\langle J_z \rangle^2}.
\end{equation}
\noindent Similarly, $C\sin\vartheta + D\cos\vartheta \leq
\sqrt{C^2+D^2}$ for all $\vartheta,C,D \in \mathbb{R}$, therefore
\begin{equation} \label{average-inequality}
\langle U_Q F U_Q^{\dag} \rangle \leq \frac{1}{4} \sqrt{\langle
(J_+^2 + J_-^2) \rangle^2 + \langle i(J_+^2 - J_-^2) \rangle^2 + 4
\langle J_z \rangle^2} + \frac{1}{2} \langle (J_x^2 + J_y^2)
\rangle.
\end{equation}
Suppose the maximum in the right hand side of
\eqref{average-inequality} is attained at some vector
$|\psi_0\rangle$, then this maximum is also attained at the vector
$|\psi_{\theta} \rangle = e^{- i J_z \theta} |\psi_0\rangle$ due
to invariance of \eqref{average-inequality} with respect to
rotations around axis $z$. On the other hand, $\langle
\psi_{\theta} | J_+ | \psi_{\theta} \rangle = e^{i\theta} \langle
\psi_{0} | J_+ | \psi_{0} \rangle$, which means that $\langle J_+
\rangle$ can always be chosen to be real, so $\langle J_+ \rangle
= \langle J_- \rangle$. In other words,
\begin{eqnarray} \label{average-inequality}
\langle U_Q F U_Q^{\dag} \rangle & \leq & \max_{\psi: \langle \psi
| \psi \rangle = 1} \frac{1}{4} \sqrt{\langle \psi | (J_+^2 +
J_-^2) | \psi \rangle^2 + 4 \langle \psi | J_z | \psi \rangle^2} +
\frac{1}{2}
\langle \psi | (J_x^2 + J_y^2) | \psi \rangle \nonumber\\
& = & \max_{\psi: \langle \psi | \psi \rangle = 1} \frac{1}{2}
\sqrt{\langle \psi | (J_x^2 - J_y^2) | \psi \rangle^2 + \langle
\psi | J_z | \psi \rangle^2} + \frac{1}{2} \langle \psi | (J_x^2 +
J_y^2) | \psi \rangle. \label{rhs-to-abc}
\end{eqnarray}
Denote $a = \langle \psi | J_x^2 | \psi \rangle$, $b = \langle
\psi | J_x^2 | \psi \rangle$, and $c = \langle \psi | J_z | \psi
\rangle^2$. The dispersion of spin projection onto axis $z$ denote
$d = \langle \psi | J_z^2 | \psi \rangle - \langle \psi | J_z |
\psi \rangle^2 = \langle \psi | J_z^2 | \psi \rangle - c$. Note
that $0 \leq a,b,c \leq j^2$ and $d \geq 0$. Since
$J_x^2+J_y^2+J_z^2 = j(j+1)I_{2j+1}$, we have $a+b+c+d = j(j+1)$.
Finally, from Lemma~\ref{lemma-3} it follows that $c \leq 9 j^2
\frac{j^2 - a}{2j-1}$ and $c \leq 9 j^2 \frac{j^2 - b}{2j-1}$.
Therefore, we simplify \eqref{rhs-to-abc} as follows:
\begin{equation} \label{abc}
\langle U_Q F U_Q^{\dag} \rangle \leq \frac{1}{2} \max_{ \small
\begin{array}{c}
a,b,c,d: \\
0 \leq a,b,c \leq j^2 \\
0 \leq d \\
(2j-1) c \leq 9
j^2 (j^2 - a) \\
(2j-1) c \leq 9
j^2 (j^2 - b) \\
a+b+c+d = j(j+1) \\
\end{array}} \sqrt{(a-b)^2+c} + a + b.
\end{equation}
Using the method of Lagrange multipliers one can readily see that
the maximum in the right hand side of \eqref{abc} is attained on
the boundary of region for parameters $a,b,c,d$. If $j>2$, then
the maximum equals $j^2$ and is attained at points with $a=j^2$
and $c=0$ or $b=j^2$ and $c=0$.
Although we have considered the cases $j=1,\frac{3}{2},2$ and
$j>2$ separately, their results can be unified, namely, $\langle
U_Q F U_Q^{\dag} \rangle \leq j^2$ if $j \geq 1$. Recalling the
fact $\langle U_Q F U_Q^{\dag} \rangle \leq \frac{1}{2}$ if
$j=\frac{1}{2}$, we obtain that $\langle U_Q F U_Q^{\dag} \rangle
\leq \max(j,j^2)$. Since this bound is valid for all normalized
states $|\psi\rangle$, we finally conclude that $\langle F \rangle
\leq \max(j,j^2)$.
\end{proof}
\begin{proposition} \label{prop-p-norm}
The maximal $p$-norm ($p \geq 1$) and the minimal output entropy
of the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ are equal to
\begin{equation} \label{p-norm-Phi}
\nu_p(\Phi) = \frac{(j^p + 1)^{1/p}}{j+1} \quad \text{and} \quad
S_{\min}(\Phi) = \log (j+1) - \frac{j}{j+1} \log j,
\end{equation}
\noindent respectively, and are attained at the state
$|j,j\rangle$.
\end{proposition}
\begin{proof}
Let $|\psi\rangle \langle \psi |$ be a pure state at which the
maximal $\infty$-norm is attained. Then $\| \Phi[|\psi\rangle
\langle \psi |] \|_{\infty} = \lambda$, where $\lambda$ is the
maximal output eigenvalue. On the other hand,
\begin{equation} \label{infinite-norm}
\lambda = \max_{\chi \neq 0} \frac{\langle \chi | \,
\Phi[|\psi\rangle \langle \psi |] \, | \chi \rangle} {\langle \chi
| \chi \rangle} = \max_{\chi \neq 0} \frac{ \sum_{\alpha=1}^3
| \langle \varphi_{\alpha} | \chi \rangle |^2 }{j(j+1) \langle \chi | \chi \rangle}, \qquad | \varphi_{\alpha} \rangle = J_{\alpha} | \psi \rangle.
\end{equation}
The vector $|\chi\rangle$ maximizing~\eqref{infinite-norm} must
belong to a linear span of vectors
$\{|\varphi_{\alpha}\rangle\}_{\alpha=1}^3$, i.e. $|\chi\rangle =
\sum_{\beta=1}^3 c_{\beta} |\varphi_{\beta}\rangle$. Introduce the
Hermitian Gram matrix $G_{\alpha\beta} = \langle \varphi_{\alpha}
| \varphi_{\beta}
\rangle = \langle \psi | J_{\alpha} J_{\beta} | \psi \rangle$ and the vector $|c\rangle = \left
\begin{array}{ccc}
c_1 & c_2 & c_3 \\
\end{array
\right)^{\top} \in \mathcal{H}_3$, then $\sum_{\alpha=1}^3
| \langle \varphi_{\alpha} | \chi \rangle |^2 = \sum_{\alpha=1}^3
\left\vert \sum_{\beta=1}^3 G_{\alpha\beta} c_{\beta}
\right\vert^2 = \langle c | G^2 | c \rangle$ and $\langle \chi |
\chi \rangle = \langle c | G | c \rangle$.
Equation~\eqref{infinite-norm} can be further simplified with the
use of vector $| c ' \rangle = \sqrt{G} | c \rangle$:
\begin{equation} \label{lambda-throgh-c}
\lambda = \frac{1}{j(j+1)} \max_{ c: \ \sqrt{G}|c\rangle \neq 0 }
\frac{\langle c | G^2 | c \rangle}{\langle c | G | c \rangle} =
\frac{1}{j(j+1)} \max_{ c' \neq 0 } \frac{\langle c' | G | c'
\rangle}{\langle c' | c' \rangle} = \frac{1}{j(j+1)}
\|G\|_{\infty}.
\end{equation}
On the other hand, the $\infty$-norm of $G$ reads
\begin{equation} \label{G-norm}
\|G\|_{\infty} = \max_{u: \, u^{\dag}u = I} \sum_{\alpha,\beta =
1}^3 \overline{u_{\alpha 1}} G_{\alpha \beta} u_{\beta 1} =
\max_{u: \, u^{\dag}u = I} \left\| \sum_{\alpha=1}^3 u_{\alpha 1}
J_{\alpha} |\psi\rangle \right\|^2.
\end{equation}
Since $u$ is a unitary matrix, the vector ${\bf u} = \left
\begin{array}{ccc}
u_{11} & u_{21} & u_{31} \\
\end{array
\right)^T = {\bf k} + i {\bf l}$, where ${\bf k},{\bf l} \in
\mathbb{R}^3$ and ${\bf u}^{\dag}{\bf u} = |{\bf k}|^2 + |{\bf
l}|^2 = 1$. By Lemma~\ref{lemma-4} for any $|\psi\rangle$ the
maximum in the right hand side of \eqref{G-norm} does not exceed
$\max(j,j^2)$. Therefore,
\begin{equation}
\lambda = \frac{1}{j(j+1)} \|G\|_{\infty} \leq \max \left(
\frac{j}{j+1},\frac{1}{j+1} \right).
\end{equation}
On the other hand, if $|\psi\rangle = |j,j\rangle$, then by
formula~\eqref{spec-phi-jm} we have $Spec \left( \Phi[| j,j
\rangle \langle j,j |] \right) = \left\{ \frac{j}{j+1},
\frac{1}{j+1}, 0, \ldots \right\}$. This implies that $\lambda =
\max(\frac{j}{j+1},\frac{1}{j+1})$ and $\| \Phi[|\psi\rangle
\langle \psi |] \|_{\infty}$ is attained at the vector
$|j,j\rangle$.
Denote $\boldsymbol{\lambda} = \left\{ \frac{j}{j+1},
\frac{1}{j+1}, 0, \ldots \right\}$. Since $\boldsymbol{\lambda}$
has only two nonzero components and the largest component is
$\nu_{\infty}(\Phi)$, then $\boldsymbol{\lambda}$ majorizes all
other output spectra $\boldsymbol{\mu}$. Here, we use the
conventional definition of majorization (Ref.~\cite{Tong},
Definition 12.1): a sequence of real numbers
$\boldsymbol{\lambda}=(\lambda_1,\lambda_2,\ldots,\lambda_n)$
majorizes another sequence of real numbers
$\boldsymbol{\mu}=(\mu_1,\mu_2,\ldots,\mu_n)$ if, after possible
renumeration, the terms of the sequences $\boldsymbol{\lambda}$
and $\boldsymbol{\mu}$ satisfy conditions $\lambda_1 \geq
\lambda_2 \geq \ldots \geq \lambda_n$, $\mu_1 \geq \mu_2 \geq
\ldots \geq \mu_n$, $\lambda_1 + \lambda_2 + \ldots + \lambda_k
\geq \mu_1 + \mu_2 + \ldots + \mu_k$ for each $k$, $1 \leq k \leq
n-1$, and $\lambda_1 + \lambda_2 + \ldots + \lambda_n = \mu_1 +
\mu_2 + \ldots + \mu_n$. In our case, $\lambda_1 =
\nu_{\infty}(\Phi)$, which guarantees $\lambda_1 \geq \mu_1$ for
all other output spectra $\boldsymbol{\mu}$. Moreover, $\lambda_1
+ \lambda_2 = 1 = \mu_1 + \mu_2 + \ldots + \mu_{2j+1} \geq \mu_1 +
\mu_2$, therefore $\boldsymbol{\lambda}$ majorizes any other
output spectrum $\boldsymbol{\mu}$. Since functions $y(x)=x{\rm
log}x$ and $y(x)=x^p$, $p\geq 1$, are convex, the Shannon entropy
$H({\bf x})=- \sum_{k=1}^n x_k {\rm log} x_k$ is a Schur-concave
function of ${\bf x} \in [0,1]^n$ and the output $p$-norm ${\cal
V}_p({\bf x}) = \left( \sum_{k=1}^{n} x_k^p \right)^{1/p}$ is a
Schur-convex function of ${\bf x} \in [0,1]^n$ (Ref.~\cite{Tong},
Definition 12.23, Theorem 12.27). Therefore, $H(\boldsymbol{\mu})
\geq H(\boldsymbol{\lambda})$ and ${\cal V}_p(\boldsymbol{\mu})
\leq {\cal V}_p(\boldsymbol{\lambda})$ for all output spectra
$\boldsymbol{\mu}$. This observation results in
formulas~\eqref{p-norm-Phi}.
\end{proof}
Corollaries~\ref{corollary-qubit-entropy}
and~\ref{corollary-qutrit-entropy} are merely consequences of
proposition~\ref{prop-p-norm}.
Consider the second tensor power $\Phi^{\otimes 2}$ of a channel
$\Phi: \mathcal{B}(\mathcal{H}) \mapsto \mathcal{B}(\mathcal{H})$.
Suppose a density operator $\rho \in \mathcal{B}(\mathcal{H})$.
Then for the factorized input $\rho^{\otimes 2}$ we have
$\Phi^{\otimes 2}[\rho^{\otimes 2}] = \left( \Phi[\rho]
\right)^{\otimes 2}$. Obviously, the purity ${\rm tr}\left[ \left(
\Phi^{\otimes 2}[\rho^{\otimes 2}] \right)^2 \right]$ of the state
$\Phi^{\otimes 2}[\rho^{\otimes 2}]$ is equal to the square of the
purity ${\rm tr}\left[ \left( \Phi[\rho] \right)^2 \right]$ of the
state $\Phi[\rho]$. However, if one uses \textit{entangled} input
states $\varrho_{\rm ent} \in \mathcal{B}(\mathcal{H}^{\otimes
2})$, then in general the purity of the state $\Phi^{\otimes
2}[\varrho_{\rm ent}]$ can be greater than the purity of all
possible factorized states $\Phi^{\otimes 2}[\rho^{\otimes 2}]$.
Therefore, in general $\nu_2(\Phi^{\otimes 2}) \geqslant \left(
\nu_2(\Phi) \right)^2$. Clearly, if $\nu_2(\Phi^{\otimes 2}) >
\left( \nu_2(\Phi) \right)^2$, then the maximal 2-norm for the
channel $\Phi^{\otimes 2}$ is attained at some entangled state.
Nevertheless, there exist some channels, for which
$\nu_2(\Phi^{\otimes 2}) = \left( \nu_2(\Phi) \right)^2$ and the
use of entangled inputs does not help to increase the output
purity~\cite{michalakis-2007}. Among such channels, there is a
class of unital qubit channels~\cite{king-2002}, so for the
Landau--Streater channel $\Phi: \mathcal{B}(\mathcal{H}_2) \mapsto
\mathcal{B}(\mathcal{H}_2)$ the multiplicativity of the maximal
2-norm holds. Since the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_3) \mapsto \mathcal{B}(\mathcal{H}_3)$
reduces to the Werner--Holevo channel, then the multiplicativity
of the maximal $p$-norm for such a channel holds for all $1
\leqslant p \leqslant 2$, Ref.~\cite{datta-2004}, and is violated
for $p
> 4.79$, Ref.~\cite{wh-2002}. The Landau--Streater channel for $j > 1$ cannot be analyzed in the
same way as the case $j=1$ and does not satisfy the known
sufficient criteria of multiplicativity of the maximal
$2$-norm~\cite{michalakis-2007}. Despite this fact, if $p=2$, our
numerical investigations of the cases $j = \frac{3}{2}$ and $j =
2$ show that the maximal $2$-norm is multiplicative within the
accuracy of calculations. We can make a conjecture that the
maximal $2$-norm is multiplicative for all Landau--Steater
channels $\Phi: \mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$.
\section{Complementary channel} \label{section-complementary}
According to the Stinespring's dilation theorem, the dual channel
$\Phi^{\dag}: \mathcal{B}(\mathcal{H}) \mapsto
\mathcal{B}(\mathcal{H})$ adopts a representation
(Ref.~\cite{holevo-2012}, Theorem 6.9)
\begin{equation}
\Phi^{\dag} [X] = V^{\dag} (X \otimes I_{\mathcal{K}}) V,
\end{equation}
\noindent where $I_{\mathcal{K}}$ is the identity operator in some
Hilbert space $\mathcal{K}$, $V$: $\mathcal{H} \mapsto \mathcal{H}
\otimes \mathcal{K}$ is an isometry operator, i.e., $V^{\dag}V =
I$.
In the case of the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ the dual channel $\Phi^{\dag}$
coincides with $\Phi$ and the corresponding Stinespring's dilation
is achieved with the help of the isometry operator $V$:
$\mathcal{H}_{2j+1} \mapsto \mathcal{H}_{2j+1} \otimes
\mathcal{H}_3$ of the form
\begin{equation}
V = \frac{1}{\sqrt{j(j+1)}} \left
\begin{array}{c}
J_x \\
J_y \\
J_z \\
\end{array
\right).
\end{equation}
Therefore, ${\rm dim}\mathcal{K} = 3$ and the Landau--Streater
channel $\Phi: \mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ can be realized via a
3-dimensional environment. In the Schr\"{o}dinger picture of
system-environment interaction (Ref.~\cite{holevo-2012}, Theorem
6.9), we have
\begin{equation} \label{channel-sch}
\Phi[\rho] = {\rm tr}_{\mathcal{K}} \left[ V \rho V^{\dag} \right]
= {\rm tr}_{\mathcal{K}} \left[ U (\rho \otimes \xi) U^{\dag}
\right],
\end{equation}
\noindent where $\xi \in \mathcal{B}(\mathcal{H}_3)$ is the pure
initial environment state, $U: \mathcal{H}_{2j+1} \otimes
\mathcal{H}_3 \mapsto \mathcal{H}_{2j+1} \otimes \mathcal{H}_3$ is
the unitary evolution operator. The general technique of finding
$U$ is described, e.g., in Ref.~\cite{nielsen-2000}, section
8.2.3.
If one replaces the partial trace over environment ${\rm
tr}_{\mathcal{K}}$ by the partial trace over system ${\rm
tr}_{\mathcal{H}}$ in formula~\eqref{channel-sch}, then one
obtains a so-called complementary channel~\cite{holevo-2005}
$\widetilde{\Phi}: \mathcal{B}(\mathcal{H}) \mapsto
\mathcal{B}(\mathcal{K})$ (also referred to as conjugate
channel~\cite{king-2007}):
\begin{equation} \label{channel-complementary}
\widetilde{\Phi}[\rho] = {\rm tr}_{\mathcal{H}} \left[ V \rho
V^{\dag} \right] = {\rm tr}_{\mathcal{H}} \left[ U (\rho \otimes
\xi) U^{\dag} \right].
\end{equation}
\noindent In the case of the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$, the complementary channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{3})$ maps spin-$j$ states into
3-dimensional environment states (also known as qutrit states). In
what follows, we use the notation $I_d$ to denote the identity
operator $I: \mathcal{H}_d \mapsto \mathcal{H}_d$.
\begin{proposition} \label{prop-complementary-on-mixed}
The channel $\widetilde{\Phi}: \mathcal{B}(\mathcal{H}_{2j+1})
\mapsto \mathcal{B}(\mathcal{H}_{3})$, which is complementary to
the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$, transforms the maximally mixed
input state $\frac{1}{2j+1} I_{2j+1}$ into the maximally mixed
output state $\frac{1}{3} I_3$.
\end{proposition}
\begin{proof}
Denote by $V_{\alpha} = [j(j+1)]^{-1/2} J_{\alpha}$, $\alpha =
1,2,3$, the Kraus operators of $\Phi$ and by $\widetilde{V}_i$,
$i=1,\ldots,2j+1$, the Kraus operators of $\widetilde{\Phi}$.
These Kraus operators are mutually related with each other by
formula (Ref.~\cite{holevo-2005}, formula (12))
\begin{equation} \label{kraus-kraus}
\langle \alpha_{\mathcal{K}} | \widetilde{V}_i = \langle
i_{\mathcal{H}}| V_{\alpha},
\end{equation}
\noindent where $\{ i_{\mathcal{H}} \}_{i=1}^{2j+1}$ is the
orthonormal basis in $\mathcal{H}$ (input) and $\{
\alpha_{\mathcal{K}} \}_{\alpha = 1}^{3}$ is the orthonormal basis
in $\mathcal{K}$ (output). Multiplying \eqref{kraus-kraus} from
the left by $| \alpha_{\mathcal{K}} \rangle$ and summing over
$\alpha$, we get
\begin{equation} \label{kraus-through-kraus}
\widetilde{V}_i = \sum_{\alpha = 1}^{3} | \alpha_{\mathcal{K}}
\rangle \langle \alpha_{\mathcal{K}} | \widetilde{V}_i =
\sum_{\alpha = 1}^{3} | \alpha_{\mathcal{K}} \rangle \langle
i_{\mathcal{H}} | V_{\alpha} = \frac{1}{\sqrt{j(j+1)}}
\sum_{\alpha = 1}^{3} | \alpha_{\mathcal{K}} \rangle \langle
i_{\mathcal{H}} | J_{\alpha}.
\end{equation}
Action of the complementary channel $\widetilde{\Phi}$ on the
maximally mixed state $(2j+1)^{-1} I_{2j+1}$ reads
\begin{eqnarray}
\frac{1}{2j+1} \widetilde{\Phi}[I_{2j+1}] &=& \frac{1}{2j+1}
\sum_{i=1}^{2j+1} \widetilde{V}_i \widetilde{V}_i^{\dag}
\nonumber\\
&=& \frac{1}{j(j+1)(2j+1)} \sum_{i=1}^{2j+1}
\sum_{\alpha,\beta=1}^3 | \alpha_{\mathcal{K}} \rangle \langle
i_{\mathcal{H}} | J_{\alpha} J_{\beta}^{\dag}
|i_{\mathcal{H}}\rangle \langle \beta_{\mathcal{K}} | \nonumber\\
&=& \frac{1}{j(j+1)(2j+1)} \sum_{\alpha,\beta=1}^3 {\rm tr}
\left[ J_{\alpha} J_{\beta}^{\dag} \right] | \alpha_{\mathcal{K}}
\rangle \langle \beta_{\mathcal{K}} |.
\end{eqnarray}
\noindent Since $SU(2)$ generators $J_{\alpha}$, $\alpha=1,2,3$,
are Hermitian and satisfy the relation ${\rm tr} \left[ J_{\alpha}
J_{\beta} \right] = \frac{1}{3} j(j+1)(2j+1) \delta_{\alpha\beta}$
(Ref.~\cite{Varshalovich}, section 2.3.4, formula (11)), then
$\widetilde{\Phi}[\frac{1}{2j+1} I_{2j+1}] = \frac{1}{3} I_3$.
\end{proof}
A channel $\Phi$ is called degradable~\cite{Cubitt} if there
exists a channel $T$ such that $\widetilde{\Phi} = T \circ \Phi$.
Conversely, a channel $\Phi$ is called
antidegradable~\cite{Cubitt} if there exists a channel $T'$ such
that $\Phi = T' \circ \widetilde{\Phi}$. The structure of
degradable and antidegradable quantum channels is studied in
Ref.~\cite{Cubitt}. Further, we explore degradability and
antidegradability of the Landau--Streater channel for various
values of $j$.
\subsection{The case $j=1/2$} \label{subsection-complementary-1/2}
If $j=\frac{1}{2}$, then the Landau--Streater channel $\Phi$ is
antidegradable but not degradable. In fact, in this case $\Phi$ is
a qubit depolarization channel with depolarization parameter
$-\frac{1}{3}$, so it is entanglement breaking~\cite{ruskai-2003}
and, consequently, antidegradable. The Kraus operators of the
complementary channel $\widetilde{\Phi}$ are calculated via
formula~\eqref{kraus-through-kraus}, which results in the
following form of $\widetilde{\Phi}$:
\begin{equation}
\widetilde{\Phi}[X] = \frac{1}{3} \left(\left(
\begin{array}{cc}
0 & 1 \\
0 & -i \\
1 & 0 \\
\end{array}
\right) X \left(
\begin{array}{ccc}
0 & 0 & 1 \\
1 & i & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
1 & 0 \\
i & 0 \\
0 & -1 \\
\end{array}
\right) X \left(
\begin{array}{ccc}
1 & -i & 0 \\
0 & 0 & -1 \\
\end{array}
\right) \right).
\end{equation}
\noindent The factoring map $T = \widetilde{\Phi} \circ \Phi^{-1}$
is well defined, and its normalized Choi matrix~\cite{Choi}
$\Omega_{T} = T \otimes {\rm Id}_{2} [ | \psi_+ \rangle \langle
\psi_+ | ]$, $| \psi_+ \rangle = \frac{1}{\sqrt{2}} \sum_{i=1}^2
|i\rangle \otimes |i\rangle$, reads
\begin{equation}
\Omega_{T} = \frac{1}{6} \left(
\begin{array}{cccccc}
1 & 0 & 3 i & 0 & 0 & 3 \\
0 & 1 & 0 & -3 i & -3 & 0 \\
-3 i & 0 & 1 & 0 & 0 & 3 i \\
0 & 3 i & 0 & 1 & 3 i & 0 \\
0 & -3 & 0 & -3 i & 1 & 0 \\
3 & 0 & -3 i & 0 & 0 & 1 \\
\end{array}
\right).
\end{equation}
\noindent Since $\Omega_{T}$ has negative eigenvalues, $T$ is not
completely positive, and $\Phi$ is not degradable.
\subsection{The case $j=1$} \label{subsection-complementary-1}
If $j=1$, then the Landau--Streater channel $\Phi$ is both
degradable and antidegradable. This follows from the fact that
$\Phi: \mathcal{B}(\mathcal{H}_{3}) \mapsto
\mathcal{B}(\mathcal{H}_{3})$ is unitarily equivalent to the
Werner--Holevo channel, which is both degradable and
antidegradable (Ref.~\cite{Cubitt}, section 2.2). For the sake of
completeness, we list the Kraus operators of the complementary
channel in this case:
\begin{equation}
\widetilde{V}_1 =
\begin{pmatrix}
0& \frac{1}{2} &0 \\
0& -\frac{ i }{2} &0 \\
\frac{1}{\sqrt{2}} &0&0
\end{pmatrix}, \quad
\widetilde{V}_2 = \begin{pmatrix}
\frac{1}{2} & 0 & \frac{1}{2} \\
\frac{ i }{2} & 0 & -\frac{ i }{2} \\
0&0&0
\end{pmatrix}, \quad
\widetilde{V}_3 = \begin{pmatrix}
0& \frac{1}{2} & 0 \\
0 & \frac{ i }{2} & 0 \\
0 & 0 & -\frac{1}{\sqrt{2}}
\end{pmatrix}.
\end{equation}
\subsection{The case $j \geq 3/2$} \label{subsection-complementary-3/2}
\begin{proposition} \label{prop-not-antidegradable}
The Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ is not antidegradable if $j
\geqslant \frac{3}{2}$.
\end{proposition}
\begin{proof}
Since doubly complementary channel $\widetilde{\widetilde{\Phi}}$
is unitarily equivalent to $\Phi$ (Ref.~\cite{holevo-2012},
Exercise 6.29), it is enough to demonstrate that
$\widetilde{\Phi}: \mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{3})$ is not degradable. The output space
for the complementary channel is 3-dimensional, so we use
Theorem~10 in Ref.~\cite{Cubitt}, which states that if
$\widetilde{\Phi}: \mathcal{B}(\mathcal{H}_{d}) \mapsto
\mathcal{B}(\mathcal{H}_{3})$ is degradable, then the Choi rank of
$\widetilde{\Phi}$ is at most 3. Choi rank is defined as the rank
of the Choi matrix~\cite{Choi} $\Omega_{\widetilde{\Phi}} =
\widetilde{\Phi} \otimes {\rm Id}_{d} [ | \psi_+ \rangle \langle
\psi_+ | ]$, with $| \psi_+ \rangle = \frac{1}{\sqrt{d}}
\sum_{i=1}^d |i\rangle \otimes |i\rangle$ being the maximally
entangled state. The Choi matrix of the complementary channel
$\widetilde{\Phi}: \mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{3})$ reads
\begin{eqnarray}
\Omega_{\widetilde{\Phi}} &=& \frac{1}{2j+1} \sum_{m,m'=-j}^j
\widetilde{\Phi} [|j,m\rangle \langle j,m'|] \otimes |j,m\rangle
\langle j,m'| \nonumber\\
&=& \frac{1}{j(j+1)(2j+1)} \sum_{\alpha,\beta=1}^3 \,
\sum_{m,m',i=-j}^j \langle j,m' | J_{\beta} | i \rangle \langle i
| J_{\alpha} | j,m \rangle \, | \alpha \rangle \langle \beta |
\otimes | j,m \rangle \langle j,m' | \nonumber\\
&=& \frac{1}{j(j+1)(2j+1)} \sum_{\alpha,\beta=1}^3 | \alpha
\rangle \langle \beta | \otimes J_{\alpha}^{\top}
J_{\beta}^{\top} \nonumber\\
&=& \frac{1}{j(j+1)(2j+1)} \left
\begin{array}{ccc}
J_x^2 & -J_x J_y & J_x J_z \\
- J_y J_x & J_y^2 & -J_y J_z \\
J_z J_x & - J_z J_y & J_z^2 \\
\end{array
\right) \nonumber\\
&=& \frac{1}{j(j+1)(2j+1)} \left
\begin{array}{c}
J_x \\
-J_y \\
J_z \\
\end{array
\right)
\left
\begin{array}{ccc}
J_x & -J_y & J_z \\
\end{array
\right), \label{sandwich}
\end{eqnarray}
\noindent where transposition is performed in the basis
$\{|j,m\rangle\}_{m=-j}^j$ and, therefore, $J_x^{\top} = J_x$,
$J_y^{\top} = - J_y$, $J_z^{\top} = J_z$. Denote $C = \left
\begin{array}{ccc}
J_x & -J_y & J_z \\
\end{array
\right)$, then $j(j+1)(2j+1) \Omega_{\widetilde{\Phi}} =
C^{\dag}C$ and ${\rm rank}\, \Omega_{\widetilde{\Phi}} \leq {\rm
rank}\,C \leq 2j+1$. On the other hand, $CC^{\dag} =
j(j+1)I_{2j+1}$ and $j(j+1)(2j+1) C \Omega_{\widetilde{\Phi}}
C^{\dag} = C C^{\dag} C C^{\dag} = j^2(j+1)^2 I_{2j+1}$. This
implies that $2j+1 = {\rm rank}\, C \Omega_{\widetilde{\Phi}}
C^{\dag} \leq {\rm rank}\, \Omega_{\widetilde{\Phi}}$. Combining
both inequalities, we get ${\rm rank}\, \Omega_{\widetilde{\Phi}}
= 2j+1$. Thus, ${\rm rank}\, \Omega_{\widetilde{\Phi}} \geq 4$ if
$j \geqslant \frac{3}{2}$. Therefore, $\widetilde{\Phi}$ is not
degradable~\cite{Cubitt} and $\Phi$ is not antidegradable.
\end{proof}
\begin{proposition}
The Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ is not degradable if $j \geqslant
\frac{3}{2}$.
\end{proposition}
\begin{proof}
To prove that the factoring map $T = \widetilde{\Phi} \circ
\Phi^{-1}$ is not completely positive, it suffices to verify
negativity of some diagonal element of the Choi matrix $\Omega_T =
T \otimes {\rm Id}_{2j+1} [ | \psi_+ \rangle \langle \psi_+ | ]$,
$| \psi_+ \rangle = \frac{1}{\sqrt{2j+1}} \sum_{m'=-j}^{j}
|j,m'\rangle \otimes |j,m'\rangle$. Consider the diagonal element
\begin{eqnarray}
&& \langle \alpha | \otimes \langle j,m | \, \Omega_T \, | \alpha
\rangle \otimes | j,m \rangle = \frac{1}{2j+1} \langle \alpha |
\widetilde{\Phi} \circ \Phi^{-1} [| j,m \rangle \langle j,m |] |
\alpha \rangle
\nonumber\\
&& = \frac{1}{2j+1} \sum_{i=1}^{2j+1} \langle \alpha |
\widetilde{V}_i \, \Phi^{-1} [| j,m \rangle \langle j,m |] \,
\widetilde{V}_i^{\dag} | \alpha \rangle = \frac{1}{2j+1}
\sum_{i=1}^{2j+1} \langle i | V_{\alpha} \, \Phi^{-1} [| j,m
\rangle \langle j,m |] \, V_{\alpha}^{\dag} | i \rangle \nonumber\\
&& = \frac{1}{j(j+1)(2j+1)} {\rm tr} \left( J_{\alpha} \,
\Phi^{-1} [| j,m \rangle \langle j,m |] \, J_{\alpha} \right) =
\frac{1}{j(j+1)(2j+1)} {\rm tr} \left(J_{\alpha}^2 \, \Phi^{-1} [|
j,m
\rangle \langle j,m |] \right) \nonumber\\
&& = \frac{1}{j(j+1)(2j+1)} {\rm tr} \left( | j,m \rangle \langle
j,m | \, \Phi^{-1} [ J_{\alpha}^2 ] \right)
\label{Phi-selfdual-property} = \frac{1}{j(j+1)(2j+1)} \langle j,m
| \, \Phi^{-1} [ J_{\alpha}^2 ] | j,m \rangle.
\end{eqnarray}
\noindent In derivation of formula \eqref{Phi-selfdual-property}
we have taken into account that $\Phi$ is a self-dual map and
${\rm tr}(X \Phi^{-1}[Y]) = {\rm tr}
\left(\Phi\left[\Phi^{-1}[X]\right] \Phi^{-1}[Y] \right) = {\rm
tr} \left( \Phi^{-1}[X] \Phi^{\dag}\left[ \Phi^{-1}[Y] \right]
\right) = {\rm tr}(\Phi^{-1}[X] Y)$.
Let us fix $\alpha = 3$ and calculate the operator $\Phi[J_z^2]$
by using formula~\eqref{Phi-on-jm} and the spectral decomposition
$J_z = \sum_{m'=-j}^{j} m' | j,m' \rangle \langle j,m' |$:
\begin{eqnarray}
\Phi[J_z^2] &=& \sum_{m'=-j}^{j} (m')^2 \Phi[| j,m' \rangle
\langle j,m' |] = \frac{1}{j(j+1)} \sum_{m'=-j}^{j} (m')^2 \bigg[
(m')^2 |j,m'\rangle\langle j,m'| \nonumber\\
&& \qquad + \frac{1}{2}\left( j(j+1)-m'(m'-1) \right)
|j,m'-1\rangle\langle j,m'-1| \nonumber\\
&& \qquad + \frac{1}{2}\left( j(j+1)-m'(m'+1)
\right) |j,m'+1\rangle\langle j,m'+1| \bigg] \nonumber\\
&=& \frac{1}{j(j+1)} \sum_{m'=-j}^{j} \bigg[
\big(j(j+1)-3\big)(m')^2 + j(j+1) \bigg] |j,m'\rangle\langle j,m'| \nonumber\\
&=& \frac{j(j+1)-3}{j(j+1)} J_z^2 + I_{2j+1}.
\end{eqnarray}
\noindent This implies that $\frac{j(j+1)}{j(j+1)-3} \Phi[J_z^2-I]
= J_z^2$ and $\Phi^{-1}[J_z^2] = \frac{j(j+1)}{j(j+1)-3}
(J_z^2-I)$. Substituting the obtained result into
formula~\eqref{Phi-selfdual-property} yields
\begin{eqnarray}
\langle 3 | \otimes \langle j,m | \, \Omega_T \, | 3 \rangle
\otimes | j,m \rangle &=& \frac{1}{j(j+1)(2j+1)} \langle j,m | \,
\Phi^{-1} [ J_z^2 ] | j,m \rangle \nonumber\\
&=& \frac{m^2-1}{(2j+1)(j^2+j-3)}.
\end{eqnarray}
\noindent If $j$ is a half-integer and $j\geq \frac{3}{2}$, then
$\langle 3 | \otimes \langle j,\frac{1}{2} | \, \Omega_T \, | 3
\rangle \otimes | j,\frac{1}{2} \rangle <0$. If $j$ is an integer
and $j\geq 2$, then $\langle 3 | \otimes \langle j,0 | \, \Omega_T
\, | 3 \rangle \otimes | j,0 \rangle <0$. Therefore, the Choi
matrix $\Omega_T$ is not positive semidefinite and $T$ is not a
channel.
\end{proof}
\section{Capacities} \label{section-capacities}
\subsection{Classical capacity}
Classical capacity~\cite{schumacher-1997,holevo-chi-1998} $C$ of a
quantum channel $\Phi$ is known to be equal to the regularized
$\chi$-capacity $C_{\chi}$, i.e., $C(\Phi) = \lim_{n \rightarrow
\infty} \frac{1}{n} C_{\chi}(\Phi^{\otimes n})$, where
$\chi$-capacity is defined by the expression
\begin{equation}
C_{\chi}(\Phi) = \sup_{\{ p_i, \rho_i \}} \left[ S \left( \sum_i
p_i \Phi[\rho_i] \right) - \sum_i p_i S \left( \Phi[\rho_i]
\right) \right]
\end{equation}
\noindent and $\{ p_i, \rho_i \}$ is an ensemble of quantum
states, in which the state $\rho_i$ is presented with the
probability $p_i$.
Further, we find $C_{\chi}(\Phi)$ for the Landau-Streater
channel~\eqref{LS-map}.
\begin{proposition} \label{prop-chi-capacity}
$\chi$-capacity $C_{\chi}(\Phi)$ of the Landau--Streater channel
$\Phi:\mathcal{H}_{2j+1} \mapsto \mathcal{H}_{2j+1}$ equals
\begin{equation} \label{C-equality}
C_{\chi}(\Phi) = \log{\frac{2j+1}{j+1}+\frac{j}{j+1}\log{j}}.
\end{equation}
\noindent If $j=1/2$, then $C(\Phi) = C_{\chi}(\Phi) = \frac{5}{3}
- \log 3$.
\end{proposition}
\begin{proof}
We exploit the fact that the Landau--Streater channel is $SU(2)$
covariant. Since the representation $U_g$ of $SU(2)$ group is
irreducible, it follows from Refs.~\cite{holevo-arxiv-2002} that
\begin{equation} \label{chi-capacity-irrep}
C_{\chi}(\Phi) = S \left( \Phi \left[\frac{1}{2j+1} I_{2j+1}
\right] \right) - \min_{\psi} S \left(\Phi[ |\psi\rangle
\langle\psi| ] \right) = \log (2j+1) - \min_{\psi} S \left(\Phi[
|\psi\rangle \langle\psi| ] \right).
\end{equation}
\noindent The minimal output entropy of $\Phi$ is given by
proposition~\ref{prop-p-norm}. Substituting~\eqref{p-norm-Phi}
into~\eqref{chi-capacity-irrep}, we get
formula~\eqref{C-equality}.
In the case $j=\frac{1}{2}$, $\chi$-capacity is known to be
additive~\cite{king-2002}, so $C(\Phi) = C_{\chi}(\Phi) =
\frac{5}{3} \log 2 - \log 3$.
\end{proof}
\subsection{Entanglement assisted capacity}
The entanglement assisted capacity $C_{\rm ea}$ quantifies the
maximal communication rate of classical information transmission
through a quantum channel $\Phi$ with the help of preshared
entanglement between the sender and receiver~\cite{bennett-1999}.
The fundamental result in quantification of the entanglement
assisted capacity is the following
formula~\cite{bennett-2002,holevo-2002}
\begin{equation} \label{c-ea-max}
C_{\rm ea}(\Phi) = \max_{\rho} \left\{ S(\rho) + S(\Phi[\rho]) -
S(\rho,\Phi) \right\},
\end{equation}
\noindent where $S(\rho,\Phi)$ is the exchange
entropy~\cite{barnum-1998}, $S(\rho,\Phi) =
S(\widetilde{\Phi}[\rho])$, with $\widetilde{\Phi}$ being the
complementary channel with respect to $\Phi$. We find the explicit
form of the entanglement assisted capacity for the
Landau--Streater channel.
\begin{proposition} \label{prop-ea-capacity}
The Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ has the entanglement-assisted
capacity $C_{\rm ea}(\Phi) = 2 \log (2j+1) - \log 3$.
\end{proposition}
\begin{proof}
Since $\Phi$ is irreducibly covariant by
proposition~\ref{prop-SU2-cov}, then it follows that the maximum
in~\eqref{c-ea-max} is attained on the maximally mixed input state
$\rho = \frac{1}{2j+1} I_{2j+1}$ (Ref.~\cite{holevo-2012},
Proposition 9.3). Recalling that $\Phi$ is unital and the
complementary channel $\widetilde{\Phi}$ transforms the maximally
mixed state $\frac{1}{2j+1} I_{2j+1}$ into the maximally mixed
qutrit state $\frac{1}{3} I_{3}$ by
proposition~\ref{prop-complementary-on-mixed}, we get $C_{\rm ea}
= 2 S[\frac{1}{2j+1} I_{2j+1}] - S(\frac{1}{3}I_3) = 2 \log (2j+1)
-\log 3$.
\end{proof}
\subsection{Quantum capacity}
The coherent information~\cite{schumacher-1996} for a channel
$\Phi: \mathcal{B}(\mathcal{H}) \mapsto \mathcal{B}(\mathcal{H})$
and state $\rho \in \mathcal{B}(\mathcal{H})$ is defined through
$I_{\rm c}(\rho,\Phi) = S(\Phi[\rho]) -
S(\widetilde{\Phi}[\rho])$. Maximizing coherent information over
states $\rho$ we get a ``single-letter'' quantum capacity
$Q_1(\Phi) = \max_{\rho} I_{\rm c}(\rho,\Phi)$. Quantum capacity
is known~\cite{devetak-2-2005} to be a regularized version of
$Q_1$, namely, $Q(\Phi) = \lim_{n \rightarrow \infty} \frac{1}{n}
Q_1(\Phi^{\otimes n})$. If $\Phi$ is degradable, than $Q(\Phi) =
Q_1(\Phi)$, Ref.~\cite{devetak-2005}. If $\Phi$ is antidegradable,
then $Q(\Phi)=0$, Ref.~\cite{giovannetti-fazio-2005}. If
$j=\frac{1}{2}$ or $j=1$, then the Landau--Streater channels
$\Phi: \mathcal{B}(\mathcal{H}_2) \rightarrow
\mathcal{B}(\mathcal{H}_2)$ and $\Phi: \mathcal{B}(\mathcal{H}_3)
\rightarrow \mathcal{B}(\mathcal{H}_3)$ are antidegradable, so
$Q(\Phi) = 0$. Since the Landau--Streater channel is not
antidegradable if $j \geqslant \frac{3}{2}$ by
proposition~\ref{prop-not-antidegradable}, one can expect that
$Q(\Phi) > 0$ if $j \geqslant \frac{3}{2}$. Note that $I_{\rm
c}(\rho^{\otimes n},\Phi^{\otimes n}) = n I_{\rm c}(\rho,\Phi)$,
Ref.~\cite{barnum-1998}, and therefore $Q_1(\Phi^{\otimes n}) =
\max_{\rho: \rho \in \mathcal{B}(\mathcal{H}^{\otimes n})} I_{\rm
c}(\rho,\Phi^{\otimes n}) \geq \max_{\rho: \rho \in
\mathcal{B}(\mathcal{H})} I_{\rm c}(\rho^{\otimes n},\Phi^{\otimes
n}) = n Q_1(\Phi)$. Consequently, $Q(\Phi) \geqslant Q_1(\Phi)
\geq I_{\rm c}(\rho_0,\Phi)$ for any density operator $\rho_0$.
This means that one can estimate the quantum capacity of the
Landau--Streater channel $\Phi: \mathcal{B}(\mathcal{H}_{2j+1})
\mapsto \mathcal{B}(\mathcal{H}_{2j+1})$ from below by $I_{\rm
c}(\rho_0,\Phi)$. In fact, if we fix the state $\rho_0 =
\frac{1}{2j+1} I_{2j+1}$, then $Q(\Phi) \geqslant I_{\rm
c}(\frac{1}{2j+1}I_{2j+1},\Phi) = \log(2j+1) - \log 3$. Thus, we
have just proved the following result.
\begin{proposition} \label{prop-q-capacity}
$Q(\Phi) = 0$ for the Landau--Streater channels $\Phi:
\mathcal{B}(\mathcal{H}_2) \mapsto \mathcal{B}(\mathcal{H}_2)$ and
$\Phi: \mathcal{B}(\mathcal{H}_3) \mapsto
\mathcal{B}(\mathcal{H}_3)$. If $j \geqslant \frac{3}{2}$, then
$Q(\Phi) \geqslant Q_1(\Phi) \geqslant \log(2j+1) - \log 3$ for
the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$.
\end{proposition}
\section{Entanglement annihilation and preservation} \label{section-entanglement}
A state $\rho \in \mathcal{B}(\mathcal{H}^{A} \otimes
\mathcal{H}^{B})$ is called separable with respect to bipartition
$A|B$ if it can be represented as the closure of convex
combination $\sum_i p_i \rho_i^{A} \otimes \rho_i^{B}$, where
$\{p_i\}$ is a probability distribution, $\rho_i^{A} \in
\mathcal{B}(\mathcal{H}^{A})$, and $\rho_i^{B} \in
\mathcal{B}(\mathcal{H}^{B})$, Ref.~\cite{werner-1989}. The
channel $\Phi: \mathcal{B}(\mathcal{H}^{A}) \mapsto
\mathcal{B}(\mathcal{H}^{A})$ is called entanglement breaking if
$\Phi \otimes {\rm Id}_k^B [\rho]$ is separable for all input
states $\rho \in \mathcal{B}(\mathcal{H}^A \otimes
\mathcal{H}^B)$, with $k = {\rm dim} \mathcal{H}^B$ being
arbitrary~\cite{horodecki-2003}. Entanglement-breaking channels
are exactly measure-and-prepare ones, and their structure is well
known~\cite{horodecki-2003}. The channel $\Lambda:
\mathcal{B}(\mathcal{H}^{A} \otimes \mathcal{H}^B) \mapsto
\mathcal{B}(\mathcal{H}^{A} \otimes \mathcal{H}^B)$ is called
entanglement annihilating if $\Lambda [\rho]$ is separable for all
input states $\rho \in \mathcal{B}(\mathcal{H}^A \otimes
\mathcal{H}^B)$~\cite{moravcikova-2010}. The structure of
entanglement annihilating channels is fully studied for local
qubit channels $\Lambda = \Phi_1 \otimes \Phi_2$~\cite{ffk-2018}
and partially studied for other classes of
channels~\cite{lami-huber-2015}. We focus on
entanglement-annihilating properties of the map $\Lambda = \Phi
\otimes \Phi$, where $\Phi$ is the Landau--Streater channel. As we
show below, $\Phi \otimes \Phi$ is not entanglement annihilating
if $j \geqslant 1$, from which it will follow that $\Phi$ is not
entanglement breaking and $\Phi \otimes \Phi$ is not absolutely
separating~\cite{fmj-2017}.
\begin{proposition} \label{prop-ent-ann}
The second tensor power of the Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$, $\Phi \otimes \Phi$, is
entanglement annihilating if $j=\frac{1}{2}$ and is not
entanglement annihilating for all $j \geqslant 1$.
\end{proposition}
\begin{proof}
The case $j=\frac{1}{2}$ corresponds to the qubit depolarizing
channel with depolarization parameter $q=-\frac{1}{3}$.
Entanglement annihilation by $\Phi \otimes \Phi$ in this case is
proved in Ref.~\cite{moravcikova-2010}.
Let $j \geqslant 1$. In what follows, we prove that $\Phi \otimes
\Phi$ is not entanglement annihilating by presenting a bipartite
entangled state, which remains entangled after the action of $\Phi
\otimes \Phi$. Consider the vector $| \phi \rangle \in
\mathcal{H}_{2j+1} \otimes \mathcal{H}_{2j+1}$ of the form
\begin{equation} \label{schmidt-2}
|\phi\rangle = \frac{1}{\sqrt{2}} \left( |j,j\rangle |j,j\rangle +
|j,-j\rangle |j,-j\rangle \right),
\end{equation}
\noindent where $|j,m\rangle$ denotes the spin-$j$ state vector
corresponding to the definite spin projection $m$ onto $z$ axis,
$J_z |j,m\rangle = m |j,m\rangle$, $m=j,j-1,\ldots,-j$. Let $\top$
be the transposition in the basis $\{ | j,m \rangle \}$. Since
$\top \circ \Phi[X] = \Phi[X^{\top}]$ for the Landau--Streater
channel $\Phi$, then the partially transposed output state
$\Phi\otimes(\top \circ \Phi) [|\phi\rangle\langle\phi|]$ is given
by formula
\begin{eqnarray}
&& 2 \, \Phi\otimes(\top \circ \Phi) [|\phi\rangle\langle\phi|] =
\Phi[|j,j\rangle\langle j,j|]\otimes\Phi[|j,j\rangle\langle j,j|]
+ \Phi[|j,-j\rangle\langle
j,-j|]\otimes\Phi[|j,-j\rangle\langle j,-j|] \nonumber\\
&& + \Phi[|j,j\rangle\langle j,-j|]\otimes\Phi[|j,-j\rangle\langle
j,j|] + \Phi[|j,-j\rangle\langle
j,j|]\otimes\Phi[|j,j\rangle\langle j,-j|].
\end{eqnarray}
\noindent Using the channel representation $\Phi[X] =
[j(j+1)]^{-1} \left( \frac{1}{2}J_- X J_+ + \frac{1}{2} J_+ X J_-
+ J_z X J_z \right)$ and formula~\eqref{J-pm}, we get
\begin{eqnarray}
\Phi[|j,\pm j\rangle \langle j, \pm j|] &=& \frac{j}{j+1} |j,\pm
j\rangle \langle j, \pm j| + \frac{1}{j+1} |j, \pm j \mp 1\rangle
\langle j,
\pm j \mp 1|, \\
\Phi[|j, \pm j\rangle\langle j, \mp j|] &=& -\frac{j}{j+1} |j, \pm
j \rangle \langle j,\mp j|.
\end{eqnarray}
\noindent If $j \geqslant 1$, then the supports of operators
$\Phi[|j,j\rangle\langle j,j|]\otimes\Phi[|j,j\rangle\langle j,j|]
+ \Phi[|j,-j\rangle\langle j,-j|]\otimes\Phi[|j,-j\rangle\langle
j,-j|]$ and $\Phi[|j,j\rangle\langle
j,-j|]\otimes\Phi[|j,-j\rangle\langle j,j|] +
\Phi[|j,-j\rangle\langle j,j|]\otimes\Phi[|j,j\rangle\langle
j,-j|]$ are orthogonal. Moreover, the operator
\begin{eqnarray}
&& \Phi[|j,j\rangle\langle j,-j|] \otimes \Phi[|j,-j\rangle\langle
j,j|] + \Phi[|j,-j\rangle\langle
j,j|] \otimes \Phi[|j,j\rangle\langle j,-j|] \nonumber\\
&& = \frac{j^2}{(j+1)^2} \left( |j, j \rangle \langle j, -j|
\otimes |j, -j \rangle \langle j, j| + |j, -j \rangle \langle j,
j| \otimes |j, j \rangle \langle j, -j| \right)
\end{eqnarray}
\noindent is not positive semidefinite as it has a the negative
eigenvalue $-\frac{j^2}{(j+1)^2}$. Therefore, the partially
transposed state $\Phi\otimes(\top \circ \Phi)
[|\phi\rangle\langle\phi|]$ is not positive semidefinite and
$\Phi\otimes\Phi [|\phi\rangle\langle\phi|]$ is entangled by the
Peres--Horodecki criterion~\cite{peres-1996,horodecki-1996}.
\end{proof}
\begin{proposition}
The Landau--Streater channel $\Phi:
\mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ is entanglement breaking if
$j=\frac{1}{2}$ and is not entanglement breaking if $j \geqslant
1$.
\end{proposition}
\begin{proof}
If $j=\frac{1}{2}$, then the Landau--Streater channel reduces to a
depolarizing qubit channel with depolarization parameter
$q=-\frac{1}{3}$. Such a channel is known to be entanglement
breaking~\cite{ruskai-2003}.
Let $j \geqslant 1$. Suppose that the Landau--Streater channel
$\Phi: \mathcal{B}(\mathcal{H}_{2j+1}) \mapsto
\mathcal{B}(\mathcal{H}_{2j+1})$ is entanglement breaking, then
$\Phi \otimes \Phi$ must be entanglement annihilating by
construction~\cite{moravcikova-2010}. By
proposition~\ref{prop-ent-ann} it is not the case. This
contradiction implies that $\Phi$ is not entanglement breaking.
\end{proof}
\section{Conclusions} \label{section-conclusions}
The channel~\eqref{LS-map} has been originally constructed as an
example of a unital completely positive and trace preserving map,
which is extremal in the set of channels if $j \geqslant 1$ and,
consequently, is not random unitary. By construction, the example
of Landau and Streater is $SU(2)$ covariant for all $j$ and,
surprisingly, is globally unitarily covariant if $j=\frac{1}{2}$
and $j=1$. We have proved that for $j>1$ the Landau--Streater
channels is not $U(2j+1)$ covariant, so global unitary covariance
is a peculiar property of spin-$\frac{1}{2}$ and spin-$1$ maps.
Using the theory of angular momentum, we have explicitly found the
spectrum of the Landau--Streater map in
proposition~\ref{prop-spectrum-LS} and pointed out that $\Phi$
always has negative eigenvalues. Negativity of those eigenvalues
indicates that $\Phi$ cannot be obtained as a result of Hermitian
Markovian quantum dynamics.
We have found the Stinespring dilation of the Landau--Streater
channel, which reveals its physical realization. The
Landau--Streater channel can be implemented as a result of the
controlled interaction between a spin-$j$ particle (system) and a
spin-$1$ particle (environment). The partial trace over
environment results in the Landau--Streater channel $\Phi$,
whereas the partial trace over system results in the complementary
channel $\widetilde{\Phi}$. The important property of the
complementary channel is its action on the maximally mixed input
state, which we have established in
proposition~\ref{prop-complementary-on-mixed}. If $j=\frac{1}{2}$,
then the Landau--Streater channel is antidegradable but not
degradable. If $j=1$, the Landau--Streater channel is unitary
equivalent to the Werner--Holevo channel, so in this case $\Phi$
is both degradable and antidegradable. For larger spins ($j
\geqslant \frac{3}{2}$) the Landau--Streater channel is neither
degradable nor antidegradable.
Using the theory of angular momentum, we find the minimal output
entropy of the Landau--Streater channel in
proposition~\ref{prop-p-norm}. Combining this result with $SU(2)$
covariance, we have managed to calculate the $\chi$-capacity
(proposition~\ref{prop-chi-capacity}) and the
entanglement-assisted capacity
(proposition~\ref{prop-ea-capacity}). Also, we have estimated the
lower bound on quantum capacity of the Landau--Streater channel
(proposition~\ref{prop-q-capacity}).
We have explored the entanglement dynamics induced by the
Landau--Streater channel. The channel is shown to be entanglement
breaking if and only if $j=\frac{1}{2}$. The channel's second
tensor power $\Phi \otimes \Phi$ does not annihilate entanglement
for any $j \geqslant 1$. We have constructed the state with
Schmidt rank $2$, formula~\eqref{schmidt-2}, which remains
entangled when affected by $\Phi \otimes \Phi$.
Finally, we have discussed the multiplicativity property of the
maximal $p$-norms for the Landau--Streater channel and conjectured
multiplicativity of the maximal $2$-norms with respect to the
second tensor power of the channel.
\begin{acknowledgements}
The authors thank the anonymous referee for helpful suggestions to
improve the quality of the paper, pointing out misprints, and
proposing a proof for the fact that ${\rm
rank}\Omega_{\widetilde{\Phi}} = 2j+1$ in Proposition~8. The study
in Sec. II was supported by Russian Science Foundation under
Project No. 16-11-00084. The results of Secs. III--VI were
obtained by S.N.F., supported by Russian Science Foundation under
Project No. 17-11-01388, and performed at the Steklov Mathematical
Institute of Russian Academy of Sciences.
\end{acknowledgements}
|
{
"timestamp": "2019-03-22T01:19:18",
"yymm": "1803",
"arxiv_id": "1803.02572",
"language": "en",
"url": "https://arxiv.org/abs/1803.02572"
}
|
\section*{Introduction}\label{sec: intro}
We define and study the bicategory of bialgebras with coverings. Its construction was,
in large part, inspired by an idea due to Grunefelder and Par\'{e} \cite{GP:87} of indexing families of algebra morphisms by coalgebras.
In this note we focus on the algebraic aspects of the theory. Combinatorial applications will be explored further in \cite{LM:xx}.
The study of this bicategory was originally motivated by applications to combinatorial properties of $\mathbb{N}$-graded connected bialgebras, but we hope that this work will also lead to further developments to the general theory of Hopf algebras. A case in point is a natural generalization of Nichols' result \cite{Nic:78}, which states that a bialgebra quotient of a Hopf algebra that is either finite dimensional or cocommutative is automatically a Hopf algebra (see Theorems \ref{th: fin-dim-cover} and \ref{th: pointed-cover}). It should also serve as a nice new example of a bicategory (and also a double category; see the remark of Section \ref{sec: bialg}).
This work began as an attempt to understand the primitives and antipode of the Hopf algebra $\bs{\Pi}$ of symmetric functions in noncommuting variables \cite{LM:11}. The formulas found there for expressing primitives and the antipode in terms of a distinguished generating set of $\bs{\Pi}$ point to the Hopf algebra $\mathrm{OMP}$ of ordered multiset partitions (see Example \ref{ex: omp over nsym}). The Hopf algebra $\mathrm{OMP}$ is the free algebra generated by all finite subsets of natural numbers and has comultiplication given by partitioning a set in all possible ways. Even though $\mathrm{OMP}$ is, in some sense, much bigger then $\bs{\Pi}$, it is much easier to get a handle on the formulas for computing its primitives and the antipode in terms of its generators (finite subsets of $\mathbb{N}$).
There is a nice family of bialgebra homomorphisms from $\mathrm{OMP}$ to $\bs{\Pi}$ that jointly covers it; and this family can then be used to transport the structure formulas from $\mathrm{OMP}$ to $\bs{\Pi}$.
This can be formalized as follows: if $B$ and $A$ are bialgebras and $C$ a coalgebra, then we say that $f\colon B\otimes C\to A$ is a partial covering if it is a measuring ({\it i.e.}, it corresponds to an algebra map from $B$ to the convolution algebra $\operatorname{Hom}(C,A)$) as well as a coalgebra map. We say that $f$ is a covering if it is also surjective. The transfer of structure discussed above can then be formalized as follows (cf. Theorem \ref{th: transfer}):
\smallskip\noindent
\textsl{Let ${f} \colon B\otimes C \to A$ be a covering. Let $\iota \colon A\to B\otimes C$ be any linear section of ${f}$, that is, ${f}\circ \iota = \operatorname{id}$.
Then the following hold.
\begin{enumerate}
\item If $p\in B$ is primitive then for any $c\in C$, ${f}(p,c)$ is primitive in $A$.
\item If $A$ and $B$ are Hopf algebras, then their antipodes are related by the formula $S_A = {f}\circ (S_B\otimes\operatorname{id})\circ \iota$.
\end{enumerate}}
Given two partial coverings $f\colon B\otimes C_f\to A$ and $g\colon B\otimes C_g\to A$, we may also consider morphisms between them.
These are given by coalgebra maps $\mathbf t\colon C_f\to C_g$ such that $f=g\circ(\operatorname{id}\otimes\mathbf t)$ and lead naturally to the bicategory of bialgebras with partial coverings. This bicategory context allows us to define a concept of a covering-equivalence of bialgebras and Hopf algebras. We hope that this will serve as a nice organizing principle in classification efforts for various classes of Hopf algebras (in particular, but not limited to, combinatorial Hopf algebras).
This paper is split into four sections:
In Section \ref{sec: measurings} we introduce several motivating examples and the bicategory of algebras with measurings ({\it i.e.}, we briefly forget about the coalgebra structure on $B$ and $A$ and we merely demand that $f\colon B\otimes C\to A$ is a measuring).
Again, this follows the seminal work of Grunenfelder and Par\'{e} \cite{GP:87}; though here we realize a bicategory structure by making $C$ a variable (in \cite{GP:87} the choice of $C$ was fixed).
In Section \ref{sec: coverings} we define the bicategories of bialgebras with coverings and partial coverings and give additional motivating examples. We adapt Sweedler's construction of the universal measuring coalgebra to construct a universal partial covering coalgebra. Additionally, we define and discuss the notion of equivalent coverings. We conclude the section by proving the theorem on transfer of structure discussed above.
In Sections \ref{sec: subcategories} and \ref{sec: Hopf}, we discuss properties of subcategories of interest.
In the former, we focus on cocommutative and graded coverings. We also identify weak initial objects for coverings of graded connected cocommutative bialgebras. In Section \ref{sec: Hopf}, we focus on coverings by Hopf algebras. We prove the following: if a Hopf algebra $B$ covers a bialgebra $A$ and if either $A$ is finite dimensional or $B$ is pointed, then $A$ is a Hopf algebra. This is the above mentioned generalization of Nichols' result.
\subsubsection*{Notation} Throughout, we suppress the unit map $u \colon \Bbbk \to A$ for algebras, identifying $1_\Bbbk$ with $1_A$ (and, e.g., writing $u\circ\varepsilon(a)$ as $\varepsilon(a)$ for bialgebra elements $a$).
\subsubsection*{Acknowledgements} We would like to thank the referee for a number of useful comments, in particular for pointing us to the references \cite{HFV:17}, \cite{PR:16} and for explaining a more categorical way of approaching some of the constructions discussed in the paper: see the discussion at the end of Section \ref{sec: measurings}, the paragraph preceding Theorem \ref{th: universal-coalg}, the remark following Proposition \ref{th: universal-bialg}, and the comment preceding Question \ref{q:cocommutative}.
\section{Measurings}\label{sec: measurings}
If $A$ and $B$ are algebras and $C$ a coalgebra, then we say that a map $f\colon B\otimes C\to A$ is a \demph{measuring} (or measures $B$ to $A$) if it corresponds to a unital algebra map from $B$ to the convolution algebra $\operatorname{Hom}_k(C,A)$. More explicitly, $f$ is a measuring if, for all $b,b'\in B$ and $c\in C$, it satisfies
\begin{eqnarray}
\label{eq: meas-prod} f(bb'\otimes c) &=& \sum_{(c)} f(b\otimes c_1)f(b'\otimes c_2), \\
\label{eq: meas-counit} f(1\otimes c) &=& \varepsilon(c).
\end{eqnarray}
Measurings were first studied by Sweedler \cite{Swe:69}.
\subsection{Motivating examples}\label{sec: examples}
It was Grunenfelder and Par\'{e} \cite{GP:87} who first observed that measurings can be viewed as $C$-indexed families of morphisms from $B$ to $A$. From this perspective, it is natural to abuse notation and write $f(b,c)$ for $f(b\otimes c)$ and we do so freely in what follows. We are most interested in the cases where the family $\left\{ f(\bs\cdot, c)\right\}_{c\in C}$ jointly spans $A$, and the most interesting of these come when no single $f(\bs\cdot,c)$ is surjective. We now illustrate with several examples (and one non example).
\begin{exam}\label{ex: A.id over A}
Let $A,B$ be algebras over $\Bbbk$, and $f\colon B\to A$ an algebra map. Give $\Bbbk$ the coalgebra structure induced by letting $1$ be group-like. Then $\overline{f}\colon B\otimes \Bbbk \to A$ given by $\overline f(b,1) = f(a)$ is a measuring. Putting $f=\operatorname{id}$, deduce that every algebra has at least one surjective measuring, albiet not a particularly interesting one.
\end{exam}
\begin{exam}\label{ex: comm over noncomm}
Identify the group algebra of $\mathbb{Z}$ with $\Bbbk[z,z^{-1}]$ and let $A$ be a Hopf algebra. Now let $C=\coalg(A)$. We define a measuring $f\colon \Bbbk[z,z^{-1}]\otimes C\to A$ by putting $f(z,\bs\cdot)=\operatorname{id}$, $f(z^{-1},\bs\cdot)=S$, and $f(z^n,\bs\cdot)=\operatorname{id}^{* n}$, where $\operatorname{id}^{* n}$ stands for the $n$-th convolution power of $\operatorname{id}\in\operatorname{Hom}_{\Bbbk}(A,A)$. In more detail, we have that
$$
f(z^n,a) = \begin{cases} a_1\cdots a_n & n>0\\
\mu\varepsilon(a) & n=0 \\
S(a_1)\cdots S(a_{-n}) & n<0\end{cases}.
$$
\end{exam}
Already we catch a glimpse of the possibilities beyond ordinary morphisms: if the algebra $A$ above is noncommutative, then we find it is jointly spanned by mappings from a commutative one. We may ask under what conditions is the $f$ above a coalgebra map. This happens if and only if $A$ is cocommutative.
(Assuming $A$ is cocommutative, checking the coalgebra map condition is straightforward; the reverse implication follows from Lemma \ref{lem: Ccoc}, noting that $\Bbbk[z,z^{-1}]$ is Hopf.)
Above and below, we employ a useful method of building measurings. If a linear map $f(\bs\cdot,c)$ is defined for generators of $B$, then we may \emph{decree} $f$ to satisfy \eqref{eq: meas-prod} and \eqref{eq: meas-counit}. Checking that this respects the relations among those generators will frequently be left to the reader.
\begin{exam}\label{ex: nsym over sym}
Let $\Bbbk[p_1,p_2,\ldots]$ and $\Bbbk\langle H_1,H_2,\ldots\rangle$ denote, respectively, the free commutative and noncommutative algebras on countable generators, one in each degree. These are presentations of what are known in algebraic combinatorics as the rings of \emph{symmetric functions} ($\mathrm{Sym}$) and \emph{noncommutative symmetric functions} ($\mathrm{NSym}$), respectively. Recall that for any positive integer $n$, we say $\lambda$ is a partition of $n$ (written $\lambda \vdash n$), if $\lambda$ is an tuple of positive integers $(\lambda_1\geq \ldots \geq \lambda_r)$ summing to $n$ (written $|\lambda| = \lambda_1+\cdots+\lambda_r = n$). We take $(0)$ to be the unique partition of $0$. Then $\mathrm{Sym}$ has basis $\{p_\lambda:=p_{\lambda_1}p_{\lambda_2}\dotsb p_{\lambda_r} \mid \lambda\vdash n, n\in\mathbb{N}\}$, identifying $p_{(0)}$ with $1$. Let each $p_k$ be primitive so that $\mathrm{Sym}$ is Hopf (being the universal envelope of a countably infinite dimensional abelian Lie algebra) and put $C=\coalg(\mathrm{Sym})$. The map $f \colon \mathrm{NSym}\otimes C \to \mathrm{Sym}$ given on generators by $f(H_k,p_\lambda) = \delta_{k,|\lambda|}\, p_\lambda$ induces a surjective measuring.
\end{exam}
In the preceding example, one can view the map $f$ as \demph{degree preserving:} taking $\deg (b\otimes c) = \deg c$, we have $\deg f(b\otimes c) = \deg(b\otimes c)$. Such maps will be important in applications of our framework to combinatorics, but we do not insist on this property at present. See Sections \ref{sec: combinat} and \ref{sec: universal}.
\begin{exam}\label{ex: poly over qsym}
Recall that for any positive integer $n$, we say $\gamma$ is a composition of $n$ (written $\gamma \vDash n$), if $\gamma$ is an ordered tuple of positive integers $(\gamma_1,\ldots,\gamma_r)$ summing to $n$ (written $|\gamma| = \gamma_1+\cdots+\gamma_r = n$). We take $(0)$ to be the unique composition of $0$. The space $\Bbbk\{ M_\gamma \mid \gamma\vDash n, n\in\mathbb{N} \}$ becomes an algebra under the shuffle product: $M_\alpha \cdot M_\beta = \sum_{\gamma\in\alpha\mathop{\sqcup\!\sqcup}\beta} M_\gamma$. See \cite{Ree:58}, \cite{Hof:00}, \cite[Ch. 7]{Sta:99} for details. (Here we identify $M_{(0)}$ with $1$.) This is a presentation of what is known as the ring $\mathrm{QSym}$ of \emph{quasisymmetric functions}. It is a graded connected Hopf algebra with (noncocommutative) deconcatenation coproduct: if $\gamma = (\gamma_1,\gamma_2,\ldots,\gamma_r)$, then $\Delta(M_\gamma) = \sum_{0\leq i\leq r} M_{(\gamma_1,\ldots,\gamma_i)} \otimes M_{(\gamma_{i+1},\ldots, \gamma_r)}$. (Here we take $\gamma_0=\gamma_{r+1}=0$.) The map $f \colon k[z] \otimes \mathrm{QSym} \to \mathrm{QSym}$ given by $f(z^n,M_\gamma) = \delta_{n,r} \, M_{\gamma_1}\cdots M_{\gamma_r}$ is a measuring.
\end{exam}
The above resembles Example \ref{ex: comm over noncomm}, but there are important differences: (1) here the mapping is a length-graded convolution power (again, useful for applications to combinatorics); and (2) the measuring algebra $B$ is now only a bialgebra, not a Hopf algebra---this distinction will become important in Proposition \ref{th: cocomm}. The above also represents our first example of a family of maps that is not jointly surjective, as $\mathrm{QSym}$ is not generated by the monomial quasisymmetric functions $M_k$. (The proper subalgebra they generate is well-known to be isomorphic to $\mathrm{Sym}$.)
\subsection{The bicategory of algebras with measurings}\label{sec: alg}
Here we define a bicategory $\mathbf{Malg}_\Bbbk$ of algebras with measurings, and collect some of its elementary properties.
\smallskip\noindent\emph{\underline{Objects:}} algebras over $\Bbbk$.
\smallskip\noindent\emph{\underline{$1$-Cells:}} morphisms $f\colon B\to A$ are identified with the set $\Meas(B,A)$ of measurings $f\colon B\otimes C_f\to A$.
\begin{description}
\item[\it Identities] $\operatorname{id}\colon A\to A$ is given by $C_{\operatorname{id}}=\Bbbk$.
\smallskip
\item[\it Composition] for $f\colon B\to A$ and $g\colon D\to B$ we define their composite $h=fg\colon D\to A$ by $C_h=C_g\otimes C_f$ and
$h=f\circ (g\otimes\operatorname{id})$.
\end{description}
\smallskip\noindent\emph{\underline{$2$-Cells:}} if $f,g\colon B\to A$ are $1$-cells, then a 2-cell $\mathbf t\colon f\to g$ is a coalgebra map $\mathbf t\colon C_f\to C_g$ making the following diagram commute.
\begin{center}
\begin{tikzpicture}[]
\matrix (m) [matrix of math nodes, row sep=2.5em,column sep=2.5em]
{ B\otimes C_f & \\
B\otimes C_{g} & A \\ };
\path[-
(m-1-1) edge[->] node[left]{$ \operatorname{id}\otimes\mathbf t $} (m-2-1)
edge[->] node[above right]{$ f $} (m-2-2)
(m-2-1) edge[->] node[below] {$ g $} (m-2-2);
\end{tikzpicture}
\end{center}
\begin{description}
\item[\it Identities] if $f\colon B\to A$ is a $1$-cell, then $\operatorname{id}\colon f\to f$ is the identity map on $C_f$.
\smallskip
\item[\it Horizontal composition] for $f,f'\in\Meas(B,A)$ and $g,g'\in\Meas(D,B)$, and $2$-cells $\mathbf s\colon f\to f'$ and $\mathbf t\colon g\to g'$, we define $\mathbf s\circ \mathbf t\colon fg\to f'g'$ so as to make the diagram at left below commute.
\smallskip
\item[\it Vertical composition] for $f,g,h\in\Meas(B,A)$ and $2$-cells $\mathbf t\colon f\to g$, $\mathbf s\colon g\to h$, we define $\mathbf{st}\colon f\to h$ so as to make the diagram at right below commute.
\begin{center}
\begin{tikzpicture}[baseline]
\matrix (m) [matrix of math nodes, row sep=3.5em,column sep=3em]
{%
D\otimes C_f \otimes C_g & B\otimes C_g & \\
D\otimes C_{f'} \otimes C_{g'} & B\otimes C_{g'} & A \\
};
\path[-
(m-1-1) edge[->] node[left]{$ \operatorname{id}\otimes\mathbf s $} (m-2-1)
edge[->] node[above right]{$ f $} (m-1-2)
(m-1-2) edge[->] node[left]{$ \operatorname{id}\otimes\mathbf t $} (m-2-2)
edge[->] node[above right]{$ g $} (m-2-3)
(m-2-1) edge[->] node[below] {$ f' $} (m-2-2)
(m-2-2) edge[->] node[below] {$ g' $} (m-2-3);
\end{tikzpicture}
\quad\quad\quad
\begin{tikzpicture}[baseline]
\matrix (m) [matrix of math nodes, row sep=2.5em,column sep=3.25em]
{B\otimes C_f & \\
B\otimes C_{g} & A \\
B\otimes C_{h} & \\
};
\path[-
(m-1-1) edge[->] node[left]{$ \operatorname{id}\otimes\mathbf t $} (m-2-1)
edge[->] node[above right]{$ f $} (m-2-2)
(m-2-1) edge[->] node[left]{$ \operatorname{id}\otimes\mathbf s $} (m-3-1)
edge[->] node[below] {$ g $} (m-2-2)
(m-3-1) edge[->] node[below] {$ h $} (m-2-2);
\end{tikzpicture}
\end{center}
\end{description}
The $1$-cell category $\Meas(B,A)$ inherits many of the properties of the category of coalgebras.
In particular, pushouts and direct sums (categorical coproducts) exist. In fact, this holds more generally for any slice category of a category with colimits, cf. \cite[Ch. V]{Mac:98}. As the theme of this work is computation, we show how to construct these objects explicitly for related categories in Section \ref{sec: bialg}.
\section{Partial Coverings and Transfer of Structure}\label{sec: coverings}
If $A,B$ are bialgebras and $C$ a coalgebra, then we say that a map $f\colon B\otimes C\to A$ is a \demph{partial covering} if it is a measuring as well as a coalgebra map. We say that $f$ is a \demph{covering} if it is also surjective. In what follows, when we wish to deemphasize the indexing coalgebra, we write $B \pmeasures A$ to indicate that $B$ partially covers $A$.
\subsection{The bicategories of bialgebras with coverings and partial coverings}\label{sec: bialg}
We define bicategories $\ensuremath \mathbf{PCbialg}$ and $\ensuremath \mathbf{Cbialg}$ of bialgebras with partial coverings and bialgebras with coverings analogous to the bicategory of algebras with measurings.
\begin{rema}
The bicategories $\mathbf{Malg}$, $\ensuremath \mathbf{PCbialg}$, and $\ensuremath \mathbf{Cbialg}$ can also be viewed as double categories with horizontal morphisms being (algebra or bialgebra) homomorphisms and vertical homomorphisms being measurings or (partial) coverings.
We do not explore this viewpoint any further here.
\end{rema}
\medskip\noindent\emph{\underline{Direct Sums:}} Let $B$, $A$ be bialgebras.
If $f \colon B\otimes C_f\to B$, $g\colon B\otimes C_g\to A$ are (partial) coverings, define the measuring $(h,C_h)$ by putting $C_h:=C_f\oplus C_g$ and $h:=f\circ \iota_f+g\circ\iota_g$. (Here, $\iota_f, \iota_g$ are the canonical injections of $C_f, C_g$ into $C_f\oplus C_g$.)
One checks that $h$ is a partial covering. If either $f$ or $g$ is a covering, then so is $h$.
\medskip\noindent\emph{\underline{Pushouts:}} Let $B$, $A$ be bialgebras, and consider coverings $f,g,h\in\PCov(B,A)$ and morphisms $\mathbf s\colon f\to g$, $\mathbf{t}\colon f\to h$ between them. The pushout (fibered coproduct) is the partial covering $(k,C_k)$ completing the diagram
\begin{center}
\begin{tikzpicture}[baseline]
\matrix (m) [matrix of math nodes, row sep=1.5em,column sep=2em]
{%
& g & \\
f & & k \\
& {h} & \\
};
\path[-
(m-1-2) edge[<-] node[xshift=-.5em, yshift=.35em]{$\mathbf s$} (m-2-1)
edge[->,dashed] node[xshift=.55em, yshift=.5em]{$\mathbf{\tilde{t}}$} (m-2-3)
(m-3-2) edge[<-] node[xshift=-.6em, yshift=-.4em]{$\mathbf{t}$} (m-2-1)
edge[->,dashed] node[xshift=.5em, yshift=-.4em]{$\mathbf{\tilde{s}}$} (m-2-3);
\end{tikzpicture},
\end{center}
where $C_k=C_g\oplus_{C_f\!} C_{h} := (C_g\oplus C_{h})/I$, where $I=\{(\mathbf s(x),-\mathbf{t}(x)) \mid x\in C_f\}$ is the pushout in the category of coalgebras. The partial covering $k\colon B\otimes C_k\to A$ is given by $k(b\otimes ((x,y)+I))=g(b\otimes x)+h(b\otimes y)$. If $f,g,h$ are coverings, then so is $k$. If $\mathbf s$ is surjective, then so is $\mathbf{\tilde{s}}$. Similarly for $\mathbf t,\mathbf{\tilde{t}}$.
\begin{xam}[\ref{ex: A.id over A}]\label{ex: morphism family}
Suppose $\Xi$ is a jointly surjective family of bialgebra maps from $B$ to $A$ and $C=k\Xi$ is the free pointed coalgebra on $\Xi$. Define a bilinear mapping $f \colon B\otimes C\to A$ by $f(b,\xi)=\xi(b)$, for $b\in B$, $\xi\in \Xi$. It is straightforward to check that $f$ is a covering. (In particular, bialgebra maps are coverings.)
\end{xam}
\begin{xam}[\ref{ex: comm over noncomm}]
Let $A$ be a group algebra $\Bbbk G$. It is straightforward to check that the measuring $f$ of Example \ref{ex: comm over noncomm} is a coalgebra map.
\end{xam}
\begin{xam}[\ref{ex: nsym over sym}]
The algebra $\mathrm{NSym}=\Bbbk\langle H_1,H_2,\ldots\rangle$ becomes a Hopf algebra with coproduct on generators given by $\Delta H_k = \sum_{i+j=k} H_i \otimes H_k$. (In fact, as shown in \cite{GKLLRT:95}, it is the graded dual of the Hopf algebra $\mathrm{QSym}$.) We check that the measuring $f$ of Example \ref{ex: nsym over sym} is a coalgebra map:
\begin{align*}
\left(f^{\otimes2} \Delta\right)\,(H_k\otimes p_\lambda) &= f^{\otimes2} \sum_{\substack{i+j=k\\ \mu\sqcup\tau=\lambda}} \left(H_i\otimes p_\mu \right) \otimes \left( H_j \otimes p_\tau\right) \\
&= \sum_{\substack{i+j=k\\ \mu\sqcup\tau=\lambda}} \delta_{i,|\mu|} \, p_\mu \otimes \delta_{j,|\tau|} \, p_\tau \\
&= \delta_{k,|\lambda|} \sum_{\mu\sqcup\tau=\lambda} p_\mu \otimes p_\tau \
= \left(\Delta f\right) (H_k \otimes p_\lambda) .
\end{align*}
\end{xam}
\begin{exam} \label{ex: poly over sym}
We give another covering of $\mathrm{Sym}$. Give $\Bbbk[x]$ the structure of Hopf algebra by letting $x$ be primitive. Let $C = \coalg(\mathrm{Sym})$, using the power sum basis $\{ p_\lambda\}$ as above. For a partition $\lambda$ of length $r$, put $f(x^n,p_\lambda) = \delta_{n,r} \, {n!}\cdot p_{\lambda}$. It is easy to check that $f \colon \Bbbk[x] \otimes C \to \mathrm{Sym}$ is a measuring. To show that $f$ is a partial covering, we verify that it is a coalgebra map.
\begin{align*}
\left(f^{\otimes2} \Delta\right)\,(x^n\otimes p_\lambda)
&= f^{\otimes2} \! \sum_{\substack{i+j=n\\ I\sqcup J= [r]}} \! \binom{n}{i} \! \left(x^i \otimes p_{\lambda_I} \right) \otimes \left( x^{j} \otimes p_{\lambda_J} \right) \\
&= \sum_{\substack{i+j=n\\ I\sqcup J = [r]}} \! \binom{n}{i} \delta_{i,|I|} \, \delta_{j,|J|} \, {i!}{j!} \cdot p_{\lambda_I} \otimes p_{\lambda_J} \\
&= \delta_{n,r} \, {n!} \sum_{I\sqcup J = [r]} p_{\lambda_I} \otimes p_{\lambda_J}
\ = \
\left(\Delta f\right) (x^n \otimes p_\lambda)
\,.
\end{align*}
\end{exam}
\subsection{Universal partial coverings}\label{sec: universal coverings}
If $A$ and $B$ are algebras then Sweedler \cite{Swe:69} constructs a universal measuring coalgebra $\mathcal M(B,A)$ and a universal measuring $\Omega\colon B\otimes \mathcal M(B,A)\to A$ satisfying the following universal property:
\begin{center}
\begin{tikzpicture}[]
\matrix (m) [matrix of math nodes, row sep=3.5em,column sep=3.5em]
B\otimes C_f & \\
B\otimes \mathcal M(B,A) & A \\
};
\path[-
(m-1-1) edge[->,dashed] node[left]{$\exists \operatorname{id}\otimes F $} (m-2-1)
edge[->] node[above right]{$ \forall f $} (m-2-2)
(m-2-1) edge[->] node[below] {\small$ \Omega $} (m-2-2);
\end{tikzpicture}
\end{center}
That is, if $f\colon B\otimes C\to A$ is a measuring, then there exists a unique coalgebra map $F\colon C\to \mathcal M(B,A)$ such that $f=\Omega\circ(\operatorname{id}\otimes F)$.
This universal object may be viewed as a generalization of Sweedler's finite dual $B^\circ$ (taking $A=\Bbbk$).
In the recent literature, one finds other generalizations \cite{HFV:17, PR:16}, the former of which shares some overlap with our work (see the remark following Proposition \ref{th: universal-bialg}). In Theorem \ref{th: universal-coalg}, we adapt Sweedler's proof to construct universal partial coverings. We remark that the universal object $\mathcal{C}(B,A)$ created there is naturally a subcoalgebra of Sweedler's $\mathcal M(B,A)$.
\begin{theo}\label{th: universal-coalg} Let $A$ and $B$ be bialgebras. Then there exists a universal covering coalgebra $\mathcal{C}(B,A)$ and a universal partial covering $\Omega\colon B\otimes \mathcal{C}(B,A)\to A$ with the following universal property: if $f\colon B\otimes C\to A$ is any partial covering, then there exits a unique coalgebra map $F\colon C\to \mathcal{C}(B,A)$ such that $F=\Omega\circ(\operatorname{id}\otimes F)$.
\end{theo}
\begin{proof}
Let $C$ denote the cofree coalgebra on the space $L(A,B)$ of linear maps from $B$ to $A$ and let $p\colon C \to L(B,A)$ be the canonical projection. We define a linear map $G\colon B\otimes C\to A$ by $G=\mathrm{ev}\circ (\operatorname{id}\otimes p)$, that is, $G(b\otimes c)=p(c)(b)$. Now we define $\mathcal{C}(B,A)$ to be the largest subcoalgebra of $C$ such that $G|_{B\otimes \mathcal{C}(B,A)}$ is a partial covering and we define $\Omega=G|_{B\otimes \mathcal{C}(B,A)}$. The existence of such a coalgebra follows from
the following observation: if $(X_i)_{i\in I}$ is a family of subcoalgebras of $C$ such that $G|_{B\otimes X_i}$ are partial coverings, then for $X=\sum_i X_i$, the map $G|_{B\otimes X}$ is also a partial covering (as being a partial covering
is described in terms of homogeneous linear equations). Now suppose that $f\colon B\otimes C_f\to A$ is a partial covering. Let $q\colon C_f\to L(B,A)$ be the corresponding map, that is, $q(c)(b)=f(b,c)$. By the cofreeness of $p\colon C\to L(B,A)$, there is a unique coalgebra map $F\colon C_f\to C$ such that $q=p\circ F$. Now note that $G|_{B\otimes F(C_f)}$ is a partial covering and hence $F$ can be viewed as a map from $C_f$ to $\mathcal{C}(B,A)$. Uniqueness of $F$ follows from the construction (coalgebra maps $F$ from $C_f$ to $\mathcal{C}(B,A)$ satisfying $f=\Omega\circ(\operatorname{id}\otimes F)$ correspond injectively to coalgebra maps from $C_f$ to $C$ satisfying $q=p\circ F$).
\end{proof}
\begin{rema}
In \cite{GM:06, GM:07}, Grunenfelder and Mastnak considered a generalization of measurings to ``bimeasurings'' in the case where $B$ is a bialgebra and the algebra $A$ is commutative. Among other things they show that the universal measuring coalgebra $\mathcal M(B,A)$ carries a natural structure of a bialgebra. Their argument readily carries over to the present setting of partial coverings. We sketch the proof here
\end{rema}
Suppose $B$ is cocommutative and $A$ is commutative. Define a partial covering $\omega_2\colon B\otimes \mathcal{C}(B,A)\otimes \mathcal{C}(B,A)\to A$ by $\omega_2 = \operatorname{m}_A(\omega\otimes\omega)(\operatorname{id}\otimes\tau\otimes\operatorname{id})(\Delta_B\otimes \operatorname{id}\otimes\operatorname{id})$ (here $\tau\colon B\otimes \mathcal{C}(B,A)\to \mathcal{C}(B,A)\otimes A$ denotes the usual twist), that is, $\omega_2(b,x\otimes y)=\Omega(b_1,x)\Omega(b_2,y)$. Check that $\omega_2$ is a partial covering (cocommutativity of $B$ is needed to ensure that $\Delta_B$ is a coalgebra map; commutativity of $A$ is needed to ensure that $m_A$ is an algebra map; conclude $\omega_2$ is a measuring). Now define $\operatorname{m}\colon \mathcal{C}(B,A)\otimes \mathcal{C}(B,A)\to \mathcal{C}(B,A)$ to be the unique coalgebra map such that $\omega_2=\Omega(\operatorname{id}\otimes\operatorname{m})$. The associativity of $\operatorname{m}$ follows from the fact that $\operatorname{m}(\operatorname{id}\otimes\operatorname{m})$ and $\operatorname{m}(\operatorname{m}\otimes\operatorname{id})$ are unique coalgebra maps such that $\omega_3 = \Omega(\operatorname{id}\otimes \operatorname{m}(\operatorname{id}\otimes\operatorname{m}))$, where $\omega_3\colon B\otimes \mathcal{C}(B,A)\otimes \mathcal{C}(B,A)\otimes \mathcal{C}(B,A)\to A$ is given by $\omega_3(b,x\otimes y\otimes z)=\Omega(b_1,x)\Omega(b_2,y)\Omega(b_3,z)$. The unit $u\colon \Bbbk\to \mathcal{C}(B,A)$ is the unique coalgebra map such that $\omega_0 = \Omega(\operatorname{id}\otimes u)$, where $\omega_0=\varepsilon\colon B\otimes \Bbbk\to A$.
This bialgebra structure on $\mathcal{C}(B,A)$ is universal in the following sense.
\begin{prop}\label{th: universal-bialg}
Let $A$ and $B$ be bialgebras with $A$ commutative and $B$ cocommutative. Then the universal covering coalgebra $\mathcal{C}(B,A)$ is a bialgebra satisfying the following universal property: if $C$ is a bialgebra and $f\colon B\otimes C\to A$ is a partial bicovering (that is, it may be viewed as either a partial covering of $A$ by $B$ or a partial covering of $A$ by $C$), then there is a unique bialgebra map $F\colon C\to \mathcal{C}(B,A)$ such that $f=\Omega\circ(\operatorname{id}\otimes F)$.
\qed
\end{prop}
\begin{rema}
The article \cite{HFV:17} may be relevant here.
In the case that $A,B$ are cocommutative, it seems our $\mathcal{C}(B,A)$ reduces to the authors' \emph{universal measuring comonoid in the category of coalgebras $P(B,A)$.} Theorem 10.11 {\it loc. cit.} provides that if $A$ is furthermore commutative and $B$ is Hopf, then $\mathcal{C}(B,A)$ is also Hopf. We note, in contrast, that $P(B,A)$ is always cocommutative, while that is not the case for $\mathcal{C}(B,A)$; cf. Example \ref{ex: cocomm-cover}.
\end{rema}
\begin{prop} Let $A,B$ be bialgebras. Then there exits a covering $F\colon B\measures A$ if and only if the universal partial covering $\Omega\colon B\otimes \mathcal{C}(B,A) \to A$ is surjective. \qed
\end{prop}
\subsection{Covering equivalence}
We define two $1$-cells in $\Cov(B,A)$ to be \demph{covering equivalent} if they factor surjectively through another $1$-cell, {\it i.e.}, $f\sim g$ if there exists a covering $h\in \Cov(B,A)$ and surjective morphisms of coverings $\mathbf t\colon f\to h$ and $\mathbf s\colon g\to h$. If instead $f,g\in\PCov(B,A)$, we further stipulate that the ranges of $f$, $g$, and $h$ coincide.
This is indeed an equivalence relation. Symmetry is obvious. Transitivity is seen by using the pushout construction: if $f_1,f_2$ factor surjectively through $g$ and $f_2, f_3$ factor surjectively through $h$, then $f_1, f_3$ factor surjectively through the pushout $g\oplus_{f_2} h$.
\begin{rema}
In Section \ref{sec: universal coverings} we show that given bialgebras $B,A$, there exists a universal partial-covering $\Omega\colon B\otimes \mathcal{C}(B,A)\to A$ that satisfies the property that any other partial covering factors uniquely through $\Omega$. It is easy to see that two partial coverings $f,g\in \PCov(B,A)$ are equivalent if and only if the images of their coalgebras $C_f, C_g$ in $\mathcal{C}(B,A)$ coincide. This observation also gives an alternative prove that the equivalence of coverings above is an equivalence relation.
\end{rema}
\begin{exam} Here we identify two partial coverings that are equivalent but not isomorphic. Let $\varphi\colon B\to A$ be a bialgebra homomorphism. Let coalgebras $C, D$ be given by $C=\Bbbk x\oplus \Bbbk y$, $D=\Bbbk z$, where $x,y,z$ are points. Define partial coverings (they are coverings if $\varphi$ is surjective) $f\colon B\otimes D\to A$, $g\colon B\otimes C\to A$ by $f(b\otimes x)=f(b\otimes y)=g(b\otimes z)=\varphi(b)$. Then $f$ factors surjectively through $g$ via the coalgebra map $\mathbf t\colon C\to D$ given by $\mathbf t(x)=\mathbf t(y)=z$.
\end{exam}
Let us call a partial covering $f$ \demph{non-degenerate} if the linear map $f(\bs\cdot,x)\colon B\to A$
is nonzero for every nonzero $x\in C_f$. Observe that if $f$ is non-degenerate and factors surjectively through $h$, then $f$ and $h$ are isomorphic. Hence two non-degenerate coverings $f,g$ are equivalent if and only if they are isomorphic.
\begin{ques}
Is everything partial covering equivalent to a non-degenerate one?
\end{ques}
\subsection{Transfer of structure}
We discuss some useful applications of coverings.
\begin{theo}\label{th: transfer}
Let ${f} \colon B\otimes C \to A$ be a partial covering, with image $A'$. Let $\iota \colon A'\to B\otimes C$ be any linear section of ${f}$, that is, ${f}\circ \iota = \operatorname{id}$.
Then the following hold.
\begin{enumerate}
\item If $p\in B$ is primitive then for any $c\in C$, ${f}(p,c)$ is primitive in $A'$.
\item If $A'$ and $B$ are Hopf algebras, then their antipodes are related by the formula $S_{A'} = {f}\circ (S_B\otimes\operatorname{id})\circ \iota$.
\end{enumerate}
\end{theo}
\begin{proof} The proof of (1) is a simple calculation:
\begin{align*}
\Delta {f}(p\otimes c) &= ({f}\otimes {f})\Delta (p\otimes c) \\
&= \sum_{(c)}{f}(p\otimes c_1)\otimes {f}(1\otimes c_2) + \sum_{(c)}{f}(1\otimes c_1)\otimes {f}(p\otimes c_2) \\
&= \sum_{(c)}{f}(p\otimes c_1)\otimes \varepsilon(c_2) + \sum_{(c)}\varepsilon(c_1)\otimes {f}(p\otimes c_2) \\
&= {f}(p\otimes c)\otimes 1 + 1\otimes {f}(p\otimes c).
\end{align*}
In the proof of (2), we assume without loss of generality that $A'=A$.
We first show that $S_A {f} = {f}(S_B\otimes\operatorname{id})$. We do this by showing that each is the convolution inverse of ${f}$ in $\mathop{\mathrm{Hom}}_k(B\otimes C, A)$.
Suppose ${f}(x)=a$. Then
\begin{align*}
(S_A{f} * {f})(x) &= \sum_{(x)}S_A({f}(x_1)) \,{f}(x_2) = \sum_{({f}(x))} S_A({f}(x)_1) \, {f}(x)_2 = \sum_{(a)} S_A(a_1) \, a_2 \\
&= \varepsilon (a) = \varepsilon(x) \,,
\end{align*}
since ${f}$ is a coalgebra map. Similarly, for $b\otimes c \in B\otimes C$, we have
\begin{align*}
({f}(S_B\otimes\operatorname{id})*{f})(b\otimes c) &= \sum_{(b),(c)} {f}(S_B(b_1)\otimes c_1){f}(b_2\otimes c_2) = \sum_{(b)} {f}(S_B(b_1)b_2\otimes c) \,, \\
&= {f}(\varepsilon(b)\otimes c) = \varepsilon(b) {f}(1\otimes c) = \varepsilon(b)\varepsilon(c) \\
& = \varepsilon(b\otimes c) \,,
\end{align*}
where the last equalities on the first and second line use the fact that ${f}$ measures $B$ to $A$. The proof that $S_A {f}$ and ${f}(S_B\otimes\operatorname{id})$ are both right convolution inverses of ${f}$ is similar.
We conclude that ${f}$ is convolution invertible and that $S_A {f} = {f}(S_B\otimes\operatorname{id})$. This implies that ${f}(S_B\otimes\operatorname{id})\iota = S_A{f}\iota = S_A$, as needed. In particular, ${f}(S_B\otimes\operatorname{id})\iota$ is well-defined and independent of the choice of $\iota$.
\end{proof}
Let ${f}\colon B\otimes C\to A$ be a covering. Let $P(B)$ and $P(A)$ denote the subspaces of primitive elements in $B$ and $A$. One of the important aspects of coverings is that ${f}(P(B)\otimes C)\subseteq P(A)$ (just proven above) and hence also ${f}(\mathcal{U}P(B)\otimes C)\subseteq \mathcal{U}P(A)$ (here $\mathcal{U}P(B)$ stands for the Hopf subalgebra of $B$ generated by $P(B)$; in characteristic $0$ it is isomorphic to the universal envelope of the Lie algebra $P(B)$). In a sense this means that ${f}$ preserves the Lie-theoretic aspects of $B$.
\begin{rema} In contrast to the preceding paragraph, coverings $f\colon B\measures A$ need not respect the group-theoretic aspects of $B$. That is, if $g$ is a grouplike element in $B$, then ${f}(g,\bs\cdot)$ may not even land in the span of the grouplikes of $A$. See Example \ref{ex: comm over noncomm} for one such example.
\end{rema}
We now argue that coverings should be viewed as moving between equations in convolution algebras. Fix a measuring $f \colon B \otimes C \to A$. Then any algebra map $\chi\in \Alg(A,\Bbbk)$ gives rise to an element ${}^{f\!}\chi \in\Alg(B,C^*)$. Indeed, putting $\langle {}^{f\!}\chi(b),c\rangle := (\chi f)(b\otimes c)$, we have
\begin{align*}
\langle {}^{f\!}\chi(bb'),c\rangle
&= (\chi f)(bb'\otimes c) = \chi\sum f(b,c_1)\,f(b',c_2) \\
&= \sum \chi f(b,c_1)\, \chi f(b',c_2)
= \langle {}^{f\!}\chi(b),c_1\rangle \langle {}^{f\!}\chi(b'),c_2\rangle \\
&= \langle {}^{f\!}\chi(b) \cdot {}^{f\!}\chi (b'), c\rangle .
\end{align*}
Now, if $A$ and $B$ are bialgebras then $\Alg(A,\Bbbk)$ and $\Alg(B,C^*)$ are monoids, under the convolution product. So properties of $\chi$ are informed by properties of ${}^{f\!}\chi$, and vice versa.
We illustrate with some explicit examples.
\begin{exam}
Let $B$ be a filtered Hopf algebra. There is a well-known formula for the antipode in the convolution algebra $\operatorname{Hom}(B,B)$ due to Takeuchi. The fact that primitives may be seen as solutions to a similar formula seems to be less appreciated: if $B$ is connected and cocommutative, then the \emph{first Eulerian idempotent} $\mathsf{e}_1 := \log(\operatorname{id} - \varepsilon) \in \operatorname{Hom}(B,B)$ is a projection onto $P(B)$. (See \cite[Ch. 4]{Lod:92} for a proof.)
\end{exam}
\begin{exam}
If ${}^{f\!}\chi$ has order $n$, then the order of $\chi$ divides $n$. Indeed, if $b$ has exponent $n$, then $f(b,c)$ has exponent dividing $n$, for all $c\in C$.
\end{exam}
\section{Cocommutative and Graded Coverings}\label{sec: subcategories}
We investigate subcategories of $\ensuremath \mathbf{PCbialg}$.
\subsection{Cocommutative coverings}\label{sec: cocomm}
We now remark that partial coverings $f\colon B\otimes C \to A$ with $C$ cocommutative are less restrictive than those with $C$ non-cocommutative.
Indeed, by the measuring and coalgebra-map properties, we have
\begin{align}
\notag \left(f^{\otimes2} \Delta\right)\,(bb'\otimes c)
&= f^{\otimes2} \left[ \left(b_1b'_1 \otimes c_1 \right) \otimes \left(b_2b'_2 \otimes c_2\right) \right] \\
\label{eq: Delta-f}
&= f(b_1,c_1)\,f(b'_1,c_2) \otimes f(b_2,c_3)\,f(b'_2,c_4) ,
\intertext{while}
\label{eq: f-Delta}
\left(\Delta f\right)(bb'\otimes c)
&= f(b_1,c_1)\,f(b'_1,c_3) \otimes f(b_2,c_2)\,f(b'_2,c_4) .
\end{align}
Note the twist $(c_2,c_3) \leadsto (c_3,c_2)$ above, so that $C$ behaves under $f$ somewhat as if it were cocommutative. If $B$ is Hopf, then it behaves precisely so, as we now show.
\begin{lemm}\label{lem: Ccoc}
If $f\colon B\otimes C\to A$ is a partial covering with $B$ Hopf, then for all $b,b'\in B$ and $c\in C$ we have that
$$
f(b',c_2)\otimes f(b, c_1)= f(b', c_1)\otimes f(b,c_2).
$$
\end{lemm}
\begin{proof} In the equality between \eqref{eq: Delta-f} and \eqref{eq: f-Delta}, the antipode offers a means to cancel the edges. Using this equality between the fourth and fifth line below, we compute:
\begin{eqnarray*}
f(b',c_1)\otimes f(b,c_2) &=& \varepsilon(b_1\otimes c_1)f(b'_1, c_2)\otimes f(b_2,c_3)\varepsilon(b'_2\otimes c_4) \\
&=& f(\varepsilon(b_1), c_1)f(b'_1,c_2)\otimes f(b_2,c_3)f(\varepsilon(b'_2), c_4) \\
&=& f(S(b_1)b_2, c_1)f(b'_1,c_2)\otimes f(b_3,c_3)f(b'_2S(b'_3), c_4)\\
&=& f(S(b_1), c_1)f(b_2, c_2)f(b'_1,c_3)\otimes f(b_1,c_4)f(b'_2, c_5)f(S(b'_3), c_6)\\
&=& f(S(b_1), c_1)f(b_2, c_2)f(b'_1,c_4)\otimes f(b_3,c_3)f(b'_2, c_5)f(S(b'_3), c_6)\\
&=& f(S(b_1)b_2, c_1)f(b'_1,c_3)\otimes f(b_3,c_2)f(b'_2S(b'_3), c_4)\\
&=& \varepsilon(b_1)\varepsilon(c_1)f(b'_1,c_3)\otimes f(b_2,c_2)\varepsilon(b'_2)\varepsilon(c_4)\\
&=& f(b', c_2)\otimes f(b,c_1).
\end{eqnarray*}
\end{proof}
\begin{prop}\label{th: cocomm}
If $B \pmeasures A$ is a partial covering with $B$ Hopf, then the image of any cocommutative element of $B$ is cocommutative in $A$.
\end{prop}
\begin{proof}
Applying Lemma \ref{lem: Ccoc}, we may write $\Delta{f}(b,c)$ as
\begin{eqnarray*}
{f}(b,c)_1\otimes {f}(b,c)_2 &=& {f}(b_1,c_1)\otimes {f}(b_2,c_2) = f(b_1,c_2)\otimes f(b_2,c_1)\\
&=& {f}(b_2,c_2)\otimes {f}(b_1,c_1) = {f}(b,c)_2\otimes {f}(b,c)_1 ,
\end{eqnarray*}
which completes the proof.
\end{proof}
Below we provide an example showing that the Hopf condition in Proposition \ref{th: cocomm} (and in Lemma \ref{lem: Ccoc}) is not vacuous.
That is, there are interesting coverings by cocommutative $B$'s and non-cocommutative $C$'s. It therefore follows (from Example \ref{ex: cocomm-cover}) that the universal covering coalgebra need not always be cocommutative. Whether this statement holds when $B$ is Hopf is an open question.
\begin{ques} \label{q:cocommutative}
Given any partial covering ${f}\colon B p\measures A$, with $B$ a Hopf algebra, does it factor through a partial covering ${f}'\colon B\otimes C'\to A$ with $C'$ cocommutative? Equivalently, is the universal covering coalgebra $\mathcal{C}(B,A)$ cocommutative whenever $B$ is Hopf?
\end{ques}
Along the same lines, we may also ask the following.
\begin{ques}
Given any partial covering ${f}\colon B\pmeasures A$, does there exist a cocommutative $C'$ and partial covering ${f}'\colon B\otimes C'\to A$ with the same range as ${f}$? What if we additionally assume that
$B$ is a Hopf algebra?
\end{ques}
\begin{exam}\label{ex: cocomm-cover}
Let $\{a,b\}$ be the semigroup with $xy = y$ for all $x,y\in\{a,b\}$, and let $M=\{e,a,b\}$ be the corresponding (nonabelian and non-cancelable) monoid. Put $A = (\Bbbk M)^*$ and $C=\coalg(A)$. Writing $\{x_i\}_{i=e,a,b}$ for the dual basis in $A$ and putting $\rho:=x_e + x_a + x_b$, one checks that
\[
x_i \cdot x_j = \delta_{ij} x_j \ \ (\rho=1_A),
\quad \Delta(x_e) = x_e\otimes x_e ,
\quad\hbox{and}\quad \Delta(x_i) = 1\otimes x_i + x_i \otimes x_e \ (i\neq e).
\]
We construct a covering $f\colon B\otimes C \twoheadrightarrow A$.
Give $B:=\Bbbk[z]$ the bialgebra structure with $z$ a point, where by \demph{point} we mean a grouplike element. (We adopt this name to avoid the connotation of invertibility.)
Define $f$ by sending $(z^n,c)$ to the Sweedler power $\operatorname{id}^{*n}(c)$ for $n\geq0$ and extending bilinearly. Then $f(z,c) = c$, meaning $f$ is onto. Note that $f$ is a measuring by fiat. We check that $f$ is a coalgebra map. In the present context, \eqref{eq: Delta-f} and \eqref{eq: f-Delta} requires the equality
\begin{gather}\label{eq: ncc-example}
(c_1c_3c_5\cdots) \otimes (c_2c_4c_{6}\cdots) = (c_1c_2c_3\cdots) \otimes (c_{n+1}c_{n+2}c_{n+3}\cdots) .
\end{gather}
This clearly holds for $c=x_e$, so we turn to $x_i$ for $i\neq e$.
The general formula for iterated coproducts is
\[
\Delta^n(x_i) = 1^{\otimes n}\otimes x_i + 1^{\otimes n-1}\otimes x_i\otimes x_e + 1^{\otimes n-2}\otimes x_i\otimes x_e^{\ot2} +
\cdots x_i \otimes x_e^{\otimes n}.
\]
Hence the only products of $n$ factors (with or without interleaving the odd and even terms) that survive in \eqref{eq: ncc-example} are $1\otimes x_i$ and $x_i\otimes x_e$, and these each occur exactly once on either side.
\end{exam}
\subsection{Graded coverings}\label{sec: combinat}
If $A,B$ are graded (bi)algebras, then a measuring $f\colon B\otimes C \to A$ is called \demph{graded} if $f(b\otimes c)_n = f(b_n \otimes c)$. We say that $f$ is \demph{locally finite} if for each fixed $c\in C$, $f(b,c) = 0$ for all $b$ of sufficiently large degree.
Finally, suppose $C$ is also graded. Then we call $f$ \demph{bigraded} if $f(b \otimes c)_n = f(b_n \otimes c_n)$.
\begin{prop}\label{th: locally finite}
Any graded partial covering $f\colon B \otimes C \to A$ of graded bialgebras may be replaced by a locally finite one.
\end{prop}
\begin{proof}
Let $\mathcal N$ denote the coalgebra $\Bbbk\{H_0,H_1,H_2,\ldots\}$ defined by $\Delta(H_k) = \sum_{i+j=k} H_i \otimes H_j$. Put $\bar C = C \otimes \mathcal N$, and $\bar f(b, c \otimes H_n) = f(b, c)_n$, the $n$th homogeneous component of $f(b,c)$. Show that this is an measuring and a coalgebra map.
\end{proof}
\begin{xam}[\ref{ex: A.id over A}]
Suppose $A$ is a graded bialgebra. Applying the construction $(f,\Bbbk) \mapsto (\bar f,\Bbbk\otimes\mathcal N)$ in the proof above, we see that $\bar f$ not only becomes locally finite, but it becomes degree-preserving as well. Thus every graded bialgebra has a degree-preserving measuring that is a covering.
\end{xam}
If $U,V, W$ are graded vector spaces, then we define the \demph{diagonal tensor product} by $U\otimes_d V := U\otimes V/(\bigoplus_{n\not=m} U_n\otimes V_m)$. The space $U\otimes_d V$ is graded by $(u\otimes_d v)_n = u_n\otimes v_n$.
Note that bigraded maps from $U\otimes V\to W$ are in bijective correspondence with graded (that is, homogeneous of degree $0$) maps from $U\otimes_d V\to W$.
\begin{lemm}
If $C,D$ are graded coalgebras, then $C\otimes_d D$ is a graded coalgebra with coproduct given by $\Delta(u\otimes_d v)=
(u_1\otimes_d v_1)\otimes (u_2\otimes_d v_2)$ and counit given by $\varepsilon(u\otimes_d v) = \varepsilon(u)\varepsilon(v)$.
\end{lemm}
\begin{proof} Observe that $\bigoplus_{n\not=m} C_n\otimes D_m$ is a coideal in $C\otimes D$.
\end{proof}
The bicategory of graded connected bialgebras with bigraded coverings $\ensuremath \mathbf{Combinat}$ is now defined in analogy to the category of bialgebras with coverings: we replace tensors by diagonal tensors and coverings by bigraded locally-finite coverings.
\begin{rema} All bicategories described above also have monoidal structures given by the tensor products of underlying vector spaces.
\end{rema}
\subsection{Universal objects in $\ensuremath \mathbf{Combinat}$}\label{sec: universal}
Any cocommutative bialgebra $A$ in $\ensuremath \mathbf{Combinat}$ has a canonical covering by $\mathrm{NSym}$, defined as follows. For all $a\in A$, put $\can(H_n,a) = a_n$ (the homogeneous component of $a$ of degree $n$), then extend multiplicatively and bilinearly to make $\can$ a measuring. Checking that $\can$ is a coalgebra map is straightforward. Note that this covering is locally finite by construction.
\begin{prop}\label{th: unique factorization}
Given a covering $f\colon \mathrm{NSym} \otimes C \to A$, there is a coalgebra map $ \bar f \colon C \to \coalg(A)$ factoring $f$ through $\can$ if and only if $C$ is a locally finite covering.
\end{prop}
\begin{proof}
Put $\bar f(c) = \sum_{n\geq0} f(H_n,c)$.
\end{proof}
\begin{exam}\label{ex: omp over nsym}
Let $\mathrm{OMP}$ (for \demph{ordered multiset partitions}\footnote{\,The reader is cautioned to interpret \emph{ordered multiset partition} as an abbreviation for ``ordered partition of a multiset into sets.'' While the blocks of a multiset partition may have repeated entries, in general, recent combinatorics literature, {\it e.g.}, \cite{HRW:18}, excludes these cases from consideration.}) denote the free algebra on finite nonempty subsets $K\subseteq \mathbb{N}$. It inherits the structure of Hopf algebra if the coproduct on generators $K$ is defined by $\Delta(K) = \sum_{I\sqcup J = K} I \otimes J$.
Identify $\emptyset$ with the unit in $\mathrm{OMP}$. We show that $\mathrm{OMP}$ covers $\mathrm{NSym}$, and hence is another weakly initial object in the subcategory of cocommutative objects in $\ensuremath \mathbf{Combinat}$.
We take $C$ to be the coalgebra $\mathcal N=\Bbbk\{H_0,H_1,H_2,\ldots\}$ from the proof of Proposition \ref{th: locally finite}. The map $f \colon \mathrm{OMP}\otimes C \to \mathrm{NSym}$ given on generators by $f(K,H_k) = \delta_{|K|,k} H_k$ induces a measuring, where $|K|$ is the cardinality of the set $K$ and $H_0$ is identified with the unit of $\mathrm{NSym}$.
We check that $f$ is a coalgebra map:
\begin{align*}
\left(f^{\otimes2} \Delta\right)\,(K\otimes H_k) &= f^{\otimes2} \sum_{\substack{I+J=K\\ i+j=k}} \left(I\otimes H_i \right) \otimes \left( J \otimes H_j\right) \\
&= \sum_{\substack{I\sqcup J = K\\i+j=k}} \delta_{|I|,i} \, H_i \otimes \delta_{|J|,j} \, H_j \\
&= \delta_{|K|,k} \sum_{i+j=k} H_i \otimes H_j \
= \left(\Delta f\right) (K \otimes H_k) .
\end{align*}
\end{exam}
We have now found two weakly initial objects for the cocommutative part of $\ensuremath \mathbf{Combinat}$.
\begin{ques} Can we drop ``weakly'' in this bicategory:
Is there a sensible equivalence relation on $\ensuremath \mathbf{Combinat}$ making $\mathrm{NSym}$ and $\mathrm{OMP}$ equivalent?
\end{ques}
\begin{ques} Are there non-cocommutative or non-graded analogs of $\mathrm{NSym}$ or $\mathrm{OMP}$?
\end{ques}
\section{A Nichols Theorem for Hopf coverings}\label{sec: Hopf}
Let $B$ be a Hopf algebra. A result of Nichols \cite{Nic:78} gives some conditions under which a bialgebra quotient $B/I$ is in fact Hopf. For example, it is Hopf if $B/I$ is commutative or if $B$ is cocommutative or pointed. We now present two analogous results for coverings. (In light of Example \ref{ex: morphism family}, these are in fact generalizations of his result.)
\begin{theo}\label{th: fin-dim-cover}
If a Hopf algebra $B$ covers a finite dimensional bialgebra $A$, then $A$ is also a Hopf algebra.
\end{theo}
\begin{proof}
We consider $A$ as a (faithfully flat, left) $A$-comodule algebra, and study the Galois map $\beta\colon A\otimes A\to A\otimes A$ given by
\[
\beta(a,a')=a_1\otimes a_2a'.
\]
A result of Schauenburg \cite{Sch:97} provides that if $\beta$ is bijective, then $A$ is Hopf.
We prove that $\beta$ is surjective and therefore bijective.
Let ${f}\colon B\otimes C\to A$ be the covering
and let $\gamma\colon B\otimes C\otimes A\to B\otimes C\otimes A$ be the linear map given by $$\gamma(b,c,a)=b_1\otimes c_1\otimes {f}(b_2,c_2)a.$$ Note that $\gamma'\colon B\otimes C\otimes A\to B\otimes C\otimes A$ given by $$\gamma'(b,c,a)=b_1\otimes c_1\otimes {f}(S(b_2),c_2)a$$ is a right composition inverse of $\gamma$, so $\gamma$ is surjective. Indeed
\begin{eqnarray*}(\gamma\circ \gamma')(b\otimes c\otimes a)&=& \gamma(b_1\otimes c_1\otimes {f}(S(b_2),c_2)a) \\
&=& b_1\otimes c_1\otimes {f}(b_2,c_2){f}(S(b_3),c_3)a =
b_1\otimes c_1\otimes {f}(b_2S(b_3), c_2)a \\
&=& b_1\otimes c_1 \otimes {f}(\varepsilon(b_2),c_2)a = b_1\otimes c_1\otimes \varepsilon(b_2)\varepsilon(c_2)a = b\otimes c\otimes a.
\end{eqnarray*}
Now note that $\beta({f}\otimes\operatorname{id}_A) = ({f}\otimes\operatorname{id}_A)\gamma$:
\begin{eqnarray*}
\beta({f}\otimes\operatorname{id}_A)(b,c,a) &=& \beta({f}(b,c)\otimes a) \\
&=&{f}(b,c)_1\otimes {f}(b,c)_2 a= {f}(b_1,c_1)\otimes {f}(b_2,c_2) a\\
&=& ({f}\otimes\operatorname{id}_A)(b_1\otimes c_1\otimes{f}(b_2,c_2)a) = ({f}\otimes\operatorname{id}_A)\gamma(b,c,a).
\end{eqnarray*}
So since $({f}\otimes\operatorname{id}_A)\gamma$ is surjective, so must be $\beta$.
\end{proof}
We now consider the case when $A$ is pointed.
\begin{lemm}\label{lem: p}
If $f\colon B\pmeasures A$ is a partial cover over an algebraically closed field, and $B$ is a pointed Hopf algebra, then its image is pointed in $A$.
\end{lemm}
\begin{proof}
Any simple subcoalgebra of $A$ is the image of $B'\otimes C'$ for some simple subcoalgebras $B',C'$. Hence $B' = \Bbbk\{z\}$ for some {grouplike} $z\in B$. Using Proposition \ref{th: cocomm}, we see that $f(B'\otimes C')$ is cocommutative. Hence $f(B\otimes C)$ is pointed.
\end{proof}
\begin{lemm}\label{lem: phi-exists}
Suppose $f\colon B\otimes C \to A$ is a partial covering and $z$ is an invertible point in $B$. Then the map $\varphi=f(z,\bs\cdot)\colon C\to A$ is convolution coalgebra map whose convolution inverse $\overline\varphi = f(z^{-1},\bs\cdot)$ is also a coalgebra map.
\end{lemm}
\begin{proof}
A simple check
\begin{align*}
(\overline\varphi\ast\varphi)(c) &= \overline\varphi(c_1)\varphi(c_2) \\
&=f(z^{-1},c_1)f(z,c_2) = f(z^{-1}z,c) = f(1,c) = \varepsilon(c).
\end{align*}
Similarly for $\varphi\ast\overline\varphi=\varepsilon$.
\end{proof}
\begin{lemm}\label{lem: phi-is-nice}
Assume that $\Bbbk$ is algebraically closed, $C$ a simple coalgebra and $A$ a pointed bialgebra. Assume that $\varphi\colon C \to A$ is a convolution invertible coalgebra map (in $\operatorname{Hom}(C,A)$) such that its convolution inverse $\overline\varphi$ is a coalgebra map. Then every point in the range of $\phi$ is invertible.
\end{lemm}
\begin{proof}
First note that for all $c\in C$ we have $\varphi(c_2)\overline\varphi(c_1)=\varepsilon(c)=\overline\varphi(c_2)\varphi(c_1)$. The later equality is proven as follows:
\begin{eqnarray*}
\varepsilon(c)&=& \overline\varphi(c_1)\varepsilon(c_2)\varphi(c_3) = \overline\varphi(c_1)(\operatorname{m}\Delta(\varepsilon(c_2)))\varphi(c_3)\\
&=&\overline\varphi(c_1)(\operatorname{m}\Delta \varphi(c_2)\overline\varphi(c_3))\varphi(c_4)\\
&=&\overline\varphi(c_1)(\varphi(c_2)\overline\varphi(c_4)\varphi(c_3)\overline\varphi(c_5))\varphi(c_6)\\
&=&(\overline\varphi(c_1)\varphi(c_2))\overline\varphi(c_4)\varphi(c_3)(\overline\varphi(c_5))\varphi(c_6))\\
&=&\varepsilon(c_1)\overline\varphi(c_3)\varphi(c_2)\varepsilon(c_4) = \overline\varphi(c_2)\varphi(c_1);
\end{eqnarray*}
the proof of the former works similarly.
Since $C$ is simple and $\Bbbk$ is algebraically closed, we have that $C$ is isomorphic to a matrix coalgebra $\Bbbk\{e^{ij}\}_{1\leq i,j\leq n} = \mathcal{M}_n(\Bbbk)^*$. Let $A'=\varphi(C)^*$. We identify $A'$ with a subalgebra of $C^*=\mathcal{M}_n(\Bbbk)$ (via $\varphi^*\colon A'\to C^*=\mathcal{M}_n(\Bbbk)$). Let $A'=S\oplus N$ be the Wederburn-Malcev decomposition of $A'$ into its semisimple part $S$ and the radical $N$. Since $A$ and therefore $\varphi(C)$ is pointed (and hence $A'$ is a basic algebra) we have that $S$ is isomorphic to a direct product of copies of the ground field. We additionally assume (using a simultaneous similarity if necessary), that $S$ is a subset of the set of diagonal matrices in $\mathcal{M}_n(\Bbbk)$, and $N$ is a subset of the set of all strictly-upper triangular matrices in $\mathcal{M}_n(\Bbbk)$
We will now show that every $\varphi(e^{ii})$ is invertible, with inverse $\overline\varphi(e^{ii})$, from whence the result follows (the above discussion shows that every point in $\varphi(C)$ is in the span of $\{\varphi (e^{ii}): i=1\,\ldots, n\}$. Since the set of all distinct points is linearly independent, we have that every point if $\varphi(C)$ is equal to some $\varphi(e^{ii})$). Since $A'$ is upper triangular we also note that for all $1\le i<j\le n$ we have that $\varphi(e^{ij})=0$. We will prove by induction on $i$ that for every $i=1,\ldots, n$, we have $\varphi(e^{ii})\overline\varphi(e^{ii})=1=\overline\varphi(e^{ii})\varphi(e^{ii})$ and for all $j>i$, $\overline\varphi(e^{ji})=0$.
\smallskip\noindent
\textbf{Base case.} Writing $\operatorname{m}$ for multiplication in $A$, we first compute
\begin{align*} 1 & = \varepsilon(e^{11}) = \operatorname{m}(\overline\varphi\otimes\varphi)\Delta(e^{11})
= \sum_{k=1}^n\overline\varphi(e^{1k})\varphi(e^{k1}) = \overline\varphi(e^{11})\varphi(e^{11}),
\intertext{and}
1& = \varepsilon(e^{11}) = \operatorname{m}(\varphi\otimes\overline\varphi)\Delta^{cop}(e^{11})
= \sum_{k=1}^n\varphi(e^{k1})\overline\varphi(e^{1k}) = \varphi(e^{11})\overline\varphi(e^{11}).
\intertext{For $j>1$, we further have}
0 & = \varepsilon(e^{j1}) = \operatorname{m}(\overline\varphi\otimes\varphi)\Delta(e^{j1})
= \sum_{i=1}^n \overline\varphi(e^{ji})\varphi(e^{i1}) = \overline\varphi(e^{j1})\varphi(e^{11}).
\end{align*}
Since $\varphi(e^{11})$ is invertible we conclude that $\overline\varphi(e^{j1})=0$.
\smallskip\noindent
\textbf{Induction step.} Assume that for some $m$, $2\le m\le n$ we have that every $i<m$
satisfies $\varphi(e^{ii})\overline\varphi(e^{ii})=1=\overline\varphi(e^{ii})\varphi(e^{ii})$ and for all $j>i$, $\overline\varphi(e^{ji})=0$. Then
\begin{align*} 1& = \varepsilon(e^{mm})=\operatorname{m}(\overline\varphi\otimes\varphi)\Delta(e^{mm})
=\sum_{k=1}^n\overline\varphi(e^{mk})\varphi(e^{km})\\
& = \sum_{k<m} \overline\varphi(e^{mk})\varphi(e^{km})+\overline\varphi(e^{mm})\varphi(e^{mm})+\sum_{k>m} \overline\varphi(e^{mk})\varphi(e^{km})\\
& = 0 + \overline\varphi(e^{mm})\varphi(e^{mm}) +0
\intertext{and}
1 & = \varepsilon(e^{mm})=\operatorname{m}(\varphi\otimes\overline\varphi)\Delta^{cop}(e^{mm})
=\sum_{k=1}^n\varphi(e^{km})\overline\varphi(e^{mk})\\
& = \sum_{k<m} \varphi(e^{km})\overline\varphi(e^{mk})+\overline\varphi(e^{mm})\varphi(e^{mm})+\sum_{k>m} \varphi(e^{km})\overline\varphi(e^{mk})\\
& = 0 + \varphi(e^{mm})\overline\varphi(e^{mm}) +0;
\intertext{and for $j>m$, we have}
0 & = \varepsilon(e^{jm}) = \operatorname{m}(\overline\varphi\otimes\varphi)\Delta(e^{jm}) \\
& = \sum_{i=1}^n \overline\varphi(e^{ji})\varphi(e^{im}) = \overline\varphi(e^{j1})\varphi(e^{11})\\
& = \sum_{i<m} \overline\varphi(e^{ji})\varphi(e^{im}) + \overline\varphi(e^{jm})\varphi(e^{mm}) +
\sum_{i>m} \overline\varphi(e^{ji})\varphi(e^{im}) \\
& = 0+\overline\varphi(e^{jm})\varphi(e^{mm})+0.
\end{align*}
Therefore, since $\varphi(e^{mm})$ is invertible, we get $\overline\varphi(e^{jm})=0$.
\end{proof}
\begin{theo}\label{th: pointed-cover}
Let $f\colon B\measures A$ be a bialgebra covering. If $B$ is a Hopf algebra with cocommutative coradical, then $A$ is a Hopf algebra as well.
\end{theo}
\begin{proof}
With no loss of generality we can and do assume that $\Bbbk$ is algebraically closed. We use Takeuchi's condition: an antipode exists on $A$ if it may be defined on the coradical of $A$.
Looking at that coradical, we use Proposition \ref{th: cocomm} to conclude that $A$ is pointed. Note that every point in $A$ is in the image, under the covering, of some
$\Bbbk\{z\}\otimes C\cong C$, where $z$ is a point in $B$ and $C$ is a simple sub-coalgebra of $C_f$. Hence by Lemmas \ref{lem: phi-exists} and \ref{lem: phi-is-nice} we have that every point in $A$ is invertible.
\end{proof}
|
{
"timestamp": "2018-09-14T02:11:56",
"yymm": "1803",
"arxiv_id": "1803.02691",
"language": "en",
"url": "https://arxiv.org/abs/1803.02691"
}
|
\section{Introduction}\label{sec:intro}
It is commonly acknowledged that domain ontologies are significant sources of semantic information \cite{Perez2004}. Hence, new ontologies are
being constructed for many different domains and they are effectively utilized within relevant applications in these domains. To name a few, in
\cite{biotop}, a domain ontology for molecular biology is proposed. In order to perform public health surveillance, an ontology for this domain
is created and described in \cite{Kucuk2017}. A domain ontology is presented in \cite{construction} for the processes in infrastructure and
construction. In \cite{materials}, an ontology for the domain of materials science and engineering is described. A domain ontology for chemical
process engineering is proposed in \cite{ontocape}. Regarding software systems, a Web service modeling ontology to describe all aspects of Web
services is presented in \cite{webont}, an ontology for scientific software metadata is proposed in \cite{gil2015ontosoft}, and an ontology to
model the variability in software product modeling is presented in \cite{kumbang}.
The domain of energy is similarly a fruitful domain for semantic applications. For different applications in the energy domain, the conceptual
modeling of the relevant energy subdomain is required and this phase will extensively benefit from domain ontologies. Similarly, ontologies will
also alleviate the interoperability problems between different applications. Yet, there are few examples of domain ontologies regarding this
domain and its subdomains. For instance, in \cite{kucuk2010pqont}, a domain ontology to model the electrical power quality parameters and events
is described. In \cite{kucuk2014wont}, a domain ontology for wind energy which was constructed through a semi-automated procedure is presented. A
high-level covering ontology for the domain of electrical energy is presented in \cite{kuccuk2015feeont} where this ontology was also aligned
with the aforementioned electrical power quality and wind energy ontologies. An ontology for energy efficiency in smart grid neighborhoods is
presented in \cite{enersip}, and finally an ontology matching system for the smart grid domain is described in \cite{smartgrid} which aims to
reduce the related interoperability issues.
In this paper, we present an improved and extended version of the wind energy ontology proposed in \cite{kucuk2014wont}, in order to increase its
understandability, coverage, and usability. We first reorganize the existing ontology hierarchy, add new concepts, attributes, and instances, in
order to arrive at a more useful and extended wind energy ontology. We make the ontology publicly-available and utilize it within a Web-based
semantic portal application for wind energy. The rest of the paper is organized as follows: In Section \ref{sec:ont}, the extended and improved
wind energy ontology is described. Section \ref{sec:system} presents the semantic portal application in which the ultimate ontology is used and
finally, Section \ref{sec:conc} concludes the paper with a summary and pointers to future work.
\section{Extended and Improved Wind Energy Ontology}\label{sec:ont}
\subsection{Initial Version of the Wind Energy Ontology}
The initial wind energy ontology has been created through a semi-automatic procedure over Wikipedia articles related to wind energy
\cite{kucuk2014wont}. In the first phase of this procedure, related Wikipedia articles are automatically processed to determine the
high-frequency ngrams from the articles, and in the second phase, the resulting ngrams are manually organized into a wind energy ontology. The
schematic representation of the concepts of this ultimate ontology with their interrelationships is given in Figure \ref{fig:wont1} as excerpted
from \cite{kucuk2014wont}. The ontology has also been publicly shared for research purposes at
\url{http://www.ceng.metu.edu.tr/~e120329/wont.owl} as a Web Ontology Language (OWL) file.
In order to include it within the larger electrical energy ontology with weighted attributes \cite{kuccuk2015feeont}, the ontology has been
extended to include weighted attributes and this extended version is again publicly shared at
\url{http://www.ceng.metu.edu.tr/~e120329/FWONT.owl}. This form of the ontology is henceforth referred to as WONT due to the name of its initial
OWL file and the electrical energy ontology \cite{kuccuk2015feeont} is henceforth referred to as FEEONT, based on the name of its
publicly-available OWL file at \url{http://www.ceng.metu.edu.tr/~e120329/FEEONT.owl}.
\subsection{Improved and Extended Wind Energy Ontology}
In the current study, we have improved and extended WONT in order to make it more useful for applications related to wind energy. The final
improved and extended wind energy ontology is henceforth referred to as OntoWind (ONTOlogy for WIND energy).
\begin{figure}
\center \scalebox{0.55}
{\includegraphics[angle=90]{wont_1.eps}}\caption{The Initial Version of the Domain Ontology for Wind Energy \cite{kucuk2014wont}, i.e.,
WONT.}\label{fig:wont1}
\end{figure}
All of the extensions and improvement efforts are carried out using the Prot\'eg\'e ontology editor \cite{protege} and the commonly-employed
ontology development methodology described in \cite{noy2001ontology} is roughly followed. In the following subsections, the main procedures
employed to construct OntoWind based on WONT are described.
\begin{figure}
\center \scalebox{0.57}
{\includegraphics{wont2_genel_v3.eps}}\caption{The Most General Concepts in OntoWind.}\label{fig:wont2_genel}
\end{figure}
\subsubsection{Restructuring}
The main concepts in WONT are restructured to better separate the different concept groups. In WONT, the concept hierarchy tree was a rather
short one, as the concept-subconcept (or, class-subclass) relations were not exhaustively specified. By taking a top-down approach, we first
determine the top general concepts of the wind energy domain and the resulting four general concepts in OntoWind are illustrated in Figure
\ref{fig:wont2_genel}. Next, the existing concepts in WONT are re-modeled by putting them under their relevant general concepts. Along the way,
we exclude some concepts like the \emph{PowerQuality} which was included in WONT to model the power quality characteristics of wind energy, as
OntoWind can be integrated with external power quality domain ontologies like PQONT \cite{kucuk2010pqont} to model these characteristics.
The ultimate concept hierarchies grouped under the four general concepts of OntoWind are provided schematically in Figure
\ref{fig:wont2_ayrinti}.
\begin{figure}
\center \scalebox{0.63}
{\includegraphics{wont2_ayrinti_v1.eps}}\caption{The Concept Hierarchies under the General Concepts in OntoWind.}\label{fig:wont2_ayrinti}
\end{figure}
The top general concepts of OntoWind are described below:
\begin{itemize}
\item \emph{WindRelatedData}: This concept and its subconcepts given as a tree structure in Figure \ref{fig:wont2_ayrinti} are used to model
the measurement and forecast data related to wind energy. For instance, meteorological forecasts obtained by running the numerical
weather prediction (NWP) models as well as meteorological measurement outputs through the related sensors are all represented by the
corresponding meteorological data concepts included in the ontology. Namely, these concepts include \emph{WindSpeed},
\emph{WindDirection}, \emph{Temperature}, \emph{Humidity}, in addition to others. The main attributes of these concepts are \emph{value}
corresponding to the actual measurement\slash forecast data and \emph{date} corresponding to the actual time that the data value belongs
to.
\item \emph{WindRelatedModel}: This concept represents the wind-related NWP and wind power forecast models. Hence, it has the corresponding
concepts of \emph{NumericalWeatherPrediction} and \emph{WindPowerForecastModel} for these models. These subconcepts, in turn, has the
related commonly-used NWP models such as ALADIN, IFS, and WRF, and wind power forecast models such as ANFIS, ANN, and SVM as their
subconcepts\footnote{The following are the open forms of the abbreviations used in the concept names:\\ALADIN: Aire Limit\'ee Adaptation
dynamique D\'eveloppement InterNational\\IFS: Integrated Forecast System\\WRF: Weather Research and Forecasting\\ANFIS: Adaptive
Neuro-Fuzzy Inference System\\ANN: Artificial Neural Network\\SVM: Support Vector Machine}.
\item \emph{WindRelatedStructuralComponent}: This concept represents the physical components within a wind power plant (WPP). Hence, it has
subconcepts such as \emph{WindPowerPlant}, \emph{WindTurbine}, and \emph{Sensor} as its subconcepts. The concepts in this category have
the necessary object attributes to represent the details of the corresponding structural components. For instance, \emph{WindPowerPlant}
has the related attributes to model the characteristics of WPPs such as its geo-referenced location information, installed capacity,
number of turbines, etc.
\item \emph{WindRelatedOrganization}: This concept represents the international and national organizations, and commercial companies related
to wind energy. Therefore, it has subconcepts such as \emph{InternationalOrganization} and \emph{NationalOrganization} to model those
organizations with finer granularity. Common organizations such as research centers, national weather services, and electricity
transmission system operators are all modeled with the concepts under this category.
\end{itemize}
\subsubsection{Extensions}
Apart from the general restructuring efforts to arrive at OntoWind, it also includes several extensions over WONT.
First of all, new relevant domain concepts are added to the ontology to increase its coverage. For instance, in order to model the national and
international organizations related to wind energy, several concepts are added under the generic concept of \emph{WindRelatedOrganization}, as
previously described. Similarly, several common models for numerical weather prediction and wind power forecasting are included as subconcepts
under the generic concept of \emph{WindRelatedModel}.
Secondly, several annotation attributes are added to the ontology to make it applicable in different semantic applications. For instance, we have
added the textual attributes of \emph{webAddress} and \emph{twitterAccount} especially for the prospective instances of the subconcepts of
\emph{WindRelatedOrganization}. The former attribute will be used to model the Web site of the wind-related energy organization, while the latter
is used to hold the address of its official Twitter account, if any. Similarly, another attribute named \emph{country} is added to hold the
country codes of the national wind-related organizations.
Finally, plausible instances (also called objects or individuals) are included in OntoWind to again make it a useful resource for Semantic Web
applications. A total of 25 instances, belonging to the subconcepts modeled under \emph{WindRelatedOrganization}, are added to OntoWind. The
numbers of instances per their immediate concept types are given in Figure \ref{fig:instances-by-type}.
\begin{figure}
\center \scalebox{0.71}
{\includegraphics{individuals_by_type.eps}}\caption{Number of Instances per Concept Names in OntoWind.}\label{fig:instances-by-type}
\end{figure}
The names of the instances and the attributes of one of them, namely, \emph{MGM} denoting \emph{Turkish State Meteorological Service}, are
illustrated in Figure \ref{fig:instances}. MGM is an instance of the \emph{NationalWeatherService} concept which is a subconcept of
\emph{GovernmentalEnergyOrganization} and it, in turn, is a subconcept of \emph{NationalOrganization} subconcept of
\emph{WindRelatedOrganization}. The readers are referred to the concept hierarchies provided in Figure \ref{fig:wont2_ayrinti} for a schematic
view.
\begin{figure}
\center \scalebox{0.71}
{\includegraphics{wont2-individuals_mgm.eps}}\caption{Instances of the Subconcepts under \emph{WindRelatedOrganization} in
OntoWind.}\label{fig:instances}
\end{figure}
Figure \ref{fig:instances} also demonstrates the use of the aforementioned new attributes of OntoWind, namely, \emph{country}, \emph{webAddress},
and \emph{twitterAccount}. The remaining ones are the already existing attributes from WONT \cite{kucuk2014wont} and FEEONT
\cite{kuccuk2015feeont}, and they are explained below again, for the purposes of completeness:
\begin{itemize}
\item \emph{label}: This attribute holds the exact concept name.
\item \emph{labelEN}: This attribute holds the most common English phrase referring to the concept.
\item \emph{membershipValueLabel}: This attribute holds the degree of membership of the value in \emph{labelEN} to the domain of wind
energy. The degree of membership is a real number between [0,1]. This attribute and similar attributes have previously been utilized to
denote the weights of the corresponding attributes (like \emph{labelEN}) within FEEONT \cite{kuccuk2015feeont}. In
\cite{kuccuk2015feeont}, Wikipedia's disambiguation pages have been used to determine the weights. In this study, we manually determine
these weights based on expert judgement instead of the order in the Wikipedia's disambiguation pages. In \cite{kuccuk2015feeont}, if
the sense related to wind energy is in the $n^{th}$ place among the entries in the disambiguation page corresponding to the value of
\emph{labelEN}, then the value of \emph{membershipValueLabel} is set to $1/n$. This is the \emph{reciprocal rank} metric used in
information retrieval and question answering literature \cite{Voorhees1999}. As previously mentioned, we have used expert judgement to
manually determine the value of \emph{membershipValueLabel} as a real number between [0,1]. The values of all attributes beginning with
\emph{membershipValue} are similarly determined. Also, the existing values in WONT for such attributes are revised in case a change of
value is needed.
\item \emph{labelTR}: This attribute holds the most common Turkish phrase referring to the concept. As it has been pointed out in related
work like \cite{kucuk2010pqont} and \cite{kuccuk2015feeont}, attributes like \emph{labelTR} facilitate the use of the ontology in
multilingual settings. For instance, to extend the ontology to other languages like Spanish or French, attributes like \emph{labelES}
and \emph{labelFR} can be added to the ontology with the corresponding values in these languages.
\item \emph{membershipValueLabelTR}: This attribute holds the degree of membership of the value in \emph{labelTR} to the domain of wind
energy.
\item \emph{synonymSet}: This attribute holds the list of English synonym phrases corresponding to this concept, if any.
\item \emph{membershipValueSynonymSet}: This attribute holds the list of the degrees of membership of the list of elements in the value
of \emph{synonymSet} to the domain of wind energy.
\item \emph{synonymSetTR}: This attribute holds the list of English synonym phrases corresponding to this concept, if any.
\item \emph{membershipValueSynonymSetTR}: This attribute holds the list of the degrees of membership of the list of elements in the value
of \emph{synonymSetTR} to the domain of wind energy.
\end{itemize}
As described above, the attributes beginning with \emph{membershipValue} hold weighted values corresponding to fuzzy membership functions, as
previously explained in \cite{kuccuk2015feeont}.
All of the 25 instances of OntoWind, shown in Figure \ref{fig:instances}, correspond to national or international organizations related to wind
energy. Apart from \emph{MGM} which models \emph{Turkish State Meteorological Service}, other instances include international organizations like
ECMWF (denoting \emph{European Centre for Medium-Range Weather Forecasts}) and WMO for \emph{World Meteorological Organization}, and national
ones like \emph{CENER} which is a Spanish institute for renewable energy research and \emph{NCAR} which corresponds to \emph{National Center for
Atmospheric Research} of the USA.
The final form of the improved and extended wind energy ontology, OntoWind, is made publicly-available at
\url{http://www.ceng.metu.edu.tr/~e120329/OntoWind.owl} as an OWL file. To summarize its characteristics again, OntoWind is an extended and
reorganized wind energy ontology which\footnote{We should note that the first two characteristics are not specific to OntoWind, as the first one
is a characteristic of both PQONT \cite{kucuk2010pqont} and FEEONT \cite{kuccuk2015feeont}, and the second one is a characteristic of FEEONT.}:
\begin{itemize}
\item supports multilinguality with its relevant attributes such as \emph{labelEN}, \emph{labelTR}, \emph{synonymSet}, and
\emph{synonymSetTR}, and the ontology can be extended with similar attributes for other languages as previously pointed out,
\item supports attributes conveying crisp as well as weighted information (such as \emph{membershipValueLabel} and
\emph{membershipValueLabelTR}) for the other attributes,
\item can also be considered as a larger knowledge base including the ontology concepts and their hierarchies as well as concept
instances (as illustrated in Figure \ref{fig:instances}).
\end{itemize}
\section{Semantic System for the Surveillance of Wind Energy Information}\label{sec:system}
In this section, we present a semantic portal for wind energy surveillance on the Web, based on OntoWind as the underlying source of semantic
information. The portal enables its users (i) to examine the concepts in OntoWind ontology, (ii) to view the wind-related research organizations
with links to their official Web sites and Twitter accounts, which correspond to the instances of OntoWind ontology, and (iii) to observe the
automatically-extracted wind-related scholarly articles. A snapshot of this Web-based semantic portal application is provided in Figure \ref{fig:portal}. The application is implemented using Java programming language, JavaServer Faces (JSF) technology \cite{jsf} and PrimeFaces JSF library \cite{primefaces} as the underlying technologies, and Java OWL API \cite{owl-api} to programmatically access OntoWind as an OWL file.
\begin{figure}
\center \scalebox{0.33}
{\includegraphics{wind-portal-son.eps}}\caption{A Snaphost of the Semantic Portal for Wind Energy Surveillance on the Web.}\label{fig:portal}
\end{figure}
On the left panel of the portal, the taxonomy of the OntoWind concepts are provided in the form of a tree. The tree is automatically generated by
processing OntoWind using Java OWL API \cite{owl-api}. The users can freely examine all of the ontology concepts through this panel.
On the right panel of the portal, OntoWind's concept instances described in the previous section are listed. All of these instances are
significant organizations related to wind energy, that are automatically extracted from OntoWind. Based on the values of the \emph{webAddress}
and \emph{twitterAccount} attributes of these instances, related buttons to access the Web sites and official Twitter account pages of these
organizations automatically appear.
The final and the most significant feature of the semantic portal for wind energy is the panel appearing at the center of the portal page. This
panel presents automatically-extracted content from the Web, using OntoWind as the underlying semantic resource. In its current form, it
basically presents automatically-extracted scholarly articles related to wind energy. We have implemented this proof-of-concept extractor system,
we have followed the categorization strategy previously proposed in \cite{kuccuk2015feeont}. That is, we have built a text categorization system
based on OntoWind, which categorizes a given text as related to the wind energy domain, if it includes at least one concept from OntoWind and the
sum of the values of the \emph{membershipValueLabel} attributes corresponding to the included concepts is at least 1.0. Otherwise, the text is
categorized as irrelevant to the wind energy domain.
In order to test our categorizer, we have used 91 articles related to the energy domain which have been published in the 134th
volume\footnote{134th volume is the most recent volume of the journal, at the time of writing this paper.} of Elsevier's \emph{Energy} journal
\cite{elsevier}. For each journal article, we have executed our categorizer on a piece of text including the title, abstract, and keywords of
this article. We have built a similar categorizer for comparison purposes, which employs WONT instead of OntoWind, and this categorizer is also
executed on the same article texts. In order to create the answer key using which the performance rates of the categorizers will be calculated,
we have manually annotated each article in the set of 91 articles as relevant or irrelevant to the wind energy domain. At the end of this
process, it is found that 12 articles are related to wind energy while 79 articles are not. Based on this answer set, the performance evaluation
results of the OntoWind-based and WONT-based categorizers are presented in Table \ref{tab:sonuc}.
\begin{table}
\caption{Evaluation Results of the OntoWind-Based and WONT-Based Classifiers.}
\label{tab:sonuc}
\centering
\begin{tabular}{|l|c|c|c|c|c|}
\hline
&\emph{True}&\emph{False}&\emph{True}&\emph{False}& \\
&\emph{Positives}&\emph{Negatives}&\emph{Negatives}&\emph{Positives}&\emph{Accuracy}\\
\hline
OntoWind-Based &&&&& \\
Categorizer & 10 & 2 & 75 & 4 & 93.4\% \\
\hline
WONT-Based &&&&& \\
Categorizer & 10 & 2 & 64 & 15 & 81.3\% \\
\hline
\end{tabular}
\end{table}
When the results in Table \ref{tab:sonuc} are examined in details, we see that both the OntoWind-based and WONT-based categorizers are good at
finding the relevant articles. Yet, WONT-based categorizer spuriously outputs irrelevant articles as relevant as revealed with the high number of
false positives. In general, it can be concluded that the performance of the categorizer based on OntoWind is highly favorable and better when
compared with the results obtained by the WONT-based categorizer. This finding provides evidence for the coverage and utility of OntoWind in semantic applications.
We have previously stated that a similar categorization experiment has been reported in \cite{kuccuk2015feeont}. In the experiment given in
\cite{kuccuk2015feeont}, the data set contains articles from many different domains while in the current study, the articles are already from the
energy domain in general which makes it harder to extract those ones relevant to the specific domain of wind energy. This difference in the data
sets also explains the performance difference of the OntoWind-based categorizer of the current study and the categorizer presented in
\cite{kuccuk2015feeont} which achieves an accuracy of 99.09\%.
Our main objectives in this categorization experiment are (i) to showcase the practical contribution of the improved and extended OntoWind
ontology over its predecessor, WONT, which is revealed with the findings given in Table \ref{tab:sonuc} and (ii) to use the
automatically-extracted articles by the OntoWind-based categorizer in our semantic wind energy portal as shown in the middle panel of the portal
shown in Figure \ref{fig:portal}. The middle panel titled ``\emph{Scholarly Articles Related to Wind Energy}" also facilitates access to the
actual publisher pages of these articles through the accompanied links.
The semantic wind energy portal application is a proof-of-concept system and an initial prototype. The following extensions to this application
are envisaged as part of future work:
\begin{itemize}
\item The portal will be extended to list recent automatically-extracted news articles related to wind energy, so that it will be a plausible
information hub for semantic information related to wind energy.
\item Similarly, recent social media content related to wind energy like relevant tweets will also be automatically extracted and presented
through the portal.
\item Currently, the semantic portal for wind energy is not publicly accessible. After the above mentioned extensions, we will make the
portal accessible through the Internet.
\end{itemize}
As wind energy and renewable energy resources in general are important research topics with considerable public impact, we believe that both our
extended and improved ontology, OntoWind, and the semantic portal application for wind energy are important contributions to the related literature.
\section{Conclusion}\label{sec:conc}
Wind energy is a ubiquitous renewable energy type and several research topics regarding wind energy still need extensive efforts to fulfill the
research needs. One of these topics is the representation and application of semantic information regarding wind energy. In this study, we
present an extended and improved wind energy ontology, called OntoWind, in order to conveniently represent the semantic information in the wind
energy domain. We restructure, improve, and extend an existing semi-automatically created ontology for this domain by adding new concepts,
attributes, and instances to arrive at OntoWind. After making it publicly-available, we have developed a semantic portal for wind energy
utilizing the final form of the ontology, hence showed its contribution to a genuine semantic application. The portal currently presents
automatically extracted scholarly articles in addition to the concept taxonomy and instances of OntoWind. As part of future work, we plan to
extend the portal to publish other semantic content related to wind energy, like recent news articles and social media content. Other directions
of future work include similarly building ontologies for other renewable energy resources and then consolidating them to make them applicable to
significant semantic applications regarding renewable energy and energy in general.
|
{
"timestamp": "2018-03-08T02:12:21",
"yymm": "1803",
"arxiv_id": "1803.02808",
"language": "en",
"url": "https://arxiv.org/abs/1803.02808"
}
|
\section{Introduction}
Attachment and detachment phenomena of bacteria, whether in biofilms
on a support \cite{C95,IWA06} or in the form of aggregates or flocs
\cite{TJF99} are well known and frequently observed in bacterial growth.
Nevertheless, it is only relatively recently that they have been
explicitly taken into account in chemostat-based mathematical models.
The Freter model \cite{FBFVC83,JKLS03}, proposed in the 1980s as a functional model
of the intestine bacterial ecosystem, is one of the very first to
explicitly distinguish planktonic biomass from attached biomass. This
model considers specific attachment and detachment terms and has been
mathematically studied in a spatialized form by introducing advection
and diffusion terms \cite{BJS08}.
Several works in the biomathematical literature consider extensions to
the chemostat model spatialized with (fixed) attachment on a wall by
\cite{BS99,JKLS03,SS00}. In general, flocculation models describe the
dynamics of the distribution of flocs sizes \cite{TJF99} and their
influence on growth dynamics \cite{HLH07}, but comparatively there are
relatively few studies of simplified models that only distinguish two biomass compartments: planktonic and attached. In \cite{HR08}, it is shown for
such models that total biomass growth follows a density-dependent
distribution, under the assumption that attachment and detachment
velocities are large compared to biological terms. This is in
accordance with experimental observations that have showed that the
kinetics of processes with attached biomass are better represented by
ratio-dependent \cite{HG07} expressions.
The purpose of the present work is to generalize the existing
results concerning these simplified models.
The majority of models of the literature consider explicit
attachment and detachment term expressions. We adopt here a more
general presentation
which does not particularize the specific attachment and detachment
kinetics terms
and thus namely includes existing models \cite{TSJ97,PW99,JKLS03}.
In every case, the assumptions
about faster growth and higher planktonic bacteria removal rate are justified by
experimental observations \cite{HMC09}. This allows us to consider reduced
models considering the total biomass instead of planktonic and
attached ones, which provides extensions of the well-know chemostat
model with unusual characteristics.
It should be observed that attachment and detachment
velocities can be of a very variable order of magnitude, according to procedures
and operating conditions \cite{BK95}, justifying the fact of considering
reduced models or not.
\section{A general formulation}
Under certain growth conditions and in some environments, microbial species may
present aggregates of microorganisms or flocs of various sizes (see
Figure \ref{figattachement}). Microorganisms can also attach themselves to the walls of
tanks, pipes, reactors, etc. (or more generally of any chemostat-based
device), and thus create biofilms with varied thicknesses. Over time,
micro-organisms, parts of flocs or of biofilms, detach and are
released in the liquid medium as isolated individuals or small-sized aggregates (see Figure \ref{figdetachement}).
These bacterial assemblages (which can be observed under the
microscope) affect the performance of chemostats at the macroscopic
level, namely regarding:
\begin{itemize}
\item the growth of biomass: bacterial individuals have differentiated
access to biotic resource (substrate)
depending on their position inside or on the periphery of assemblies. In addition,
microorganism secretions of polymers that enable the attachment are generally achieved
to the detriment of their growth.
\item the disappearance of biomass: flocs and biofilms are most often less likely to be
dragged away by the chemostat outflow, comparatively to isolated
individuals.
\end{itemize}
The appearance and evolution mechanisms of these assemblies, which at the same
time relate to biology, mechanics and hydrodynamics, are complex, partially understood
and difficult to be modeled at a microscopic scale. Our objective is
to study how the conventional model of the chemostat can be enriched with considerations
reflecting the effects of biomass attachment and detachment at the macroscopic
level (in other words, without representing all the refinements that a description would
bring at the microscopic level).
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.4\textwidth]{attachement.pdf}
\caption{Isolated individuals may aggregate to form
a floc, or else attach to an already formed aggregate. \label{figattachement}}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.4\textwidth]{detachement.pdf}
\caption{Individuals can detach from an aggregate.
An aggregate can be split into smaller aggregates.
\label{figdetachement}}
\end{center}
\end{figure}
We consider that the total biomass of a given species is decomposed
into "planktonic" (or "free") biomass made up of non-attached
microorganisms (or at least that behave as such; which may still be
the case of small assemblies) and "aggregate" biomass (without accurately taking account of the shape and of the size of assemblies).
Thus, we write the concentration $x$ of the total biomass as the sum
of concentrations $u$ and $v$ of planktonic and aggregate biomass,
respectively:
\begin{equation}
\label{somme}
x=u+v \ .
\end{equation}
This distinction allows us to take into account different growth and death characteristics
according to whether microorganisms are attached or not. We thus denote
respectively by $\mu_{u}(\cdot)$, $D_{u}$ and $\mu_{v}(\cdot)$, $D_{v}$
the specific growth and removal rates of
planktonic and aggregate compartments.
$D_{u}$ and $D_{v}$ are positive numbers and $\mu_{u}(\cdot)$,
$\mu_{v}(\cdot)$ are smooth functions that verify
$\mu_{u}(0)=\mu_{v}(0)=0$ and positive away from zero.
On the other hand, we denote the specific
velocities of attachment of planktonic biomass by $\alpha(\cdot)$ and
by $\beta(\cdot)$ the ones of detachment
of the attached biomass. As a result, we obtain the following
chemostat model, where $s$ denotes the substrate concentration:
\begin{equation}
\label{chem_attach}
\left\{
\begin{array}{lll}
\displaystyle \frac{ds}{dt} = & \displaystyle D(S_{in}-s)-\mu_{u}(s)u-\mu_{v}(s)v\\[4mm]
\displaystyle \frac{du}{dt} = & \mu_{u}(s)u-D_{u}u
-\alpha(u,v)u+\beta(v)v\\[4mm]
\displaystyle \frac{dv}{dt} = & \mu_{v}(s)v-D_{v}v
+\alpha(u,v)u-\beta(v)v
\end{array}\right.
\end{equation}
The positive parameters $D$ and $S_{in}$ denote the dilution rate and
input concentration of the substrate.
As usual in chemostat models, we take unit yield coefficients without loss of generality. The simplicity
of this representation, which does not account for the richness of forms and
possible sizes of aggregates, should be regarded as the considering of an average microorganism
behavior within aggregates or biofilms, which differs from that of isolated
microorganisms. Since it is difficult to obtain or to justify precise expressions of the
attachment and detachment terms for this type of model, our purpose is to understand
and qualitatively predict the possible effects of these terms on the dynamics of the
system (to this end, we will merely consider simple expressions as possible representatives).
It should be noted that the attachment and detachment terms depend on the
operating conditions (in particular the flow rate), that we
consider here to be fixed.
\medskip
We first show that the solutions of system (\ref{chem_attach}) stay non-negative
and bounded, as in the classical chemostat model.
\begin{lemma}
The non-negative orthant $\mathbb{R}_{+}^3$ is forwardly invariant by the dynamics (\ref{chem_attach})
and any solution in this domain is bounded.
\end{lemma}
\begin{proof}
At $s=0$, one has $\dot s=DS_{in}>0$. Therefore $s$ stays positive.
One has $\frac{d}{dt}(u+v) \geq (\mu_{v}(s)-D_{u})(u+v)$, which
shows that $x=u+v$ stay positive. At $u=0$, resp. $v=0$, one
has $\frac{d}{dt}u \geq \beta(\cdot)x\geq 0$, resp. $\frac{d}{dt}v \geq
\alpha(\cdot)x \geq 0$. Therefore the variables $u$ and $v$ stay
non-negative. Finally, on has $\frac{d}{dt}(s+u+v)\leq
DS_{in}-D_{v}(s+u+v)$ which shows that the quantity $s+u+v$ is
bounded, and a consequence, $s$, $u$ and $v$ also.
\end{proof}
Hereafter, we consider the following assumptions, which reflect the
considerations discussed in the introduction:
\begin{assumptions}
The kinetics functions $\mu_{u}(\cdot)$, $\mu_{v}(\cdot)$,
$\alpha(\cdot)$, $\beta(\cdot)$ and parameters $D$, $D_{u}$, $D_{v}$ fulfill the following properties.
\label{hypagreg}
\begin{itemize}
\item[i.] The specific growth kinetics $\mu_{u}(\cdot)$ and
$\mu_{v}(\cdot)$ are smooth increasing functions, null at zero,
that verify:
\begin{equation}
\label{hypo_mu}
\mu_{u}(s) > \mu_{v}(s), \quad \forall s>0
\end{equation}
\item[ii.] The removal rates of aggregate and planktonic biomass
verify:
\begin{equation}
\label{hypoD}
D \geq D_{u} \geq D_{v}>0
\end{equation}
\item[iii.] The function $\alpha$ only depends on concentrations $u$ and $v$ in an increasing manner
and such that:
\[
u>0 \; \Rightarrow \alpha(u,0)>0
\]
with
\[
\frac{\partial\alpha}{\partial u}(u,v) \geq
\frac{\partial\alpha}{\partial v}(u,v), \quad \forall (u,v) .
\]
\item[iv.] The function $\beta$ depends only on the concentration $v$ in a decreasing manner
and such that $v\mapsto \beta(v)v$ is increasing with:
\[
v>0 \; \Rightarrow \; \beta(v)>0 .
\]
\end{itemize}
\end{assumptions}
Typical instances of functions $\mu_{u}$, $\mu_{v}$ are given by
the Monod expression
\[
\mu_{\max}\,\frac{s}{K_{s}+s}
\]
(with distinct values of the parameters
$\mu_{\max}$, $K_{s}$ for planktonic and attached bacteria), that is
quite popular in microbiology.
Assumption i. expresses the observation that attached bacteria have
generally a more difficult
acces to substrate.
With Assumption ii, we first neglect the mortality of planktonic bacteria,
compared to the removal rate $D$, and considered that the
substrate is the reactant that is removed most easily because of the
the size of its molecules (that is usually much smaller that
micro-organisms, justifying the assumption $D_{u}\leq D$). In a similar way, the
attachment slows down the effective removal rate of the attached
bacteria compared to the planktonic ones (which is represented by
the inequality $D_{v}\leq D_{u}$).
Typically, it can be considered that the specific attachment velocity $\alpha(u,v)$ can be
decomposed into a sum of two terms $\alpha_{u}(u)$ and $\alpha_{v}(v)$ that reflect the two possible
types of attachments: on free bacteria or on bacteria already in flocs. Considering
that free bacteria mainly attach on the surface of flocs, and that when the size of flocs
increases, the ratio surface over volume does not increase as quickly as the volume,
it can be expected that the function $\alpha_{v}$ increases more slowly than $\alpha_{u}$, which is then
reflected by $\alpha_{u}'(u)\geq \alpha_{v}'(v)$ for all $(u,v)$, justifying Assumption iii.
In general, it is expected that the detachment velocity $v \mapsto \beta(v)v$ increases with
the density $v$ of the attached biomass, but when the flocs size increases, the ratio surface
over volume increases more slowly than the volume, which results in a decrease of the
function $v \mapsto \beta(v)v$, thus justifying Assumption iv.
\section{Study of the coexistence between the two forms}
We assume that
\[
D=D_{u}=D_{v} ,
\]
(the more general case of different removal rates is discussed
in Section \ref{SectionDiffD}), which allows to consider the variable $z(t) = s(t) + x(t)$, a solution
of the differential equation :
\[
\frac{dz}{dt}=D(S_{in}-z) .
\]
whose solutions converge exponentially to $S_{in}$.
Therefore, the system (\ref{chem_attach}) has a cascade structure in
the $(z,u,v)$ coordinates :
\begin{equation}
\label{3dsys}
\begin{array}{l}
\displaystyle \frac{dz}{dt} = f_{0}(z)\\[3mm]
\displaystyle \frac{du}{dt} = f_{1}(z,u,v) , \; \frac{dv}{dt} = f_{2}(z,u,v)
\end{array}
\end{equation}
and the local stability analysis of its equilibriums is given by the local
stability of the equilibriums of the reduced dynamics :
\begin{equation}
\label{2dsys}
\frac{du}{dt} = f_{1}(S_{in},u,v) , \;
\displaystyle \frac{dv}{dt} = f_{2}(S_{in},u,v)
\end{equation}
The global behavior of the solutions of the system (\ref{3dsys}) is more
delicate to be deduced from the global behavior of the reduced system
(\ref{2dsys}) and relies on the theory of asymptotically autonomous
systems \cite{MST95}. However, we recall the well-known result when the reduced
system (\ref{2dsys})has a unique globally asymptotically stable
equilibrium, that states that any bounded solution of (\ref{3dsys})
converge to the unique equilibrium of (\ref{3dsys}).
We consider in the following the reduced dynamics of
(\ref{chem_attach}) for $z=S_{in}$:
\begin{equation}
\label{chem_attach_reduit}
\left\{
\begin{array}{lll}
\displaystyle \frac{du}{dt} = & \mu_{u}(S_{in}-u-v)u-Du
-\alpha(u,v)u+\beta(v)v\\[4mm]
\displaystyle \frac{dv}{dt} = & \mu_{v}(S_{in}-u-v)v-Dv
+\alpha(u,v)u-\beta(v)v
\end{array}\right.
\end{equation}
We study the possible positive steady-states $(u^\star,v^\star)$ of this system, that is to say, the positive
solutions of the system:
\begin{equation}
\label{sys-equ}
\left\{
\begin{array}{l}
\displaystyle \mu_{u}(S_{in}-u-v)u-Du -\alpha(u,v)u+\beta(v)v=0\\[4mm]
\displaystyle \mu_{v}(S_{in}-u-v)v-Dv +\alpha(u,v)u-\beta(v)v=0
\end{array}\right.
\end{equation}
It can be immediately noticed that $u^\star=0$ implies
$\beta(v^\star)v^\star=0$ and $v^\star=0$,
$\alpha(u^\star,0)u^\star=0$. The assumptions \ref{hypagreg} that we consider on terms $\alpha(\cdot)$ and
$\beta(\cdot)$
then allow us to infer that there is no steady-state where only one of the two forms
would be present.
\subsection{Coexistence steady-state}
\label{sec_coex_agreg}
Adding equations (\ref{sys-equ}), we obtain $(u^\star,v^\star)$
as a solution of the system:
\[
\left\{
\begin{array}{rcrll}
(\mu_{u}(s)-D)u & + & (\mu_{v}(s)-D)v & = & 0\\
u & + & v & = & S_{in}-s
\end{array}\right.
\]
Consequently, a coexistence steady-state (if it exists) verifies:
\begin{equation}
\label{xpxa}
u^\star=(S_{in}-s^\star)\frac{D-\mu_{v}(s^\star)}{\mu_{u}(s^\star)-\mu_{v}(s^\star)},
\quad
v^\star=(S_{in}-s^\star)\frac{\mu_{u}(s^\star)-D}{\mu_{u}(s^\star)-\mu_{v}(s^\star)}
\end{equation}
with $s^\star=S_{in}-u^\star-v^\star$.
According to hypothesis (\ref{hypo_mu}), we obtain the following
necessary condition:
\[
\mu_{u}(s^\star)>D>\mu_{v}(s^\star) .
\]
By defining the break-even concentration by $\lambda_{u}$,
$\lambda_{v}$ for the dilution rate $D$ (that is that verify
$\mu_{u}(\lambda_{u})=\mu_{v}(\lambda_{v})=D$ with $\lambda_{v}>
\lambda_{u}$, see \cite{SW95,HLRS17}), we deduce
that a coexistence steady-state must verify:
\[
s^\star \in (\lambda_{u},\lambda_{v}) .
\]
Thus, a necessary condition for the existence of a coexistence
steady-state is:
\begin{equation}
\label{condlambda}
\lambda_{u}<S_{in} .
\end{equation}
At this stage, it is difficult to prove the existence of solutions without specifying
attachment and detachment functions $\alpha(\cdot)$
and $\beta(\cdot)$. If we consider that we are only
dealing with flocs of small size, as a first approximation it is possible to assume that
$\alpha$ is a function of $x = u+v$ (that is, functions $\alpha_{u}$
and $\alpha_{v}$ are identical), which will be
chosen as linear (to simplify), and that the function $\beta$ does not
depend of $v$:
\begin{equation}
\label{alpha_beta_simples}
\alpha(u,v)=a(u+v)=ax, \quad \beta(v)=b
\end{equation}
where $a$ and $b$ are two positive constants. Thereby, the hypotheses
\ref{hypagreg} are correctly verified.
\begin{proposition}
\label{propexist}
For growth functions $\mu_{u}$, $\mu_{v}$ that verify point i) of Assumptions
\ref{hypagreg} and attachment and detachment functions $\alpha(\cdot)$, $\beta(\cdot)$ of the form (\ref{alpha_beta_simples}), there
exists a unique coexistence steady-state of system
(\ref{chem_attach}) if and only if the condition :
\begin{equation}
\label{condexist}
D <\mu_{u}(S_{in})
\end{equation}
is verified.
\end{proposition}
\begin{proof}
As mentioned previously, it is enough to show the existence of
a positive equilibrium of the reduced dynamics (\ref{chem_attach_reduit}).
$I$ denotes the interval :
$$I= ]\lambda_u,\lambda_v[.$$
To simplify the writing, the following notations are introduced:
$$
\varphi_u(s)=\mu_u(s)-D \quad \mbox{and} \quad \varphi_v(s)=\mu_v(s)-D.
$$
For all $s\in I$, we have $\varphi_u(s)>0>\varphi_v(s)$. The steady-states $(s^*,u^*,v^*)$ are
given by:
\begin{eqnarray} \label{IsoFlocGen}
\left\{
\begin{array}{lll}
0=\varphi_u(s^*)u^*-a(u^*+v^*)u^*+b v^*\\[1mm]
0=\varphi_v(s^*)v^*+a(u^*+v^*)u^*-b v^*.
\end{array}
\right.
\end{eqnarray}
If $u^*=0$ then, from the first equation, it can be deduced
that $v^*=0$. Similarly, if $v^*=0$ then, from the second equation it can be deduced that
$u^*=0$. Consequently, the steady-states are the washout $E_0=(S_{in},0,0)$ or a
steady-state of the form:
$$E^*=(s^*,u^*,v^*)$$
with $u^*>0$ , $v^*>0$ and $s^*=S_{in}-u^*-v^*$.
In order to solve Equations (\ref{IsoFlocGen}), one uses a method
similar to the characteristic at steady-state method. This
method consists in determining the steady-states of the system formed by the 2nd
and 3rd equations of (\ref{chem_attach}), where the variable $s$ is considered to be an input of the
system. In other words the aim is to solve the system formed by the first and the
second equation of (\ref{IsoFlocGen}), in which $u^*$ and $v^*$ are the unknowns and $s^*$ is considered
as being a parameter. It thus yields :
$$u^*=U(s^*),\qquad v^*=V(s^*).$$
If $u^*$ and $v^*$ are replaced by these expressions in the first equation of (\ref{chem_attach}), an
equation of the single variable $s^*$ is obtained of the form:
$$D(S_{in}-s^*)=H(s^*)\quad\mbox{with}\quad
H(s^*)=\mu_u(s^*)U(s^*)+\mu_v(s^*)V(s^*) $$
that is solved, see Figure \ref{figsol}, to find a positive solution $s^*$ . This solution gives a
positive steady-state, provided that $U(s^*)$ and $V(s^*)$ be positive. In the following, the
functions U, V and H are determined and the conditions are given in order for the
solution $s^*$ to exist.
\begin{figure}[ht]
\setlength{\unitlength}{1.0cm}
\begin{center}
\begin{picture}(6.7,6)(0,0)
\put(0,6.5){\rotatebox{-90}{\includegraphics[width=6cm,height=8cm]{characteristic.pdf}}}
\put(5.5,4.7){{ ${H(s)}$}}
\put(2.9,3.9){{ ${D(S_{in}-s)}$}}
\put(0.5,5.2){{ $DS_{in}$}}
\put(4.7,2.2){{ ${E^*}$}}
\put(5.9,0.9){{ ${E_0}$}}
\put(5.7,0.4){{ $S_{in}$}}
\put(3.1,0.4){{ ${\lambda_u}$}}
\put(6.3,0.4){{ ${\lambda_v}$}}
\put(6.5,0.7){{ $s$}}
\end{picture}
\end{center}
\caption{Existence of a unique positive steady-state.} \label{figsol}
\end{figure}
By summing the 1st and 2nd equations (\ref{IsoFlocGen}), we obtain:
\begin{equation} \label{eqPhiPsi}
\varphi_u(s^*)u^* + \varphi_v(s^*)v^*=0.
\end{equation}
This equation admits a positive solution if and only if $\varphi_u(s^*)$ and
$\varphi_v(s^*)$ are of
opposite signs, that is, if and only if $s^*\in I$. If this equation admits a solution in this
interval then Equation (\ref{eqPhiPsi}) can be written as follows :
\begin{equation} \label{eqV-U}
v^*=-\frac{\varphi_u(s^*)}{\varphi_v(s^*)}u^*.
\end{equation}
By replacing $v^*$ by Expression (\ref{eqV-U}) in the first equation
of (\ref{IsoFlocGen}), it yields:
\begin{equation} \label{equ}
u^*=U(s^*) \quad \mbox{with} \quad
U(s)=\frac{\varphi_u(s)(\varphi_v(s)-b)}{a[\varphi_v(s)-\varphi_u(s)]}.
\end{equation}
Note that $u^*$ defined by (\ref{equ}) is positive because $s^*\in
I$. By replacing $u^*$ by (\ref{equ}) in (\ref{eqV-U}), we get:
\begin{equation} \label{eqv}
v^*=V(s^*) \quad \mbox{with} \quad
V(s)=-\frac{\varphi_u^2(s)(\varphi_v(s)-b)}{a[\varphi_v(s)-\varphi_u(s)]\varphi_v(s)}.
\end{equation}
Substituting the expressions of $U(s^*)$ and $V(s^*)$ given by
(\ref{equ}) and (\ref{eqv}) in
the expression of $H(s^*)$ yields a characterization of $s^*$:
\begin{equation} \label{eqH(S)}
D(S_{in}-s^*)=H(s^*)\quad \mbox{with} \quad
H(s)=D\frac{\varphi_u(s)(\varphi_v(s)-b)}{a\varphi_v(s)} .
\end{equation}
Note that for all $s\in I$ , $U(s)>0$, $V(s)>0$ and $H(s)>0$ and that :
$$\lim_{s\to\lambda_u}H(s)=0,\quad \lim_{s\to\lambda_v}H(s)=+\infty .$$
In addition, function $H$ is strictly increasing on $I$. Indeed, we have:
$$H'(s)=\frac{D}{a}
\frac{\varphi_v(s)(\varphi_v(s)-b)\varphi_u'(s)+b\varphi_u(s)\varphi_v'(s)}{\varphi_v^2(s)}>0 .$$
Consequently, Equation (\ref{eqH(S)}) admits a unique solution
$s^*\in I=]\lambda_u,\lambda_v[$ if and
only if $S_{in}>\lambda_u$, which is equivalent to $\mu_u(S_{in})>D$.
\end{proof}
\subsection{Study of stability}
Under the conditions of stability and global attractiveness of the
washout steady-state of the chemostat model in which only the
planktonic biomass would be considered (see \cite{SW95,HLRS17}):
\begin{equation}
\label{condlessivage}
D \geq \mu_{u}(S_{in})
\end{equation}
one can easily check that the washout $(S_{in},0,0)$ is also the only steady-state of the
system (\ref{chem_attach}), stable
and globally attractive. As a matter of fact, by considering the reduced model (\ref{chem_attach_reduit}),
under this assumption we have:
\[
x \in ]0,S_{in}] \; \Rightarrow \;
\frac{dx}{dt}=(\mu_{u}(S_{in}-x)-D)u+(\mu_{v}(S_{in}-x)-D)v
<0
\]
which demonstrates that $x(\cdot)$ asymptotically converges towards
$0$ for any initial condition.
As any solution of system
(\ref{chem_attach}) is bounded, we deduce that it converges to the washout equilibrium.
According to the study conducted in Section \ref{sec_coex_agreg}, a positive steady-state exists
as soon as the condition (\ref{condexist}) is verified and is unique. By particularizing the attachment
and detachment functions as we did in Section \ref{sec_coex_agreg}, the following stability
result is obtained (the case in which $D_{u}$ and $D_{v}$ are
different from $D$ is addressed in \cite{FHCRS13}).
\begin{proposition}
Under the assumptions of Proposition \ref{propexist} the coexistence
steady-state is a locally exponentially stable of system (\ref{chem_attach}).
\end{proposition}
\begin{proof}
As mentioned previously, it is enough to study the local
stability for the reduced dynamics (\ref{chem_attach_reduit}).
The Jacobian matrix of (\ref{chem_attach_reduit}) for the steady-state $(u^*,v^*)$, which corresponds
to the positive equilibrium $E^*=(s^*,u^*,v^*)$ of (\ref{chem_attach}), is equal to:
$$
J^*=
\left[
\begin{array}{ll}
-u^*\varphi_u'(s^*)+\varphi_u(s^*)-a(2u^*+v^*) & -u^*\varphi_u'(s^*)-au^*+b \\[2mm]
-v^*\varphi_v'(s^*)+a(2u^*+v^*) &-v^*\varphi_v'(s^*)+\varphi_v(s^*)+au^*-b
\end{array}
\right]
$$
The trace of this matrix is equal to:
$${\rm Tr}J^*=-u^*\varphi_u'(s^*)-v^*\varphi_v'(s^*)+\varphi_u(s^*)-a(u^*+v^*)+\varphi_v(s^*)-b$$
Note that based on Equations (\ref{IsoFlocGen}), it can be deduced
that:
\begin{equation}
\varphi_u(s^*)-a(u^*+v^*)=-b\frac{v^*}{u^*}<0,\qquad \varphi_v(s^*)-b=-a\frac{(u^*+v^*)u^*}{v^*}<0
\label{equv}
\end{equation}
Further, as $\varphi_u'(s^*)>0$ and $\varphi_v'(s^*)>0$,
it can be deduced that ${\rm Tr}J^*<0$.
The determinant of this matrix is equal to:
$${\rm Det}J^*=Au^*\varphi_u'(s^*)+Bv^*\varphi_v'(s^*)+C$$
with:
$$
A=a(u^*+v^*)+b-\varphi_v(s^*),
\quad
B=a(u^*+v^*)+b-\varphi_u(s^*),
$$
and:
$$
C=\varphi_u(s^*)\varphi_v(s^*)+\varphi_u(s^*)(au^*-b)-\varphi_v(s^*)a(2u^*+v^*)
$$
By using Expressions (\ref{equv}), it yields that:
$$
A=a\frac{(u^*+v^*)^2}{v^*}>0,
\quad
B=b\frac{u^*+v^*}{u^*}>0
$$
Moreover, we have:
$$
C=\varphi_u(s^*)\left(\varphi_v(s^*)-b\right)+a\left(u^*\varphi_u(s^*)-v^*\varphi_v(s^*)\right)-2au^*\varphi_v(s^*)
$$
Utilizing (\ref{eqPhiPsi}), we get:
$$
C=\varphi_u(s^*)\left(\varphi_v(s^*)-b\right)+
2au^*\varphi_u(s^*)-2au^*\varphi_v(s^*)
$$
Utilizing (\ref{equv}), we have:
$$
au^*\left(\varphi_u(s^*)-\varphi_v(s^*)\right)=-\varphi_u(s^*)\left(\varphi_v(s^*)-b\right)$$
Consequently:
$$
C=-\varphi_u(s^*)\left(\varphi_v(s^*)-b\right)>0
$$
Thereof, it can be deduced that ${\rm Det}J^*>0$, and as a
consequence, the real parts of the eigenvalues of $J^*$ are strictly negative.
\end{proof}
\section{The case of fast attachments/detachments}
\label{SecFlocsLentRapide}
Depending on species and on hydrodynamic conditions, attachment and detachment
velocities may prove to be large compared to growth kinetics and to dilution
rate. In this case, it is possible to consider that the attachment and detachment terms,
$\alpha(\cdot)$ and $\beta(\cdot)$ respectively, can be rewritten in
the form:
\[
\frac{\alpha(\cdot)}{\varepsilon}, \quad
\frac{\beta(\cdot)}{\varepsilon}
\]
where $\varepsilon$ is a positive number supposed to be small, and
functions $\alpha(\cdot)$, $\beta(\cdot)$ verify the
same assumptions \ref{hypagreg}. Thus, the model (\ref{chem_attach}) is written as:
\begin{equation}
\label{chem_attachLR}
\left\{
\begin{array}{lll}
\displaystyle \frac{ds}{dt} & = & \displaystyle D(S_{in}-s)-\mu_{u}(s)u-\mu_{v}(s)v\\[4mm]
\displaystyle \frac{du}{dt} & = & \displaystyle \mu_{u}(s)u-Du
-\frac{1}{\epsilon}\left(\alpha(u,v)u-\beta(v)v\right)\\[4mm]
\displaystyle \frac{dv}{dt} & = & \displaystyle \mu_{v}(s)v-Dv
+\frac{1}{\epsilon}\left(\alpha(u,v)u-\beta(v)v\right)
\end{array}\right.
\end{equation}
It is convenient to write this dynamic by replacing the variables $u$
and $v$ by $x=u+v$ and $p=u/x$
\begin{equation}
\label{chem_attach_xp}
\left\{
\begin{array}{lll}
\displaystyle \frac{ds}{dt} & = & \displaystyle D(S_{in}-s)-\bar\mu(s,p)x\\[4mm]
\displaystyle \frac{dx}{dt} & = & \displaystyle \bar\mu(s,p)x-Dx\\[4mm]
\displaystyle \frac{dp}{dt} & = & \displaystyle
\left(\mu_{u}(s)-\mu_{v}(s)\right)p(1-p)-\frac{1}{\epsilon}\left(\alpha(px,(1-p)x)p-\beta((1-p)x)(1-p)\right)
\end{array}\right.
\end{equation}
by defining:
\[
\bar\mu(s,p):=p\,\mu_{u}(s)+(1-p)\,\mu_{v}(s) .
\]
Observe that this dynamic system is of the form:
\[
\left\{
\begin{array}{lll}
\displaystyle \frac{ds}{dt} & = & \displaystyle f_{s}(s,x,p)\\[4mm]
\displaystyle \frac{dx}{dt} & = & \displaystyle f_{x}(s,x,p)\\[4mm]
\displaystyle \frac{dp}{dt} & = & \displaystyle \frac{1}{\epsilon}\left[\epsilon f_{p}(s,p)+g(x,p)\right]
\end{array}\right.
\]
where we posit:
\[
g(x,p):=-\alpha(px,(1-p)x)p+\beta((1-p)x)(1-p) .
\]
When $\epsilon$ is small and the terms $f_{s}(s,x,p)$, $f_{x}(s,x,p)$
and $\epsilon f_{p}(s,p)+g(x,p)$ are
of the same order of magnitude, the velocity $\frac{dp}{dt}$
is then very large compared to velocities $\frac{ds}{dt}$, $\frac{dx}{dt}$.
Variables $s$ and $x$ can then be considered as almost constant and the
approximation of the dynamics of variable $p$ as "fast":
\begin{equation}
\label{dyn_p_reduit}
\frac{dp}{dt} = \frac{1}{\epsilon}g(x,p)
\end{equation}
where $s$ is considered as a constant parameter (the term $\epsilon f_{p}(s,p)$ being negligible
with regard to $g(x,p)$). If for any $x$, the differential equation (\ref{dyn_p_reduit}) admits a unique
steady-state $\bar p(x)$, then this expression can be carried to
the system (\ref{chem_attach_xp}) to obtain the "slow" approximation of the dynamics of the variables
$s$ and $x$:
\begin{equation}
\label{chem_attach_red}
\left\{
\begin{array}{lll}
\displaystyle \frac{ds}{dt} & = & \displaystyle D(S_{in}-s)-\mu(s,x)x\\[4mm]
\displaystyle \frac{dx}{dt} & = & \displaystyle \mu(s,x)x-Dx
\end{array}\right.
\end{equation}
by defining:
\[
\mu(s,x)=\bar\mu(s,\bar p(x)) .
\]
This reduction technique (which consists in replacing $\epsilon$ by $0$) is well known in
physics under the name of quasi-steady state approximation method. At the mathematical
level, the rigorous proof of the convergence of the solutions of the system
(\ref{chem_attach_xp}) towards those of the reduced system (\ref{chem_attach_red}) makes use of the theory of singular
perturbations (see for instance \cite{K96}).
When the slow manifold is globally attractive, that is when $\bar
p(x)$ is a globally asymptotically stable of the dynamics
$dp/d\tau=g(x,p)$ for any fixed $x>0$ (where $\tau=t/\epsilon$ is the
``fast'' time), then Tikhonov's Theorem
applies.
Recall that this Theorem asserts that for any initial condition of
(\ref{chem_attach_xp}) with $x(0)>0$ and any time interval $[0,T]$ with $T>0$,
the solution $s(\cdot)$, $x(\cdot)$ of (\ref{chem_attach_xp}) converge uniformly on
$[0,T]$ to the solution of (\ref{chem_attach_red}). Furthermore, when the solution of
the reduced dynamics (\ref{chem_attach_red}) converges to an asymptotically stable
equilibrium, then one can take $T=+\infty$ (see for instance \cite{LST98}).
The Proposition below shows that the existence and the global asymptotic
stability of the slow manifold, under
Assumptions \ref{hypagreg}.
\begin{proposition}
\label{propbarp}
Under Assumptions \ref{hypagreg}, there exists a unique function
$\bar p: \mathbb{R}_{+} \mapsto [0,1]$ $C^1$, strictly decreasing, such
that $g(x,\bar p(x))=0$ for all $x>0$. In
addition, $\bar p(x)$ is the unique globally asymptotically stable
steady-state of the scalar equation (\ref{dyn_p_reduit}), for all $x>0$.
\end{proposition}
\begin{proof}
For any $x>0$, we have $g(x,0)=\beta(x)>0$ and $g(x,1)=-\alpha(x,0)<0$
(following Assumptions \ref{hypagreg}). According to the intermediate value theorem,
there therefore exists $\bar p(x) \in ]0,1[$ such that $g(x,\bar p(x))=0$. Let us determine the
partial derivatives of the function $g$:
\[
\begin{array}{lll}
\displaystyle \frac{\partial g}{\partial x} & = &
\displaystyle
-\left[\left(\frac{\partial \alpha}{\partial
u}(u,v)p+\frac{\partial\alpha}{\partial v}(u,v)(1-p)\right)p-\beta'(v)(1-p)^2\right]_{u=px,v=(1-p)x}\\[5mm]
\displaystyle \frac{\partial g}{\partial p} & = &
\displaystyle -\left[\left(\frac{\partial\alpha}{\partial
u}(u,v)-\frac{\partial\alpha}{\partial v}(u,v)\right)u+\alpha(u,v)+\frac{1}{u+v}\frac{d}{dv}(\beta(v)v)\right]_{u=px,v=(1-p)x}
\end{array}
\]
For $x>0$, Assumptions \ref{hypagreg} guarantee $\frac{\partial
g}{\partial x}<0$ and $\frac{\partial g}{\partial p}<0$.
Thus, the function
$p \mapsto g(x,p)$ is strictly decreasing, guaranteeing the uniqueness of the solution $\bar p(x)$
of $g(x,p)=0$. According to the implicit function theorem, the function $\bar p$ is also
differentiable for any $x>0$ and its derivative is written as:
\[
\bar p'(x)=-\frac{\displaystyle\frac{\partial g}{\partial x}(x,\bar p(x))}{\displaystyle\frac{\partial g}{\partial
p}(x,\bar p(x))} <0 .
\]
The function $\bar p$ is thus $C^1$ on $\mathbb{R}_{+}\setminus\{0\}$ and strictly decreasing.
Thereby, for all fixed $x>0$, $\bar p(x)$ is the unique steady-state of the differential
equation (\ref{dyn_p_reduit}), and since $\frac{\partial g}{\partial
p}<0$ for every $(x,p)$, it can be thereof deduced that the
steady-state $\bar p(x)$ is globally asymptotically stable for the
scalar dynamics (\ref{dyn_p_reduit}).
\end{proof}
For instance, for functions considered in (\ref{alpha_beta_simples}),
we get:
\begin{equation}
\label{exp}
\bar p(x)=\frac{1}{\displaystyle 1+\frac{a}{b}x} .
\end{equation}
Figure \ref{figSF} presents simulations with functions
(\ref{alpha_beta_simples}) and compares the solutions (in plain line)
of the original system (\ref{chem_attach_xp}) with the ones (in
dashed line) of the reduced dynamics (\ref{chem_attach_red}).
It shows that the slow-fast approximation is good even for value of
$\epsilon$ that are not so small.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{ll}
\begin{minipage}{6cm}
\includegraphics[width=7cm,height=6cm]{eps2.pdf}
\end{minipage}
&
\begin{minipage}{6cm}
\includegraphics[width=7cm,height=6cm]{eps0point5.pdf}
\end{minipage}
\end{tabular}
\end{center}
\caption{\label{figSF}
Simulations for $\mu_{u}(s)=\frac{s}{1+s}$,
$\mu_{v}(s)=\frac{0.7s}{1+s}$, $S_{in}=2$, $D=0.5$, $a=1$, $b=0.5$
with $\epsilon=2$ (left) and $\epsilon=0.5$ (right)
}
\end{figure}
\begin{remark}
Thanks to Assumptions \ref{hypagreg}, it yields that:
\[
\frac{\partial \mu}{\partial x}(s,x)= \frac{\partial \bar\mu}{\partial
p}(s,p)\vert_{p=\bar p(x)}.\bar p'(x)=(\mu_{u}(s)-\mu_{v}(s)).\bar p'(x)<0
\]
and thus the model (\ref{chem_attach_red}) for the total biomass $x$
has a density-dependent growth, decreasing with respect $x$.
\end{remark}
\subsection{Consideration of several species}
When several species are in competition, we can similarly decompose the biomass
of each species $i$ into planktonic biomass $u_{i}$ and attached biomass $v_{i}$ (without differentiating
the composition of flocs which can mix individuals from different
species):
\[
\left\{
\begin{array}{lll}
\displaystyle \frac{ds}{dt} = & \displaystyle D(S_{in}-s)-\sum_{j=1}^n\mu_{u_{j}}(s)u_{j}-\sum_{j=1}^n\mu_{v_{j}}(s)v_{j}\\[4mm]
\displaystyle \frac{du_{i}}{dt} = & \mu_{u_{i}}(s)u_{i}-Du_{i}
-\alpha_{i}(u_1,\cdots,u_{n},v_{1},\cdots,v_{n})u_{i}+\beta(v_{1},\cdots,v_{n})v_{i}\\
& & \hfill (i=1\cdots n)\\
\displaystyle \frac{dv_{i}}{dt} = & \mu_{v_{i}}(s)v_{i}-Dv_{i}
+\alpha_{i}(u_1,\cdots,u_{n},v_{1},\cdots,v_{n})u_{i}-\beta(v_{1},\cdots,v_{n})v_{i}
\end{array}\right.
\]
The specific attachment functions $\alpha_{i}$ then depend (a priori) on all others quantities $u_{j}$, $v_{j}$ since a free individual of species $i$ can attach to free biomass or biomass with any
species attached. Analogously, the specific detachment functions $\beta_{i}$ depend a priori
on all quantities $v_{j}$ of biomass attached where an individual $i$
could have attached.
To simplify, it will be possible, for example, to assume that
the $\alpha_{i}$ are functions of the total
planktonic and attached biomass $u=\sum_{j}u_{j}$ and $v=\sum_{v}v_{j}$,
and the $\beta_{i}$ functions
of $v$ only, with the same Assumptions (\ref{hypagreg}). The combinatorics of the possible specific
cases makes the mathematical study much more complicated, but when the attachment
and detachment velocities can be considered to be fast, the quasi-steady state approximation
makes it possible to write a dynamic system for biomass $x_{i}=u_{i}+v_{i}$ by
expressing the terms $u_{i}$ and $v_{i}$ according to all the $x_{j}$ on the "slow" manifold defined by
the system of equations:
\[
\alpha_{i}(u_1,\cdots,u_{n},v_{1},\cdots,v_{n})u_{i}-\beta_{i}(v_{1},\cdots,v_{n})v_{i}=0
\qquad i=1\cdots n .
\]
For example, by considering simple functions like we did in
(\ref{alpha_beta_simples}):
\[
\alpha_{i}(x_{1},\cdots,x_{n})=\sum_{j=1}^na_{ij}x_{j}, \quad \beta_{i}=b_{i}
\]
where parameters $a_{ij}$ reflect how easily an individual of species $i$ attaches to an
individual of species $j$, the following expressions are obtained for
the proportions
$q_{i}=u_{i}/x_{i}$ on the slow manifold, which is uniquely
defined by
\[
\bar q_{i}(x_{1},\cdots,x_{n})=\frac{1}{\displaystyle 1+\frac{1}{b_{i}}\sum_{j=1}^na_{ij}x_{j}}
\]
as in Section \ref{SecFlocsLentRapide} (under the assumption of fast attachments and detachments), and
the reduced system is then written as:
\[
\left\{\begin{array}{lll}
\displaystyle \frac{ds}{dt} & = & \displaystyle D(S_{in}-s)-\sum_{j=1}^n\mu_{j}(s,x_{1},\cdots,x_{n})x_{j}\\[4mm]
\displaystyle \frac{dx_{i}}{dt} & = & \displaystyle
\mu_{i}\left(s,x_{1},\cdots,x_{n}\right)x_{i}-Dx_{i} \qquad (i=1\cdots
n)
\end{array}\right.
\]
by setting:
\[
\mu_{i}(s,x)=\bar q_{i}(x_{1},\cdots,x_{n})\mu_{u_{i}}(s)+(1-\bar q_{i}(x_{1},\cdots,x_{n}))\mu_{v_{i}}(s)
\]
The dynamics of the fast variables $q_{i}$ is given by the system
\[
\frac{dq_{i}}{d\tau}=-\alpha_{i}(x)q_{i}+b_{i}(1-q_{i}) \qquad
(i=1\cdots n)
\]
(where $\tau=t/\epsilon)$ for which $(\bar q_{1},\cdots,\bar q_{n})$ is clearly the unique
globally asymptotically stable equilibrium, for any fixed
$(x_{1},\cdots, x_{n})$. Therefore Thikonov's Theorem applies.
Notice that $\mu_{i}$ are density-dependent growth functions,
decreasing with respect to the $x_{i}$. This then
exactly corresponds to the context of density-dependent competition model, which shows that a coexistence
between species is possible \cite{LMR05,ADLS06}.
It is thus concluded that a mechanism of (fast)
attachment and detachment of biomass is a possible (theoretical) explanation for the
maintaining of biodiversity in a chemostat.
\section{Consideration of distinct removal rates}
\label{SectionDiffD}
In this Section, we consider that the removal rates of planktonic and
attached bacteria are distinct, and accordingly to Assumptions
(\ref{hypagreg}) one has $D_{v}<D_{u}\leq D$.
This Section follows part of the work \cite{F13,FHCRS13}.
The reduction technique we use in Section
\ref{SecFlocsLentRapide} gives the following reduced model:
\begin{equation}
\label{chem_attach_red_D}
\left\{
\begin{array}{lll}
\displaystyle \frac{ds}{dt} & = & \displaystyle D(S_{in}-s)-\mu(s,x)x\\[4mm]
\displaystyle \frac{dx}{dt} & = & \displaystyle \mu(s,x)x-d(x)x
\end{array}\right.
\end{equation}
where we posit:
\[
d(x)=\bar p(x)D_{u} +(1-\bar p(x))D_{v} .
\]
Notice that the dynamics of the fast variable $p$ is given by
equation (\ref{dyn_p_reduit}), exactly as in Section \ref{SecFlocsLentRapide}.
Therefore, Proposition \ref{propbarp} applies.
Let us underline that having a density dependent removal rate in the
chemostat model has not being considered (and justified) before in the
literature.
\medskip
As in Section \ref{sec_coex_agreg}, we consider break-even
concentrations $\lambda_{u}$, $\lambda_{v}$ associated to functions
$\mu_{u}$ and $\mu_{v}$ but here for the distinct removal rates
$D_{u}$, $D_{v}$ (which are numbers that verify
$\mu_{u}(\lambda_{u})=D_{u}$ and $\mu_{v}(\lambda_{v})=D_{v}$).
Differently to the case of identical
removal rates, for which Assumptions \ref{hypagreg} implies the inequality
$\lambda_{u}<\lambda_{v}$, this later inequality is no longer
necessarily satisfied, as depicted on Figure \ref{figlambdas}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=.4\textwidth]{lambdas1.pdf}
\hspace{2mm}
\includegraphics[width=.4\textwidth]{lambdas2.pdf}
\caption{One can have $\lambda_{u}>\lambda_{v}$ (left) as well as
$\lambda_{u} <\lambda_{v}$ (right).
\label{figlambdas}}
\end{center}
\end{figure}
\medskip
The model (\ref{chem_attach_red_D}) admits clearly the washout $(S_{in},0)$ as
an equilibrium, and let us study the possibility for the system to have
another steady state.
A positive equilibrium $(s^\star,x^\star)$ of dynamics (\ref{chem_attach_red_D})
has to fulfill
\begin{equation}
s^\star = \gamma(x^\star) := S_{in} -\frac{x^\star d(x^\star)}{D}
\end{equation}
and
\begin{equation}
\label{equmud}
\mu(s^\star,x^\star)=d(x^\star)
\end{equation}
Notice that when $s<\min(\lambda_{u},\lambda_{v})$,
resp. $s>\max(\lambda_{u},\lambda_{v})$, one has $\mu(s,x)<d(x)$,
resp. $\mu(s,x)>d(x)$, for any $x$. Therefore, one has
\[
s^\star \in
[\min(\lambda_{u},\lambda_{v}),\max(\lambda_{u},\lambda_{v})] .
\]
Since the functions $\mu_{u}$ and $\mu_{v}$ are increasing, the map $s
\mapsto \mu(s,x)$ is increasing for any $x$ and by the Implicit
Function Theorem, we deduce the existence of an unique solution of
(\ref{equmud}) as $s^\star=\phi(x^\star)$.
Therefore, a positive equilibrium (if it exists) has to fulfill
\[
\Gamma(x^\star):=\gamma(x^\star)-\phi(x^\star)=0 .
\]
Notice that one has $\Gamma(0)=S_{in}-\lambda_{u}$ and
$\Gamma(+\infty)=-\infty$. Therefore, the existence of a positive
equilibrium is guaranteed when $\lambda_{u}<S_{in}$.
Notice that this last condition is exactly the one that guarantees the
existence of a positive equilibrium for the chemostat model without
attachment:
\[
\left\{\begin{array}{lll}
\displaystyle \frac{ds}{dt} & = & \displaystyle D(S_{in}-s)-\mu_{u}(s)u\\[4mm]
\displaystyle \frac{du}{dt} & = & \displaystyle \mu_{u}(s)u-D_{u}u
\end{array}\right.
\]
We examine now the possibilities of having more than one positive equilibrium.
The function $\gamma$ is such that $\gamma(0)=S_{in}$ and
$\gamma(+\infty)=-\infty$. So, it has to decrease somewhere on the
interval $[0,+\infty)$.
From the Implicit Function Theorem, we can write
\[
\phi'(x) =
\frac{d'(x)-\frac{\partial\mu}{\partial
x}(\phi(x),x)}{\frac{\partial\mu}{\partial s}(\phi(x),x)}
= \frac{\bar
p'(x)}{\frac{\partial\mu}{\partial s}(\phi(x),x)}\left(D_{u}-D_{v}-\mu_{u}(\phi(x))+\mu_{v}(\phi(x))\right)
\]
When $\lambda_{u}<\lambda_{v}$, one has $\mu_{u}(s)\geq D_{u}$ and
$\mu_{v}(s)<D_{v}$ for any $s \in [\lambda_{u},\lambda_{v})$.
As $\bar p'(x)<0$ (see Proposition \ref{propbarp}) and
$\frac{\partial\mu}{\partial s}(\phi(x),x)>0$, we deduce
$\phi'(x)>0$ for any $x$ such that $\phi(x) \in
[\lambda_{u},\lambda_{v})$.
At the opposite, when $\lambda_{u}>\lambda_{v}$, one has $\phi'(x)<0$
for any $x$ such that $\phi(x) \in [\lambda_{v},\lambda_{u})$.
This leaves open the possibility of having the functions $\gamma$ and $\phi$
simultaneously decreasing with more than one intersection of their
graphs (and then having the function $\Gamma$ non-monotonic with
alternate signs of $\Gamma'(x^\star)$ at the solutions $x^\star$).
At a positive equilibrium $E^*=(s^*,x^*)$, the Jacobian matrix is:
\[
J(E^*)=\left[\begin{array}{cc}
\displaystyle -D-x^*\frac{\partial \mu}{\partial s}(s^*,x^*) & \displaystyle -x^*\frac{\partial
\mu}{\partial x}(s^*,x^*)-d(x^*)\\[4mm]
\displaystyle x^*\frac{\partial \mu}{\partial s}(s^*,x^*) & \displaystyle x^*\frac{\partial
\mu}{\partial x}(s^*,x^*) -x^*d'(x^*)
\end{array}\right]
\]
with determinant:
\[
det J(E^*)=Dx^*\left(d'(x^*)-\frac{\partial
\mu}{\partial x}(s^*,x^*)\right)+x^*\frac{\partial \mu}{\partial
s}(s^*,x^*)
\frac{d}{dx}[xd(x)](x^*) .
\]
One can easily check that it can be also written as
\[
det J(E^*)=-Dx^*\frac{\partial \mu}{\partial x}(s^*,x^*)\Gamma'(x^*)
\]
which shows an alternation of stability of the equilibriums $E^*$
depending on the sign of $\Gamma'(x^*)$.
We illustrate the possibility of having multiple-stability in the case
$\lambda_{v}<\lambda_{u}<S_{in}$ with the functions $\alpha$, $\beta$ given in
(\ref{alpha_beta_simples}), that provide the simple expression
(\ref{exp}) of the function $\bar p(\cdot)$, and Monod expressions
for functions $\mu_{u}$, $\mu_{v}$.
Even in this simple case, the expression of the function $\Gamma$
is too complicated to conduct an analytic study. Figure
\ref{figbistability} presents the phase portrait of the reduced dynamics
(\ref{chem_attach_red_D}) and shows its bi-stability for the numerical values of the parameters
that have been chosen.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=8.5cm]{phaseportrait.pdf}
\caption{\label{figbistability} Example of bi-stability with
$\mu_{u}(s)=\frac{2s}{1+s}$, $\mu_{v}(s)=\frac{1.5 s}{0.8+s}$,
$D_{u}=1$, $D_{v}=0.5$, $S_{in}=0.9$, $D=1$, $a/b=4$.}
\end{center}
\end{figure}
In the reference \cite{F13}, it is shown that under the additional assumption that the
map $x^* \mapsto x^*\bar p(x^*)$ is increasing, the multiplicity can
indeed occur only when $\lambda_{u}>\lambda_{v}$, and that generically
each equilibrium is necessarily either a stable node or a saddle
point.
Therefore, Tikhonov's Theorem, that has been recalled in Section
\ref{SecFlocsLentRapide}, allows to claim that for any initial condition of the system
(\ref{chem_attach}) such that
$(s_{0},x_{0})$ does not belong to the stable manifold of a saddle
equilibrium of the reduced dynamics (\ref{chem_attach_red_D}), the solution $s(\cdot)$,
$x(\cdot)$ converges to the solution of the reduced dynamics on the
$[0,+\infty)$ time interval, that is for almost any initial condition.
\medskip
Finally, this shows that multiple stability can occur in the chemostat
model with attachment and distinct removal rates, even though the
growth functions are monotonically increasing.
This fact is quite remarkable comparing to the classical chemostat
model (i.e. without attachment) for which a multiple stability is
possible only for non-monotonic growth functions (see for instance
\cite{HLRS17}).
Nevertheless, the analysis of all the generic behaviors of the solutions of the model with
several species (and different removal rates) remains today an open
problem. Dynamics in dimension higher than two potentially reserve a richness
of possible behaviors. In particular, the possibility of having
unstable nodes leave open the possibilities of having limit cycles, as
illustrated in \cite{FRS16}.
\section{Conclusion}
In this work we have proposed a generic framework of chemostat models
with free and attached biomass compartments. Under
general assumptions, we have shown that a coexistence of the two forms
is possible and leads to a unique positive equilibrium which is
moreover globally asymptotically stable.
When the assumptions about fast attachment and detachment are
justified, we have also shown that reduced
models with the total biomass instead of planktonic and
attached ones provide natural extensions of the classical chemostat
model with a density-dependent growth function,
such as in the Contois model \cite{C59}. This allows coexistence of multiple
species when each of them can be present in the two forms: planktonic
and attached (with same or different species).
We have also shown that the consideration of different removal rates
for the free and attached biomass could lead to some non-intuitive
behaviors, such as multiple stability, that is today widely not well
understood in presence of several species.
\begin{acknowledgement}
This work has been initiated in the ``DISCO'' project funded by the
French National Research Agency (ANR) in the SYSCOMM program.
The author warmly thanks T. Sari, C. Lobry, J. Harmand and R. Fekih-Salem,
whose PhD work having inspired the present paper.
\end{acknowledgement}
\bibliographystyle{plain}
|
{
"timestamp": "2018-03-08T02:08:05",
"yymm": "1803",
"arxiv_id": "1803.02634",
"language": "en",
"url": "https://arxiv.org/abs/1803.02634"
}
|
\section{Introduction\label{intro}}
Young massive stars in OB associations photoionize the surrounding gas, creating an HII region, and their powerful stellar winds can inflate a bubble around the star cluster. Magnetic fields are important to the dynamics of these structures \citep{Tomisaka:1990,Ferriere:1991,Vallee:1993,Tomisaka:1998,Haverkorn:2004,Sun:2008,Stil:2009}, and they can elongate the cavity preferentially in the direction of the magnetic field and thicken the shell perpendicular to the field \citep{Ferriere:1991,deAvillez:2005,Stil:2009}, causing deviations from the classical structure of the \citet{Weaver:1977} wind-blown bubble. Knowledge of the magnitude and direction of the magnetic field within stellar bubbles and HII regions is important for simulations and for understanding how the magnetic field interacts with and modifies these structures.
In previous work (i.e., \citealt{Savage:2013} and \citealt{Costa:2016}), we investigated whether the Galactic magnetic field is amplified in the shell of the Rosette Nebula, an HII region and stellar bubble associated with NGC 2244 ($\ell$ = 206.5\ddeg, $b$ = --2.1\ddeg). Other similar work investigating magnetic fields near massive star clusters has been done by \citet{Harvey:2011} and \citet{Purcell:2015}. In this work, we continue our investigation of how HII regions and stellar bubbles modify the ambient Galactic magnetic field by considering another example of a young star cluster and an HII region that appears to be formed into a shell by the effect of stellar winds.
\subsection{Faraday Rotation and Magnetic Fields in the Interstellar Medium}
Faraday rotation measurements probe the line of sight (LOS) component of the magnetic field in ionized parts of the interstellar medium (ISM), provided there is an independent estimate of the electron density. Faraday rotation is the rotation in the plane of polarization of a wave as it passes through magnetized plasma and is described by the equation
\begin{equation}
\chi=\chi_{0}+\left[\left(\frac{e^{3}}{2\pi m_{e}^{2}c^{4}}\right)\int{n_e\ \mathbf{B}\cdot \textrm{d}\mathbf{s}}\right]\lambda^{2},
\label{eq:rmorg}
\end{equation}
where $\chi$ is the polarization position angle, $\chi_0$ is the intrinsic polarization position angle, the quantities in the parentheses are the usual standard physical constants in cgs units, \textit{n$_{\textrm{e}}$} is the electron density, \textbf{B} is the vector magnetic field, d\textbf{s} is the incremental path length interval along the LOS, and $\lambda$ is the wavelength. We define the terms in the square bracket as the rotation measure, RM, and we can express the RM in mixed but convenient interstellar units as
\begin{equation}
\textnormal{RM}=0.81\in
n_{e} \ (\text{cm$^{-3}$}) \ \mathbf{B} \ (\mu\text{G})\cdot \textrm{d}\mathbf{s} \text{ (pc) rad m$^{-2}$.}
\label{eq:rmprat}
\end{equation}
\subsection{The HII Region and Stellar Bubble Associated with the W4 Complex\label{sec:structure}}
The HII region and stellar bubble of interest for the present study is IC 1805, which is located in the Perseus Arm. The star cluster responsible for the HII region and stellar bubble is OCl 352, which is a young cluster (1--3 Myr) \citep{Basu:1999}. OCl 352 has 60 OB stars \citep{Shi:1999}. Three of these are the O stars HD 15570, HD 15558, and HD 15629, and they have mass loss rates between 10$^{-6}$ and 10$^{-5}$ \textrm{M}$_{\odot}$ yr$^{-1}$~\citep{Massey:1995} and terminal wind velocities of 2200 -- 3000 km s$^{-1}$~\citep{Garmany:1988,Groenewegen:1989,Bouret:2012}. We adopt the nominal center of the star cluster to be R.A.(J2000) = 02$^h$ 23$^m$ 42$^s$, decl.(J2000) = +61\ddeg 27$'$ 0$''$ ($\ell$ = 134.73, \textit{b} = +0.92) \citep{Guetter:1989} and a distance of 2.2 kpc to IC 1805 to conform with previous studies of the region (e.g., \citealt{Normandeau:1996,Dennison:1997,Reynolds:2001,Terebey:2003,Gao:2015}). In the literature, other distance values include: 2.35 kpc \citep{Massey:1995,Basu:1999,West:2007,Lagrois:2012}, 2 kpc \citep{Dickel:1980}, 2.04 kpc \citep{Feigelson:2013,Townsley:2014}, and 2.4 $\pm$ 0.1 kpc \citep{Guetter:1989}.
We refer to the HII region between --0.2\ddeg~$<$ \textit{b} $<$ 2\ddeg~as IC 1805. This structure is also known as the Heart Nebula for its appearance at optical wavelengths. We differentiate this region from the northern latitudes that constitute the W4 Superbubble \citep{Normandeau:1996,West:2007,Gao:2015}, and we use the nomenclature of W4 to describe the entire region, which includes IC 1805 and the W4 Superbubble. Below we summarize the structure of IC 1805 and Figure \ref{fig:cartoon} is a cartoon diagram of the structure described here.
\begin{itemize}
\item[--] \textit{South.} On the southern portion of IC 1805, there is a loop structure of ionized material at 134\ddeg $<$ $\ell$ $<$ 136\ddeg, \textit{b} $<$ 1\ddeg, which we call the southern loop. \citet{Terebey:2003} find that at far infrared and radio wavelengths, the shell structure is well defined and ionization bounded, since the ionized gas lies interior to the dust shell. However, they also find that there is warm dust that extends past the southern loop and a faint ionized halo (see their Figure 6). \citet{Terebey:2003} argue that the shell is patchy and inhomogeneous in density, which allows ionizing photons to escape. \citet{Gray:1999} discuss extended emission surrounding IC 1805 and suggest that it may be evidence of an extended HII region \citep{Anantharamaiah:1985}. Also surrounding IC 1805 are patchy regions of HI \citep{Braunsfurth:1983,Hasegawa:1983,Sato:1990} and CO \citep{Heyer:1998,Lagrois:2009}.
\citet{Terebey:2003} model the structure of the southern loop using radio continuum data. They assume a spherical shell and place OCl 352 at the top edge of the bubble instead of at the center to accommodate spherical symmetry (see their Figures 4 and 5). The center of their shell model is at ($\ell$, $b$) = (135.02\ddeg, 0.42\ddeg). They find an inner radius of 30 arcmin (19 pc) and a shell thickness of 10 arcmin (6 pc) and 2.5 arcmin (2 pc) for a thick and thin shell model, respectively. \citet{Terebey:2003} report electron densities of 10 cm$^{-3}$~and 20 cm$^{-3}$~for the thick and thin shell models, respectively (see Section 3.5 and Table 3 of \citealt{Terebey:2003}). While we utilize and discuss these models in the following sections, the center position of the shell in \citet{Terebey:2003} was selected to fit the ionized shell, and as such, the shell parameters should only be used to describe the bottom of IC 1805. For latitudes near the star cluster, the model fails, as the star cluster is at the top edge of the bubble instead of at the center.
\item[--] \textit{East.} On the eastern edge of IC 1805 ($\ell$ $>$ 134.6\ddeg, \textit{b} $<$ 0.9\ddeg), \citet{Terebey:2003} find that warm dust extends outside the loop boundary and suggest that if the warm dust is associated with the ionized gas, then the bubble has blown out on the eastern side of IC 1805.
At the Galactic latitude equal to the star cluster, the ionized gas appears to be pinched \citep{Basu:1999}, which is usually caused by higher densities. There is a clump of CO emission in the vicinity of the eastern pinch at ($\ell$, $b$) = (135.2\ddeg, 1.0\ddeg) \citep{Lagrois:2009}, and there is HI emission on the eastern edge at ($\ell$, $b$) $\geq$ (136\ddeg, 0.5\ddeg) (see Figure 1 of \citealt{Sato:1990}).
\item[--] \textit{West.} On the western edge of IC 1805 is the W3 molecular cloud and the W3 complex, which hosts a number of compact HII regions and young stellar objects (see \citealt{Bik:2012} and their Figure 1). \citet{Dickel:1980} modeled the structure of W3, which is thought to be slightly in front of W4, and they argue that the advancement of the IC 1805 ionization front and shock front into the W3 molecular cloud may have triggered star formation. \citet{Moore:2007} similarly conclude that the W3 molecular cloud has been compressed on one side by the expansion of IC 1805. While infrared sources nestled between the western edge of IC 1805 and eastern edge of the W3 molecular cloud are thought to be the product of this interaction , W3 Main, W3 (OH), and W3 North are thought to be sites of triggered star formation from IC 1795, which is part of W3 as well and not from the expansions of the ionization front \citep{Nakano:2017, Jose:2016,Kiminki:2015}. There is therefore uncertainty regarding a physical connection between W3 and IC 1805.
\item[--] \textit{North.} North of OCl 352, the bubble opens up into what is called the W4 Superbubble \citep{Normandeau:1997,Dennison:1997,West:2007,Gao:2015}, which is a sealed ``egg-shaped'' structure that extends up to \textit{b} $\sim$ 7\ddeg~ \citep{Dennison:1997,West:2007}. At the latitude of the star cluster, \citet{Lagrois:2009} estimate the distance between the eastern and western shell to be $\sim$ 1.2\ddeg~(46 pc) that increases in size up to 1.6\ddeg~(61 pc) at \textit{b} = 1.8\ddeg~(see Figure 11 of \citealt{Lagrois:2009}). At higher latitudes, \citet{Dennison:1997} model the thickness of the shell to be between 10--20 pc (16 -- 31 arcminutes) from H$\alpha$ observations.
The ``v''-shaped feature seen in Figure \ref{fig:w4} at ($\ell$, $b$) $\sim$ (134.8\ddeg, 1.35\ddeg) is prominent in the ionized emission, and \citet{Heyer:1996} report a cometary-shaped molecular cloud near ($\ell$, $b$) $\sim$ (134.8\ddeg, 1.35\ddeg). The alignment of the cometary cloud, as it is pointed towards IC 1805, suggests that the UV photons from the star cluster are responsible for the ``v'' shaped feature in the ionized emission on the side closest to the star cluster \citep{Dennison:1997,Taylor:1999}. \citet{Lagrois:2009} argue, from radial velocity measurements, that the cloud is located on the far side of the bubble wall, and while it may appear to be a cap to the bubble connecting to the southern loop, it is simply a projection effect. As such, the ridge of ionized material directly north of OCl 352 is not the outer radius of the shell but is part of the rear bubble wall.
\item [--] \textit{PDR.} The HI and molecular emission near the southern ($\ell$ $<$ 0.9\ddeg) portions of IC 1805 suggest that a Photodissociation Region (PDR){} has formed exterior to the HII region. PDRs are the transition layer between the fully ionized HII region and molecular material, where far UV photons can propagate out and photodissociate molecules. We discuss the importance and observational evidence of a PDR in Section \ref{sec:pdr}.
\end{itemize}
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.6\textwidth]{f1.pdf}
\caption[]{Cartoon of structure of IC 1805 and the surrounding region, which includes the W3 molecular cloud, W3 (blue filled circle), and the W4 Superbubble. The solid black lines represent the bright ionized shell in Figure \ref{fig:w4}, the dotted lines show molecular material from \citet{Lagrois:2009}, and the gray shading represents the extended halo or PDR. The star represents the center of the exciting star cluster, OCl 352.}
\label{fig:cartoon}
\end{figure}
There is an extensive literature on the W4 region and its relationship to W3, dealing with the morphology \citep{Dickel:1980,Dickel:1980b,Braunsfurth:1983,Normandeau:1996,Dennison:1997,Heyer:1998,Taylor:1999,Basu:1999, Terebey:2003,Lagrois:2009,Lagrois:2009b,Stil:2009} and star formation history \citep{Carpenter:2000,Oey:2005}. In the following paragraphs, we summarize those results from the literature that are most relevant to our polarimetric study and inferences on magnetic fields in this region.
Measurements of the total intensity and polarization of the Galactic nonthermal emission in the vicinity of HII regions are of interest because the HII regions and environs act as a Faraday-rotating screen inserted between the Galactic emission behind the HII region and that in front. Few radio polarimetric studies exist in the literature to date of the IC 1805 stellar bubble. \citet{Gray:1999} present their polarimetric results of the W3/W4 region at 1420 MHz with the Dominion Radio Astrophysical Observatory (DRAO) Synthesis Telescope. They find zones of strong depolarization near the HII regions, particularly in the south, where there is a halo of extended emission around IC 1805.
They conclude that RM values on order 10$^3$ rad m$^{-2}$~and spatial RM gradients must exist to explain the depolarization near the HII region. More recently, \citet{Hill:2017} present results of their polarimetric study of the Fan region ($\ell$ $\sim$ 130\ddeg, --5\ddeg $\leq$ \textit{b} $\leq$ +10\ddeg), which is a large structure in the Perseus arm that includes W3/W4. While the focus of their study was not on W4 specifically, they find similar results to \citet{Gray:1999} in that there is sufficient Faraday rotation to cause beam depolarization in the regions of extended emission.
In the W4 Superbubble, \citet{West:2007} determined the LOS magnetic field strength by estimating depolarization effects along adjacent lines of sight. Using estimates of the shell thickness and the electron density from \citet{Dennison:1997}, \citet{West:2007} estimate B$_{\textrm{LOS}}$~$\sim$ 3.4 -- 9.1 $\mu$G for lines of sight at \textit{b} $>$ 5\ddeg. \citet{Gao:2015} also report B$_{\textrm{LOS}}$~estimates in the W4 Superbubble by assuming a passive Faraday screen model \citep{Sun:2007} and measuring the polarization angle for lines of sight interior and exterior to the screen. For the western shell ($\ell$ $\sim$ 132.5\ddeg, 4\ddeg $<$ \textit{b} $<$ 6\ddeg) and the eastern shell ($\ell$ $\sim$ 136\ddeg, 6\ddeg $<$ \textit{b} $<$ 7.5\ddeg) in the superbubble, \citet{Gao:2015} report negative RMs between --70 and --300 rad m$^{-2}$~in the western shell and positive RMs on order +55 rad m$^{-2}$~ in the eastern shell. \citet{Gao:2015} conclude that the sign reversal is expected in the case of the Galactic magnetic field being lifted out of the plane by the expanding bubble. With H$\alpha$ estimates from \citet{Dennison:1997} for the electron density and geometric arguments for the shell radii of the W4 Superbubble, \citet{Gao:2015} estimate $|$B$_{\textrm{LOS}}$$|$ $\sim$ 5 $\mu$G.
\citet{Stil:2009} compare their magnetohydrodynamic simulations of superbubbles to the W4 Superbubble. In general, they find that the largest Faraday rotation occurs in a thin region around the cavity, and inside the cavity, it would be smaller. They also present two limiting cases for the orientation of the Galactic magnetic field with respect to the line of sight, and the consequences for the RMs through the shell. If the Galactic magnetic field is perpendicular to the observer's line of sight, then the contributions to the RM from the front and rear bubble wall would be of equal but opposite magnitude, except for small asymmetries which would lead to low RMs ($\sim$~20 rad m$^{-2}$) through the cavity. This requires the magnetic field to be bent by the bubble to have a non-zero line of sight component. If the Galactic magnetic field is parallel to the line of sight, then the RMs through the front and rear bubble wall reinforce each other, and there are high RMs for lines of sight through the shell. In this case, there are higher RMs ($\sim${} 3 $\times$ 10$^3$ rad m$^{-2}$) everywhere.
There are also studies of the magnetic field for W3. From HI Zeeman observations, \citet{vanderWerf:1990} conclude that the B$_{\textrm{LOS}}$~has small-scale structures that can vary on order of 50 $\mu$G over $\sim$ 9 arcsec scales. \citet{Roberts:1993} report values of the LOS magnetic field from HI Zeeman observations towards three resolved components of W3. The three components are near ($\ell$, $b$) $\sim$ (133.7\ddeg, 1.21\ddeg), with a maximum separation of 1.5 arcmin, and the LOS magnetic field is between --50 $\mu$G and +100 $\mu$G. \citet{Balser:2016} observed carbon radio recombination line (RRL) widths to estimate the total magnetic field strength in the photodissociation region (see \citealt{Roshi:2007} for details). They report B$_{\textrm{tot}}$ = 140 -- 320 $\mu$G near W3A (133.72\ddeg, 1.22\ddeg) and argue that for a random magnetic field, B$_{\textrm{tot}}$ = 2 $|$B$_{\textrm{LOS}}$$|$, which would then be consistent with the \citet{Roberts:1993} estimates of the B$_{\textrm{LOS}}$. It should be noted that these magnetic field strengths are substantially larger than those inferred for the W4 Superbubble on the basis of polarimetry of the Galactic background (see text above).
In this paper, we present new Faraday rotation results for IC 1805 to investigate the role of the magnetic field in the HII region and stellar bubble. As in \citet{Savage:2013} and \citet{Costa:2016}, we utilize an arguably simpler and more direct method of inferring the LOS component of the magnetic field in HII regions. This is the measurement of the Faraday rotation of nonthermal background sources (usually extragalactic radio sources) whose lines of sight pass through the HII region and its vicinity. In Section \ref{sec:obs}, we describe the instrumental configuration and observations, including source selection. Section \ref{sec:dataredux} details the data reduction process, including the methods used to determine RM values. In Section \ref{sec:obsres}, we report the results of the RM analysis and discuss Faraday rotation through the W4 complex in Section \ref{sec:fr}. We present models for the RM within the HII region and stellar bubble in Section \ref{sec:models}. We discuss our observational results and their significance for the nature of IC 1805 in Section \ref{sec:results} and compare the results of this study with our previous study of the Rosette nebula in Section \ref{sec:rosette}. We discuss future research in Section \ref{sec:fut}, and present our conclusions and summary in Section \ref{sec:sum}.
\section{Observations}\label{sec:obs}
\subsection{Source Selection\label{sec:sourceselect}}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.9\textwidth]{f2.pdf}
\caption[Radio Continuum map at 1.42GHz of W4]{Mosaic of IC 1805 from the Canadian Galactic Plane Survey at 1.42 GHz, with Galactic longitude and latitude axes. The lines of sight listed in Table \ref{tab:sources} are the red and blue symbols, where positive RMs are blue and negative RMs are red. The green and purple symbols are RM values from \citet{Taylor:2009} or \citet{Brown:2003}, where positive RMs are green and negative RMs are purple. We utilize the naming scheme from Table \ref{tab:sources} for the RM values from the literature for ease of reference, but we omit the ``W4-'' prefix in this image for clarity. The size of the plotted symbols is proportional to the $|$RM$|$ value.}
\label{fig:w4}
\end{figure}
Our criteria for source selection were identical to \citet{Savage:2013} and \citet{Costa:2016} in that we searched the National Radio Astronomy Observatory Very Large Array Sky Survey (NVSS, \citealt{Condon:1998}) database for point sources within 1\ddeg~of OCl 352 (the ``I'' sources) with a minimum flux density of 20 mJy. We also searched in an annulus centered on the star cluster with inner and outer radii of 1\ddeg~and 2\ddeg~for outer sources (``O'') to measure the background RM due to the general ISM. We identified 31 inner sources and 26 outer sources in the region. We then inspected the NVSS postage stamps to ensure that they were point sources at the resolution of the NVSS ($\sim${} 45 arcseconds). We discarded sources that showed extended structure similar to Galactic sources. We selected 24 inner sources and 8 outer sources from this final list.
\begin{table}
\centering
\caption{ List of Sources Observed}
\begin{threeparttable}
\centering
\small
\begin{tabular}{ccccccccc}
\hline
Source & $\alpha$(J2000) & $\delta$(J2000) & $\emph{l}$ & $\emph{b}$ & $\xi$\tnote{a} & S$_{4.33\textrm{GHz}}$ & m \\
Name & h m s & $^o$ $'$ $''$ & ($^o$) & ($^o$) & (arcmin) & (mJy) & ($\%$)\\
\hline
W4-I1 & 02 30 16.2 & +62 09 37.9 & 134.19 & 1.47 & 46.0 & 77 & 4 \\
W4-I2 & 02 38 34.2 & +61 08 46.6 & 135.49 & 0.91 & 46.2 & 5 & 12 \\
W4-I3 & 02 36 45.5 & +60 55 48.8 & 135.38 & 0.63 & 42.8 & 82 & 3 \\
W4-I4 & 02 27 59.8 & +62 15 44.0 & 133.91 & 1.47 & 58.9 & 46 & 10 \\
W4-I5\tnote{b} & 02 28 01.6 & +62 02 16.7 & 133.99 & 1.26 & 48.4 & --- & --- \\
W4-I6 & 02 38 19.9 & +61 08 03.5 & 135.47 & 0.89 & 44.8 & 12 & 11 \\
W4-I7\tnote{b} & 02 27 33.8 & +61 55 58.1 & 133.98 & 1.14 & 46.6 & -- & --- \\
W4-I8 & 02 38 10.1 & +62 08 57.0 & 135.05 & 1.81 & 57.1 & 47 & 9 \\
W4-I9 & 02 28 21.6 & +61 28 36.5 & 134.23 & 0.75 & 31.1 & --- & --- \\
W4-I10\tnote{b} & 02 29 13.0 & +61 00 53.4 & 134.50 & 0.36 & 36.2 & --- & --- \\
W4-I11 & 02 25 15.2 & +61 19 14.4 & 133.94 & 0.47 & 54.0 & 39 & 3 \\
W4-I12 & 02 35 20.6 & +62 16 02.3 & 134.70 & 1.79 & 52.5 & 77 & 2 \\
W4-I13 & 02 28 25.1 & +60 56 20.2 & 134.44 & 0.25 & 43.5 & 16 & 6 \\
W4-I14 & 02 36 19.2 & +61 44 05.5 & 135.01 & 1.35 & 31.4 & 2 & 16 \\
W4-I15 & 02 33 36.1 & +60 37 40.4 & 135.14 & 0.20 & 49.8 & 27 & 4 \\
W4-I16 & 02 36 56.8 & +61 57 58.6 & 134.99 & 1.59 & 43.3 & 35 & 0 \\
W4-I17 & 02 34 08.8 & +61 40 35.5 & 134.80 & 1.19 & 17.1 & 9 & 16 \\
W4-I18 & 02 31 56.3 & +61 25 50.9 & 134.65 & 0.87 & 5.6 & 24 & 2 \\
W4-I19\tnote{c} & 02 40 31.7 & +61 13 45.9 & 135.68 & 1.09 & 57.8 & 11 & 3 \\
W4-I20 & 02 27 03.9 & +61 52 24.9 & 133.94 & 1.07 & 47.6& 457 & 0 \\
W4-I21 & 02 30 44.5 & +61 05 30.2 & 134.64 & 0.50 & 25.9 & 5 & 7 \\
W4-I22\tnote{b} & 02 26 07.8 & +61 56 43.7 & 133.82 & 1.09 & 55.4 & --- & --- \\
W4-I23 & 02 40 30.9 & +61 47 10.1 & 135.45 & 1.59 & 59.3 & 3 & 0 \\
W4-I24 & 02 37 45.1 & +60 37 31.4 & 135.61 & 0.40 & 61.5 & 20 & 10 \\
W4-O1 & 02 41 33.9 & +61 26 29.5 & 135.70 & 1.33 & 63.5 & 377 & 0 \\
W4-O2 & 02 35 37.8 & +59 56 29.5 & 135.64 & -0.33 & 93.0 & 107 & 0 \\
W4-O4 & 02 44 57.7 & +62 28 06.5 & 135.64 & 2.43 & 105.8 & 747 & 0 \\
W4-O5 & 02 21 52.6 & +60 10 03.2 & 133.96 & -0.75 & 110.3 & 94 & 0 \\
W4-O6 & 02 31 59.2 & +62 50 34.1 & 134.12 & 2.18 & 83.7 & 120 & 4 \\
W4-O7 & 02 43 35.6 & +61 55 54.6 & 135.72 & 1.88 & 82.7 & 52 & 0 \\
W4-O8 & 02 23 04.5 & +60 58 19.6 & 133.82 & 0.05 & 75.2 & 40 & 0 \\
W4-O10 & 02 20 26.2 & +61 34 46.2 & 133.31 & 0.51 & 88.0 & 64 & 3 \\
\hline
\end{tabular}
\label{tab:sources}
\begin{tablenotes}
\item[a] Angular distance between the line of sight and a line of sight through the center of the star cluster.
\item[b] NVSS position. No source detected in the Stokes I map in any frequency bin.
\item[c] High Mass X-Ray Binary LSI +61\ddeg303.
\end{tablenotes}
\end{threeparttable}
\end{table}
The sources are listed in Table \ref{tab:sources}, where the first column lists the source name in our nomenclature. The second and third columns list the right ascension ($\alpha$) and declination ($\delta$) of the observed sources. The positions are determined with the {\sc imfit} task in CASA, which fits a 2D Gaussian to the intensity distribution at 4.33 GHz. Columns four and five give the Galactic longitude ($\ell$), Galactic latitude (\textit{b}), which is converted from $\alpha$ and $\delta$ using the Python \textit{Astropy} package, and the angular separation from the center of the nebula ($\xi$) is given in column six. Column seven lists the flux density at 4.33 GHz calculated with {\sc imfit}, and column eight gives the mean percent linear polarization (m = \textit{P}/\textit{I}) as measured across the eight 128 MHz maps and assuming a \textit{Faraday simple}{} source. Figure \ref{fig:w4} is a radio continuum mosaic from the Canadian Galactic Plane Survey (CGPS) \citep{Taylor:2003,Landecker:2010} with the location of the sources, along with the names, indicated with filled circles.
\subsection{VLA Observations\label{sec:vlaobs}}
\begin{table}[!htb]
\centering
\begin{threeparttable}
\caption{Log of Observations \label{tab:logofobs}}
\begin{tabular}{p{0.54\linewidth}
p{0.3\linewidth}}
\hline
VLA Project Code & 13A-035 \\
Date of Observations & 2013 July 10, 13, 16, and 17\\
Number of Scheduling Blocks & 4 \\
Duration of Scheduling Blocks (h) & 4\\
Frequencies of Observation\tnote{a}~ (GHz) & 4.850; 7.250\\
Number of Frequency Channels per IF & 512\\
Channel Width (MHz) & 2 \\
VLA array & C \\
Restoring Beam (diameter) & 4\farcs81\\
Total Integration Time per Source & 18--25 minutes\tnote{b}\\
RMS Noise in Q and U Maps ($\mu$Jy/beam) & 39\tnote{c}\\
RMS Noise in RM Synthesis Maps ($\mu$Jy/beam) & 23\tnote{d} \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] The observations had 1.024 GHz wide intermediate frequency bands (IFs) centered on the frequencies listed, each composed of eight 128 MHz wide subbands.
\item [b] The ``O'' sources (see Table \ref{tab:sources}) averaged 18 minutes, and the ``I'' sources, being weaker, were between 22--25 minutes.
\item [c] This number represents the average rms noise level for all the Q and U maps.
\item[d] Polarized sensitivity of the combined RM Synthesis maps.
\end{tablenotes}
\end{threeparttable}
\end{table}
We observed 32 radio sources with the NSF's Karl G. Jansky Very Large Array (VLA)\footnote{\footnotesize{The Karl G. Jansky Very Large Array is an instrument of the National Radio Astronomy
Observatory (NRAO). The NRAO is a facility of the National Science Foundation,
operated under cooperative agreement with Associated Universities, Inc.}} whose lines of sight pass through or near to the shell of the IC 1805 stellar bubble. Table \ref{tab:logofobs} lists details of the observations. Traditionally, polarization observations require observing a polarization calibrator source frequently over the course of an observation to acquire at least 60\ddeg~of parallactic angle coverage. This is done to determine the instrumental polarization (D-factors, leakage solutions). Since the completion of the upgraded VLA, shorter scheduling blocks, typically less than 4 hours in duration, have become a common mode of observation. It is difficult, if not impossible, with very short scheduling blocks to acquire enough parallactic angle coverage to measure the instrumental calibration with a polarized source. Another method of determining the instrumental polarization is to observe a single scan of an unpolarized source. This technique can be used with shorter scheduling blocks.
In this project we calibrated the instrumental polarization using both techniques. We used the source J0228+6721, observed over a wide range of parallactic angle, and also made a single scan of the unpolarized source 3C84. Use of the CASA task \textsc{polcal}~on the J0228+6721 data solved for the instrumental polarization, determined by the antenna-specific D factors \citep{Bignell:1982}, which are complex, as well as the source polarization (\textit{Q} and \textit{U} fluxes). In the case of 3C84, \textsc{polcal}~solves only for the D factors.
We find no significant deviations between these two calibration methods, indicating accurate values for the instrumental polarization parameters.
3C138 and 3C48 functioned as both flux density and polarization position angle calibrators. J0228+6721 was used to determine the complex gain of the antennas as a function of time as well to as serve as a check, as described above, for the D-factors. We observed the program sources for 5 minute intervals and interleaved the observations of J0228+6721. There was one observation of 3C138, 3C48, and 3C84 each. For our final data products, we utilized 3C84 as the primary leakage calibrator and 3C138 as the flux density and polarization position angle calibrator.
\section{Data Reduction\label{sec:dataredux}}
The data were reduced and imaged using the NRAO Common Astronomy Software Applications (CASA)\footnote{\footnotesize{For further reference on data reduction, see the NRAO Jansky VLA tutorial ``EVLA Continuum Tutorial 3C391'' (http$://$casaguides.nrao.edu$/$index.php$?$title$=$EVLA$_{-}$Continuum$_{-}$Tutorial$_{-}$3C391)}} version 4.5. The procedure for the data reduction as described in Section 3 of \citet{Costa:2016} is identical to the procedure we employed in this study. The only difference for the current data set is that in the CASA task \textsc{clean}, we utilized \textit{Briggs} weighting with the ``robust'' parameter set to 0.5, which adjusts the weighting to be slightly more \textit{natural} than \textit{uniform}. \textit{Natural} weighting has the best signal/noise ratio at the expense of resolution, while \textit{uniform} is the opposite. \textit{Briggs} weighting allows for intermediate options. As in our previous work, we also implemented a cutoff in the (\textit{u}, \textit{v}) plane for distances $<$ 5000 wavelengths to remove foreground nebular emission.
Similar to \citet{Costa:2016}, we had two sets of data products after calibration and imaging. The first set of images consisted of radio maps (see Figures \ref{fig:I18} and \ref{fig:I24}) of each Stokes parameter, formed over a 128 MHz wide subband for each source. These images were inputs to the $\chi$($\lambda^{2}$)~analysis (Section \ref{sec:chilam}), and there were typically 14 individual maps for each source per Stokes parameter.
The second set of images consisted of maps of \textit{I}, \textit{Q}, and \textit{U} in 4 MHz wide steps across the entire bandwidth using the \textsc{clean}~mode ``channel'', which averages two adjacent 2 MHz channels.
Ideally, changes in \textit{Q}{} and \textit{U}{} should only be due to Faraday rotation; however, the spectral index can affect \textit{Q}{} and \textit{U}{} independently of the RM, which can be interpreted as depolarization. RM Synthesis does not, by default, account for the spectral index, so a correction must be applied prior to performing RM Synthesis (see Section 3 of \citealt{Brentjens:2005}). We first determine the spectral index, $\alpha$, of each source from a least-squares fit to the log of the flux density, $S_\nu$, and the log of the frequency, $\nu$. We adopt the convention that $S_\nu$ $\sim$~ $\nu^{-\alpha}$. We use the center frequency, $\nu_c$, of the band and the measured value of \textit{Q} and \textit{U} at each frequency, $\nu$, to find \textit{Q$_o$} and \textit{U$_o$} using the relationship \[Q = Q_o\left(\frac{\nu}{\nu_c}\right)^{-\alpha} \textrm{ and } \ U = U_o\left(\frac{\nu}{\nu_c}\right)^{-\alpha}. \] The final images consisted of approximately 336 maps per source, per Stokes parameter, as inputs for the RM Synthesis analysis (Section \ref{sec:rmsyn}).
\begin{figure}[!htb]
\centering
\subfloat[\label{fig:I18}]{
\includegraphics[width=0.4\textwidth]{f3a.pdf}}
\quad
\subfloat[\label{fig:I24}]{
\includegraphics[width=0.42\textwidth]{f3b.pdf}}
\caption[CASA Map]{Map of (a) W4-I118 and (b) W4-I24 at 4913 MHz. The circle in the lower left is the restoring beam. The gray scale is the linear polarized intensity, \textit{P}, the vectors show the polarization position angle, $\chi$, and the contours are the Stokes \textit{I} intensity with levels of -2, -1 , 2, 10, 20, 40, 60, and 80$\%$ of the peak intensity, 21.5 mJy beam$^{-1}$ and 11.7 mJy beam$^{-1}$ for W4-I18 and W4-I24a, respectively. }
\label{fig:MAP}
\end{figure}
\subsection{Rotation Measure Analysis via a Least-Squares Fit to $\chi$ vs $\lambda^2$ \label{sec:chilam}}
The output of the CASA task \textsc{clean}~produces images in Stokes \textit{I, Q, U,} and \textit{V}. From these images, we generated maps with the task \textsc{immath}~of the linear polarized intensity \textit{P}, \[P = \sqrt{Q^2+U^2} \] and the polarization position angle $\chi$,
\begin{equation*}
\chi=\frac{1}{2}\tan^{-1}{\left(\frac{U}{Q}\right)}
\end{equation*}
for each source over a 128 MHz subband. Data that are below the threshold of 5$\sigma_{\textrm{Q}}$ are masked in the \textit{P} and $\chi$ maps, where $\sigma_{\textrm{Q}}$ = $\sigma_{\textrm{U}}$ is the rms noise in the \textit{Q} data. This threshold prevents noise in the \textit{Q} and \textit{U} data from generating false structure in the \textit{P} and $\chi$ maps. Examples of images are shown in Figure \ref{fig:MAP}, which displays the total intensity, polarized intensity, and polarization position angle for sources W4-I18 and W4-I24. W4-I18 is an example of a point source, or slightly resolved source. Twelve of the sources in Table \ref{tab:sources} were of this type and unresolved to the VLA in C array. Eight sources were like W4-I24, showing extended structure in the observations and potentially yielding RM values on more than one line of sight.
\begin{figure}[hbt!]
\centering
\subfloat[][\label{fig:I1chi}]{
\includegraphics[width=0.48\textwidth]{f4a.pdf}} \quad
\subfloat[][\label{fig:I18chi}]{
\includegraphics[width=0.48\textwidth]{f4b.pdf}}\\
\caption[Plot of $\chi(\lambda^{2})$]{Plot of the polarization position angle as a function of the square of the wavelength, $\chi(\lambda^{2})$, for the source (a) W4-I18, RM= +514 $\pm$ 12 rad m$^{-2}$, and (b) W4-I24a, RM = --658 $\pm$ 5 rad m$^{-2}$. Each plotted point results from a measurement in a single 128 MHz-wide subband. The gap in $\lambda^2${} coverage is due to the observation configuration that consisted of two 1.024 GHz wide IFs separated by $\sim${} 1.4 GHz.}
\label{fig:newpol}
\end{figure}
In the case of a single foreground magnetic-ionic medium responsible for the rotation of an incoming radio wave, the relation between $\chi$ and $\lambda^2$ is linear, and we calculate the RM through a least-squares fit of $\chi$($\lambda^{2}$). To measure $\chi$, we select the pixel that corresponds to the highest value of \textit{P} on the source in the 4338 MHz map, and we then measure $\chi$ at that location in each subsequent 128 MHz wide subband. Figure \ref{fig:newpol} shows two examples of the least-squares fit to $\chi$($\lambda^{2}$). The $\chi$ errors are \(\sigma_{\chi} = \frac{\sigma_{Q}}{2P},\) (\citealt{Everett:2001}, Equation 12).
\subsection{Rotation Measure Synthesis \label{sec:rmsyn}}
In additional to the least-squares fit to $\chi$($\lambda^{2}$), we performed Rotation Measure Synthesis \citep{Brentjens:2005}. The inputs to RM Synthesis are images in Stokes \textit{I}, \textit{Q}, and \textit{U} across the entire observed spectrum in 4 MHz spectral intervals. We refer the reader to Section 3.1.2 of \citet{Costa:2016} for a detailed account of our procedure, which follows the implementation of RM Synthesis as developed by \citet{Brentjens:2005}. The goal of RM synthesis is to recover the Faraday dispersion function $F(\phi)$. Here $\phi$, the Faraday depth, is a variable which is Fourier-conjugate to $\lambda^2$ (see \citealt{Costa:2016}, Equations 3 and 4), and has units of rad m$^{-2}$. We also refer to $F(\phi)$ as the ``Faraday spectrum''.
\begin{table}[htb!]
\centering
\caption{Rotation Measure Synthesis Parameters\label{tab:rmpar}}
\begin{threeparttable}
\centering
\begin{tabular}{p{2cm} p{4cm} p{9.5cm}}
\hline
$\Delta \lambda^2$ & 3.2 $\times$ 10$^{-3}$ (m$^2$) & Total bandwidth\tnote{a}. \\
$\lambda^2_{min}$ & 1.5 $\times$ 10$^{-3}$ (m$^2$) & Shortest observed wavelength squared. \\
$\delta \lambda^2$ & 4.8 $\times$ 10$^{-6}$ (m$^2$) & Width of a channel; Eq (35) \citet{Brentjens:2005}. \\
$\delta \phi$ & 1072\tnote{b}~ (rad m$^{-2}$) & FWHM of RMSF; Eq (61) \citet{Brentjens:2005}. \\
\\
\multirow{2}{*}{max-scale} & \multirow{2}{*}{2098 (rad m$^{-2}$)} & Sensitivity to extended Faraday structures; Eq (62) \citet{Brentjens:2005}. \\
\\
\multirow{3}{*}{$|\phi_{max}|$} & \multirow{3}{*}{3.6 $\times$ 10$^{5}$ (rad m$^{-2}$)} & Maximum detectable Faraday depth before bandwidth depolarization; Eq (63) \citet{Brentjens:2005}. \\ \\ \hline
\end{tabular}
\begin{tablenotes}
\item[a] This bandwidth includes the frequencies not observed that lie between our two IFs. They are set to 0 via the weighting function, W($\lambda^2$).
\item[b] Since flagging for RFI and bad antennas were done individually for each scheduling block, the FWHM of the RMSF can vary slightly from source to source. However, these slight variations are not significant in our interpretation of the RM values report in this paper.
\end{tablenotes}
\end{threeparttable}
\end{table}
\textit{F($\phi$)}~is recovered via an \textsc{rmclean}~algothrim \citep{Heald:2009,Bell:2012}, and we applied a 7$\sigma$ cutoff, which is above the amplitude at which peaks due to noise are likely to arise \citep{Brentjens:2005,Macquart:2012,Anderson:2015}. The \textsc{rmsynthesis} algorithm initially searched for peaks in the Faraday spectrum using a range of $\phi$ $\pm$ 10,000 rad m$^{-2}$~at a resolution of 40 rad m$^{-2}$~to determine if there were significant peaks at large values of $|\phi|$. Then, we performed a finer search at $\phi$ = $\pm$ 3000 rad m$^{-2}$~at a resolution of 10 rad m$^{-2}$. The RM Synthesis parameters, such as the full-width-at-half-maximum (FWHM) of the rotation measure spread function (RMSF) and the maximum detectable Faraday depth, are given in Table \ref{tab:rmpar}.
\begin{figure}[hbt!]
\centering
\subfloat[\label{fig:psynmap}]{
\includegraphics[width=0.48\textwidth]{f5a.pdf}}
\quad
\subfloat[\label{fig:rmsynmap}]{
\includegraphics[width=0.48\textwidth]{f5b.pdf}}
\caption[Plot of RM Synthesis Map]{(a) Linear polarization map and (b) RM map of W4-I24 from the RM Synthesis analysis. In both images, the data cube was flattened over the Faraday depth axes, and a threshold of 7$\sigma$ was applied. W4-I24 has extended structure, so there are two peaks, which are also present in the CASA maps (Figure \ref{fig:I24}).}
\end{figure}
As in \citet{Costa:2016}, we utilized an IDL code for the \textsc{rmsynthesis} and \textsc{rmclean}~algorithms. The output of the IDL code is a data cube in Faraday depth space that is equal in range to the range of $\phi$ that was searched over in the \textsc{rmsynthesis} algorithm. The data cube contains, for example, 500 maps of the polarized intensity as a function of spatial coordinates and $\phi$, which ranges between $\pm$ 10,000 rad m$^{-2}$~at intervals of 40 rad m$^{-2}$. Initially, we generated these maps for a 1024 $\times$ 1024 pixel image. We then used the Karma package \citep{Gooch:1995} tool \textsc{kvis} to review the maps to search for sources or source components away from the phase center that, while being too weak to detect in the 128 MHz maps, may be detectable in the RM Synthesis technique since it uses the entire bandwidth to determine the Faraday spectrum\footnote{private communication, L. Rudnick}. However, no such sources were identified above the cutoff. From the 1024 $\times$ 1024 maps of the Faraday spectrum, we identified the \textit{P$_{\textrm{max}}$} for the observed sources and extracted the Faraday spectrum at that location. Figure \ref{fig:psynmap} shows an example of a \textit{P$_{\textrm{max}}$} map that has been flattened along the $\phi$ axis, i.e., the gray scale in the image represents the full range of $\phi$. From this map, it is easy to identify the spatial location of \textit{P$_{\textrm{max}}$} for the source, which agrees with the location of the peak linear polarized intensity in the $\chi$($\lambda^{2}$)~analysis. We obtained this same result in \citet{Costa:2016} for the Rosette Nebula.
To determine the RM, we fit a 2 degree polynomial to the Faraday spectrum at each pixel in the 1024 x 1024 image above the 7$\sigma$ cutoff. The gray scale in Figure \ref{fig:rmsynmap} shows the RM value from the fit to each pixel. The image is zoomed and centered on the source. While we can mathematically determine the RM at each pixel, the sources are not resolved, so we only select the RM at the spatial location of \textit{P$_{\textrm{max}}$}. Figure \ref{fig:RMSYN} plots the Faraday spectrum and \textsc{rmclean}~components for W4-I18, and Figure \ref{fig:RMSF} shows the RMSF.
\citet{Anderson:2015} describe two cases for the behavior of the Faraday spectrum. A source is considered \textit{Faraday simple}~when \textit{F($\phi$)}~is non-zero at only one value of $\phi$, \textit{Q} and \textit{U} as a function of $\lambda^2$~vary sinusoidally with equal amplitude, and \textit{P($\lambda^2$)}~is constant. The \textit{Faraday simple}~case has the physical meaning of a uniform Faraday screen in the foreground that is responsible for the Faraday rotation, and $\chi$ is linearly dependent on $\lambda^2$. If a source is \textit{Faraday simple}, then \textit{F($\phi$)}~is a delta function at a Faraday depth equal to the RM. The second behavior \citet{Anderson:2015} describe for the Faraday spectrum is a \textit{Faraday complex}~source, which is any spectrum that deviates from the criteria set for the \textit{Faraday simple}~case. A \textit{Faraday complex}~spectrum can be the result of depolarization in form of beam depolarization, internal Faraday dispersion, multiple interfering Faraday rotating components, etc. \citep{Sokoloff:1998}.
\begin{figure}[htb!]
\centering
\subfloat[\label{fig:RMSYN}]{
\includegraphics[width=0.45\textwidth]{f6a.pdf}}
\quad
\subfloat[\label{fig:RMSF}]{
\includegraphics[width=0.45\textwidth]{f6b.pdf}}
\caption[]{Plot of (a) cleaned Faraday dispersion function, F($\phi$), for W4-I18, where $\phi_{peak}$ = 501 $\pm$ 33 rad m$^{-2}$~and (b) the RMSF (R($\phi$)). The 7$\sigma$ cutoff is shown in the red, dashed, horizontal line. The vertical red lines are the clean components, and the black curves represent the \textit{P} (solid), \textit{Q} (dot dashed), and U (dashed) components of the spectrum. }
\label{fig:rmsynrmsf}
\end{figure}
\section{Observational Results\label{sec:obsres}}
\subsection{Measurements of Radio Sources Viewed Through the W4 Complex\label{sec:obs2}}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.6\textwidth]{f7.pdf}
\caption[]{ Plot of RM values derived from the $\chi$($\lambda^{2}$)~analysis vs the RM Synthesis analysis. The blue markers are the RM from primary component and the red are the secondary component. The straight line represents perfect agreement between the two sets of measurements.}
\label{fig:comprm}
\end{figure}
We measured 27 RM values for 20 lines of sight, including secondary components, through or near to IC 1805. In Table \ref{tab:results}, the first column lists the source name using our naming scheme, and column two gives the component, if the source had multiple resolved components for which we could determine a RM value.
Columns three and four list the RM value from the least-squares method and the reduced chi-squared value, respectively. Column five lists the RM value determined from the RM Synthesis technique and the associated error (Equation 7 of \citealt{Mao:2010}).
Figure \ref{fig:comprm} shows the agreement between the two techniques for determining the RM. As in \citet{Costa:2016}, we find good agreement between the two techniques, and the good agreement between the results using the two techniques gives us confidence in our RM measurements.
\begin{table}[!hbtp]
\centering
\begin{threeparttable}
\caption{Faraday Rotation Measurement Values through the W4 Complex \label{tab:results}}
\begin{tabular}{ccccccc}
\hline
\multirow{2}{*}{Source} & \multirow{2}{*}{Component} & RM\tnote{a} & Reduced & RM\tnote{c} & $\xi$\tnote{d} & $\xi$\tnote{e}\\
& & (rad m$^{-2}$) & $\chi^{2}$\tnote{b} & (rad m$^{-2}$) & (pc) & (pc)\\
\hline
W4-I1 & a & -277 $\pm$ 1 & 29 & -258 $\pm$ 3 & 29 & 53\\ \hline
\multirow{2}{*}{W4-I2} & a & -1042 $\pm$ 7 & 1.5 & -930 $\pm$ 30 & \multirow{2}{*}{30}& \multirow{2}{*}{26}\\
& b & -935 $\pm$ 6 & 1.5 & -954 $\pm$ 11 & & \\ \hline
W4-I3 & a & -876 $\pm$ 2 & 1.9 & -878 $\pm$ 8 & 27 & 16 \\ \hline
\multirow{2}{*}{W4-I4} & a & -139 $\pm$ 3 & 1.9 & -153 $\pm$ 12 & \multirow{2}{*}{38} & \multirow{2}{*}{59}\\
& b & -91 $\pm$ 6 & 2 & -68 $\pm$ 15 & & \\ \hline
W4-I6 & a & -990 $\pm$ 8 & 1.3 & -961 $\pm$ 23 & 29 & 25\\ \hline
W4-I8 & a & -276 $\pm$ 2 & 4.4 & -337 $\pm$ 8 & 37& 54\\ \hline
W4-I11 & a & -377 $\pm$ 8 & 12 & -141 $\pm$ 18 & 35 & 41\\ \hline
W4-I12 & a & -315 $\pm$ 4 & 2.8 & -306 $\pm$ 10 &34 & 54\\ \hline
\multirow{2}{*}{W4-I13} & a & -777 $\pm$ 8 & 1.2 & -801 $\pm$ 24 & \multirow{2}{*}{28} & \multirow{2}{*}{36}\\
& b & -701 $\pm$ 28 & 0.5 & -772 $\pm$ 66 & & \\ \hline
W4-I14 & a & -678 $\pm$ 27 & 0.6 & -666 $\pm$ 66 & 20 & 36\\ \hline
W4-I15 & a & -157 $\pm$ 9 & 0.8 & -124 $\pm$ 14 & 32 & 10\\ \hline
\multirow{2}{*}{W4-I17}& a & -492 $\pm$ 8 & 1.6 & -440 $\pm$ 40 & \multirow{2}{*}{11} & \multirow{2}{*}{31}\\
& b & -509 $\pm$ 15 & 0.6 & -464 $\pm$ 40 & & \\ \hline
W4-I18 & a & +514 $\pm$ 12 & 1.1 & +501 $\pm$ 33 & 4 & 22\\ \hline
W4-I19 & a & -407 $\pm$ 14 & 0.3 & -431 $\pm$ 36 & 37 & 34\\ \hline
\multirow{3}{*}{W4-I21} & a & -53 $\pm$ 26 & 1.4 & -167 $\pm$ 67 & \multirow{3}{*}{17} & \multirow{3}{*}{15}\\
& b & -98 $\pm$ 28 & 0.5 & -79 $\pm$ 62& & \\
& c & -173 $\pm$ 34 & 0.5 & -232 $\pm$ 70 & & \\ \hline
\multirow{2}{*}{W4-I24}& a & -658 $\pm$ 5 & 1.4 & -678 $\pm$ 14 & \multirow{2}{*}{39} & \multirow{2}{*}{23}\\
& b & -675 $\pm$ 12 & 0.4 & -716 $\pm$ 30 & & \\ \hline
W4-O4 & a & -31 $\pm$ 12 & 24 & -178 $\pm$ 18 &68 &81 \\ \hline
W4-O6 & a & -95 $\pm$ 1 & 3.6 & -96 $\pm$ 4 & 54 & 76\\ \hline
W4-O7 & a & -175 $\pm$ 24 & 0.8 & -256 $\pm$ 56 & 53 & 62\\ \hline
W4-O10 & a & -379 $\pm$ 5 & 1.8 & -343 $\pm$ 16 & 56 & 66\\ \hline
\end{tabular}
\begin{tablenotes}
\item[a] RM value obtained from a least-squares linear fit to $\chi(\lambda^2)$. The errors are 1$\sigma$.
\item[b] Reduced $\chi^{2}$ for the $\chi$($\lambda^{2}$)~fit.
\item[c] Effective RM derived from RM Synthesis.
\item[d] Distance from center of OCl 352.
\item[e] Distance from \citet{Terebey:2003} center.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Report on Faraday Complexity and Unpolarized Lines of Sight\label{sec:depol}}
In the last paragraph of Section \ref{sec:rmsyn}, we discuss Faraday complexity. If a source is \textit{Faraday simple}, then the RM is equal to a delta function in \textit{F($\phi$)}~at the Faraday depth. If a source is \textit{Faraday complex}, then the interpretation of the RM is not as straightforward. There is extensive literature (e.g. \citealt{Farnsworth:2011}, \citealt{OSullivan:2012}, \citealt{Anderson:2015}, \citealt{Sun:2015}, \citealt{Purcell:2015}) to understand Faraday complexity
One indicator of a \textit{Faraday complex}~source is a decreasing fractional polarization, \textit{p} = \textit{P}/\textit{I}, as a function of $\lambda^2$. The ways in which this can arise are discussed at the end of Section \ref{sec:rmsyn}. Although depolarization does not necessarily lead to a net rotation of the source $\chi$, its presence indicates the potential for a $\chi$ rotation independent of the plasma medium through which the radio waves subsequently propagate. This could result in an error in our deduced RMs. Nine of the sources, W4-I1, -I3, -I11, -I15, I21b, -O4, -O6, -O7, and -O10 show a decreasing \textit{p} with increasing $\lambda^2$.
A rough estimate of the potential position angle rotation associated with depolarization may be obtained using the analysis in \citet{Cioffi:1980}. These calculations assume that depolarization arises from Faraday rotation within the synchrotron radiation source, and we can estimate the effect of internal depolarization from the changes in fraction polarization. Given the fractional polarization at the shortest and longest wavelength, we obtain the corresponding polarization angle change from Figure 1 of \citet{Cioffi:1980} and then calculate a RM due to internal depolarization. If the calculated RM due to depolarization (RM$_{depol}$) is larger than the observed RM, then the RM is potentially affected by depolarization.
W4-I1, -O6, -O7, and -O10 show RM$_{depol}$ $\sim$~ RM$_{obs}$, which indicates that internal depolarization could affect the observed RM. The observed RM of W4-I3 is $\sim$~3 times larger than RM$_{depol}$, so it is not affected by internal depolarization.
Depolarization due to internal Faraday rotation makes predictions for the form of $\chi (\lambda^2)$ which would not have $\chi \propto \lambda^2$ \citep{Cioffi:1980}. For all of the sources mentioned above, we compared the observed behavior of $\chi$($\lambda^{2}$)~to the predicted behavior (Equation 4b of \citealt{Cioffi:1980}). Within the errors, only W4-O7 is consistent with the non-linear behavior of a RM affected by internal depolarization. We interpret this result as meaning that our deduced RM values for most of the sources are not significantly in error due to internal depolarization, and we consider the measurement of depolarization as providing a cautionary flag.
We also considered whether our measurements could have been affected by bandwidth depolarization or beam depolarization. Bandwidth depolarization occurs when the polarization angle varies over frequency averaged bins, $\Delta\nu$. For example in this study, we use values of $\chi$ in 128 MHz wide bins (Section \ref{sec:chilam}) and 4 MHz (Section \ref{sec:rmsyn}). For a center frequency of the lowest frequency bin, we use $\nu_c$ = 4466 MHz, and the relationship between the change in polarization position angle, $\Delta\chi$, is
\begin{equation}
|\Delta\chi| = 2|\textrm{RM}| \; c^2 \; \frac{\Delta\nu}{\nu_c^3},
\label{eq:bandwidth}
\end{equation}
where c is the speed of light. This formula shows that even for $|$RM$|$ = 10$^4$ rad m$^{-2}$, which is far larger than any RMs we measure, the Faraday rotation across the band is 0.41 radians. This is insufficient to cause substantial bandwidth depolarization. Beam depolarization occurs when there are small scale variations of the electron density or the magnetic field within a beam. It is unlikely that the RMs are affected by beam depolarization as the beam at 6 cm for the VLA in C array is $\sim$~5 arcseconds.
We interpret these RMs as a characteristic value due to the plasma medium (primarily the Galactic ISM) between the source and the observer. In the analysis that follows, we choose the RM values from the RM Synthesis method.
When the data were mapped and inspected, we found that a few sources that had passed our criteria for flux density and compactness to the VLA D array at L band (1.42 GHz) were completely unpolarized. W4-I16 and W4-O8 are not polarized at any frequency, and the RM Synthesis technique does not show significant ($>$ 7$\sigma$) peaks at any $\phi$. Three of the lines of sight, W4-I5, W4-I10, and W4-I22, have no source in the field. Despite appearing to be point sources in the NVSS postage stamps (see Section \ref{sec:obs}), we do not observe a source at these locations, and they may have been clumpy foreground nebular emission that was filtered out during the imaging process.
Subsequent investigations determined that some of the selected sources were previously cataloged ultra compact HII regions associated with the W3 star formation region. These sources are W4-I7 (W3(OH)-C), W4-I9 (AFGL 333), and W4-I20 (W3(OH)-A) \citep{Feigelson:2008,Navarete:2011,Roman:2015}. The W4-I7 field has no source at the observed $\alpha$ and $\delta$, despite it being identified as W3(OH)-C. We do not observe a source at this location in any frequency bin. W3(OH)-A, however, is observed and is a point source in our maps at all frequencies. Similarly, W4-I9 is detected in each frequency bin and is an extended source. These sources are unpolarized and do not feature in our subsequent analysis.
\subsection{A Unique Line of Sight Through the W4 Region: LSI +61\ddeg 303\label{sec:HMXB}}
W4-I19 has a spectrum which is inconsistent with an optically-thin extragalactic radio source. It is linearly polarized, and we measure RM = --431 $\pm$ 36 rad m$^{-2}$. Investigation of this source during the data analysis phase revealed that it is not an extragalactic source, although it passed our selection criteria for flux and compactness. W4-I19 is the high mass X-ray binary (HMXB) LSI +61\ddeg303 \citep{Gregory:1979,Bignami:1981}, which is notable for being one of five known gamma ray binary systems \citep{Frail:1991}. This system has been extensively studied, and as a result, much is known about the nature of the compact object \citep{Massi:2004,Massi:2004b,Dubus:2006,Paredes:2007,Massi:2017}, the stellar companion \citep{Casares:2005,Dubus:2006,Paredes:2007}, orbital period \citep{Gregory:2002}, radio structure \citep{Albert:2008}, and radial velocity \citep{Gregory:1979,Lestrade:1999}.
The spatial location of LSI +61\ddeg303 is important for understanding the RM we determined for this source. \citet{Frail:1991} argue that since signatures of the Perseus arm shock are present in the absorption spectrum to LSI +61\ddeg303 but not the post-shock gas from the Perseus arm, LSI +61\ddeg303 must lie between the two features at a distance of 2.0 $\pm$ 0.2 kpc. They also report that they do not see absorption features due to the IC 1805 ionization front and shock front. The estimated distance to LSI +61\ddeg303{} is consistent with distance estimates to OCl 352. The position relative to the nebula has consequences for the interpretation of the RM that we measure. The possibilities are:
\begin{enumerate}
\item LSI +61\ddeg303 is in front of the stellar bubble and HII region, so it is exterior to a region modified by OCl 352. The RM is then an estimate of the foreground ISM between us and the nebula.
\item If LSI +61\ddeg303 is at the same distance as IC 1805 or slightly behind (greater distance), then the RM is unique among our sources in that it is not affected by Faraday rotation from material in the outer Galaxy. The RM is then probing at least a part of the Faraday rotating material due to the nebula.
\end{enumerate}
To further determine the position of LSI +61\ddeg303{} with respect to IC 1805, we review the current state of knowledge on the subject from the literature. \citet{Dhawan:2006} observed LSI +61\ddeg303{} with the Very Long Baseline Array (VLBA) and report a proper motion of ($\mu_{\alpha}$, $\mu_{\delta}$) = (-0.30 $\pm$ 0.07, -0.26 $\pm$ 0.05) mas yr$^{-1}$. \citet{Aragona:2009} report a radial velocity for LSI +61\ddeg303~of $V_{rad}$ = --41.4 $\pm$ 0.6 km s$^{-1}$, which agrees with previous estimates by \citet{Casares:2005}. For OCl 352, \citet{Dambis:2001} estimate the radial velocity to be --41 $\pm$ 3 km s$^{-1}$, and more recent estimates by \citet{Kharchenko:2005} ($V_{rad}$ = --47 $\pm$ 18 km s$^{-1}$) agree within the errors. Both LSI +61\ddeg303~and OCl 352 have similar radial velocities, and the proper motion estimates by \citet{Dhawan:2006} indicate that LSI +61\ddeg303~is moving similarly on the plane of the sky to OCl 352, which has a proper motion of ($\mu_{\alpha}$, $\mu_{\delta}$) = (--1.0 $\pm$ 0.4, --0.9 $\pm$ 0.4) mas yr$^{-1}$ \citep{Dambis:2001}.
From proper motion and radial velocity estimates, LSI +61\ddeg303~appears to be moving in relatively the same direction and speed as OCl 352. Using a distance of 2 kpc to LSI +61\ddeg303~and 2.2 kpc to OCl 352, the transverse velocities are $\sim$~3~km s$^{-1}${} and $\sim$~14 km s$^{-1}$, respectively. If LSI +61\ddeg303~originally belonged to OCl 352, then it is unlikely that it is in front of IC 1805, given that both are moving at the same radial velocity. While LSI +61\ddeg303~appears to be outside the obvious shell structure of IC 1805, it is more likely that it is probing material modified by OCl 352. We discuss this possibility further in Section \ref{sec:pdr}.
If LSI +61\ddeg303~did not originate in OCl 352, then it is possible to still be in front of the nebula, despite the similar velocities. In such a case, the RM we obtained for this line of sight is due to the ISM between us and IC 1805.
The RM value we find for LSI +61\ddeg303~is nearly 3 times larger than the background RM, which we discuss in Section \ref{sec:bkgrm}. This would require a magneto-ionic medium between the observer and the nebula capable of producing $\sim$~400 rad m$^{-2}$~along this line of sight. As may be seen from Table \ref{tab:results} and Figure \ref{fig:w4}, other lines of sight near IC 1805, but exterior to the shell, do not have as large of RM values (e.g. W4-O26, -O19, -O7, -I11). It therefore seems most probable that the RM for W-I19 (LSI +61\ddeg303) is dominated by plasma in W4
In summary, there is evidence in the literature that suggests LSI +61\ddeg303~may lie within a region modified by OCl 352, particularly if LSI +61\ddeg303~did indeed once belong to OCl 352. If this is the case, then the RM we find is unaffected by the ISM in the outer galaxy and is due to the material near IC 1805.
\section{Results on Faraday Rotation Through the W4 Complex\label{sec:fr}}
\subsection{The Rotation Measure Sky in the Direction of W4}
\begin{figure}[!htb]
\centering
\subfloat[][\label{fig:northb}]{
\includegraphics[width=0.48\textwidth]{f8a.pdf}}
\quad
\subfloat[][\label{fig:southb}]{
\includegraphics[width=0.48\textwidth]{f8b.pdf}}
\caption[]{Plot of RM vs distance from the center of OCl 352 for (a) lines of sight that pass through the W4 Superbubble and (b) lines of sight near and close to the southern loop. The solid line represents the estimate of the background RM using sources in this study and in the literature, and the dashed line is the predicted background RM from the \citet{vanEck:2011} model of the Galactic magnetic field.}
\label{fig:rmvsxi}
\end{figure}
\citet{Whiting:2009}, \citet{Savage:2013}, and \citet{Costa:2016} compared observations to a model of the ionized shell in which the RM depended only on $\xi$, the impact parameter, or closest approach of a line of sight to the center of the shell. In anticipation of a similar analysis in this study, we show Figures \ref{fig:northb} and \ref{fig:southb}, which plot the RM versus distance from the center of star cluster for the lines of sight through the W4 Superbubble (W4-I1, -I4, -I8, -I12, -I14, -I17, -O4, -O6, and -O7) and the ones through or close to the southern loop (W4-I2, -I3, -I6, -I11, -I13, -I15, -I18, -I19, -I21, -I24, -O10).
In Section \ref{sec:structure}, we discussed the morphology of the region around IC 1805 and made the distinction between the southern latitudes and the northern latitudes, so in the following sections, we address each region near IC 1805 separately.
\subsection{The Galactic Background RM in the Direction of W4\label{sec:bkgrm}}
In \citet{Savage:2013}, we determined the background RM in the vicinity of the Rosette Nebula ($\ell$ $\sim$ 206\ddeg) by finding the median value of the RM for sources outside the obvious shell structure of the Rosette. Determining the background RM near IC 1805 is difficult, however, due to proximity of W3, the W3 molecular cloud, and the W4 Superbubble. Given the morphological difference between the northern and southern parts of IC 1805, we assume that sources south of OCl 352 (\textit{b} $<$ 0.9\ddeg) should be modeled independently of the northern sources, since the W4 Superbubble extends up to \textit{b} $\sim$ 7\ddeg~\citep{West:2007}. The lines of sight north of the star cluster are intersecting the W4 Superbubble and are not probing the RM due to the general ISM independent of IC 1805. Therefore, the only lines of sight that are potentially probing the RM in the vicinity of IC 1805 are those exterior to the shell structure of the southern loop.
\begin{table}[htb!]
\centering
\caption{List of Sources with RM values from Catalogs\label{tab:taylor}}
\begin{threeparttable}
\centering
\begin{tabular}{cccc}
\hline
Source & $\alpha$(J2000) & $\delta$(J2000) & RM\tnote{a} \\
Name & h m s & $^o$ $'$ $''$ & (rad m$^2$) \\
\hline
W4-O3 & 02 35 43.0 & +63 22 33.0 & --138\tnote{b} $\pm$ 18 \\
W4-O19 & 02 46 23.9 & +61 33 19.9 & --157\tnote{c} $\pm$ 15 \\
W4-O26 & 02 42 32.3 & +60 02 31.0 & +61 $\pm$ 41 \\
W4-O27 & 02 25 48.7 & +59 53 52.0 & --145 $\pm$ 22 \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] RM values from \citet{Brown:2003} unless otherwise noted.
\item[b] \citet{Taylor:2009} give --75 $\pm$ 9 rad m$^{-2}$~for this line of sight.
\item[c] RM value from \citet{Taylor:2009}.
\end{tablenotes}
\end{threeparttable}
\end{table}
If we apply the thick shell model from \citet{Terebey:2003} (see Section \ref{sec:structure} for details), then the lines of sight with RM values exterior to the shell are W4-I2, -I11, and -O10. For the thin shell case, W4-I6, -I13 and -I24 are also exterior sources. The mean RM value for the background using these sources is --554 rad m$^{-2}$~and --670 rad m$^{-2}$~for the thick or thin shell, respectively. In Table \ref{tab:taylor}, we list RM values from the literature for lines of sight near IC 1805 that we include in our estimate of the background RM. The mean RM value for these sources (excluding W4-O3 for being in the superbubble) is --80 rad m$^{-2}$. The sources W4-I2, -I6, -I13, and -I24 are seemingly outside the obvious ionized shell structure; however, they are also the lines of sight for which we measure some of the highest RM values. This is a surprising result, and one we did not observe in the case of the Rosette Nebula. It strongly suggests that the lines of sight to W4-I2, -I6, -I13, and -I24 have RMs that are dominated by the W4 complex, despite the fact that they are outside the obvious ionized shell of IC 1805. We discuss this further in the next section.
For the present discussion, we exclude these sources from the estimate of the background. Using W4-I11, -O10, -O19, -O26, and -O27, we find a mean value for the background RM due to the ISM of --145 rad m$^{-2}$. While this value is similar in magnitude to the value of the background RM we found in our studies on the Rosette Nebula, we have significantly fewer lines of sight, and only two of the lines of sight were observed in this study.
Due to a low number of lines of sight exterior to IC 1805, we utilize the model of a Galactic magnetic field by \citet{vanEck:2011} to estimate the background RM due to the ISM. From their Figure 6, they find the Galactic magnetic field is best modeled by an almost purely azimuthal, clockwise field. \citet{vanEck:2011} use their model to predict the RM values in the Galaxy, and in the vicinity of IC 1805, their model predicts RMs of order --100 rad m$^{-2}$. Using this as an estimate of the background RM, we find an excess RM due to IC 1805 of +600 to --860 rad m$^{-2}$.
\subsection{High Faraday Rotation Through Photodissociation Regions\label{sec:pdr}}
The lines of sight with the highest RM values, W4-I2, -I6, and -I24, appear to be outside the obvious shell of the southern loop. These sources are very near to the bright ionized shell. \citet{Terebey:2003} and \citet{Gray:1999} discuss a halo of ionized gas that surrounds IC 1805, which may be causing the high RM values. \citet{Gray:1999} speculate that the diffuse extended structure is an extended HII envelope as suggested by \citet{Anantharamaiah:1985}. Another possibility is that these high RMs arise in the PDR surrounding the IC 1805 HII region.
PDRs are the regions between ionized gas, which is fully ionized by photons with \textit{h$\nu$} $>$ 13.6 eV, and neutral or molecular material. PDRs can be partially ionized and heated by far-ultraviolet photons (6 eV $<$ \textit{h$\nu$} $<$ 13.6 eV) \citep{Tielens:1985,Hollenbach:1999}. Typically, the PDR consists of neutral hydrogen, ionized carbon, and neutral oxygen nearest to the ionization front, and with increasing distance, molecular species (e.g., CO, H$_2$, and O$_2$) dominate the chemical composition of a PDR \citep{Hollenbach:1999}. One tracer of PDRs is polycyclic aromatic hydrocarbon (PAH) emission at infrared (IR) wavelengths. \citet{Churchwell:2006} identify more than 300 bubbles at IR wavelengths in the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE), and 25$\%$ of these bubbles coincide with known HII regions. \citet{Watson:2008} examine three bubbles from the \citet{Churchwell:2006} catalog with the \textit{Spitzer} Infrared Array Camera (IRAC) bands 4.5, 5.8, and 8.0 \micron~and the 24 $\mu$m band from the \textit{Spitzer} Multiband Imaging Photometer (MIPS) to determine the extent of the PDR around three young HII regions. One of their main results is that the 8 \micron~emission, which is due to PAHs, encloses the 24 \micron~emission, which traces hot dust. \citet{Kerton:2013} discuss similar observations near the W 39 HII region. \citet{Watson:2008} use ratios between the 4.5, 5.8, and 8.0 \micron~bands to determine the extent of the PDRs, as the 4.5 \micron~emission does not include PAHs but the 5.8 and 8.0 \micron~bands do (see their Section 1 for details).
To determine the presence and extend of a potential PDR around IC 1805, we analyze Wide-field Infrared Survey Explorer (WISE) data from the IPAC All-Sky Data Release\footnote{http://wise2.ipac.caltech.edu/docs/release/allsky/} at 3.6, 4.6, 12, and 22 $\mu$m. The 4.6 $\mu$m~WISE bands is similar in bandwidth and center frequency to the IRAC 4.5 \micron~band, and the WISE 22 \micron~band is also similar to the MIPS 24 \micron~band \citep{Anderson:2014}. The 12 \micron~WISE band does not overlap with the 8.0 \micron~band of IRAC, but the WISE band traces PAH emission at 11.2 and 12.7 \micron. \citet{Anderson:2012} note, however, that the 12 \micron~flux is on average lower than the 8.0 \micron~IRAC band, which is most likely due to the WISE band sampling different wavelengths of PAH emission instead of the 7.7 and 8.6 \micron~PAH emission in the IRAC band.
Figure \ref{fig:wise} is a RGB image of the southern loop of IC 1805 at 4.6 \micron~(blue), 12 \micron~(green), and 22 $\mu$m (red). The 1.42 GHz radio continuum emission is shown in the white contours at 8.5, 9.5, and 10 K, and the lines of sight that intersect this region are labeled as well. Similar to the results of \citet{Watson:2008}, the majority of the 22 \micron~emission is located inside the bubble. The radio contours trace the ionized shell of the HII region, which show a patchy ionized shell. Outside of the radio contours, there is a shell of 12 \micron~(green) PAH emission that encloses the 22 \micron~emission as well. In the northeastern portion of the image, there is extended 22 \micron~(hot dust) emission, which is spatially coincident with a CO clump \citep{Lagrois:2009}.
The PDR model predicts the presence of neutral hydrogen and molecular CO (see Figure 3 of \citealt{Hollenbach:1999}) at increasing distance from the exciting star cluster. Figure 1 of \citet{Sato:1990} and Figure 2 of \citet{Hasegawa:1983} show HI contours in the vicinity of IC 1805, and the HI emission appears to completely enclose the southern loop except
near 135.5\ddeg{} $\leq$ $\ell$ $\leq$ 136\ddeg, 0.2\ddeg $\leq$ $b$ $\leq$ 0.9\ddeg.
\citet{Braunsfurth:1983} report HI emission near IC 1805, and he notes that the hole could be due to cold HI gas or the lack of gas if the winds have sufficiently swept the material away or ionized it. Figure 6 of \citet{Digel:1996} shows the CO emission, with the W3 molecular cloud on the western side of IC 1805, CO emission along the southern loop of IC 1805, and the molecular material associated with the W5 ($\ell$ = 137.1, $b$ = +0.89) HII region on the eastern side of IC 1805. We interpret the WISE data, the radio contours, and the CO and HI maps as a patchy ionized shell surrounded by a PDR.
If there is a PDR surrounding IC 1805, then the highest RM values from our data set, RM = --954 rad m$^{-2}${} and --961 rad m$^{-2}${} for W4-I2 and -I6, respectively, lie outside the ionized shell of the HII region and in the PDR. Similarly, the sources W4-I19 and -I24 are also outside the radio continuum contours but appear to be within the 12 \micron~(green) emission. This is a surprising result compared with our results from the Rosette Nebula, where we found the highest RM values for lines of sight that pass through the ionized shell. \citet{Gray:1999} note zones of depolarization near the southern portion of IC 1805, which require RMs on order 10$^3$ rad m$^{-2}$, and spatial RM gradients. The RMs for W4-I2 and -I6 are on this order, but those for W4-I24 and -I19 are not, and we do not find that these lines of sight are affect by depolarization. W4-I24 has two components for which we measure RMs, and the components are separated by $\sim$~18 arcseconds. The $\Delta$RM, which is the difference in RM between the two components is 38 rad m$^{-2}$, which is not a large change in the RM and is consistent within the errors.
The presence of the PDR is complicated, however, by the extended diffusion ionized emission reported by \citet{Terebey:2003} and \citet{Gray:1999}. At lower contours, the high RM sources do lie within the radio continuum emission. To fully understand the presence and extent of a PDR or an extended HII envelope, observations of radio recombination lines on the eastern side of IC 1805 would clarify the structure as well as observations of other tracers of PDRs (e.g., fine structure lines of C and C$^+$, H$_2$, and CO). It may be the case that the ionized shell is patchy along the shell wall, which allows photons $>$ 13.6 eV to escape the shell at places, but the shell is sufficiently ionization-bounded at other places such that a PDR can form.
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.95\textwidth]{f9.pdf}
\caption[WISE image of W4]{Inset from Figure \ref{fig:w4}. A RGB image of archive WISE data at 4.6 \micron~(blue), 12 \micron~(green), and 22 \micron~(red) with CGPS contours at 8.5, 9.5, and 10 K in white. The lines of sight from the present study are shown with circles and are labeled according to Table \ref{tab:sources}.}
\label{fig:wise}
\end{figure}
\subsection{Faraday Rotation Through the Cavity and Shell of the Stellar Bubble}
There are four lines of sight through the cavity of the stellar bubble, assuming an inner radius from the \citet{Terebey:2003} model. The sources W4-I3, -I15, -I18, and -I21 are through the cavity, and including multiple components, we find 6 RM values. W4-I3 has a high RM (--878 $\pm$ 18 rad m$^{-2}$), and W4-I15 and -I21 have comparatively low RM values ( --79 to --232 rad m$^{-2}$). Examination of Figures \ref{fig:w4} and \ref{fig:wise} does not reveal enhanced emission near W4-I3 in comparison to W4-I21. W4-I15, however, is in a region of relatively low emission, which may explain why W4-I15 has a RM value at least 4 times smaller than W4-I3.
W4-I13 is outside the shell, assuming a shell radius from either \citet{Terebey:2003} model. From Figure \ref{fig:w4}, it does appear to be outside the ionized shell. However W4-I13 is within a 8.5 K contour on the 1.42 GHz radio continuum map, which may indicate that it is probing the ionized shell. We find a high RM for both components of this source, which is similar to the RM values for W4-I2 and -I6.
Across IC 1805, we observe negative RM values for all lines of sight except one: W4-I18, which is 5.6 arcmin (4 pc) from the center of the star cluster. The absolute value of the RM for W4-I18 is also large (+501 $\pm$ 33 rad m$^{-2}$), indicating a large change in RM along this line of sight relative to other lines of sight in this part of the sky. This line of sight is probing the space close to the massive O and B stars responsible for IC 1805. In the \citet{Weaver:1977} model for a stellar bubble, the hypersonic stellar wind dominates the region between the star responsible for the bubble and the inner termination shock. Equation (12) of \citet{Weaver:1977} states that the distance of the inner shock, $R_t$ is
\begin{equation}
R_{\textrm{t}} = 0.90\; \alpha^{3/2} \left(\frac{1}{\rho_0}\frac{\textrm{d}M_w}{\textrm{d}t}\right)^{3/10} \ V_w^{1/10} \ t^{2/5},
\label{eq:weaver}
\end{equation}
where $\alpha$ is a constant equal to 0.88, $\rho_0$ is the mass density in the external ISM, d$M_w$/d$t$ is the mass loss rate, $V_w$ is the terminal wind speed, and $t$ is time. For a rough estimate of the inner shock distance, we utilize general stellar parameters for OCl 352 of d$M_w/dt$ = 10$^{-5}$ \textrm{M}$_{\odot}$ yr$^{-1}$, $t$ = 10$^6$ yr, and V$_w$ = 2200 km s$^{-1}$ (see Section \ref{sec:structure} or Table \ref{tab:stellarpar}). From the discussion in Section \ref{sec:ferriere}, we adopt $n_0$ = 4.5 cm$^{-3}$~for $\rho_0$ = $n_0 m_p$, where $m_p$ is the mass of a proton. With these values in the appropriate SI units, $R_t$ $\sim$ 6 pc. It is possible that the line of sight to W4-I18 passes inside the inner shock, and the large, positive RM is due to material modified by the hypersonic stellar wind and not the shocked interstellar material.
Because the inner shock is interior to the contact discontinuity between the stellar wind and the ambient ISM, the magnetic field close to the star cluster may be oriented in any direction relative to the exterior (upstream) field. With a positive value of the RM for W4-I18, the line of sight component of the field points toward us while the remaining lines of sight in the cavity are negative, meaning B$_{\textrm{LOS}}$~points away.
\subsection{Low Rotation Measure Values Through the W4 Superbubble}
North of IC 1805 is the W4 Superbubble, which is an extended ``egg-shaped'' structure closed at $b$ $\sim$ 7\ddeg{} \citep{West:2007}. \citet{Basu:1999} utilize an H$\alpha$ map to define the shape, which would include the southern loop (134\ddeg $<$ $\ell$ $<$ 136\ddeg, \textit{b} $<$ 0.5\ddeg); \citet{Normandeau:1996} examine the HI distribution, however, and place the base of the structure at OCl 352. Similarly, \citet{West:2007} place an offset bottom of the ``egg'' at OCl 352. The southern loop of IC 1805 is seemingly sufficiently different from the northern latitudes, as it is often not included in the discussion of the W4 Superbubble in spite of the fact that OCl 352 is thought to be responsible for the formation of both structures \citep{Terebey:2003,West:2007}.
Nine lines of sight in the present study are north of OCl 352 in the W4 Superbubble. These sources are W4-I1, -I4, -I8, -I12, -I14, -I17, -O4, -O6, and -O7, and they have a mean RM of --293 rad m$^{-2}$~and a standard deviation of 178 rad m$^{-2}$. Of these sources, W4-I14 and -I17 have the largest RM values, --666 rad m$^{-2}$~and --460 rad m$^{-2}$, respectively, and they are close to OCl 352, with distances of 31 arcminutes (20 pc) and 17 arcminutes (11 pc), respectively. As discussed in Section \ref{sec:structure}, \citet{Lagrois:2009} argue that the ionized ``v'' structure north of OCl 352 is part of the bubble wall and not a cap to southern loop structure, but examination of Figure \ref{fig:w4} suggests that the bubble walls are denser, or thicker, at latitudes $<$ 1.5\ddeg~than higher latitudes, which may explain the larger RM associated with W4-I14 and -I17. The remaining lines of sight, however, in the W4 Superbubble have some of the lowest RM values in the data set and are consistently lower RM values than the lines of sight through the PDR.
At higher latitudes, \citet{Gao:2015} modeled the polarized emission and applied a Faraday screen model to the W4 Superbubble. They report RMs on the western side of W4 ($\ell$ $\sim$ 132\ddeg, $b$ $\sim$ 4.8\ddeg) between --70 and --300 rad m$^{-2}$~and $\sim$ +55 rad m$^{-2}$~for the eastern shell ($\ell$ $\sim$ 136\ddeg, $b$ $\sim$ 7\ddeg). \citet{Gao:2015} argue that since W4 is tilted at an angle towards the observer \citep{Normandeau:1997}, a change in the sign of the RM is consistent with a scenario in which the superbubble lifts up a clockwise running Galactic magnetic field \citep{Han:2006} out of the Galactic plane. The magnetic field would go up the eastern side of the superbubble and then down the western side, resulting in the field being pointed toward the observer in the east and away from the observer in the west. While the lines of sight reported in this paper are at $b$ $<$ 2\ddeg, we find a similar range of RM values as reported by \citet{Gao:2015} for the western side. However, we measure RM values 3 -- 4.5 times higher on the eastern side, and we do not observe a sign reversal on the eastern side as suggested by \citet{Gao:2015}.
\citet{West:2007} report positive values of the magnetic field for the western side from a change in polarization position angle of $\sim$ 60\ddeg~at 21 cm, which gives a RM value on order of 20 rad m$^{-2}$. We do not observe RM values this low for any of our lines of sight through the northern latitudes. Our lines of sight, however, do not probe the same regions as the \citet{West:2007} and \citet{Gao:2015} studies
The line of sight W4-I4 is arguably within the W4 Superbubble; however, it is also $\sim$~8 arcmin (5 pc) on the sky from W3-North (G133.8 +1.4), which is a star forming region within W3. W4-I4 has two components, separated by 15 arcsec (0.2 pc), and a difference in RM values between the two components of $\Delta$RM = 85 rad m$^{-2}$. The RM values for both components are low (--153 rad m$^{-2}$~and --68 rad m$^{-2}$) despite being in the superbubble and near to W3, which may have variable but potentially large magnetic fields \citep{vanderWerf:1990,Roberts:1993} (see Section \ref{sec:structure}).
\section{Models for the Structure of the HII region and Stellar Bubble\label{sec:models}}
\subsection{\citet{Whiting:2009} Model of the Rotation Measure in the Shell of a Magnetized Bubble\label{sec:whitingmod}}
\citet{Whiting:2009} developed a simple analytical shell model intended to represent the Faraday rotation due to a \citet{Weaver:1977} solution for a wind-blown bubble. We employed this model in \citet{Savage:2013} and \citet{Costa:2016} to model the magnitude of the RM in the shell of the Rosette Nebula as a function of distance from the exciting star cluster. Figure 6 of \citet{Whiting:2009} and their Section 5.1 give the details of the model, and Sections 4.1 of \citet{Savage:2013} and 5 of \citet{Costa:2016} describe the application of the model to the Rosette Nebula. This model takes as inputs the general interstellar magnetic field ($\textbf{B}$) in $\mu$G, the inner ($R_1$) and outer ($R_0$) radii of the shell in parsecs, and the electron density in the shell, $n_e$ (cm$^{-3}$). $R_0$ represents the shock between the ambient ISM and the shocked, compressed ISM, and $R_1$ separates the shocked ISM from the hot, diffuse stellar wind in the cavity. Only the component of the ambient interstellar magnetic field that is perpendicular to the shock normal is amplified by the density compression ratio, X. The resulting expression for the RM through the shell is
\begin{equation}
\textrm{RM}=C\, n_{e}\, L(\xi)\, B_{0z} \left(1+(X-1)\left(\frac{\xi}{R_{0}}\right)^{2}\right),
\label{eq:rmmodelW}
\end{equation}
where L($\xi$) is the cord length through the shell in parsecs (see Equation 10 in \citealt{Whiting:2009} or Equation 6 in \citealt{Costa:2016}), and $B_{0z}$ is the z-component of \textbf{B$_0$}, the magnetic field in the ISM. If $n_e$ has units of cm$^{-3}$, $B_{0z}$ is in $\mu$G, and $L$ is in parsecs, $C=0.81$ (see Equation \ref{eq:rmprat}). $B_{0z}$ is at an angle $\Theta$ with respect to the LOS and is written as
\begin{equation}
B_{0z}=B_{0}\cos{\Theta}.
\end{equation}
In our previous work, we presented two cases for the behavior of the magnetic field in the shell. The first is that the magnetic field is amplified by a factor of 4 in the shell. The second case, in which there is not an amplification of the magnetic field in the shell, sets X = 1. Equation \ref{eq:rmmodelW} then simplifies to
\begin{equation}
\textrm{RM}(\xi)=0.81\, n_{e} \, L(\xi) \, B_{0z}.
\label{eq:rmmodelH}
\end{equation}
In \citet{Costa:2016}, we employed a Bayesian analysis to determine which of the two models better reproduces the observed dependence of the RM as a function of distance. We found that neither model was strongly favored in the case of the Rosette. The model given in Equation (\ref{eq:rmmodelW}) is subject to the criticism that it applies shock jump conditions for \textbf{B} over a large volume of a shell, and that the outer radius of an observed HII region need not be the outer shock of a Weaver bubble (see remarks in Section 5.1.1 of \citealt{Costa:2016}). It is worth including this model, however, in our analysis of IC 1805 for completeness and in order to compare our results to those of the Rosette Nebula.
In Section \ref{sec:structure}, we discussed the the structure of IC 1805, and we present evidence from the literature that north of OCl 352 is part of the W4 Superbubble. Thus, lines of sight north of OCl 352 may have different model parameters for the shell radii and electron density than the southern loop. For the southern latitudes, we utilize the \citet{Terebey:2003} thin and thick shell values for the shell radii and electron density. The remaining parameters in Equation (\ref{eq:rmmodelH}) are $B_0$ and $\Theta$. As in \citet{Savage:2013} and \citet{Costa:2016}, we adopt $B_0$ = 4 $\mu$G for the general Galactic field in front of the HII region. The angle $\Theta$ is calculated as follows.
Assuming a distance of 8.5 kpc to the Galactic center, a distance to OCl 352 of 2.2 kpc, and given a Galactic longitude of 135\ddeg, the angle between the line of sight and an azimuthal magnetic field is $\Theta$ = 55\ddeg. We discuss our comparison of this model with the data in Section \ref{sec:results}.
\begin{figure}[htb!]
\centering
\subfloat[\label{fig:southshell-thick}]{
\includegraphics[width=0.48\textwidth]{f11a.pdf}}
\quad
\subfloat[\label{fig:southshell-thin}]{
\includegraphics[width=0.48\textwidth]{f11b.pdf}}
\caption[]{ Plots of LOS versus distance with the \citet{Whiting:2009} model for X = 4 (solid) and X = 1 (dashed) for $b$ $<$ +0.9\ddeg{} using (a) the thick shell and (b) the thin shell parameters from \citet{Terebey:2003}. The background RM = --100 rad m$^{-2}$~from the \citet{vanEck:2011} model. The errors on the RM values include the measurement errors and an expected deviation of $\sim${} 67 rad m$^{-2}${} from the \citet{vanEck:2011} model. See Table \ref{tab:model} for model parameters.}
\label{fig:shell}
\end{figure}
\begin{table}[hbtp]
\centering
\begin{threeparttable}
\caption{Model Parameters \label{tab:model}}
\begin{tabular}{ccccccc}
\hline
\multicolumn{7}{c}{\citet{Whiting:2009} Model} \\
\hline
Center\tnote{a}\phantom{2} & R$_i$ & $R_o$ & $n_e$ & X\tnote{b} & $\Theta$ & Figure \\
(\ddeg) & (pc) & (pc) & (cm$^{-3}$) & & (\ddeg) & \\
(135, +0.42) & 19 & 25 & 10 & 1, 4 & 55 & \ref{fig:southshell-thick} \\
(135, +0.42) & 19 & 21 & 20 & 1, 4 & 55 & \ref{fig:southshell-thin} \\
\hline \hline
\multicolumn{7}{c}{\citet{Ferriere:1991} Model} \\
\hline
Center & $\Delta R$ & $R_s$ & $n_s$ & $\epsilon$ & $\Theta$ &Figure \\
(\ddeg) & (pc) & (pc) & (cm$^{-3}$) & & (\ddeg) & \\
(135, +0.42) & 6 & 25 & 10 & 0.25 & 55 & \ref{fig:steve2}\\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] Position of model center in Galactic coordinates in the format of ($\ell$, $b$).
\item[b] The model uses either X = 1 or X = 4.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Analytical Approximation to Magnetized Bubbles of \citet{Ferriere:1991}\label{sec:ferriere}}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.6\textwidth]{f10.pdf}
\caption{Illustration of a simplified version of the shell and cavity produced by a stellar wind, as discussed by \citet{Ferriere:1991}. The z direction is that of the interstellar magnetic field, and $\Theta$ is the angle between the magnetic field and the line of sight. A line of sight passes at a closest distance $\xi$ from the center of the cavity (the ``impact parameter''). Other parameters in the figure are defined in the text. A Faraday rotation measurement is along a line of sight offset a linear distance $\xi$ from the center of the bubble. The quantity d$s$ represents an incremental spatial interval along the LOS. }
\label{fig:steve1}
\end{figure}
\citet{Ferriere:1991} presented a semi-analytic discussion of the evolution of a stellar bubble in a magnetized interstellar medium. The theoretical object discussed by \citet{Ferriere:1991} could describe a shock wave produced by a supernova explosion or energy input due to a stellar wind. The main features of the model were an outer boundary (e.g. outer shock) which was the first interface between the undisturbed ISM and the bubble, and an inner contact discontinuity between ISM material, albeit modified by the bubble, and matter that originated from the central star or star cluster.
The main feature of the model is that plasma passing through the outer boundary is concentrated in a region between the outer boundary and the contact discontinuity. In what follows, we will refer to this region as the shell of the bubble. The equation of continuity then indicates that there will be higher plasma density in the shell, and the law of magnetic flux conservation indicates that there will be an increase in the strength of the magnetic field in the shell relative to the general ISM field. \citet{Ferriere:1991} were interested in the structure of the bubble, and their results have been corroborated by the fully numerical studies of \citet{Stil:2009}. However, \citet{Ferriere:1991} did not calculate the Faraday rotation measure through their model for diagnostic purposes. \citet{Stil:2009} explicitly considered the model RMs from their calculations, but only for a couple of cases and for two values of bubble orientation. It is our goal in this section to use a simplified, fully analytic approximation of the results of \citet{Ferriere:1991}, that permits RM profiles RM($\xi$) for a wide range of bubble parameters and orientation with respect to the LOS.
The geometry of the bubble is shown in Figure \ref{fig:steve1}, which is an adaptation of, and approximation to Figure 1 and Figure 4 from \citet{Ferriere:1991}. An important feature of Figure \ref{fig:steve1}, not present in \citet{Ferriere:1991}, is the orientation of the line of sight at an angle $\Theta$ with respect to the ISM magnetic field at the position of the bubble, and the impact parameter $\xi$ indicating the separation of the LOS from the center of the bubble. The region interior to the contact discontinuity is referred to as the cavity, and for the purposes of our discussion will be considered a vacuum. Another important shell parameter is the thickness $\Delta R$ $\equiv$ $R_s$ - $R_i$, where $R_s$ and $R_i$ are the outer radius of the bubble and the radius of the contact discontinuity, respectively (see Figure \ref{fig:steve1}). We also define and use the dimensionless shell thickness
\begin{equation}
\epsilon \equiv \frac{\Delta R}{R_s}
\label{eq:steve8}
\end{equation}
A major simplification that we adopt, based on an approximation of the results of \citet{Ferriere:1991}, is that the magnetic field in the shell ($\textbf{B}_s$) is entirely in the azimuthal direction, and that we ignore radial variations within the shell, i.e.
\begin{equation}
\textbf{B}_s(r, \theta) \equiv \pm B_s(\theta) \hat{e}_{\theta}
\label{eq:steve9}
\end{equation}
where $\hat{e}_{\theta}$ is a unit vector in the azimuthal direction, and the $\pm$ is selected by the polarity of the interstellar field at the bubble.
We need expressions for the electron density and vector magnetic field within the shell, as well as the geometry of the line of sight. The most important aspect of the \citet{Ferriere:1991} theory is the conservation of magnetic flux as the magnetic field in the external medium is swept up and accumulated in the shell. This results in the azimuthal component of the magnetic field increasing as $\theta$ increases from $0$ to $\frac{\pi}{2}$, as given by Equation (40) of \citet{Ferriere:1991}.
In \citet{Ferriere:1991} the shell thickness also depends on $\theta$ (Equation 46 of \citealt{Ferriere:1991}), and as a consequence, so does the plasma density in the shell $n_s$ (Equation 38 of that paper).
In the initial version of this paper, we calculated the RM through model bubbles in which $B_s$, $n_s$, and $\Delta$ R all varied with $\theta$ as prescribed by \citet{Ferriere:1991}. These calculations utilized an approximate form for lines of sight that intersected the bubble in two segments (passing through the central cavity between), and a form that contained a numerically-evaluated expression for lines of sight that remained within the shell from ingress to egress. The algebraic distinction between these 2 cases is discussed below (Sections \ref{sec:walls} and \ref{sec:allshell}). These expressions for RM($\xi$), including a comparison with our RM measurements, are given in \citet{Costa:2018phd}.
After examining the results of these calculations, it was decided to simplify our bubble model to that of a spherical shell with constant $\epsilon$. The motivation for this suggestion was the very limited success of the more general model in representing our data, which did not justify the extensive algebraic presentation and non-compact expressions that resulted. The calculations with the approximation of constant $\epsilon$ are presented below.
Due to magnetic flux conservation, the expanding shell (now approximated as spherical) will have a magnetic field that is larger than in the external medium, and increases with $\theta$, as in the original discussion of \citet{Ferriere:1991}. For our spherical case, it may be shown that the magnetic field in the shell is
\begin{equation}
B_s(\theta) = \frac{B_0}{2 \epsilon} \sin \theta
\label{eq:steve10}
\end{equation}
where $B_0$ is the magnitude of the magnetic field in the external medium. The dimensionless shell thickness $\epsilon$ remains a free parameter of the model, or one that can be determined by observations. Finally, the plasma density in the shell, determined by mass conservation, is
\begin{equation}
n_s = \frac{n_0}{3 \epsilon}
\label{eq:steve11}
\end{equation}
where $n_0$ is the plasma density in the external medium. \equa{steve11} is a valid approximation for $\epsilon \ll 1$.
\subsubsection{RM Calculation for Lines of Sight Through the Walls of the Shell\label{sec:walls}}
In evaluating the integral Equation (\ref{eq:rmorg}) or (\ref{eq:rmprat}) through the model shell shown in Figure \ref{fig:steve1}, we consider two cases. The first calculation is for lines of sight that pass through a portion of the shell, emerge into the cavity, and then reenter the shell on the opposite side before exiting the shell entirely. This is the case illustrated in Figure \ref{fig:steve1}. The incremental RM for a spatial interval d$s$ along the line of sight is
\begin{equation}
\textrm{d(RM)} = \pm \, C \, n_s \, B_s(\theta) \, (\hat{e}_s \cdot \hat{e}_{\theta})\, \textrm{d}s
\label{eq:steve12}
\end{equation}
The $\pm$ in front of the RHS indicates that the polarity of the field in the external medium determines the sign of the measured RM. We introduce the variable $s$ as a coordinate along the line of sight; d$s$ is an incremental vector along the line of sight from the source to the observer, and $\hat{e}_s$ is the corresponding unit vector. The constant $C$ is the same as introduced in Equation (\ref{eq:rmmodelW}).
It is convenient to change the variable of integration over the LOS from $s$ to $\phi$, an angle defined in Figure \ref{fig:steve1}. With the introduction of this variable, the term $(\hat{e}_s \cdot \hat{e}_{\theta}) = -\sin \phi$. Integration through the shell segments along the line of sight then corresponds to an appropriate integration over $\phi$. The shell segment closest to the observer corresponds to an integration from $\phi_1$ to $\phi_2$, and the segment furthest from the observer is given by an integration from $\phi_3$ to $\phi_4$.
Substitution of Equations (\ref{eq:steve10}) and (\ref{eq:steve11}) into (\ref{eq:steve12}), followed by integration over $\phi$ and straightforward algebraic manipulation yields the following expression for the RM
\begin{equation}
\textrm{RM}(x) = \pm \left( \frac{C \, n_0\, B_0\, R_s}{3 \epsilon^2} \right) x \left[ \arcsin \left(\frac{x}{1 - \epsilon}\right) - \arcsin (x) \right] \cos \Theta
\label{eq:steve13}
\end{equation}
where the new dependent variable is the normalized impact parameter $x \equiv \frac{\xi}{R_s}$. The identity (\ref{eq:steve11}) may be used to convert Equation (\ref{eq:steve13}) into a form in which the observed plasma density in the shell ($n_s$) is the density parameter rather than that in the external medium ($n_0$). This substitution makes Equation (\ref{eq:steve13}) more directly comparable to Equation (\ref{eq:rmmodelW}).
\subsubsection{RM for Lines of Sight Entirely Within the Shell \label{sec:allshell}}
If the ``impact parameter'' $\xi$ is sufficiently large, the entire line of sight is within the shell from the point of ingress to that of egress. From Figure \ref{fig:steve1}, it can be seen that this occurs if
\begin{equation}
x \equiv \frac{\xi}{R_s}\, \geq\, x_{min} = 1 -\epsilon
\label{eq:steve14}
\end{equation}
The RM in this case is a simple generalization of the algebra involved in obtaining Equation (\ref{eq:steve13}) via an integration over the angular variable $\phi$; the upper limit of integration in the segment closest to the observer $\phi_2 \rightarrow \frac{\pi}{2}$, and the lower limit of integration for the shell segment further from the observer $\phi_3 \rightarrow \frac{\pi}{2}$.
\begin{equation}
\textrm{RM}(x) = \pm \left( \frac{C\, n_0\, B_0\, R_s}{3 \epsilon^2} \right)\, x\, \left[ \frac{\pi}{2} - \arcsin (x) \right] \cos \Theta \mbox{ , if: } x_{min} \leq x \leq 1
\label{eq:steve15}
\end{equation}
A plot of the expression RM($x$) given by (\ref{eq:steve13}) and (\ref{eq:steve15}) is shown in Figure \ref{fig:steve2} for a set of parameters that are representative for the IC 1805 HII region (see Table \ref{tab:model}). The curve is very similar in form to the Whiting model, for the case of no magnetic compression, Equation (\ref{eq:rmmodelW}) with $X = 1$ or Equation (\ref{eq:rmmodelH}). The model expression for RM($x$) is dependent on $n_0$ (or the shell density $n_s$), $B_0$, $R_s$, $\Theta$, and $\epsilon$, the shell thickness parameter. For comparison with observations, we also need to specify the background Galactic rotation measure, RM$_{off}$.
Our simple model contained in Equations (\ref{eq:steve13}) and (\ref{eq:steve15}) immediately accounts for one of the main results emergent from the numerical simulations of \citet{Stil:2009}. The RM through a bubble is maximized when the LOS is parallel to \textbf{B}$_0$ ($\cos \Theta = 1$) and small or zero when the LOS is $\perp \mbox{ to } \textbf{B}_0$ ($\cos \Theta = 0$).
\begin{figure}[htb!]
\centering
\includegraphics[width=0.6\textwidth]{f12.pdf}
\caption{Model for the analytic approximation to the bubble model of \citet{Ferriere:1991}, Equations (\ref{eq:steve13}) and (\ref{eq:steve14}). The model RM is function of the normalized impact parameter $x = \frac{\xi}{R_s}$. The plotted points represent measured RMs presented in this paper. }
\label{fig:steve2}
\end{figure}
\section{Discussion of Observational Results\label{sec:results}}
\subsection{Comparison of Models with Observations in the HII Region}
In this section we discuss the results of the two models presented in Sections \ref{sec:whitingmod} and \ref{sec:ferriere}. In both cases, we adopt the \citet{Terebey:2003} center for geometric ease and spherical symmetry as well as the parameters given in their Table 3 for a thick shell.
Figures \ref{fig:southshell-thick} and \ref{fig:southshell-thin} show model RM values for lines of sight south of IC 1805 (\textit{b} $<$ 0.9\ddeg) with the \citet{Whiting:2009} model for the RM as a function of distance and the shell parameters from \citet{Terebey:2003}. Table \ref{tab:model} gives the values of the center of the bubble, the shell radii, the electron density, X, and $\Theta$ for Figure \ref{fig:shell}. Neither model reliably reproduces the observed RM as a function of distance, and as in \citet{Costa:2016}, the model can not account for the dispersion of RM values at similar distances. Generally, the lines of sight in the cavity are low and are more consistent with the background RM. In the thin shell approximation, the largest RM values are associated with lines of sight outside the shell. While the model without amplification of the magnetic field in the shell can marginally account for the magnitude of the RM, the model with amplification (Equation \ref{eq:rmmodelW}) predicts far too high values for the RM for $\Theta$ = 55\ddeg. The analysis contained here mildly supports a result from \citet{Costa:2016} for the Rosette Nebula; Faraday rotation values through these HII regions do not permit a substantial increase in $|$B$|$ over the general Galactic field.
To reproduce the observed RM in the shell at $\xi$ $\sim${} 20 pc, the angle between the magnetic field and the observer would need to be tilted more into the plane of the sky for the X = 4 case or into the line of sight for the X = 1 case. For the former case, an angle of $\sim${} 75\ddeg{} would reproduce the magnitude of the RM in the shell; such an angle is greater than that expected from a geometric argument, even accounting for a magnetic field pitch angle of $\sim${} 8\ddeg. Also, no one angle can account for the range of the RM values in the cavity.
With our analytic solution for the RM due to a magnetized bubble as described by \citet{Ferriere:1991}, we can examine the dependence of RM on $\Theta$ as well as $\xi$. The most obvious choice for the latter parameter is $\Theta = 55^{\circ}$, based on the geometry as described in Section \ref{sec:whitingmod}. Figure \ref{fig:steve2} shows our model RM(x) for $\Theta = 55^{\circ}$, with other parameters given in Table 6. Data for sources south of IC 1805 are superposed on the model. Although the model obviously does not reproduce the measurements in detail, it can describe the overall scale of the ``rotation measure anomaly'' associated with W4, as well as the approximate magnitude of the largest measured RMs ($|$RM$|$ $\sim$ 1000 rad m$^{-2}$). The peak model RM values shown in Figure 12 do not significantly exceed the measured values, unlike the case for the Whiting model with X = 4 (see Figure \ref{fig:shell}). It should be kept in mind that the shell modeled in Figure \ref{fig:steve2} is the ``thick shell model'' of \citet{Terebey:2003}; the center of that shell is not the star cluster OCl 352, as might be expected.
\citet{Stil:2009} carried out numerical MHD simulations of the Ferri\`ere bubbles, which are obviously more accurate than our analytic approximations. Furthermore, they specifically consider and calculate the Faraday rotation through their models. However, \citet{Stil:2009} only consider $\Theta$ = 0\ddeg{} and $\Theta$ = 90\ddeg, so the calculations reported in that paper can not explore the changes in RM structure with $\Theta$. Furthermore, the Faraday rotation calculation of \citet{Stil:2009} is done when the outer radius $R_s$ $\sim${} 200 pc (see Figure 14 of \citealt{Stil:2009}), which is much larger than the structure we are modeling in Section \ref{sec:ferriere} of this paper. In what follows, we compare our observations with the results presented in Section 6 of \citet{Stil:2009}.
If LOS $||$ \textbf{B}$_{\textrm{ext}}$, then the highest values of RM will be through the shell closest to the Galactic plane, but the mean RM across the region will be similar to the mean RM exterior to the bubble (see Figure 14 of \citealt{Stil:2009}). Out of the Galactic plane, the RM is 20 -- 30$\%$ of the mean RM exterior to the bubble. Effectively, the largest RMs will always be found in the Galactic plane, and different lines of sight through the bubble will have varying RM values.
In comparing the simulations of \citet{Stil:2009} to our observational results, we find low RM measures for lines of sight through the cavity, though not always low (e.g., W4-I14 and -I17 vs -I15 and -I21). Lines of sight through the shell have generally large RMs, which is inconsistent with a \textbf{B}$_{\textrm{ext}}${} perpendicular to the LOS. The case of LOS $||$ \textbf{B$_{\textrm{ext}}$} is inconsistent as well because far from the bubble, the RM is low (e.g., W4-O26 vs -I24) even at similar latitudes, and the lines of sight at $b$ $>$ 1\ddeg{} are consistent with the background RM instead of being reduced by 70 -- 80$\%$. Unsurprisingly, our results indicate a case somewhere between these two predictions. As a reminder, we note that the largest values of the RM are for lines of sight exterior to the shell, which is not a prediction from \citet{Stil:2009}, most likely due to their simulations modeling the ionized bubble and not a PDR structure.
\subsection{Magnetic Fields in the PDR}
In Section \ref{sec:pdr}, we examine evidence for a PDR outside the southern loop of IC 1805. \citet{Brogan:1999}, \citet{Troland:2016}, and \citet{Pellegrini:2007} report large ($\sim${} 150 $\mu$G) magnetic fields in PDRs associated with the Orion Veil and M17. In the analysis that follows, we attempt to understand the large RM values for lines of sight through the IC 1805 PDR.
If we consider the PDR and the HII region to be in pressure equilibrium and include magnetic pressure in the PDR, then
\begin{equation}
P^{\textrm{H}\,\textsc{ii}}_{\textrm{th}} = P^{\textrm{PDR}}_{\textrm{th}} \; + P^{\textrm{PDR}}_{\textrm{mag}},
\label{eq:pbal}
\end{equation}
where $P^{\textrm{H}\,\textsc{ii}}_{\textrm{th}}${} and $P^{\textrm{PDR}}_{\textrm{th}}${} are the thermal pressures in the HII region and PDR, respectively, and \(P^{\textrm{PDR}}_{\textrm{mag}}=\frac{B^2}{8\pi} \) is the magnetic pressure in the PDR. In the HII region, \(P^{\textrm{H}\,\textsc{ii}}_{\textrm{th}} = 2n_e^{\textrm{H}\,\textsc{ii}}\, k \,T_{\textrm{H}\,\textsc{ii}} \), where $n_e^{\textrm{H}\,\textsc{ii}}$ and T$_{\textrm{H}\,\textsc{ii}}$ are the electron density and temperature, $k$ is the Boltzmann constant, and the factor of 2 accounts for the contribution from both ions and electrons. For P$^{\textrm{PDR}}_{\textrm{th}}$ = $N_{\textrm{PDR}}$ $k$ $T_{\textrm{PDR}}$, $N_{\textrm{PDR}}$ and $T_{\textrm{PDR}}$ are the neutral hydrogen density and the temperature in the PDR.
Near the interface of the PDR and the HII region, the electron density in the PDR is governed by photoioniziation of carbon \citep{Tielens:1985}, so we estimate $n_e^{\textrm{PDR}}$ by \[n_e^{\textrm{PDR}} = N_{\textrm{PDR}}X_C,\] where $X_C$ is the cosmic abundance of carbon given in Table 1.4 of \citet{Draine:2011} ($X_C$ $\sim${} 2.95 $\times$ 10$^{-4}$). Solving for $B$ in \equa{pbal} gives
\begin{equation}
B = \sqrt{8\pi \; k(2\; n_e^{\textrm{H}\,\textsc{ii}} \;T_{\textrm{H}\,\textsc{ii}} - N_{\textrm{PDR}}\;T_{\textrm{PDR}})},
\label{eq:bpbal}
\end{equation}
and inserting it into \equa{rmprat}, we express the RM in the PDR as
\begin{equation}
RM = 0.81\; L \; X_C N_{\textrm{PDR}} \sqrt{8\pi \; k \; (2\;n_e^{\textrm{H}\,\textsc{ii}} T_{\textrm{H}\,\textsc{ii}} - N_{\textrm{PDR}}\; T_{\textrm{PDR}})}.
\label{eq:rmpdr}
\end{equation}
It should be emphasized that this RM estimate is in the nature of an upper limit to the rotation measure through the PDR. The reason is that it is obtained from a value of $B$, given by Equation (\ref{eq:rmpdr}), which is based on the magnetic pressure $\frac{B^2}{8 \pi}$. The magnetic pressure includes contributions from turbulent fluctuations on all scales, as well as that from a mean or large scale field that produces the net Faraday rotation. In general then, a magnetic field value obtained from an estimate of the magnetic pressure will exceed that obtained from a Faraday rotation measurement.
We differentiate \equa{rmpdr} with respect to $N_{\textrm{PDR}}$ to find the value of $N_{\textrm{PDR}}$ that maximizes the RM, which is
\begin{equation}
N_{\textrm{PDR}} = \frac{4}{3}\frac{n_e^{\textrm{H}\,\textsc{ii}}\;T_{\textrm{H}\,\textsc{ii}}}{T_{\textrm{PDR}}}.
\label{eq:maxN}
\end{equation}
Inserting values of $T_{\textrm{H}\,\textsc{ii}}$ = 8000 K, $n_e^{\textrm{H}\,\textsc{ii}}$ = 10 cm$^{-3}${} \citep{Terebey:2003}, and $T_{\textrm{PDR}}$ = 100 K \citep{Tielens:1985}, gives $N_{\textrm{PDR}}$ $\sim${} 1000 cm$^{-3}${}, $B$ $\sim${} 14 $\mu$G (Eq \ref{eq:bpbal}), and RM $\sim${} 100 rad m$^{-2}$. The electron density in the HII region is governing the maximum $B$ expected in the PDR given pressure balance. For the IC 1805 HII region, $n_e$ is low compared to M17 ($n_e$ $\sim${} 560 cm$^{-3}$) \citep{Pellegrini:2007}, which suggests that a high density (pressure) HII region is needed to explain large magnetic fields in the PDR.
Our analysis suggests that a simple pressure balance analysis predicts low RM values from the PDR that are inconsistent with our observations. It appears that a different mechanism is required to achieve the magnetic fields strengths observed in \citet{Brogan:1999}.
\citet{Terebey:2003} discuss an extended halo of ionized emission around the southern loop, which may indicate that there are more free electrons present outside the obvious ionized shell as seen in Figure \ref{fig:w4}. This may account for the larger values of the RM we observe. It is clear that knowing the electron density in this region and determining the presence of a PDR through observations, such as carbon radio recombination lines, is necessary to understand how the magnetic field is modified in this complex region.
\section{A Comparison of IC 1805 and the Rosette Nebula as ``Rotation Measure Anomalies'' \label{sec:rosette}}
\begin{table}[htb!]
\centering
\caption{Stellar Parameters \label{tab:stellarpar}}
\begin{threeparttable}
\centering
\begin{tabular}{llllll}
\hline
\multicolumn{1}{c}{Star Cluster} & \multicolumn{1}{c}{Star} & \multicolumn{1}{c}{Type} & \multicolumn{1}{c}{\ML} & \multicolumn{1}{c}{V$_{\ensuremath{\infty}}$} & \multicolumn{1}{c}{ L$_{W}$=$\frac{1}{2}\dot{M}$v$_{\ensuremath{\infty}}^{2}$}\tnote{a} \\
\multicolumn{1}{c}{ } &\multicolumn{1}{c}{ } &\multicolumn{1}{c}{ } & \multicolumn{1}{c}{ (\Msun yr$^{-1}$) } & \multicolumn{1}{c}{(km/s)} & \multicolumn{1}{c}{ (erg s$^{-1}$)} \\
\hline
{\multirow{4}{*}{NGC 2244}} & HD 46223 & O4V(f)\tnote{b} & 1.6$\times$10$^{-6}$ \tnote{c} & 3100\tnote{d} & 4.8$\times$10$^{36}$ \\
& HD 46150 & O5.5V\tnote{e} & 2.0$\times$10$^{-6}$ \tnote{c} & 3100\tnote{d} & 6.0$\times$10$^{36}$ \\
& HD 46202 & O9V(f)\tnote{b} & 6.3$\times$10$^{-8}$ \tnote{c} & 1150\tnote{d} & 2.6$\times$10$^{34}$ \\
& HD 46149 & O8.5V(f)\tnote{b} & 2.0$\times$10$^{-7}$ \tnote{c} & 1700\tnote{f} & 1.8$\times$10$^{35}$\\
\hline
{\multirow{3}{*}{OCl 352}} & HD15570 & O4I\tnote{b} & 1.0$\times$10$^{-5}$ \tnote{c} & 2200\tnote{g} & 1.5$\times$10$^{37} $ \\
& HD15558 & O4III\tnote{b} & 6.3$\times$10$^{-6}$ \tnote{c} & 3000\tnote{f} & 1.8$\times$10$^{37} $ \\
& HD 15629 & O5V\tnote{b} & 2.0$\times$10$^{-6}$ \tnote{c} & 2900\tnote{h} & 5.3$\times$10$^{36}$ \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] Calculated mechanical wind luminosity based on cited mass loss rates and terminal velocities.
\item[b] \citet{Massey:1995}
\item[c] \citet{Howarth:1989}
\item[d] \citet{Chlebowski:1991}
\item[e] \citet{Roman:2008}
\item[f] \citet{Garmany:1988}
\item[g] \citet{Bouret:2012}
\item[h] \citet{Groenewegen:1989}
\end{tablenotes}
\end{threeparttable}
\end{table}
We are interested in how the Galactic magnetic field is modified by OB associations via their stellar winds and ionizing photons, and we started our study with the Rosette Nebula, where we found large ($\sim$~10$^{3}$ rad m$^{-2}$) RM measurements through the ionized shell of the HII region \citep{Costa:2016}
In the case of the Rosette, we find positive RM across the region, and for IC 1805, we find negative values. If the Galactic magnetic field follows the spiral arms in a clockwise direction, then we would expect the LOS magnetic field component to be pointed towards us (positive B) for $\ell$ $>$ 180\ddeg, and pointed away from us (negative B) for $\ell$ $<$ 180\ddeg. Except for one line of sight in each nebula, we find that the polarity of the Galactic magnetic field is preserved across each nebula and is consistent with the large scale field through the arm.
In our study of the Rosette, we investigated whether the magnetic field is amplified in the shell of the nebula. We found that the model without amplification was weakly favored over the case when the magnetic field is amplified in the shell. When we applied the same model to IC 1805, however, it is difficult to conclude in favor of either model, but in both cases, the model with an enhanced magnetic field overpredicts the RM. From inspection of Figures \ref{fig:southshell-thick} and \ref{fig:southshell-thin}, it seems that the model without amplification better accounts for the magnitude of the observed RMs, but the observations do not conform to the model prediction of RM($\xi$), and the model can not account for the wide range in observed values of RM at a given $\xi$.
In the present study, we find the highest RMs for lines of sight outside the obvious shell structure, though one line of sight (W4-I13) does appear to intersect the ionized shell and it has a large RM. These lines of sight may be probing the magnetic field within the PDR. In the case of the Rosette, we found that the highest RM values were for lines of sight through the bright ionized shell. However with our work on IC 1805 and the PDR associated with it, we have briefly revisited our results in the Rosette, particularly Figure 1 from \citet{Costa:2016}. There are a few lines of sight with RM of order a few 10$^2$ rad m$^{-2}${} that appear to be outside the ionized shell. These lines of sight were included in the background estimate for the Rosette, but if the Rosette also has a PDR, then these lines of sight may actually be probing that material.
Table \ref{tab:stellarpar} lists spectral type, mass loss rate, terminal wind velocity, and calculated wind luminosity from the literature for O stars with the largest wind luminosities in both NGC 2244, which is associated with the Rosette Nebula, and OCl 352. The sum of the wind luminosities of the three main stars in OCl 352 is 3.8 $\times$ 10$^{37}$ ergs s$^{-1}$, while the corresponding number for NGC 2244 (4 stars) is 1.1 $\times$ 10$^{37}$ ergs s$^{-1}$. In addition, OCl 352 appears to have more luminous stars. As such, OCl 352 might be expected to produce a more energetic stellar bubble than NGC 2244. Our Faraday rotation measurements show no indication of this, in that the largest RMs observed are similar for the two objects. In fact, higher RMs were measured for the Rosette than for any line of sight through IC 1805. A number of factors can control the impact a star cluster has on the ISM. If some relationship exists between the total wind luminosity of a star cluster and properties of an interstellar bubble that can be measure with Faraday rotation, it will apparently require a large sample of clusters/ HII regions to reveal it.
\section{Future Research\label{sec:fut}}
In the future, we will continue our investigation of HII regions and how they modify their surroundings and the Galactic magnetic field. An immediate investigation will be centered on observations of the HII region IC 1396. This will provide a third HII region with different age, stellar content, and Galactic location. The observations are similar to those we have made of the Rosette Nebula and IC 1805. The observations of IC 1396 have been made with the VLA and are awaiting analysis. By adding more HII regions to our study, we can begin to address questions such as
\begin{enumerate}
\item Since the electron density distributions in HII regions are known from radio continuum observations, we can inquire what conditions would result in an RM $>$$>$ 10$^3$ rad m$^{-2}$~through the shell of an HII region.
\item Is it a general property of HII regions and stellar bubbles that the polarity of the Galactic magnetic field is preserved within the region? The answer to this question has implications for the amplitude of MHD turbulence in the ISM on scales of the order of the HII regions, $\sim 10 - 30$ pc.
\item Do PDRs around other nebulae produce high RMs? What is the magnitude of the RM due to the PDR relative to that of the shell of an HII region?
\end{enumerate}
In addition to increasing the number of HII regions, understanding Faraday complexity and how to interpret the associated RM measurements is important to studies of Galactic magnetic fields, particularly with large polarization surveys like the VLA Sky Survey (VLASS){} and Polarisation Sky Survey of the Universe's Magnetism (POSSUM){} with the Australian Square Kilometre Array Pathfinder (ASKAP){} in the near future.
\section{Summary and Conclusions\label{sec:sum}}
\begin{enumerate}
\item We performed polarimetric observations using the VLA for 27 lines of sight through or near the shell of the HII region and stellar bubble associated with the OB association OCl 352.
\item We obtain RM measurements for 20 sources using two methods. The first is through the traditional least-squares fit to $\chi$($\lambda^{2}$), and the second is using RM Synthesis. Including components that are resolved, we report 27 RM values, and we find good agreement between the two methods. We find the same sign of the RM across the entire region with the exception of one source, W4-I18. We estimate a background RM due to the general ISM of --145 rad m$^{-2}${} in this part of the Galactic plane.
We measure an excess of RM of $\sim$~+600 to --800 rad m$^{-2}$~due to W4.
\item Only one line of sight has a positive RM value, W4-I18. It has a RM of +501 $\pm$ 33 rad m$^{-2}$, and it is located 5.6 arcminutes from the center of OCl 352. This line of sight may be probing the material close to the massive stars.
The orientation of the line of sight component of the magnetic field is directed towards the observer, whereas in the rest of the region, the magnetic field is directed away.
\item We find that some of the lines of sight with the largest RM values occur just outside the obvious ionized shell of IC 1805 and are potentially probing the magnetic field in the PDR. The lines of sight through the cavity of the bubble have lower RM values than those through the shell. In the W4 Superbubble, which is north of OCl 352, we find RM values consistent with the background RM.
\item We discuss two shell models to reproduce the magnitude of the RM and its dependency on distance from the center of the star cluster. We employed the first of these models in \citet{Savage:2013} and \citet{Costa:2016}, and it is based on the \citet{Weaver:1977} solution for a stellar bubble, which includes a shock expanding into an ambient medium. The second model uses magnetic flux conservation to describe how the magnetic field is modified in the shell and consists of a simplified analytic approximation to the results presented by \citet{Ferriere:1991}. Neither of these simplified models satisfactorily accounts for the dependence of RM on spatial location within the shell, although the Whiting model without field amplification (X = 1) and the simplified Ferri\`ere model approximately reproduce the magnitude of the largest RMs. However, both models predict a single-valued dependence of RM on $\xi$, the separation of the line of sight from the center of the nebula, whereas the observations show a large range of RM for sources with similar values of $\xi$.
\item Because we have independent information on the electron density from radio continuum observations of both IC 1805 and the Rosette Nebula, our observations can limit the magnitude of the magnetic field in the HII regions. Our RM measurements indicate that the field does not greatly exceed the value in the general ISM.
\item We compare our results from the current study of IC 1805 and our previous study of the Rosette Nebula. Notably, we find the same order of magnitude for the RM for the two nebulae, but the sign of the RM in each region is opposite. Since IC 1805 and the Rosette are at different Galactic longitudes and on either side of $b$ = 180\ddeg, the sign difference between the two nebula is consistent with a Galactic magnetic field that follows the spiral arm structure in a clock-wise direction, as suggested in models \citep{vanEck:2011}.
\end{enumerate}
\acknowledgments
This research was partially supported at the University of Iowa by grants AST09-07911 and ATM09-56901 from the National Science Foundation. This publication makes use of data products from the Wide-field Infrared Survey Explorer \citep{2010AJ....140.1868W}, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Additionally, the research presented in this paper uses data from the Canadian Galactic Plane Survey, a Canadian project with international partners, supported by the Natural Sciences and Engineering Research Council. This research also uses the Python packages Astropy, a community-developed core Python package for Astronomy \citep{2013A&A...558A..33A} and NumPy \citep{van2011numpy}. Finally, we thank the referee of this paper for a helpful and collegial review.
\clearpage
\newpage
\section{Introduction\label{intro}}
Young massive stars in OB associations photoionize the surrounding gas, creating an HII region, and their powerful stellar winds can inflate a bubble around the star cluster. Magnetic fields are important to the dynamics of these structures \citep{Tomisaka:1990,Ferriere:1991,Vallee:1993,Tomisaka:1998,Haverkorn:2004,Sun:2008,Stil:2009}, and they can elongate the cavity preferentially in the direction of the magnetic field and thicken the shell perpendicular to the field \citep{Ferriere:1991,deAvillez:2005,Stil:2009}, causing deviations from the classical structure of the \citet{Weaver:1977} wind-blown bubble. Knowledge of the magnitude and direction of the magnetic field within stellar bubbles and HII regions is important for simulations and for understanding how the magnetic field interacts with and modifies these structures.
In previous work (i.e., \citealt{Savage:2013} and \citealt{Costa:2016}), we investigated whether the Galactic magnetic field is amplified in the shell of the Rosette Nebula, an HII region and stellar bubble associated with NGC 2244 ($\ell$ = 206.5\ddeg, $b$ = --2.1\ddeg). Other similar work investigating magnetic fields near massive star clusters has been done by \citet{Harvey:2011} and \citet{Purcell:2015}. In this work, we continue our investigation of how HII regions and stellar bubbles modify the ambient Galactic magnetic field by considering another example of a young star cluster and an HII region that appears to be formed into a shell by the effect of stellar winds.
\subsection{Faraday Rotation and Magnetic Fields in the Interstellar Medium}
Faraday rotation measurements probe the line of sight (LOS) component of the magnetic field in ionized parts of the interstellar medium (ISM), provided there is an independent estimate of the electron density. Faraday rotation is the rotation in the plane of polarization of a wave as it passes through magnetized plasma and is described by the equation
\begin{equation}
\chi=\chi_{0}+\left[\left(\frac{e^{3}}{2\pi m_{e}^{2}c^{4}}\right)\int{n_e\ \mathbf{B}\cdot \textrm{d}\mathbf{s}}\right]\lambda^{2},
\label{eq:rmorg}
\end{equation}
where $\chi$ is the polarization position angle, $\chi_0$ is the intrinsic polarization position angle, the quantities in the parentheses are the usual standard physical constants in cgs units, \textit{n$_{\textrm{e}}$} is the electron density, \textbf{B} is the vector magnetic field, d\textbf{s} is the incremental path length interval along the LOS, and $\lambda$ is the wavelength. We define the terms in the square bracket as the rotation measure, RM, and we can express the RM in mixed but convenient interstellar units as
\begin{equation}
\textnormal{RM}=0.81\in
n_{e} \ (\text{cm$^{-3}$}) \ \mathbf{B} \ (\mu\text{G})\cdot \textrm{d}\mathbf{s} \text{ (pc) rad m$^{-2}$.}
\label{eq:rmprat}
\end{equation}
\subsection{The HII Region and Stellar Bubble Associated with the W4 Complex\label{sec:structure}}
The HII region and stellar bubble of interest for the present study is IC 1805, which is located in the Perseus Arm. The star cluster responsible for the HII region and stellar bubble is OCl 352, which is a young cluster (1--3 Myr) \citep{Basu:1999}. OCl 352 has 60 OB stars \citep{Shi:1999}. Three of these are the O stars HD 15570, HD 15558, and HD 15629, and they have mass loss rates between 10$^{-6}$ and 10$^{-5}$ \textrm{M}$_{\odot}$ yr$^{-1}$~\citep{Massey:1995} and terminal wind velocities of 2200 -- 3000 km s$^{-1}$~\citep{Garmany:1988,Groenewegen:1989,Bouret:2012}. We adopt the nominal center of the star cluster to be R.A.(J2000) = 02$^h$ 23$^m$ 42$^s$, decl.(J2000) = +61\ddeg 27$'$ 0$''$ ($\ell$ = 134.73, \textit{b} = +0.92) \citep{Guetter:1989} and a distance of 2.2 kpc to IC 1805 to conform with previous studies of the region (e.g., \citealt{Normandeau:1996,Dennison:1997,Reynolds:2001,Terebey:2003,Gao:2015}). In the literature, other distance values include: 2.35 kpc \citep{Massey:1995,Basu:1999,West:2007,Lagrois:2012}, 2 kpc \citep{Dickel:1980}, 2.04 kpc \citep{Feigelson:2013,Townsley:2014}, and 2.4 $\pm$ 0.1 kpc \citep{Guetter:1989}.
We refer to the HII region between --0.2\ddeg~$<$ \textit{b} $<$ 2\ddeg~as IC 1805. This structure is also known as the Heart Nebula for its appearance at optical wavelengths. We differentiate this region from the northern latitudes that constitute the W4 Superbubble \citep{Normandeau:1996,West:2007,Gao:2015}, and we use the nomenclature of W4 to describe the entire region, which includes IC 1805 and the W4 Superbubble. Below we summarize the structure of IC 1805 and Figure \ref{fig:cartoon} is a cartoon diagram of the structure described here.
\begin{itemize}
\item[--] \textit{South.} On the southern portion of IC 1805, there is a loop structure of ionized material at 134\ddeg $<$ $\ell$ $<$ 136\ddeg, \textit{b} $<$ 1\ddeg, which we call the southern loop. \citet{Terebey:2003} find that at far infrared and radio wavelengths, the shell structure is well defined and ionization bounded, since the ionized gas lies interior to the dust shell. However, they also find that there is warm dust that extends past the southern loop and a faint ionized halo (see their Figure 6). \citet{Terebey:2003} argue that the shell is patchy and inhomogeneous in density, which allows ionizing photons to escape. \citet{Gray:1999} discuss extended emission surrounding IC 1805 and suggest that it may be evidence of an extended HII region \citep{Anantharamaiah:1985}. Also surrounding IC 1805 are patchy regions of HI \citep{Braunsfurth:1983,Hasegawa:1983,Sato:1990} and CO \citep{Heyer:1998,Lagrois:2009}.
\citet{Terebey:2003} model the structure of the southern loop using radio continuum data. They assume a spherical shell and place OCl 352 at the top edge of the bubble instead of at the center to accommodate spherical symmetry (see their Figures 4 and 5). The center of their shell model is at ($\ell$, $b$) = (135.02\ddeg, 0.42\ddeg). They find an inner radius of 30 arcmin (19 pc) and a shell thickness of 10 arcmin (6 pc) and 2.5 arcmin (2 pc) for a thick and thin shell model, respectively. \citet{Terebey:2003} report electron densities of 10 cm$^{-3}$~and 20 cm$^{-3}$~for the thick and thin shell models, respectively (see Section 3.5 and Table 3 of \citealt{Terebey:2003}). While we utilize and discuss these models in the following sections, the center position of the shell in \citet{Terebey:2003} was selected to fit the ionized shell, and as such, the shell parameters should only be used to describe the bottom of IC 1805. For latitudes near the star cluster, the model fails, as the star cluster is at the top edge of the bubble instead of at the center.
\item[--] \textit{East.} On the eastern edge of IC 1805 ($\ell$ $>$ 134.6\ddeg, \textit{b} $<$ 0.9\ddeg), \citet{Terebey:2003} find that warm dust extends outside the loop boundary and suggest that if the warm dust is associated with the ionized gas, then the bubble has blown out on the eastern side of IC 1805.
At the Galactic latitude equal to the star cluster, the ionized gas appears to be pinched \citep{Basu:1999}, which is usually caused by higher densities. There is a clump of CO emission in the vicinity of the eastern pinch at ($\ell$, $b$) = (135.2\ddeg, 1.0\ddeg) \citep{Lagrois:2009}, and there is HI emission on the eastern edge at ($\ell$, $b$) $\geq$ (136\ddeg, 0.5\ddeg) (see Figure 1 of \citealt{Sato:1990}).
\item[--] \textit{West.} On the western edge of IC 1805 is the W3 molecular cloud and the W3 complex, which hosts a number of compact HII regions and young stellar objects (see \citealt{Bik:2012} and their Figure 1). \citet{Dickel:1980} modeled the structure of W3, which is thought to be slightly in front of W4, and they argue that the advancement of the IC 1805 ionization front and shock front into the W3 molecular cloud may have triggered star formation. \citet{Moore:2007} similarly conclude that the W3 molecular cloud has been compressed on one side by the expansion of IC 1805. While infrared sources nestled between the western edge of IC 1805 and eastern edge of the W3 molecular cloud are thought to be the product of this interaction , W3 Main, W3 (OH), and W3 North are thought to be sites of triggered star formation from IC 1795, which is part of W3 as well and not from the expansions of the ionization front \citep{Nakano:2017, Jose:2016,Kiminki:2015}. There is therefore uncertainty regarding a physical connection between W3 and IC 1805.
\item[--] \textit{North.} North of OCl 352, the bubble opens up into what is called the W4 Superbubble \citep{Normandeau:1997,Dennison:1997,West:2007,Gao:2015}, which is a sealed ``egg-shaped'' structure that extends up to \textit{b} $\sim$ 7\ddeg~ \citep{Dennison:1997,West:2007}. At the latitude of the star cluster, \citet{Lagrois:2009} estimate the distance between the eastern and western shell to be $\sim$ 1.2\ddeg~(46 pc) that increases in size up to 1.6\ddeg~(61 pc) at \textit{b} = 1.8\ddeg~(see Figure 11 of \citealt{Lagrois:2009}). At higher latitudes, \citet{Dennison:1997} model the thickness of the shell to be between 10--20 pc (16 -- 31 arcminutes) from H$\alpha$ observations.
The ``v''-shaped feature seen in Figure \ref{fig:w4} at ($\ell$, $b$) $\sim$ (134.8\ddeg, 1.35\ddeg) is prominent in the ionized emission, and \citet{Heyer:1996} report a cometary-shaped molecular cloud near ($\ell$, $b$) $\sim$ (134.8\ddeg, 1.35\ddeg). The alignment of the cometary cloud, as it is pointed towards IC 1805, suggests that the UV photons from the star cluster are responsible for the ``v'' shaped feature in the ionized emission on the side closest to the star cluster \citep{Dennison:1997,Taylor:1999}. \citet{Lagrois:2009} argue, from radial velocity measurements, that the cloud is located on the far side of the bubble wall, and while it may appear to be a cap to the bubble connecting to the southern loop, it is simply a projection effect. As such, the ridge of ionized material directly north of OCl 352 is not the outer radius of the shell but is part of the rear bubble wall.
\item [--] \textit{PDR.} The HI and molecular emission near the southern ($\ell$ $<$ 0.9\ddeg) portions of IC 1805 suggest that a Photodissociation Region (PDR){} has formed exterior to the HII region. PDRs are the transition layer between the fully ionized HII region and molecular material, where far UV photons can propagate out and photodissociate molecules. We discuss the importance and observational evidence of a PDR in Section \ref{sec:pdr}.
\end{itemize}
\begin{figure}[htb!]
\centering
\includegraphics[width = 0.6\textwidth]{f1.pdf}
\caption[]{Cartoon of structure of IC 1805 and the surrounding region, which includes the W3 molecular cloud, W3 (blue filled circle), and the W4 Superbubble. The solid black lines represent the bright ionized shell in Figure \ref{fig:w4}, the dotted lines show molecular material from \citet{Lagrois:2009}, and the gray shading represents the extended halo or PDR. The star represents the center of the exciting star cluster, OCl 352.}
\label{fig:cartoon}
\end{figure}
There is an extensive literature on the W4 region and its relationship to W3, dealing with the morphology \citep{Dickel:1980,Dickel:1980b,Braunsfurth:1983,Normandeau:1996,Dennison:1997,Heyer:1998,Taylor:1999,Basu:1999, Terebey:2003,Lagrois:2009,Lagrois:2009b,Stil:2009} and star formation history \citep{Carpenter:2000,Oey:2005}. In the following paragraphs, we summarize those results from the literature that are most relevant to our polarimetric study and inferences on magnetic fields in this region.
Measurements of the total intensity and polarization of the Galactic nonthermal emission in the vicinity of HII regions are of interest because the HII regions and environs act as a Faraday-rotating screen inserted between the Galactic emission behind the HII region and that in front. Few radio polarimetric studies exist in the literature to date of the IC 1805 stellar bubble. \citet{Gray:1999} present their polarimetric results of the W3/W4 region at 1420 MHz with the Dominion Radio Astrophysical Observatory (DRAO) Synthesis Telescope. They find zones of strong depolarization near the HII regions, particularly in the south, where there is a halo of extended emission around IC 1805.
They conclude that RM values on order 10$^3$ rad m$^{-2}$~and spatial RM gradients must exist to explain the depolarization near the HII region. More recently, \citet{Hill:2017} present results of their polarimetric study of the Fan region ($\ell$ $\sim$ 130\ddeg, --5\ddeg $\leq$ \textit{b} $\leq$ +10\ddeg), which is a large structure in the Perseus arm that includes W3/W4. While the focus of their study was not on W4 specifically, they find similar results to \citet{Gray:1999} in that there is sufficient Faraday rotation to cause beam depolarization in the regions of extended emission.
In the W4 Superbubble, \citet{West:2007} determined the LOS magnetic field strength by estimating depolarization effects along adjacent lines of sight. Using estimates of the shell thickness and the electron density from \citet{Dennison:1997}, \citet{West:2007} estimate B$_{\textrm{LOS}}$~$\sim$ 3.4 -- 9.1 $\mu$G for lines of sight at \textit{b} $>$ 5\ddeg. \citet{Gao:2015} also report B$_{\textrm{LOS}}$~estimates in the W4 Superbubble by assuming a passive Faraday screen model \citep{Sun:2007} and measuring the polarization angle for lines of sight interior and exterior to the screen. For the western shell ($\ell$ $\sim$ 132.5\ddeg, 4\ddeg $<$ \textit{b} $<$ 6\ddeg) and the eastern shell ($\ell$ $\sim$ 136\ddeg, 6\ddeg $<$ \textit{b} $<$ 7.5\ddeg) in the superbubble, \citet{Gao:2015} report negative RMs between --70 and --300 rad m$^{-2}$~in the western shell and positive RMs on order +55 rad m$^{-2}$~ in the eastern shell. \citet{Gao:2015} conclude that the sign reversal is expected in the case of the Galactic magnetic field being lifted out of the plane by the expanding bubble. With H$\alpha$ estimates from \citet{Dennison:1997} for the electron density and geometric arguments for the shell radii of the W4 Superbubble, \citet{Gao:2015} estimate $|$B$_{\textrm{LOS}}$$|$ $\sim$ 5 $\mu$G.
\citet{Stil:2009} compare their magnetohydrodynamic simulations of superbubbles to the W4 Superbubble. In general, they find that the largest Faraday rotation occurs in a thin region around the cavity, and inside the cavity, it would be smaller. They also present two limiting cases for the orientation of the Galactic magnetic field with respect to the line of sight, and the consequences for the RMs through the shell. If the Galactic magnetic field is perpendicular to the observer's line of sight, then the contributions to the RM from the front and rear bubble wall would be of equal but opposite magnitude, except for small asymmetries which would lead to low RMs ($\sim$~20 rad m$^{-2}$) through the cavity. This requires the magnetic field to be bent by the bubble to have a non-zero line of sight component. If the Galactic magnetic field is parallel to the line of sight, then the RMs through the front and rear bubble wall reinforce each other, and there are high RMs for lines of sight through the shell. In this case, there are higher RMs ($\sim${} 3 $\times$ 10$^3$ rad m$^{-2}$) everywhere.
There are also studies of the magnetic field for W3. From HI Zeeman observations, \citet{vanderWerf:1990} conclude that the B$_{\textrm{LOS}}$~has small-scale structures that can vary on order of 50 $\mu$G over $\sim$ 9 arcsec scales. \citet{Roberts:1993} report values of the LOS magnetic field from HI Zeeman observations towards three resolved components of W3. The three components are near ($\ell$, $b$) $\sim$ (133.7\ddeg, 1.21\ddeg), with a maximum separation of 1.5 arcmin, and the LOS magnetic field is between --50 $\mu$G and +100 $\mu$G. \citet{Balser:2016} observed carbon radio recombination line (RRL) widths to estimate the total magnetic field strength in the photodissociation region (see \citealt{Roshi:2007} for details). They report B$_{\textrm{tot}}$ = 140 -- 320 $\mu$G near W3A (133.72\ddeg, 1.22\ddeg) and argue that for a random magnetic field, B$_{\textrm{tot}}$ = 2 $|$B$_{\textrm{LOS}}$$|$, which would then be consistent with the \citet{Roberts:1993} estimates of the B$_{\textrm{LOS}}$. It should be noted that these magnetic field strengths are substantially larger than those inferred for the W4 Superbubble on the basis of polarimetry of the Galactic background (see text above).
In this paper, we present new Faraday rotation results for IC 1805 to investigate the role of the magnetic field in the HII region and stellar bubble. As in \citet{Savage:2013} and \citet{Costa:2016}, we utilize an arguably simpler and more direct method of inferring the LOS component of the magnetic field in HII regions. This is the measurement of the Faraday rotation of nonthermal background sources (usually extragalactic radio sources) whose lines of sight pass through the HII region and its vicinity. In Section \ref{sec:obs}, we describe the instrumental configuration and observations, including source selection. Section \ref{sec:dataredux} details the data reduction process, including the methods used to determine RM values. In Section \ref{sec:obsres}, we report the results of the RM analysis and discuss Faraday rotation through the W4 complex in Section \ref{sec:fr}. We present models for the RM within the HII region and stellar bubble in Section \ref{sec:models}. We discuss our observational results and their significance for the nature of IC 1805 in Section \ref{sec:results} and compare the results of this study with our previous study of the Rosette nebula in Section \ref{sec:rosette}. We discuss future research in Section \ref{sec:fut}, and present our conclusions and summary in Section \ref{sec:sum}.
\section{Observations}\label{sec:obs}
\subsection{Source Selection\label{sec:sourceselect}}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.9\textwidth]{f2.pdf}
\caption[Radio Continuum map at 1.42GHz of W4]{Mosaic of IC 1805 from the Canadian Galactic Plane Survey at 1.42 GHz, with Galactic longitude and latitude axes. The lines of sight listed in Table \ref{tab:sources} are the red and blue symbols, where positive RMs are blue and negative RMs are red. The green and purple symbols are RM values from \citet{Taylor:2009} or \citet{Brown:2003}, where positive RMs are green and negative RMs are purple. We utilize the naming scheme from Table \ref{tab:sources} for the RM values from the literature for ease of reference, but we omit the ``W4-'' prefix in this image for clarity. The size of the plotted symbols is proportional to the $|$RM$|$ value.}
\label{fig:w4}
\end{figure}
Our criteria for source selection were identical to \citet{Savage:2013} and \citet{Costa:2016} in that we searched the National Radio Astronomy Observatory Very Large Array Sky Survey (NVSS, \citealt{Condon:1998}) database for point sources within 1\ddeg~of OCl 352 (the ``I'' sources) with a minimum flux density of 20 mJy. We also searched in an annulus centered on the star cluster with inner and outer radii of 1\ddeg~and 2\ddeg~for outer sources (``O'') to measure the background RM due to the general ISM. We identified 31 inner sources and 26 outer sources in the region. We then inspected the NVSS postage stamps to ensure that they were point sources at the resolution of the NVSS ($\sim${} 45 arcseconds). We discarded sources that showed extended structure similar to Galactic sources. We selected 24 inner sources and 8 outer sources from this final list.
\begin{table}
\centering
\caption{ List of Sources Observed}
\begin{threeparttable}
\centering
\small
\begin{tabular}{ccccccccc}
\hline
Source & $\alpha$(J2000) & $\delta$(J2000) & $\emph{l}$ & $\emph{b}$ & $\xi$\tnote{a} & S$_{4.33\textrm{GHz}}$ & m \\
Name & h m s & $^o$ $'$ $''$ & ($^o$) & ($^o$) & (arcmin) & (mJy) & ($\%$)\\
\hline
W4-I1 & 02 30 16.2 & +62 09 37.9 & 134.19 & 1.47 & 46.0 & 77 & 4 \\
W4-I2 & 02 38 34.2 & +61 08 46.6 & 135.49 & 0.91 & 46.2 & 5 & 12 \\
W4-I3 & 02 36 45.5 & +60 55 48.8 & 135.38 & 0.63 & 42.8 & 82 & 3 \\
W4-I4 & 02 27 59.8 & +62 15 44.0 & 133.91 & 1.47 & 58.9 & 46 & 10 \\
W4-I5\tnote{b} & 02 28 01.6 & +62 02 16.7 & 133.99 & 1.26 & 48.4 & --- & --- \\
W4-I6 & 02 38 19.9 & +61 08 03.5 & 135.47 & 0.89 & 44.8 & 12 & 11 \\
W4-I7\tnote{b} & 02 27 33.8 & +61 55 58.1 & 133.98 & 1.14 & 46.6 & -- & --- \\
W4-I8 & 02 38 10.1 & +62 08 57.0 & 135.05 & 1.81 & 57.1 & 47 & 9 \\
W4-I9 & 02 28 21.6 & +61 28 36.5 & 134.23 & 0.75 & 31.1 & --- & --- \\
W4-I10\tnote{b} & 02 29 13.0 & +61 00 53.4 & 134.50 & 0.36 & 36.2 & --- & --- \\
W4-I11 & 02 25 15.2 & +61 19 14.4 & 133.94 & 0.47 & 54.0 & 39 & 3 \\
W4-I12 & 02 35 20.6 & +62 16 02.3 & 134.70 & 1.79 & 52.5 & 77 & 2 \\
W4-I13 & 02 28 25.1 & +60 56 20.2 & 134.44 & 0.25 & 43.5 & 16 & 6 \\
W4-I14 & 02 36 19.2 & +61 44 05.5 & 135.01 & 1.35 & 31.4 & 2 & 16 \\
W4-I15 & 02 33 36.1 & +60 37 40.4 & 135.14 & 0.20 & 49.8 & 27 & 4 \\
W4-I16 & 02 36 56.8 & +61 57 58.6 & 134.99 & 1.59 & 43.3 & 35 & 0 \\
W4-I17 & 02 34 08.8 & +61 40 35.5 & 134.80 & 1.19 & 17.1 & 9 & 16 \\
W4-I18 & 02 31 56.3 & +61 25 50.9 & 134.65 & 0.87 & 5.6 & 24 & 2 \\
W4-I19\tnote{c} & 02 40 31.7 & +61 13 45.9 & 135.68 & 1.09 & 57.8 & 11 & 3 \\
W4-I20 & 02 27 03.9 & +61 52 24.9 & 133.94 & 1.07 & 47.6& 457 & 0 \\
W4-I21 & 02 30 44.5 & +61 05 30.2 & 134.64 & 0.50 & 25.9 & 5 & 7 \\
W4-I22\tnote{b} & 02 26 07.8 & +61 56 43.7 & 133.82 & 1.09 & 55.4 & --- & --- \\
W4-I23 & 02 40 30.9 & +61 47 10.1 & 135.45 & 1.59 & 59.3 & 3 & 0 \\
W4-I24 & 02 37 45.1 & +60 37 31.4 & 135.61 & 0.40 & 61.5 & 20 & 10 \\
W4-O1 & 02 41 33.9 & +61 26 29.5 & 135.70 & 1.33 & 63.5 & 377 & 0 \\
W4-O2 & 02 35 37.8 & +59 56 29.5 & 135.64 & -0.33 & 93.0 & 107 & 0 \\
W4-O4 & 02 44 57.7 & +62 28 06.5 & 135.64 & 2.43 & 105.8 & 747 & 0 \\
W4-O5 & 02 21 52.6 & +60 10 03.2 & 133.96 & -0.75 & 110.3 & 94 & 0 \\
W4-O6 & 02 31 59.2 & +62 50 34.1 & 134.12 & 2.18 & 83.7 & 120 & 4 \\
W4-O7 & 02 43 35.6 & +61 55 54.6 & 135.72 & 1.88 & 82.7 & 52 & 0 \\
W4-O8 & 02 23 04.5 & +60 58 19.6 & 133.82 & 0.05 & 75.2 & 40 & 0 \\
W4-O10 & 02 20 26.2 & +61 34 46.2 & 133.31 & 0.51 & 88.0 & 64 & 3 \\
\hline
\end{tabular}
\label{tab:sources}
\begin{tablenotes}
\item[a] Angular distance between the line of sight and a line of sight through the center of the star cluster.
\item[b] NVSS position. No source detected in the Stokes I map in any frequency bin.
\item[c] High Mass X-Ray Binary LSI +61\ddeg303.
\end{tablenotes}
\end{threeparttable}
\end{table}
The sources are listed in Table \ref{tab:sources}, where the first column lists the source name in our nomenclature. The second and third columns list the right ascension ($\alpha$) and declination ($\delta$) of the observed sources. The positions are determined with the {\sc imfit} task in CASA, which fits a 2D Gaussian to the intensity distribution at 4.33 GHz. Columns four and five give the Galactic longitude ($\ell$), Galactic latitude (\textit{b}), which is converted from $\alpha$ and $\delta$ using the Python \textit{Astropy} package, and the angular separation from the center of the nebula ($\xi$) is given in column six. Column seven lists the flux density at 4.33 GHz calculated with {\sc imfit}, and column eight gives the mean percent linear polarization (m = \textit{P}/\textit{I}) as measured across the eight 128 MHz maps and assuming a \textit{Faraday simple}{} source. Figure \ref{fig:w4} is a radio continuum mosaic from the Canadian Galactic Plane Survey (CGPS) \citep{Taylor:2003,Landecker:2010} with the location of the sources, along with the names, indicated with filled circles.
\subsection{VLA Observations\label{sec:vlaobs}}
\begin{table}[!htb]
\centering
\begin{threeparttable}
\caption{Log of Observations \label{tab:logofobs}}
\begin{tabular}{p{0.54\linewidth}
p{0.3\linewidth}}
\hline
VLA Project Code & 13A-035 \\
Date of Observations & 2013 July 10, 13, 16, and 17\\
Number of Scheduling Blocks & 4 \\
Duration of Scheduling Blocks (h) & 4\\
Frequencies of Observation\tnote{a}~ (GHz) & 4.850; 7.250\\
Number of Frequency Channels per IF & 512\\
Channel Width (MHz) & 2 \\
VLA array & C \\
Restoring Beam (diameter) & 4\farcs81\\
Total Integration Time per Source & 18--25 minutes\tnote{b}\\
RMS Noise in Q and U Maps ($\mu$Jy/beam) & 39\tnote{c}\\
RMS Noise in RM Synthesis Maps ($\mu$Jy/beam) & 23\tnote{d} \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] The observations had 1.024 GHz wide intermediate frequency bands (IFs) centered on the frequencies listed, each composed of eight 128 MHz wide subbands.
\item [b] The ``O'' sources (see Table \ref{tab:sources}) averaged 18 minutes, and the ``I'' sources, being weaker, were between 22--25 minutes.
\item [c] This number represents the average rms noise level for all the Q and U maps.
\item[d] Polarized sensitivity of the combined RM Synthesis maps.
\end{tablenotes}
\end{threeparttable}
\end{table}
We observed 32 radio sources with the NSF's Karl G. Jansky Very Large Array (VLA)\footnote{\footnotesize{The Karl G. Jansky Very Large Array is an instrument of the National Radio Astronomy
Observatory (NRAO). The NRAO is a facility of the National Science Foundation,
operated under cooperative agreement with Associated Universities, Inc.}} whose lines of sight pass through or near to the shell of the IC 1805 stellar bubble. Table \ref{tab:logofobs} lists details of the observations. Traditionally, polarization observations require observing a polarization calibrator source frequently over the course of an observation to acquire at least 60\ddeg~of parallactic angle coverage. This is done to determine the instrumental polarization (D-factors, leakage solutions). Since the completion of the upgraded VLA, shorter scheduling blocks, typically less than 4 hours in duration, have become a common mode of observation. It is difficult, if not impossible, with very short scheduling blocks to acquire enough parallactic angle coverage to measure the instrumental calibration with a polarized source. Another method of determining the instrumental polarization is to observe a single scan of an unpolarized source. This technique can be used with shorter scheduling blocks.
In this project we calibrated the instrumental polarization using both techniques. We used the source J0228+6721, observed over a wide range of parallactic angle, and also made a single scan of the unpolarized source 3C84. Use of the CASA task \textsc{polcal}~on the J0228+6721 data solved for the instrumental polarization, determined by the antenna-specific D factors \citep{Bignell:1982}, which are complex, as well as the source polarization (\textit{Q} and \textit{U} fluxes). In the case of 3C84, \textsc{polcal}~solves only for the D factors.
We find no significant deviations between these two calibration methods, indicating accurate values for the instrumental polarization parameters.
3C138 and 3C48 functioned as both flux density and polarization position angle calibrators. J0228+6721 was used to determine the complex gain of the antennas as a function of time as well to as serve as a check, as described above, for the D-factors. We observed the program sources for 5 minute intervals and interleaved the observations of J0228+6721. There was one observation of 3C138, 3C48, and 3C84 each. For our final data products, we utilized 3C84 as the primary leakage calibrator and 3C138 as the flux density and polarization position angle calibrator.
\section{Data Reduction\label{sec:dataredux}}
The data were reduced and imaged using the NRAO Common Astronomy Software Applications (CASA)\footnote{\footnotesize{For further reference on data reduction, see the NRAO Jansky VLA tutorial ``EVLA Continuum Tutorial 3C391'' (http$://$casaguides.nrao.edu$/$index.php$?$title$=$EVLA$_{-}$Continuum$_{-}$Tutorial$_{-}$3C391)}} version 4.5. The procedure for the data reduction as described in Section 3 of \citet{Costa:2016} is identical to the procedure we employed in this study. The only difference for the current data set is that in the CASA task \textsc{clean}, we utilized \textit{Briggs} weighting with the ``robust'' parameter set to 0.5, which adjusts the weighting to be slightly more \textit{natural} than \textit{uniform}. \textit{Natural} weighting has the best signal/noise ratio at the expense of resolution, while \textit{uniform} is the opposite. \textit{Briggs} weighting allows for intermediate options. As in our previous work, we also implemented a cutoff in the (\textit{u}, \textit{v}) plane for distances $<$ 5000 wavelengths to remove foreground nebular emission.
Similar to \citet{Costa:2016}, we had two sets of data products after calibration and imaging. The first set of images consisted of radio maps (see Figures \ref{fig:I18} and \ref{fig:I24}) of each Stokes parameter, formed over a 128 MHz wide subband for each source. These images were inputs to the $\chi$($\lambda^{2}$)~analysis (Section \ref{sec:chilam}), and there were typically 14 individual maps for each source per Stokes parameter.
The second set of images consisted of maps of \textit{I}, \textit{Q}, and \textit{U} in 4 MHz wide steps across the entire bandwidth using the \textsc{clean}~mode ``channel'', which averages two adjacent 2 MHz channels.
Ideally, changes in \textit{Q}{} and \textit{U}{} should only be due to Faraday rotation; however, the spectral index can affect \textit{Q}{} and \textit{U}{} independently of the RM, which can be interpreted as depolarization. RM Synthesis does not, by default, account for the spectral index, so a correction must be applied prior to performing RM Synthesis (see Section 3 of \citealt{Brentjens:2005}). We first determine the spectral index, $\alpha$, of each source from a least-squares fit to the log of the flux density, $S_\nu$, and the log of the frequency, $\nu$. We adopt the convention that $S_\nu$ $\sim$~ $\nu^{-\alpha}$. We use the center frequency, $\nu_c$, of the band and the measured value of \textit{Q} and \textit{U} at each frequency, $\nu$, to find \textit{Q$_o$} and \textit{U$_o$} using the relationship \[Q = Q_o\left(\frac{\nu}{\nu_c}\right)^{-\alpha} \textrm{ and } \ U = U_o\left(\frac{\nu}{\nu_c}\right)^{-\alpha}. \] The final images consisted of approximately 336 maps per source, per Stokes parameter, as inputs for the RM Synthesis analysis (Section \ref{sec:rmsyn}).
\begin{figure}[!htb]
\centering
\subfloat[\label{fig:I18}]{
\includegraphics[width=0.4\textwidth]{f3a.pdf}}
\quad
\subfloat[\label{fig:I24}]{
\includegraphics[width=0.42\textwidth]{f3b.pdf}}
\caption[CASA Map]{Map of (a) W4-I118 and (b) W4-I24 at 4913 MHz. The circle in the lower left is the restoring beam. The gray scale is the linear polarized intensity, \textit{P}, the vectors show the polarization position angle, $\chi$, and the contours are the Stokes \textit{I} intensity with levels of -2, -1 , 2, 10, 20, 40, 60, and 80$\%$ of the peak intensity, 21.5 mJy beam$^{-1}$ and 11.7 mJy beam$^{-1}$ for W4-I18 and W4-I24a, respectively. }
\label{fig:MAP}
\end{figure}
\subsection{Rotation Measure Analysis via a Least-Squares Fit to $\chi$ vs $\lambda^2$ \label{sec:chilam}}
The output of the CASA task \textsc{clean}~produces images in Stokes \textit{I, Q, U,} and \textit{V}. From these images, we generated maps with the task \textsc{immath}~of the linear polarized intensity \textit{P}, \[P = \sqrt{Q^2+U^2} \] and the polarization position angle $\chi$,
\begin{equation*}
\chi=\frac{1}{2}\tan^{-1}{\left(\frac{U}{Q}\right)}
\end{equation*}
for each source over a 128 MHz subband. Data that are below the threshold of 5$\sigma_{\textrm{Q}}$ are masked in the \textit{P} and $\chi$ maps, where $\sigma_{\textrm{Q}}$ = $\sigma_{\textrm{U}}$ is the rms noise in the \textit{Q} data. This threshold prevents noise in the \textit{Q} and \textit{U} data from generating false structure in the \textit{P} and $\chi$ maps. Examples of images are shown in Figure \ref{fig:MAP}, which displays the total intensity, polarized intensity, and polarization position angle for sources W4-I18 and W4-I24. W4-I18 is an example of a point source, or slightly resolved source. Twelve of the sources in Table \ref{tab:sources} were of this type and unresolved to the VLA in C array. Eight sources were like W4-I24, showing extended structure in the observations and potentially yielding RM values on more than one line of sight.
\begin{figure}[hbt!]
\centering
\subfloat[][\label{fig:I1chi}]{
\includegraphics[width=0.48\textwidth]{f4a.pdf}} \quad
\subfloat[][\label{fig:I18chi}]{
\includegraphics[width=0.48\textwidth]{f4b.pdf}}\\
\caption[Plot of $\chi(\lambda^{2})$]{Plot of the polarization position angle as a function of the square of the wavelength, $\chi(\lambda^{2})$, for the source (a) W4-I18, RM= +514 $\pm$ 12 rad m$^{-2}$, and (b) W4-I24a, RM = --658 $\pm$ 5 rad m$^{-2}$. Each plotted point results from a measurement in a single 128 MHz-wide subband. The gap in $\lambda^2${} coverage is due to the observation configuration that consisted of two 1.024 GHz wide IFs separated by $\sim${} 1.4 GHz.}
\label{fig:newpol}
\end{figure}
In the case of a single foreground magnetic-ionic medium responsible for the rotation of an incoming radio wave, the relation between $\chi$ and $\lambda^2$ is linear, and we calculate the RM through a least-squares fit of $\chi$($\lambda^{2}$). To measure $\chi$, we select the pixel that corresponds to the highest value of \textit{P} on the source in the 4338 MHz map, and we then measure $\chi$ at that location in each subsequent 128 MHz wide subband. Figure \ref{fig:newpol} shows two examples of the least-squares fit to $\chi$($\lambda^{2}$). The $\chi$ errors are \(\sigma_{\chi} = \frac{\sigma_{Q}}{2P},\) (\citealt{Everett:2001}, Equation 12).
\subsection{Rotation Measure Synthesis \label{sec:rmsyn}}
In additional to the least-squares fit to $\chi$($\lambda^{2}$), we performed Rotation Measure Synthesis \citep{Brentjens:2005}. The inputs to RM Synthesis are images in Stokes \textit{I}, \textit{Q}, and \textit{U} across the entire observed spectrum in 4 MHz spectral intervals. We refer the reader to Section 3.1.2 of \citet{Costa:2016} for a detailed account of our procedure, which follows the implementation of RM Synthesis as developed by \citet{Brentjens:2005}. The goal of RM synthesis is to recover the Faraday dispersion function $F(\phi)$. Here $\phi$, the Faraday depth, is a variable which is Fourier-conjugate to $\lambda^2$ (see \citealt{Costa:2016}, Equations 3 and 4), and has units of rad m$^{-2}$. We also refer to $F(\phi)$ as the ``Faraday spectrum''.
\begin{table}[htb!]
\centering
\caption{Rotation Measure Synthesis Parameters\label{tab:rmpar}}
\begin{threeparttable}
\centering
\begin{tabular}{p{2cm} p{4cm} p{9.5cm}}
\hline
$\Delta \lambda^2$ & 3.2 $\times$ 10$^{-3}$ (m$^2$) & Total bandwidth\tnote{a}. \\
$\lambda^2_{min}$ & 1.5 $\times$ 10$^{-3}$ (m$^2$) & Shortest observed wavelength squared. \\
$\delta \lambda^2$ & 4.8 $\times$ 10$^{-6}$ (m$^2$) & Width of a channel; Eq (35) \citet{Brentjens:2005}. \\
$\delta \phi$ & 1072\tnote{b}~ (rad m$^{-2}$) & FWHM of RMSF; Eq (61) \citet{Brentjens:2005}. \\
\\
\multirow{2}{*}{max-scale} & \multirow{2}{*}{2098 (rad m$^{-2}$)} & Sensitivity to extended Faraday structures; Eq (62) \citet{Brentjens:2005}. \\
\\
\multirow{3}{*}{$|\phi_{max}|$} & \multirow{3}{*}{3.6 $\times$ 10$^{5}$ (rad m$^{-2}$)} & Maximum detectable Faraday depth before bandwidth depolarization; Eq (63) \citet{Brentjens:2005}. \\ \\ \hline
\end{tabular}
\begin{tablenotes}
\item[a] This bandwidth includes the frequencies not observed that lie between our two IFs. They are set to 0 via the weighting function, W($\lambda^2$).
\item[b] Since flagging for RFI and bad antennas were done individually for each scheduling block, the FWHM of the RMSF can vary slightly from source to source. However, these slight variations are not significant in our interpretation of the RM values report in this paper.
\end{tablenotes}
\end{threeparttable}
\end{table}
\textit{F($\phi$)}~is recovered via an \textsc{rmclean}~algothrim \citep{Heald:2009,Bell:2012}, and we applied a 7$\sigma$ cutoff, which is above the amplitude at which peaks due to noise are likely to arise \citep{Brentjens:2005,Macquart:2012,Anderson:2015}. The \textsc{rmsynthesis} algorithm initially searched for peaks in the Faraday spectrum using a range of $\phi$ $\pm$ 10,000 rad m$^{-2}$~at a resolution of 40 rad m$^{-2}$~to determine if there were significant peaks at large values of $|\phi|$. Then, we performed a finer search at $\phi$ = $\pm$ 3000 rad m$^{-2}$~at a resolution of 10 rad m$^{-2}$. The RM Synthesis parameters, such as the full-width-at-half-maximum (FWHM) of the rotation measure spread function (RMSF) and the maximum detectable Faraday depth, are given in Table \ref{tab:rmpar}.
\begin{figure}[hbt!]
\centering
\subfloat[\label{fig:psynmap}]{
\includegraphics[width=0.48\textwidth]{f5a.pdf}}
\quad
\subfloat[\label{fig:rmsynmap}]{
\includegraphics[width=0.48\textwidth]{f5b.pdf}}
\caption[Plot of RM Synthesis Map]{(a) Linear polarization map and (b) RM map of W4-I24 from the RM Synthesis analysis. In both images, the data cube was flattened over the Faraday depth axes, and a threshold of 7$\sigma$ was applied. W4-I24 has extended structure, so there are two peaks, which are also present in the CASA maps (Figure \ref{fig:I24}).}
\end{figure}
As in \citet{Costa:2016}, we utilized an IDL code for the \textsc{rmsynthesis} and \textsc{rmclean}~algorithms. The output of the IDL code is a data cube in Faraday depth space that is equal in range to the range of $\phi$ that was searched over in the \textsc{rmsynthesis} algorithm. The data cube contains, for example, 500 maps of the polarized intensity as a function of spatial coordinates and $\phi$, which ranges between $\pm$ 10,000 rad m$^{-2}$~at intervals of 40 rad m$^{-2}$. Initially, we generated these maps for a 1024 $\times$ 1024 pixel image. We then used the Karma package \citep{Gooch:1995} tool \textsc{kvis} to review the maps to search for sources or source components away from the phase center that, while being too weak to detect in the 128 MHz maps, may be detectable in the RM Synthesis technique since it uses the entire bandwidth to determine the Faraday spectrum\footnote{private communication, L. Rudnick}. However, no such sources were identified above the cutoff. From the 1024 $\times$ 1024 maps of the Faraday spectrum, we identified the \textit{P$_{\textrm{max}}$} for the observed sources and extracted the Faraday spectrum at that location. Figure \ref{fig:psynmap} shows an example of a \textit{P$_{\textrm{max}}$} map that has been flattened along the $\phi$ axis, i.e., the gray scale in the image represents the full range of $\phi$. From this map, it is easy to identify the spatial location of \textit{P$_{\textrm{max}}$} for the source, which agrees with the location of the peak linear polarized intensity in the $\chi$($\lambda^{2}$)~analysis. We obtained this same result in \citet{Costa:2016} for the Rosette Nebula.
To determine the RM, we fit a 2 degree polynomial to the Faraday spectrum at each pixel in the 1024 x 1024 image above the 7$\sigma$ cutoff. The gray scale in Figure \ref{fig:rmsynmap} shows the RM value from the fit to each pixel. The image is zoomed and centered on the source. While we can mathematically determine the RM at each pixel, the sources are not resolved, so we only select the RM at the spatial location of \textit{P$_{\textrm{max}}$}. Figure \ref{fig:RMSYN} plots the Faraday spectrum and \textsc{rmclean}~components for W4-I18, and Figure \ref{fig:RMSF} shows the RMSF.
\citet{Anderson:2015} describe two cases for the behavior of the Faraday spectrum. A source is considered \textit{Faraday simple}~when \textit{F($\phi$)}~is non-zero at only one value of $\phi$, \textit{Q} and \textit{U} as a function of $\lambda^2$~vary sinusoidally with equal amplitude, and \textit{P($\lambda^2$)}~is constant. The \textit{Faraday simple}~case has the physical meaning of a uniform Faraday screen in the foreground that is responsible for the Faraday rotation, and $\chi$ is linearly dependent on $\lambda^2$. If a source is \textit{Faraday simple}, then \textit{F($\phi$)}~is a delta function at a Faraday depth equal to the RM. The second behavior \citet{Anderson:2015} describe for the Faraday spectrum is a \textit{Faraday complex}~source, which is any spectrum that deviates from the criteria set for the \textit{Faraday simple}~case. A \textit{Faraday complex}~spectrum can be the result of depolarization in form of beam depolarization, internal Faraday dispersion, multiple interfering Faraday rotating components, etc. \citep{Sokoloff:1998}.
\begin{figure}[htb!]
\centering
\subfloat[\label{fig:RMSYN}]{
\includegraphics[width=0.45\textwidth]{f6a.pdf}}
\quad
\subfloat[\label{fig:RMSF}]{
\includegraphics[width=0.45\textwidth]{f6b.pdf}}
\caption[]{Plot of (a) cleaned Faraday dispersion function, F($\phi$), for W4-I18, where $\phi_{peak}$ = 501 $\pm$ 33 rad m$^{-2}$~and (b) the RMSF (R($\phi$)). The 7$\sigma$ cutoff is shown in the red, dashed, horizontal line. The vertical red lines are the clean components, and the black curves represent the \textit{P} (solid), \textit{Q} (dot dashed), and U (dashed) components of the spectrum. }
\label{fig:rmsynrmsf}
\end{figure}
\section{Observational Results\label{sec:obsres}}
\subsection{Measurements of Radio Sources Viewed Through the W4 Complex\label{sec:obs2}}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.6\textwidth]{f7.pdf}
\caption[]{ Plot of RM values derived from the $\chi$($\lambda^{2}$)~analysis vs the RM Synthesis analysis. The blue markers are the RM from primary component and the red are the secondary component. The straight line represents perfect agreement between the two sets of measurements.}
\label{fig:comprm}
\end{figure}
We measured 27 RM values for 20 lines of sight, including secondary components, through or near to IC 1805. In Table \ref{tab:results}, the first column lists the source name using our naming scheme, and column two gives the component, if the source had multiple resolved components for which we could determine a RM value.
Columns three and four list the RM value from the least-squares method and the reduced chi-squared value, respectively. Column five lists the RM value determined from the RM Synthesis technique and the associated error (Equation 7 of \citealt{Mao:2010}).
Figure \ref{fig:comprm} shows the agreement between the two techniques for determining the RM. As in \citet{Costa:2016}, we find good agreement between the two techniques, and the good agreement between the results using the two techniques gives us confidence in our RM measurements.
\begin{table}[!hbtp]
\centering
\begin{threeparttable}
\caption{Faraday Rotation Measurement Values through the W4 Complex \label{tab:results}}
\begin{tabular}{ccccccc}
\hline
\multirow{2}{*}{Source} & \multirow{2}{*}{Component} & RM\tnote{a} & Reduced & RM\tnote{c} & $\xi$\tnote{d} & $\xi$\tnote{e}\\
& & (rad m$^{-2}$) & $\chi^{2}$\tnote{b} & (rad m$^{-2}$) & (pc) & (pc)\\
\hline
W4-I1 & a & -277 $\pm$ 1 & 29 & -258 $\pm$ 3 & 29 & 53\\ \hline
\multirow{2}{*}{W4-I2} & a & -1042 $\pm$ 7 & 1.5 & -930 $\pm$ 30 & \multirow{2}{*}{30}& \multirow{2}{*}{26}\\
& b & -935 $\pm$ 6 & 1.5 & -954 $\pm$ 11 & & \\ \hline
W4-I3 & a & -876 $\pm$ 2 & 1.9 & -878 $\pm$ 8 & 27 & 16 \\ \hline
\multirow{2}{*}{W4-I4} & a & -139 $\pm$ 3 & 1.9 & -153 $\pm$ 12 & \multirow{2}{*}{38} & \multirow{2}{*}{59}\\
& b & -91 $\pm$ 6 & 2 & -68 $\pm$ 15 & & \\ \hline
W4-I6 & a & -990 $\pm$ 8 & 1.3 & -961 $\pm$ 23 & 29 & 25\\ \hline
W4-I8 & a & -276 $\pm$ 2 & 4.4 & -337 $\pm$ 8 & 37& 54\\ \hline
W4-I11 & a & -377 $\pm$ 8 & 12 & -141 $\pm$ 18 & 35 & 41\\ \hline
W4-I12 & a & -315 $\pm$ 4 & 2.8 & -306 $\pm$ 10 &34 & 54\\ \hline
\multirow{2}{*}{W4-I13} & a & -777 $\pm$ 8 & 1.2 & -801 $\pm$ 24 & \multirow{2}{*}{28} & \multirow{2}{*}{36}\\
& b & -701 $\pm$ 28 & 0.5 & -772 $\pm$ 66 & & \\ \hline
W4-I14 & a & -678 $\pm$ 27 & 0.6 & -666 $\pm$ 66 & 20 & 36\\ \hline
W4-I15 & a & -157 $\pm$ 9 & 0.8 & -124 $\pm$ 14 & 32 & 10\\ \hline
\multirow{2}{*}{W4-I17}& a & -492 $\pm$ 8 & 1.6 & -440 $\pm$ 40 & \multirow{2}{*}{11} & \multirow{2}{*}{31}\\
& b & -509 $\pm$ 15 & 0.6 & -464 $\pm$ 40 & & \\ \hline
W4-I18 & a & +514 $\pm$ 12 & 1.1 & +501 $\pm$ 33 & 4 & 22\\ \hline
W4-I19 & a & -407 $\pm$ 14 & 0.3 & -431 $\pm$ 36 & 37 & 34\\ \hline
\multirow{3}{*}{W4-I21} & a & -53 $\pm$ 26 & 1.4 & -167 $\pm$ 67 & \multirow{3}{*}{17} & \multirow{3}{*}{15}\\
& b & -98 $\pm$ 28 & 0.5 & -79 $\pm$ 62& & \\
& c & -173 $\pm$ 34 & 0.5 & -232 $\pm$ 70 & & \\ \hline
\multirow{2}{*}{W4-I24}& a & -658 $\pm$ 5 & 1.4 & -678 $\pm$ 14 & \multirow{2}{*}{39} & \multirow{2}{*}{23}\\
& b & -675 $\pm$ 12 & 0.4 & -716 $\pm$ 30 & & \\ \hline
W4-O4 & a & -31 $\pm$ 12 & 24 & -178 $\pm$ 18 &68 &81 \\ \hline
W4-O6 & a & -95 $\pm$ 1 & 3.6 & -96 $\pm$ 4 & 54 & 76\\ \hline
W4-O7 & a & -175 $\pm$ 24 & 0.8 & -256 $\pm$ 56 & 53 & 62\\ \hline
W4-O10 & a & -379 $\pm$ 5 & 1.8 & -343 $\pm$ 16 & 56 & 66\\ \hline
\end{tabular}
\begin{tablenotes}
\item[a] RM value obtained from a least-squares linear fit to $\chi(\lambda^2)$. The errors are 1$\sigma$.
\item[b] Reduced $\chi^{2}$ for the $\chi$($\lambda^{2}$)~fit.
\item[c] Effective RM derived from RM Synthesis.
\item[d] Distance from center of OCl 352.
\item[e] Distance from \citet{Terebey:2003} center.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Report on Faraday Complexity and Unpolarized Lines of Sight\label{sec:depol}}
In the last paragraph of Section \ref{sec:rmsyn}, we discuss Faraday complexity. If a source is \textit{Faraday simple}, then the RM is equal to a delta function in \textit{F($\phi$)}~at the Faraday depth. If a source is \textit{Faraday complex}, then the interpretation of the RM is not as straightforward. There is extensive literature (e.g. \citealt{Farnsworth:2011}, \citealt{OSullivan:2012}, \citealt{Anderson:2015}, \citealt{Sun:2015}, \citealt{Purcell:2015}) to understand Faraday complexity
One indicator of a \textit{Faraday complex}~source is a decreasing fractional polarization, \textit{p} = \textit{P}/\textit{I}, as a function of $\lambda^2$. The ways in which this can arise are discussed at the end of Section \ref{sec:rmsyn}. Although depolarization does not necessarily lead to a net rotation of the source $\chi$, its presence indicates the potential for a $\chi$ rotation independent of the plasma medium through which the radio waves subsequently propagate. This could result in an error in our deduced RMs. Nine of the sources, W4-I1, -I3, -I11, -I15, I21b, -O4, -O6, -O7, and -O10 show a decreasing \textit{p} with increasing $\lambda^2$.
A rough estimate of the potential position angle rotation associated with depolarization may be obtained using the analysis in \citet{Cioffi:1980}. These calculations assume that depolarization arises from Faraday rotation within the synchrotron radiation source, and we can estimate the effect of internal depolarization from the changes in fraction polarization. Given the fractional polarization at the shortest and longest wavelength, we obtain the corresponding polarization angle change from Figure 1 of \citet{Cioffi:1980} and then calculate a RM due to internal depolarization. If the calculated RM due to depolarization (RM$_{depol}$) is larger than the observed RM, then the RM is potentially affected by depolarization.
W4-I1, -O6, -O7, and -O10 show RM$_{depol}$ $\sim$~ RM$_{obs}$, which indicates that internal depolarization could affect the observed RM. The observed RM of W4-I3 is $\sim$~3 times larger than RM$_{depol}$, so it is not affected by internal depolarization.
Depolarization due to internal Faraday rotation makes predictions for the form of $\chi (\lambda^2)$ which would not have $\chi \propto \lambda^2$ \citep{Cioffi:1980}. For all of the sources mentioned above, we compared the observed behavior of $\chi$($\lambda^{2}$)~to the predicted behavior (Equation 4b of \citealt{Cioffi:1980}). Within the errors, only W4-O7 is consistent with the non-linear behavior of a RM affected by internal depolarization. We interpret this result as meaning that our deduced RM values for most of the sources are not significantly in error due to internal depolarization, and we consider the measurement of depolarization as providing a cautionary flag.
We also considered whether our measurements could have been affected by bandwidth depolarization or beam depolarization. Bandwidth depolarization occurs when the polarization angle varies over frequency averaged bins, $\Delta\nu$. For example in this study, we use values of $\chi$ in 128 MHz wide bins (Section \ref{sec:chilam}) and 4 MHz (Section \ref{sec:rmsyn}). For a center frequency of the lowest frequency bin, we use $\nu_c$ = 4466 MHz, and the relationship between the change in polarization position angle, $\Delta\chi$, is
\begin{equation}
|\Delta\chi| = 2|\textrm{RM}| \; c^2 \; \frac{\Delta\nu}{\nu_c^3},
\label{eq:bandwidth}
\end{equation}
where c is the speed of light. This formula shows that even for $|$RM$|$ = 10$^4$ rad m$^{-2}$, which is far larger than any RMs we measure, the Faraday rotation across the band is 0.41 radians. This is insufficient to cause substantial bandwidth depolarization. Beam depolarization occurs when there are small scale variations of the electron density or the magnetic field within a beam. It is unlikely that the RMs are affected by beam depolarization as the beam at 6 cm for the VLA in C array is $\sim$~5 arcseconds.
We interpret these RMs as a characteristic value due to the plasma medium (primarily the Galactic ISM) between the source and the observer. In the analysis that follows, we choose the RM values from the RM Synthesis method.
When the data were mapped and inspected, we found that a few sources that had passed our criteria for flux density and compactness to the VLA D array at L band (1.42 GHz) were completely unpolarized. W4-I16 and W4-O8 are not polarized at any frequency, and the RM Synthesis technique does not show significant ($>$ 7$\sigma$) peaks at any $\phi$. Three of the lines of sight, W4-I5, W4-I10, and W4-I22, have no source in the field. Despite appearing to be point sources in the NVSS postage stamps (see Section \ref{sec:obs}), we do not observe a source at these locations, and they may have been clumpy foreground nebular emission that was filtered out during the imaging process.
Subsequent investigations determined that some of the selected sources were previously cataloged ultra compact HII regions associated with the W3 star formation region. These sources are W4-I7 (W3(OH)-C), W4-I9 (AFGL 333), and W4-I20 (W3(OH)-A) \citep{Feigelson:2008,Navarete:2011,Roman:2015}. The W4-I7 field has no source at the observed $\alpha$ and $\delta$, despite it being identified as W3(OH)-C. We do not observe a source at this location in any frequency bin. W3(OH)-A, however, is observed and is a point source in our maps at all frequencies. Similarly, W4-I9 is detected in each frequency bin and is an extended source. These sources are unpolarized and do not feature in our subsequent analysis.
\subsection{A Unique Line of Sight Through the W4 Region: LSI +61\ddeg 303\label{sec:HMXB}}
W4-I19 has a spectrum which is inconsistent with an optically-thin extragalactic radio source. It is linearly polarized, and we measure RM = --431 $\pm$ 36 rad m$^{-2}$. Investigation of this source during the data analysis phase revealed that it is not an extragalactic source, although it passed our selection criteria for flux and compactness. W4-I19 is the high mass X-ray binary (HMXB) LSI +61\ddeg303 \citep{Gregory:1979,Bignami:1981}, which is notable for being one of five known gamma ray binary systems \citep{Frail:1991}. This system has been extensively studied, and as a result, much is known about the nature of the compact object \citep{Massi:2004,Massi:2004b,Dubus:2006,Paredes:2007,Massi:2017}, the stellar companion \citep{Casares:2005,Dubus:2006,Paredes:2007}, orbital period \citep{Gregory:2002}, radio structure \citep{Albert:2008}, and radial velocity \citep{Gregory:1979,Lestrade:1999}.
The spatial location of LSI +61\ddeg303 is important for understanding the RM we determined for this source. \citet{Frail:1991} argue that since signatures of the Perseus arm shock are present in the absorption spectrum to LSI +61\ddeg303 but not the post-shock gas from the Perseus arm, LSI +61\ddeg303 must lie between the two features at a distance of 2.0 $\pm$ 0.2 kpc. They also report that they do not see absorption features due to the IC 1805 ionization front and shock front. The estimated distance to LSI +61\ddeg303{} is consistent with distance estimates to OCl 352. The position relative to the nebula has consequences for the interpretation of the RM that we measure. The possibilities are:
\begin{enumerate}
\item LSI +61\ddeg303 is in front of the stellar bubble and HII region, so it is exterior to a region modified by OCl 352. The RM is then an estimate of the foreground ISM between us and the nebula.
\item If LSI +61\ddeg303 is at the same distance as IC 1805 or slightly behind (greater distance), then the RM is unique among our sources in that it is not affected by Faraday rotation from material in the outer Galaxy. The RM is then probing at least a part of the Faraday rotating material due to the nebula.
\end{enumerate}
To further determine the position of LSI +61\ddeg303{} with respect to IC 1805, we review the current state of knowledge on the subject from the literature. \citet{Dhawan:2006} observed LSI +61\ddeg303{} with the Very Long Baseline Array (VLBA) and report a proper motion of ($\mu_{\alpha}$, $\mu_{\delta}$) = (-0.30 $\pm$ 0.07, -0.26 $\pm$ 0.05) mas yr$^{-1}$. \citet{Aragona:2009} report a radial velocity for LSI +61\ddeg303~of $V_{rad}$ = --41.4 $\pm$ 0.6 km s$^{-1}$, which agrees with previous estimates by \citet{Casares:2005}. For OCl 352, \citet{Dambis:2001} estimate the radial velocity to be --41 $\pm$ 3 km s$^{-1}$, and more recent estimates by \citet{Kharchenko:2005} ($V_{rad}$ = --47 $\pm$ 18 km s$^{-1}$) agree within the errors. Both LSI +61\ddeg303~and OCl 352 have similar radial velocities, and the proper motion estimates by \citet{Dhawan:2006} indicate that LSI +61\ddeg303~is moving similarly on the plane of the sky to OCl 352, which has a proper motion of ($\mu_{\alpha}$, $\mu_{\delta}$) = (--1.0 $\pm$ 0.4, --0.9 $\pm$ 0.4) mas yr$^{-1}$ \citep{Dambis:2001}.
From proper motion and radial velocity estimates, LSI +61\ddeg303~appears to be moving in relatively the same direction and speed as OCl 352. Using a distance of 2 kpc to LSI +61\ddeg303~and 2.2 kpc to OCl 352, the transverse velocities are $\sim$~3~km s$^{-1}${} and $\sim$~14 km s$^{-1}$, respectively. If LSI +61\ddeg303~originally belonged to OCl 352, then it is unlikely that it is in front of IC 1805, given that both are moving at the same radial velocity. While LSI +61\ddeg303~appears to be outside the obvious shell structure of IC 1805, it is more likely that it is probing material modified by OCl 352. We discuss this possibility further in Section \ref{sec:pdr}.
If LSI +61\ddeg303~did not originate in OCl 352, then it is possible to still be in front of the nebula, despite the similar velocities. In such a case, the RM we obtained for this line of sight is due to the ISM between us and IC 1805.
The RM value we find for LSI +61\ddeg303~is nearly 3 times larger than the background RM, which we discuss in Section \ref{sec:bkgrm}. This would require a magneto-ionic medium between the observer and the nebula capable of producing $\sim$~400 rad m$^{-2}$~along this line of sight. As may be seen from Table \ref{tab:results} and Figure \ref{fig:w4}, other lines of sight near IC 1805, but exterior to the shell, do not have as large of RM values (e.g. W4-O26, -O19, -O7, -I11). It therefore seems most probable that the RM for W-I19 (LSI +61\ddeg303) is dominated by plasma in W4
In summary, there is evidence in the literature that suggests LSI +61\ddeg303~may lie within a region modified by OCl 352, particularly if LSI +61\ddeg303~did indeed once belong to OCl 352. If this is the case, then the RM we find is unaffected by the ISM in the outer galaxy and is due to the material near IC 1805.
\section{Results on Faraday Rotation Through the W4 Complex\label{sec:fr}}
\subsection{The Rotation Measure Sky in the Direction of W4}
\begin{figure}[!htb]
\centering
\subfloat[][\label{fig:northb}]{
\includegraphics[width=0.48\textwidth]{f8a.pdf}}
\quad
\subfloat[][\label{fig:southb}]{
\includegraphics[width=0.48\textwidth]{f8b.pdf}}
\caption[]{Plot of RM vs distance from the center of OCl 352 for (a) lines of sight that pass through the W4 Superbubble and (b) lines of sight near and close to the southern loop. The solid line represents the estimate of the background RM using sources in this study and in the literature, and the dashed line is the predicted background RM from the \citet{vanEck:2011} model of the Galactic magnetic field.}
\label{fig:rmvsxi}
\end{figure}
\citet{Whiting:2009}, \citet{Savage:2013}, and \citet{Costa:2016} compared observations to a model of the ionized shell in which the RM depended only on $\xi$, the impact parameter, or closest approach of a line of sight to the center of the shell. In anticipation of a similar analysis in this study, we show Figures \ref{fig:northb} and \ref{fig:southb}, which plot the RM versus distance from the center of star cluster for the lines of sight through the W4 Superbubble (W4-I1, -I4, -I8, -I12, -I14, -I17, -O4, -O6, and -O7) and the ones through or close to the southern loop (W4-I2, -I3, -I6, -I11, -I13, -I15, -I18, -I19, -I21, -I24, -O10).
In Section \ref{sec:structure}, we discussed the morphology of the region around IC 1805 and made the distinction between the southern latitudes and the northern latitudes, so in the following sections, we address each region near IC 1805 separately.
\subsection{The Galactic Background RM in the Direction of W4\label{sec:bkgrm}}
In \citet{Savage:2013}, we determined the background RM in the vicinity of the Rosette Nebula ($\ell$ $\sim$ 206\ddeg) by finding the median value of the RM for sources outside the obvious shell structure of the Rosette. Determining the background RM near IC 1805 is difficult, however, due to proximity of W3, the W3 molecular cloud, and the W4 Superbubble. Given the morphological difference between the northern and southern parts of IC 1805, we assume that sources south of OCl 352 (\textit{b} $<$ 0.9\ddeg) should be modeled independently of the northern sources, since the W4 Superbubble extends up to \textit{b} $\sim$ 7\ddeg~\citep{West:2007}. The lines of sight north of the star cluster are intersecting the W4 Superbubble and are not probing the RM due to the general ISM independent of IC 1805. Therefore, the only lines of sight that are potentially probing the RM in the vicinity of IC 1805 are those exterior to the shell structure of the southern loop.
\begin{table}[htb!]
\centering
\caption{List of Sources with RM values from Catalogs\label{tab:taylor}}
\begin{threeparttable}
\centering
\begin{tabular}{cccc}
\hline
Source & $\alpha$(J2000) & $\delta$(J2000) & RM\tnote{a} \\
Name & h m s & $^o$ $'$ $''$ & (rad m$^2$) \\
\hline
W4-O3 & 02 35 43.0 & +63 22 33.0 & --138\tnote{b} $\pm$ 18 \\
W4-O19 & 02 46 23.9 & +61 33 19.9 & --157\tnote{c} $\pm$ 15 \\
W4-O26 & 02 42 32.3 & +60 02 31.0 & +61 $\pm$ 41 \\
W4-O27 & 02 25 48.7 & +59 53 52.0 & --145 $\pm$ 22 \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] RM values from \citet{Brown:2003} unless otherwise noted.
\item[b] \citet{Taylor:2009} give --75 $\pm$ 9 rad m$^{-2}$~for this line of sight.
\item[c] RM value from \citet{Taylor:2009}.
\end{tablenotes}
\end{threeparttable}
\end{table}
If we apply the thick shell model from \citet{Terebey:2003} (see Section \ref{sec:structure} for details), then the lines of sight with RM values exterior to the shell are W4-I2, -I11, and -O10. For the thin shell case, W4-I6, -I13 and -I24 are also exterior sources. The mean RM value for the background using these sources is --554 rad m$^{-2}$~and --670 rad m$^{-2}$~for the thick or thin shell, respectively. In Table \ref{tab:taylor}, we list RM values from the literature for lines of sight near IC 1805 that we include in our estimate of the background RM. The mean RM value for these sources (excluding W4-O3 for being in the superbubble) is --80 rad m$^{-2}$. The sources W4-I2, -I6, -I13, and -I24 are seemingly outside the obvious ionized shell structure; however, they are also the lines of sight for which we measure some of the highest RM values. This is a surprising result, and one we did not observe in the case of the Rosette Nebula. It strongly suggests that the lines of sight to W4-I2, -I6, -I13, and -I24 have RMs that are dominated by the W4 complex, despite the fact that they are outside the obvious ionized shell of IC 1805. We discuss this further in the next section.
For the present discussion, we exclude these sources from the estimate of the background. Using W4-I11, -O10, -O19, -O26, and -O27, we find a mean value for the background RM due to the ISM of --145 rad m$^{-2}$. While this value is similar in magnitude to the value of the background RM we found in our studies on the Rosette Nebula, we have significantly fewer lines of sight, and only two of the lines of sight were observed in this study.
Due to a low number of lines of sight exterior to IC 1805, we utilize the model of a Galactic magnetic field by \citet{vanEck:2011} to estimate the background RM due to the ISM. From their Figure 6, they find the Galactic magnetic field is best modeled by an almost purely azimuthal, clockwise field. \citet{vanEck:2011} use their model to predict the RM values in the Galaxy, and in the vicinity of IC 1805, their model predicts RMs of order --100 rad m$^{-2}$. Using this as an estimate of the background RM, we find an excess RM due to IC 1805 of +600 to --860 rad m$^{-2}$.
\subsection{High Faraday Rotation Through Photodissociation Regions\label{sec:pdr}}
The lines of sight with the highest RM values, W4-I2, -I6, and -I24, appear to be outside the obvious shell of the southern loop. These sources are very near to the bright ionized shell. \citet{Terebey:2003} and \citet{Gray:1999} discuss a halo of ionized gas that surrounds IC 1805, which may be causing the high RM values. \citet{Gray:1999} speculate that the diffuse extended structure is an extended HII envelope as suggested by \citet{Anantharamaiah:1985}. Another possibility is that these high RMs arise in the PDR surrounding the IC 1805 HII region.
PDRs are the regions between ionized gas, which is fully ionized by photons with \textit{h$\nu$} $>$ 13.6 eV, and neutral or molecular material. PDRs can be partially ionized and heated by far-ultraviolet photons (6 eV $<$ \textit{h$\nu$} $<$ 13.6 eV) \citep{Tielens:1985,Hollenbach:1999}. Typically, the PDR consists of neutral hydrogen, ionized carbon, and neutral oxygen nearest to the ionization front, and with increasing distance, molecular species (e.g., CO, H$_2$, and O$_2$) dominate the chemical composition of a PDR \citep{Hollenbach:1999}. One tracer of PDRs is polycyclic aromatic hydrocarbon (PAH) emission at infrared (IR) wavelengths. \citet{Churchwell:2006} identify more than 300 bubbles at IR wavelengths in the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE), and 25$\%$ of these bubbles coincide with known HII regions. \citet{Watson:2008} examine three bubbles from the \citet{Churchwell:2006} catalog with the \textit{Spitzer} Infrared Array Camera (IRAC) bands 4.5, 5.8, and 8.0 \micron~and the 24 $\mu$m band from the \textit{Spitzer} Multiband Imaging Photometer (MIPS) to determine the extent of the PDR around three young HII regions. One of their main results is that the 8 \micron~emission, which is due to PAHs, encloses the 24 \micron~emission, which traces hot dust. \citet{Kerton:2013} discuss similar observations near the W 39 HII region. \citet{Watson:2008} use ratios between the 4.5, 5.8, and 8.0 \micron~bands to determine the extent of the PDRs, as the 4.5 \micron~emission does not include PAHs but the 5.8 and 8.0 \micron~bands do (see their Section 1 for details).
To determine the presence and extend of a potential PDR around IC 1805, we analyze Wide-field Infrared Survey Explorer (WISE) data from the IPAC All-Sky Data Release\footnote{http://wise2.ipac.caltech.edu/docs/release/allsky/} at 3.6, 4.6, 12, and 22 $\mu$m. The 4.6 $\mu$m~WISE bands is similar in bandwidth and center frequency to the IRAC 4.5 \micron~band, and the WISE 22 \micron~band is also similar to the MIPS 24 \micron~band \citep{Anderson:2014}. The 12 \micron~WISE band does not overlap with the 8.0 \micron~band of IRAC, but the WISE band traces PAH emission at 11.2 and 12.7 \micron. \citet{Anderson:2012} note, however, that the 12 \micron~flux is on average lower than the 8.0 \micron~IRAC band, which is most likely due to the WISE band sampling different wavelengths of PAH emission instead of the 7.7 and 8.6 \micron~PAH emission in the IRAC band.
Figure \ref{fig:wise} is a RGB image of the southern loop of IC 1805 at 4.6 \micron~(blue), 12 \micron~(green), and 22 $\mu$m (red). The 1.42 GHz radio continuum emission is shown in the white contours at 8.5, 9.5, and 10 K, and the lines of sight that intersect this region are labeled as well. Similar to the results of \citet{Watson:2008}, the majority of the 22 \micron~emission is located inside the bubble. The radio contours trace the ionized shell of the HII region, which show a patchy ionized shell. Outside of the radio contours, there is a shell of 12 \micron~(green) PAH emission that encloses the 22 \micron~emission as well. In the northeastern portion of the image, there is extended 22 \micron~(hot dust) emission, which is spatially coincident with a CO clump \citep{Lagrois:2009}.
The PDR model predicts the presence of neutral hydrogen and molecular CO (see Figure 3 of \citealt{Hollenbach:1999}) at increasing distance from the exciting star cluster. Figure 1 of \citet{Sato:1990} and Figure 2 of \citet{Hasegawa:1983} show HI contours in the vicinity of IC 1805, and the HI emission appears to completely enclose the southern loop except
near 135.5\ddeg{} $\leq$ $\ell$ $\leq$ 136\ddeg, 0.2\ddeg $\leq$ $b$ $\leq$ 0.9\ddeg.
\citet{Braunsfurth:1983} report HI emission near IC 1805, and he notes that the hole could be due to cold HI gas or the lack of gas if the winds have sufficiently swept the material away or ionized it. Figure 6 of \citet{Digel:1996} shows the CO emission, with the W3 molecular cloud on the western side of IC 1805, CO emission along the southern loop of IC 1805, and the molecular material associated with the W5 ($\ell$ = 137.1, $b$ = +0.89) HII region on the eastern side of IC 1805. We interpret the WISE data, the radio contours, and the CO and HI maps as a patchy ionized shell surrounded by a PDR.
If there is a PDR surrounding IC 1805, then the highest RM values from our data set, RM = --954 rad m$^{-2}${} and --961 rad m$^{-2}${} for W4-I2 and -I6, respectively, lie outside the ionized shell of the HII region and in the PDR. Similarly, the sources W4-I19 and -I24 are also outside the radio continuum contours but appear to be within the 12 \micron~(green) emission. This is a surprising result compared with our results from the Rosette Nebula, where we found the highest RM values for lines of sight that pass through the ionized shell. \citet{Gray:1999} note zones of depolarization near the southern portion of IC 1805, which require RMs on order 10$^3$ rad m$^{-2}$, and spatial RM gradients. The RMs for W4-I2 and -I6 are on this order, but those for W4-I24 and -I19 are not, and we do not find that these lines of sight are affect by depolarization. W4-I24 has two components for which we measure RMs, and the components are separated by $\sim$~18 arcseconds. The $\Delta$RM, which is the difference in RM between the two components is 38 rad m$^{-2}$, which is not a large change in the RM and is consistent within the errors.
The presence of the PDR is complicated, however, by the extended diffusion ionized emission reported by \citet{Terebey:2003} and \citet{Gray:1999}. At lower contours, the high RM sources do lie within the radio continuum emission. To fully understand the presence and extent of a PDR or an extended HII envelope, observations of radio recombination lines on the eastern side of IC 1805 would clarify the structure as well as observations of other tracers of PDRs (e.g., fine structure lines of C and C$^+$, H$_2$, and CO). It may be the case that the ionized shell is patchy along the shell wall, which allows photons $>$ 13.6 eV to escape the shell at places, but the shell is sufficiently ionization-bounded at other places such that a PDR can form.
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.95\textwidth]{f9.pdf}
\caption[WISE image of W4]{Inset from Figure \ref{fig:w4}. A RGB image of archive WISE data at 4.6 \micron~(blue), 12 \micron~(green), and 22 \micron~(red) with CGPS contours at 8.5, 9.5, and 10 K in white. The lines of sight from the present study are shown with circles and are labeled according to Table \ref{tab:sources}.}
\label{fig:wise}
\end{figure}
\subsection{Faraday Rotation Through the Cavity and Shell of the Stellar Bubble}
There are four lines of sight through the cavity of the stellar bubble, assuming an inner radius from the \citet{Terebey:2003} model. The sources W4-I3, -I15, -I18, and -I21 are through the cavity, and including multiple components, we find 6 RM values. W4-I3 has a high RM (--878 $\pm$ 18 rad m$^{-2}$), and W4-I15 and -I21 have comparatively low RM values ( --79 to --232 rad m$^{-2}$). Examination of Figures \ref{fig:w4} and \ref{fig:wise} does not reveal enhanced emission near W4-I3 in comparison to W4-I21. W4-I15, however, is in a region of relatively low emission, which may explain why W4-I15 has a RM value at least 4 times smaller than W4-I3.
W4-I13 is outside the shell, assuming a shell radius from either \citet{Terebey:2003} model. From Figure \ref{fig:w4}, it does appear to be outside the ionized shell. However W4-I13 is within a 8.5 K contour on the 1.42 GHz radio continuum map, which may indicate that it is probing the ionized shell. We find a high RM for both components of this source, which is similar to the RM values for W4-I2 and -I6.
Across IC 1805, we observe negative RM values for all lines of sight except one: W4-I18, which is 5.6 arcmin (4 pc) from the center of the star cluster. The absolute value of the RM for W4-I18 is also large (+501 $\pm$ 33 rad m$^{-2}$), indicating a large change in RM along this line of sight relative to other lines of sight in this part of the sky. This line of sight is probing the space close to the massive O and B stars responsible for IC 1805. In the \citet{Weaver:1977} model for a stellar bubble, the hypersonic stellar wind dominates the region between the star responsible for the bubble and the inner termination shock. Equation (12) of \citet{Weaver:1977} states that the distance of the inner shock, $R_t$ is
\begin{equation}
R_{\textrm{t}} = 0.90\; \alpha^{3/2} \left(\frac{1}{\rho_0}\frac{\textrm{d}M_w}{\textrm{d}t}\right)^{3/10} \ V_w^{1/10} \ t^{2/5},
\label{eq:weaver}
\end{equation}
where $\alpha$ is a constant equal to 0.88, $\rho_0$ is the mass density in the external ISM, d$M_w$/d$t$ is the mass loss rate, $V_w$ is the terminal wind speed, and $t$ is time. For a rough estimate of the inner shock distance, we utilize general stellar parameters for OCl 352 of d$M_w/dt$ = 10$^{-5}$ \textrm{M}$_{\odot}$ yr$^{-1}$, $t$ = 10$^6$ yr, and V$_w$ = 2200 km s$^{-1}$ (see Section \ref{sec:structure} or Table \ref{tab:stellarpar}). From the discussion in Section \ref{sec:ferriere}, we adopt $n_0$ = 4.5 cm$^{-3}$~for $\rho_0$ = $n_0 m_p$, where $m_p$ is the mass of a proton. With these values in the appropriate SI units, $R_t$ $\sim$ 6 pc. It is possible that the line of sight to W4-I18 passes inside the inner shock, and the large, positive RM is due to material modified by the hypersonic stellar wind and not the shocked interstellar material.
Because the inner shock is interior to the contact discontinuity between the stellar wind and the ambient ISM, the magnetic field close to the star cluster may be oriented in any direction relative to the exterior (upstream) field. With a positive value of the RM for W4-I18, the line of sight component of the field points toward us while the remaining lines of sight in the cavity are negative, meaning B$_{\textrm{LOS}}$~points away.
\subsection{Low Rotation Measure Values Through the W4 Superbubble}
North of IC 1805 is the W4 Superbubble, which is an extended ``egg-shaped'' structure closed at $b$ $\sim$ 7\ddeg{} \citep{West:2007}. \citet{Basu:1999} utilize an H$\alpha$ map to define the shape, which would include the southern loop (134\ddeg $<$ $\ell$ $<$ 136\ddeg, \textit{b} $<$ 0.5\ddeg); \citet{Normandeau:1996} examine the HI distribution, however, and place the base of the structure at OCl 352. Similarly, \citet{West:2007} place an offset bottom of the ``egg'' at OCl 352. The southern loop of IC 1805 is seemingly sufficiently different from the northern latitudes, as it is often not included in the discussion of the W4 Superbubble in spite of the fact that OCl 352 is thought to be responsible for the formation of both structures \citep{Terebey:2003,West:2007}.
Nine lines of sight in the present study are north of OCl 352 in the W4 Superbubble. These sources are W4-I1, -I4, -I8, -I12, -I14, -I17, -O4, -O6, and -O7, and they have a mean RM of --293 rad m$^{-2}$~and a standard deviation of 178 rad m$^{-2}$. Of these sources, W4-I14 and -I17 have the largest RM values, --666 rad m$^{-2}$~and --460 rad m$^{-2}$, respectively, and they are close to OCl 352, with distances of 31 arcminutes (20 pc) and 17 arcminutes (11 pc), respectively. As discussed in Section \ref{sec:structure}, \citet{Lagrois:2009} argue that the ionized ``v'' structure north of OCl 352 is part of the bubble wall and not a cap to southern loop structure, but examination of Figure \ref{fig:w4} suggests that the bubble walls are denser, or thicker, at latitudes $<$ 1.5\ddeg~than higher latitudes, which may explain the larger RM associated with W4-I14 and -I17. The remaining lines of sight, however, in the W4 Superbubble have some of the lowest RM values in the data set and are consistently lower RM values than the lines of sight through the PDR.
At higher latitudes, \citet{Gao:2015} modeled the polarized emission and applied a Faraday screen model to the W4 Superbubble. They report RMs on the western side of W4 ($\ell$ $\sim$ 132\ddeg, $b$ $\sim$ 4.8\ddeg) between --70 and --300 rad m$^{-2}$~and $\sim$ +55 rad m$^{-2}$~for the eastern shell ($\ell$ $\sim$ 136\ddeg, $b$ $\sim$ 7\ddeg). \citet{Gao:2015} argue that since W4 is tilted at an angle towards the observer \citep{Normandeau:1997}, a change in the sign of the RM is consistent with a scenario in which the superbubble lifts up a clockwise running Galactic magnetic field \citep{Han:2006} out of the Galactic plane. The magnetic field would go up the eastern side of the superbubble and then down the western side, resulting in the field being pointed toward the observer in the east and away from the observer in the west. While the lines of sight reported in this paper are at $b$ $<$ 2\ddeg, we find a similar range of RM values as reported by \citet{Gao:2015} for the western side. However, we measure RM values 3 -- 4.5 times higher on the eastern side, and we do not observe a sign reversal on the eastern side as suggested by \citet{Gao:2015}.
\citet{West:2007} report positive values of the magnetic field for the western side from a change in polarization position angle of $\sim$ 60\ddeg~at 21 cm, which gives a RM value on order of 20 rad m$^{-2}$. We do not observe RM values this low for any of our lines of sight through the northern latitudes. Our lines of sight, however, do not probe the same regions as the \citet{West:2007} and \citet{Gao:2015} studies
The line of sight W4-I4 is arguably within the W4 Superbubble; however, it is also $\sim$~8 arcmin (5 pc) on the sky from W3-North (G133.8 +1.4), which is a star forming region within W3. W4-I4 has two components, separated by 15 arcsec (0.2 pc), and a difference in RM values between the two components of $\Delta$RM = 85 rad m$^{-2}$. The RM values for both components are low (--153 rad m$^{-2}$~and --68 rad m$^{-2}$) despite being in the superbubble and near to W3, which may have variable but potentially large magnetic fields \citep{vanderWerf:1990,Roberts:1993} (see Section \ref{sec:structure}).
\section{Models for the Structure of the HII region and Stellar Bubble\label{sec:models}}
\subsection{\citet{Whiting:2009} Model of the Rotation Measure in the Shell of a Magnetized Bubble\label{sec:whitingmod}}
\citet{Whiting:2009} developed a simple analytical shell model intended to represent the Faraday rotation due to a \citet{Weaver:1977} solution for a wind-blown bubble. We employed this model in \citet{Savage:2013} and \citet{Costa:2016} to model the magnitude of the RM in the shell of the Rosette Nebula as a function of distance from the exciting star cluster. Figure 6 of \citet{Whiting:2009} and their Section 5.1 give the details of the model, and Sections 4.1 of \citet{Savage:2013} and 5 of \citet{Costa:2016} describe the application of the model to the Rosette Nebula. This model takes as inputs the general interstellar magnetic field ($\textbf{B}$) in $\mu$G, the inner ($R_1$) and outer ($R_0$) radii of the shell in parsecs, and the electron density in the shell, $n_e$ (cm$^{-3}$). $R_0$ represents the shock between the ambient ISM and the shocked, compressed ISM, and $R_1$ separates the shocked ISM from the hot, diffuse stellar wind in the cavity. Only the component of the ambient interstellar magnetic field that is perpendicular to the shock normal is amplified by the density compression ratio, X. The resulting expression for the RM through the shell is
\begin{equation}
\textrm{RM}=C\, n_{e}\, L(\xi)\, B_{0z} \left(1+(X-1)\left(\frac{\xi}{R_{0}}\right)^{2}\right),
\label{eq:rmmodelW}
\end{equation}
where L($\xi$) is the cord length through the shell in parsecs (see Equation 10 in \citealt{Whiting:2009} or Equation 6 in \citealt{Costa:2016}), and $B_{0z}$ is the z-component of \textbf{B$_0$}, the magnetic field in the ISM. If $n_e$ has units of cm$^{-3}$, $B_{0z}$ is in $\mu$G, and $L$ is in parsecs, $C=0.81$ (see Equation \ref{eq:rmprat}). $B_{0z}$ is at an angle $\Theta$ with respect to the LOS and is written as
\begin{equation}
B_{0z}=B_{0}\cos{\Theta}.
\end{equation}
In our previous work, we presented two cases for the behavior of the magnetic field in the shell. The first is that the magnetic field is amplified by a factor of 4 in the shell. The second case, in which there is not an amplification of the magnetic field in the shell, sets X = 1. Equation \ref{eq:rmmodelW} then simplifies to
\begin{equation}
\textrm{RM}(\xi)=0.81\, n_{e} \, L(\xi) \, B_{0z}.
\label{eq:rmmodelH}
\end{equation}
In \citet{Costa:2016}, we employed a Bayesian analysis to determine which of the two models better reproduces the observed dependence of the RM as a function of distance. We found that neither model was strongly favored in the case of the Rosette. The model given in Equation (\ref{eq:rmmodelW}) is subject to the criticism that it applies shock jump conditions for \textbf{B} over a large volume of a shell, and that the outer radius of an observed HII region need not be the outer shock of a Weaver bubble (see remarks in Section 5.1.1 of \citealt{Costa:2016}). It is worth including this model, however, in our analysis of IC 1805 for completeness and in order to compare our results to those of the Rosette Nebula.
In Section \ref{sec:structure}, we discussed the the structure of IC 1805, and we present evidence from the literature that north of OCl 352 is part of the W4 Superbubble. Thus, lines of sight north of OCl 352 may have different model parameters for the shell radii and electron density than the southern loop. For the southern latitudes, we utilize the \citet{Terebey:2003} thin and thick shell values for the shell radii and electron density. The remaining parameters in Equation (\ref{eq:rmmodelH}) are $B_0$ and $\Theta$. As in \citet{Savage:2013} and \citet{Costa:2016}, we adopt $B_0$ = 4 $\mu$G for the general Galactic field in front of the HII region. The angle $\Theta$ is calculated as follows.
Assuming a distance of 8.5 kpc to the Galactic center, a distance to OCl 352 of 2.2 kpc, and given a Galactic longitude of 135\ddeg, the angle between the line of sight and an azimuthal magnetic field is $\Theta$ = 55\ddeg. We discuss our comparison of this model with the data in Section \ref{sec:results}.
\begin{figure}[htb!]
\centering
\subfloat[\label{fig:southshell-thick}]{
\includegraphics[width=0.48\textwidth]{f11a.pdf}}
\quad
\subfloat[\label{fig:southshell-thin}]{
\includegraphics[width=0.48\textwidth]{f11b.pdf}}
\caption[]{ Plots of LOS versus distance with the \citet{Whiting:2009} model for X = 4 (solid) and X = 1 (dashed) for $b$ $<$ +0.9\ddeg{} using (a) the thick shell and (b) the thin shell parameters from \citet{Terebey:2003}. The background RM = --100 rad m$^{-2}$~from the \citet{vanEck:2011} model. The errors on the RM values include the measurement errors and an expected deviation of $\sim${} 67 rad m$^{-2}${} from the \citet{vanEck:2011} model. See Table \ref{tab:model} for model parameters.}
\label{fig:shell}
\end{figure}
\begin{table}[hbtp]
\centering
\begin{threeparttable}
\caption{Model Parameters \label{tab:model}}
\begin{tabular}{ccccccc}
\hline
\multicolumn{7}{c}{\citet{Whiting:2009} Model} \\
\hline
Center\tnote{a}\phantom{2} & R$_i$ & $R_o$ & $n_e$ & X\tnote{b} & $\Theta$ & Figure \\
(\ddeg) & (pc) & (pc) & (cm$^{-3}$) & & (\ddeg) & \\
(135, +0.42) & 19 & 25 & 10 & 1, 4 & 55 & \ref{fig:southshell-thick} \\
(135, +0.42) & 19 & 21 & 20 & 1, 4 & 55 & \ref{fig:southshell-thin} \\
\hline \hline
\multicolumn{7}{c}{\citet{Ferriere:1991} Model} \\
\hline
Center & $\Delta R$ & $R_s$ & $n_s$ & $\epsilon$ & $\Theta$ &Figure \\
(\ddeg) & (pc) & (pc) & (cm$^{-3}$) & & (\ddeg) & \\
(135, +0.42) & 6 & 25 & 10 & 0.25 & 55 & \ref{fig:steve2}\\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] Position of model center in Galactic coordinates in the format of ($\ell$, $b$).
\item[b] The model uses either X = 1 or X = 4.
\end{tablenotes}
\end{threeparttable}
\end{table}
\subsection{Analytical Approximation to Magnetized Bubbles of \citet{Ferriere:1991}\label{sec:ferriere}}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.6\textwidth]{f10.pdf}
\caption{Illustration of a simplified version of the shell and cavity produced by a stellar wind, as discussed by \citet{Ferriere:1991}. The z direction is that of the interstellar magnetic field, and $\Theta$ is the angle between the magnetic field and the line of sight. A line of sight passes at a closest distance $\xi$ from the center of the cavity (the ``impact parameter''). Other parameters in the figure are defined in the text. A Faraday rotation measurement is along a line of sight offset a linear distance $\xi$ from the center of the bubble. The quantity d$s$ represents an incremental spatial interval along the LOS. }
\label{fig:steve1}
\end{figure}
\citet{Ferriere:1991} presented a semi-analytic discussion of the evolution of a stellar bubble in a magnetized interstellar medium. The theoretical object discussed by \citet{Ferriere:1991} could describe a shock wave produced by a supernova explosion or energy input due to a stellar wind. The main features of the model were an outer boundary (e.g. outer shock) which was the first interface between the undisturbed ISM and the bubble, and an inner contact discontinuity between ISM material, albeit modified by the bubble, and matter that originated from the central star or star cluster.
The main feature of the model is that plasma passing through the outer boundary is concentrated in a region between the outer boundary and the contact discontinuity. In what follows, we will refer to this region as the shell of the bubble. The equation of continuity then indicates that there will be higher plasma density in the shell, and the law of magnetic flux conservation indicates that there will be an increase in the strength of the magnetic field in the shell relative to the general ISM field. \citet{Ferriere:1991} were interested in the structure of the bubble, and their results have been corroborated by the fully numerical studies of \citet{Stil:2009}. However, \citet{Ferriere:1991} did not calculate the Faraday rotation measure through their model for diagnostic purposes. \citet{Stil:2009} explicitly considered the model RMs from their calculations, but only for a couple of cases and for two values of bubble orientation. It is our goal in this section to use a simplified, fully analytic approximation of the results of \citet{Ferriere:1991}, that permits RM profiles RM($\xi$) for a wide range of bubble parameters and orientation with respect to the LOS.
The geometry of the bubble is shown in Figure \ref{fig:steve1}, which is an adaptation of, and approximation to Figure 1 and Figure 4 from \citet{Ferriere:1991}. An important feature of Figure \ref{fig:steve1}, not present in \citet{Ferriere:1991}, is the orientation of the line of sight at an angle $\Theta$ with respect to the ISM magnetic field at the position of the bubble, and the impact parameter $\xi$ indicating the separation of the LOS from the center of the bubble. The region interior to the contact discontinuity is referred to as the cavity, and for the purposes of our discussion will be considered a vacuum. Another important shell parameter is the thickness $\Delta R$ $\equiv$ $R_s$ - $R_i$, where $R_s$ and $R_i$ are the outer radius of the bubble and the radius of the contact discontinuity, respectively (see Figure \ref{fig:steve1}). We also define and use the dimensionless shell thickness
\begin{equation}
\epsilon \equiv \frac{\Delta R}{R_s}
\label{eq:steve8}
\end{equation}
A major simplification that we adopt, based on an approximation of the results of \citet{Ferriere:1991}, is that the magnetic field in the shell ($\textbf{B}_s$) is entirely in the azimuthal direction, and that we ignore radial variations within the shell, i.e.
\begin{equation}
\textbf{B}_s(r, \theta) \equiv \pm B_s(\theta) \hat{e}_{\theta}
\label{eq:steve9}
\end{equation}
where $\hat{e}_{\theta}$ is a unit vector in the azimuthal direction, and the $\pm$ is selected by the polarity of the interstellar field at the bubble.
We need expressions for the electron density and vector magnetic field within the shell, as well as the geometry of the line of sight. The most important aspect of the \citet{Ferriere:1991} theory is the conservation of magnetic flux as the magnetic field in the external medium is swept up and accumulated in the shell. This results in the azimuthal component of the magnetic field increasing as $\theta$ increases from $0$ to $\frac{\pi}{2}$, as given by Equation (40) of \citet{Ferriere:1991}.
In \citet{Ferriere:1991} the shell thickness also depends on $\theta$ (Equation 46 of \citealt{Ferriere:1991}), and as a consequence, so does the plasma density in the shell $n_s$ (Equation 38 of that paper).
In the initial version of this paper, we calculated the RM through model bubbles in which $B_s$, $n_s$, and $\Delta$ R all varied with $\theta$ as prescribed by \citet{Ferriere:1991}. These calculations utilized an approximate form for lines of sight that intersected the bubble in two segments (passing through the central cavity between), and a form that contained a numerically-evaluated expression for lines of sight that remained within the shell from ingress to egress. The algebraic distinction between these 2 cases is discussed below (Sections \ref{sec:walls} and \ref{sec:allshell}). These expressions for RM($\xi$), including a comparison with our RM measurements, are given in \citet{Costa:2018phd}.
After examining the results of these calculations, it was decided to simplify our bubble model to that of a spherical shell with constant $\epsilon$. The motivation for this suggestion was the very limited success of the more general model in representing our data, which did not justify the extensive algebraic presentation and non-compact expressions that resulted. The calculations with the approximation of constant $\epsilon$ are presented below.
Due to magnetic flux conservation, the expanding shell (now approximated as spherical) will have a magnetic field that is larger than in the external medium, and increases with $\theta$, as in the original discussion of \citet{Ferriere:1991}. For our spherical case, it may be shown that the magnetic field in the shell is
\begin{equation}
B_s(\theta) = \frac{B_0}{2 \epsilon} \sin \theta
\label{eq:steve10}
\end{equation}
where $B_0$ is the magnitude of the magnetic field in the external medium. The dimensionless shell thickness $\epsilon$ remains a free parameter of the model, or one that can be determined by observations. Finally, the plasma density in the shell, determined by mass conservation, is
\begin{equation}
n_s = \frac{n_0}{3 \epsilon}
\label{eq:steve11}
\end{equation}
where $n_0$ is the plasma density in the external medium. \equa{steve11} is a valid approximation for $\epsilon \ll 1$.
\subsubsection{RM Calculation for Lines of Sight Through the Walls of the Shell\label{sec:walls}}
In evaluating the integral Equation (\ref{eq:rmorg}) or (\ref{eq:rmprat}) through the model shell shown in Figure \ref{fig:steve1}, we consider two cases. The first calculation is for lines of sight that pass through a portion of the shell, emerge into the cavity, and then reenter the shell on the opposite side before exiting the shell entirely. This is the case illustrated in Figure \ref{fig:steve1}. The incremental RM for a spatial interval d$s$ along the line of sight is
\begin{equation}
\textrm{d(RM)} = \pm \, C \, n_s \, B_s(\theta) \, (\hat{e}_s \cdot \hat{e}_{\theta})\, \textrm{d}s
\label{eq:steve12}
\end{equation}
The $\pm$ in front of the RHS indicates that the polarity of the field in the external medium determines the sign of the measured RM. We introduce the variable $s$ as a coordinate along the line of sight; d$s$ is an incremental vector along the line of sight from the source to the observer, and $\hat{e}_s$ is the corresponding unit vector. The constant $C$ is the same as introduced in Equation (\ref{eq:rmmodelW}).
It is convenient to change the variable of integration over the LOS from $s$ to $\phi$, an angle defined in Figure \ref{fig:steve1}. With the introduction of this variable, the term $(\hat{e}_s \cdot \hat{e}_{\theta}) = -\sin \phi$. Integration through the shell segments along the line of sight then corresponds to an appropriate integration over $\phi$. The shell segment closest to the observer corresponds to an integration from $\phi_1$ to $\phi_2$, and the segment furthest from the observer is given by an integration from $\phi_3$ to $\phi_4$.
Substitution of Equations (\ref{eq:steve10}) and (\ref{eq:steve11}) into (\ref{eq:steve12}), followed by integration over $\phi$ and straightforward algebraic manipulation yields the following expression for the RM
\begin{equation}
\textrm{RM}(x) = \pm \left( \frac{C \, n_0\, B_0\, R_s}{3 \epsilon^2} \right) x \left[ \arcsin \left(\frac{x}{1 - \epsilon}\right) - \arcsin (x) \right] \cos \Theta
\label{eq:steve13}
\end{equation}
where the new dependent variable is the normalized impact parameter $x \equiv \frac{\xi}{R_s}$. The identity (\ref{eq:steve11}) may be used to convert Equation (\ref{eq:steve13}) into a form in which the observed plasma density in the shell ($n_s$) is the density parameter rather than that in the external medium ($n_0$). This substitution makes Equation (\ref{eq:steve13}) more directly comparable to Equation (\ref{eq:rmmodelW}).
\subsubsection{RM for Lines of Sight Entirely Within the Shell \label{sec:allshell}}
If the ``impact parameter'' $\xi$ is sufficiently large, the entire line of sight is within the shell from the point of ingress to that of egress. From Figure \ref{fig:steve1}, it can be seen that this occurs if
\begin{equation}
x \equiv \frac{\xi}{R_s}\, \geq\, x_{min} = 1 -\epsilon
\label{eq:steve14}
\end{equation}
The RM in this case is a simple generalization of the algebra involved in obtaining Equation (\ref{eq:steve13}) via an integration over the angular variable $\phi$; the upper limit of integration in the segment closest to the observer $\phi_2 \rightarrow \frac{\pi}{2}$, and the lower limit of integration for the shell segment further from the observer $\phi_3 \rightarrow \frac{\pi}{2}$.
\begin{equation}
\textrm{RM}(x) = \pm \left( \frac{C\, n_0\, B_0\, R_s}{3 \epsilon^2} \right)\, x\, \left[ \frac{\pi}{2} - \arcsin (x) \right] \cos \Theta \mbox{ , if: } x_{min} \leq x \leq 1
\label{eq:steve15}
\end{equation}
A plot of the expression RM($x$) given by (\ref{eq:steve13}) and (\ref{eq:steve15}) is shown in Figure \ref{fig:steve2} for a set of parameters that are representative for the IC 1805 HII region (see Table \ref{tab:model}). The curve is very similar in form to the Whiting model, for the case of no magnetic compression, Equation (\ref{eq:rmmodelW}) with $X = 1$ or Equation (\ref{eq:rmmodelH}). The model expression for RM($x$) is dependent on $n_0$ (or the shell density $n_s$), $B_0$, $R_s$, $\Theta$, and $\epsilon$, the shell thickness parameter. For comparison with observations, we also need to specify the background Galactic rotation measure, RM$_{off}$.
Our simple model contained in Equations (\ref{eq:steve13}) and (\ref{eq:steve15}) immediately accounts for one of the main results emergent from the numerical simulations of \citet{Stil:2009}. The RM through a bubble is maximized when the LOS is parallel to \textbf{B}$_0$ ($\cos \Theta = 1$) and small or zero when the LOS is $\perp \mbox{ to } \textbf{B}_0$ ($\cos \Theta = 0$).
\begin{figure}[htb!]
\centering
\includegraphics[width=0.6\textwidth]{f12.pdf}
\caption{Model for the analytic approximation to the bubble model of \citet{Ferriere:1991}, Equations (\ref{eq:steve13}) and (\ref{eq:steve14}). The model RM is function of the normalized impact parameter $x = \frac{\xi}{R_s}$. The plotted points represent measured RMs presented in this paper. }
\label{fig:steve2}
\end{figure}
\section{Discussion of Observational Results\label{sec:results}}
\subsection{Comparison of Models with Observations in the HII Region}
In this section we discuss the results of the two models presented in Sections \ref{sec:whitingmod} and \ref{sec:ferriere}. In both cases, we adopt the \citet{Terebey:2003} center for geometric ease and spherical symmetry as well as the parameters given in their Table 3 for a thick shell.
Figures \ref{fig:southshell-thick} and \ref{fig:southshell-thin} show model RM values for lines of sight south of IC 1805 (\textit{b} $<$ 0.9\ddeg) with the \citet{Whiting:2009} model for the RM as a function of distance and the shell parameters from \citet{Terebey:2003}. Table \ref{tab:model} gives the values of the center of the bubble, the shell radii, the electron density, X, and $\Theta$ for Figure \ref{fig:shell}. Neither model reliably reproduces the observed RM as a function of distance, and as in \citet{Costa:2016}, the model can not account for the dispersion of RM values at similar distances. Generally, the lines of sight in the cavity are low and are more consistent with the background RM. In the thin shell approximation, the largest RM values are associated with lines of sight outside the shell. While the model without amplification of the magnetic field in the shell can marginally account for the magnitude of the RM, the model with amplification (Equation \ref{eq:rmmodelW}) predicts far too high values for the RM for $\Theta$ = 55\ddeg. The analysis contained here mildly supports a result from \citet{Costa:2016} for the Rosette Nebula; Faraday rotation values through these HII regions do not permit a substantial increase in $|$B$|$ over the general Galactic field.
To reproduce the observed RM in the shell at $\xi$ $\sim${} 20 pc, the angle between the magnetic field and the observer would need to be tilted more into the plane of the sky for the X = 4 case or into the line of sight for the X = 1 case. For the former case, an angle of $\sim${} 75\ddeg{} would reproduce the magnitude of the RM in the shell; such an angle is greater than that expected from a geometric argument, even accounting for a magnetic field pitch angle of $\sim${} 8\ddeg. Also, no one angle can account for the range of the RM values in the cavity.
With our analytic solution for the RM due to a magnetized bubble as described by \citet{Ferriere:1991}, we can examine the dependence of RM on $\Theta$ as well as $\xi$. The most obvious choice for the latter parameter is $\Theta = 55^{\circ}$, based on the geometry as described in Section \ref{sec:whitingmod}. Figure \ref{fig:steve2} shows our model RM(x) for $\Theta = 55^{\circ}$, with other parameters given in Table 6. Data for sources south of IC 1805 are superposed on the model. Although the model obviously does not reproduce the measurements in detail, it can describe the overall scale of the ``rotation measure anomaly'' associated with W4, as well as the approximate magnitude of the largest measured RMs ($|$RM$|$ $\sim$ 1000 rad m$^{-2}$). The peak model RM values shown in Figure 12 do not significantly exceed the measured values, unlike the case for the Whiting model with X = 4 (see Figure \ref{fig:shell}). It should be kept in mind that the shell modeled in Figure \ref{fig:steve2} is the ``thick shell model'' of \citet{Terebey:2003}; the center of that shell is not the star cluster OCl 352, as might be expected.
\citet{Stil:2009} carried out numerical MHD simulations of the Ferri\`ere bubbles, which are obviously more accurate than our analytic approximations. Furthermore, they specifically consider and calculate the Faraday rotation through their models. However, \citet{Stil:2009} only consider $\Theta$ = 0\ddeg{} and $\Theta$ = 90\ddeg, so the calculations reported in that paper can not explore the changes in RM structure with $\Theta$. Furthermore, the Faraday rotation calculation of \citet{Stil:2009} is done when the outer radius $R_s$ $\sim${} 200 pc (see Figure 14 of \citealt{Stil:2009}), which is much larger than the structure we are modeling in Section \ref{sec:ferriere} of this paper. In what follows, we compare our observations with the results presented in Section 6 of \citet{Stil:2009}.
If LOS $||$ \textbf{B}$_{\textrm{ext}}$, then the highest values of RM will be through the shell closest to the Galactic plane, but the mean RM across the region will be similar to the mean RM exterior to the bubble (see Figure 14 of \citealt{Stil:2009}). Out of the Galactic plane, the RM is 20 -- 30$\%$ of the mean RM exterior to the bubble. Effectively, the largest RMs will always be found in the Galactic plane, and different lines of sight through the bubble will have varying RM values.
In comparing the simulations of \citet{Stil:2009} to our observational results, we find low RM measures for lines of sight through the cavity, though not always low (e.g., W4-I14 and -I17 vs -I15 and -I21). Lines of sight through the shell have generally large RMs, which is inconsistent with a \textbf{B}$_{\textrm{ext}}${} perpendicular to the LOS. The case of LOS $||$ \textbf{B$_{\textrm{ext}}$} is inconsistent as well because far from the bubble, the RM is low (e.g., W4-O26 vs -I24) even at similar latitudes, and the lines of sight at $b$ $>$ 1\ddeg{} are consistent with the background RM instead of being reduced by 70 -- 80$\%$. Unsurprisingly, our results indicate a case somewhere between these two predictions. As a reminder, we note that the largest values of the RM are for lines of sight exterior to the shell, which is not a prediction from \citet{Stil:2009}, most likely due to their simulations modeling the ionized bubble and not a PDR structure.
\subsection{Magnetic Fields in the PDR}
In Section \ref{sec:pdr}, we examine evidence for a PDR outside the southern loop of IC 1805. \citet{Brogan:1999}, \citet{Troland:2016}, and \citet{Pellegrini:2007} report large ($\sim${} 150 $\mu$G) magnetic fields in PDRs associated with the Orion Veil and M17. In the analysis that follows, we attempt to understand the large RM values for lines of sight through the IC 1805 PDR.
If we consider the PDR and the HII region to be in pressure equilibrium and include magnetic pressure in the PDR, then
\begin{equation}
P^{\textrm{H}\,\textsc{ii}}_{\textrm{th}} = P^{\textrm{PDR}}_{\textrm{th}} \; + P^{\textrm{PDR}}_{\textrm{mag}},
\label{eq:pbal}
\end{equation}
where $P^{\textrm{H}\,\textsc{ii}}_{\textrm{th}}${} and $P^{\textrm{PDR}}_{\textrm{th}}${} are the thermal pressures in the HII region and PDR, respectively, and \(P^{\textrm{PDR}}_{\textrm{mag}}=\frac{B^2}{8\pi} \) is the magnetic pressure in the PDR. In the HII region, \(P^{\textrm{H}\,\textsc{ii}}_{\textrm{th}} = 2n_e^{\textrm{H}\,\textsc{ii}}\, k \,T_{\textrm{H}\,\textsc{ii}} \), where $n_e^{\textrm{H}\,\textsc{ii}}$ and T$_{\textrm{H}\,\textsc{ii}}$ are the electron density and temperature, $k$ is the Boltzmann constant, and the factor of 2 accounts for the contribution from both ions and electrons. For P$^{\textrm{PDR}}_{\textrm{th}}$ = $N_{\textrm{PDR}}$ $k$ $T_{\textrm{PDR}}$, $N_{\textrm{PDR}}$ and $T_{\textrm{PDR}}$ are the neutral hydrogen density and the temperature in the PDR.
Near the interface of the PDR and the HII region, the electron density in the PDR is governed by photoioniziation of carbon \citep{Tielens:1985}, so we estimate $n_e^{\textrm{PDR}}$ by \[n_e^{\textrm{PDR}} = N_{\textrm{PDR}}X_C,\] where $X_C$ is the cosmic abundance of carbon given in Table 1.4 of \citet{Draine:2011} ($X_C$ $\sim${} 2.95 $\times$ 10$^{-4}$). Solving for $B$ in \equa{pbal} gives
\begin{equation}
B = \sqrt{8\pi \; k(2\; n_e^{\textrm{H}\,\textsc{ii}} \;T_{\textrm{H}\,\textsc{ii}} - N_{\textrm{PDR}}\;T_{\textrm{PDR}})},
\label{eq:bpbal}
\end{equation}
and inserting it into \equa{rmprat}, we express the RM in the PDR as
\begin{equation}
RM = 0.81\; L \; X_C N_{\textrm{PDR}} \sqrt{8\pi \; k \; (2\;n_e^{\textrm{H}\,\textsc{ii}} T_{\textrm{H}\,\textsc{ii}} - N_{\textrm{PDR}}\; T_{\textrm{PDR}})}.
\label{eq:rmpdr}
\end{equation}
It should be emphasized that this RM estimate is in the nature of an upper limit to the rotation measure through the PDR. The reason is that it is obtained from a value of $B$, given by Equation (\ref{eq:rmpdr}), which is based on the magnetic pressure $\frac{B^2}{8 \pi}$. The magnetic pressure includes contributions from turbulent fluctuations on all scales, as well as that from a mean or large scale field that produces the net Faraday rotation. In general then, a magnetic field value obtained from an estimate of the magnetic pressure will exceed that obtained from a Faraday rotation measurement.
We differentiate \equa{rmpdr} with respect to $N_{\textrm{PDR}}$ to find the value of $N_{\textrm{PDR}}$ that maximizes the RM, which is
\begin{equation}
N_{\textrm{PDR}} = \frac{4}{3}\frac{n_e^{\textrm{H}\,\textsc{ii}}\;T_{\textrm{H}\,\textsc{ii}}}{T_{\textrm{PDR}}}.
\label{eq:maxN}
\end{equation}
Inserting values of $T_{\textrm{H}\,\textsc{ii}}$ = 8000 K, $n_e^{\textrm{H}\,\textsc{ii}}$ = 10 cm$^{-3}${} \citep{Terebey:2003}, and $T_{\textrm{PDR}}$ = 100 K \citep{Tielens:1985}, gives $N_{\textrm{PDR}}$ $\sim${} 1000 cm$^{-3}${}, $B$ $\sim${} 14 $\mu$G (Eq \ref{eq:bpbal}), and RM $\sim${} 100 rad m$^{-2}$. The electron density in the HII region is governing the maximum $B$ expected in the PDR given pressure balance. For the IC 1805 HII region, $n_e$ is low compared to M17 ($n_e$ $\sim${} 560 cm$^{-3}$) \citep{Pellegrini:2007}, which suggests that a high density (pressure) HII region is needed to explain large magnetic fields in the PDR.
Our analysis suggests that a simple pressure balance analysis predicts low RM values from the PDR that are inconsistent with our observations. It appears that a different mechanism is required to achieve the magnetic fields strengths observed in \citet{Brogan:1999}.
\citet{Terebey:2003} discuss an extended halo of ionized emission around the southern loop, which may indicate that there are more free electrons present outside the obvious ionized shell as seen in Figure \ref{fig:w4}. This may account for the larger values of the RM we observe. It is clear that knowing the electron density in this region and determining the presence of a PDR through observations, such as carbon radio recombination lines, is necessary to understand how the magnetic field is modified in this complex region.
\section{A Comparison of IC 1805 and the Rosette Nebula as ``Rotation Measure Anomalies'' \label{sec:rosette}}
\begin{table}[htb!]
\centering
\caption{Stellar Parameters \label{tab:stellarpar}}
\begin{threeparttable}
\centering
\begin{tabular}{llllll}
\hline
\multicolumn{1}{c}{Star Cluster} & \multicolumn{1}{c}{Star} & \multicolumn{1}{c}{Type} & \multicolumn{1}{c}{\ML} & \multicolumn{1}{c}{V$_{\ensuremath{\infty}}$} & \multicolumn{1}{c}{ L$_{W}$=$\frac{1}{2}\dot{M}$v$_{\ensuremath{\infty}}^{2}$}\tnote{a} \\
\multicolumn{1}{c}{ } &\multicolumn{1}{c}{ } &\multicolumn{1}{c}{ } & \multicolumn{1}{c}{ (\Msun yr$^{-1}$) } & \multicolumn{1}{c}{(km/s)} & \multicolumn{1}{c}{ (erg s$^{-1}$)} \\
\hline
{\multirow{4}{*}{NGC 2244}} & HD 46223 & O4V(f)\tnote{b} & 1.6$\times$10$^{-6}$ \tnote{c} & 3100\tnote{d} & 4.8$\times$10$^{36}$ \\
& HD 46150 & O5.5V\tnote{e} & 2.0$\times$10$^{-6}$ \tnote{c} & 3100\tnote{d} & 6.0$\times$10$^{36}$ \\
& HD 46202 & O9V(f)\tnote{b} & 6.3$\times$10$^{-8}$ \tnote{c} & 1150\tnote{d} & 2.6$\times$10$^{34}$ \\
& HD 46149 & O8.5V(f)\tnote{b} & 2.0$\times$10$^{-7}$ \tnote{c} & 1700\tnote{f} & 1.8$\times$10$^{35}$\\
\hline
{\multirow{3}{*}{OCl 352}} & HD15570 & O4I\tnote{b} & 1.0$\times$10$^{-5}$ \tnote{c} & 2200\tnote{g} & 1.5$\times$10$^{37} $ \\
& HD15558 & O4III\tnote{b} & 6.3$\times$10$^{-6}$ \tnote{c} & 3000\tnote{f} & 1.8$\times$10$^{37} $ \\
& HD 15629 & O5V\tnote{b} & 2.0$\times$10$^{-6}$ \tnote{c} & 2900\tnote{h} & 5.3$\times$10$^{36}$ \\
\hline
\end{tabular}
\begin{tablenotes}
\item[a] Calculated mechanical wind luminosity based on cited mass loss rates and terminal velocities.
\item[b] \citet{Massey:1995}
\item[c] \citet{Howarth:1989}
\item[d] \citet{Chlebowski:1991}
\item[e] \citet{Roman:2008}
\item[f] \citet{Garmany:1988}
\item[g] \citet{Bouret:2012}
\item[h] \citet{Groenewegen:1989}
\end{tablenotes}
\end{threeparttable}
\end{table}
We are interested in how the Galactic magnetic field is modified by OB associations via their stellar winds and ionizing photons, and we started our study with the Rosette Nebula, where we found large ($\sim$~10$^{3}$ rad m$^{-2}$) RM measurements through the ionized shell of the HII region \citep{Costa:2016}
In the case of the Rosette, we find positive RM across the region, and for IC 1805, we find negative values. If the Galactic magnetic field follows the spiral arms in a clockwise direction, then we would expect the LOS magnetic field component to be pointed towards us (positive B) for $\ell$ $>$ 180\ddeg, and pointed away from us (negative B) for $\ell$ $<$ 180\ddeg. Except for one line of sight in each nebula, we find that the polarity of the Galactic magnetic field is preserved across each nebula and is consistent with the large scale field through the arm.
In our study of the Rosette, we investigated whether the magnetic field is amplified in the shell of the nebula. We found that the model without amplification was weakly favored over the case when the magnetic field is amplified in the shell. When we applied the same model to IC 1805, however, it is difficult to conclude in favor of either model, but in both cases, the model with an enhanced magnetic field overpredicts the RM. From inspection of Figures \ref{fig:southshell-thick} and \ref{fig:southshell-thin}, it seems that the model without amplification better accounts for the magnitude of the observed RMs, but the observations do not conform to the model prediction of RM($\xi$), and the model can not account for the wide range in observed values of RM at a given $\xi$.
In the present study, we find the highest RMs for lines of sight outside the obvious shell structure, though one line of sight (W4-I13) does appear to intersect the ionized shell and it has a large RM. These lines of sight may be probing the magnetic field within the PDR. In the case of the Rosette, we found that the highest RM values were for lines of sight through the bright ionized shell. However with our work on IC 1805 and the PDR associated with it, we have briefly revisited our results in the Rosette, particularly Figure 1 from \citet{Costa:2016}. There are a few lines of sight with RM of order a few 10$^2$ rad m$^{-2}${} that appear to be outside the ionized shell. These lines of sight were included in the background estimate for the Rosette, but if the Rosette also has a PDR, then these lines of sight may actually be probing that material.
Table \ref{tab:stellarpar} lists spectral type, mass loss rate, terminal wind velocity, and calculated wind luminosity from the literature for O stars with the largest wind luminosities in both NGC 2244, which is associated with the Rosette Nebula, and OCl 352. The sum of the wind luminosities of the three main stars in OCl 352 is 3.8 $\times$ 10$^{37}$ ergs s$^{-1}$, while the corresponding number for NGC 2244 (4 stars) is 1.1 $\times$ 10$^{37}$ ergs s$^{-1}$. In addition, OCl 352 appears to have more luminous stars. As such, OCl 352 might be expected to produce a more energetic stellar bubble than NGC 2244. Our Faraday rotation measurements show no indication of this, in that the largest RMs observed are similar for the two objects. In fact, higher RMs were measured for the Rosette than for any line of sight through IC 1805. A number of factors can control the impact a star cluster has on the ISM. If some relationship exists between the total wind luminosity of a star cluster and properties of an interstellar bubble that can be measure with Faraday rotation, it will apparently require a large sample of clusters/ HII regions to reveal it.
\section{Future Research\label{sec:fut}}
In the future, we will continue our investigation of HII regions and how they modify their surroundings and the Galactic magnetic field. An immediate investigation will be centered on observations of the HII region IC 1396. This will provide a third HII region with different age, stellar content, and Galactic location. The observations are similar to those we have made of the Rosette Nebula and IC 1805. The observations of IC 1396 have been made with the VLA and are awaiting analysis. By adding more HII regions to our study, we can begin to address questions such as
\begin{enumerate}
\item Since the electron density distributions in HII regions are known from radio continuum observations, we can inquire what conditions would result in an RM $>$$>$ 10$^3$ rad m$^{-2}$~through the shell of an HII region.
\item Is it a general property of HII regions and stellar bubbles that the polarity of the Galactic magnetic field is preserved within the region? The answer to this question has implications for the amplitude of MHD turbulence in the ISM on scales of the order of the HII regions, $\sim 10 - 30$ pc.
\item Do PDRs around other nebulae produce high RMs? What is the magnitude of the RM due to the PDR relative to that of the shell of an HII region?
\end{enumerate}
In addition to increasing the number of HII regions, understanding Faraday complexity and how to interpret the associated RM measurements is important to studies of Galactic magnetic fields, particularly with large polarization surveys like the VLA Sky Survey (VLASS){} and Polarisation Sky Survey of the Universe's Magnetism (POSSUM){} with the Australian Square Kilometre Array Pathfinder (ASKAP){} in the near future.
\section{Summary and Conclusions\label{sec:sum}}
\begin{enumerate}
\item We performed polarimetric observations using the VLA for 27 lines of sight through or near the shell of the HII region and stellar bubble associated with the OB association OCl 352.
\item We obtain RM measurements for 20 sources using two methods. The first is through the traditional least-squares fit to $\chi$($\lambda^{2}$), and the second is using RM Synthesis. Including components that are resolved, we report 27 RM values, and we find good agreement between the two methods. We find the same sign of the RM across the entire region with the exception of one source, W4-I18. We estimate a background RM due to the general ISM of --145 rad m$^{-2}${} in this part of the Galactic plane.
We measure an excess of RM of $\sim$~+600 to --800 rad m$^{-2}$~due to W4.
\item Only one line of sight has a positive RM value, W4-I18. It has a RM of +501 $\pm$ 33 rad m$^{-2}$, and it is located 5.6 arcminutes from the center of OCl 352. This line of sight may be probing the material close to the massive stars.
The orientation of the line of sight component of the magnetic field is directed towards the observer, whereas in the rest of the region, the magnetic field is directed away.
\item We find that some of the lines of sight with the largest RM values occur just outside the obvious ionized shell of IC 1805 and are potentially probing the magnetic field in the PDR. The lines of sight through the cavity of the bubble have lower RM values than those through the shell. In the W4 Superbubble, which is north of OCl 352, we find RM values consistent with the background RM.
\item We discuss two shell models to reproduce the magnitude of the RM and its dependency on distance from the center of the star cluster. We employed the first of these models in \citet{Savage:2013} and \citet{Costa:2016}, and it is based on the \citet{Weaver:1977} solution for a stellar bubble, which includes a shock expanding into an ambient medium. The second model uses magnetic flux conservation to describe how the magnetic field is modified in the shell and consists of a simplified analytic approximation to the results presented by \citet{Ferriere:1991}. Neither of these simplified models satisfactorily accounts for the dependence of RM on spatial location within the shell, although the Whiting model without field amplification (X = 1) and the simplified Ferri\`ere model approximately reproduce the magnitude of the largest RMs. However, both models predict a single-valued dependence of RM on $\xi$, the separation of the line of sight from the center of the nebula, whereas the observations show a large range of RM for sources with similar values of $\xi$.
\item Because we have independent information on the electron density from radio continuum observations of both IC 1805 and the Rosette Nebula, our observations can limit the magnitude of the magnetic field in the HII regions. Our RM measurements indicate that the field does not greatly exceed the value in the general ISM.
\item We compare our results from the current study of IC 1805 and our previous study of the Rosette Nebula. Notably, we find the same order of magnitude for the RM for the two nebulae, but the sign of the RM in each region is opposite. Since IC 1805 and the Rosette are at different Galactic longitudes and on either side of $b$ = 180\ddeg, the sign difference between the two nebula is consistent with a Galactic magnetic field that follows the spiral arm structure in a clock-wise direction, as suggested in models \citep{vanEck:2011}.
\end{enumerate}
\acknowledgments
This research was partially supported at the University of Iowa by grants AST09-07911 and ATM09-56901 from the National Science Foundation. This publication makes use of data products from the Wide-field Infrared Survey Explorer \citep{2010AJ....140.1868W}, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. Additionally, the research presented in this paper uses data from the Canadian Galactic Plane Survey, a Canadian project with international partners, supported by the Natural Sciences and Engineering Research Council. This research also uses the Python packages Astropy, a community-developed core Python package for Astronomy \citep{2013A&A...558A..33A} and NumPy \citep{van2011numpy}. Finally, we thank the referee of this paper for a helpful and collegial review.
\clearpage
\newpage
|
{
"timestamp": "2018-09-24T02:12:23",
"yymm": "1803",
"arxiv_id": "1803.02878",
"language": "en",
"url": "https://arxiv.org/abs/1803.02878"
}
|
\section{Introduction}
\label{sec:Introduction}
In natural language processing (NLP), access to data is critical in the development of high-quality tools. Indeed, \citet{halevy2009unreasonable} remarked on the ``unreasonable effectiveness of data'' insofar as simple models using sufficient data often outperform more complex or clever models using less data. This realization presents a unique challenge within the clinical domain.
Electronic Health Record (EHR) notes are an important source of information for improving our current understanding of patients~\citep{ghassemi2014unfolding}. However, text typically contains sensitive protected health information (PHI) and consequently cannot be shared easily among researchers due to legal limitations established to ensure patient privacy. These restrictions have led to NLP systems in the clinical domain that often lag behind those in the general domain, even for tasks that are essentially considered solved~\citep{boag2015cliner}. Improving clinical NLP, while retaining patient privacy, requires de-identified clinical notes, those from which PHI has been redacted.
Manual approaches to de-identification are unsuitable as modern EHRs are rapidly growing. Automatic solutions benefit from scalability, but existing systems are hamstrung by a set of mutually dependent circumstances: developing de-identification systems requires access to notes, but these very same notes cannot be made available prior to de-identification. With this in mind \citet{uzuner2007evaluating} created the 2006 Informatics for Integrating Biology and the Bedside (i2b2) challenge for automatic de-identification of clinical records. The shared task oversaw the creation of a corpus of $889$ discharge summaries in which PHI was annotated, removed, and replaced with out-of-vocabulary PHI surrogates.
The 2014 i2b2 challenge revisited the task of de-identification, providing the largest known corpus of $1304$ longitudinal records for $296$ patients~\citep{stubbs2015annotating}.
While exceptional in quality, the i2b2 corpora together provide just over $2000$ records. \citet{ferrandez2012evaluating} note that access to sufficient data for machine learning approaches stands to provide improved precision relative to prior systems based on pattern matching and rules. Indeed, recent work by \citet{dernoncourt2016identification} demonstrates the potential of systems with access to substantially more data. Using this motivation, the present work makes the following two contributions:
\begin{enumerate}
\item Provide first steps toward a corpus of clinical notes that have been synthetically-identified, meaning they have been de-identified and PHI has been replaced with reasonable surrogates.
\item Evaluate a baseline de-identification system against this data in order to understand advantages and shortcomings of this approach.
\end{enumerate}
\begin{comment}
http://dspace.mit.edu/handle/1721.1/28460
The creation of systems for assembling and analyzing medical data is currently one of the major factors in advancing the speed of medical research. To ensure patient privacy, legal limitations have been placed on these systems. The Health Insurance Portability and Accountability Act requires that certain types potential identifiers be removed from the data before it can be shared freely. The process of removing the identifiers is called de-identification. The purpose of this project is to create a de-identification filter for the MIMIC database, a system that retrieves and organizes data from the intensive care unit at the Beth Israel Deaconess Medical Center.
\end{comment}
\begin{comment}
(From Franck and Jenny's RNN for De-id paper)
Patient notes in electronic health records (EHRs) may contain critical information for medical
investigations. However, the vast majority of medical investigators can only access de-identified notes, in
order to protect the confidentiality of patients. In the United States, the Health Insurance Portability and
Accountability Act (HIPAA) defines 18 types of protected health information (PHI) that needs to be removed
to de-identify patient notes. Manual de-identification is impracical given the size of EHR databases,
the limited number of researchers with access to the non-de-identified notes, and the frequent mistakes of
human annotators. A reliable automated de-identification system would consequently be of high value.
\end{comment}
\begin{comment}
http://francky.me/doc/2016deid.pdf
Medical investigations may
greatly benefit from the resulting increasingly large
EHR datasets. One of the key components of EHRs
is patient notes: the information they contain can
be critical for a medical investigation because much
information present in texts cannot be found in the
other elements of the EHR. However, before patient
notes can be shared with medical investigators, some
types of information, referred to as protected health
information (PHI), must be removed in order to preserve
patient confidentiality. In the United States,
the Health Insurance Portability and Accountability
Act (HIPAA) (Office for Civil Rights, 2002) de-
fines 18 different types of PHI, ranging from patient
names to phone numbers. Table 1 presents the exhaustive
list of PHI types as defined by HIPAA.
========================
MOTIVATION: Basically, all of their arguments for why DeID should be automated are the same reasons for why we need a big DeID dataset.
THE REAL QUESTION: How much is *too much** to copy?
========================
Deidentification
can be either manual or automated.
Manual de-identification means that the PHI are labeled
by human annotators. There are three main
shortcomings of this approach. First, only a restricted
set of individuals is allowed to access the
identified patient notes, thus the task cannot be
crowdsourced. Second, humans are prone to mistakes.
(Neamatullah et al., 2008) asked 14 clinicians
to detect PHI in approximately 130 patient notes: the
results of the manual de-identification varied from
clinician to clinician, with recall ranging from 0.63
to 0.94. (Douglass et al., 2005; Douglas et al., 2004)
reported that annotators were paid US$50 per hour
and read 20,000 words per hour at best. As a matter
of comparison, the MIMIC dataset (Goldberger et
al., 2000; Saeed et al., 2011), which contains data
from 50,000 intensive care unit (ICU) stays, consists
of 100 million words. This would require 5,000
hours of annotation, which would cost US$250,000
at the same pay rate. Given the annotators’ spotty
performance, each patient note would have to be annotated
by at least two different annotators, so it
would cost at least US$500,000 to de-identify the
notes in the MIMIC dataset.
Common statistical
methods include decision trees (Szarvas et
al., 2006), log-linear models, support vector machines
(Guo et al., 2006; Uzuner et al., 2008; Hara,
2006), and conditional random fields (Aberdeen et
al., 2010), the latter being employed in most of the
state-of-the-art systems. For a thorough review of
existing systems, see (Meystre et al., 2010; Stubbs
et al., 2015). All these methods share two downsides:
they require a decent sized labeled dataset
and much feature engineering. As with rules, quality
features are challenging and time-consuming to
develop.
two different datasets for patient
notes, the i2b2 2014 challenge dataset and the
MIMIC dataset.
===================================
I need to figure out what is wrong with these existing datasets?
(1) too private (access is VERY tight)?
(2) too small (can't train nice models)?
===================================
i2b2 2014 and MIMIC de-identification datasets. The
i2b2 2014 dataset was released as part of the 2014
i2b2/UTHealth shared task Track 1 (Stubbs et al.,
2015). It is the largest publicly available dataset
for de-identification. Ten teams participated in this
shared task, and 22 systems were submitted. As a
result, we used the i2b2 2014 dataset to compare our
models against state-of-the-art systems.
The MIMIC de-identification dataset was created
for this work as follows. The MIMIC-III
dataset (Johnson et al., 2016; Goldberger et al.,
2000; Saeed et al., 2011) contains data for 61,532
ICU stays over 58,976 hospital admissions for
46,520 patients, including 2 million patient notes.
In order to make the notes publicly available, a rulebased
de-identification system (Douglass, 2005;
Douglass et al., 2005; Douglas et al., 2004) was
written for the specific purpose of de-identifying patient
notes in MIMIC, leveraging dataset-specific information
such as the list of patient names or addresses.
The system favors recall over precision:
there are virtually no false negatives, while there are
numerous false positives. To create the gold standard
MIMIC de-identification dataset, we selected
1,635 discharge summaries, each belonging to a different
patient, containing a total of 60.7k PHI instances.
We then annotated the PHI instances detected
by the rule-based system as true positives or
false positives. We found that 1
detected by the rule-based system were false
positives.
\end{comment}
\section{Methods}
\label{sec:Methods}
\subsection{Synthetic-Identification}
MIMIC-III is an publicly available dataset developed by the MIT Lab for Computational Physiology, containing data for $61,532$ ICU stays for $46,520$ patients~\citep{mimic3}. MIMIC-III contains over 2 million de-identified clinical notes with over 12.5 million instances of PHI among nearly 500 million tokens as shown in Table~\ref{tab:mimic3notes}. To make these notes publicly available, \citet{neamatullah2008automated} developed a rule-based system tailored to the notes that appear in MIMIC.
Synthetically-identified notes were created by substituting reasonable surrogate values for annotated PHI. E.g., replacing ``[**Patient Name**] visited [**Hospital**]'' with ``Mary Smith visited MGH.'' Though patient identity coference was not performed for this initial work, we hope to incorporate it into the notes by reverse engineering the naming conventions from the original de-identification script. Names were chosen proportional to their frequency from 2005 U.S. Census~\footnote{Most Common First and Last Names in the U.S.: http://names.mongabay.com/} and hospitals were chosen uniformly from a list of U.S. hospitals~\footnote{List of Hospitals in the U.S.: en.wikipedia.org/wiki/Lists\_of\_hospitals\_in\_the\_United\_States}.
While many types of PHI exist in MIMIC, patient names and hospitals are abundant; therefore, initial efforts have been directed toward these two categories -- the focus of our subsequent experiments.
\begin{table}[t]
\caption{Distribution of de-identified protected health information (PHI) in MIMIC-III v1.4 notes by category. Particularly noteworthy are the \~12.5 million instances of PHI among \~500 million tokens.}
\label{tab:mimic3notes}
\centering
\begin{tabular}{r|llr|llr}
\toprule
Category & Notes & \multicolumn{2}{c}{Contain PHI} & Tokens & \multicolumn{2}{c}{PHI Instances} \\
\midrule
Case Management & 967 & 954 & (98.65\%) & 131806 & 9860 & (7.48\%)\\
Consult & 98 & 98 & (100\%) & 71453 & 1843 & (2.58\%)\\
Discharge summary & 59652 & 59651 & (99.99\%) & 80986971 & 2632527 & (3.25\%)\\
ECG & 209051 & 133146 & (63.69\%) & 5856486 & 135048 & (2.31\%)\\
Echo & 45794 & 45794 & (100\%) & 14817189 & 127233 & (0.86\%)\\
General & 8301 & 5200 & (62.64\%) & 1688905 & 36923 & (2.19\%)\\
Nursing & 223556 & 188691 & (84.40\%) & 56107626 & 1048996 & (1.87\%)\\
Nursing/other & 822497 & 561187 & (68.23\%) & 104063367 & 1718441 & (1.65\%)\\
Nutrition & 9418 & 9196 & (97.64\%) & 3068351 & 204730 & (6.67\%)\\
Pharmacy & 103 & 96 & (93.20\%) & 34466 & 1100 & (3.19\%)\\
Physician & 141624 & 141047 & (99.59\%) & 115484159 & 3475738 & (3.01\%)\\
Radiology & 522279 & 522278 & (99.99\%) & 102460089 & 3097379 & (3.02\%)\\
Rehab Services & 5431 & 5010 & (92.24\%) & 2125724 & 52955 & (2.49\%)\\
Respiratory & 31739 & 10395 & (32.75\%) & 4717416 & 14662 & (0.31\%)\\
Social Work & 2670 & 2609 & (97.72\%) & 779550 & 38691 & (4.96\%)\\
\midrule
{\bf Total} & 2083180 & 1685352 & (80.90\%) & 492393558 & 12596126 & (2.56\%) \\
\bottomrule
\end{tabular}
\end{table}
\subsection{De-Identification}
De-identification was performed with a Conditional Random Field (CRF), a statistical modeling method which has proven to be effective for the de-identification problem~\citep{ferrandez2012evaluating,uzuner2007evaluating}. The system was trained to identify a variety of PHI tags, including: names, hospitals, locations, dates, and identifying numbers (e.g. SSN).
For each word, $w_i$, the CRF model made use of the following features:
\begin{enumerate}
\item Presence/Absence of each word in the training vocabulary (one hot where only $w_i$ is on)
\item Presence/Absence of previous three words (only $w_{i-1}$, $w_{i-2}$, and $w_{i-3}$ are on)
\item Presence/Absence of next three words (only $w_{i+1}$, $w_{i+2}$, and $w_{i+3}$ are on)
\item Whether $w_i$ occurred in lists of common male names, female names, surnames, or hospitals (4 features in total)
\end{enumerate}
Rather than padding the beginning and end of sentences with special START/STOP symbols for those features which inspect previous or next tokens, the features were instead generated by reading from adjacent lines in the note (i.e., only the beginning and end of each note was padded rather than each statement). This was done because the free-form structure of the various notes--including alternating prose and nonprose unpredictably, insertion of line-breaks midway through prose, and variety of note types--are difficult to parse without special consideration.
Instead, we provide the model with the ability to learn these relationships automatically.
\section{Results}
\label{sec:Results}
Evaluation was performed at the token-level, so that multi-token PHI instances are treated separately during evaluation. The motivation for this is that it gives a good sense for what percentage of PHI is being detected: {\it exact span} metrics cannot differentiate between a decent method that gets most-but-not-all tokens in a span and {\it inexact span} metrics might give too much partial credit to systems that are not identifying all relevant PHI. We felt that token-level precision, recall, and F1 provided the most interpretable and reasonable metric.
Unsurprisingly, our experiments show a simple trend: the more data, the better. We can see from Table~\ref{tab:results-all} that as we increase the number of files used for training, we see improvements in all scores for both patient name and hospital retrieval. In particular, we can see that although precision is already very good with a small training data, the recall continues to show sizable improvements as we train on more data, as indicated by Figure~\ref{fig:n-patient-recall}.
These results demonstrate the benefits that a large-scale dataset can offer, especially for complex de-identification models involving deep learning approaches
\begin{figure}
\centering
\caption{The effect that the number of training files has on HOSPITAL recall.}
\includegraphics[width=70mm]{img-recall-n.png}
\label{fig:n-patient-recall}
\end{figure}
\begin{table}[ht]
\caption{The results for patient name and hospital PHI identification as we add more training data.}
\begin{center}
\begin{tabular}{ccc}
\footnotesize
\begin{minipage}{.5\linewidth}
\begin{tabular}{|l|l|l|l|}
\hline
\multicolumn{4}{|c|}{PATIENT\_NAME}\\ \hline
\# training files & precision & recall & f1 \\ \hline
100 & 0.89 & 0.78 & 0.83 \\ \hline
200 & 0.90 & 0.87 & 0.88 \\ \hline
500 & 0.91 & 0.93 & 0.92 \\ \hline
1000 & 0.92 & 0.96 & 0.94 \\ \hline
\hline
\end{tabular}
\end{minipage} &
\footnotesize
\begin{minipage}{.45\linewidth}
\begin{tabular}{|l|l|l|l|}
\hline
\multicolumn{4}{|c|}{HOSPITAL}\\ \hline
\# training files & precision & recall & f1 \\ \hline
100 & 0.92 & 0.72 & 0.81 \\ \hline
200 & 0.93 & 0.79 & 0.86 \\ \hline
500 & 0.94 & 0.87 & 0.91 \\ \hline
1000 & 0.95 & 0.89 & 0.92 \\ \hline
\hline
\end{tabular}
\end{minipage} &
\end{tabular}
\end{center}
\label{tab:results-all}
\end{table}
\section{Conclusions}
\label{sec:Conclusions}
While this work represents an important first step toward the creation of a large corpus of synthetically-identified clinical notes, it suffers from several limitations. Notably, \citet{dernoncourt2016identification} note that the system created by \citet{neamatullah2008automated} favors recall over precision as it introduces virtually no false negatives, while there are numerous false positives accounting for up to 15\% of the PHI instances detected. This suggests that creation of the synthetically-identified dataset directly from the identified data is necessary so as not to unnecessarily introduce meaningless surrogate data.
Further, we recognize that the using surrogate data may constrain the natural variety of names, places, and other entities that appear in notes (e.g., through misspellings); thus, making the task slightly easier. Again, it is the hope that with such a large volume of PHI annotations this will become decreasingly important relative to the potential improvement in tools possible.
Nevertheless, this work marks a first attempt to providing the community with a synthetically-identified corpus of notes. The authors hope that this may be subsequently improved through collaboration with the MIT Laboratory of Computational Physiology to provide a higher-quality version of synthetically-identified notes. While the issue of co-reference makes this problem more difficult that the usual task of de-identification (which most closely resembles named entity recognition), providing synthetically-identified notes makes the identification of errors less conspicuous to the human reader. It may therefore be a preferable means of distributing notes as well.
\subsubsection*{Acknowledgments}
This research was funded in part by the Intel Science and Technology Center for Big Data, the National Library of Medicine Biomedical Informatics Research Training grant (NIH/NLM 2T15 LM007092-22).
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 1122374. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
\bibliographystyle{abbrvnat}
|
{
"timestamp": "2018-03-08T02:10:10",
"yymm": "1803",
"arxiv_id": "1803.02728",
"language": "en",
"url": "https://arxiv.org/abs/1803.02728"
}
|
\section{Introduction}
The two-dimensional electron system (2DES) formed at the interface of
the band insulators LaAlO$_3$ (LAO) and SrTiO$_3$ (STO) displays many
intriguing features such as superconductivity, spin-orbit interaction (SOI)
and multiple quantum criticality \cite{1,2,3}, and has thus made LAO/STO a
prototypical system for studying low-dimensional strongly correlated electron
systems. Magnetic properties are reported alike \cite{4}, however, believed to
arise rather from extrinsic sources like oxygen vacancies and strain. \\
In (001) oriented LAO/STO the sheet carrier concentration $n_s$ can be tuned by
electric field gating through a Lifshitz transition \cite{5} occurring at a
critical sheet carrier concentration $n_c \approx 1.7\times10^{13} $ cm$^{-2}$,
where itinerant electrons change from populating only Ti derived 3d
$t_{2g}$ orbitals with $d_{xy}$ symmetry to occupying also the
$d_{xz}$, $d_{yz}$ orbitals. These bands result in a highly elliptical
Fermi
surface oriented along crystalline directions and may give reason for the
observation of crystalline anisotropic electronic properties. In addition,
localized magnetic moments, pinned to specific $d_{xy}$ orbitals may lead to
crystalline
anisotropy as well and may complicate anisotropic electronic transport. The
coexistence of localized charge carriers close to the interface and itinerant
$d$ electrons may lead to fascinating phenomena such as non-isotropic
magnetotransport or magnetic exchange. However, it is not clear whether
interaction between these localized magnetic moments and mobile charge
carriers really happens. \\
The SOI in (001) LAO/STO results in a non-crystalline two-fold anisotropic
in-plane magnetoresistance ($AM\hspace{-0.5mm}R$) \cite{6}. Interestingly,
for $n_s > n_c$ sometimes
a more complex $AM\hspace{-0.5mm}R$ with a four-fold crystalline anisotropy
is reported which
is discussed in terms of a tunable coupling between itinerant
electrons and electrons localized in
$d_{xy}$ orbitals at Ti vacancies\cite{7}. However, the appearance of a
crystalline $AM\hspace{-0.5mm}R$ with increasing $n_s$ is not always evident
and rises question
about its microscopic origin. More recently, a giant crystalline
$AM\hspace{-0.5mm}R$ of up
to 100\% was reported in (110) oriented LAO/STO \cite{8}. Here, $n_s$ was
about
$2\times10^{13}$ cm$^{-2}$. However, the $AM\hspace{-0.5mm}R$ was
supposed to be related to strong
anisotropic spin-orbit field and the anisotropic band structure of (110)
LAO/STO.\\
With respect to both, namely fundamental aspects such as the possible
simultaneous
appearance of magnetism and superconductivity and applications in the field
of spintronics, a more fundamental knowledge about the origin of anisotropic
magnetotransport is highly desired. Measurements of the $AM\hspace{-0.5mm}R$
in a rotating
in-plane magnetic field are well suited to probe crystalline anisotropy and
symmetry of a 2DES and are a promising tool to elucidate magnetic properties
because of its high sensitivity towards spin-texture and spin-orbit
interaction \cite{9}. \\
In order to investigate the microscopic origin of the anisotropic electronic
properties of the 2DES of STO-based heterostructures we studied in detail
the electronic transport of the 2DES formed at the interface of
spinel-type Al$_2$O$_{3-\delta}$ and (110) oriented STO (AO/STO). The presence
of oxygen vacancies \cite{10} promoting localized $d_{xy}$ electrons in
combination
with the anisotropic band structure of (110) STO surface \cite{11} makes (110)
AO/STO very suitable for these experiments. The heterostructures were produced
by standard pulsed laser deposition, and characterization of the electronic
transport was done by sheet resistance measurements. Band structure
calculations
were carried out using linear combination of atomic orbitals (LCAO)
approximation to model band structure and Fermi surface properties of
(110) AO/STO. $AM\hspace{-0.5mm}R$ was deduced using
semi-classical Boltzmann theory. Surprisingly, despite the anisotropy of the
electronic band structure and SOI, and the presumably
large content of oxygen vacancies as compared to LAO/STO, we did not observe
indications for a crystalline $AM\hspace{-0.5mm}R$ in (110) AO/STO. The
$AM\hspace{-0.5mm}R$ displays two-fold non-crystalline anisotropic behavior.
Contributions to the electronic transport from the different Fermi surface
sheets as well as the anisotropy of the Fermi surfaces itself are sensitively
affected by $n_s$.
\section{Experimental}
Sample preparation has been carried out by depositing Al$_2$O$_{3-\delta}$ films
onto $(110)$ oriented STO substrates
with a thickness of about 15 nm at a substrate temperature of
$T_s = 250^{\circ}C$ by pulsed laser deposition \cite{12}. In order to achieve
an atomically flat, single-type terminated substrate surface, the substrates
are annealed at $T = 950^{\circ}C$ for 5h in flowing oxygen.
The $(110)$ STO
surface can be terminated by a SrTiO or an oxygen layer, see
Fig. \ref{fig1} (a), where
the cation composition at the interface should be always the same in case of
single-type termination. Annealing results in a stepped surface topography
with a step height of about $2.7$\AA $\;$and a step width of 80 nm, see Fig.
\ref{fig1} (b).
Oxygen partial pressure during Al$_2$O$_{3-\delta}$ deposition and
cool-down process was
$p(O_2) = 10^{-6}$ mbar. Prior to the deposition, microbridges with
a
length of $100 \mu$m and a width of $20 \mu$m in Hall bar geometry have been
patterned along specific crystallographic directions using a CeO$_2$ hard mask
technique \cite{13}, see Fig. \ref{fig1} (c).
The microbridges are labeled from A to E, with angle
$\varphi = 0^{\circ}, 22.5^{\circ}, 45^{\circ}, 67.5^{\circ}$, and $90^{\circ}$
towards the $[1\bar{1}0]$ direction, i.e.,
A and E parallel to $[1\bar{1}0]$ and $[001]$ direction, respectively.\\
\begin{figure}
\includegraphics[width=0.9\columnwidth]{Figure_1.eps}
\caption{\label{fig1}
(a) Schematic of the crystal structure of STO. In case of $(110)$
orientation, the surface can be terminated with a SrTiO or oxygen layer.
The spacing of the cation layers is $a/\sqrt{2}$, where $a = 3.905$\AA$\;$ is
the cubic lattice parameter
of STO. Crystallographic directions and atom labels are indicated. (b) Surface
topography before Al$_2$O$_{3-\delta}$ deposition characterized by atomic force
microscopy. The image was taken on microbridge A. (c) Optical micrograph of a
patterned sample. Sharp contrast between AO/CeO$_2$ (dark) and AO/STO (bright)
enables identifying microbridges labeled alphabetically from A$-$E. }
\end{figure}
The sheet resistance $R_s$ was measured using a physical property measurement
system (PPMS) from Quantum Design in the temperature and magnetic field ranges
$2$ K $\le T \le 300$K and $0 \le B \le 14$T. To avoid charge carrier
activation by
light \cite{14,15}, alternating current measurements ($I_{ac} = 3 \mu$A) were
started
not before 12 hours after loading the samples to the PPMS. The
magnetoresistance, $M\hspace{-0.5mm}R = [R_s(B) - R_s(0)]/R_s(0)$, and the
$AM\hspace{-0.5mm}R =[R_s(B_{ip},\phi) - R_s(B_{ip},0)]/R_s(B_{ip},0)$,
have been measured with magnetic
field normal ($B$) and parallel ($B_{ip}$) to the interface, respectively. For
measuring
$AM\hspace{-0.5mm}R$ with rotating in-plane magnetic field $B_{ip}(\phi)$, a
sample rotator was used.
The angle $\phi$ between $B_{ip}$ and $[001]$-direction was varied from
$0^{\circ} - 360^{\circ}$.
Special care has been taken to minimize sample wobbling in the apparatus.
Residual
tilts ($1^{\circ} - 2^{\circ}$) of the surface normal with respect to the rotation
axis which
produces a perpendicular field component oscillating in sync with $\phi$ could
be identified by comparison of $R_s(B_{ip},\phi)$ for different microbridges and
could therefore be corrected properly.
\section{Results and Discussion}
\subsection{Temperature dependence of the anisotropic electronic transport}
The anisotropic electronic band structure of the 2DES found in $(110)$ oriented
LAO/STO heterostructures \cite{8} and at the reconstructed surface of $(110)$
oriented STO \cite{11} obviously lead to anisotropic electronic transport
\cite{16,17}. The lowest electronic subbands along the $[1\bar{1}0]$
direction (along $\Gamma-M$) display much weaker dispersion and smaller
band-width compared to
the $[001]$ direction (along $\Gamma-Z$) which typically results in larger
resistance
for current $I$ direction along the $[1\bar{1}0]$ direction \cite{17,8}. The
electronic transport in $(110)$ AO/STO displays distinct anisotropy as well.
The $T$-dependence of the sheet resistance $R_s$ along different
crystallographic
directions is shown in Fig. \ref{fig2} (a). For all the microbridges,
$R_s$ decreases
with deceasing $T$ nearly $\propto T^2$ down to about 100 K and shows a shallow
minimum
around 20 K. The $T$-dependence is very similar to that observed in $(001)$
AO/STO
and is likely explained by strong renormalization due to electron-phonon
interaction and impurity scattering \cite{18,19}. The resistivity ratio
between 300 K and 10 K amounts to about 20, which is nearly the same as
that of $(001)$ AO/STO. $R_s$ steadily decreases from A
($I \parallel [1\bar{1}0]$) to E ($I \parallel [001]$) with
increasing $\varphi$ at constant $T$
throughout the complete $T$-range. Obviously, anisotropic transport is not only
restricted to low temperatures $T < 10 K$, where usually impurity
scattering dominates $R_s$. Moreover, the anisotropy between A and E,
$[R_s(A) - R_s(E)]/R_s(E)$ is largest at $T = 300$K amounting to 47\% and
decreases with decreasing $T$ to 29\% at $T = 5$K. This rather small
$T$-dependence
indicates that the intrinsic anisotropic electronic band structure is very
likely the dominant source for the anisotropic transport. In contrast,
anisotropic transport in $(001)$ AO/STO is extrinsic in nature and
is found only at low $T$ where it is
caused mainly by anisotropic impurity scattering
due to an inhomogeneous distribution
of $<110> $lattice dislocations \cite{20}. With respect to these results,
anisotropy in $(110)$ AO/STO may be diminished at low $T$ and intrinsic
anisotropy would be even larger. \\
In order to extract sheet carrier density $n_s$ and mobility $\mu$,
Hall measurements have been carried out in a magnetic field
$-14$T$\le B \le 14$T
applied normal to the interface for $2$K$\le T \le 300$K. For $T < 30$K, the
Hall resistance $R_{xy}$ becomes slightly nonlinear, indicating multi-type
carrier transport. However, $n_s$ which we determined from the asymptotic
value of $R_{xy}$ at high fields, i.e., the total $n_s$, usually deviates by
less than 10\% from $n_s$ extracted from $R_{xy}$ in the limit of $B = 0$.
In Fig. \ref{fig2} (b) the total $n_s$ and the Hall mobility, calculated by
$\mu = (R_s(B = 0)\times n_s\times e)^{-1}$, where $e$ is the elementary charge,
are shown as functions of $T$. $n_s$ decreases with decreasing $T$ from about
$1.3\times 10^{14} $ cm$^{-2}$ at $T = 300$K to $2.5\times 10^{13}$ cm$^{-2}$ at
$T = 5$K and is well comparable to that of $(110)$ LAO/STO \cite{10}. \\
In contrast to the $T$-dependence of $n_s$, $\mu$ increases from about
$2.5$ cm$^2/($Vs$)$ with decreasing $T$ to $150$ cm$^2/($Vs$)$. The
$T$-dependence of
$n_s$ and $\mu$ is well comparable to that observed in 2DES of $(001)$ STO
based heterostructures \cite{21,22,20}. As expected from $R_s$ the maximum
anisotropy of $n_s$ and $\mu$ is observed at $T = 300$K amounting to 16\% and
65\%, respectively, and decreases to 2\% and 34\% at $T = 5$K. Therefore, the
anisotropy of $R_s$ at low $T$ is mainly caused by the anisotropy of $\mu$,
whereas $n_s$
for the different microbridges A$-$E are roughly the same. The superior role
of $\mu$ with respect to electronic anisotropy is demonstrated in
Fig. \ref{fig2} (c)
where $n_s$ and $\mu$ are plotted for A$-$E at $T$ = 5K. $n_s$ differs only
little
for the different microbridges. In contrast, $\mu$ steadily increases from A
to E with increasing $\varphi$ and shows the highest mobility for bridge E,
i.e., along the $[001]$ direction. The results are reasonable with respect
to the anisotropic band structure and Fermi surface of $(110)$ AO/STO, which
will be discussed in more detail in III. C. \\
\begin{figure}
\includegraphics[width=\columnwidth]{Figure_2.eps}
\caption{
(a) Sheet resistance $R_s$ versus $T$ for microbridges A $-$ E (from top
to bottom) with an angle
$\varphi = 0^{\circ}, 22.5^{\circ}, 45^{\circ}, 67.5^{\circ}$, and $90^{\circ}$
towards the $[1\bar{1}0]$ direction, i.e., $A \parallel [1\bar{1}0]$ and
$E \parallel [001]$. (b) Sheet carrier density $n_s$, left scale, and Hall
mobility $\mu$, right scale, versus $T$ for A $-$ E. (c) $n_s$, left scale,
and $\mu$, right scale for bridge A$-$E at $T = 5$K.}
\label{fig2}
\end{figure}
\subsection{Magnetotransport}
Measurements of the $M\hspace{-0.5mm}R$ and $AM\hspace{-0.5mm}R$ with magnetic
field direction normal or parallel
to the interface, respectively, are often used to characterize SOI in
low-dimensional electron systems. In STO-based 2DES, the Rashba-type SOI
usually leads to a weak antilocalization (WAL) of the charge carrier transport
at low $T$ \cite{2}, resulting in a logarithmic $T$-dependence of
$R_s$ \cite{23}.
However, the quantum coherence can be destroyed by applying moderate magnetic
fields leading to a distinct positive $M\hspace{-0.5mm}R$ \cite{24}.\\
For $T\ge 50 $K the $M\hspace{-0.5mm}R$ of $(110)$ AO/STO is rather small,
less than 2\%, and
displays no distinct anisotropy with respect to the crystallographic direction.
For $T < 50$K, $M\hspace{-0.5mm}R$ starts to increase with respect to
amplitude and anisotropy.
In Fig. \ref{fig3}, $M\hspace{-0.5mm}R$ is shown for the microbridges A$-$E,
for $T = 10 $K and $2$K.
For $T = 10 $K, $M\hspace{-0.5mm}R$ is positive and amounts to about 10\%.
The $B$-dependence of
$M\hspace{-0.5mm}R$ indicates orbital motion of free carriers due to the
Lorentz force, i.e.,
classical Lorentz scattering (LS) as the dominant scattering mechanism, where
$M\hspace{-0.5mm}R$ is well described by the Kohler form:
$M\hspace{-0.5mm}R \propto (1/R_0) \times
\left(\frac{B}{w}\right)^2/(1+\left(\frac{B}{w}\right)^2)$,
with the zero-field resistance $R_0$\cite{25}. Fits to
the Kohler form are shown by solid lines in Fig. \ref{fig3} (a).
$M\hspace{-0.5mm}R$ displays clear anisotropic behavior with respect to the
microbridges, showing a systematic increase from A to E. This is very likely
related to the decrease of the zero-field resistance from A to E, see
Fig. 2 (a). \\
For $T = 2 $K, an additional contribution to the positive $M\hspace{-0.5mm}R$
appears. However, significant changes to $M\hspace{-0.5mm}R$ are restricted
to the low field region, $B < 8 $T.
As mentioned above, in 2DES charge transport in the diffusive regime is well
described by the 2D WAL theory\cite{23}. The quantum corrections to the
conductivity arise from the interference of
electron waves scattered along closed paths in opposite directions. Phase
coherence is destroyed if the applied magnetic field which results in a phase
shift between the corresponding amplitudes
exceeds a critical value. An estimation for the field limit
$B^* = \hbar/(2el_m^2)$ can be deduced from the electron mean free path
$l_m=\frac{\hbar}{e}\sqrt{2\pi n_s}\mu$
of the sample. For our sample we obtain $l_m = 12$nm which results in a field
limit of about 2 T. \\
Zeeman corrections to the WAL are taken into account by the Maekawa and
Fukuyama (MF)
theory \cite{24}, which is usually used to describe the $B$-dependence of the
$M\hspace{-0.5mm}R$ in LAO/STO and AO/STO \cite{2,20}. The parameters of the
MF-expression are
the inelastic field $B_i$, the spin-orbit field $B_{so}$, and the electron
g-factor which enters into the Zeeman corrections.\\
For $B \le 2$T, $M\hspace{-0.5mm}R$ at $2$K is well described by LS and WAL.
Fits to the data, using the MF-based expression given in \cite{2} in
combination with a Kohler term, are shown in Fig. \ref{fig3} (b) and (c)
by solid lines.\\
The WAL fitting results in parameters $B_i \approx 180$mT and
$B_{so} \approx 0.6$T. Within the experimental resolution and the limited field range WAL effect appears to be nearly the same for all the microbridges.
Zeeman corrections to
$M\hspace{-0.5mm}R$ have been found to play only a minor role for the applied
magnetic fields. The magnitude of $B_i$ and $B_{so}$ are well comparable to
those found in $(001)$ AO/STO and LAO/STO,
where Rashba-type SOI has been identified as the dominant source of spin orbit
coupling. \\
In comparison to WAL, contributions from LS to $M\hspace{-0.5mm}R$ at
$2$K are rather small for
$B < 2$T. However, for $B > 8$T, where WAL can usually be neglected,
LS dominates $M\hspace{-0.5mm}R$ again.
Interestingly, in comparison to the anisotropy of $M\hspace{-0.5mm}R$ with
respect to the
microbridges for $B > 8$T and at $T = 10$K, the anisotropy of
$M\hspace{-0.5mm}R$ at 2 K, is
slightly decreased. In contrast, the anisotropy of $R_0$, $\mu$, and $n_s$
with respect
to the current direction, are well comparable for $T = 10$K and 2K, or even
slightly larger at 2K and likely do not explain that behavior.
It might be suggested, that Rashba-type SOI not only influences
$M\hspace{-0.5mm}R$ by
WAL at low magnetic fields but also at higher fields, where WAL
should be absent.\\
Anisotropic Rashba splitting was indeed observed by angle resolved
photoemission (ARPES) experiments on $(110)$ STO surfaces \cite{11} and
discussed for (110) LAO/STO
heterostructures \cite{8,16}.
The influence of SOI and Rashba effect on magnetotransport can be studied
more specific,
if the magnetic field is applied parallel to the interface where changes of
the $M\hspace{-0.5mm}R$ by WAL are negligible. \\
\begin{figure}
\includegraphics[width=0.6\columnwidth]{Figure_3.eps}
\caption{\label{fig3}
Magnetoresistance $M\hspace{-0.5mm}R$ for the different microbridges
A$-$E at $T = 10 $K
(a) and $2$K (b) versus magnetic field $B$ applied perpendicular to the
interface. (c) $M\hspace{-0.5mm}R$ at $T = 2$K in the low-field range for
$B < B^*$.
Fits to the data with respect to the Kohler form and MF-expression
are shown by solid lines, see text.}
\end{figure}
Applying the magnetic field parallel to the interface at an angle $\phi$ with
respect to the $[001]$ direction, $B_{ip}(\phi)$, results in a strong
field-induced directional anisotropy of the resistance $R_s(B,\phi)$, i.e.,
an $AM\hspace{-0.5mm}R$. Fig. \ref{fig4} (a) and (b) document
$R_s(B_{ip},\phi)$ versus $\phi$ for
different $B_{ip}$ at $T = 2$K for bridge A and E, respectively.
$R_s(B_{ip},\phi)$ shows a sinusoidal oscillating two-fold anisotropic behavior
with maxima at $\phi \approx 90^{\circ}/270^{\circ}$ and $0^{\circ}/180^{\circ}$
for A and E, respectively. Obviously, the anisotropy depends much stronger on
the angle between the direction of the bridge, i.e., the direction of
current $I$ and $B_{ip}$ than on the crystallographic direction. Maxima of
$R_s(B_{ip},\phi)$ are always observed for $B_{ip}$ parallel to the microbridge,
i.e., current flow direction. \\
Similar anisotropic behavior was also found in $(001)$ LAO/STO and AO/STO.
In the framework of the Drude-Boltzmann theory it was shown, that a
Rashba-type SOI in $(001)$ LAO/STO induces a two-fold non-crystalline
anisotropy in the magnetoconductance \cite{6}, i.e.,
$\Delta\sigma = [\sigma(B_{ip},\Theta)-\sigma(B_{ip},0) ] \propto
sin^2(\Theta)$, where the amplitude of the oscillations should scale for
moderate field strength with the square of the spin-orbit energy, i.e.,
$\Delta\sigma(\Theta=90^{\circ})/\sigma_0 \propto \Delta_{so}^2$, where
$\sigma_0 = \sigma(B_{ip},0)$ and $\Theta$ the angle between $I$ and $B_{ip}$.
Therefore, it is very likely, that the observed anisotropy of
$R_s(B_{ip},\phi)$ is caused by Rashba-type SOI alike. \\
The amplitude of the oscillations of $R_s(B_{ip},\phi)$ increases with
increasing magnetic field reaching an $AM\hspace{-0.5mm}R$ of about 1\% for A, i.e., along
the $[1\bar{1}0]$ direction and 1.4\% for E, parallel to the $[001]$ direction
for $B_{ip} = 14 $T. The different amplitudes likely indicate an anisotropic Rashba-type SOI.
Note that the $AM\hspace{-0.5mm}R$ is about one order of magnitude smaller
as compared to the $M\hspace{-0.5mm}R$.
Interestingly, $R_s$ first increases with increasing $B_{ip}$ up to about $5$T
and then decreases again. This behavior is documented in more detail in
Fig. \ref{fig4} (c) and (d) where the in-plane magnetoresistance
$M\hspace{-0.5mm}R_{ip} = [R_s(B_{ip},\Theta) - R_s(0,\Theta)]
/ R_s(0,\Theta)$ is plotted for
A and E versus $B_{ip}$ for field direction parallel ($\Theta = 0^{\circ}$) and
perpendicular ($\Theta = 90^{\circ}$) to the current $I$ direction. \\
The $M\hspace{-0.5mm}R_{ip}$ for $\Theta = 0^{\circ}$ is only slightly
larger compared to $\Theta = 90^{\circ}$. With increasing $B_{ip}$, the
$M\hspace{-0.5mm}R_{ip}$ first
increases, displaying a maximum positive magnetoresistance around $5$T. Then,
the $M\hspace{-0.5mm}R_{ip}$ decreases and even becomes negative for $B_{ip}$
above about $10$T. The negative $M\hspace{-0.5mm}R_{ip}$ at large
fields possibly results from spin-polarized bands due to Zeeman effect,
leading to a suppression of interband scattering with increasing
$B$ \cite{27}.\\
Fig. \ref{fig4} (e) shows the anisotropic magnetoconductance
$AMC= [\sigma(B_{ip},\Theta)-\sigma(B_{ip},0)]/\sigma(B_{ip},0)$ versus $\Theta$
for $I \parallel [1\bar{1}0]$ and $I \parallel [001]$ at $B_{ip} = 14 $T and
$T = 2 $K. The maxima of the magnetoconductance oscillations $AMC_{max}$ always
appear at $\Theta = 90^{\circ}$ and $270^{\circ}$, i.e., $B_{ip}$ perpendicular
to the current direction. For the $[001]$ direction $AMC_{max}$ amounts to
about 1.3\% and is distinctly larger compared to that of the $[1\bar{1}0]$
direction ($\approx 0.8\%$).
The field dependence of the amplitude $AMC_{max}$ for the two orthogonal
directions is shown in Fig. \ref{fig4} (f). Measurable magnetoconductance
appears
for $B_{ip} > 3$T and increases with field to 1.2\% and 0.9\% at $14$T for
the $[001]$ and $[1\bar{1}0]$ direction, respectively.
Rashba effect seems to increase with increasing $B_{ip}$ and to be anisotropic with respect to crystallographic direction.\\
In contrast to the non-crystalline $AM\hspace{-0.5mm}R$ of $(110)$ AO/STO
shown here, a giant crystalline $AM\hspace{-0.5mm}R$ was reported for $(110)$
LAO/STO - displaying comparable $n_s$ and $\mu$ - with resistance maxima for
$B_{ip}$ along the $[1\bar{1}0]$ direction, independent of current
direction \cite{8}. \\
\begin{figure}
\includegraphics[width=\columnwidth]{Figure_4.eps}
\caption{\label{fig4}
$AM\hspace{-0.5mm}R$ of $(110)$ AO/STO at 2 K. $R_s$ versus in-plane angle
$\phi$ for
different strengths of $B_{ip}$ $(0, 1, 3, 5, 8, 10, 12$, and $14$T$)$ for (a)
bridge A, i.e., $I$ parallel to the $[1\bar{1}0]$ direction, and (b) bridge E,
$I$ being parallel to the $[001]$ direction. The field strength $B_{ip}$ is
indicated
exemplarily. In-plane magnetoresistance $M\hspace{-0.5mm}R_{ip}$ versus
$B_{ip}$ for field
direction parallel ($\Theta = 0^{\circ}$) and perpendicular
($\Theta = 90^{\circ}$) to current flow direction for (c) bridge A and (d)
bridge E. (e) Anisotropic magnetoconductance $AM\hspace{-0.5mm}C$
versus $\Theta$, the angle
between current $I$ and $B_{ip}$ for the $[1\bar{1}0]$ and $[001]$ direction
for
$B = 14$ T and $T=2K$.
(f) The amplitude of the magnetoconductance oscillations $AMC_{max}$
versus $B_{ip}$ for the $[1\bar{1}0]$ and the $[001]$ direction. The minima
of the
magnetoconductance oscillations were found always at
$\Theta = 0^{\circ}, 180^{\circ}$, i.e.,
$B_{ip}$ parallel to the current direction.}
\end{figure}
\subsection{Theoretical modeling of the electronic band structure and
magnetotransport}
In order to obtain a better understanding of the measured electronic transport,
especially the $AM\hspace{-0.5mm}R$ behavior, we carried out
tight-binding calculations to model the electronic subbandstructure of
$(110)$ AO/STO. Details to the linear combination of atomic orbitals (LCAO)
calculations are given in the Appendix. The calculation yields the
energy bands $E_{\nu,{\bf k}}$ where $\nu$ is the band index and
${\bf k}$ the wave vector in the rectangular Brillouin zone.\\
\begin{figure}
\includegraphics[width=0.7\columnwidth]{Figure_5.eps}
\caption{\label{fig5} LCAO Band structure for the $(110)$ AO/STO
interface. (a) Band structure without spin-orbit
coupling and symmetry-breaking electric field.
All bands have two-fold spin degeneracy, the horizonzal line
is the Fermi energy for $n_e=0.4$/unit-cell. (b) Band
structure including spin-orbit
coupling and symmetry-breaking electric field, but $B=0$.
(c) Closeup of the box indicated in (b). In (b) and (c) the horizontal dashed
line is the Fermi energy for $n_e=0.05$/unit-cell.}
\end{figure}
Figure \ref{fig5} shows the band structure obtained in this way.
The topmost panel shows the band structure
in the absence of spin orbit coupling and symmetry breaking electric field,
which roughly agrees with the band structure obtained by Wang {\em et al.}
from a fit to their ARPES data\cite{11} (the reason for the deviation
is our modification of the nearest neighbor hopping $t$ for bonds
in $[001]$-direction, see the discussion in the Appendix).
Since there is no mixing between the three $t_{2g}$ orbitals
the bands can be classified according to the type of $d$-orbital
from which they are composed, whereby the bands derived from the
$d_{xz}$ and $d_{yz}$ orbitals are degenerate.
The dashed horizontal line gives the Fermi energy
for an electron density of $0.40$/unit-cell or
$1.8\times 10^{14} \;$ cm$^{-2}$. This is considerably
higher than the electron densities studied here but
corresponds roughly to the experiments by Wang {\em et al.}\cite{11}.
In the Figure one can identify
the various subbands generated by the confinement of the electrons
perpendicular to the interface. This hierarchy of subbands in fact extends to
considerably higher energies than shown in the Figure.\\
The two lower panels show the band structure for finite spin-orbit coupling
and symmetry breaking electric field, but $B=0$.
One can recognize the two different manifestations of the Rashba effect
discussed already by Zhong {\em et al.} \cite{Zhong}: the splitting of bands
near $\Gamma$ which can be either $\propto |{\bf k}|$ or $\propto |{\bf k}|^3$
(see below) and the opening of gaps. The formation of gaps
is particularly obvious at $\Gamma$ where
the lowest $d_{xy}$-derived band along $\Gamma-\bar{M}$ combines
with one of the $d_{xz}/d_{yz}$-derived bands along $\Gamma-\bar{Z}$
to form a mixed band whose minimum is shifted upward
by $\approx 20 $ meV.
The dashed horizontal line in the lower two panels gives $E_F$
for an electron density of $n_e=0.05$/unit-cell or $2.3\times 10^{13}\;$cm$^{-2}$
which is roughly appropriate for our experiment. We have verified that varying
the density in the range $0.04$/unit-cell $\le n_e \le 0.07$/unit-cell
does not have a significant influence on the magnetoresistance
discussed below. From now
on the labeling of bands is according to their energy i.e.
the lowest band is labeled 1 and so on.
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Figure_6.eps}
\caption{\label{fig6} Rashba-induced band splittings
$\Delta_{\nu,\nu'}({\bf k})$ near $\Gamma$ along the two symmetry lines
of the Brillouin zone.}
\end{figure}
Figure \ref{fig6} shows the differences
$\Delta_{\nu,\nu'}({\bf k})= E_{\nu,{\bf k}}-E_{\nu',{\bf k}}$
and demonstrates
the power-law behaviour of the Rashba-induced band splitting
for small $|{\bf k}|$. Thereby the splitting along $[1\bar{1}0]$
is linear, i.e. $\Delta_{\nu,\nu'}({\bf k})=C_{\nu,\nu'}\frac{ka}{\pi}$
with $C_{2,1}=60.8$ meV and $C_{4,3}=23.4$ meV, whereas
along $[001]$ the splitting between
bands $3$ and $4$ still has this form with $C_{4,3}=42.4$ meV
whereas the splitting between bands $1$ and $2$ now is cubic,
$\Delta_{2,1}(k)=2720$meV$\left(\frac{ka}{\pi}\right)^3$.
This highlights the anisotropy of the Rashba-effect at the
$(110)$ AO/STO interface.\\
\begin{figure}
\includegraphics[width=\columnwidth]{Figure_7.eps}
\caption{\label{fig7} The four Fermi sheets
for $n_e=0.05/$unit-cell in zero magnetic field and a magnetic field of
$14$ T.
The field direction is $[001]$ for the four topmost panels
and $[1\bar{1}0]$ for the four bottom panels (see arrows).}
\end{figure}
Figure \ref{fig7} compares the Fermi surfaces for $n_e=0.05$/unit
cell in zero magnetic field and a field of $14$ Tesla.
All panels show the square
$[-\frac{\pi}{4}:\frac{\pi}{4}]\otimes[-\frac{\pi}{4}:\frac{\pi}{4}]$,
the field direction is along $[001]$ (along $[1\bar{1}0]$)
in the top four (bottom four) panels.
In the absence of SOC and electric field the Fermi surface
would consist of two elliptical sheets centered at $\Gamma$,
each of them two-fold (spin-)degenerate. The
ellipse derived from the $d_{xz}/d_{yz}$ orbitals is elongated
along the $[1\bar{1}0]$ (or $\Gamma-\bar{M}$) direction, whereas the
ellipse derived from the $d_{xy}$ orbitals is elongated
along the $[001]$ (or $\Gamma-\bar{Z}$) direction. The Rashba effect
splits and mixes these bands and creates the more
complicated $4$-sheet Fermi surface in Figure \ref{fig7}.\\
Switching on the magnetic field results in an area change of
the various Fermi surface sheets as well as
a displacement perpendicular to the field direction whereby
pairs of bands are shifted in opposite direction, namely bands $1$ and $2$
and bands $3$ and $4$. This displacement is considerably
more pronounced for the magnetic field in $[001]$-direction
and barely visible for magnetic field in $[1\bar{1}0]$-direction.
Qualitatively this behaviour can be derived from
the simplified single-band model\cite{Raimondi}:
\begin{eqnarray}
H=\frac{p^2}{2m} + \alpha \;{\bm \tau}\cdot ({\bm p}\times{\bm e}_z) -
\omega_s {\bm \tau}\cdot{\bm B}.
\label{raiham}
\end{eqnarray}
Here $\alpha$ is the strength of the Rashba coupling,
$\omega_s=\mu_BB$ and ${\bm \tau}$ the vector
of Pauli matrices.
The magnetic field ${\bm B}=B {\bm e}_B$ is in the $(x,y)$-plane
and it is assumed
that $p_F^2/2m \gg \alpha p_F, \omega_s$ where $p_F$ is the Fermi momentum.
The eigenvalues are
\begin{eqnarray*}
E_{\bm p}^{(\pm)}&=&\frac{p^2}{2m} \pm
|\; \alpha {\bm p} + \omega_s {\bm e}_\perp |\nonumber \\
&\approx& \left\{ \begin{array}{l c}
\frac{p^2}{2m} \pm \omega_s \pm\;\;
\alpha {\bm p}\cdot {\bm e}_\perp,& \alpha p_F \ll \omega_s\\
&\\
\frac{p^2}{2m} \pm \alpha p \pm \frac{\omega_s}{p}
{\bm p}\cdot {\bm e}_\perp,& \omega_s\ll \alpha p_F
\end{array} \right.
\end{eqnarray*}
with ${\bm e}_\perp={\bm e}_B\times{\bm e}_z$.
The Fermi momenta for the two sheets can be parameterized by
the angle $\varphi \in [0,2\pi]$:
\begin{eqnarray*}
{\bm p}_F(\varphi) &=&
\left\{ \begin{array}{l c}
\pm m\alpha\; {\bm e}_\perp
+ \left(\;p_F\pm \frac{m\omega_s}{p_F}\;\right) {\bm e}_p,
& \alpha p_F \ll \omega_s\nonumber \\
&\nonumber \\
\pm \frac{m\omega_s}{p_F}\;{\bm e}_\perp
+ \left(\;p_F\pm m\alpha\;\right) {\bm e}_p,& \omega_s\ll \alpha p_F
\end{array} \right.
\end{eqnarray*}
where $p_F=\sqrt{2m E_F}$ and
${\bm e}_p=(\cos(\varphi),\sin(\varphi))$.
In both limiting cases these are two circular sheets with slightly different
radii, displaced in the direction perpendicular to the magnetic field.
More precisely, when looking along ${\bm B}$, for $\alpha > 0$ the larger
(smaller) circle is displaced to the right (left).
For $\alpha < 0$, on the other hand, the
larger (smaller) circle is displaced to the left (right).
Figure \ref{fig7} shows that for ${\bm B}\parallel [001]$
the bands $1$ and $2$ as well as the bands $3$ and $4$ form two such pairs of
Fermi surface sheets which are displaced in opposite direction. Thereby the
direction of displacement indicates that the sheets $1$ and $2$
appear to have an effective $\alpha < 0$ whereas the two inner
sheets $3$ and $4$ have $\alpha > 0$.
As already mentioned the displacement is much smaller
for ${\bm B}\parallel[1\bar{1}0]$ than for ${\bm B}\parallel[001]$
which again shows the pronounced anisotropy of the Rashba-effect in
the more realistic LCAO-Hamiltonian.\\
Using the energy bands $E_{\nu,{\bf k}}$ we calculated the
$2\times 2$ conductivity tensor using the semiclassical
expression
\begin{eqnarray}
\sigma_{\alpha\beta} &=&e^2\;\sum_\nu\;\tau_\nu\;I_{\alpha,\beta}^{(\nu)}\nonumber \\
I_{\alpha,\beta}^{(\nu)} &=&\frac{1}{4\pi^2}\;
\int\;d{\bm k}\;\delta(E_{\nu,{\bf k}} - E_F)\;v_{\nu,\alpha} v_{\nu,\beta}\nonumber \\
&=& \frac{1}{4\pi^2}\;\int_0^{2\pi}d\varphi\;k_{F,\nu}(\varphi)\;
\frac{v_{\nu,\alpha}(\varphi) v_{\nu,\beta}(\varphi) }
{\nabla_{\bm k} E_{\nu}(\varphi)\cdot{\bf e}_{\bf k} }.
\label{semiclassical}
\end{eqnarray}
Here $\alpha,\beta\in\{x,y\}$,
${\bf k}_{F,\nu}(\varphi)$ is the Fermi momentum of the $\nu^{th}$ sheet along the
direction ${\bf e}_{\bf k}=(\cos(\varphi),\sin(\varphi))$, and
${\bf v}_\nu(\varphi)$ is the velocity
$\hbar^{-1}{\bf \nabla}_{\bm k} E_{\nu{\bm k}}$
evaluated at ${\bf k}_{F,\nu}(\varphi)$.
Moreover, $\tau_\nu$ denotes the lifetime
of the electrons in band $\nu$ which
we assume independent of $\varphi$ for simplicity.\\
Figure \ref{fig8} then shows the variation of the `band resolved'
conductivities
with the angle $\phi$ between magnetic field and $[001]$-axis.
\begin{figure}
\includegraphics[width=\columnwidth]{Figure_8.eps}
\caption{\label{fig8}
Variation of $\sigma_{[1\bar{1}0]}$ (left) and $\sigma_{[001]}$ (right)
with the angle $\phi$ between ${\bf B}$ and
the $[001]$ direction. Thereby $B=14T$.
A $\phi$-independent constant has been subtracted
to make the variations visible, for the labeling of the Fermi surface
sheets see Figure \ref{fig7}.}
\end{figure}
More precisely the figure shows the contributions
of different bands $\nu$ in (\ref{semiclassical})
to the two diagonal elements
$\sigma_{[1\bar{1}0]}$ and $\sigma_{[001]}$ of $\sigma$.
Thereby these contributions are actually summed
over pairs of bands as suggested by Figure \ref{fig7},
which shows that the two sheets belonging to
one pair have similar Fermi surface geometry and shift in opposite
direction in a magnetic field. The variation of the conductivity
with field direction has the form
\begin{eqnarray}
\sigma &\approx& A_0 + A_2\cos(2\Theta) + A_4\cos(4\Theta),
\label{raisimp}
\end{eqnarray}
where $\Theta$ again is the angle between magnetic field and current
direction.
For bands $1$ and $2$ the constant $A_2$ is negative
and substantially larger than $A_4$ so that
the conductivity is minimal for ${\bm j}\parallel {\bm B}$.
This behaviour can be reproduced qualitatively already in the framework
of the generic model (\ref{raiham}).
In the limit $\alpha p_F \ll E_F, \omega_s$ evaluation of
(\ref{semiclassical}) yields
\begin{eqnarray}
\sigma&=&e^2\tau\;\pi^{-1}\;\left[\; E_F - m\alpha^2\;\sin^2(\Theta)\;\right].
\label{raicond}
\end{eqnarray}
Numerical evaluation shows that this result is quite general, i.e. $\sigma$
has the form (\ref{raisimp}) with $A_2<0$ and $A_4=0$ for any $\alpha$ or
$\omega_s$. Figure \ref{fig9} shows the numerical values of $A_2/A_0$ versus
$\omega_s$. Nonvanishing magnetoresistance occurs only
above a threshold value $\omega_s^{(min)}$ which depends on $\alpha$.
This was found previously by Raimondi {\em et al.}\cite{Raimondi}
although these authors did not consider the detailed
variation with field direction.\\
\begin{figure}
\includegraphics[width=0.7\columnwidth]{Figure_9.eps}
\caption{\label{fig9} Ratio of Fourier coefficients
$A_2/A_0$ in (\ref{raisimp}) calculated numerically for the simplified model
(\ref{raiham}).}
\end{figure}
The behaviour of the contribution from
bands $3$ and $4$ differs strongly from the prediction of the simple model.
First, the variation of $\sigma_{[1\bar{1}0]}$ has a substantial admixture
of the higher angular harmonic $\cos(4\Theta)$. Second, while the variation of
$\sigma_{[001]}$ does have a predominant $\cos(2\Theta)$ behaviour,
one now has $A_2 > 0$. The deviating behaviour for this pair of bands
is hardly surprising in that Figure \ref{fig7}
shows that the displacement of the Fermi surface is practically
zero for ${\bm B}\parallel [1\bar{1}0]$ but quite strong for
${\bm B}\parallel [001]$ which suggests that for these two bands
the effective Rashba parameter $\alpha$ depends on
the direction of the magnetic field.\\
Despite an extensive search we were unable to find
a set of LCAO parameters such that the
sheet resistivities (obtained by inversion of the $2\times 2$
conductivity matrix (\ref{semiclassical})) obtained with a single,
band independent
relaxation time $\tau$ match the experimental $R_s$ vs. $\phi$
curves in Figure \ref{fig4}. Agreement with experiment could be achieved only
by choosing a band-dependent relaxation time, more precisely
the relaxation time $\tau_{1,2}$ for the bands $1$ and $2$ had to be chosen
larger by roughly a factor $4$ as compared to $\tau_{3,4}$ for bands
$3$ and $4$. The relaxation times obtained by fitting the experimental data
are shown in Figure \ref{fig10}. They have the expected
order of magnitude and their monotonic and smooth variation with magnetic field
is a few percent. Using the Fermi surface averages
of the Fermi velocity of $\approx 5\times 10^4\;\frac{m}{s}$
the mean free paths are
$l_{1,2}\approx 9$nm and $l_{3,4}\approx 2.5$nm.\\
\begin{figure}
\includegraphics[width=0.7\columnwidth]{Figure_10.eps}
\caption{\label{fig10}
Variation of the lifetimes $\tau_{1,2}$ and $\tau_{3,4}$
with magnetic field.}
\end{figure}
The resulting $\phi$-dependence of the sheet resistance
is compared to the experimental data
in Figure \ref{fig11}. While the agreement for
current along $[001]$ is good, there is some discrepancy
for current along $[1\bar{1}0]$ in that
the experimental curves have wide minima and sharp maxima,
whereas this is opposite for the calculated curves.
\begin{figure}
\includegraphics[width=\columnwidth]{Figure_11.eps}
\caption{\label{fig11} Calculated
sheet resistance (lines) versus angle $\phi$ between
magnetic field and $[001]$ direction compared to the
experimental data (squares), c.f. Figure \ref{fig4}.}
\end{figure}
This may indicate the limitations of the
quasiclassical Boltzmann approach as already discussed in Ref. \cite{27}
where a full solution of the Boltzmann equation was necessary to reproduce
the experimental data. For completeness
we note that a band dependent relaxation time
has been observed experimentally in materials such
as MgB$_2$\cite{Yelland} and some iron-pnictide
superconductors\cite{Terashima2,Terashima}.\\
Summarizing this section we may say that while a detailed
fit of the experimental data in the framework of the relaxation time
approximation to semiclassical Boltzmann theory is not entirely successful,
the overall behavior observed in experiment - a variation of the
conductivity with magnetic field direction
predominantly according to $\sigma=A_0 + A_2\;\cos(2\Theta)$ i.e. a
noncrystalline anisotropy, is quite generic and can be reproduced
qualitatively already in the simplest model (\ref{raiham}). Interestingly
the considerably more complicated and
anisotropic band structure does not change this significantly.
On the other hand, the results show that the
Rashba effect in the (110) surface is strongly anisotropic so that
some deviations from this simple behaviour are not surprising.
The calculation moreover shows that at least within the framework of the
relaxation time approximation the lifetime for electrons in the
two inner Fermi surface sheets must be chosen shorter.
\section{Summary}
Anisotropic electronic transport of the 2DES in $(110)$ AO/STO was
characterized by temperature and magnetic-field dependent 4-point resistance
measurements along different crystallographic directions.
Anisotropic
behavior of $R_s$ is evident over the complete measured $T$-range
($2$K$\le T \le 300$K)
with lowest sheet resistance and largest electron mobility along the $[001]$
direction. The anisotropy
of $\mu$ is mainly responsible for the anisotropic behavior of the normal
magnetotransport $M\hspace{-0.5mm}R$ for $30$K$ > T > 5 $K, where lorentz
scattering dominates magnetotransport.
At $2$K and $B < 2$T, $M\hspace{-0.5mm}R$ is dominated by weak
antilocalization. The spin
orbit field deduced from WAL is well comparable to that found in (001)
AO/STO and LAO/STO and seems to depend not on specific crystallographic
direction.\\
Tight-binding
calculations were carried out to model the electronic subbandstructure,
confirming the anisotropy of $\mu$. Despite the high anisotropy
of the Fermi surfaces, the $AM\hspace{-0.5mm}R$ shows a
non-crystalline behavior with
resistance maxima for in-plane magnetic field parallel to current direction.
Semi-classical Boltzmann theory was used to calculate conductivity and
$AM\hspace{-0.5mm}R$
confirming the rather unexpected experimental result of a non-crystalline
$AM\hspace{-0.5mm}R$, despite strong anisotropic Fermi surface sheets
and Rashba coupling which however lead to
a strong sensitivity of the $AM\hspace{-0.5mm}R$
behavior of $(110)$ AO/STO on $E_F$ as already observed for $(001)$ LAO/STO.
On the other side, electronic subband-engineering by, e. g., epitaxial strain,
may also provide possibilities to tune $AM\hspace{-0.5mm}R$ behavior which
might be interesting with respect to spintronics.\\
ACKNOWLEDGEMENTS\\
Part of this paper was supported by the Deutsche Forschungsgemeinschaft (DFG)
Grant No. FU 457/2-1. We are grateful to R. Thelen and the Karlsruhe Nano
Micro Facility (KNMF) for technical support. We also acknowledge D. Gerthsen
and M. Meffert from the laboratory for electron microscopy (LEM) for
transmission electron microscopy analysis of our samples.
\section{Appendix: LCAO calculation}
We describe bulk SrTiO$_3$ as a simple cubic lattice of Ti atoms with lattice
constant unity
at the positions ${\bm R}_i$ and retain only the three $t_{2g}$ orbitals
of each Ti atom.
The Hamiltonian is most easily formulated in a coordinate system
with axes parallel to Ti-Ti bonds which we call the bulk coordinate
system. In the following
$\alpha,\beta,\gamma \in\{x,y,z\}$ always refer to the
bulk coordinate system, are assumed to be pairwise
unequal, and
${\bm e}_\alpha$ denotes the lattice vector in $\alpha$-direction.
Following Wang {\em et al.}\cite{11}
we use a tight-binding parameterization of the Hamiltonian
with hopping integrals
\begin{eqnarray}
\langle d_{\alpha\beta}({\bm R}_i \pm {\bm e}_\alpha)|H| d_{\alpha\beta}({\bm R}_i)
\rangle &=& t,\nonumber \\
\langle d_{\alpha\beta}({\bm R}_i \pm {\bm e}_\gamma)|H|d_{\alpha\beta}({\bm R}_i)
\rangle &=& t_1,\nonumber\\
\langle d_{\alpha\beta}({\bm R}_i \pm {\bm e}_\alpha\pm {\bm e}_\beta)|H|
d_{\alpha\beta}({\bm R}_i)
\rangle &=& t_2.
\label{lcao}
\end{eqnarray}
Following Zhong {\em et al.}\cite{Zhong}
we model the interface as a hemispace of bulk SrTiO$_3$
with surface perpendicular to the unit vector
${\bm e}_n=\frac{1}{\sqrt{2}}(1,1,0)$ and the origin of the
coordinate system coinciding with some atom on the surface.
Accordingly, only atoms with ${\bm R}_i \cdot {\bm e}_n \ge 0$ are retained.
The electrons are confined to the interface by a wedge-shaped
electrostatic potential which gives an extra energy
$\epsilon_i= eE {\bm R}_i \cdot {\bm e}_n$ with $E> 0$ for
all three $t_{2g}$ orbitals on the Ti atom at ${\bm R}_i$.\\
The unit-cell of the resulting $(1,1,0)$ surface is a rectangle
with edges $\sqrt{2} \parallel [1\bar{1}0]$ and $1\parallel [001]$.
The Brillouin zone has the extension $\sqrt{2}\pi$ in
$[1\bar{1}0]$-direction and $2\pi$ in $[001]$-direction and we define
$\bar{M}=(\pi/\sqrt{2},0)$ and $\bar{Z}=(0,\pi)$. \\
Using the model described so far, Wang {\em et al.}
obtained an excellent fit to their ARPES band structure at
the SrTiO$_3$ $(110)$ surface by using the values
$t=-277$ meV, $t_1=-31$ meV, $t_2=-76$ meV
and $eE=10\;meV/\sqrt{2}$. Using these values, the conductivity
calculated within the Boltzmann equation formalism (as described in the
main text)
shows a rather strong anisotropy,
$\sigma_{[001]}/\sigma_{[1\bar{1}0]}\approx 3.5$,
much larger than the experimental value
$\sigma_{[001]}/\sigma_{[1\bar{1}0]} \approx 1.3$.
The reason is that for the low electron densities of $\approx 0.05$/unit
cell in our experiment the $d_{xy}$-derived band which has small effective
mass along the $[1\bar{1}0]$ direction (see Figure \ref{fig5})
is almost empty. This can be changed, however,
by reducing $t\rightarrow-269.5$ meV for all bonds in $[001]$-direction.
This reduction by $2.5\%$ might be the consequence of a slight distortion
of the lattice in the neighborhood of the interface. The same reduction
of the anisotropy could also be obtained by lowering the energy of the
$d_{xy}$-orbital by $\approx 10$ meV. Both modifications
shift the minimum of the $d_{xy}$-derived band to lower energy
and thus increase its filling.\\
To discuss the magnetoconductance
we extended the model of Wang {\em et al.} by including the Rashba effect -
that means the combined effect of spin-orbit coupling in the Ti 3d shell
and the confining electric field - as well as an external magnetic field.
First, the nonvanishing matrix elements
of the of orbital angular momentum operator
${\bm L}$ within the subspace of the $t_{2g}$ orbitals are
\begin{eqnarray*}
\langle d_{xz}|L_x |d_{xy}\rangle &=& i\hbar,
\end{eqnarray*}
plus two more equations obtained by cyclic permutations of $(x,y,z)$.
Choosing the basis on each Ti atom
as $(d_{xy,\uparrow},d_{xy,\downarrow},d_{xz,\uparrow},d_{xz,\downarrow},
d_{yz,\uparrow},d_{yz,\downarrow})$
one thus finds
\begin{eqnarray*}
L_x&=&\hbar\;\left(\begin{array}{r r r}
0 &-i & 0\\
i & 0 & 0\\
0 & 0 & 0 \\
\end{array}\right)\otimes \tau_0,\\
L_y&=&\hbar\;\left(\begin{array}{r r r}
0 & 0 & i\\
0 & 0 & 0\\
-i & 0 & 0 \\
\end{array}\right)\otimes \tau_0,\\
L_z&=&\hbar\;\left(\begin{array}{r r r}
0 & 0 & 0\\
0 & 0 &-i\\
0 & i & 0 \\
\end{array}\right)\otimes \tau_0,\\
\end{eqnarray*}
where $\tau_0$ is the unit matrix in spin space.
The Hamiltonian for the spin orbit-coupling then is\cite{Zhong}
\begin{eqnarray*}
H_{SO} &=&\lambda_{SO}\;{\bm L}\cdot {\bm S}\\
&=&\frac{\lambda_{SO}\;i\hbar^2}{2}\;\left(\;
\left(\;|xz\rangle\langle xy| - |xy\rangle\langle xz|\;\right)\;\tau_x
+ c.p.\;\right)\\
&=&\frac{\lambda_{SO}\;\hbar^2}{2}\;
\left(\begin{array}{r r r}
0 & -i\tau_x & i\tau_y\\
i\tau_x & 0 &-i\tau_z\\
-i\tau_y & i\tau_z & 0
\end{array}\right).
\end{eqnarray*}
Here $c.p.$ denotes two more terms obtained by cyclic
permutation of $x,y,z$ and ${\bm \tau}$ is the vector
of Pauli matrices.
We use $\lambda_{SO}\hbar^2 =20$ meV.
The coupling to an external magnetic field ${\bm B}$ is
\begin{eqnarray*}
H_B&=&\mu_B({\bf L} + g \;{\bf S})\cdot {\bm B}
\end{eqnarray*}
with the Bohr magneton $\mu_B$ and
we use $g=5$\cite{fete}.\\
In addition to the matrix elements (\ref{lcao}) the confining electric field
gives rise to small but nonvanishing hopping elements, which
would vanish due to symmetry in the bulk.
The respective term in the Hamiltonian is
$H_E=|e|{\bf E}_\perp\cdot {\bf r}$ where
${\bf E}_\perp$ is the component of the electric field perpendicular
to the bond. As shown by Zhong {\em et al.}\cite{Zhong} the respective
matrix elements can be written as ($\alpha$, $\beta$ and $\gamma$
refer to the bulk system and are pairwise unequal)
\begin{eqnarray*}
\langle d_{\alpha\beta}({\bm R}_i \pm {\bm e}_\gamma) | H_E |
d_{\beta\gamma}({\bm R}_i)\rangle &=& \pm |e|\;E_\alpha\;V_E,\\
\langle d_{\beta\gamma}({\bm R}_i \pm {\bm e}_\gamma) | H_E |
d_{\alpha\beta}({\bm R}_i)\rangle &=& \mp|e|\;E_\alpha\;V_E,
\end{eqnarray*}
and we used the value $|e|\;E\;V_E =5$ meV. The sign of
$V_E$ is positive if one really considers only
two $d$-orbitals at the given distance. This might change if one
really considers hopping via the oxygen-ion between the two
Ti ions in the true crystal structure of SrTiO$_3$.
We have verified, however,that inverting the sign of $V_E$ does not
change the angular dependence of the magnetoresistance in
Figure \ref{fig11}, although it does in fact
change the direction of the shift of the Fermi surface sheets in
Figure \ref{fig3}, that means the sign of the effective
$\alpha$. In fact, as can be seen from (\ref{raicond})
the sign of $\alpha$ does not influence the angular variation
of the conductivity.\\
We neglect any matrix elements of the electric field
between orbitals centered on atoms more distant than
nearest neighbors. The interplay between these additional hopping
matrix elements and the spin orbit coupling gives rise to the
Rashba splitting of the bands.
Adding the respective terms to the tight-binding Hamiltonian
we obtain the band structure and its variation with a magnetic field.
We have verified that slight variations of $\lambda_{SO}$, $|e|EV_E$
or $g$ do not lead to qualitative changes of the results reported in the
main text.
|
{
"timestamp": "2018-03-08T02:08:20",
"yymm": "1803",
"arxiv_id": "1803.02646",
"language": "en",
"url": "https://arxiv.org/abs/1803.02646"
}
|
\section*{Discussion}
The inverse problem in topological design is solved by a supervised machine learning regression technique. We employ a self-consistent procedure to rule out unphysical solutions enabling tailored engineering of protected edge-states.
We successfully tackle multivalued functions introducing categorical features, as the trend, which tags training data according to their gradient's sign. Discontinuous domains are effectively treated by adopting multiple independent neural networks each one specific to its domain.
Our general method can be extensively applied - well beyond the example considered in this work - and may also be exploited for other physical systems in topological science, as polaritonics \cite{Kartashov, Mihalache}, quantum technologies and ultra-cold atoms \cite{PhysRevX.7.031057,Mancini1510}.
The method is scalable to very complex structures involving hundreds of topological devices, as those recently considered for large scale synchronization \cite{PartoSegev}, and frequency comb generation \cite{Laura}, eventually including non-hermitian systems \cite{Longhi2018,Zeuner}. Further applications include 2D and 3D topological systems \cite{Bahari} and quantum sources and simulations \cite{zilberberg_photonic_2018,lohse_exploring_2018}.
\section*{Methods}
\textit{TensorFlow ---}
Tensorflow is Google's versatile open-source multiplatform dataflow library capable of efficiently performing machine learning tasks such as implementing neural networks. Multidimensional data arrays, referred to as ``tensors'' are executed on the basis of stateful dataflow graphs, hence the name TensorFlow. For our final code implementation Tensorflow version 1.3 with python API bindings was used.
The nature of our problem is such that there is a discontinuity in $\xi = 0$ which cannot be correctly handled by a single NN bridging this point; this is relevant to both the inverse and direct cases. Breaking up the data-set into two parts to be used for two separate NNs is the simplest solution to this problem.
Another interesting aspect is related to the fact that the feature set in our inverse and direct NNs contain both continuous and discrete variables. The discrete variables can either be treated as such or handled by constructing multiple NNs each relative to a specific value of the discrete variable. The trend variable which has two possible values is one such case as is the mode number. In our code we have implemented a flexible system which allows one to decide which discrete variables are to be included in each NN, the others being broken up into arrays of NNs one for each value of the variable. Once the bookkeeping issues have been tackled this generalized approach allows one to tailor the problem to the given data-set.
\textit{Transfer matrix ---} Given the stepped and periodic dielectric function of period $D=qd_o$:
\[
\varepsilon _\phi (z) = \left\{ \begin{array}{l}
\varepsilon _A \quad z_n - L_A /2 \le z \le z_n + L_A /2 \\
\varepsilon _B \quad z_n + L_A /2 \le z \le z_{n + 1} - L_A /2 \\
\end{array} \right.
\]
in each layer the electric field can be represented as the superposition of a left- and a right-traveling wave.
Applying the boundary conditions, the matrices
\[
M_{\alpha \gamma } = \frac{{q_\gamma + q_\alpha }}{{2q_\gamma }}\left( {\begin{array}{*{20}c}
1 & {r_{\alpha \gamma } } \\
{r_{\alpha \gamma } } & 1 \\
\end{array}} \right)
\]
with $\alpha, \gamma$ = A or B and $
r_{\alpha \gamma } = \frac{{q_\gamma - q_\alpha }}{{q_\gamma + q_\alpha }}$, describe the light propagation through the interfaces,
having introduced $q_{\alpha}=(\omega/c)\sqrt{\epsilon_\alpha}$, while the propagation within each layer A and B is given by:
\[
T_A = \left( {\begin{array}{*{20}c}
{e^{iq_A d_o\xi } } & 0 \\
0 & {e^{ - iq_A d_o\xi } } \\
\end{array}} \right), T_{B_n} = \left( {\begin{array}{*{20}c}
{e^{iq_B d_os_n } } & 0 \\
0 & {e^{ - iq_B d_os_n } } \\
\end{array}} \right)
\]
where $s_n =[z_{n + 1} - z_n - L_A]/d_o$ are the normalized thicknesses of the B layers.
From these we obtain the transfer matrix for the single period $
T^{(1)}(\omega)$, the matrix connecting the fields in the left side of the elementary cell to the ones in the right side:
\[
T^{(1)} = \prod\limits_{i = 0}^{q - 1} {T_{B(q - i)} M}
\]
with $M = M_{AB} T_A M_{BA}$.
The quantity $\rho=-\frac{1}{2}TrT^{(1)} (\omega, \phi,\xi )$ allows one to locate bulk bands in the regions where $\rho^2\leqslant 1$, and gaps where $\rho^2>1$. Alternatively, the amplitude $\left| {r_\infty (\omega, \phi, \xi )} \right|^2 $ of the reflection coefficient of the structure~\cite{Posha}
\begin{equation}
r_\infty (\omega, \phi, \xi ) = \frac{{e^{ik(\omega )D} - T_{11}^{(1)} (\omega, \phi, \xi )}}{{T_{12}^{(1)} (\omega, \phi, \xi)}}, \label{eq:r_inf}
\end{equation}
where $e^{ik(\omega )D}$ is an eigenvalue of the matrix $T_{}^{(1)} (\omega, \phi,\xi )$, can also be used to locate the gaps of the system.
\textit{Band structure of the unmodulated system ---}
The unmodulated structure ($\eta=0$) features stopbands at ${\tilde \omega} _0=\omega _0 d_0/c =\pi /(\sqrt {\varepsilon _{A} } +(1-\xi)\sqrt {\varepsilon _{B} })$, where $\xi=L_A/d_o$ is the characteristic size ratio.
\textit{$Q(\omega,\phi,\xi)$ function---} To determine the existence of the edge states one needs to specify the boundary conditions on each edge of the structure.
For the left edge this condition is given by:
\[0=(q_b+q_a)A_1+(q_b-q_a)B_1\]
where $A_1$ and $B_1$ are the amplitudes of the right and left-travelling waves in the first layer of the structure.
This condition can be reformulated as \[det(b_1,a_1)=0\] with $b_1=((q_a-q_b),(q_a+q_b))^T$ and $a_1=(A_1,B_1)^T$, and together with the eigenvalues $\lambda_\pm$ and eigenvectors $v_\pm=(T_{12}^{(1)},\lambda_\pm-T_{11}^{(1)})$ of the transfer matrix $T^{(1)}$ it is possible to determine existence and dispersion of edge states.
Following \cite{Hatsugai,Tauber} it can be in fact shown that a proportionality relation exists between the boundary vector $b_1$ and the eigenvectors $v_\pm$ of the transfer matrix. So the condition for the existence of the edge states is given by $det(b_1,v_\pm)=0$ in a gap where $|\lambda_\pm|<1$.
This entails searching for the zeros of the function $F_{l,\pm}=(q_A-q_B)(\lambda_\pm-T_{11}^{(1)})-T_{12}^{(1)}(q_A+q_B)$.
Specifically, the real part of $F_{l,\pm}=0$ yields the function $Q(\omega,\phi,\xi)=Re\{T_{12}^{(1)}(q_A+q_B)-(q_A-q_B)(T_{22}^{(1)}-T_{11}^{(1)})/2\}$ and, as shown in Fig.~\ref{fig:f1}c, this implies that edge states exist only in the gaps where $|\rho|>1$ and $Q(\omega,\phi,\xi)\cdot\rho>0$. At the same time, edge states cannot exist in gaps where $Q(\omega,\phi,\xi)$ does not change sign.
Moreover, due to a bulk-boundary correspondence \cite{Graf}, the number of these edge modes is equal to the modulus of the associated topological invariant $|\nu_{ij}|$, given by the winding number of the reflection coefficient:
\begin{equation}
\nu_{ij} = \frac{1}{{2\pi i}}\int\limits_{ - \pi }^\pi {d\chi \frac{{\partial ln(r_\infty(\omega ,\chi ))}}{{\partial \chi }}},
\label{eq:nu}
\end{equation}
i.e., the extra phase (divided by $2\pi$) of $r_\infty(\omega,\chi)$ when $\chi$ varies in the range ($-\pi,\pi$) with $\omega$ in the stop band\cite{PoshaAr}.
\section*{Acknowledgements}
We acknowledge support from the Templeton foundation (grant number 58277), the PRIN2015 NEMO project (2015KEZNYM grant), the H2020 QuantERA project QUOMPLEX (grant number 731473), the Italian MAE project NECST.
We thank Dr. Alexander Poshakinskiy for the fruitful comments regarding the training dataset generation.
\section*{Author contribution}
C.C. conceived the initial idea and supervised the project. F.F. expanded the concept and developed the code. L.P., G.M. and C.C. developed the theoretical part. F.F. and L.P. carried out the simulations. F.F., L.P. and G.M. contribute to data analysis and figure preparation. All the authors contributed to the manuscript writing.
|
{
"timestamp": "2018-03-09T02:01:27",
"yymm": "1803",
"arxiv_id": "1803.02875",
"language": "en",
"url": "https://arxiv.org/abs/1803.02875"
}
|
\section{Introduction}
\label{sec:intro}
A large amount of human-generated information is available online in the form of text exchanged between individuals at specific times.
Examples include social network sites, online forums and emails.
The public accessibility of several of these sources allows us
to observe our society at various scales, from focused conversations among small groups of individuals to broad political discussions involving heterogeneous audiences from large geographical areas~\cite{Hristova2017:Socialbehaviour,Nerghes2014:IntroPoliticalEconomical}.
This information is undoubtedly very valuable, as shown for example by the large revenues of big Internet companies and by its usage during political campaigns, but it is also very complex because of its joint textual, structural and temporal nature.
To cope with this complexity, researchers have typically focused on either the topology of the network, as commonly done in Network Science, or the text exchanged among individuals, using methods from Computational Linguistics.
In some cases time has also been taken into consideration as in, respectively, the fields of Temporal Networks and Temporal Information Retrieval.
However, despite this broad interest in human information networks, only a limited number of works have been developed to address text, network topology and time \hlc{in an integrated way and using a common data model}. In our opinion, this is partly a result of the over-specialization of today's academia, and the fragmented and discipline-specific development of network research. Unfortunately, omitting any of the three basic elements of temporal text networks may lead to significant information loss and prevent a deeper understanding of the information system, as exemplified in the next section.
\subsection{A motivating example}
\begin{figure*}[h!]
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{mike_net.pdf}
\caption{Topology}
\label{fig:mike_topology}
\end{subfigure}
~
\begin{subfigure}{0.3\textwidth}
\begin{minipage}{\textwidth}
\begin{center}
\mbox{}\vspace{1.05cm}\mbox{}
Mike passed away!
\mbox{}\vspace{.3cm}\mbox{}
Bye grandpa Mike
\mbox{}\vspace{.3cm}\mbox{}
R.I.P.
\mbox{}\vspace{.3cm}\mbox{}
How has television changed?
\mbox{}\vspace{.3cm}\mbox{}
R.I.P.
\mbox{}\vspace{.3cm}\mbox{}
\dots
\mbox{}\vspace{.2cm}\mbox{}
\end{center}
\end{minipage}
\caption{Text}
\label{fig:mike_text}
\end{subfigure}
~
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{mike_time.pdf}
\caption{Time}
\label{fig:mike_time}
\end{subfigure}
\begin{subfigure}{\textwidth}
\mbox{}\hfill%
\includegraphics[width=.6\textwidth]{mike_all.pdf}%
\hfill{}\mbox{}%
\caption{Topology, text and time}
\label{fig:mike_all}
\end{subfigure}
\caption{~\textbf{Three elements of an online human information network}:~\emph{a)} The topology, where each edge represents an observed information propagation path: user A writes a post about some news, user B reads the post and writes herself about it, for example by commenting on it;~\emph{b)} the text exchanged between users, that is, the text of posts and comments;~\emph{c)} the number of comments over time;~\textbf{d) Topology, time and text combined into a temporal text network}. Only details about two posts are shown.}
\label{fig:mike}
\end{figure*}
One typical usage of social media data in research is to study how information propagates online. In one of the many studies on this topic, the authors have analyzed different aspects of the propagation process considering the online reactions generated by the death of a well-known Italian TV anchorman \cite{Magnani2010}.
In Figure~\ref{fig:mike} we have reproduced (a) the information propagation network, showing which posts contained information obtained by which others, (b) the text of some of the posts generated about this event, and (c) a temporal pattern indicating the number of comments per day.
While each of these pieces of information alone reveals something, putting them together into a temporal text network (Figure~\ref{fig:mike_all}) we obtain a much more comprehensive understanding of the process. \hlc{On the one hand, w}e can see that for the posts representing explicit attempts to propagate information (e.g., \textit{Mike passed away}) publication time is fundamental to determine their success, and only the first of this type of posts generated a large and sudden burst of reactions in a very short time; \hlc{on the other hand,} conversational posts evolving from it (e.g., \textit{How has television changed?}) can appear later and still create long but less dense chains of reactions. Other posts not present in the information propagation network neither explicitly give the news nor ask for an answer, generating no or few reactions, but still have the role of re-activating the information cascade so that even the latecomers can find a trace of it; some of these posts (e.g., \textit{Bye granpa Mike!}, or \textit{R.I.P.}) form what has been called an online mourning ritual.
In summary, time, text and topology together can lead to a deeper understanding of how this information network evolved into its current structure and how information propagated through it.
\subsection{Contribution and outline}
In this work we introduce a simple but expressive and easily extensible model for temporal text networks, and define two main approaches to analyze this type of data. We also show how existing primitive data manipulation operations for multilayer networks can be composed to easily construct new algorithms for temporal text networks.
Our claim is that such a model can play a similar role of other recent attempts to unify related areas of network science, such as multilayer networks, which have boosted research in already existing fields (e.g., multiplex network analysis) by showing that results in one area could be directly applied to other types of data now expressed using a uniform terminology and mathematical form. \textbf{Our objective is to define an essential model, with a minimal number of features, so that several existing models can be unified into it without a significant increase in model complexity}.
We also believe that a unified model will promote the development of software libraries providing different data analysis functions for temporal text networks inside a single system, from centrality measures to community detection and generative models.
The article is organized as follows. In the next section we present an overview of related work, highlighting how a large \hlc{amount of research} has been produced to analyze human information networks\hlc{. As the main objective of this article is to introduce a data model for temporal text networks, our overview of the state of the art focuses on the data models already introduced in the literature, to allow a precise comparison with our model.} In Section~\ref{sec:model} we define our model as a simple attributed bipartite network. We also show how this simple model can be used to represent many existing types of text\hlc{-based} interactions, such as direct messages, multicast and broadcast. \hlc{In addition}, we show how to express \hlc{different types of information networks} using our model, and how to extend it with additional features. \hlc{Finally, we provide a detailed comparison of our model with the ones presented in the state of the art, showing how some existing models can be expressed using ours, while others can be obtained by applying some lossy processing to ours, e.g., replacing the exchanged text with a bag of words, a set of topics, a sentiment, etc. } Section~\ref{sec:analysis} explains how the model can be used in data analysis. We show how the \emph{direct} manipulation of the model can be complemented by two additional types of analysis: \emph{continuous} and \emph{discrete}. In the continuous case, time and text are treated as points in a metric space, and analysis operations are based on the computation of similarities between these points. In the discrete case, discretization operations (such as time slicing and topic modeling) are applied, encoding text and time into multiple discrete layers and enabling the direct application of the large number of methods already available for multilayer networks. In Section~\ref{sec:case} we present a practical example of our model and analysis strategies applied to Twitter data.
\section{Related work}
\label{sec:sota}
Our concept of temporal text network is a combination of text, network topology and time.
\hlc{In the literature there is a large number of models supporting one or more of these aspects, and the objective of this section is to characterize existing models from a common viewpoint. In this way, in the next section we will be able to provide a precise comparison between our proposal and existing work, showing that our model is more expressive but at the same time consistent with existing approaches, reusing existing modeling constructs when possible. In particular, we will show that we can express existing models using ours, but not vice-versa.}
\hlc{Notice that t}here are \hlc{entire} well-established disciplines developed to address \hlc{text, network topology and time} in isolation, and we do not review these here as they are widely covered by text books \cite{Newman2010,Baeza-Yates:1999:MIR:553876}, described in numerous research papers (see for example \cite{Blei:2003:LDA:944919.944937} and subsequent extensions), and included in several software packages and systems. \hlc{Instead} we describe recent research efforts combining at least two of these aspects.
\hlc{Table~\ref{tab:sota} presents a summary of the models selected for this review, also including our proposed model (core and extended temporal text network), organized according to four main criteria: (1) the type of graph used to represent the topological portion of the data, (2) the type(s) of nodes allowed in the graph, (3) the way in which text is represented in the model and (4) the way in which time is represented in the model. In Section~\ref{subsec:comparison} these criteria will be used for a comparison with our model. As our aim is to comprehensively list models, not papers, and the number of works using some of the models is very large, we have sometimes arbitrarily and unavoidably chosen a key set of references based on our knowledge and personal selection. Therefore, please notice that in the table we only indicate selected representative references; additional references are included in the text.
Figure~\ref{fig:models} complements Table~\ref{tab:sota} providing a visual intuition of the reviewed models and of the new models introduced in this article.}
\input{table_sota}
\begin{figure*}[h!]
\begin{center}
\includegraphics[width=.8\textwidth]{models.eps}
\caption{\textbf{\hlc{A visual gallery of models for time text and networks}.}}
\label{fig:models}
\end{center}
\end{figure*}
\subsection{Time \& Topology}
\hlc{The most basic family of models including both time and topology is the \emph{contact sequence} \cite{Holme2012:Temporal,Gauvin2013}.}
\hlc{This is the most popular model for representing time and relations as a simple network structure. Mathematically, the model can be represented as a directed multi-graph $G = (V, E, T)$ with attributed edges. The set of vertices $V$ represent actors (e.g., individuals, companies) and the set of edges $E$ represent the interactions among the actors. When used in practice~\cite{Lambiotte2013burstiness,Gauvin2013,Paranjape2017:TemporalMotifs} the duration of the interactions is sometimes considered negligible and hence represented as a single scalar $t \in T$, while in other occasions the temporal information is represented as time intervals $t = (t_s, t_e)$ indicating when the contact between two actors starts ($t_s$) and ends ($t_e$) \cite{viard2016:temporal}. }
Contact sequences have been \hlc{typically} used to study information spreading~\cite{Lambiotte2013burstiness,Cheng2016:MethodsCascades}, \hlc{and existing concepts such as motifs and triadic closure have been re-defined to study the evolving structure of these networks}~\cite{Paranjape2017:TemporalMotifs,viard2016:temporal,Kim2017:TimeTriadicClousure}.
\hlc{
Differently from contact sequences, where interactions are time-annotated one by one, other types of models use sequences of time-annotated graphs, where each graph is sometimes also called layer. In \emph{time-sliced} models, also known as time-aggregated models, time is expressed as an interval and an edge indicates that an interaction has happened at some point during the time interval associated to the graph \cite{Mucha876}. These models are typically obtained starting from a contact sequence and aggregating edges by time. In}
\emph{longitudinal networks} relationships about the same or similar actors are detected at different \hlc{points in} time \cite{snijders_2005,Snijders2014}. \hlc{From a data modeling point of view, time slicing and longitudinal networks are very similar, and in practice the main difference lies in the nature of the time annotation associated to each slice, where in time slicing adjacent slices are typically associated with adjacent time intervals while in longitudinal network studies adjacent layers represent network snapshots obtained at specific points in time. Different types of time annotations are described for example in \cite{batagelj2016:temporal}.}
\hlc{Memory models provide a different view over a temporal network, where ordered tuples of two or more actors are represented as single nodes \cite{scholtes2014:temporal,rosvall2014:temporal,lambiotte2015:temporal,peixoto2017:temporal}. For example, second order memory networks~\cite{scholtes2014:temporal,rosvall2014:temporal} can model the impact of one predecessor edge.
For example, if an actor $v_i$ is receiving one message from $v_j$ and one from $v_k$, and is later sending a message to $v_j$ and one to $v_k$, a contact sequence loses information on whether $v_i$ is replying to $v_j$ and $v_k$ ($j$ $\rightarrow$ $i$ $\rightarrow$ $j$, $k$ $\rightarrow$ $i$ $\rightarrow$ $k$) or forwarding the messages ($j$ $\rightarrow$ $i$ $\rightarrow$ $k$, $k$ $\rightarrow$ $i$ $\rightarrow$ $j$). A first-order memory model will contain nodes for each pair of users and have an edge between two nodes if the corresponding pairs appear on consecutive paths. In our example, if $v_i$ is replying we will have two edges in the memory model: ($\overrightarrow{ji}$, $\overrightarrow{ij}$) and ($\overrightarrow{ki}$, $\overrightarrow{ik}$), while if $v_i$ is forwarding the messages we will have the edges ($\overrightarrow{ji}$, $\overrightarrow{ik}$) and ($\overrightarrow{ki}$, $\overrightarrow{ij}$).
Higher order memory networks also exist~\cite{lambiotte2015:temporal}, although they are not as common, to represent causality effects between pathways consisting of 3 or more nodes. Deciding the order of the model is not trivial as specific patterns can be revealed only on a specific subset of memory models. To solve this problem, Scholtes et al. \cite{scholtes2017:topology} introduced a multilayer memory network, composed of multiple memory networks of different order hierarchically connected between them (e.g., each node in the 2nd-order layer $v_{ij}$ is connected with all nodes in the 3rd-order layer whose path $v_klm$ contains the leg $\overrightarrow{ij}$, so $\overrightarrow{ij} \subseteq \overrightarrow{klm}$).}
\hlc{Time often plays an important role when networks are concerned, because networks often represent dynamical systems. However, in Table~\ref{tab:sota} we have only listed distinct data models explicitly providing time annotations. As an example,
}
\emph{growing network models} \cite{Newman2010} such as preferential attachment \cite{Barabasi99emergenceScaling} aim at explaining the observed topology of empirical networks based on how they evolve in time from an initial small network. \hlc{Even if nodes and edges join the network one after the other, there is no explicit representation of time in the final model. Similarly, we have not listed papers about methods not explicitly introducing new data models, such as~\cite{lentz2013:temporal}.}
\subsection{Time \& Text}
Time is often present inside text, and commercial systems handling large human information networks from Google mail to common text messaging applications on smart phones can automatically identify \hlc{the messages} and annotate the text with temporal information.
In research, text and time are studied together in the field known as temporal information retrieval \cite{Alonso:2007:VTI:1328964.1328968,Kanhabua:2015:TIR:2864701.2864702}. This is an active area, also represented at the TREC conference where state-of-the-art information retrieval methods compete on various practical tasks. Time can be present in the text, as in the examples above or as metadata, expressed as absolute or relative time and it can also be specified in queries used to express information requirements \cite{BrucatoMontesi2014temporal}.
Another set of studies has focused on how text evolves in time, and in particular sentiment, with case studies ranging from tweets \cite{OConnorBRS10ICWSMsentiment} to songs, blogs and presidential speeches \cite{Dodds2010}. \hlc{Text and time are also studied across data sources,} for example to correlate texts from online news to trends emerging in time series such as financial data \cite{Lavrenko00miningof}. \hlc{However, no specific data model is used for this type of tasks, but only time-annotated documents (understood in a broad sense, including words, etc.) and time series.}
\subsection{Text \& Topology}
Text and networks have been studied together in various areas, either without considering time or using networks to represent relationships between texts.
Models where nodes represent parts of a document have been used in structured information retrieval, which was a particularly active research area when hypertexts and markup languages became popular
\cite{Kotsakis:2002:SIR:508791.508919}. Text is often contained inside some structure \hlc{(}e.g., a title, sections, sub-sections, etc.\hlc{)} and queries can be tuned to return specific parts of a document instead of a full one. As an example, if the searched keyword is contained inside Subsections 3.1 and 3.3 of a document, a query may return either the two subsections, or the whole Section 3, depending on the method.
\hlc{More relevant for this article are document networks, that are graphs whose nodes represent text documents \cite{journals/jmlr/ChangB09,Menczer5261}. These network models can be classified in different groups depending on whether they include time or not; later in this section we refer to citation networks as a type of directed document network where time is also typically present.} Text mining, and in particular clustering, \hlc{can be applied to document networks to identify groups of documents that are similar not only because of their text but also because of their connections, as summarized in a} recent article about clustering attributed graphs \cite{Bothorel2015}.
Several works have focused on networks extracted from text, \hlc{and we can broadly classify them into models representing the text itself, aimed at characterizing language, and models representing actors and concepts mentioned in the text.}
Networks where nodes represent words have been used to model both text documents and languages \cite{sole2010language}. For example, a document can be modeled as a network where words are connected by an edge when they are contiguous, or appear in the same sentence, paragraph, etc. Similarly whole languages can be modeled focusing on the relationships between words, as in WordNet or BabelNet.
\hlc{With regard to the second class of models for networks extracted from text,} Named Entity Recognition methods \hlc{are typically used} to identify the nodes and co-occurrence (or other language analysis approaches) to create edges among them \cite{Diesner2004RevealingSS,Chang:2009:CLA:1557019.1557044}. In this case, the output network connects different portions of a text document, or concepts extracted from the text.
A model that has been used to represent the relationships extracted from texts is \hlc{known as} heterogeneous information network (HIN)~\cite{Shi2017:HCISurvey,Ren:2016:AER:2872518.2891065}. \hlc{HINs are defined as attributed directed graphs $G = (V, E, A, R)$ with an object type mapping function $V \rightarrow A$ and a link type mapping function $E \rightarrow R$, so that each object in the network (vertices and edges) belongs to a single type and if two edges belong to the same relation type $R$, the two edges share the same starting object type as well as the ending object type. For example, HINs have been used in the past to model co-occurrence relations between entities (e.g., famous characters, sports, companies) in Wikipedia articles~\cite{Kralj2016:EnrichedHCIN}. In~\cite{Chang:2009:CLA:1557019.1557044} vertices represent either famous characters from the text or bags of words, while the edges connect words that best explain the contexts where two or more famous characters appear together in the text. Document-phrase graphs as defined in \cite{Ren2017:CDA} are also HIN-based models, and more in detail probabilistic bipartite networks $B = (V, U, E, W)$ where the vertices in one partition $V$ represent documents from a large document collection, the vertices in $U$ represent salient phrases which are semantically relevant to one or more documents in $V$, and edges $E$ indicate the relevance of each sentence for each document. }
\hlc{
HINs are not limited to represent relations within documents, text and concepts; but they can also model relations between actors and text. The most common use of HIN is actually to represent co-author or citation networks. In~\cite{Wang2017:Models}, for example, the authors use an heterogeneous information network to describe the relations between scientific articles, their authors, and the venues where they were published.}
One of the concerns recently raised against using methods from social network analysis to analyze social media is their intrinsic actor-centered approach (e.g., people, companies, stakeholders), focusing on social interactions without properly characterizing other aspects of the communication~\cite{Roth2017:FCA}.
A similar argument can be used against the use of just Natural Language Processing or semantic networks~\cite{sowa2014principles}.
Following this reasoning, a recent stream of research focused on combining structural and semantic data simultaneously,
which led to the formalization of the socio-semantic network model~\cite{Roth2010:SemanticCoevolution,Roth2017:FCA,Loet2017:SocioSemanticNetworkModel}. Originally, socio-semantic networks were just bipartite graphs interconnecting \emph{agents} (also known as actors in Social Network Analysis) with semantic objects called \emph{concepts}, corresponding for example to terms, n-grams, or lexical tags.
During the last decade the socio-semantic network model has been extended to extract more valuable knowledge from social media. An illustrative example of such extension can be found in~\cite{Loet2017:SocioSemanticNetworkModel} where the authors propose to combine the aforementioned social and socio-semantic networks into a single model. In short, they use a single matrix representation where the diagonal sub-matrices represent the relation between the same type of entities (agents and concepts) and the off-diagonal matrices represent the relation between different ones (agent/concept and concept/agent). \hlc{From the point of view of data modeling, HINs are very related to socio-semantic network models, even though HINs have been introduced as more general modeling tools while socio-semantic networks have emerged and are used in a specific application context.}
A final work worth mentioning in this class is \cite{Rosen-Zvi:2004:AMA:1036843.1036902}, where topic modeling is performed using an extended model considering not only the association between topics, words and documents, but also the association between documents and their authors. \hlc{However, this has not been included in our summary table because it introduces a generative model to summarize the data in the form of parameters indicating the probability that a given actor produces a given set of words, but not to represent the empirical data showing which actors have written what text.}
\subsection{Time \& text \& topology}
Many works in the literature have dealt with time, text and topology using ad hoc models specifically designed to capture relevant aspects of specific platforms such as Twitter.
For example, in~\cite{Tamine2016:IntroInterestText} a communication network is built in three steps: (1) conversation trees are extracted from the dataset by inversely following the chain of Twitter user interactions (replies, mentions and retweets); (2) the trees are pruned based on the time elapsed between the root \hlc{t}weet and the overlap of tweets and participants in the tree; (3) finally, all trees are merged to generate a simple weighted graph of interactions between authors.
\hlc{A related model is the so-called polyadic conversation~\cite{Magnani2012:ConversationRetrieval}, designed to describe user interactions in microblogging sites as a series of related conversations --- also called polyadic interactions. A polyadic interaction is a tuple $i = (v, U, m, t)$ where $v \in V$ is the sender of the message $m \in M$, $U \subseteq V$ is the set of receivers and $t \in T$ is the timestamp of the communication act. A polyadic conversation is then defined as a chronologically ordered tree $G = (I, E)$ where $I$ is a set of polyadic interactions and $E \subset I \times I$. }
In~\cite{Roth2010:SemanticCoevolution} a temporal model was used to compare the co-growth of two epistemic networks, a Twitter dataset and a set of related blogs, with the underlying social network of contacts. The temporal information attached to the edges of the network is, afterwards, used to compare the order of formation of epistemic and social communities.
\hlc{Citation networks have received a lot of attention, and include text documents, directed edges between them and also time annotations
\cite{institute1964use,DBLP:journals/corr/cs-DL-0309023}. In addition, when author co-citation analysis is performed \cite{white1981author}, the underlying data model must also contain information about who authored which documents.}
\hlc
Information diffusion processes are often modeled including the diffused information item (meme, blog post, etc.), the actors propagating it, and the times of propagation. This is for example the case for the model used in \cite{Leskovec2007}. However, the majority of these models do not use text to perform the data analysis, but (sometimes) to define the links between documents. Time can also be used to infer network structure based on the observation of propagation events. For example, the observation of a group of individuals repeatedly re-sharing common tweets in the same temporal order may suggest that these people are connected, and that information (tweets, in this case) passes through these hidden connections \cite{GomezRodriguez:2010:IND:1835804.1835933}. In \cite{Salehi2015survey} existing theoretical diffusion models for interconnected networks are reviewed, extending concepts in information diffusion to a multilayer model.}
In order to preserve as much original information as possible, \v{S}\'{c}epanovi\'{c} et al.~\cite{Goncalves2017:ModelsMethods} use a more generic process to build the network, mixing techniques from social network and semantic analysis. In their work, the communication network \hlc{is} modeled as a simple, temporal graph using the Twitter ``replies'' to relate actors with each other. Then, they appl\hlc{y} several semantic analysis procedures to generate supporting networks that describe the text\hlc{-}related features. A comparative analysis between the communication network and a subset of the semantic networks \hlc{is} used to study several aspects of the overall system such as semantic homophily and its evolution. \hlc{However, from a modeling point of view text is not explicitly represented in this model, but coded inside the semantic layers. We will later use a related approach to exemplify how to use our model for data analysis.}
Some attention has also been devoted to models describing co-evolutionary networks \cite{GrossBlasius2008coevolutionary,Magnani2013}. Some of these models allow the representation of a status associated to each node. Statuses can be used for example to represent the political affiliation of the person represented by the node. In growing network models, the status can influence the evolution of the network for example by increasing the probability that people will create connections with other individuals sharing the same political affiliation \cite{Kimura2008coevolutionary,Lee2005affinities}. \hlc{As for the case of simple network growing models, time is not typically kept at the end of the growing process, and in addition status has not been used to model text to the best of our knowledge. Therefore, we have not included these works in our summary table, even if we consider them potentially relevant for this field if extended in the future.}
\section{Modeling temporal text networks}
\label{sec:model}
In our opinion, a good model for temporal text networks should be general enough to be able to represent a wide range of systems, but also contain a minimal number of modeling constructs, to make the model easier to use and study. In other terms, a good compromise should be found between expressiveness and simplicity.
In addition, given the large number of existing models that have been used for a long time to describe specific aspects of temporal text networks, we believe that both the modeling constructs and the terminology used in our model should be as aligned with previous work as possible.
Following these design principles, we propose the following definition of temporal text networks:
\begin{definition}[Temporal text network]
\label{def:txt}
A temporal text network is a triple $(G, x, t)$ where:
\begin{enumerate}
\item $G = (A, M, E)$ is a directed bipartite graph, where, $A$ is a set of actors, $M$ is a set of messages, and $E \subseteq (A \times M) \cup (M \times A)$.
\item $x: M \rightarrow X$, where $X$ is a set of sequences of characters (texts).
\item $t: E \rightarrow T$, where $T$ is an ordered set of time annotations.
\end{enumerate}
and where the following constraints are satisfied:
\begin{enumerate}
\item $\forall m \in M, \textrm{in-degree}(m)=1$.
\item $(a_i,m), (m,a_j) \in E \Rightarrow t(a_i,m) \leq t(m,a_j)$.
\end{enumerate}
\end{definition}
In our model edge directionality indicates the flow of text in the network: $(a_i, m_j) \in E$ indicates that actor $a_i$ has produced text $m_j$, while $(m_j, a_i) \in E$ indicates that actor $a_i$ is the recipient of text $m_j$.
Actors with out-degree larger than \hlc{0} are information producers, actors with in-degree greater than \hlc{0} are information consumers, and actors with both positive in- and out-degree are information prosumers.
Text is represented as a combination of a text container ($m \in M$), and a textual content ($x(m)$). As a consequence, actors in our model do not only generate text, but produce text messages. Two text messages (for example, two tweets, or two emails) may be different messages even if they contain the same text and have been exchanged between the same actors at the same timestamp.
The third key component of temporal text networks is the time attribute $t$. In our model, time is defined based on a generic set of ordered time annotations $T$. This enables the adoption of several ways of representing time: as an absolute date-time, as a relative date-time, as a timestamp with an arbitrary format or as a discrete time interval if time has been sliced into time windows as it often happens when temporal networks are analyzed \hlc{(See Table~\ref{tab:sota})}.
When writing about the model's elements, we will sometimes use a concise notation. For example, we will sometimes write an edge and its time together, as in: $(a_i, m_j, t_q)$, where $t_q = t(a_i, m_j)$, and we will sometimes write a message by also indicating its sender, its recipients and its text, as in: $(a_s, m_j, \{a_{r_1}, \dots, a_{r_n}\},\textrm{``text''})$, where $\textrm{``text''} = x(m_j)$. Finally, when all the timestamps on the edges adjacent to a message are equal, we can also add a time to the previous notation, as in: $(a_s, m_j, \{a_{r_1}, \dots, a_{r_n}\},\textrm{``text''},t_q)$.
\subsection{Applicability}
\label{subsec:models}
While very simple, the model introduced above can be used to represent a range of different forms of communication and data from different sources. \hlc{In particular, by explicitly dividing the network nodes into \emph{actors} and \emph{messages}, their relations implicitly carry more information. For example, w}hether the type of communication implemented by a message is unicast, multicast or broadcast is
indicated by the out-degree of the message.
With \textbf{unicast}
a message such as a handwritten letter is sent from a single source to a specific target. This form of written communication has been preserved to the present day through
instant messaging services such as those offered by Twitter, Facebook Messenger or Whatsapp \hlc{and, more traditionally, using the electronic email}.
Unicast communication allows to keep some text private between two actors, but it can have a large overhead if the same text must be sent to multiple sources because it requires an individual message for every recipient. In order to reach a larger population it is sometimes preferable to use \textbf{broadcasting} or \textbf{multicasting}. In the former, the message is transmitted to all possible receivers\footnote{For simplicity we use the expression ``all possible receivers'' to refer to the community in which the information is spread, independent of whether the community is the whole Internet, the whole world or a set of members registered to a private site.}, while
when the information is addressed to a group of people but not to all possible receivers, such as a post on a Facebook wall, the communication is called multicast.
Fig.~\ref{fig:com} shows these different types of communication represented using our model.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{communication_Fig2.pdf}%
\caption{~\textbf{Models for different types of communication}.~\textit{a)} unicast from A to C;~\textit{b)} unicast from A to B, C and D;~\textit{c)} broadcast from A --- which can also be implemented as in the previous case if $x(M_1)=x(M_2)=x(M_3)$ and~\textit{c)} multicast from A to C and D.}
\label{fig:com}
\end{figure}
\begin{figure*}[th!]
\centering
\includegraphics[width=0.95\textwidth]{mail_example.pdf}%
\caption{\hlc{\textbf{Model of a multicast email as a temporal text network}. The entire text content of the email (including the subject line and the body) are encoded as a single message $M_1$. The sender of the email~\textit{(User A)} and the two friends (\textit{User B}, ~\textit{User C}) are modeled as individual actors. In this case, the ingoing and outgoing edges of the message contain a different time, indicating the delivery and reception timestamps registered in the email servers.}}
\label{fig:mail_ttn}
\end{figure*}
Figure~\ref{fig:mail_ttn} shows an example of how \hlc{a multicast communication through email}
can be modeled as part of a temporal text network. The resulting network includes the sender of the message (\emph{User A}) and two other actors (\emph{\hlc{User B}} and \emph{\hlc{User C}}) who where \hlc{explicit recipients}
of the message. The fourth vertex $M_1 \in M$ represents the \hlc{email}
and $x(M_1)$ corresponds to its text content \hlc{(the subject line and the body content).}
In this case, \hlc{the time attribute associated to each one of the edges represents the time when the message was delivered or received by the SMTP and POP3 servers allowing us not only to represent the communication flow, but also the effect of the channel and/or medium.}
Representing multiple \hlc{emails}
as in the example above would lead to a full temporal text network.
\hlc{In the next section we describe how to express other human information networks by extending our core model.}
\subsection{Model extensions}
\label{subsec:extension}
\begin{figure*}[th!]
\centering
\includegraphics[width=0.95\textwidth]{blog_example.pdf}%
\caption{\hlc{\textbf{Model of a blog post as a temporal text network}. The original data set contains a blog post $M_1$ and three comments ($M_2, M_3, M_4$); which are encoded as three individual messages. The three participants on the discussion (\textit{User A}, ~\textit{Follower B}, ~\textit{Follower C}) are modeled as individual producers. In this case, the edges of the messages indicate the relation between their content.}}
\label{fig:blog_ttn}
\end{figure*}
\begin{figure*}[th!]
\centering
\includegraphics[width=0.95\textwidth]{twitter_example.pdf}%
\caption{\hlc{\textbf{Model of a Twitter network as a temporal text network}. The entire content of each tweet (including hashtags, urls and retweeted content) are encoded as messages. Senders (\textit{@A}, ~\textit{@D}) and mentioned users (\textit{@B}, ~\textit{@C}, ~\textit{@D}) are modeled as individual actors. In this case, both the ingoing and outgoing edges of the message contain the same time, which indicates when the tweet has been sent. The edge between $M_3$ and $M_2$ indicates the retweet relation between both tweets.}}
\label{fig:twitter_ttn}
\end{figure*}
One of the design principles we used to define our model was simplicity, to make it tractable and general. On top of the basic model defined above, we can also easily add extensions to fit context-specific requirements.
With regard to the structure, we can straightforwardly add edges between \hlc{messages} to represent either information available from the data such as retweets on Twitter, or information deduced from the analysis of the data such as links indicating that one message is probably an answer to another, if we want to study information flows. \hlc{Figure~\ref{fig:blog_ttn} shows, for example, the modeling process of a blog post $M_1$ and the associated comments from the readers $\{M_2, M_3, M_4\}$. In this particular case, we know the identity of each one of the authors, because they are authenticated in the web platform, but we do not know exactly who are the recipients of their comments. While we can assume by context that the blog post $M_1$ was read by follower $B$ and that her message was then read by the blog owner $A$, it is uncertain what the third user (follower $C$) has read. We only know that the text produced by user $C$ is a reply to the previous comment $M_3$, but we cannot infer if he has or has not read the previous messages $M_1$ and $M_2$. One possible way to model such scenario is to represent the relation between messages instead of the relation between messages \emph{and} receivers. Similarly, in the example of Figure~\ref{fig:twitter_ttn} the edges between messages are used to represent retweets on a micro-blogging platform.}
As we discuss in the next sections, this type of extension would nicely fit our analysis framework where one main class of operations transforms the data into a multilayer representation. Similarly, we may add edges between
\hlc{actors indicating other types of relations relevant for the analysis of the human information network such as indirect recipients. Figure~\ref{fig:twitter_ttn} shows the modeling process of Twitter as a temporal text network. Unlike the previous communication channels we discussed, in Twitter the recipients of the information are encoded in the text of the messages rather than being explicit in the metadata (e.g., the edge ($M_1, B, t_1$) exists because actor $A$ mentions $B$ in the first message of the data set). In addition, Twitter users can also see messages from other users they are following, which in our model is represented by the actor-to-actor relations. This difference between intra- and inter-layer relations allows us to differentiate between direct and indirect communication in many social platforms.}
In our basic model $x$ represents a generic string of characters over some alphabet, whose interpretation will depend on the source of the data and the context of the analysis. For example, while the symbol \emph{\#} usually denotes the start of a filtering tag in online social networks such as Twitter or Instagram, in other media sites it is just an acronym for the word ``number''. Therefore, for specific application contexts additional attributes can be added for example to messages by providing special information, such as the hashtags included in the text in the case of Twitter (See Figure~\ref{fig:twitter_ttn}). In particular, we can think of having three types of information associated to each message:
\begin{enumerate}
\item The text, as in our basic model,
\item Metadata that is available in the specific data source used for the analysis, such as links to other resources (webpages, other tweets or multimedia content), like and retweet counts, or hashtags.
\item Additional information not directly available from the data source but obtained analyzing the text, for example through topic analysis.
\end{enumerate}
Different types of temporal information have been used in existing works on temporal networks and temporal text analysis (See Section~\ref{sec:sota}). For example, time can represent actions from the users such as the time when a message is posted and\hlc{/or} the time when it is read \hlc{as we did in the Twitter example}. Alternatively, \hlc{times can be used to represent a physical property of the channel, as it happens in computer networks when there can be a transmission delay from the source to the destination of a message (See Figure~\ref{fig:mail_ttn})}. Finally, time can also be associated to the message, indicating for example the time interval when the message exists. Furthermore, this information can be complete or incomplete, so that if only the initial time of the interval exists we must assume the message is still valid at the time of analysis \hlc{as we did when we describe the blog posts}; it can be private (accessible only to specific actors) or universally accessible by everyone.
\hlc{
\subsection{A comparison with the state of the art}
\label{subsec:comparison}
Our core and extended models of temporal text networks allow us to describe a variety of human information networks ranging from person-to-person email communication to complex interactions in social media sites. In Section~\ref{sec:sota} we summarized other models from the literature, that have been used in the past to partially support similar scenarios. In this section we provide a comparative review between our models and the ones described in Table~\ref{tab:sota} and Figure~\ref{fig:models}, emphasizing how they can be used to describe human information networks.
}
\hlc{
All models based only on time and topology (See Figs.~\ref{fig:models}\emph{a-e}) do not include information about messages, documents or text. A simple extension adding a text attribute to the edges would still be less expressive than our model, because this simpler solution would not be able to differentiate between different types of communication such as unicast, multicast and broadcast. These are instead allowed in our model exploiting the presence of nodes representing text messages, and thus justifying the adoption of a bipartite model instead of the simple graphs used in contact sequences. Single time annotations are also unable to distinguish between production/consumption or sending/receiving time. In summary, contact sequence models (Fig~\ref{fig:models}\emph{a}) can be expressed using our model by representing edges as edge-message-edge triples, but contact sequences cannot represent all the information that we can express using our model.
Time-slices (Fig~\ref{fig:models}\emph{b}) and longitudinal models (Fig~\ref{fig:models}\emph{c})
can also be obtained starting from our model, as we do not make any assumption about how the time is represented on the edges. It is thus possible to represent both time-slices and longitudinal models as temporal text networks by just creating a new message $m_j$ and a sequence of edges $(v_i, m_j, l), (m_j, v_k, l)$ for each original edge $e = (v_i, v_k, l)$ in the layer $l$ of the sliced network.
Finally, when only time and structure are concerned, memory models (Figs.~\ref{fig:models}\emph{d-e}) are usually constructed from contact sequence models by aggregating the edges conditional on preceding pathways. While the original temporal information is partially preserved during the creation of the memory model, it is impossible to preserve more information from our temporal text network such as messages or network attributes.
Therefore, we can think of our model as a way to represent raw and complete information about the temporal interactions and memory models as a way to emphasize information provenance. However, to represent provenance we need to allow edges between messages, and for this reason only our extended temporal text network model is able to express all the information present in memory models (in addition to text, multicasting and production/consumption times, as for all the other models not based on bipartite graphs).
}
\hlc{
The absence of relations makes it difficult to describe human information networks using just time and text (Figs.~\ref{fig:models}\emph{f-g}),
and despite their versatility to analyze text documents, strictly speaking none of the models only focusing on text and topology without actors (Figs.~\ref{fig:models}\emph{h-j}) allows us to represent human-information networks as they do not contain any representation of the consumers and producers of the text. When also actors are represented, as in some HIN-based models (Fig~\ref{fig:models}\emph{k}) and in socio-semantic networks (Figs.~\ref{fig:models}\emph{l-m}), our model adds directionality, which is necessary to represent text sender/receiver and producer/consumer relationships. Time is also not typically used in these models, but a temporal extension of existing HIN-based and basic socio-semantic models is straightforward and has in fact already appeared in the literature (Fig~\ref{fig:models}\emph{m}). The application of socio-semantic networks are also limited if compared with our model, as they contain already processed information (concepts) rather than text. With this we do not mean that our model is superior, as it can be useful to process the text into concepts, but this shows how we can go from our model to a socio-semantic model but not the other way round.
}
\hlc{
Citation networks and author-citation networks (Figs. \ref{fig:models}\emph{n-o}) can represent relationships between messages, and thus require our extended model to express their information. However, they cannot express communication, because even the more expressive author-citation network model (Fig~\ref{fig:models}\emph{o}) only focuses on the production of text. In particular, there are no edges between documents and authors, but only (implicit) edges between authors end documents. Spreading processes (Fig~\ref{fig:models}\emph{p}) also share the same limitations of either contact sequences or author-citation networks, depending on whether messages and/or authors are represented in the specific model, in addition of not (typically) keeping the text content, which is however a minor problem as text can be easily added to the nodes representing the shared items.
Compared with our core model, polyadic conversations (Fig~\ref{fig:models}\emph{q}) can express almost the same information: both can express unicast, multicast and broadcast relations between messages and actors, both differentiate between information producers and consumers and contain the raw textual information. However, while in our model each individual edge connecting messages and consumers can have a different temporal attribute, in the polyadic conversation model each polyadic interaction has one single temporal value.
}
\section{Analyzing temporal text networks}
\label{sec:analysis}
One reason to adopt a common model instead of defining ad hoc models for each application is to reuse existing analysis methods. While our model can be analyzed \emph{directly}, for example studying dynamical processes such as text propagation in a similar way as in our motivating example, we can consider other strategies. Here we define two more approaches that can be used to analyze temporal text networks: we call them \emph{continuous} and \emph{discrete}.
The practical benefit of using these two approaches is that instead of developing new algorithms the analyst can focus on defining mapping functions encoding the model in a way that fits the data and analysis at hand. Then, these functions automatically generate model views of which existing algorithms can be computed.
\subsection{Continuous analysis}
\label{subsec:continuous}
The main idea behind this approach is to map the elements of the network (e.g., actors, messages, content, etc.) into an asymmetric metric space. This means that it is possible to compute distances between them.
Once distances are available, one can directly reuse existing data analysis methods for metric spaces, such as traditional distance-based and density-based algorithms (k-means, db-scan, etc.). Distances can also be used to retrieve relevant information from large temporal text networks, specifying an information query as an element of the metric space and retrieving those elements that are the closest. We present an example of this last type of analysis in the next section.
The first way of doing this is to use a network embedding method \cite{GoyalF17embedding}. While network embedding was initially defined for simple graphs, more recent algorithms can be directly applied to attributed graphs \cite{Huang:2017:LIA:3018661.3018667}. Meanwhile, we foresee the definition of special versions of these algorithms that are specific for temporal text networks. Figure~\ref{fig:continuous} shows an example of this first type of translation, where messages are the target of the analysis. The same approach can also be used to study other structures and elements in the temporal text network such as actors or combinations of actors and messages.
\begin{figure}[th!]
\centering
\includegraphics[width=0.45\textwidth]{continuous_Fig4.pdf}%
\caption{\textbf{Continuous approach: embedding}. (\textit{left}) A temporal text network with 6 actors --- circles --- and 5 messages --- squares; (\textit{right}) the messages have been grouped into two clusters based on their topological, temporal and textual distance. The point marked with $q$ represents a user's information requirements; in this example the left cluster $(m_1, m_2, m_3)$ contains nodes that are more relevant for the user.}
\label{fig:continuous}
\end{figure}
The second way to use the continuous approach is to directly define a distance function, without any explicit embedding into a coordinate system, so that the points form a metric space but have not an explicit position: only their relationships are defined. This approach is represented in Figure~\ref{fig:continuous:distance}.
\begin{figure}[th!]
\centering
\includegraphics[width=0.45\textwidth]{distance_FigE.pdf}%
\caption{\textbf{Continuous approach: distance-based}. (\textit{left}) A temporal text network with 6 actors --- circles --- and 5 messages --- squares; (\textit{right}) a messages' distance matrix is obtained from the network \hlc{topology and time attributes}.}
\label{fig:continuous:distance}
\end{figure}
The two approaches may look similar: in both cases algorithms use distances, which can be computed after an embedding or are directly defined in the distance matrix. In practice, however, there can be relevant differences. For example, after embedding it is easier to index the data so that not all distances must be computed when algorithms are executed, leading to lower computation time. On the other hand, the direct usage of a distance function is more natural if distances are asymmetric, e.g., when $d(M1,M2) \neq d(M2,M1)$. Asymmetric distances often appear in temporal and directed networks, that are both features of our model.
\subsection{Discrete analysis}
\label{subsec:discrete}
The main idea behind this approach is to encode temporal and textual information into network structures, in particular layers in a multilayer network, so that methods from multilayer network analysis can be directly applied \cite{Kivela2014,DickisonMagnaniRossi2016}. This can be done by defining a mapping function from time and text into a discrete set of classes that are relevant for the analysis. Then, topic-and-time-based user centrality, topic-and-time-based relevance, as well as community detection algorithms can be used. An example of this last type of analysis on real data follows in the next section.
\emph{Textual discretization} is typically performed using
methods from Natural Language Processing such as topic, sentiment or semantic analysis. The main objective of the procedure is to group together messages whose contents have similar characteristics.
\emph{Time discretization} is apparently simpler, because only the cutting points between time slices must be indicated. However, also time discretization presents many options. First, there are often many ways of defining the cutting points, leading to different results. Second, after the cutting points have been defined there can still be different ways of distributing network structures into the slices. For example, if we want to discretize messages,
we can place a message $m_i$ in a specific interval $(t_a, t_b)$ either if the incoming edge $e = (v_j, m_i, t)$ exists in the interval $(t_a, t_b)$, if all the edges from/to $m_i$ exist in the interval, if at least one of the out-going edges $e = (m_i, v_j, t)$ exist in the interval, etc.
Finally, we use the term \emph{multiple discretization} when both textual and time discretization are applied together to generate the different groups.
\begin{figure}[th!]
\centering
\includegraphics[width=0.45\textwidth]{discretization_Fig5.pdf}%
\caption{\textbf{Textual discretization}. (\textit{left}) A temporal text network with 6 actors --- circles --- and 5 messages --- squares; (\textit{right}) the network has been discretized into two clusters --- the top one with 2 messages, the bottom one with 4 --- based on the topic of the messages.}
\label{fig:discretization}
\end{figure}
Under this procedure, our model would produce a k-partite network with one partition for each new cluster of messages and one partition for the actors. The procedure to generate such network is straightforward once the discretization function is defined.
Figure~\ref{fig:discretization} shows an example of textual discretization
where the resulting 3-partite network contains the original layer of actors $A$, and two message layers with 2 and 4 messages each grouping together messages about the same topic. In this particular example, $x(M_4)$ was related to both topics, therefore the message $M_4$ appears in both layers. \hlc{A similar network structure will emerge from time discretization.}
An additional operation on multilayer networks that can be applied to the discretized data is projection,
creating edges in one layer based on the information present in another layer. In the resulting multilayer network, a new edge $e_{ij}^{[l]} = (v_i, v_j)$ is created if there is a message $m_k$ in the partition $l \in L$ of the original network with: a) an edge $(v_i, m_k)$ from actor $v_i$ to message $m_k$ and b) an edge $(m_k, v_j)$ from message $m_k$ to actor $v_j$. Weights can also be added to the new edges, using various methods. Figure~\ref{fig:projection} shows one possible projection from the network in Figure~\ref{fig:discretization}.
In this example the content of the messages (and more in general also the time) are now encoded into the relations between actors.
\begin{figure}[th!]
\centering
\includegraphics[width=0.45\textwidth]{projection_Fig6.pdf}%
\caption{\textbf{Projection}. (\textit{left}) A projection of the message layers into the actor layer in the original bipartite network in Figure~\ref{fig:discretization}-\emph{left}. The projected multilayer network has 6 actors, 12 nodes and 5 weighted edges; (\textit{right}) a similar projection using the 3-partite network described in Figure~\ref{fig:discretization}-\emph{right} which generates a multilayer network with 6 actors, 18 nodes and 7 weighted edges.}
\label{fig:projection}
\end{figure}
The main advantage of using a projected multilayer network to analyze temporal text networks is the vast available literature that has targeted this type of data.
In Section~\ref{subsec:discrete_analysis} we use the approach described above together with a clustering algorithm for multilayer networks
to find communities of actors discussing about the same topics during the same time spans.
\section{A case study}
\label{sec:case}
In this section, we apply the model and approaches introduced in Sections~\ref{sec:model} and~\ref{sec:analysis} to a real temporal text network.
In particular, we focus on
\hlc{using} the discretization approach introduced in Section~\ref{subsec:discrete} to analyze the formation and evolution of communities of actors and messages.
The objective of this section is two-fold. First, we want to give \hlc{a} concrete example of the abstract type of analysis described in the previous section\hlc{.}
Second, we want to show in practice how a new type of analysis can be easily built as a composition of the \hlc{transformations} introduced in the previous section and an existing algorithm (Section
\ref{subsec:discrete_analysis}).
\subsection{Dataset}
\label{subsec:dataset}
Our initial dataset consists of 247,399 public tweets with the hashtag \emph{\#iot} (Internet of Things) or some of its variants (e.g., \#IoT, \#IOT, etc.) automatically collected using the Twitter streaming API in June, 2017. The dataset contains mentions (tweets including \emph{@username}), retweets (tweets starting with \emph{RT @username}), other tweets that are neither mentions nor retweets, and the 51,369 users involved in the aforementioned communications.
In order to improve the homogeneity of the collected data we further filtered our dataset by keeping only the tweets using at least one of thirty-two hashtags
selected by domain experts as representative of main topics in this domain. This operation removed for example tweets containing the string \emph{\#iot} but not concerning the Internet of Things.
In the following experiments we focus on the network obtained starting from the tweets containing mentions (about 5\% of the initial tweets), built by coding each tweet as in Figure~\ref{fig:twitter_ttn}.
\input{table_networks}
The resulting temporal text network contains about one third of the users in the initial dataset (15,717) and the 13,210 messages exchanged between them (See Table~\ref{tab:datasets}). We call this the \emph{original network}, and use it as a the starting point for both the following experiments.
\subsection{Discrete analysis}
\label{subsec:discrete_analysis}
Social interactions within a group of participants can form a community if they occur more frequently within the group than with other members of the network. In temporal text networks, those interactions are the result of the exchange of messages between actors. In this example we show how our model can be used to find communities of actors discussing about the same topics during the same weeks. Following the method described in Section~\ref{subsec:discrete} we first transform our network to a multilayer network preserving information about interactions between users, topics and time, so that we can then apply an existing clustering algorithm.
The \emph{discretized k-partite network} is built following the procedure explained in Section~\ref{subsec:discrete}. In this particular example, we first split the original layer of messages using their hashtags as an indication of the topic, then we further discretize based on the week when messages are posted.
The second discretization uses the posting time to create hashtag-week-specific layers.
Finally, we build the \emph{multilayer network} by projecting each one of the layers containing messages into the actors' layer. Two actors in this network are connected in a given layer $L = (h, w)$ if at least one of them has sent a message to the other using the hashtag $h$ during the week $w$. If multiple messages have been exchanged between two actors in the same layer, only a single edge is generated during the projection. At this step all edges are undirected and unweighted to fit the community detection algorithm we used. Table~\ref{tab:datasets} describes the main properties of the original temporal text network, the projected k-partite and the final multilayer network used during the analysis.
Using the multilayer network and the clique percolation mechanism described in~\cite{Mucha876}, we proceed to detect communities of actors across the whole network.
Figure~\ref{fig:com} shows the communities with more than 3 actors formed in the multilayer network. Communities contain users and topics, and both users and topics can overlap across communities. The number of users is indicated by the size of the community, while the layers representing the topics of interest of the actors are annotated next to each community. The smallest community in the diagram has 4 actors in the same layer, while the largest community contains 27 different actors and 3 layers. The edges between communities in different weeks indicate that at least one third of the users in the second community were also present in its predecessor. The thicker the line, the more users are shared between them.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{communities_Fig9.pdf}%
\caption{\textbf{Evolution of communities in the IoT space}. The size of the communities is indicated by the size of the nodes --- representing the number of actors --- and the annotated hashtags. The thickness of the edges between two communities indicates the number of common actors between them.}
\label{fig:com2}
\end{figure}
We can observe that some of the hashtags, in particular \emph{artificial intelligence} (\#ai), \emph{augmented reality} (\#ar) and \emph{virtual reality} (\#ai), are very popular in the IoT space, with several groups of interest of different sizes forming around one or more of them. However, while the three topics are present across the whole month, the communities they form are very volatile. Only one of the smallest community with just 4 actors, for example, is preserved in time without changing its members or the topics they discuss. The largest communities formed during the first week, instead, disappear in week 2. Later on, some of the same users form new communities but with less members and a higher variance of topics.
Less frequent hashtags such as \#machinelearning, \#security, \#sensors, \#smartcity and \#blockchain also form groups of interest, usually smaller and with no or a few connections with the groups of users discussing the most common topics.
Overall these results suggest that the IoT space is very fragmented in this Twitter dataset. None of the found communities was big enough to become the main arena to develop a long-standing conversation on a specific topic. Instead, users organize themselves in smaller groups that change over time. Without combining topology, text and time we would find bigger communities, that would however include users talking about different things and at different times.
In summary, this example shows how a new analysis method can be easily constructed using our model and the approaches described in the previous section. In addition, also the results of this experiment highlight the value of using all the elements of the temporal text network in the analysis.
\section{Discussion and conclusions}
In this work
we introduce a general model to represent temporal text networks based on the principles of expressiveness, simplicity and tractability. Our model is expressive and simple enough to encode the key components of human information networks (topology, time and text) into a single bipartite network, so that we can represent a range of different forms of communication and data sources spanning from postal services to online social media.
We additionally show how the model can be analyzed either directly or indirectly, to perform a variety of mining tasks. In particular, we define various transformations for two approaches that we call continuous and discrete. Using such transformations, we can map the data into existing models, allowing to reuse part of the machinery already developed to analyze complex data. While we do not describe each one of the possibilities enabled by our model in detail, in the experimental section we show two concrete experiments using the aforementioned transformations to analyze a set of communication messages exchanged in the Twitter platform during
June 2017.
During the past century, the research community has demonstrated a huge interest in studying human information networks. As a consequence, researchers from different disciplines have devoted a considerable time to develop new models and methods to describe aspects of interest in this scenario. However, as we have shown in our review, there has been none or few successful attempts to unify the literature under a common framework: several models and algorithms have been proposed, but only for a subset of the aspects we consider in this article or they have been developed ad hoc to address a specific problem. So, results in one area cannot be directly applied to other types of data.
We believe that our work can play a key role in the process of consolidating existing efforts from different disciplines under a common framework,
in the establishment of a common terminology and in the
development of new analytical software able to cope with the complexity of such data. \\
\textbf{Acknowledgements.} We would like to thank Luca Rossi and Irina Shklovski for the selection of the hashtags used in the experiments.
\bibliographystyle{elsarticle-num}
|
{
"timestamp": "2018-06-26T02:04:45",
"yymm": "1803",
"arxiv_id": "1803.02592",
"language": "en",
"url": "https://arxiv.org/abs/1803.02592"
}
|
\section{Introduction}
\label{intro}
The configurations of multiquark states were proposed by Gell-Mann \cite{1964-Gell-Mann-p214-215} and Zweig \cite{1964-Zweig-p-} at the birth of quark model (QM). In the past fifty years, it has been an extremely intriguing
research issue of searching for multiquark matter. The light tetraquark $qq\bar q\bar q$ state has been used to
investigate the scalar mesons below 1 GeV \cite{1977-Jaffe-p281-281}. Since 2003, plenty of charmoniumlike
states have been observed and the hidden-charm $qc\bar q\bar c$ tetraquark fomalism is extensively
discussed to explain the nature of these new XYZ states \cite{2016-Chen-p1-121,2011-Chen-p34010-34010,2013-Chen-p45027-45027,2016-Esposito-p1-97,2017-Lebed-p143-194,2017-Guo-p-,2015-Chen-p54002-54002,2017-Chen-p160-160}.
The doubly hidden-charm/bottom tetraquark $QQ\bar Q\bar Q$ is composed of four heavy quarks.
Such tetraquark states did not receive much attention in both experimental and theoretical aspects
\cite{1975-Iwasaki-p492-492,1981-Chao-p317-317,1982-Ader-p2370-2370,1983-Ballot-p449-451,
1985-Heller-p755-755,2004-Lloyd-p14009-14009,1992-Silvestre-Brac-p2179-2189,1993-Silvestre-Brac-p273-282,
2006-Barnea-p54004-54004,2012-Berezhnoy-p34004-34004}.
Recently, there are some discussions about the masses and decays of the $QQ\bar Q\bar Q$ states
\cite{2017-Chen-p247-251,2017-Karliner-p34011-34011,2016-Bai-p-,2016-Wu-p-,2016-Brambilla-p54002-54002,2017-Wang-p432-432,2017-Anwar-p-,2017-Debastiani-p-,2017-Eichten-p-,2017-Hughes-p-}.
The masses of these $QQ\bar Q\bar Q$ tetraquarks
are far away from the mass regions of the conventional $Q\bar Q$ mesons and the XYZ states. It will be very easy to distinguish them from the XYZ and $Q\bar Q$ states in the spectroscopy. On the other
hand, the $QQ\bar Q\bar Q$ states favor the compact tetraquark configuration than the loosely bound hadron
molecular configuration, since the light mesons can not be exchanged between the two charmonium/bottomonium
states.
In this paper, we develop a moment QCD sum rule method to calculate the mass spectra for the doubly
hidden-charm/bottom $cc\bar c\bar c$ and $bb\bar b\bar b$ tetraquark states.
\section{QCD sum rules}
\label{sec-qsr}
In this section we briefly introduce the method of QCD sum rules \cite{1979-Shifman-p385-447,1985-Reinders-p1-1,2000-Colangelo-p1495-1576}. Comparing to the traditional SVZ QCD sum rules, we use another version of QCD sum rules, the moment QCD sum rules in our analyses for the doubly hidden-charm/bottom $QQ\bar Q\bar Q$ tetraquark systems. The moment QCD sum rules have been very successfully used for studying the charmonium and bottomonium mass
spectra \cite{1979-Shifman-p385-447,1979-Shifman-p448-518,1983-Nikolaev-p526-526,1981-Reinders-p109-109,1985-Reinders-p1-1} and determining the heavy quark masses and the strong coupling constant \cite{1997-Jamin-p334-352,2001-Eidemuller-p203-210,2001-Kuhn-p588-602}.
We start by considering the following two-point correlation functions
\begin{equation}
\begin{split}
\Pi(q)&= i \int d^4xe^{iq\cdot x}\langle 0|T[J(x)J^{\dag}(0)]|0\rangle\, , \\
\Pi_{\mu\nu}(q)&=i\int d^4x e^{iq\cdot x}\langle 0|T [J_\mu(x)J_\nu^\dagger(0)]|0\rangle\, , \\
\Pi_{\mu\nu, \,\rho\sigma}(q)&=i\int d^4x e^{iq\cdot x}\langle 0|T [J_{\mu\nu}(x)J_{\rho\sigma}^\dagger(0)]|0\rangle\, ,
\label{Piq}
\end{split}
\end{equation}
in which the interpolating currents $J(x)$, $J_{\mu}(x)$ and $J_{\mu\nu}(x)$ couple to the scalar, vector and
tensor states respectively.
To study the doubly hidden-charm/bottom tetraquarks, we construct the $QQ\bar Q\bar Q$ interpolating currents with four heavy quarks in the compact diquark-antidiquark configuration.
We use all diquark fields $Q^T_a CQ_b$, $Q^T_a C\gamma_5Q_b$, $Q^T_aC\gamma_\mu\gamma_5Q_b$,
$Q^T_aC\gamma_\mu Q_b$, $Q^T_a C\sigma_{\mu\nu}Q_b$ and $Q^T_aC\sigma_{\mu\nu}\gamma_5Q_b$
and consider the Pauli principle to determine the color and flavor structures for the tetraquark operators. Following Refs. \cite{2013-Du-p14003-14003,2017-Chen-p247-251}, we obtain the $QQ\bar Q\bar Q$ tetraquark interpolating currents
as the following.
The interpolating currents with $J^{PC}=0^{++}$ are
\begin{equation}
\begin{split}
J_1&=Q^T_aC\gamma_5Q_b\bar{Q}_a\gamma_5C\bar{Q}_b^T\, , \\
J_2&=Q^T_aC\gamma_\mu\gamma_5Q_b\bar{Q}_a\gamma^\mu \gamma_5C\bar{Q}_b^T\, , \\
J_3&=Q^T_aC\sigma_{\mu\nu}Q_b\bar{Q}_a\sigma^{\mu\nu}C\bar{Q}^T_b\, , \label{currents1} \\
J_4&=Q^T_aC\gamma_\mu Q_b\bar{Q}_a\gamma^\mu C\bar{Q}_b^T\, , \\
J_5&=Q^T_aCQ_b\bar{Q}_aC\bar{Q}_b^T\, ,
\end{split}
\end{equation}
where $J_1, J_2, J_5$ belong to the symmetric $[\mathbf{6_c}]_{QQ}\otimes[\mathbf{\bar 6_c}]_{\bar Q\bar Q}$
color structure while $J_3, J_4$ belong to the antisymmetric
$[\mathbf{\bar 3_c}]_{QQ}\otimes[\mathbf{3_c}]_{\bar Q\bar Q}$ color structure.
The interpolating currents with $J^{PC}=0^{-+}$ and $0^{--}$ are
\begin{equation}
\begin{split}
J_1^{\pm}&=Q^T_aCQ_b\bar{Q}_a\gamma_5C\bar{Q}_b^T\pm Q^T_aC\gamma_5Q_b\bar{Q}_aC\bar{Q}_b^T\, , \\
J_2^+&=Q^T_aC\sigma_{\mu\nu}Q_b\bar{Q}_a\sigma^{\mu\nu}\gamma_5C\bar{Q}^T_b\, , \label{currents2}
\end{split}
\end{equation}
in which $J^+_1$ and $J_2^+$ couple to the states with $J^{PC}=0^{-+}$, and $J_1^-$ couples to the states with $J^{PC}=0^{--}$. The currents $J_1^{\pm}$ belong to the symmetric color structure while $J_2^+$ belongs to antisymmetric
color structure.
The interpolating currents with $J^{PC}=1^{++}$ and $1^{+-}$ are
\begin{equation}
\begin{split}
J_{1\mu}^{\pm}&=Q^T_aC\gamma_\mu\gamma_5 Q_b\bar{Q}_aC\bar{Q}_b^T
\pm Q^T_aCQ_b\bar{Q}_a\gamma_\mu\gamma_5 C\bar{Q}_b^T\, , \label{currents3}
\\
J_{2\mu}^{\pm}&=Q^T_aC\sigma_{\mu\nu}\gamma_5 Q_b\bar{Q}_a\gamma^\nu C\bar{Q}^T_b
\pm Q^T_aC\gamma^\nu Q_b\bar{Q}_a\sigma_{\mu\nu}\gamma_5C\bar{Q}^T_b\, ,
\end{split}
\end{equation}
in which $J_{1\mu}^{+}$ and $J_{2\mu}^{+}$ couple to the states with $J^{PC}=1^{++}$, and $J_{1\mu}^{-}$ and $J_{2\mu}^{-}$ couple to the states with $J^{PC}=1^{+-}$. The currents $J_{1\mu}^{\pm}$ belong to the symmetric color structure while $J_{2\mu}^{\pm}$ belongs to antisymmetric color structure. The interpolating currents with $J^{PC}=1^{-+}$ and $1^{--}$ are
\begin{equation}
\begin{split}
J_{1\mu}^{\pm}&=Q^T_aC\gamma_\mu\gamma_5 Q_b\bar{Q}_a\gamma_5C\bar{Q}_b^T
\pm Q^T_aC\gamma_5Q_b\bar{Q}_a\gamma_\mu\gamma_5 C\bar{Q}_b^T \, ,\\
J_{2\mu}^{\pm}&=Q^T_aC\sigma_{\mu\nu}Q_b\bar{Q}_a\gamma^\nu C\bar{Q}^T_b
\pm Q^T_aC\gamma^\nu Q_b\bar{Q}_a\sigma_{\mu\nu}C\bar{Q}^T_b \, ,\label{currents4}
\end{split}
\end{equation}
in which $J_{1\mu}^{+}$ and $J_{2\mu}^{+}$ couple to the states with $J^{PC}=1^{-+}$, and $J_{1\mu}^{-}$ and $J_{2\mu}^{-}$ couple to the states with $J^{PC}=1^{--}$. The currents $J_{1\mu}^{\pm}$ belong to the symmetric color structure while $J_{2\mu}^{\pm}$ belongs to antisymmetric color structure.
The interpolating currents with $J^{PC}=2^{++}$ are
\begin{equation}
\begin{split}
J_{1\mu\nu}&=Q^T_aC\gamma_\mu Q_b\bar{Q}_a\gamma_\nu C\bar{Q}_b^T
+Q^T_aC\gamma_\nu Q_b\bar{Q}_a\gamma_\mu C\bar{Q}_b^T\, , \label{currents5}
\\
J_{2\mu\nu}&=Q^T_aC\gamma_\mu\gamma_5 Q_b\bar{Q}_a\gamma_\nu\gamma_5 C\bar{Q}_b^T
+Q^T_aC\gamma_\nu\gamma_5Q_b\bar{Q}_a\gamma_\mu\gamma_5 C\bar{Q}_b^T\, ,
\end{split}
\end{equation}
where current $J_{1\mu\nu}$ belongs to the antisymmetric color structure while $J_{2\mu\nu}$ belongs to symmetric color structure.
At the hadronic level, the correlation functions in Eq.\eqref{Piq} can be described by the dispersion relation
\begin{align}
\Pi(q^2)=\frac{(q^2)^N}{\pi}\int_{M_H^2}^{\infty}\frac{\mbox{Im}\Pi(s)}{s^N(s-q^2-i\epsilon)}ds+\sum_{n=0}^{N-1}b_n(q^2)^n\, ,
\label{dispersionrelation}
\end{align}
where $M_H$ is the hadron mass and $b_n$ are unknown subtraction constants. A narrow resonance approximation is usually used to describe the spectral function
\begin{align}
\nonumber
\rho(s)=\frac{1}{\pi}\text{Im}\Pi(s)&=\sum_n\delta(s-m_n^2)\langle0|J|n\rangle\langle
n|J^{\dagger}|0\rangle+\cdots \\
&=f_X^2\delta(s-m_X^2)+\cdots\, , \label{Imaginary}
\end{align}
where ``$\cdots$" represents the excited higher states and continuum contributions and
$f_X$ is a coupling constant between the interpolating current and hadron state
\begin{equation}
\begin{split}
\langle0|J|X\rangle&=f_X\, ,
\\
\langle0|J_{\mu}|X\rangle&=f_X\epsilon_{\mu}\, ,
\\
\langle0|J_{\mu\nu}|X\rangle&=f_X\epsilon_{\mu\nu}\, , \label{coupling parameters}
\end{split}
\end{equation}
in which $\epsilon_{\mu}$ and $\epsilon_{\mu\nu}$ are the polarization vector and tensor, respectively.
To pick out the contribution of the lowest lying resonance in Eq. \eqref{Imaginary}, we define moments
in Euclidean region $Q^2=-q^2>0$ \cite{1985-Reinders-p1-1,2014-Chen-p201-215}:
\begin{align}
M_n(Q^2_0)=\frac{1}{n!}\bigg(-\frac{d}{dQ^2}\bigg)^n\Pi(Q^2)|_{Q^2=Q_0^2}
&=\int_{16m_Q^2}^{\infty}\frac{\rho(s)}{(s+Q^2_0)^{n+1}}ds\, \label{moment}
\\
&=\frac{f_X^2}{(m_X^2+Q_0^2)^{n+1}}\big[1+\delta_n(Q_0^2)\big]\,, \label{Phemoment}
\end{align}
in which $\delta_n(Q_0^2)$ contains the contributions of higher states and continuum.
It tends to zero as $n$ goes to infinity. We consider the following ratio to eliminate $f_X$
in Eq.~\eqref{Phemoment}
\begin{align}
r(n,Q_0^2)\equiv\frac{M_{n}(Q_0^2)}{M_{n+1}(Q_0^2)}=\big(m_X^2+Q_0^2\big)
\frac{1+\delta_{n}(Q_0^2)}{1+\delta_{n+1}(Q_0^2)}. \label{ratio}
\end{align}
One expects $\delta_{n}(Q_0^2)\cong\delta_{n+1}(Q_0^2)$ for sufficiently large $n$
to suppress the contributions of higher states and continuum \cite{1985-Reinders-p1-1}.
Then hadron mass of the lowest lying resonance $m_X$ is then extracted as
\begin{align}
m_X=\sqrt{r(n,Q_0^2)-Q_0^2}\, . \label{mass}
\end{align}
Using the operator production expansion (OPE) method, the two-point function can also be evaluated
at the quark-gluonic level as a function of various QCD parameters. In the fully heavy tetraquark
systems, we only need to calculate the perturbative term and the gluon condensate contributions
to the correlation functions. One can find the results of the moments $M_n(Q^2_0)$ in
Ref. \cite{2017-Chen-p247-251}.
\section{Numerical results}
We perform the numerical analyses by using the following values of parameters \cite{2014-Olive-p90001-90001,1996-Narison-p162-172,2003-Ioffe-p229-241,2010-Narison-p559-559}
\begin{equation}
\begin{split}
m_c(\overline{\rm MS})&=1.27\pm0.03~\text{GeV}, \,
\\
\label{parameters}
m_b(\overline{\rm MS})&=4.18\pm0.03~\text{GeV}\, ,
\\
\langle g_s^2GG\rangle&=(0.48\pm0.14)~\text{GeV}^4\, .
\end{split}
\end{equation}
To provide reliable moment sum rule analyses, one needs to find suitable working regions of the two parameters
$n$ and $Q_0^2$ in the ratio $r(n,Q_0^2)$. We define $\xi=Q^2_0/16m_c^2$ for $cc\bar c\bar c$ and
$Q^2_0/m_b^2$ for $bb\bar b\bar b$ systems for convenience. These two parameters will affect the pole
contribution and the OPE convergence. For small value of $\xi$, the high dimension condensates
in OPE will give large contributions, and thus leading to bad OPE convergence \cite{1981-Reinders-p109-109,
1985-Reinders-p1-1}. However, a larger value of $\xi$ means slower convergence of $\delta_{n}(Q_0^2)$
in Eq. \eqref{Phemoment}. Such behavior can be compensated by $n$: the OPE convergence becomes
good for small $n$ while $\delta_{n}(Q_0^2)$ tends to zero for sufficiently large $n$.
One needs to find suitable working regions for $(n, \xi)$ where the lowest lying resonance dominates the
moments and the OPE converges well.
\begin{figure}[hbt]
\begin{center}
\scalebox{0.65}{\includegraphics{bbbbmass0++_n_J1.eps}}
\caption{Hadron mass $m_{X_b}$ for $J_{1}(bb\bar b\bar b)$ with $J^{PC}=0^{++}$, as a function of $n$
for different value of $\xi$.} \label{figbbbb0++}
\end{center}
\end{figure}
As an example, we use the interpolating current $J_1$ with $J^{PC}=0^{++}$ in Eq. \eqref{currents1} to perform
numerical analyses. Requiring the perturbative term to be larger than the gluon condensate term, we obtain
upper limits $n_{max}$, which increases with respect to the value of $\xi$.
We show the hadron mass $m_{X_b}$ as a function of $n$ for $\xi=0.2, 0.4, 0.6, 0.8$ in Fig.~\ref{figbbbb0++}.
One notes that the mass curves have plateaus which provide stable mass prediction
\begin{equation}
m_{X_b}=(18.45 \pm 0.15) \,\mbox{GeV}\, ,
\end{equation}
in which the error comes from the uncertainties of $\xi$, the heavy quark mass and
the gluon condensate in Eq.~\eqref{parameters}. Using the interpolating currents in Eqs.~\eqref{currents1}--\eqref{currents5}, we perform numerical analyses for all $cc\bar c\bar c$ and $bb\bar b\bar b$ systems
with various quantum numbers. We collect the numerical results in Table \ref{tablemass}.
It is shown that the negative parity states ($J^{PC}=0^{-+}, 0^{--}, 1^{-+}, 1^{--}$) are slightly heavier than
the positive parity states ($J^{PC}=0^{++}, 1^{++}, 1^{+-}, 2^{++}$).
\begin{table}
\begin{center}
\begin{tabular*}{8.4cm}{cccc}
\hline
~~~~$J^{PC}$ ~~~~& Currents &~~~~ $m_{X_{c}}$\mbox{(GeV)} ~~~~&~~~~ $m_{X_{b}}$\mbox{(GeV)} ~~~~ \\
\hline
$0^{++}$ & $J_1$ & $6.44\pm0.15$ & $18.45\pm0.15$ \\
& $J_2$ & $6.59\pm0.17$ & $18.59\pm0.17$ \\
& $J_3$ & $6.47\pm0.16$ &$18.49\pm0.16$ \\
& $J_4$ & $6.46\pm0.16$ &$18.46\pm0.14$ \\
& $J_5$ & $6.82\pm0.18$ & $19.64\pm0.14$
\vspace{4pt}\\
$0^{-+}$ & $J_1^+$ & $6.84\pm0.18$ & $18.77\pm0.18$ \\
& $J_2^+$ & $6.85\pm0.18$ & $18.79\pm0.18$
\vspace{4pt}\\
$0^{--}$ & $J_1^-$ & $6.84\pm0.18$ & $18.77\pm0.18$
\vspace{4pt}\\
$1^{++}$ & $J_{1\mu}^+$ & $6.40\pm0.19$ & $18.33\pm0.17$ \\
& $J_{2\mu}^+$ & $6.34\pm0.19$ & $18.32\pm0.18$
\vspace{4pt}\\
$1^{+-}$ & $J_{1\mu}^-$ & $6.37\pm0.18$ & $18.32\pm0.17$ \\
& $J_{2\mu}^+$ & $6.51\pm0.15$ & $18.54\pm0.15$
\vspace{4pt}\\
$1^{-+}$ & $J_{1\mu}^+$ & $6.84\pm0.18$ & $18.80\pm0.18$ \\
& $J_{2\mu}^+$ & $6.88\pm0.18$ & $18.83\pm0.18$
\vspace{4pt}\\
$1^{--}$ & $J_{1\mu}^-$ & $6.84\pm0.18$ & $18.77\pm0.18$ \\
& $J_{2\mu}^-$ & $6.83\pm0.18$ & $18.77\pm0.16$
\vspace{4pt}\\
$2^{++}$ & $J_{1\mu\nu}$ & $6.51\pm0.15$ & $18.53\pm0.15$ \\
& $J_{2\mu\nu}$ & $6.37\pm0.19$ & $18.32\pm0.17$ \\
\hline
\end{tabular*}
\caption{Mass spectra for the $cc\bar c\bar c$ and $bb\bar b\bar b$
tetraquarks. \label{tablemass}}
\end{center}
\end{table}
\begin{figure}[hbt]
\begin{center}
\scalebox{0.5}{\includegraphics{QQQQSpectrum.eps}}
\end{center}
\caption{Summary of the doubly hidden-charm/bottom tetraquark spectra labelled by $J^{PC}$.
The green and red solid (dashed) lines indicate the $\eta_c(1S)\eta_c(1S)$ ($\eta_b(1S)\eta_b(1S)$) and $J/\psi J/\psi$
($\Upsilon(1S)\Upsilon(1S)$) thresholds, respectively.} \label{fig:spectra}
\end{figure}
It is interesting to compare the mass spectra with the corresponding two-meson mass thresholds.
As shown in Fig.~\ref{fig:spectra}, the masses of $bb\bar b\bar b$ tetraquarks are below the $\eta_b(1S)\eta_b(1S)$
threshold while all $cc\bar c\bar c$ tetraquarks lie above the $\eta_c(1S)\eta_c(1S)$ threshold.
The two bottomonium mesons decays for the $bb\bar b\bar b$ tetraquarks are thus forbidden by the kinematics.
For the doubly hidden-charm $cc\bar c\bar c$ tetraquarks, they can decay via the
spontaneous dissociation mechanism by considering the restrictions of the symmetries.
In Table \ref{ccccdecay}, we collect the possible $S$-wave and $P$-wave dissociation decay channels for the $cc\bar c\bar c$ states.
In principle, the $bb\bar b\bar b$ tetraquark can also decay into $B^{(\ast)}\bar B^{(\ast)}$ via a heavy quark pair annihilation and a light quark pair creation processes. The suppression by the annihilation of a heavy quark pair
will be compensated by the large phase space factor. Such $B^{(\ast)}\bar B^{(\ast)}$ decay modes may dominate
the total width of the doubly hidden-bottom $bb\bar b\bar b$ tetraquark state.
\begin{table}[hbt]
\begin{center}
\begin{tabular*}{9.2cm}{ccc}
\hline
$J^{PC}$ & S-wave & P-wave \\
\hline
$0^{++}$ & $\eta_c(1S)\eta_c(1S)$, $ J/\psi J/\psi$ & $\eta_c(1S)\chi_{c1}(1P)$, $ J/\psi h_c(1P)$
\vspace{6pt}\\
$0^{-+}$ & $\eta_c(1S)\chi_{c0}(1P)$, $ J/\psi h_c(1P)$ & $ J/\psi J/\psi$
\vspace{6pt}\\
$0^{--}$ & $ J/\psi\chi_{c1}(1P)$ & $ J/\psi\eta_c(1S)$
\vspace{6pt}\\
$1^{++}$ & $-$ & $ J/\psi h_c(1P)$, $\eta_c(1S)\chi_{c1}(1P)$, \\
& & $\eta_c(1S)\chi_{c0}(1P)$
\vspace{6pt}\\
$1^{+-}$ & $ J/\psi\eta_c(1S)$ & $ J/\psi\chi_{c0}(1P)$, $ J/\psi\chi_{c1}(1P)$, \\
& & $\eta_c(1S) h_c(1P)$
\vspace{6pt}\\
$1^{-+}$ & $ J/\psi h_c(1P)$, $\eta_c(1S)\chi_{c1}(1P)$ & $-$
\vspace{6pt}\\
$1^{--}$ & $ J/\psi\chi_{c0}(1P)$, $ J/\psi\chi_{c1}(1P)$, & $ J/\psi\eta_c(1S)$ \\
& $\eta_c(1S) h_c(1P)$ &
\\ \hline
\end{tabular*}
\caption{Possible decay modes of the $cc\bar c\bar c$ states by spontaneous dissociation
into two charmonium mesons. \label{ccccdecay}}
\end{center}
\end{table}
\section{Summary}
In this paper, we have calculated the mass spectra for the doubly hidden-charm/bottom $cc\bar c\bar c$
and $bb\bar b\bar b$ tetraquark states by using the moment QCD sum rule method.
Our results show that the $cc\bar c\bar c$ tetraquarks lie above the two charmonium spontaneous dissociation
thresholds and thus can mainly decay into two charmonium mesons. We suggest to search for these doubly
hidden-charm $cc\bar c\bar c$ states in the $J/\psi J/\psi$ and $\eta_c(1S)\eta_c(1S)$ channels.
For the $bb\bar b\bar b$ tetraquarks, their masses are lower than the $\eta_b(1S)\eta_b(1S)$ threshold
so that the two bottomonium mesons decays are kinematical forbidden. These $bb\bar b\bar b$ tetraquark,
if exist, may be very narrow and stable. In the near future, these doubly hidden-charm/bottom $cc\bar c\bar c$
and $bb\bar b\bar b$ tetraquark states can be searched for at facilities such as LHCb, CMS, RHIC and the
forthcoming BelleII.
\section*{Acknowledgments}
This project is supported by the Chinese National Youth Thousand Talents Program; the Natural Sciences and Engineering Research Council of Canada (NSERC); the National Natural Science
Foundation of China under Grants No. 11475015, 373
No. 11375024, No. 11222547, No. 11175073, 374 No. 11575008, and No. 11621131001; the 973 program; 375 the Ministry of Education of China (SRFDP under Grant 376 No. 20120211110002 and the Fundamental Research 377 Funds for the Central Universities); and the National 378 Program for Support of Top-Notch Youth Professionals.
|
{
"timestamp": "2018-03-14T01:08:45",
"yymm": "1803",
"arxiv_id": "1803.02522",
"language": "en",
"url": "https://arxiv.org/abs/1803.02522"
}
|
\section{Introduction}\label{int}
Process algebras are well-known formal theories to capture computational concepts in computer science, especially parallelism and concurrency. CCS \cite{CCS}, CSP \cite{CSP} and ACP \cite{ACP} are three process algebras, and CSP is used widely in verify the behaviors of computer systems for its rich expressive power.
CSP only permits a process to engage in one event on a moment and records this single event into the traces of the process. CSP cannot process events simultaneously, it treat the events occurred simultaneously as one single event. In this paper, we modify CSP to process the events occurred simultaneously, this work is called communicating concurrent processes (CCP). The main difference between CCP and CSP is the treatment of concurrency.
This paper is organized as follows. In section \ref{pct}, we introduce the concepts and laws of processes and concurrent traces in CCP. In section \ref{concurrency}, we give the concepts and laws of concurrency in CCP. we discuss the rest contents of CCP corresponding to CSP briefly, for they are almost same. And we conclude this paper in section \ref{con}.
\section{Processes and Concurrent Traces}\label{pct}
In this section, we modify processes and traces to be suitable for CCP. Some concepts are modified for this situation, the others are just retype from CSP, including concepts of processes and traces.
\subsection{Processes}
The intuitions of the concept events, event names, alphabet, processes, are the same as CSP. We also use the following conventions:
\begin{itemize}
\item Words in lower-case letters denote distinct events;
\item Words in upper-case letters denote specific defined processes;
\item The letters $x,y,z$ are variables denoting events;
\item The letters $A,B,C$ denote sets of events;
\item The letters $X,Y$ are variables denoting processes;
\item The alphabet of process $P$ is denoted $\alpha P$;
\item The process with alphabet $A$ which never actually engages in any of the events of $A$ is called $STOP_A$;
\item The process with alphabet $A$ which can engage in any event of $A$ is called $RUN_A$.
\end{itemize}
\begin{definition}[Prefix]
Let $x_1,\cdots,x_n$ ($n\in\mathbb{N}$) be events and let $P$ be a process. Then the prefix
\[(\{x_1,\cdots,x_n\}\rightarrow P)\]
denotes a process engaged in the events $x_1,\cdots,x_n$ simultaneously firstly, and then behaving as $P$, with $\alpha(\{x_1,\cdots,x_n\}\rightarrow P)=\alpha P$, if $x_1,\cdots, x_n\in \alpha P$.
\end{definition}
To describe infinite behaviors, we also introduce recursion.
\begin{definition}[Recursion]
For a prefix guarded expression $F(X)$ containing the the process name $X$, with the alphabet $A$ of $X$, then the recursive equation
\[X=F(X)\]
has a unique solution denoted as $\mu X:A\bullet F(X)$.
\end{definition}
\begin{definition}[Choice]
If $\{x_1,\cdots, x_n\}\neq\{y_1,\cdots,y_m\}$ with $m,n\in\mathbb{N}$, then the choice
\[(\{x_1,\cdots,x_n\}\rightarrow P \mid \{y_1,\cdots,y_m\}\rightarrow Q)\]
will initially engage in either of the events $\{x_1,\cdots,x_n\}$ or $\{y_1,\cdots, y_m\}$ simultaneously, with $\alpha(\{x_1,\cdots,x_n\}\rightarrow P \mid \{y_1,\cdots,y_m\}\rightarrow Q)=\alpha P = \alpha Q$.
\end{definition}
\begin{proposition}[Laws of Processes]\label{LoP}
We have the following laws of processes.
\begin{itemize}
\item L1. $(\{x_1,\cdots,x_n\}:A\rightarrow P(x_1,\cdots,x_n))=(\{y_1,\cdots,y_m\}: B\rightarrow Q(y_1,\cdots,y_m))\equiv(m=n\wedge A=B\wedge\forall x_i:A\bullet P(x_i)=Q(x_i))$
\begin{itemize}
\item L1A. $STOP\neq(\{d_1,\cdots, d_n\}\rightarrow P)$
\item L1B. $(\{c_1,\cdots,c_n\}\rightarrow P)\neq(\{d_1,\cdots,d_m\}\rightarrow Q)$ if $\{c_1,\cdots,c_n\}\neq\{d_1,\cdots,d_m\}$
\item L1C. $(\{c_1,\cdots,c_n\}\rightarrow P\mid\{d_1,\cdots,d_m\}\rightarrow Q)=(\{d_1,\cdots,d_m\}\rightarrow Q\mid \{c_1,\cdots,c_n\}\rightarrow P)$
\item L1D. $(\{c_1,\cdots,c_n\}\rightarrow P)=(\{c_1,\cdots,c_n\}\rightarrow Q)\equiv P=Q$
\end{itemize}
\item L2. If $F(X)$ is a guarded expression, $(Y=F(Y))\equiv(Y=\mu X\bullet F(X))$.
\begin{itemize}
\item L2A. $\mu X\bullet F(X)=F(\mu X\bullet F(X))$
\end{itemize}
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definitions of processes.
\end{proof}
The processes also can be implemented by LISP as in CSP.
\subsection{Concurrent Traces}
\begin{definition}[Concurrent Traces]
A trace is a sequence of symbols, separated by commas and enclosed in angular brackets. A concurrent trace is a trace within a pair of commas, there may be a set of symbols.
\begin{itemize}
\item $\langle\{x_1,\cdots,x_n\},\{y_1,\cdots,y_m\}\rangle$ is a concurrent trace contains two event sets, $\{x_1,\cdots,x_n\}$ followed by $\{y_1,\cdots,y_m\}$;
\item $\langle\rangle$ is the empty trace.
\end{itemize}
In the following, we also call a concurrent trace as a trace. And we write $s,t,u$ for traces, $S,T,U$ for sets of traces, and $f,g,h$ for functions.
\end{definition}
\begin{definition}[Catenation]
The catenation of a pair of traces $s$ and $t$ is defined as $s\smallfrown t = \langle s, t\rangle$.
\end{definition}
\begin{proposition}[Laws of Catenation]
The laws of catenation for concurrent traces are the same for traces in CSP, we retype them as follows.
\begin{itemize}
\item L1. $s\smallfrown\langle\rangle = \langle\rangle\smallfrown s = s$
\item L2. $s\smallfrown(t\smallfrown u) = (s\smallfrown t)\smallfrown u$
\item L3. $s\smallfrown t = s\smallfrown u\equiv t = u$
\item L4. $s\smallfrown t = u\smallfrown t\equiv s=u$
\item L5. $s\smallfrown t=\langle\rangle\equiv s=\langle\rangle\wedge t=\langle\rangle$
\item L6. For $t^n$ is $n$ copies of $t$ catenated with each other, $t^0=\langle\rangle$
\item L7. $t^{n+1}=t\smallfrown t^n$
\item L8. $t^{n+1}=t^n\smallfrown t$
\item L9. $(s\smallfrown t)^{n+1}=s\smallfrown(t\smallfrown s)^n\smallfrown t$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of catenation.
\end{proof}
\begin{definition}[Restriction]
The restriction $(t\upharpoonright A)$ of the trace $t$ restricted to the set of symbols $A$, is obtained by omitting all symbols outside $A$ from $t$.
\end{definition}
\begin{proposition}[Laws of Restriction]
The laws of restriction for concurrent traces are the same as for traces in CSP, we also retype them as follows.
\begin{itemize}
\item L1. $\langle\rangle\upharpoonright A=\langle\rangle$
\item L2. $(s\smallfrown t)\upharpoonright A=(s\upharpoonright A)\smallfrown (t\upharpoonright A)$
\item L3. $\langle \{x_1,\cdots,x_n\}\rangle\upharpoonright A = \langle\{x_1,\cdots,x_n\}$, if $x_1,\cdots,x_n\in A$
\item L4. $\langle\{y_1,\cdots, y_m\}\rangle\upharpoonright A=\langle\rangle$, if $y_1,\cdots, y_m\notin A$
\item L5. $s\upharpoonright\{\}=\langle\rangle$
\item L6. $(s\upharpoonright A)\upharpoonright B = s\upharpoonright(A\cup B)$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of restriction.
\end{proof}
\begin{definition}[Head and Tail]
For a nonempty trace $s$, the head of $s$ is the first sequence denoted $s_0$, the left sequence is the tail of $s$ denoted $s'$.
\end{definition}
\begin{proposition}[Laws of Head and Tail]
The laws of head and tail for concurrent traces are as follows.
\begin{itemize}
\item L1. $(\langle\{x_1,\cdots,x_n\}\rangle\smallfrown s)_0=\{x_1,\cdots,x_n\}$
\item L2. $(\langle\{x_1,\cdots,x_n\}\rangle\smallfrown s)'=s$
\item L3. $s=(\langle s_0\rangle\smallfrown s')$, if $s\neq\langle\rangle$
\item L4. $s=t\equiv(s=t=\langle\rangle\vee(s_0=t_0\wedge s'=t'))$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of head and tail.
\end{proof}
\begin{definition}[Star]
The star $A^\ast$ of $A$ is defined as follows. $A^\ast=\{s|s\upharpoonright A = s\}$
\end{definition}
\begin{proposition}[Laws of Star]
The laws of star for concurrent traces are as follows.
\begin{itemize}
\item L1. $\langle\rangle\in A^\ast$
\item L2. $\langle\{x_1,\cdots,x_n\}\rangle\in A^\ast\equiv\{x_1,\cdots,x_n\}\in A$
\item L3. $(s\smallfrown t)\in A^\ast\equiv s\in A^\ast\wedge t\in A^\ast$
\item L4. $A^\ast=\{t|t=\langle\rangle\vee(t_0\in A\wedge t'\in A^\ast)\}$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of star.
\end{proof}
\begin{definition}[Ordering]
For traces $s,u,t$, such that $s\smallfrown u = t$, The ordering relation of concurrent traces is defined as follows. $s\leq t = (\exists u\bullet s\smallfrown u = t)$.
\end{definition}
\begin{proposition}[Laws of Ordering]
The laws of the ordering relation for concurrent traces are the same for traces in CSP, we retype them as follows.
\begin{itemize}
\item L1. $\langle\rangle\leq s$
\item L2. $s\leq s$
\item L3. $s\leq t\wedge t\leq s\Rightarrow s=t$
\item L4. $s\leq t\wedge t\leq u\Rightarrow s\leq u$
\item L5. $(\langle\{x_1,\cdots,x_n\}\rangle\smallfrown s)\leq t\equiv t\neq\langle\rangle \wedge \{x_1,\cdots,x_n\}=t_0\wedge s\leq t'$
\item L6. $s\leq u\wedge t\leq u\Rightarrow s\leq t\vee t\leq s$
\item L7. $s$ in $t=(\exists u,v\bullet t=u\smallfrown s\smallfrown v)$
\item L8. $(\langle\{x_1,\cdots,x_n\}\rangle\smallfrown s)$ in $t\equiv t\neq\langle\rangle\wedge((t_0=\{x_1,\cdots,x_n\}\wedge s\leq t')\vee(\langle\{x_1,\cdots,x_n\}\rangle\smallfrown s$ in $t'))$
\item L9. $s\leq t\Rightarrow (s\upharpoonright A)\leq(t\upharpoonright A)$
\item L10. $t\leq u\Rightarrow(s\smallfrown t)\leq(s\smallfrown u)$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of the ordering relation.
\end{proof}
\begin{definition}[Length]
The length of a trace $s$ is defined as the number of commas plus 1, and denoted $\sharp s$.
\end{definition}
\begin{proposition}[Laws of Length]
The laws of length for concurrent traces are as follows.
\begin{itemize}
\item L1. $\sharp\langle\rangle = 0$
\item L2. $\sharp\langle\{x_1,\cdots,x_n\}\rangle = 1$
\item L3. $\sharp(s\smallfrown t)=(\sharp s)+ (\sharp t)$
\item L4. $\sharp(t\upharpoonright(A\cup B))=\sharp(t\upharpoonright A)+ \sharp(t\upharpoonright B)-\sharp(t\upharpoonright(A\cap B))$
\item L5. $s\leq t\Rightarrow \sharp s\leq \sharp t$
\item L6. $\sharp(t^n)= n\times(\sharp t)$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of the length.
\end{proof}
The concurrent traces can also be implemented by LISP as traces in CSP.
\begin{definition}[Concurrent Traces of a Process]
The complete set of all possible concurrent traces of a process $P$ can be known in advance, and is denoted by a function $traces(P)$.
\end{definition}
\begin{proposition}[Laws of concurrent Traces of a Process]
The laws of concurrent traces of a process are as follows.
\begin{itemize}
\item L1. $traces(STOP)=\{t|t=\langle\rangle\}=\{\langle\rangle\}$
\item L2. $traces(\{c_1,\cdots,c_n\}\rightarrow P)=\{t|t=\langle\rangle \vee (t_0=\{c_1,\cdots,c_n\}\wedge t'\in traces(P))\}=\{\langle\rangle\}\cup\{\langle \{c_1,\cdots,c_n\}\rangle\smallfrown t|t\in traces(P)\}$
\item L3. $traces(\{c_1,\cdots,c_n\}\rightarrow P\mid \{d_1,\cdots,d_m\}\rightarrow Q )=\{t|t=\langle\rangle \vee (t_0=\{c_1,\cdots,c_n\}\wedge t'\in traces(P)) \vee(t_0=\{d_1,\cdots,d_m\}\wedge t'\in traces(Q))\}$
\item L4. $traces(\{x_1,\cdots,x_n\}:B\rightarrow P(x_1,\cdots,x_n))=\{t|t=\langle \rangle \vee (t_0\in B\wedge t'\in traces(P(t_0)))\}$
\item L5. $traces(\mu X:A\bullet F(X))=\bigcup_{n\geq 0}traces(F^n(STOP_A))$
\item L6. $\langle\rangle\in traces(P)$
\item L7. $s\smallfrown t\in traces(P)\Rightarrow s\in traces(P)$
\item L8. $traces(P)\subseteq (\alpha P)^\ast$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of concurrent traces of a process.
\end{proof}
The concurrent traces of a process can also be implemented by LISP.
\begin{definition}[After]
If $s\in traces(P)$, $P$ after $s$ denoted $P/s$ is behaves the same as $P$ after $P$ has executed all events in $s$.
\end{definition}
\begin{proposition}[Laws of After]
The laws of after for concurrent traces are as follows.
\begin{itemize}
\item L1. $P/\langle\rangle = P$
\item L2. $P/(s\smallfrown t)=(P/s)/t$
\item L3. $(\{x_1,\cdots,x_n\}:B\rightarrow P(x_1,\cdots,x_n))/\langle\{c_1,\cdots, c_n\rangle=P(c_1,\cdots,c_n)\}$, if $c_1,\cdots,c_n\in B$
\begin{itemize}
\item L3A. $(\{c_1,\cdots,c_n\}\rightarrow P)/\langle\{c_1,\cdots,c_n \}\rangle = P$
\end{itemize}
\item $traces(P/\langle s\rangle)=\{t|s\smallfrown t\in traces(P)\}$, if $s\in traces(P)$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of after.
\end{proof}
\begin{definition}[Change of Symbol]
The change of symbol for concurrent traces is a function $f^\ast:A^\ast\rightarrow B^\ast$ with $f:A\rightarrow B$.
\end{definition}
\begin{proposition}[Laws of Change of Symbol]
The laws of change of symbol for concurrent traces are the same as for traces in CSP, we retype them as follows.
\begin{itemize}
\item L1. $f^\ast(\langle\rangle)=\langle\rangle$
\item L2. $f^\ast(\langle \{x_1,\cdots,x_n\}\rangle)=\langle f(x_1,\cdots,x_n)\rangle$
\item L3. $f^\ast(s\smallfrown t)=f^\ast(s)\smallfrown f^\ast(t)$
\item L4. $f^\ast(s)_0=f(s_0)$, if $s\neq\langle\rangle$
\item L5. $\sharp f^\ast(s)=\sharp s$
\item L6. $f^\ast(s\upharpoonright A)=f^\ast(s)\upharpoonright f(A)$, if $f$ is an injection.
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of change of symbol.
\end{proof}
\begin{definition}[Another Catenation]
If $s$ is a sequence and all its elements are also sequences, another catenation $\smallfrown/s$ is defined as catenating all the elements together in the original order.
\end{definition}
\begin{proposition}[Laws of Another Catenation]
The laws of another catenation for concurrent traces are the same as for traces in CSP, we retype them as follows.
\begin{itemize}
\item L1. $\smallfrown/\langle\rangle=\langle\rangle$
\item L2. $\smallfrown/\langle s\rangle=s$
\item L3. $\smallfrown/(s\smallfrown t)=(\smallfrown/s)\smallfrown(\smallfrown/t)$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of another catenation.
\end{proof}
\begin{definition}[Concurrent Composition]
A sequence $s$ is a concurrent composition ($\textsf{cc}$) of two sequences $t$ and $u$, if it can be composed one by one from $t$ and $u$.
\end{definition}
\begin{proposition}[Laws of Concurrent Composition]
The laws of concurrent composition for concurrent traces are as follows.
\begin{itemize}
\item L1. $\langle\rangle \textsf{ cc } s\equiv s$
\item L2. $s\textsf{ cc } \langle\rangle\equiv s$
\item L3. $(\langle x \rangle)\smallfrown s \textsf{ cc } \langle y\rangle\smallfrown t\equiv (\langle\{x,y\}\rangle\smallfrown(s\textsf{ cc } t))$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of concurrent composition.
\end{proof}
\begin{definition}[Subscription]
The $i^{th}$ element of the sequence $s$ denoted $s[i]$ for $=\leq i\leq \sharp s$.
\end{definition}
\begin{proposition}[Laws of Subscription]
The laws of subscription for concurrent traces are the same as for traces in CSP, we retype them as follows.
\begin{itemize}
\item L1. $s[0]=s_0\wedge s[i+1]=s'[i]$, if $s\neq\langle\rangle$
\item L2. $(f^\ast(s))[i]=f(s[i])$, for $i<\sharp s$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of subscription.
\end{proof}
\begin{definition}[Reversal]
The reversal of a sequence $s$ denoted $\overline{s}$, is obtained by taking its elements in reverse order.
\end{definition}
\begin{proposition}[Laws of Reversal]
The laws of reversal for concurrent traces are as follows.
\begin{itemize}
\item L1. $\overline{\langle\rangle}=\langle\rangle$
\item L2. $\overline{\langle\{x_1,\cdots,x_n\}\rangle}=\langle\{x_1,\cdots,x_n\}\rangle$
\item L3. $\overline{s\smallfrown t}=\overline{t}\smallfrown\overline{s}$
\item L4. $\overline{\overline{s}}=s$
\item L5. $\overline{s}[i]=s[\sharp s-i-1]$, for $i\leq\sharp s$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of reversal.
\end{proof}
\begin{definition}[Selection]
The selection $s\downarrow \{x_1,\cdots, x_n\}$ of a sequence of pairs $s$ is obtained by replacing each pair by its second element, if its first element is $\{x_1,\cdots, x_n\}$. \end{definition}
\begin{proposition}[Laws of Selection]
The laws of selection for concurrent traces are as follows.
\begin{itemize}
\item L1. $\langle\rangle\downarrow \{x_1,\cdots,x_n\}=\langle\rangle$
\item L2. $(\langle\{y_1,\cdots,y_m\}.\{z_1,\cdots,z_k\}\rangle\smallfrown t\downarrow\{x_1,\cdots,x_n \})=t\downarrow\{x_1,\cdots,x_n\}$, if $m\neq n$ or $y_i\neq x_i(1\leq i\leq m=n)$
\item L3. $(\langle\{x_1,\cdots,x_n\}.\{z_1,\cdots,z_k\}\rangle\smallfrown t)\downarrow\{x_1,\cdots,x_n\}=\langle\{z_1,\cdots,z_k\}\rangle\smallfrown(t\downarrow \{x_1,\cdots,x_n\})$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of selection.
\end{proof}
\begin{definition}[Composition]
The composition of sequences $s$ and $t$ denoted $(s;t)$, means that when $s$ is successfully terminated (denoted $\surd$), $s$ starts.
\end{definition}
\begin{proposition}[Laws of Composition]
The laws of composition for concurrent traces are as follows.
\begin{itemize}
\item L1. $s;t=s$, if $\neg(\langle\surd\rangle$ in $s)$
\item L2. $(s\smallfrown\langle\surd\rangle);t=s\smallfrown t$, if $\neg(\langle\surd\rangle$ in $s)$
\begin{itemize}
\item L2A. $(s\smallfrown\langle\surd\rangle\smallfrown u);t=s\smallfrown t$, if $\neg(\langle\surd\rangle$ in $s)$
\end{itemize}
\item L3. $s;(t;u)=(s;t);u$
\item L4A. $s\leq t\Rightarrow ((u;s)\leq(u;t))$
\item L4B. $s\leq t\Rightarrow ((s;u)\leq(t;u))$
\item L5. $\langle\rangle;t=\langle\rangle$
\item L6. $\langle\surd\rangle;t=t$
\item L7. $s;\langle\surd\rangle=s$, if $\neg(\langle\surd\rangle$ in $(\overline{s})')$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of composition.
\end{proof}
\begin{definition}[Specification]
A specification of a product $P$ is a description of the way it is intended to behave, which contains free variables standing some observable behaviors. A specification is denoted by $\mathcal{S}(tr)$, where $tr$ are free variables.
\end{definition}
\begin{definition}[Satisfaction]
A product $P$ satisfies a specification $\mathcal{S}$ when $P$ meets $\mathcal{S}$, denoted by $P \textsf{ sat }\mathcal{S}$.
\end{definition}
\begin{proposition}[Laws of Specification and Satisfaction]
The laws of specification and satisfaction for concurrent traces are the same as for traces in CSP, and we retype them as follows.
\begin{itemize}
\item L1. $P\textsf{ sat }true$
\item L2A. If $P\textsf{ sat }\mathcal{S}$ and $P\textsf{ sat }\mathcal{T}$, then $P\textsf{ sat }(\mathcal{S}\wedge\mathcal{T})$
\item L2. If $\forall n\bullet(P\textsf{ sat }\mathcal{S}(n))$, then $P\textsf{ sat }(\forall n\bullet\mathcal{S}(n))$, if $P$ has not relation to $n$
\item L3. If $P\textsf{ sat }\mathcal{S}$ and $\mathcal{S}\Rightarrow\mathcal{T}$, then $P\textsf{ sat }\mathcal{T}$
\item L4A. $STOP\textsf{ sat }(tr=\langle\rangle)$
\item L4B. If $P\textsf{ sat }\mathcal{S}(tr)$, then $(\{c_1,\cdots, c_n\}\rightarrow P)\textsf{ sat }$ $(tr=\langle\rangle\vee (tr_0=\{c_1,\cdots,c_n\}\wedge\mathcal{S} (tr')))$
\item L4C. If $P\textsf{ sat }\mathcal{S}(tr)$, then $(\{c_1,\cdots,c_n\}\rightarrow \{d_1,\cdots,d_m\}\rightarrow P)\textsf{ sat }$ $(tr\leq \langle \{c_1,\cdots,c_n\} ,\{d_1,\cdots,d_m\}\rangle\vee (tr\geq\langle \{c_1,\cdots,c_n\},\{d_1,\cdots,d_m\}\rangle\wedge\mathcal{S}(tr'')))$
\item L4D. If $P\textsf{ sat }\mathcal{S}(tr)$ and $Q\textsf{ sat }\mathcal{T}(tr)$, then $(\{c_1,\cdots,c_n\}\rightarrow P\mid\{d_1,\cdots,d_m\}\rightarrow Q)\textsf{ sat }$ $(tr=\langle\rangle \vee (tr_0=\{c_1,\cdots,c_n\}\wedge \mathcal{S}(tr')) \vee (tr_0=\{d_1,\cdots,d_m\}\wedge\mathcal{T}(tr')))$
\item L4. If $\forall \{x_1,\cdots,x_n\}:B\bullet(P(x_1,\cdots,x_n)\textsf{ sat }\mathcal{S}(tr,x_1,\cdots,x_n))$, then $(\{x_1,\cdots,x_n\}:B\rightarrow P(x_1,\cdots,x_n))\textsf{ sat }$ $(tr=\langle\rangle \vee(tr_0\in B\wedge \mathcal{S}(tr',tr_0)))$
\item L5. If $P\textsf{ sat }\mathcal{S}(tr)$ and $s\in traces(P)$, then $P/s\textsf{ sat }\mathcal{S}(s\smallfrown tr)$
\item L6. If $F(X)$ is guarded and $STOP\textsf{ sat }\mathcal{S}$ and $((X\textsf{ sat }\mathcal{S})\Rightarrow(F(X)\textsf{ sat }\mathcal{S}))$, then $\mu X\bullet F(X)\textsf{ sat }\mathcal{S}$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definitions of specification and satisfaction.
\end{proof}
\section{Concurrency}\label{concurrency}
In this section, we will inspect concurrency for CCP, including interaction and concurrency.
\subsection{Interaction}
\begin{definition}[Interaction]
Two processes interact with each other, and with the same alphabet, denoted $P\parallel Q$.
\end{definition}
\begin{proposition}[Laws of Interaction]
The laws of interaction of CCP are almost the same as those of CSP, we retype them as follows.
\begin{itemize}
\item L1. $P\parallel Q = Q\parallel P$
\item L2. $P\parallel (Q\parallel R)=(P\parallel Q)\parallel R$
\item L3A. $P\parallel STOP_{\alpha P}=STOP_{\alpha P}$
\item L3B. $P\parallel RUN_{\alpha P}=P$
\item L4A. $(c\rightarrow P)\parallel (c\rightarrow Q)=(c\rightarrow(P\parallel Q))$
\item L4B. $(c\rightarrow P)\parallel (d\rightarrow Q)=STOP$, if $c\neq d$
\item L4. $(x:A\rightarrow P(x))\parallel (y:B\rightarrow Q(y))=(z:(A\cap B)\rightarrow (P(z)\parallel Q(z)))$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of interaction.
\end{proof}
For the implementation of interaction operator $\parallel$ by LISP, we omit it.
\begin{proposition}[Laws of Traces of Interaction Operator]
The laws of traces of interaction operator are as follows.
\begin{itemize}
\item L1. $traces(P\parallel Q)=traces(P)\cap traces(Q)$
\item L2. $(P\parallel Q)/s=(P/s)\parallel(Q/s)$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definitions of interaction and concurrent traces.
\end{proof}
\subsection{Concurrency}
\begin{definition}[Concurrency]
Concurrency is a generalization of the case of interaction, while $P$ and $Q$ may have different alphabets, and it is also denoted by $P\parallel Q$.
\end{definition}
\begin{proposition}[Laws of Concurrency]
The laws of concurrency of CCP are quite different to those of CSP, we give them as follows.
\begin{itemize}
\item L1. $P\parallel Q = Q\parallel P$
\item L2. $P\parallel (Q\parallel R)=(P\parallel Q)\parallel R$
\item L3A. $P\parallel STOP_{\alpha P}=STOP_{\alpha P}$
\item L3B. $P\parallel RUN_{\alpha P}=P$
\item L4. $(c\rightarrow P)\parallel (c\rightarrow Q)=(c\rightarrow(P\parallel Q))$
\item L5. $(c\rightarrow P)\parallel (d\rightarrow Q)=(\{c,d\}\rightarrow(P\parallel Q))$, if $c\neq d$
\item L6. $(x:A\rightarrow P(x))\parallel (y:B\rightarrow Q(y))=(z:(A\cup B)\rightarrow (P\parallel Q))$, $z=x=y$, otherwise, z=\{x,y\}
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definition of concurrency.
\end{proof}
The concurrency operator also can be implemented by LISP, we omit the implementation.
\begin{proposition}[Laws of Traces of Concurrency Operator]
The laws of traces of concurrency operator are as follows.
\begin{itemize}
\item L1. $traces(P\parallel Q)=traces(P)\cap traces(Q)$, if $\alpha P=\alpha Q$
\item L2. $traces(P\parallel Q)=traces(P)\textsf{ cc } traces(Q)$, if $\alpha P\cap\alpha Q=\{\}$
\item L3. $(P\parallel Q)/s=(P/(s\upharpoonright\alpha P))\parallel(Q/(s \upharpoonright\alpha Q))$
\item L4. $traces(P\parallel Q)=\{t|t\in(\alpha P\cup\alpha Q)^\ast\}$
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definitions of concurrency and concurrent traces.
\end{proof}
\subsection{Specification}
Let $P$ and $Q$ be processes, $tr\in traces(P\parallel Q)$, and $P\textsf{ sat }\mathcal{S}(tr)$, $Q\textsf{ sat }\mathcal{T}(tr)$. So $(tr\upharpoonright\alpha P)\in traces(P)$
and $P\textsf{ sat }\mathcal{S}(tr\upharpoonright\alpha P)$, and also $tr\upharpoonright\alpha Q\in traces(Q)$ and $Q\textsf{ sat }\mathcal{T}(tr\upharpoonright\alpha Q)$.
\begin{proposition}[Laws of Specification for Concurrency]
The laws of specification for concurrency operator are as follows.
\begin{itemize}
\item L1. If $P\textsf{ sat }\mathcal{S}(tr)$, and $Q\textsf{ sat }\mathcal{T}(tr)$, then
$(P\parallel Q)\textsf{ sat }(\mathcal{S}(tr\upharpoonright\alpha P)\wedge \mathcal{T}(tr\upharpoonright\alpha Q))$
\item L2. If $P$ and $Q$ never stop, then $P\parallel Q$ never stops.
\end{itemize}
\end{proposition}
\begin{proof}
These laws can be proven straightforwardly from the related definitions of concurrency and specification.
\end{proof}
\subsection{Mathematical Theory of Deterministic Processes}
The laws in the above two sections are true in intuition, to make them strict, it is needed to give those denotational semantics.
Fortunately, the denotational semantics of each operator of deterministic processes are the same as those in CSP, including the interaction operator and concurrency operator.
And the proofs of the existence of solution for recursive equation in the law Proposition \ref{LoP}.L2 and the uniqueness of the solution by use of Scott's fix-point theory,
are also the same as those of CSP, we omit them (please refer to CSP \cite{CSP}).
\section{Other Contents of CCP}\label{occ}
We inspect the rest of CCP corresponding to CSP, including nondeterminism, communication, etc, and we find that the rest of CCP is almost the same as the corresponding parts of CSP. So, there is not necessary to retype the quite long contents again. Indeed, the main difference between CCP and CSP is the treatment of concurrency.
\subsection{Nondeterminism}
The nondeterminism is defined by internal choice operator, general choice operator, refusals, concealment, but no interleaving operator. The nondeterminism of CCP are almost the same as that of CSP, we omit it.
\subsection{Communication}
The communication is defined by input and output, the communication of CCP is also almost the same as CSP, we also omit it.
\section{Conclusions}\label{con}
We modify CSP to CCP, and the main difference between the two algebras is the treatment of concurrency. CSP only permits a process to engage in one event on a moment and record only this one event in the process traces, while CCP permits a process to engage in multi events simultaneously and record simultaneously these events in the process traces.
|
{
"timestamp": "2018-03-08T02:08:44",
"yymm": "1803",
"arxiv_id": "1803.02659",
"language": "en",
"url": "https://arxiv.org/abs/1803.02659"
}
|
\section{Conclusion}
\label{conclusion}
Along the article we have studied the implications of adapting a wired bridging protocol to a wireless environment. Apart from the specificities of the implementation, we have discovered to main drawbacks during the migration:
\begin{enumerate}
\item \textbf{Frame format}: Wired bridging protocols are mainly Ethernet-based, while in wireless the frame format is diverse and we had to choose one to continue the implementation (i.e. moving from one frame format to other might not be necessarily straightforward). This affects our protocol as is based on layer 2, but it would not affect layer 3 routing protocols.
\item \textbf{Flooding and scalability}: While broadcast in wired networks might be relatively useful in some scenarios, in wireless networks might be totally unacceptable. More specifically, wireless networks always flood the information per se (the radio signal is received by all nodes in the range), so adding an overhead to the forwarding protocol should imply --at least-- saving time in processing the frame, i.e. avoiding broadcasting the frame whenever possible.
\end{enumerate}
Additionally, some mechanisms such as path repair are not applicable to wireless networks, which imply redesigning parts of the protocol (e.g. to define some type of keep-alive or mobility-awareness mechanism).
Therefore, although ARP-Path is a simple and efficient protocol for wired bridging, wARP-Path did not show the equivalent benefits for wireless networks. The main conclusion is that wireless bridging protocols should be more efficient than routing protocols in different aspects (for example, drastically decreasing table size), with a groundbreaking approach, otherwise protocols as AODV or even applying SDN might still be more suitable for wireless networks.
\section{Discussion}
\label{discussion}
Finally, we discuss different aspects of the wARP-Path protocol. Particularly, in some of the topics we compare wARP-Path to AODV, which we consider the most similar protocol, as they are both reactive routing protocols for layer 2 and layer 3, respectively. \\
\textbf{Layer 2 vs. Layer 3} \\
wARP-Path acts on Layer 2, and since this protocol tries to find the shortest paths with the least load and delay, so it can as well take advantages of this layer on the network, such as higher speed and lack of processing latency due to layer 3 routing. \\
Since on-demand routings are based on query reply, they endure a delay to find a route~\cite{Iwata99}. However, ARP-Path uses standard ARP Request and ARP Reply packets for exploring a path, it does not include this delay. \\
\textbf{Scalability} \\
In on-demand protocols, when a node wants to communicate with a destination, the route is computed. Therefore, they do not store route information for all destinations permanently, so this feature increases their scalability to be used in large networks~\cite{Iwata99}. Despite this common feature in both wARP-Path and AODV, each protocol has unique features which cause differences in their scalability. In spite of the mentioned advantages of Layer 2 for wARP-Path, this protocol uses the flat addressing structure in layer 2, and AODV uses the hierarchical structure of IP addressing in Layer 3. Hierarchical routing increasingly reduces the size of routing tables and processing overhead~\cite{Iwata99}, whereas using the flat addressing structure raises the problem of increasing the number of entries in the forwarding tables. \\
In general, we found that scalability was the first issue when migrating the ARP-Path protocol to wireless networks. \\
\textbf{Stability vs Mobility} \\
Wired networks are more stable than wireless networks due to their nature in using fixed nodes and links. This is one of the reasons why the protocols used in these two types of networks are different. Since the All-Path protocols have been designed for wired networks, they are inherently stable. ARP-Path, as one of these protocols, tries to maintain the stability of the path until the end of each communication unless network physical stability is lost. In this case, by sending the path recovery messages, it starts to discover the path in the unstable parts of the network. The stability of paths is maintained by refreshing the lifetime of the entries in the forwarding tables until the end of a communication. \\
But in AODV, designed for using wireless networks, this bound of stability seen in the ARP-Path is not seen here. The discovered routes in AODV are not necessarily kept to the end of communication, and life time of the routes in the routing tables are not updated by transferring data packets, they are updated only by transferring protocol packets such as RREQ and RREP. The existence of this feature in AODV increases its efficiency to overcome the unstable state of nodes and links in wireless networks. \\
\textbf{Path repair} \\
Another issue found when migrating the protocol was path repair. In wired networks, link failure detection is direct, as usually the physical layer provides this feature. However, link failure detection in wireless networks requires additional mechanisms, not only to probe if neighbors are still available, but also to guarantee if packets are effectively reaching their destinations. \\
Additionally, the path repair mechanism in ARP-Path requires broadcasting and might be too costly for wireless networks, especially when their nodes have mobility. So simple methods such as broadcasting might be more efficient. That is the main reason why we did not implement and test path repair in wARP-Path. \\
\textbf{Wireless frame format} \\
ARP-Path leverages the fact that the most common frame format in wired networks is Ethernet. However, in wireless, there is a wide range of layer 2 frames and protocols, such as WiFi, WiFi-Direct, Bluetooth, Bluetooth SMART or BLE~\cite{Gomez12}, LR-WPAN (802.15.4) or Zigbee. \\
This diversity affects the implementation of wARP-Path, which might have variations depending on the layer 2 implemented. \\
\textbf{Software-Defined Networking} \\
Software-Defined Networking (SDN)~\cite{Kreutz15} is flourishing rapidly and, although the initial deployments were based on wired networks, wireless networks are also targeted as part of the SDN spectrum, which makes harder the appearance of new distributed protocols. \\
Thus, wARP-Path is not good enough to beat the advantages of SDN and a more disruptive approach to wireless networks should be applied, instead of a simple migration of a protocol from wired to wireless.
\section{Evaluation}
\label{evaluation}
This chapter is devoted to evaluate wARP-Path. An initial thought was to compare it with AODV, but we found a bug in the implementation of AODV in the INET framework \footnote{More specifically, some \texttt{RREQ} packets are deleted, and it has a negative impact on the end-to-end delay and quality of discovered routes of AODV.}. Therefore, we finally decided to exclude the evaluation of AODV (as results were not reliable) from our analysis.
In this evaluation, we first define the test cases and scenarios. We then briefly compare wARP-Path and ARP-Path, and finally we analyze wARP-Path in terms of \emph{goodput ratio} and \emph{average end-to-end delay}.
\subsection{Definition of test cases and scenarios}
In order to evaluate the wARP-Path protocol, our model was inspired by~\cite{perkins1999aodv} and~\cite{broch1998performance}. We defined 50 nodes uniformly distributed within a fixed-size area of 1500 * 1500. To manage centralized behaviors such as: (1) the uniform election of a source and destination node between all nodes to start a session, (2) the selection of traffic for a session based on Table~\ref{table:traffic}, or (3) the computation of the average metrics for all nodes and flows in the network, we defined a module called \texttt{flowGenerator} in the simulation. Interval time of the simulation is 600s and the first session is started at time 0.2s. Next session is
started after 10 seconds and other sessions are
started after the same time, periodically.
By using a fixed time (i.e. 10s) instead of a random time in an interval, we adopted enough interval between the flows so that we can analyze the effect of the ARP Request messages on other flows. The total number of sessions in the simulation is 10.
\begin{table}[!h]
\centering
\caption{Traffic Parameters}
\label{table:traffic}
\begin{tabular}{|l|l|l|}
\hline
\textbf{Parameter}& \textbf{S\_DATA}& \textbf{VOICE}\\
\hline
Transport protocol& UDP& UDP\\
Session interval& Geometric (mean 900)& Geometric (mean 600)\\
Packet size& 64 Bytes& 160 Bytes\\
Packet send interval & 20 ms& 20 ms\\
\hline
\end{tabular}
\end{table}
Two types of traffic will be analyzed: S\_DATA (that stands for \textit{small data}) and VOICE, as defined in~\cite{perkins1999aodv}. \\
To generate S\_DATA traffic, as it can be seen in Table~\ref{table:traffic}, data packets contain 64 bytes, and inter-arrival time of data packets is 20 ms. Therefore, this traffic is produced with a rate of 25,6 Kbps in each selected host as a source per session. \\
To generate VOICE traffic, as also illustrated in Table~\ref{table:traffic}, we suppose that quality of voice is telephony, so sampling frequency is 8 KHz. Since each sample is expressed with 1 byte, each selected host as a source per session produces a traffic with rate of 64 Kbps. Given that each packet is generated every 20 ms, the packet size is 160 bytes.
Each node has one transmitter and one receiver, and all nodes use the same channel model. Their physical and MAC layers are based on IEEE 802.11~\cite{ieee80211}, considering the \emph{Distributed Coordination Function (DCF)} mode, whose properties are shown in Table~\ref{table:simulation}.
\begin{table}[!h]
\centering
\caption{Simulation Parameters}
\label{table:simulation}
\begin{tabular}{|l|l|}
\hline
\textbf{Parameter}& \textbf{Value}\\
\hline
Simulation time& 600 s\\
Traffic generation start time& 0.2 s \\
Traffic generation start time& 600 s\\
\hline
\multicolumn{2}{|c|}{\textbf{wARP-Path}}
\\\hline
LT aging time& 120 s\\
BT blocking time& 1 s\\
MAXJITTER& 5 ms\\
Jitter& uniform(0, MAXJITTER)\\
\hline
\multicolumn{2}{|c|}{\textbf{ARP protocol}}
\\\hline
ARP retry count& 5\\
ARP retry timeout& 200 ms\\
ARP cache timeout& 120 s\\
\hline
\multicolumn{2}{|c|}{\textbf{MAC layer}}
\\\hline
IEEE802.11 type& IEEE802.11 g\\
Maximum size of queue& 14\\
MAC retry limit& 7\\
CW min (for S\_DATA)& 15 time slots\\
CW min (for VOICE)& 20 time slots\\
\hline
\multicolumn{2}{|c|}{\textbf{Phy layer}}
\\\hline
Carrier frecuency& 2.4 GHz\\
Bandwidth& 2 MHz\\
Modulation scheme& BPSK + DSSS\\
Bit rate& 1 Mbps\\
Transmit Power& 2 mW\\
Receiver sensitivity& -85 dBm\\
SINR threshold& 4dB\\
Energy detection threshold& -85 dBm\\
Path loss type& Free Space Path Loss\\
Background noise power& -110 dBm\\
\hline
\end{tabular}
\end{table}
\subsection{Comparison with ARP-Path}
The wARP-Path protocol is quite similar to ARP-Path. But in wARP-Path, since intermediate nodes are hosts (not only switches) that can create a new session to each destination, they can use the paths explored by each node that is not an intermediate node. According to computation of forwarding state in the ARP-Path protocol~\cite{Rojas15}, in the worst case of ARP-Path, when all nodes (non-intermediate) communicated with each other, the paths between some non-intermediate nodes and some intermediate nodes had been creating. Now, in wARP-Path, since intermediate nodes have the capability to start a session with non-intermediate nodes, they can start some additional sessions without adding new entries in the forwarding tables, and this amount of communications without inserting a new entry in tables is a payoff of wARP-Path against ARP-Path protocol. Briefly, with the same number of entries in forwarding tables, wARP-Path might have a higher number of active communications than ARP-Path.
\subsection{Goodput Ratio}
\begin{figure*}
\centering
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth] {figs/good_sdata_cw15.png}
\caption{S\_DATA}
\label{fig:good:sdata}
\end{subfigure}%
\quad
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/good_voice_cw20.png}
\caption{VOICE}
\label{fig:good:voice}
\end{subfigure}
\caption{Achieved Goodput Ratio}\label{fig:good}
\end{figure*}
\begin{table*}[!h]
\centering
\caption{Generated flows}
\label{table:flows}
\begin{tabular}{|l|l|l|l|l|l|l|l|c|}
\hline
\multicolumn{9}{|c|}{\textbf{S\_DATA Traffic}}
\\\hline
\textbf{Session \#}& \textbf{Source}& \textbf{Destination}& \textbf{Amount of Traffic}& \textbf{Start Time}& \textbf{ End Time}& \textbf{\# ARP Requests}& \textbf{\# ARP Replies}& \textbf{Path Discovered}\\
\hline
1& Host 42& Host 24& 3.5424e+006 B& 0.2 s& 1107.2 s& 163& 19& \checkmark\\
2& Host 3& Host 8& 2.0576e+006 B& 10.2 s& 653.2 s& 5& 5& \checkmark\\
3& Host 21& Host 19& 6.5216e+006 B& 20.2 s& 2058.2 s& 3165 (flows 3, 5)& 0& -\\
4& Host 43& Host 41& 665600 B& 30.2 s& 238.2 s& 2& 2& \checkmark\\
5& Host 21& Host 38& 1.6704e+006 B& 40.2 s& 562.2 s& 3165 (flows 3, 5)& 29& \checkmark\\
6& Host 20& Host 44& 451200 B& 50.2 s& 191.2 s& 51& 7& \checkmark\\
7& Host 39& Host 14& 3.0144e+006 B& 60.2 s& 1002.2 s& 71& 16& \checkmark\\
8& Host 26& Host 17& 1.2544e+006 B& 70.2 s& 462.2 s& 2000& 0& -\\
9& Host 22& Host 2& 2.5696e+006 B& 80.2 s& 883.2 s& 6& 6& \checkmark\\
10& Host 1& Host 26& 860800 B& 90.2 s& 359.2 s& 1375& 0& -\\
\hline
\multicolumn{9}{|c|}{\textbf{VOICE Traffic}}
\\\hline
\textbf{Session \#}& \textbf{Source}& \textbf{Destination}& \textbf{Amount of Traffic}& \textbf{Start Time}& \textbf{ End Time}& \textbf{\# ARP Requests}& \textbf{\# ARP Replies}& \textbf{Path Discovered}\\
\hline
1& Host 42& Host 24& 5.904e+006 B& 0.2 s& 738.2 s& 18& 5& \checkmark\\
2& Host 3& Host 8& 3.424e+006 B& 10.2 s& 438.2 s& 2& 2& \checkmark\\
3& Host 21& Host 19& 1.0864e+007 B& 20.2 s& 1378.2 s& 2818 (flows 3, 5)& 0& -\\
4& Host 43& Host 41& 1.112e+006 B& 30.2 s& 169.2 s& 1& 1& \checkmark\\
5& Host 21& Host 38& 2.784e+006 B& 40.2 s& 388.2 s& 4 (flows 3, 5)& 4& \checkmark\\
6& Host 20& Host 44& 752000 B& 50.2 s& 144.2 s& 16& 3& \checkmark\\
7& Host 39& Host 14& 5.016e+006 B& 60.2 s& 687.2 s& 54& 8& \checkmark\\
8& Host 26& Host 17& 2.088e+006 B& 70.2 s& 331.2 s& 525& 0& -\\
9& Host 22& Host 2& 4.28e+006 B& 80.2 s& 615.2 s& 2& 2& \checkmark\\
10& Host 1& Host 26& 1.432e+006 B& 90.2 s& 269.2 s& 360& 0& -\\
\hline
\end{tabular}
\end{table*}
The goodput ratio is calculated over the entire network when a host generates or receives a packet on the network. Therefore, goodput ratio is a function of time in the simulation time interval, and we denote it as follows:
$${GoodputRatio(t)} = {\frac {n^r_t} {n^g_t} \times 100}$$ \\
where $n^g_t$ is the number of bytes in packets generated by the application layer of hosts from the start time of the simulation until time $t$, and $n^r_t$ is the number of bytes in packets received by the application layer of hosts from the start time of simulation until time $t$. The reason of using number of bytes instead of the number of packets is because the last packet of each session might vary in size, depending of the session, although this situation did not occur in our traffic.
Figures~\ref{fig:good:sdata} and~\ref{fig:good:voice} respectively show the achieved goodput ratio as a function of time for S\_DATA and VOICE traffic with 10 sessions. As depicted in the figures, at the initial moments of the simulation, goodput ratio is low since source hosts only generated packets, and no destination host received packets. After this small interval, once destinations received packets, the goodput ratio increasingly grows near to 100\%. This situation is stable until the number of sessions and broadcasts increase. \\
Although using \emph{jitter} is a good approach to overcome broadcast problems, this problem can still cause collision with other broadcast (simultaneously forwarding), \emph{RTS} (simultaneously forwarding), \emph{CTS} (when the \emph{RTS} sender is hidden from broadcast sender), or even \emph{ACK} or \emph{data} (when \texttt{CTS} has collided and a node that is hidden from the \texttt{RTS} sender sends a broadcast) packets in dense and high broadcast scenarios. Therefore, when the number of broadcasts increases in the network, as mentioned in previous section, it negatively affects on the channel capacity and quality of the explored paths (and even a path can not be discovered though it exists), which causes reduction of the goodput ratio. \\
Time interval between two consecutive packets is the same in both traffic types, but since VOICE packets are bigger than S\_DATA packets, the probability of collision increases with VOICE packets, and the loss of a packet has a greater impact on its respective goodput ratio.
The traffic is \texttt{UDP}, connection-less and unreliable transport protocol, and there is no hand shaking, so traffic is continuously injected to lower layer. As shown in Table~\ref{table:flows} (i.e. sessions 3, 8, and 10 in both traffic types), a lot of traffic is wasted before a path is found (or there is no path at all), which causes that the goodput ratio stays at the same low value. For S\_DATA traffic, they produce 3970560 bytes until the end of simulation. If we reduce this value (i.e. total sent bytes in Table~\ref{table:results}), the goodput ratio is increased up to 48.12\%.
As shown in Table~\ref{table:flows}, another impact of the injected traffic to lower layers are the ARP Request messages, which are frequently sent to the network. Collisions resulting from these broadcasts will negatively affect the goodput ratio and the delay. Collision of the broadcast packets with control and data packets will cause the MAC layer to increase CW and select a back-off value in a larger range. Therefore, delay increases in the interval in which ARP Request messages enter the network, and also using the jitter will cause a delay in starting a transmission.
\subsection{End-to-end Delay}
\begin{figure*}
\centering
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/e2e_intrvl_sdata_cw15.png}
\caption{Average end-to-end delay computed in each interval for all hosts}
\label{fig:e2e:S-DATA:invl}
\end{subfigure}%
\quad
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/e2e_all_sdata_cw15.png}
\caption{Average end-to-end delay of all hosts}
\label{fig:e2e:S-DATA:all}
\end{subfigure}
\quad
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/e2e_hosts_sdata_cw15.png}
\caption{Average end-to-end delay of each host}
\label{fig:e2e:S-DATA:host}
\end{subfigure}
\caption{Achieved end-to-end delay for S\_DATA traffic}\label{fig:e2e:S-DATA}
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/e2e_intrvl_voice_cw20.png}
\caption{Average end-to-end delay computed in each interval for all hosts}
\label{fig:e2e:VOICE:invl}
\end{subfigure}%
\quad
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/e2e_all_voice_cw20.png}
\caption{Average end-to-end delay of all hosts}
\label{fig:e2e:VOICE:all}
\end{subfigure}
\quad
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/e2e_hosts_voice_cw20.png}
\caption{Average end-to-end delay of each host}
\label{fig:e2e:VOICE:host}
\end{subfigure}
\caption{Achieved end-to-end delay for VOICE traffic}\label{fig:e2e:VOICE}
\end{figure*}
\begin{table}[!h]
\centering
\caption{Results at the end of simulation}
\label{table:results}
\begin{tabular}{|l|l|l|}
\hline
& \textbf{S\_DATA}& \textbf{VOICE}\\
\hline
\textbf{Goodput ratio}& 34.22\%& 26.99\%\\
\textbf{Average end-to-end delay}& 0.0431 s& 0.0579 s\\
\textbf{Average end-to-end delay last intvl}& 0.0081 s& 0.0316 s\\
\textbf{Total sent bytes}& 1.3742848E7& 2.95056E7\\
\textbf{Total sent packets}& 214732& 184410\\
\textbf{Total received bytes}& 4702784& 7964160\\
\textbf{Total received packets}& 73481& 49776\\
\hline
\end{tabular}
\end{table}
Additionally, we computed the average end-to-end delay of the network based on the following equation for both each host and all possible destinations in the network.
$${\overline {Delay_{e2e}}(t)} = {\frac {\sum_{i\in \{p \mid t_{r_p} \le t\}} d_i} {n_{set}}}, {d_i = t_{r_i} - t_{g_i}}$$
Considering the equation, $p$ is a packet received to the host for which we want to calculate average end-to-end delay, $t_{r_p}$ is the arrival time of packet $p$ at the application layer, $d_i$ denotes the end-to-end delay of packet $i$ which is obtained by subtracting the arrival time of the packet ($t_{r_i}$) from the generation time of the packet ($t_{g_i}$), and ${n_{set}}$ is the number of elements in the set. In case of calculating the average end-to-end delay of all the network, $p$ is the packet received by one host.
Figures~\ref{fig:e2e:S-DATA:host} and~\ref{fig:e2e:VOICE:host} respectively show the average end-to-end delay of each host for the traffics of S\_DATA and VOICE for 10 sessions. Figures~\ref{fig:e2e:S-DATA:all} and~\ref{fig:e2e:VOICE:all} respectively show the average end-to-end delay of all the network for the traffics of S\_DATA and VOICE with 10 sessions.
The values just mentioned give us good information about the overall and current state of the network. To calculate end-to-end delay, since there are peaks at the end-to-end delay of the network and in order to balance these peaks, we obtain the average end-to-end delay in the small intervals (for example, this interval we will denoted by $\Delta t$, and it is 1 second in our simulation) based on the following equation, instead of pure end-to-end delay (as shown in Figures~\ref{fig:e2e:S-DATA:invl} and~\ref{fig:e2e:VOICE:invl}).
$${\overline {IntervalDelay_{e2e}}(t)} = {\frac {\sum_{i\in \{p \mid \Delta t \lfloor \frac {t}{\Delta t} \rfloor \le t_{r_p} < \Delta t \lfloor \frac {t}{\Delta t} \rfloor + \Delta t\}} d_i} {n_{set}}}$$
$$, {d_i = t_{r_i} - t_{g_i}}$$
\section{Implementation}
\label{implementation}
\begin{figure*}[hbt]
\centering
\includegraphics[width=0.9\textwidth]{figs/AdhocHostAPB.png}
\caption{Implementation of the \texttt{AdhocHostAPB} module in OMNeT++ 4.2.2 and INET 2.0.0}
\label{fig:adhochostapb}
\end{figure*}
As mentioned in the introduction, the ARP-Path protocol had been previously implemented in different platforms. However, most of them are intended for wired networks. Therefore, we decided to implement wARP-Path in the OMNeT++ simulator, which was the fastest alternative to check the suitability of the protocol for wireless networks.
\subsection{Implementation in OMNeT++ 4.2.2 and INET 2.0.0}
The wARP-Path was first developed by modifying some parts of the modules defined for ad hoc networks in the INET framework for OMNeT++, specifically the modification was done in the so-called \texttt{AdHocHost} module which was converted into the a new module called \texttt{AdHocHostAPB} by adding a relay submodule in it, as it can be seen in Fig.~\ref{fig:adhochostapb}.
The \texttt{AdHocHost} module implements, as it name recalls, a host for wireless ad hoc communications. This module is composed of several submodules. For instance, in the lower part, we can see several submodules directly related to the physical layer (\texttt{wlan}, \texttt{eth}, \texttt{ppp}, etc), while in the upper part there are modules associated to the application (\texttt{tcpApp}, \texttt{udpApp}, etc) and transport (\texttt{tcp}, \texttt{udp}, etc) layer. At the same time, most of the submodules are connected to the network layer submodule (called \texttt{networkLayer}), which is responsible of routing decisions and it can apply different protocols for it, such as AODV.
The module developed for implementing the wARP-Path PoC was called \texttt{AdHocHostAPB}, which is shown in Fig.~\ref{fig:adhochostapb}, and it is an extension of the \texttt{AdHocHost}, which simply adds and intermediate module between the network layer and the physical layer, and it is called relayUnit. The relayUnit submodule applies learning and forwarding based on the frames received from the wlan submodule, specifically it applies the pseudocode shown before in Listing~\ref{pseudocode} for any frame received from wlan.
\subsection{Implementation in OMNeT++ 5.2 and INET 3.6.3}
Lately, the wARP-Path protocol was reimplemented using the latest version of OMNeT++ and INET framework (code available in~\cite{WARP-Path}). The reason behind is that INET is quickly updated, and it covers a wide variety of protocols and components, and also, it is taken as a base for several other simulation frameworks~\cite{INET}.
Most of the differences in the two implementations are due to the evolution of the INET framework. In the recent versions of INET framework, the forwarding table (\texttt{macTable}) has been separated from the MAC relay unit (\texttt{MACRelayUnit}). Based on this, in this new implementation, the Learning/Lookup Table (\texttt{LT}) and Blocking/Broadcast Table (\texttt{BT}) are separated from the \texttt{MACRelayUnit} and placed beside the \texttt{MACRelayUnit} as two independent modules to provide the service to the \texttt{MACRelayUnit}, as it can be seen in Fig.~\ref{fig:adhochostapb}. All steps of the algorithm excluding line 05 (as shown in Listing~\ref{pseudocode}) are implemented in the \texttt{Ieee80211MgmtAdhocAPB} module, and line 05 is implemented in the \texttt{MACRelayUnitWAPB} module.
\begin{figure*}[hbt]
\centering
\includegraphics[width=0.65\textwidth]{figs/AdhocHostAPBnew.png}
\caption{Implementation of the \texttt{AdhocHostAPB} module in OMNeT++ 5.2 and INET 3.6.3}
\label{fig:adhochostapb}
\end{figure*}
There are two addresses for forwarding between two sequential hops (\textit{physical} addresses) and two other addresses to indicate the beginning and end of the path (\textit{logical} addresses). All four address fields embedded in the IEEE 802.11 MAC frame format (as shown in Fig.~\ref{fig:adr:ieeeff}) are used, which are receiver address as physical receiver, transmitter address as physical transmitter, destination address as logical receiver, and source address as logical transmitter respectively (as shown in Fig.~\ref{fig:adr:wap}). The physical addresses change in each hop, and the logical addresses do not change in the intermediate nodes, but in the final destination. To achieve the transparency required for the upper layer, the logical addresses are put in the physical address space before decapsulating the frame.
\begin{figure*}
\centering
\begin{subfigure}[htb]{0.80\textwidth}
\includegraphics[width=\textwidth]{figs/frameformat.png}
\caption{IEEE 802.11 frame format}
\label{fig:adr:ieeeff}
\end{subfigure}%
\quad
\begin{subfigure}[htb]{0.48\textwidth}
\includegraphics[width=\textwidth]{figs/adhocadr.png}
\caption{Interpretation of the MAC Addresses in the Ad-hoc mode
}
\label{adhocadr}
\end{subfigure}
\begin{subfigure}[htb]{0.48\textwidth}
\includegraphics[width=\textwidth]{figs/wapadr.png}
\caption{Interpretation of the MAC Addresses in wARP-Path
}
\label{fig:adr:wap}
\end{subfigure}
\caption{Applying the address fields of IEEE 802.11 frame format in wARP-Path}\label{fig:adr}
\end{figure*}
\subsection{Regarding OMNeT++/INET and their implementation of media access control in wireless networks}
In OMNeT++, frames are usually flooded at the same exact simulated time. Accordingly, when there is more than a single path to reach destination and the frame is being flooded for path exploration as in wARP-Path, it is high probability that some nodes will simultaneously broadcast this frame, e.g. an ARP Request. Since there is no unique receiver for sending the broadcast packets, no control packets (\texttt{Request-to-Send (RTS)} and \texttt{Clear-to-Send (CTS)}) are used and only carrier sense on the transmitter is performed because of the lack of coordination of the receivers and the possibility of a collision between the \texttt{CTSs}~\cite{MACAW}. If a broadcast packet collides, in addition to wasting a further capacity of the channel since the broadcast packets are larger than the control packets, it also has a negative impact on the quality of the explored paths. Therefore, a solution to this problem is necessary.
In the AODV implementation of INET framework, using a random jitter before sending protocol control packets such as \texttt{hello}, \texttt{Route Request (RREQ)}, and \texttt{Route Reply (RREP)} messages according to RFC 5148~\cite{jitter} has been adopted to overcome this problem. In this way, two random jitter has been used. First, when AODV generates the periodic \texttt{hello} messages in INET framework, for this type of simultaneity in RFC 5148, a random value \texttt{(jitter)} is subtracted from the time interval between two consecutive transmission of the same type messages\texttt{(MESSAGE\_INTERVAL)}. Using subtract instead of sum prevents excessive delay in receiving messages. Second, when AODV generates the \texttt{RREQ} and \texttt{RREP} messages in INET framework, since these messages are not periodic messages, there is not any \texttt{(MESSAGE\_INTERVAL)} to calculate the delay. In this type of simultaneity, RFC 5148 introduces jitter in an interval between zero and \texttt{MAXJITTER}.
We use the same approach to overcome the simultaneously broadcast problem in wARP-Path, with the difference that the control messages in this protocol are the standard ARP packets. Therefore, the mechanisms mentioned above apply to these packets similarly.
\section{Related Work}
\label{related}
There is a vast literature related to routing protocols applied to wireless networks. More specifically, one of the most prominent research fields that has thoroughly studied this problem is the one linked to \textbf{ad hoc networks}, and more specially \textbf{wireless mesh networks (WMNs)}. A seminal paper related to routing strategies in wireless networks is~\cite{Iwata99}, where authors propose and evaluate two competitive proposals for routing, called fisheye state routing and hierarchical
state routing. However, probably the most well-known routing protocols for routing in wireless networks are Ad hoc On-Demand Distance Vector (AODV) and Dynamic Source routing (DSR), as they are both standardized by IETF. \\
\textbf{AODV} is described in RFC 3561~\cite{aodv} and is based in using destination sequence numbers to avoid loops even in such situations where there are anomalous delivery of routing control messages. To our purpose, it is also very interesting the work in~\cite{aodv-paper}, as it describes some of the most interesting evolutions of AODV that have improved issues like performance, robustness or scalability and sheds light on future evolutions for the protocols. In fact, AODV features are the grounds of the All-Path family protocols. \\
The second standardized protocol is \textbf{DSR} protocol and is defined in RFC 4728~\cite{dsr}. DSR is a distributed protocol able to work in multi-hop wireless networks and it discovers and maintains routes by means of two mechanisms: route discovery and route maintenance. More specifically, route discovery in DSR is based on source routing and route caches, maintaining multiple routes per destination. A performance comparison between AODV and DSR can be found in~\cite{Das00}.
Other routing protocols that focus on improving different issues have been proposed in the literature --since the proposal of AODV and DSR-- are Source Node Compute Routing (SNCR)~\cite{He10} or Protection AODV (P-AODV)~\cite{Zhu13}. In the first case, \textbf{SNCR} aims to improve overhead and efficiency proposing a quick computation of the best metric for arriving from the source node to any destination, combining proactive and on-demand modes to adapt to different traffic settings. On the other hand, \textbf{P-AODV} focuses on improving reliability in wireless networks by means of building a protection path. Another important issue in routing in wireless routing is related to the link quality evaluation. In this sense, it must be highlighted the work in~\cite{Hong15}, where authors propose a link quality prediction (LQP) model to sense the link state.
Finally, the comparison of \textbf{bridging and routing techniques in wireless networks} is also a topic aligned with our proposal.
In~\cite{Suliman04}, authors compare wireless bridging and routing, concluding that bridging performs better than routing in terms of throughput both in TCP and UDP. Moreover, authors in~\cite{Maurina10} study the consequences of using pure bridging based solutions in wireless networks and present an enhanced bridged-based implementation for providing dynamic, self-configuration and self-healing features avoiding a routing protocol.
\section{Introduction}
\label{introduction}
The All-Path family~\cite{Rojas15} comprises diverse routing protocols running on layer 2, leveraging the well-known advantages of Ethernet in wired networks. All-Path protocols are based on the simple and basic mechanism of backward address learning used by bridges/switches, extended with a \textit{lock} mechanism to prevent loops. Paths are discovered and built based on minimum latency and may be created per destination host (ARP-Path), per communication flow or host pair (Flow-Path), per destination bridge (Bridge-Path) or even combining parameters from different layers, such as TCP (TCP-Path~\cite{tcppath}).
The first protocol, and origin of the whole All-Path family, was ARP-Path~\cite{IbanezCL}, already implemented in multiple and diverse platforms like Linux, NetFPGA, OpenFlow, or OMNeT++~\cite{Ibanez10}. Its main competitors are protocols using layer 2 variants of the IS-IS protocol such as SPB~\cite{Allan12} or TRILL RBridges~\cite{RBridges,Perlman11}. The approach for ARP-Path is to discover low latency paths via network exploration with broadcast frames, while in SPB and TRILL paths are computed based on the network topology obtained after an initial exchange of data at link level.
In its origins, the All-Path family was inspired by the Ad hoc On-Demand Distance Vector (AODV) routing~\cite{aodv,aodv-paper}, designed for wireless networks. Both create paths following a reactive routing approach. Therefore, one of the challenges was to adapt this family of bridging protocols from a wired environment to a wireless one, in order to analyze and compare them to AODV.
This paper is organized as follows. Chapter~\ref{introduction} introduces the topic, followed by the related work in Chapter~\ref{related}. Afterwards, Chapter~\ref{fromto} explains the implication of moving ARP-Path to a wireless environment and defines wARP-Path. Chapter~\ref{implementation} describes the implementation in the OMNeT++ simulator, later on evaluated in Chapter~\ref{evaluation}. Finally, Chapter~\ref{discussion} discusses the topic and Chapter~\ref{conclusion} concludes the article.
\section{From ARP-Path to wARP-Path}
\label{fromto}
In this section, we evaluate the key aspects for the transition from ARP-Path (wired and strictly Ethernet-based) to wARP-Path (wireless and potentially implemented using different layer-two protocols). For this purpose, we start by summarizing ARP-Path, then explaining the basics of wARP-Path (with emphasis on the differences), and finally analyze the implications of the frame format.
\subsection{ARP-Path}
\label{sec:ap}
The ARP-Path protocol creates minimum latency paths at request, based on path exploration instead of computation~\cite{IbanezCL,Rojas15}. ARP-Path does not require any modification of the Ethernet frame, and it only needs a small new feature in standard switches: a \textit{lock}. The \textit{lock} is a mechanism that prevents the switch from learning more than once (in a certain period of time) the same MAC address, which in the end prevents network routing loops~\cite{Rojas15} and allows any broadcast frame to explore the whole network as a probe, creating a source-based routing tree in its way.
Fig.~\ref{fig:ap} summarizes the operation of ARP-Path in an example Ethernet network that connects two hosts $A$ and $B$. Before any communication in IPv4, $A$ sends and ARP Request, which is leveraged to explore the topology and create the shortest paths. The difference with a standard switch is shown in switch 6, which only saves the first MAC address arriving --the one from switch 3-- and \textit{locks} the association of this MAC to the input port. Therefore, when the ARP Request arrives at a different port --from switch 5--, the frame is discarded, thus avoiding the potential loop, which would not happen in a standard Ethernet switch. Accordingly, only the fastest copy of the ARP Request arrives at $B$.
The learning process continues from $B$ to $A$ with the ARP Reply. This time, the unicast ARP Reply is forwarded towards $A$, already known by the different hops --switches-- in the network. Hence it crosses switches 6, 3, 2 and 1, respectively, until arriving at $A$. In the meantime, these switches save the input port for the ARP Reply, eventually generating the path towards $B$.
\begin{figure}
\centering
\begin{subfigure}[htb]{0.48\textwidth}
\includegraphics[width=\textwidth]{figs/ap1.png}
\caption{Learning process for path to $A$ when the ARP Request is broadcast}
\label{ap1}
\end{subfigure}%
\quad
\begin{subfigure}[htb]{0.48\textwidth}
\includegraphics[width=\textwidth]{figs/ap2.png}
\caption{Learning process for path to $B$ when the ARP Reply is sent from $B$ to $A$}
\label{ap2}
\end{subfigure}
\caption{ARP-Path operation in an Ethernet-based network}\label{fig:ap}
\end{figure}
Although ARP-Path takes it name from the ARP protocol and bases its exploration in the ARP frames, it could be applied to non-ARP based networks, such as IPv6 networks. The only difference is that ARP-Path should use a specific frame for the exploration in that case.
\subsection{wARP-Path}
\label{sec:wap}
The wARP-Path protocol follows the same basics for creating paths to reach final hosts than ARP-Path. However, there are three main differences:
\begin{enumerate}
\item \textbf{Locking mechanism:} The ARP-Path protocol locks the input port with the source MAC address in the arriving ARP message. In wireless networks there are no links and therefore no ports, so the locking mechanism saves the \texttt{\{MAC address, next hop's MAC address\}} tuple instead of the \texttt{\{MAC address, port\}} one. Basically, wARP-Path \textit{locks} nodes instead of input ports.
\item \textbf{Flooding:} To explore the network and reach the destination, the ARP-Path protocol broadcast frames through all ports but the input one. However, this concept is not directly applicable to wireless forwarding devices, which always broadcast frames in the whole radio coverage area.
\item \textbf{Forwarding nodes:} The ARP-Path protocol is used in bridge-based networks. However, wARP-Path can be applied to any ad hoc wireless network. For this reason, intermediate nodes can be final hosts at the same time, and therefore they might implement more communication layers than bridges (up to layer two only). This causes that ARP messages already have a default processing by intermediate nodes that should be slightly modified, i.e. intermediate nodes should discard only sARP messages not directed to them and not all of them (which is done in ARP-Path by default).
\end{enumerate}
Fig.~\ref{fig:wap} shows and example of a network, consisting of six intermediate nodes and two final hosts, in which wARP-Path can be applied. A circular dotted line represents the range or coverage area of each forwarding node. The range of final hosts is not represented, but we considered they reach only the closest node, for the sake of simplicity.
\begin{figure}
\centering
\begin{subfigure}[htb]{0.48\textwidth}
\includegraphics[width=\textwidth]{figs/wap1.png}
\caption{Learning process for path to $A$ when the ARP Request is broadcast}
\label{wap1}
\end{subfigure}%
\quad
\begin{subfigure}[htb]{0.48\textwidth}
\includegraphics[width=\textwidth]{figs/wap2.png}
\caption{Learning process for path to $B$ when the ARP Reply is sent from $B$ to $A$}
\label{wap2}
\end{subfigure}
\caption{wARP-Path operation in a wireless network}\label{fig:wap}
\end{figure}
In this example, there are two possible paths between host $A$ and host $B$, same ones than in the example in Fig.~\ref{fig:ap}.
When host $A$ emits the ARP Request message to start the communication, it first reaches node 1 which saves the tuple \texttt{A's MAC address, A's MAC address} since transmitter address of the frame received from host $A$ is regarded as the next hop address in host 1(i.e. in this case, the next hop is directly A). Later on, other nodes receive the frame, such as node 3, which saves the tuple \texttt{A's MAC address, 2's MAC address}, indicating that node 2 is locked as the next hop for the path to reach $A$. Eventually, several frames arrive to node 6, which only locks the MAC address of the first node from which it received the frame and discards the rest.
Differently to ARP-Path, in wARP-Path many nodes receive back multiple copies of the frame and need to discard them. This is because wireless nodes emit in their range and cannot avoid \textit{emitting back} to the previous sender as in wired networks, where it is possible to flood through all ports but the incoming one.
Finally, host $B$ receives the ARP Request message and emits the ARP Reply message with destination $A$. This ARP Reply message can follow the path just created to $A$ and, at the same time, it creates the path to $B$. Therefore the communication between $A$ and $B$ can start now, which will use the path involving nodes 1-2-3-6.
The pseudocode of the wARP-Path protocol is summarized in Listing~\ref{pseudocode}.
\begin{lstlisting}[basicstyle=\scriptsize, frame=single, caption={Pseudocode of the wARP-Path protocol}, label=pseudocode]
When a wARP-Path node receives a frame from another node:
01:if (src_mac == node's MAC address) then
02: discard frame
03:else
04: eth_frame = convertToEthernetFrame(frame)
05: next_hop = wARP-Path table(eth_frame, input_hop)
06: If (dst_mac == node's MAC address) then
07: send frame to upper layers of the node
08: else if (dst_mac == BCAST && next_hop) then
09: send frame to upper layers of the node
10: If (next_hop) then
11: send frame to the next hop
12: else
13: discard frame
\end{lstlisting}
\subsection{From a wired to a wireless frame format }
\label{sec:frame}
Apart from the implications previously mentioned, the implementation of wARP-Path requires another one related to the frame format. \\
ARP-Path leverages the fact that Ethernet is the most commonly used layer 2 protocol, and reuses the Ethernet frame for its purpose. However, wARP-Path depends on the type of network that relies beneath and their corresponding frame format.
Therefore, for the sake of simplicity, we considered wireless local area networks, following the standards defined by IEEE802.11~\cite{ieee80211}.
\section*{Acknowledgment}
This work has been supported by Comunidad de Madrid through project TIGRE5-CM (S2013/ICE-2919).
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-03-08T02:06:44",
"yymm": "1803",
"arxiv_id": "1803.02593",
"language": "en",
"url": "https://arxiv.org/abs/1803.02593"
}
|
\section{Introduction}
A fascinating consequence of the discovery of the Higgs
\cite{ATLAS:2012ae,Chatrchyan:2012tx}, is that the standard model
vacuum appears to be metastable \cite{Degrassi:2012ry,
Gorsky:2014una,Bezrukov:2014ina,Ellis:2015dha,Blum:2015rpa}
(see also earlier work \cite{Krive:1976sg,1982Natur.298,
Sher:1988mj,Isidori:2001bm,EliasMiro:2011aa}). Although it was originally thought
that this would not be an issue due to the extremely long half-life
predicted by the classic bubble nucleation arguments of
Coleman et al.\ \cite{coleman1977,callan1977,CDL}, (see also
\cite{Kobzarev:1974cp}), recent work by two of us
\cite{GMW,BGM1,BGM2,BGM3,Gregory:2016xix} indicates that
the situation may not be quite so rosy.
In \cite{GMW}, we developed
a description of vacuum decay catalysed by black holes,
with the result that the strong local spacetime curvature of
small black holes catalyses vacuum decay and
dramatically changes the prediction for the lifetime of the
universe\footnote{Some of these results were examined in \cite{Tetradis:2016vqb},
however without explicitly computing the Euclidean instanton action.}.
Tunnelling is initiated by a black hole seed in the the false vacuum
that decays into a remnant black hole surrounded by Higgs fields
which have overcome the potential barrier and lie in a lower energy
state. The tunnelling rate is determined by the difference in action between
the remnant black hole-instanton combination and the seed black hole
false vacuum configuration that turns out to be proportional to the difference
in horizon area of the seed and remnant black holes. Because of
this dependence on black hole area, enhancement occurs
only for very small black holes, the obvious candidates being
primordial black holes in our universe, indeed, there is an
interesting thermal interpretation of our result, see for example
\cite{Chen:2017suz,Gorbunov:2017fhq,Mukaida:2017bgd}.
There is however another possible scenario in which small black
holes could occur, and that is in particle collisions. If we have a situation
where our four dimensional Planck scale is derived from a higher
dimensional Planck mass close to the standard model scale
\cite{ArkaniHamed:1998rs,Antoniadis:1998ig,Randall:1999ee,
Randall:1999vf}, then
it is easier to form black holes in particle collisions
\cite{Giddings:2001bu,Dimopoulos:2001hw,Landsberg:2003br,Harris:2004xt}.
Such higher dimensional
theories are dubbed \emph{Large Extra Dimension} scenarios,
and the premise is that we live
on a four dimensional ``brane'' in a higher dimensional spacetime.
Our relatively high Planck scale, $M_p = 1/\sqrt{8\pi G_N}$, is the result of a
geometric hierarchy coming from an integration over the extra dimensions.
Since the true Planck scale is the higher dimensional one,
it is easier to form black holes in high energy processes,
leading to the possibility of black holes being
produced at the LHC (for a review see \cite{Park:2012fe}).
Given this exciting possibility for producing small black holes,
we should revisit our four dimensional black hole instanton
calculations and explore the impact of large extra dimensions.
As a first step in looking at vacuum decay with extra dimensions,
we considered the impact of dimensionality on our toy model thin
wall calculations in \cite{BGM2}, finding that extra dimensions seemed
to impede vacuum decay, however, these estimates were predicated
on a rather crude higher dimensional generalization that did not take
the braneworld aspect of the Large Extra Dimension models into account.
In this paper, we revisit the role of large
extra dimensions in vacuum decay, explicitly modelling the brane black
hole and finding exact solutions for the instanton on the brane.
We also make a more careful estimation of the black hole
Hawking radiation rate on the brane. We find that, while for a
given seed mass the higher dimensional tunnelling rate is
indeed lower than the four dimensional one, what we gain from
higher dimensions is that lower seed masses are allowed due to the
lower value of the fundamental Planck scale, $M_D$.
The layout of the paper is as follows: in the next section, we review
the status of constructing instantons both in four dimensions
with black holes, and for braneworlds in five dimensions without
black holes, and discuss the problems involved in introducing
a black hole to the higher dimensional calculation. In section
\ref{sec:action} we discuss the calculation of the action of an
approximate black hole instanton, showing that, as in four
dimensions, the static instanton action is the difference in black hole
horizon areas. In section \ref{sec:bubble} we solve for the brane
scalar field and find the instantons and their actions numerically.
In section \ref{sec:disc} we conclude.
\section{Braneworlds and Black Holes}
It is perhaps worth recalling the various challenges in finding
an instanton for vacuum decay in a braneworld setting.
The braneworld paradigm describes our universe as an effective
submanifold of a higher dimensional manifold, with standard
model fields living only on the four-dimensional braneworld, but
with gravity propagating throughout all of the dimensions, leading
to the renormalization of Newton's constant. For one extra
dimension we can consistently solve for the spacetime geometry
using the Israel approach \cite{Israel:1966}, giving the standard
Randall-Sundrum (RS) braneworld \cite{Randall:1999vf}, a
paradigm for warped compactifications.
For higher codimension, there is no unique ``delta-function'' limit for
a thin braneworld \cite{Geroch:1987qn}, and typically one resorts
to approximate hybrid Kaluza-Klein/warped descriptions for gravity
on a lower-dimensional brane. Thus, for a concrete gravitational
description in this paper we will remain within the RS model.
The RS model supposes that we have one extra
dimension, and that the higher dimensional spacetime, or bulk,
has a negative cosmological constant. The braneworld has
a positive tension, and the vacuum brane has an energy-momentum
tensor that is parallel to the brane with energy and tension equal.
The original solution presented by Randall and Sundrum had the tension
tuned to give a flat brane:
\begin{equation}
ds^2 = e^{-2|z|/\ell} \eta_{\mu\nu} dx^\mu dx^\nu - dz^2
\end{equation}
where the cusp in the warp factor at $z=0$ corresponds to the
brane. The local negative curvature of the bulk supports the brane
tension that is easily calculated from the Israel junction conditions:
\begin{equation}
{\cal K}^{(+)}_{\mu\nu} = -\frac1\ell \eta_{\mu\nu} \;\; \Rightarrow\;\;\;
8\pi G_5 \sigma = \Delta {\cal K}_{\mu\nu} - \Delta {\cal K} \eta_{\mu\nu}
= \frac6\ell \eta_{\mu\nu}
\end{equation}
and is tuned to fit with the cosmological constant $\Lambda_5 = -6/\ell^2$.
De-tuned branes, with tension greater or less than this critical
value may also be embedded within the bulk AdS spacetime,
although the natural embeddings now become either space- or time-like
\cite{Chamblin:1999ya,Kaloper:1999sm,Kraus:1999it,Binetruy:1999ut,
BCG,Karch:2000gx}, but as long as the brane energy-momentum
is approximately homogeneous (i.e.\ having a spatially isotropic
pressure term only) the bulk solution can be fully integrated, and
the brane trajectory found \cite{BCG}.
For a brane black hole solution, we must break this spatial homogeneity,
but even with the added benefit of having only one codimension, the
exact solution for a brane black hole has been extremely elusive
\cite{Gregory:2008rf,Kanti:2009sz}.
The natural geometry of a Schwarzschild black hole that
extends off the brane into a black string, found by Chamblin,
Hawking and Reall \cite{Chamblin:1999by},
has the problem that it is neither representative of matter
localised on the brane, nor is it stable, suffering from a Gregory-Laflamme type
of instability \cite{Gregory:1993vy,Gregory:2000gf}. A lower dimensional
analogue of the brane black hole was found by Emparan et al.\
\cite{Emparan:1999wa,Emparan:1999fd} by taking a $2+1$ dimensional
brane through the equatorial plane of a $4$-dimensional AdS C-metric
\cite{Kinnersley:1970zw,Plebanski:1976gy}.
The black hole would be expected to be accelerating from the perspective
of the bulk, since an observer hovering at fixed distance from the brane
is in fact undergoing uniform acceleration towards it. Unfortunately,
there is no known exact solution for a C-metric in more than 4 dimensions,
thus no template for constructing a braneworld black hole plus bulk analytically.
To maintain an analytic approach one can explore the
effective brane gravitational equations using the approach of
Shiromizu et al.\ \cite{Shiromizu:1999wj}, leading to the tidal solution
that we will use in this paper \cite{Dadhich:2000am}. (One can also
explore braneworlds with additional matter, either on the brane or in the
bulk, to support analyticity of the brane embedding, see e.g.\
\cite{Galfard:2005va,Creek:2006je,Dai:2010jx,Kanti:2013lca})
Alternately, one can take a numerical approach; the equations of motion
to be solved are an elliptic system \cite{Wiseman:2001xt}, with
the brane junction conditions and asymptotic Poincare
horizon providing the boundary conditions. The solutions for small
black holes were found in \cite{Kudoh:2003xz}, although the
large black hole solutions have been far more tricky to
determine due to the nonlinearity of the Einstein equations and
the impact of the bulk warping of the horizon, however there has
been some interesting recent work in this direction
\cite{Figueras:2011gd,Wang:2016nqi}.
\begin{wrapfigure}[23]{r}{0.45\textwidth}
\centering
\vskip -5mm
\includegraphics[width=0.3\textwidth]{simplcdladsa.pdf}
\caption{
The braneworld instanton for decay of a Minkowski false vacuum
brane to a sub-critical AdS brane from \cite{Gregory:2001dn}. }
\label{fig:simplecdl}
\end{wrapfigure}
Now let us consider the instanton from a higher dimensional perspective.
The decay of a metastable false vacuum was first computed by Coleman
and collaborators in a series of papers \cite{coleman1977,callan1977,CDL},
in which a Euclidean approach was used to find an instanton solution
interpolating between the true and false vacua. A convenient approximation,
extremely useful for visualisation, is to take the region over which the
vacuum interpolates to be very narrow in comparison with the interior
of the bubble. This ``thin wall'' then has a straightforward generalisation
to gravity, as described in the paper with de Luccia \cite{CDL} (CDL).
While this thin wall description is not appropriate for the Higgs
vacuum decay \cite{BGM3}, where the vacuum interpolation is
very wide and relatively gentle, it nonetheless provides an excellent
shorthand for visualising the process of decay.
The CDL picture however, is very symmetric, and assumes that both the
initial and final states are completely devoid of features and are
homogeneous. If instead one relaxes this assumption, minimally, by
allowing for an inhomogeneity in the form of a black hole, the analytic
approach of CDL can be preserved, and the equations of motion for
the instanton are only minimally altered \cite{GMW,BGM1,BGM2,BGM3},
however, the impact on the action of the instanton can be quite
significant, and particularly for the thick scalar domain walls
appropriate to the Higgs potential \cite{BGM3}, tunnelling
turns out to be significantly enhanced to the extent that if there
are primordial black holes, false vacuum decay will happen.
Let us now consider how these arguments might lift to higher
dimensions. In \cite{Gregory:2001dn}, the equivalent of the CDL
instantons on a Randall-Sundrum braneworld were constructed,
the 5D instanton being geometrically akin to the 4D representations
of the CDL instantons. Sub- and super- critical branes follow
spherical trajectories in the AdS bulk, so the tunneling of a Minkowski
false vacuum to an AdS true vacuum is represented by a flat
brane with a bubble sticking out, as shown in figure \ref{fig:simplecdl}.
As is usual with the RS model, two copies of the picture are identified,
and the ``bubble wall'' is the sharp edge between the spherical and
flat parts of the braneworld,
appearing roughly as a codimension two object.
\begin{wrapfigure}[21]{r}{0.45\textwidth}
\centering
\vskip -5mm
\includegraphics[width=0.25\textwidth]{bhcdla.pdf}
\caption{A sketch of the expected geometry of braneworld
vacuum decay with a braneworld black hole.}
\label{fig:bhcdl}
\end{wrapfigure}
Ideally, one would like to construct a similar instanton, but with a
black hole, however, at this point the lack of an exact brane black
hole solution becomes problematic. Even if we drop a dimension
to have a 2+1 dimensional braneworld, for which the brane black
hole solution is constructed via the C-metric \cite{Emparan:1999wa},
we have the problem that the C-metric has a unique slicing for the
braneworld \cite{Kudoh:2004ub}, so we cannot patch together two
different braneworld trajectories such as an equatorial sub-critical
slice matching to a flat brane further away as suggested in figure \ref{fig:bhcdl}.
Indeed, slicing a bulk Schwarzschild metric induces additional energy
momentum on the brane \cite{Galfard:2005va,Creek:2006je},
(except for the uniform radius ``cosmological'' brane solutions).
Thus as a direct approach to finding the instanton seems problematic,
we follow a more pragmatic approach, and rather than seeking an exact
analytic solution, instead consider what a black hole instanton might
approximately look like. From the intuition gleaned in the 4D black hole
instantons, we expect that small black holes are the most dangerous,
and that the dominant instanton will be the static instanton \cite{BGM3}.
Then, analogous to the modelling of collider black hole phenomenology
\cite{Harris:2003db}, we
use the higher dimensional Schwarzschild-AdS solution as an
approximation to the local bulk black hole: this allows us to
construct a method of calculating the instanton action formally.
Finally, in order to correctly identify the asymptotics of our instanton,
we need a way of interpolating between the near horizon
and far-field brane solution, which we expect to have a 4D
Schwarzschild $G_NM/r$ behaviour. This final step requires
a choice for the braneworld solution, and we use the
tidal brane solution of Dadhich et al.\ \cite{Dadhich:2000am},
found by considering vacuum solutions with a non-vanishing
bulk Weyl tensor in the formalism of Shiromizu et al.\ \cite{Shiromizu:1999wj}.
The tidal solution has the attractive feature that it has
the correct asymptotic form at large brane radius, but looks like
the five dimensional Schwarzschild potential for small radius,
indeed, it is similar to the Reissner-Nordstrom black hole,
although the ``tidal charge'' term $-r_Q^2/r^2$ is negative. This
tidal charge was not related to the mass in \cite{Dadhich:2000am},
but left as an arbitrary degree of freedom, therefore part of our task
in section \ref{sec:bubble} will be to relate the tidal charge to the
mass of the black hole.
Our strategy is then as follows: we first take our brane black hole,
approximately modelled by the 5D Sch-AdS solution, and continue
to Euclidean time. We then compute the action of this solution
in a rather general way, using the approach of Hawking and Horowitz
\cite{Hawking:1995fd};
as per usual, the direct way of computing the action leads to an
apparent divergence that we cannot in this case regulate directly
by introducing a cut-off as we will explain. Nonetheless, however
we choose to regulate the action, the same method will apply for
the false vacuum black hole and the instanton bubble
solution, thus we simply subtract the seed and bubble actions
to get the final amplitude for vacuum decay. Crucially, this turns
out to be simply the difference in areas of the seed and remnant
black hole horizon geometries. Finally, we integrate the scalar
equations of motion on the brane to obtain the brane bubble solution,
and use the tidal metric to relate the near horizon and asymptotic
geometries. The nett result is an amplitude for brane black hole seeded
vacuum decay that we can compare to the higher dimensional
brane black hole evaporation rate to explore whether brane vacuum
metastability is an issue.
\section{The Euclidean Brane Black Hole Action}
\label{sec:action}
In this section we will show that, just like in four dimensions, the
Euclidean action of any static black hole solution can be expressed
entirely by surface terms. This is a remarkable result, because it not
only applies to the vacuum black hole, it also applies with a cosmological
constant, with matter and even with a conical singularity at the horizon.
We begin by recalling the properties of the
Euclidean Schwarzschild black hole in four dimensions
\begin{equation}
ds^2=f(r)d\tau^2+f(r)^{-1}dr^2+r^2d\Omega_{I\!I}^2,
\end{equation}
where
\begin{equation}
f(r)=1-\frac{2G_NM}{r}
\end{equation}
In order to explore the geometry near the `horizon' $r_h=2G_NM$,
we expand using a new coordinate ${\varrho}$, defined by
\begin{equation}
{\varrho}=\sqrt{{2(r-r_h)\over\kappa}}
\label{varrho4d}
\end{equation}
where $\kappa$ is the surface gravity, $\kappa=f'(r_h)/2$.
To leading order $f(r)= \kappa^2{\varrho}^2+O({\varrho}^4)$, and
close to the horizon,
\begin{equation}
ds^2=d{\varrho}^2+{\varrho}^2d(\kappa\tau)^2
+r_h^2d\Omega_{I\!I}^2+O({\varrho}^4),
\end{equation}
For small ${\varrho}\ge 0$, the metric is geometrically the product of a
disc with a sphere, provided that $\kappa\tau$ is taken to be an angular
coordinate with the usual range $2\pi$. If $\kappa\tau$
has a different range, then the manifold has a conical singularity at
$r_h$. Note that the Euclidean section is perfectly regular other than
this, but only covers the exterior region of the original black hole.
The \emph{event} horizon of the original Lorentzian black hole is
encoded in the topology of the Euclidean solution: the surface
${\varrho}=0$ is a 2-sphere of radius $r_h$.
For the brane black hole in five dimensions, the metric is extended
into an additional direction, parametrised by $\chi$ in Kudoh et al.\
\cite{Kudoh:2003xz}, who numerically constructed small brane black
holes with horizon size less than the AdS radius $\ell$. In
\cite{Kudoh:2003xz}, the metric was written in the form
\begin{equation}
ds^2 = \frac{1}{(1+\frac{\rho}{\ell}\cos\chi)^2} \left [ T^2(\rho,\chi) d\tau^2
+ e^{2B(\rho,\chi)} (d\rho^2+\rho^2d\chi^2)
+ e^{2C(\rho,\chi)} \rho^2 \sin^2\!\chi d\Omega_{I\!I}^2 \right]\,,
\label{fivemetric}
\end{equation}
where the brane sits at $\chi=\pi/2$, and $\chi\leq\pi/2$ is kept as the bulk.
Clearly, in the small black hole limit, $\ell \to \infty$, we have the five
dimensional Schwarzschild black hole:
\begin{equation}
ds^2 = \left (\frac{\rho^2-\rho_h^2}{\rho^2+\rho_h^2} \right)^2 d\tau^2
+ \left (\frac{\rho^2+\rho_h^2}{\rho^2} \right)^2 \left [
d\rho^2 + \rho^2 d\Omega_{I\!I\!I}^2 \right]
\end{equation}
written here in homogeneous co-ordinates, rather than the area
gauge. The local Euclidean horizon coordinate is
${\varrho}=2(\rho-\rho_h)$, and the horizon has area
${\cal A} = 4\rho_h^2$, and surface gravity
\begin{equation}
\kappa = e^{-B(\rho_h)} T'
\label{Kudohkappa}
\end{equation}
The black hole is corrected at order $\rho/\ell$ by the conformal factor,
and at order $\rho_h/\ell$ in the other metric functions close to the horizon.
Kudoh and collaborators integrated the functions $T, B$ and $C$
numerically, and found that the $T$ function to a very good approximation
extends hyperspherically off the brane. Although $B$ and $C$ are not
precisely the same, their difference is roughly of order $\rho_h/\ell$
as expected. At large $\rho$, $T,B,C\to1$, and
the metric is asymptotically AdS in the Poincar\'e patch.
We do not use the explicit form of the metric, however, the features
we require from the solutions of \cite{Kudoh:2003xz} are that the
event horizon is topologically hyperspherical with
constant surface gravity, and that the
braneworld black hole asymptotes the Poincar\'e patch of AdS.
The coordinate transformation between the local black hole
coordinates and the Poincar\'e RS coordinates is
\begin{equation}
\rho^2 = r^2 + \ell^2 (e^{|z|/\ell}-1)^2,\qquad
\tan\chi = \frac{r}{\ell(e^{|z|/\ell}-1)}\,,
\end{equation}
and we expect that the `trajectory' of the brane in the
black hole metric will bend slightly in response to the black
hole at $\rho_h$, giving rise to a four dimensional Newtonian
potential as described in \cite{Garriga:1999yh}. From the
perspective of the $\{\rho,\chi\}$ coordinates, in which the
brane sits at $\chi=\pi/2$, this will show up as a $1/\rho$ correction
to $T,B,C$. We therefore take our asymptotic metric to be of the form
\begin{equation}
ds^2 =e^{-2|z|/\ell}\left [ F(r,z)d\tau^2
+ F(r,z)^{-1}dr^2+r^2d\Omega^2\right]+dz^2,
\label{asympmetric}
\end{equation}
where $F\sim1-2G_NM(z)/r+O(r^{-2})$. We can think of $M(z)$ as
coming from the brane bending term of $M/\rho$ in the original coordinates.
\subsection{Computing the Action}
The action of the black hole instanton combination diverges and has to
be regulated in some way. We do this by truncating the five dimensional
manifold at large distances from the black hole, taking a surface at
large radius $R$ on the brane, and extending this along geodesics
in the $\pm z$ directions orthogonal to the brane to produce the outer
boundary surface ${\partial\cal M}_R$ as indicated in the cartoon in
figure \ref{fig:cartoon}. The interior is denoted by ${\cal M}_R$
and the intersection of ${\cal M}_R$ with the brane world is
denoted by ${\cal B}$.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\textwidth]{cartoon.pdf}~~
\includegraphics[width=0.4\textwidth]{cartoontau.pdf}
\caption{A cartoon of the Euclidean tidal black hole and the cut-off surfaces.
On the left, the $\tau, \theta$ coordinates are suppressed, and the cut-off
surface is indicated relative to the brane and bulk black hole horizon.
Only one half of the $\mathbb{Z}_2$ symmetric solution is shown.
On the right, the Euclidean $\tau$ coordinate is shown but the bulk and angular
coordinates are suppressed, and the ``black hole cigar'' geometry is indicated.
Two circles denote the boundary ${\partial \cal H}$ of the region just outside the
horizon and the boundary ${\partial \cal M}_r$ at large radius.}
\label{fig:cartoon}
\end{center}
\end{figure}
The Euclidean action for this truncated instanton or black hole
solution is
\begin{equation}
I_R=-\frac{1}{16\pi G_5}\int_{{\cal M}_R}(R_5-2\Lambda_5)
+\int_{{\cal B}}{\cal L}_m\sqrt{g}
+\frac{1}{8\pi G_5}\int_{{\partial\cal M}_R} K\sqrt{h},
\end{equation}
where $K$ denotes the extrinsic curvature of the boundary surface
$\partial {\cal M}_R$ defined with an \emph{inward} pointing normal
to the bulk manifold ${\cal M}_R$. The matter Lagrangian ${\cal L}_m$ includes
the contribution from any nontrivial Higgs field profile, as well as
the brane stress-energy tensor. The bulk integral is understood to
range across all $z$, and includes the $\delta-$function curvature
at the brane source in the spirit of the Israel approach.
Note that the gravitational constant in five dimensions is related to
Newton's constant in four dimensions by $G_5=\ell G_N$.
We now show that the tunnelling exponent, given by the difference
between the actions of the instanton geometry with a remnant black
hole, and the false vacuum geometry with the seed black hole:
$B=I_{\text{inst}}-I_{\text{FV}}$, is finite in the limit $R\to\infty$.
The first step is to introduce a small ball, ${\cal H}$, extending a
proper distance of order ${\cal O}(\varepsilon)$ out from the black
hole event horizon, to formally deal with any conical deficits arising
from a generic periodicity in Euclidean time. This splits the action
calculation into two terms,
\begin{equation}
I_R=I_{R}^{\rm hor}+I_{R}^{\rm ext},
\end{equation}
where\footnote{Note, the extrinsic curvature in the Gibbons-Hawking term
is computed with an inward pointing normal, hence the \emph{same} sign
for that term in each expression.}
\begin{equation}
I_R^{\rm hor}=-\frac{1}{16\pi G_5}\int_{\cal H}(R_5-2\Lambda_5)+
\int_{{\cal B}_{\cal H}}{\cal L}_m\sqrt{g} +
\frac{1}{8\pi G_5}\int_{\partial\cal H} K\sqrt{h},\hskip 15mm
\end{equation}
\begin{equation}
\begin{aligned}
I_R^{\rm ext}&=-\frac{1}{16\pi G_5}\int_{{\cal M}_R-{\cal H}}(R_5-2\Lambda_5)+
\int_{{\cal B}-{\cal B}_{\cal H}}{\cal L}_m\sqrt{g}
+\frac{1}{8\pi G_5}\int_{\partial\cal H} K\sqrt{h}\\
&\qquad+\frac{1}{8\pi G_5}\int_{{\partial\cal M}_R} K\sqrt{h},
\end{aligned}
\label{ir}
\end{equation}
and ${\cal B}_{\cal H}= {\cal B}\cap{\cal H}$ is the intersection of the
event horizon cap with the brane.
In order to deal with the near-horizon contribution, we transform
\eqref{fivemetric} to local horizon coordinates,
analogous to the Euclidean Schwarzschild transformation,
\eqref{varrho4d}, so that
\begin{equation}
ds^2\approx d\varrho^2+A^2(\varrho,\xi)d\tau^2
+D^2(\varrho,\xi) d\Omega^2_{I\!I}+N^2(\varrho,\xi)d\xi^2,
\label{nearhor}
\end{equation}
where $\varrho<\varepsilon$ inside ${\cal H}$.
Comparing to \eqref{fivemetric}, we see $A = T/(1+\frac\rho\ell\cos\chi)$,
$D = \rho \sin\chi e^C/(1+\frac\rho\ell\cos\chi)$, with $\varrho \approx
(\rho-\rho_h)/(1+\frac{\rho_h}\ell\cos\chi)$
and $\xi = \chi + {\cal O}(\varrho^2)$. The brane sits at $\xi=\pi/2$,
and on the horizon, $\xi\in[0,\pi]$.
As with the four dimensional Euclidean Schwarzschild, there is
a natural periodicity of $\tau$ for which the Euclidean metric
is nonsingular; this periodicity is $\beta_0=2\pi/\kappa$, where
$\kappa$ is the surface gravity of the black hole given in the original
coordinates by \eqref{Kudohkappa}, and in the horizon coordinates
by $\partial A/\partial\varrho$. From nonsingularity of the geometry, we deduce
$N\sim N_0(\xi) + {\cal O}(\varrho^2)$, $D\sim D_0(\xi)+{\cal O}(\varrho^2)$,
and $A \sim \kappa \varrho+ {\cal O}(\varrho^2)$. Now let us consider a
general periodicity $\beta$ for the Euclidean time $\tau$, then we will have
a conical singularity at $\varrho=0$. In order to compute the action, we smooth
this out by modifying the $A$ function so that $A'(\varepsilon,\xi) = \kappa$,
but $A'(0,\xi) = \kappa \beta_0/\beta$. Computing the curvature for this
smoothed metric gives
\begin{equation}
\sqrt{g} (R-2\Lambda_5) = - 2 N_0(\xi) C_0(\xi)^2 A'' (\varrho) + {\cal O}(\varrho)
\end{equation}
which gives the bulk contribution to $I_R^{\rm hor}$ as
\begin{equation}
\begin{aligned}
-\frac{1}{16\pi G_5}\int_{\cal H}(R_5-2\Lambda_5)+
\int_{{\cal B}_{\cal H}}{\cal L}_m\sqrt{g}&=
\frac{\beta}{2} [A'(\varepsilon)-A'(0)] \int N_0D_0^2 d\xi
+ {\cal O}(\varepsilon^2)\\
&= \frac{\kappa}{8\pi} [\beta-\beta_0] {\cal A}_5
\end{aligned}
\end{equation}
where $ {\cal A}_5=4\pi \int N_0D_0^2 d\xi $ is the area of the braneworld
black hole horizon extending into the bulk (on both sides of the brane).
To compute the Gibbons-Hawking boundary term we note that the normal to
${\partial \cal H}$ is $n = -d\varrho$, hence the extrinsic curvature is
\begin{equation}
K=-A^{-1}A_{,\varrho}+{\cal O}(\varepsilon)
\end{equation}
and
\begin{equation}
\frac{1}{8\pi G_5} \int_{\partial\cal H} K\sqrt{h}
= -\frac{\kappa \beta}{2G_5} \int N_0D_0^2 d\xi
= -\frac{\kappa \beta {\cal A}_5}{8\pi G_5}
\end{equation}
Thus the contribution to the action from the horizon region is
\begin{equation}
I_R^{\rm hor}=-\frac{\kappa\beta_0 {\cal A}_5}{8\pi G_5}
= -\frac{{\cal A}_5}{4G_5}
\end{equation}
In appendix \ref{Appaction}, we show that the external part
$I_R^{\rm ext}$ can be simplified by taking a canonical decomposition
based on a foliation of the manifold by surfaces of constant $\tau$,
$\Sigma_\tau$, and the part of the action outside the horizon cylinder
reduces to simple surface terms,
\begin{equation}
I_R^{\rm ext} = \frac{1}{8\pi G_5} \int_0^\beta d\tau
\left( \int_{C_R} {}^3\!K\, \sqrt{h} + \int_{C_{\cal H}} {}^3\!K\, \sqrt{h} \right).
\label{baction}
\end{equation}
where ${}^3\!K$ are the extrinsic curvatures of codimension two
surfaces of constant $r$, regarded as submanifolds of
surfaces of constant $\tau$, $\Sigma_\tau$ as described in appendix
\ref{Appaction}.
Close to the horizon, we use the metric \eqref{nearhor} and find
\begin{equation}
{}^3\!K=2D^{-1}D_{,\varrho}+N^{-1}N_{,\varrho}\to 0,
\end{equation}
at the horizon $\varrho=0$ for the behaviour of the metric coefficients
$D(\varrho,\xi)$ and $N(\varrho,\xi)$ given earlier. There is no contribution
to the action from this boundary term.
At large distances, the metric approaches the perturbed Poincar\'e
form \eqref{asympmetric}, and we find
\begin{equation}
{}^3\!K=-\frac{2}{R}e^{|z|/\ell} F^{1/2},\qquad
\sqrt{h}=R^2e^{-3|z|/\ell} F^{1/2}.
\end{equation}
hence
\begin{equation}
I_R^{\rm ext}=-\frac{\beta}{G_N\ell}\int_0^\infty \! dz
e^{-2z/\ell}\left(2R-4G_NM(z)+O(R^{-1})\right).
\end{equation}
Ideally, we would like to regularise this action either by background
subtraction, or adding in boundary counterterms along the lines of
\cite{Balasubramanian:1999re,Emparan:1999pm}, however, the
counterterms of \cite{Emparan:1999pm} do not regulate this action,
and one cannot replace the interior of ${\cal M}_R$ with a pure RS
braneworld, due to the variation of $M(z)$ along $\partial {\cal M}_R$.
Instead, we note that the Higgs fields on the brane in any instanton
solution will die off exponentially for large $r$, so from the intuition
that $M(z)/r \sim M_\infty/\rho = M_\infty/\sqrt{r^2 + \ell^2 (e^{|z|/\ell}-1)^2}$,
we then deduce that the mass function $M(z)$ will be the same at leading
order for both the false vacuum with the seed brane black hole, and the
instanton solution, therefore the exterior terms will cancel when we take the
difference between the instanton action and the false vacuum action:
\begin{equation}
B = I_{\text{inst}} - I_{\text{FV}} = \lim_{R\to\infty} \left [
I_R^{\rm ext} \Big|_{\text{inst}} -I_R^{\rm ext} \Big|_{\text{FV}} \right]
-\frac{{\cal A}_5^{\text{inst}}}{4G_5}
+\frac{{\cal A}_5^{\text{FV}}}{4G_5}
= \frac{{\cal A}_5^{\rm seed}}{4G_5}
-\frac{{\cal A}_5^{\rm rem}}{4G_5}
\label{bterm}
\end{equation}
where ${\cal A}_5^{\rm seed}$ and ${\cal A}_5^{\rm rem}$ refer to the areas of the seed
and remnant black hole horizon areas respectively.
This is simply the reduction in entropy $-\Delta S$ caused by the decay
process, and the tunnelling rate is recognisable as the probability of an
entropy reduction $\propto \exp(\Delta S)$.
The difficulty we face when applying (\ref{bterm}) is that we have to relate the
black hole area to the mass of the black hole triggering the vacuum decay and the
physical parameters in the Higgs potential. This requires explicit solutions for the
gravitational and Higgs fields.
\section{Tidal black hole bubbles}
\label{sec:bubble}
As we reviewed, the main obstacle to finding tunnelling instantons is the
lack of any analytic brane black hole solutions. The brane-vacuum
equations are complicated by the reduced symmetry of the expected
static, brane-rotationally symmetric geometry. Although we
have numerical brane black hole solutions, once we introduce Higgs
profiles on the brane, these would be modified, and a new full
numerical brane$+$bulk solution would have to be computed --
a formidable task. Instead, we adopt a more practical alternative,
based on the tidal black hole solutions of Dadhich et al.\ \cite{Dadhich:2000am}.
As described, for example, by Maartens \cite{Maartens:2000fg}, one can take
an approach of solving purely the brane ``Einstein equations'',
i.e.\ the induced Einstein equations on the brane found by the Gauss
Codazzi projection of the Einstein tensor in Shiromizu et al.\
\cite{Shiromizu:1999wj} (SMS). These equations are similar to the four
dimensional Einstein equations, but contain additional terms involving
the square of the energy momentum of any matter on the brane, and
an additional so-called Weyl tensor, ${\cal E}_{\mu\nu}$,
coming from a projection of the bulk Weyl tensor onto the brane.
The Weyl tensor for the tidal black hole satisfies the equations
${\cal E}_\mu{}^\mu=0$ and $\nabla^\mu{\cal E}_{\mu\nu}=0$.
Following \cite{Maartens:2000fg}, one uses the symmetry of the
physical set up to write the Weyl tensor as
\begin{equation}
{\cal E}^\mu_\nu = \text{diag} \left ( {\cal U} , -\frac{({\cal U} + 2\Pi)}{3},
\frac{\Pi - {\cal U}}{3} \right)
\end{equation}
This is manifestly tracefree, and the `Bianchi' identity
implies a conservation equation for ${\cal U}, \Pi$. For the spherically
symmetric static brane metric
\begin{equation}
ds^2_{\text{brane}} = f(r) e^{2\delta(r)}d\tau^2
+f^{-1}(r) dr^2 + r^2 d\Omega^2_{I\!I},
\label{genbranemet}
\end{equation}
the conservation equation implies
\begin{equation}
\left ( {\cal U} + 2\Pi \right) ' + \left ( \frac{f'}{f} + 2\delta' \right)
\left ( 2{\cal U} + \Pi \right) + \frac{6\Pi }{r} = 0\,.
\label{weyleos}
\end{equation}
Even for the vacuum brane this is not a closed system, but if
one assumes an equation of state, one can find an induced
brane solution \cite{Gregory:2004vt}. The tidal black hole
corresponds to the choice $\Pi = -2{\cal U}$, for which
\eqref{weyleos} is easily solved by ${\cal U} \propto 1/r^4$.
The tidal black hole of Dadhich et al.\ \cite{Dadhich:2000am},
has $\delta(r) \equiv 0$,
\begin{equation}
f(r)=1-\frac{2G_NM}{r}-\frac{r_Q^2}{r^2}\,,
\label{vbh}
\end{equation}
and
\begin{equation}
{\cal E}_{\mu\nu}dx^\mu dx^\nu=-\frac{r_Q^2}{ r^4}\left(
f(r) d\tau^2+f^{-1}(r) dr^2-r^2d\Omega^2\right),
\label{conf}
\end{equation}
where $r_Q$ is a constant parameter related to the tidal charge $Q$
of \cite{Dadhich:2000am} by $r_Q^2 = -Q$. The motivation
for this solution is clear: at large distances, the Newtonian potential
of a mass source has the conventional $G_NM/r$ behaviour due to
a ``brane-bending'' term identified by Garriga and Tanaka \cite{Garriga:1999yh};
the interpretation being that the brane shifts relative to the bulk in
response to matter on the brane. At small distances on the other hand
we would expect the higher dimensional Schwarzschild potential to be
more appropriate, hence the $-r_Q^2/r^2$ term. The event horizon is
distorted by the Weyl tensor, hence the name.
Other choices for the Weyl tensor
lead to different brane solutions \cite{Gregory:2004vt}, however these
tend to have either wormholes or singularities (or both), therefore we
do not consider these here.
For our bubble solution, we will need to find the fully coupled
Higgs plus brane SMS-gravitational equations of motion in the
spherically symmetric gauge \eqref{genbranemet}, and we
will use the same \emph{tidal} Ansatz for the equation of state
of the Weyl tensor: $\Pi = -2{\cal U}$. The beauty of the tidal Ansatz
is that even with the Higgs fields taking a nontrivial bubble profile,
the conservation equation for the Weyl tensor \eqref{weyleos}
is still solved by ${\cal U} = - r_Q^2/r^4$.
We also have some limited information about the form of the tidal
black hole solution away from the brane from an expansion in the
fifth coordinate. According to Maartens and Koyama, \cite{Maartens:2010ar}
the metric parallel to the brane at proper distance $z$ from the brane is
\begin{equation}
\tilde g_{\mu\nu}(z) = g_{\mu\nu}(0) - ( 8\pi G_5 S_{\mu\nu}) \, z
+\left [ (4\pi G_5)^2 S_{\mu\sigma}S^\sigma{}_\nu
- 8\pi G_N S_{\mu\nu} - {\cal E}_{\mu\nu} \right] z^2 +\dots
\end{equation}
where $S_{\mu\nu}=T_{\mu\nu}-\frac13 T g_{\mu\nu}$ is composed of
the energy momentum tensor of brane matter.
In the false vacuum state, we have $T_{\mu\nu}=0$ and the metric
expansion away from the brane reduces to
\begin{equation}
\begin{aligned}
ds^2&\approx e^{-2|z|/\ell}\left(g_{\mu\nu}-{\cal E}_{\mu\nu}z^2\right)+dz^2\\
& \approx e^{-2|z|/\ell}\left\{
\left(1 + \frac{r_Q^2z^2}{r^4}\right) \left( fd\tau^2+f^{-1}dr^2\right)
+ \left(1-\frac{r_Q^2z^2}{r^4}\right) r^2d\Omega^2_{I\!I} \right\}+dz^2,
\end{aligned}
\end{equation}
which shows clearly how the horizon area decreases in the
$z$ direction. The horizon forms into a true bulk black hole when
the area vanishes for some value of $z$ of order $r_h^2/r_Q$.
Although this tidal black hole has many attractive features, the
main difficulty that has to be overcome when finding the bubble
solutions is that the tidal constant $r_Q$ is undetermined.
Clearly a nonsingular brane black hole, if approximately tidal,
should have a relation between the asymptotic mass
measured on the brane, $M$, and the tidal charge $r_Q^2$.
For very large black holes, we expect the horizon radius to be
predominantly determined by $M$, and this ambiguity is not
relevant, however for the small black holes we are interested in,
the horizon radius is primarily dependent on $r_Q$, and we
must confront this ambiguity.
We start by noting that the tidal black hole solution should be
identical to the five dimensional Schwarzschild black hole
in the limit that the AdS radius $\ell\to\infty$, as the brane
stress-energy tensor, which is tuned to the cosmological constant,
vanishes in this limit, and full $SO(4)$ rotational symmetry is restored.
Since $G_N=G_5/\ell$, \eqref{vbh} implies that $r_Q^2\to r_h^2$
in this limit. Intuitively, we also expect that for small black holes, the
bulk AdS scale should also be subdominant, and the black hole should
look (near the horizon at least) mainly like a five dimensional black
hole, i.e.\ $r_Q^2\to r_h^2$ as $r_h\to 0$. We will therefore assume
analyticity in $r_h/\ell$ and write
\begin{equation}
r_Q^2=r_h^2\left(1-b{r_h\over\ell}
+ {\cal O} \left({r_h^2\over\ell^2}\right)\right)\label{rq}
\end{equation}
for small $r_h/\ell$, where $b$ is some constant independent of $r_h$
and $\ell$, expected to be roughly of order unity. For the tidal black hole,
a trivial rewriting of \eqref{vbh} gives the relation
\begin{equation}
M = \frac{b r_h^2}{2G_5}
\label{mtorh}
\end{equation}
in other words, we have expressed the ambiguity in the tidal parameter
for small black holes by the parameter $b$, and the relationship
between the asymptotic mass of the black hole as measured on the brane
and the horizon radius explicitly factors in this ambiguity. As we now see,
this uncertainty can be absorbed into a redefinition of the low energy
Planck scale in the tunnelling rate.
The tunnelling process starts with the uniform false vacuum $\phi_v$
and a seed black hole with mass $M_S$. This false vacuum configuration
resembles the tidal black hole on the brane, and a slightly perturbed
5D Schwarzschild solution in the bulk \cite{Kudoh:2003xz}.
The bubble solution represents the decay process to another state
with the field asymptoting the same false vacuum at large distances
but with the field approaching its true vacuum near the horizon of a
remnant black hole with mass $M_R$, which remains after tunnelling.
In the previous section we showed that the tunnelling exponent is given by
\begin{equation}
B=\frac{1}{4G_5}\left({\cal A}_S-{\cal A}_R\right),
\end{equation}
where $S$ represents the seed black hole area
and $R$ that of the remnant black hole (recall, this
area is the full five dimensional area of the horizon
extending into the bulk).
To leading order in $r_h/\ell$, the small black hole horizon has
an approximately hyperspherical shape, therefore the area will
be well approximated by $2\pi^2 r^3$, hence
\begin{equation}
B= \frac{\pi^2}{2G_5}\left(r_S^3-r_R^3\right)
= \frac{\pi^2 r_S^3}{2G_5}
\left[1- \left (\frac{M_R}{M_S} \right)^{\frac32}\right]
\end{equation}
using \eqref{mtorh}. In the limit that the difference in seed and
remnant black hole masses is small, $(M_S-M_R)/M_S =
\delta M/M_s \ll 1$, we finally arrive at
\begin{equation}
B\approx \frac34 \left(\frac{\pi M_S}{ b M_5}\right)^{3/2}
\frac{\delta M}{M_S},
\label{Bapprox}
\end{equation}
where $M_5=(8\pi G_N\ell)^{-1/2}$ is the low energy Planck scale. Fortuitously,
the uncertainty in the value of the tidal charge parameter
$b$ can be absorbed into our uncertainty in the low energy Planck scale, and
so we let $bM_5\to M_5$.
\subsection{Higgs bubbles on the brane}
The Higgs bubble will correspond to a solution of the brane SMS
equations with an energy momentum tensor derived from the
(Euclidean) scalar field Lagrangian\footnote{Note that we have
defined the Euclidean Lagrangian to contain $+V$, meaning that
the false vacuum solution will have energy-momentum $-V g_{\mu\nu}$,
but that our 4D Einstein equations will have the conventional sign
for the energy-momentum, i.e.\ $G_{\mu\nu} =
8\pi G_N T_{\mu\nu}+ \dots$.}
\begin{equation}
{\cal L}_m=\frac12g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}+V(\phi).
\end{equation}
where $V(\phi)$ has a metastable false vacuum. The SMS equations
for the bubble, assuming the general form \eqref{genbranemet}
are derived in appendix \ref{app:braneq}, and are
\begin{align}
&f\phi''+f'\phi'+\frac2r f\phi'+f\delta'\phi'-V_{,\phi}=0,\label{eq1}\\
&\mu'=4\pi r^2\left\{ \frac12 f\phi'{}^2+V-
\frac{2\pi G_N}{3}\ell^2(\frac12 f\phi'{}^2-V)
(\frac32 f\phi'{}^2+V)\right\},\label{eq2}\\
&\delta'=4\pi G_N r\phi'{}^2\left\{1-\frac{4\pi G_N}{3}\ell^2
(\frac12 f\phi'{}^2-V)\right\}\label{eq3}.
\end{align}
where, for comparison with the vacuum case (\ref{vbh}), we
have defined a ``mass'' function $\mu(r)$ by
\begin{equation}
f(r)=1-\frac{2G_N\mu(r)}{r} - \frac{r_Q^2}{r^2}.
\label{vbub}
\end{equation}
These are integrated numerically from the black hole horizon $r_h$
to $r\to\infty$ where $\phi$ is in the false vacuum.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\textwidth]{fit.pdf}
\caption{The Higgs potential calculated numerically at one loop order for top
quark mass $M_t=172\,{\rm GeV}$ and the
approximate potential using (\ref{lambdaeff}) with values of $g$ and
$\Lambda_\phi$ chosen
for the best fit.}
\label{fig:potential}
\end{figure}
The numerical results contained in this section are based on a Higgs-like potential,
assuming that the standard model holds for energy scales up to the low energy
Planck mass $M_5$. The detailed form of the potential is determined by
renormalisation group methods and depends on low-energy particle masses,
with strong dependence on the Higgs and top quark masses. Of these, the
top quark mass is less well known, and for masses in the range
$171-174\, {\rm GeV}$, Higgs instability sets in at scales from
$10^{10}-10^{18}\,{\rm GeV}$.
The Higgs potential is usually expressed in the form
\begin{equation}
V(\phi)=\frac14\lambda_{\rm eff}(\phi)\phi^4.
\end{equation}
with a running coupling constant $\lambda_{\rm eff}(\phi)$ that
becomes negative at some crossover scale $\Lambda_\phi$.
Vacuum decay depends on the shape of the potential barrier
in the Higgs potential around this instability scale, and in order
to explore the likelihood of decay it is useful to use an analytic
fit to $\lambda_{\rm eff}$. In \cite{BGM3}, we used a two
parameter fit to $\lambda_{\rm eff}$, where one of the parameters
was closely related to the crossover scale. We found that the dependence
of the instanton action on the potential was strongly dependent on
this parameter, but very weakly dependent on the second parameter,
which was more related to the shape of the potential at low energy.
For clarity therefore, here we take a one parameter analytic fit to
$\lambda_{\rm eff}$, where the single parameter is the crossover scale
$\Lambda_\phi$:
\begin{equation}
\lambda_{\rm eff}=g(\Lambda_\phi) \left\{\left(\ln \frac{\phi}{M_p}\right)^4
-\left(\ln \frac{\Lambda_\phi}{M_p}\right)^4\right\}
\label{lambdaeff}
\end{equation}
and $g(\Lambda_\phi)$, chosen to fit the high energy asymptote of
$\lambda_{\rm eff}$, varies very little across the range of $\Lambda_\phi$
of relevance to the Standard Model $\lambda_{\rm eff}$. Figure
\ref{fig:potential} shows a sample of our analytic fit for the Higgs
potential to the actual $\lambda_{\rm eff}$ computed for $M_t = 172$GeV.
In four dimensions, we can have a Higgs instability scale very close to
the Planck scale, however with large extra dimensions, new physics
could potentially enter at the low-energy Planck scale $M_5$, thus to be
consistent, we should restrict our parameters to the range $\Lambda_\phi<M_5<M_p$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\textwidth]{field.pdf}
\includegraphics[width=0.9\textwidth]{mass.pdf}
\caption{Profiles for the bubble and the mass term $\mu(r)$ outside
the horizon $r_h$ with $M_5=10^{15}$GeV, $\Lambda_{\phi}=10^{12}$GeV
and $r_h=20000/M_p$.}
\label{fig:field}
\end{figure}
Figure \ref{fig:field} gives profiles for the bubble centered on the black
hole after tunnelling and for the mass term $\mu(r)$ beyond the horizon
radius $r_h$. The solutions are shown for an instanton with action $B=4.3$.
The field is in the true vacuum at the horizon and approaches the false
vacuum as $r\to\infty$ with a characteristic thick wall profile.
The bubble radius greatly exceeds the horizon of the black hole.
The change in the mass term is given by $\Delta\mu(r)=\mu(r)-\mu(r_h)$.
Near the horizon, $\Delta \mu(r)$ is negative due to the negative potential
$V$ in equation \ref{eq2}.
$\mu(r)$ becomes positive at large $r$ where there is a positive contribution
from the kinetic term and hence $\Delta M$ is positive.
\subsection{Branching ratios}
The calculation of the vacuum decay rate assumes a stationary background
which only makes sense when the decay rate exceeds the Hawking evaporation rate.
The brane black hole can radiate in the brane or into the extra dimension, but
if we consider a scenario as close as possible to the standard model then most
of the radiation will be in the form of quarks and leptons radiated into the brane,
simply because these are the most numerous particles. (For a review of Hawking
evaporation rates in higher dimensions see \cite{Kanti:2014vsa}.)
Black hole radiation is similar to the radiation from a black body with the same area
as the black hole horizon and at the Hawking temperature,
but with additional `grey body' factors representing the effects of back-scattering of
the radiation from the space-time curvature around the black hole.
Following \cite{Kanti:2014vsa}, we can express the energy loss rate due to evaporation
as $\dot E$, where on dimensional grounds (since $r_h$ is the only relevant dimensionful
parameter)
\begin{equation}
|\dot E|=\gamma \,r_h^{-2},
\end{equation}
for some constant $\gamma$. The Hawking decay rate of the black hole $\Gamma_H$, using
(\ref{mtorh}) to eliminate the radius, is
\begin{equation}
\Gamma_H=\frac{|\dot E|}{M_S}=\frac{4\pi\gamma M_5^3}{M_S^2}
\end{equation}
The vacuum decay rate is given by
\begin{equation}
\Gamma_D=Ae^{-B}.
\end{equation}
The pre-factor $A$ contains a factor $(B/2\pi)^{1/2}$ from a zero mode
and a vacuum polarisation term from the other modes, whose characteristic
length scale is the bubble radius $r_b$. We estimate
\begin{equation}
\Gamma_D\approx\left(\frac{B}{2\pi}\right)^{1/2}\frac{1}{r_b}e^{-B}.
\end{equation}
The branching ratio of the two is
\begin{equation}
{\Gamma_D\over\Gamma_H}\approx{1\over\gamma}\left(\frac{B}{2\pi}\right)^{1/2}
\left(\frac{M_S}{M_5}\right)^{3/2}\left(\frac{r_h}{r_b}\right)e^{-B}
\end{equation}
Vacuum decay is important when this ratio is larger than one.
In the case of small $r_h/\ell$, the five-dimensional black hole has a temperature
\begin{equation}
T\approx \frac{1}{2\pi r_h},
\end{equation}
which is double the temperature of a black hole solely in four dimensions.
We would therefore expect to have energy flux on the brane roughly
$\propto T^4\sim 16$ times the flux solely in four dimensions. Numerical
results actually give a factor of 14.2 for fermion fields, which give the
largest contribution to the decay \cite{Harris:2003eg}. The energy
loss due to a fermion in four dimensions contributes a factor of
$7.88\times 10^{-4}$ for each degree of freedom to $\gamma$,
giving a total for $90$ standard model fermion degrees of freedom of
\begin{equation}
\gamma\approx 14.2\times 90\times 7.88\times 10^{-4}=0.10.
\end{equation}
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\textwidth]{Rates.pdf}
\caption{
The branching ratio of the false vacuum nucleation rate to
the Hawking evaporation rate as a function
of the seed mass for a selection of Higgs models with $M_5=10^{15}{\rm GeV}$.}
\label{fig:ratio}
\end{figure}
The branching ratio is plotted in figure \ref{fig:ratio} for $M_5=10^{15}{\rm GeV}$ and
Higgs instability scale around $10^{12}\,{\rm GeV}$ (corresponding to a top quark
mass of $172\,{\rm GeV}$). Note that the decay rates in this parameter range are
larger than $M_5^3/M_S^2$, i.e.\ they are extremely fast. The figure shows an example
where black holes with masses between $10^{17}\,{\rm GeV}$ and $10^{20}\,{\rm GeV}$,
or $10^{-7}\,{\rm g}$ to $10^{-4}\,{\rm g}$, would seed rapid Higgs vacuum decay.
\section{Conclusions}
\label{sec:disc}
In this paper we have explored the impact of large extra dimensions
on black hole seeded vacuum decay. We used the Randall-Sundrum
set-up as a concrete example for warped extra dimensions, and
numerically computed the Higgs profile on the brane for vacuum decay
assuming a \emph{tidal Ansatz} for the Weyl tensor on the brane.
Although the solution for a brane black hole is not known analytically, we
were nonetheless able to construct an argument that the action for tunnelling
would still be the difference in areas of the black hole horizons. In order
to estimate these areas, we focussed on small brane black holes (expected
to be the most relevant for vacuum decay), and used qualitative features of
the numerical solutions to argue the black hole area would be very well
approximated by the hyperspherical result $2\pi^2r_h^3$. We then used the
tidal model for a brane black hole (in keeping with the tidal Ansatz for the
Weyl tensor), expanded for small masses, to relate the 4D brane mass of the
black hole, the $1/r$ fall-off of the Newtonian potential, to the horizon radius.
This then allowed us to compute the amplitude for tunnelling.
Since a black hole can also radiate, we then have to consider whether
the evaporation rate is so fast that the tunnelling amplitude is irrelevant,
or whether the tunnelling probability becomes so high for small black holes
(as was the case for purely four dimensional black holes \cite{BGM3}) that
the black hole always initiates decay. We therefore estimated the nett
evaporation rate by taking the integrated flux from \cite{Harris:2003eg},
which is dominated by the fermion radiation, and summing up the effect
from the standard model particles. The branching ratio plot of figure \ref{fig:ratio}
demonstrates that, just as in 4D, small black holes in higher dimensions
are overwhelmingly likely to initiate vacuum decay once they have
radiated away sufficient mass to enter this danger range. As with pure 4D,
any small black hole, formed either in the early universe, or in a high energy
cosmic ray collision, will radiate, lose mass, then become sufficiently light
that it seeds decay with a rate of order $10^{3-5} T_5$.\footnote{Here,
$T_5 = (c^3/8\pi G_5 \hbar)^{1/3}$ is the 5D Planck time.} What is interesting
here is that what we mean by \emph{small} is now very different to the
pure 4D case.
With large extra dimensional scenarios, we generate a high 4D Planck
scale geometrically, having a renormalization of the Newton constant
coming from the `volume' of the internal dimensions. Thus, in 4D, where
the typical black hole seeding vacuum decay for the Higgs was
in the range $10^5-10^9 M_p \simeq 1$g$-10$ tonnes, these black
holes could only be primordial in origin, having far too high a mass to be
produced in a particle collision. Here however, our Planck mass can be much
lower, so $10^5 M_5$ can potentially be sufficiently low that the black hole
could be produced in cosmic ray collision. For example,
the highest energy cosmic ray collisions
\cite{Linsley:1963km,Nagano:2000ve,ThePierreAuger:2015rha} observed
have an energy in excess of $10^{11}$GeV.
Hut and Rees \cite{Hut:1983xa} have shown that there are
at least $10^5$ collisions with centre of mass energy exceeding $10^{11}$ GeV in our
past light cone. Thus, provided the higher dimensional
Planck scale were below $M_5\lesssim 10^9$GeV, black holes could
be formed in a cosmic ray collision that would be sufficiently light to
catalyse vacuum decay.
In the context of the Higgs field, the standard model potential is only
valid at best for energy scales below the scale of new physics, $M_5$,
therefore the instability scale should satisfy $\Lambda_\phi<M_5$.
The lowest possible value for the instability scale consistent with experimental
limits on the top quark mass is around $10^8\,{\rm GeV}$, thus we cannot use
our standard model Higgs decay results unless $M_5\gg10^8\,{\rm GeV}$,
well outside the range probed by the LHC.
As an example, consider an instability scale $\Lambda_\phi\sim 10^{8}\,{\rm GeV}$,
and Planck scale $M_5\sim10^{9}\,{\rm GeV}$, then black holes of mass
$M_S\sim10^{11}\,{\rm GeV}$ could cause Higgs vacuum decay.
These values are below those for which we were able
to obtain numerical results, but we can make a rough approximation by
taking the exponent for vacuum decay $B$ from \eqref{Bapprox}, and
the mass of the instanton $\delta M\sim \Lambda_\phi$. For these
values we estimate $B=O(1)$ and rapid Higgs decay would take place.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\textwidth]{Rotation.pdf}
\caption{
The branching ratio of the false vacuum nucleation rate to
the Hawking evaporation rate as a function of the seed mass
for a selection of Higgs models with $M_5=10^{15}{\rm GeV}$,
and $\Lambda_\phi=5\times 10^{12}\,{\rm GeV}$}
\label{fig:rotating}
\end{figure}
Black holes produced by high energy collisions would be likely to
be rotating. Rotating tidal black hole solutions \cite{Aliev:2005bi}
can be used as the basis for these black hole seeds.
The bubble solutions about these rotating holes will become
distorted, however the profile of the bubble solution (fig.\ \ref{fig:field})
indicates that much of the variation of the bubble fields occurs at large
radii compared to the horizon size of the black hole. This
suggests that the distortion will be localised in the
small part of the bubble near the black hole,
leaving the effective mass $\delta M$ in the field configuration
relatively unaffected.
In this case, we can use our earlier result (\ref{Bapprox}) but
replacing the horizon area with the area ${\cal A}_{MP}$ of a rotating
Myers-Perry black hole in flat space \cite{Myers:1986un} when $r_h\ll \ell$,
\begin{equation}
B\approx \frac{{\cal A}_{MP}}{4G_5}\frac{3\delta M}{2M_S}.
\end{equation}
The area depends on two rotation parameters $a_1$ and $a_2$,
but for a rotation axis aligned to the brane we can take $a_2=0$.
In this case
\begin{equation}
{\cal A}_{MP}=2\pi^2r_0^3\left(1-\frac{a^2}{r_0^2}\right)^{1/2},
\end{equation}
where $r_0$ is the horizon radius of the non-rotating black hole solution,
\begin{equation}
r_0^2=\frac{8G_5M_S}{3\pi}.
\end{equation}
The area is smaller than the non-rotating case.
Furthermore, the Hawking temperature is reduced, since
\begin{equation}
T_H=T_0\left(1- \frac{a^2}{r_0^2}\right)^{1/2}
\end{equation}
The numerical results for vacuum decay are shown in figure \ref{fig:rotating}.
The vacuum decay rate $Ae^{-B}$ with rotating seeds is larger
than than with non-rotating seeds due to the reduced area.
While this is a rather rough argument, the basic intuition that the
branching ratio will be enhanced both by the larger decay rate and
the reduced Hawking evaporation rate is likely to be correct. In other
words, if the existence of large extra dimensions does not destroy the
vacuum metastability of the standard model Higgs, then ultra high energy
particle collisions risk producing black hole seeds that will catalyse the
decay of the vacuum.
\acknowledgments
We are grateful for the hospitality of the Perimeter Institute, where part
of this research was undertaken. This work was supported in part by the
Leverhulme grant \emph{Challenging the Standard Model with Black Holes}
and in part by STFC consolidated grant ST/P000371/1.
LC acknowledges financial support from CONACyT,
RG is supported in part by the Perimeter Institute for Theoretical Physics,
and KM is supported by an STFC studentship.
Research at Perimeter Institute is supported by the Government of
Canada through the Department of Innovation, Science and Economic
Development Canada and by the Province of Ontario through the
Ministry of Research, Innovation and Science.
|
{
"timestamp": "2018-12-12T02:18:48",
"yymm": "1803",
"arxiv_id": "1803.02871",
"language": "en",
"url": "https://arxiv.org/abs/1803.02871"
}
|
\section{Harris Corner Detection PolyMage DSL Code}
\lstset{ %
basicstyle=\rmfamily\scriptsize,
keywordstyle=\textbf,
language=Python,
numbers=none,
numberstyle=\footnotesize,
stepnumber=1,
numbersep=3pt,
backgroundcolor=\color{white},
showspaces=false,
showstringspaces=false,
showtabs=false,
frame=false,
tabsize=1,
captionpos=b,
breaklines=true,
breakatwhitespace=true,
escapeinside={(*}{*)},
framerule=0pt,
morekeywords={Domain,Function,Interval,Accumulator,Variable,
Constant,Parameter,Kernel2D,Correlate,Min,Max,Sum,
Image,Case,Default,Upsample,Downsample,Replicate,
Reduce, UChar, Char, Float, Double, Int, UInt,
Short, UShort, Long, ULong, Condition, Stencil,
Accumulate}
}
\begin{lstlisting}[language=python,label={listing:hcd},basicstyle=\scriptsize\ttfamily,frame=lines,captionpos=b,caption=PolyMage
DSL code for Harris Corner Detection,float=h]
R, C = Parameter(Int, "R"), Parameter(Int, "C")
I = Image(Float, "I", [R+2, C+2])
x, y = Variable("x"), Variable("y")
row, col = Interval(0, R+1, 1), Interval(0, C+1, 1)
c = Condition(x, '>=', 1) & Condition(x, '<=', R) & \
Condition(y, '>=', 1) & Condition(y, '<=', C)
cb = Condition(x, '>=', 2) & Condition(x, '<=', R-1) & \
Condition(y, '>=', 2) & Condition(y, '<=', C-1)
Iy = Function(varDom = ([x, y], [row, col]), Float, "Iy")
Iy.defn = [ Case(c, Stencil(I(x, y), 1.0/12, \
[[-1,-2,-1], \
[ 0, 0, 0], \
[ 1, 2, 1]])) ]
Ix = Function(varDom = ([x, y], [row, col]), Float, "Ix")
Ix.defn = [ Case(c, Stencil(I(x, y), 1.0/12, \
[[-1, 0, 1], \
[-2, 0, 2], \
[-1, 0, 1]])) ]
Ixx = Function(varDom = ([x, y], [row, col]), Float, "Ixx")
Ixx.defn = [ Case(c, Ix(x, y) * Ix(x, y)) ]
Iyy = Function(varDom = ([x, y], [row, col]), Float, "Iyy")
Iyy.defn = [ Case(c, Iy(x, y) * Iy(x, y)) ]
Ixy = Function(varDom = ([x, y], [row, col]), Float, "Ixy")
Ixy.defn = [ Case(c, Ix(x, y) * Iy(x, y)) ]
Sxx = Function(varDom = ([x, y], [row, col]), Float, "Sxx")
Syy = Function(varDom = ([x, y], [row, col]), Float, "Syy")
Sxy = Function(varDom = ([x, y], [row, col]), Float, "Sxy")
for pair in [(Sxx, Ixx), (Syy, Iyy), (Sxy, Ixy)]:
pair[0].defn = [ Case(cb, Stencil(pair[1], 1, \
[[1, 1, 1], \
[1, 1, 1], \
[1, 1, 1]])) ]
Det = Function(varDom = ([x, y], [row, col]), Float, "det")
d = Sxx(x, y) * Syy(x, y) - Sxy(x, y) * Sxy(x, y)
Det.defn = [ Case(cb, d) ]
Trace = Function(varDom = ([x, y], [row, col]), Float, "trace")
Trace.defn = [ Case(cb, Sxx(x, y) + Syy(x, y)) ]
Harris = Function(varDom = ([x, y], [row, col]), Float, "harris")
coarsity = Det(x, y) - 0.04 * Trace(x, y) * Trace(x, y)
Harris.defn = [ Case(cb, coarsity) ]
\end{lstlisting}
\end{appendices}
\section{Acknowledgments}
\label{sec:acknowledgments}
We would like to gratefully acknowledge the Science and Engineering Research Board (SERB), Government of India for funded this research work in part through a grant under its EMR program (EMR/2016/008015).
\section{Background}
\label{sec:background}
\begin{figure}[tb]
\centering
\overview
\vskip 5pt
\caption{PolyMage high-level compilation for FPGAs.}
\label{fig:overview}
\end{figure}
In this section, we briefly explain the architecture of the PolyMage-HLS
compilation framework; and further introduce the basics of interval and
affine arithmetic necessary to understand how range analysis techniques based
on them can be seamlessly integrated into our PolyMage-HLS compiler.
\subsection{PolyMage DSL and Compilation Framework}
In this paper, we use the PolyMage DSL and its compiler infrastructure
to implement and evaluate our automatic bitwidth analysis approach.
The PolyMage compiler infrastructure, when it was first
proposed~\cite{mullapudi2015asplos}, comprised an optimizing
source-to-source translator that generated OpenMP C++ code from an input
PolyMage DSL program. Chugh et al.~\cite{chugh16pact} developed a
backend for PolyMage targeting FPGAs by generating
HLS C++ code. The generated C++ code is translated into a hardware design
expressed in a Hardware Description Language (HDL) such as VHDL or Verilog
using High Level Synthesis compiler.
Figure~\ref{fig:overview}
shows the entire design flow using the PolyMage-HLS compiler and Xilinx Vivado
tool chain. For syntactic details and code examples, we refer the reader to
the PolyMage webpage~\cite{polymage-web}.
\subsection{Interval and Affine arithmetic}
With interval analysis, one estimates the range of an output
signal $z \leftarrow f\left(x, y\right)$ based on the range of the input signals $x$
and $y$, and the function $f$. For example, if the range of $x$ and $y$ are
$[\low{x},\high{x}]$ and $[\low{y}, \high{y}]$ respectively, and
$z\leftarrow x+y$, then the range of $z$ is
$[\low{x}+\low{y}, \high{x} +\high{y}]$. Such range estimation functions have
to be defined for different operations that are applied iteratively to
obtain the ranges of different intermediate and output signals involved in
the computation. Although interval arithmetic is simple and easy
to use in to practice, it suffers from the problem of range over-estimation.
For example, if the range of a signal $x$ is $[5, 10]$, then the interval
arithmetic estimates the range of the expression $x - x$ as $[-5, 5 ]$ whereas
the actual range is $[0, 0]$. This is due to the fact that the interval
arithmetic ignores the correlations between the operand signals if there were any.
\iffalse
Let the range of the two input signals $x$
and $y$ be $[\low{x}, \high{x}]$ and $[\low{y}, \high{y}]$ respectively. Then
range of the resulting signal $z = x~op~y$ where $op\in\{+,-,*,/, \verb|^| \}$ is
obtained by applying standard rules of interval arithmetic as follows.
\begin{enumerate}
\item {$\mathbf{z=x+y:}$} $[\low{z},\high{z}] = [\low{x}+\low{y}, \high{x}+\high{y}]$
\item {$\mathbf{z=x-y:}$} $[\low{z},\high{z}] = [\low{x}-\high{y}, \high{x}-\low{y}]$
\item {$\mathbf{z=x*y:}$} $[\low{z},\high{z}] = [t_1, t_2 ]$ where \\$t_1 = \min\left(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y}\right)$
and \\ $t_2 = \max\left(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y}\right)$.
\item $\mathbf{z=x/y:}$$[\low{z},\high{z}] = [\low{x}, \high{x}] * [1/\high{y}, 1/\low{y}]$ if $0 \not \in [\low{y}, \high{y}]$
\item $\mathbf{z = x^n}$
\begin{enumerate}
\item $n$ is odd: $[\low{z},\high{z}] = [\low{x}^n, \high{x}^n ]$
\item $n$ is even :
$\begin{aligned}[t]
[\low{z},\high{z}] & = & [\low{x}^n, \high{x}^n ] ~\textrm{if}~\low{x} \geq 0 \\
& = & [\high{x}^n, \low{x}^n ] ~\textrm{if}~\high{x} < 0\\
& = & [0, max\{\low{x}^n, \high{x}^n\} ]~\textrm{otherwise}
\end{aligned}$
\end{enumerate}
\iffalse
\begin{enumerate}
\item {\bf If} $\mathbf{\low{y} > 0}$ {\bf or} $\mathbf{\high{y}<0}$: \\$[\low{z},\high{z}] = [t_1, t_2 ]$ where \\ $t_1 = \min\left(\low{x}\mbox{\small/}\low{y}, \low{x}\mbox{\small/}\high{y}, \high{x}\mbox{\small/}\low{y}, \high{x}\mbox{\small/}\high{y}\right)$ and \\$t_2 = \max\left(\low{x}\mbox{\small/}\low{y}, \low{x}\mbox{\small/}\high{y}, \high{x}\mbox{\small/}\low{y}, \high{x}\mbox{\small/}\high{y}\right)$.
\item {\bf Otherwise:} $[\low{z}, \high{z}] = [\low{p}, \high{p}]$ where $\low{p}$ and $\high{p}$ are obtained via profiling information as explained in Section~\ref{sec:profile}.
\end{enumerate}
\fi
\end{enumerate}
Note that if an operand $x$ (or $y$) is constant, we get a tight range for
the result since $x=\low{x}=\high{x}$. In the above four cases, division
operation requires special care. If $0$ is present in the range $[\low{y},
\high{y}]$ of the denominator, then the theoretical range of $z$ can
potentially tend to infinity. Although interval arithmetic is simple and easy
to use in practice, it suffers from the problem of range over-estimation.
For example, if the range of a signal $x$ is $[5, 10]$, then the interval
arithmetic estimates the range of the expression $x - x$ as $[-5, 5 ]$ whereas
the actual range is $[0, 0]$. This is due to the fact that the interval
arithmetic ignores the correlation between the operand signals if there is
any.
\fi
With affine arithmetic analysis, a signal $x$ is
represented in an affine form as $x=x_0 + \sum_{i=1}^{n} x_i \epsilon_i$
where $\epsilon_i \in [-1, 1]$ are interpreted as independent noise signals
and their respective coefficients $x_i$'s are treated as the weights
associated with them. The interval of the signal $x$ from its affine form
can be inferred as $[x_0-r, x_0+r]$ where $r=\sum_{i=1}^{n} |x_i|$. The
addition
and subtraction operations on two input signals is defined as
$z=x\pm y = \left(x_0 \pm y_0\right) + \sum_{i=1}^{n}
\left(x_i \pm y_i\right)\epsilon_i$ and yields the resulting signal in its affine form.
The correlation between the signals
$x$ and $y$ is captured by sharing the independent noise signals $\epsilon_i,
1\leq i \leq n $ in their affine forms either partially or totally.
Now, when we perform a computation $x-x$ by considering the signal $x$ in its
affine form, the resulting range will be zero as against the over-estimated
range which we get in interval arithmetic analysis. Thus the techniques based
on affine arithmetic
arrive at better bounds when compared with interval analysis based techniques by
taking into account cancellation effects in computations involving
correlated signals.
However, note that if the operation is multiplication,
then the resulting signal contains quadratic terms and hence has to be
approximated to an affine form. A detailed discussion on affine arithmetic
analysis is beyond the scope of this paper and we recommend the reader to
Stolfi and Figueiredo~\cite{TEMA352} for the same.
\section{Bitwidth Analysis}\label{sec:bitwidth}
In this section, we present the main technical contributions of this paper which
are summarized below.
\begin{enumerate}
\item A simple interval arithmetic based range analysis algorithm
illustrating how DSLs facilitate practical and efficient program analysis
techniques when compared with C/C++ kind of languages
(Section~\ref{sec:ra}).
\item A software architecture for range analysis in DSL compilers in which variants of interval and affine arithmetic based approaches
can be easily deployed (Section~\ref{sec:ca}).
\item An SMT solver based approach for range analysis
which substantially improves the accuracy of range estimates and contains the
propagation of estimation errors across iterations. Again, such an SMT solver
based approach would have been hard to realize if not for the DSL compiler
framework (Section~\ref{sec:smt}).
\item A profile driven approach for range analysis (Section~\ref{sec:profile}).
\item A greedy heuristic search technique for precision estimation
(Section~\ref{sec:precision}).
\end{enumerate}
\subsection{Variable Width Fixed-Point Data Types}
A fixed-point data type is specified by a tuple $(\alpha, \beta)$ where
$\alpha$ and $\beta$ denote the number of bits allocated for representing
the integral and fractional parts respectively. The total bitwidth of the
data type is $\alpha + \beta$. The decimal value associated with a
fixed-point binary number $x=b_{\alpha-1}\ldots b_0 . b_{-1} \ldots
b_{-\beta}$ depends on whether it is interpreted as an unsigned integer or a
two's complement signed integer, and is given as follows:
\begin{equation*}
value(x) = \begin{cases}
\sum^{\alpha-1}_{i=-\beta} 2^i b_i & \text{unsigned} \\
-2^{\alpha-1} b_{\alpha-1} + \sum_{i=-\beta}^{\alpha-2} 2^i b_i & \text{2's
complement.}
\end{cases}
\end{equation*}
This gives us the ranges $[0, 2^{\alpha}-2^{-\beta}]$ and $[-2^{\alpha-1} ,
2^{\alpha-1}-2^{-\beta}]$ for unsigned and signed fixed-point numbers
respectively. The parameter $\alpha$ gives the range of values that can be
represented and the parameter $\beta$ indicates that the values in the range
can be represented at a resolution of $2^{-\beta}$. Hence, the range and
precision can be improved, by increasing $\alpha$ and $\beta$ respectively.
In this paper, we overload the term precision to also mean the entire data
type $(\alpha, \beta)$, and this can be disambiguated based on context.
Fixed-point data types are useful in image processing applications where the
range of values produced during computations is usually limited
and the precision requirements are less demanding when compared to many
other numerical algorithms. The data type (range and precision) requirement
at a stage depends on the input data type and the nature of local
computations carried out at that particular stage. Further, overflows during
computations can be addressed by using saturation mode arithmetic instead of
the conventional wrap around arithmetic operations performed in CPUs and GPUs.
\iffalse
This feature is
particularly useful in handling image processing pipelines using the
approximate computing paradigm.
\fi
The complexity of arithmetic operations on
fixed-point data type $(\alpha, \beta)$ is very similar to that of integer
operations on bitwidth $\alpha+\beta$.
\subsection{Range ($\alpha$) Analysis Algorithm}\label{sec:ra}
The number of integer bits required at a stage $I$ denoted as $\alpha_I$ is
a direct function of the bitwidth of the input data and the operations it
performs on them. The input data here refers to the data supplied to the
stage by its predecessor stages in the DAG. Further, the computations on
the pixel signals at each stage of DAG are identical and hence their
corresponding ranges would be the same. This information is implicitly provided
by a PolyMage DSL program and is hard to elicit from C like programs.
We use this insight to group all the pixel signals at a stage and perform a
combined range analysis using interval arithmetic. If the range of the
data produced at a stage is $[\low{x}, \high{x}]$, then the number of
integral bits $\alpha_I$ required to store the data without overflow is as
follows:
\begin{equation*}
\alpha=
\begin{cases}
\max(\ceil{\log_2(\ceil{|\low{x}|})},\ceil{\log_2(\floor{|\high{x}|}+1)})+1 & \text{if } \low{x} <0\\
\ceil{\log_2(\floor{\high{x}}+1)}, & \text{otherwise}.
\end{cases}%
\end{equation*}
The number of fractional bits
$\beta_I$ required at a stage depends on the application-specific error
tolerance or quality metric, and we propose a profile-driven estimation
technique in Section~\ref{sec:profile}.
The range analysis algorithm iterates over the stages of a DAG in a
topologically sorted order. At each stage, an equivalent expression tree
for the computations (point-wise or stencil) is built. Then the range of the
pixel signals is estimated by recursively performing interval arithmetic on the
expression tree using one of following five interval arithmetic rules.
\begin{enumerate}
\item {$\mathbf{z=x+y:}$} $[\low{z},\high{z}] = [\low{x}+\low{y}, \high{x}+\high{y}]$
\item {$\mathbf{z=x-y:}$} $[\low{z},\high{z}] = [\low{x}-\high{y}, \high{x}-\low{y}]$
\item {$\mathbf{z=x*y:}$} $[\low{z},\high{z}] = [t_1, t_2 ]$ where \\$t_1 = \min(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y})$
and \\ $t_2 = \max(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y})$.
\item $\mathbf{z=x/y:}$$[\low{z},\high{z}] = [\low{x}, \high{x}] * [1/\high{y}, 1/\low{y}]$ if $0 \not \in [\low{y}, \high{y}]$
\item $\mathbf{z = x^n}$
\begin{enumerate}
\item $n$ is odd: $[\low{z},\high{z}] = [\low{x}^n, \high{x}^n ]$
\item $n$ is even :
$\begin{aligned}[t]
[\low{z},\high{z}] & = & [\low{x}^n, \high{x}^n ] ~\textrm{if}~\low{x} \geq 0 \\
& = & [\high{x}^n, \low{x}^n ] ~\textrm{if}~\high{x} < 0\\
& = & [0, max\{\low{x}^n, \high{x}^n\} ]~\textrm{otherwise}
\end{aligned}$
\end{enumerate}
\end{enumerate}
\iffalse
. Algorithm~\ref{alg:precision}
shows the complete description of the bitwidth estimation algorithm applied
at each stage of the DAG computation. We do not show how an expression such
as $x ^ n$, where $n$ is a compile-time constant, is handled in
Algorithm~\ref{alg:precision} for the sake of simplicity. The interval
arithmetic for exponentiation operation, $[\low{x}, \high{x}]^n$ , is defined
as follows.
\begin{eqnarray*}
[\low{z},\high{z}] & = & [\low{x}^n, \high{x}^n ] ~\textrm{if}~\low{x} \geq 0 \\
& = & [\high{x}^n, \low{x}^n ] ~\textrm{if}~\high{x} < 0\\
& = & [0, max\{\low{x}^n, \high{x}^n\} ]~\textrm{otherwise}
\end{eqnarray*}
Our compiler maps simple expressions such as $x*x$ into $x^2$ as this results
in better range estimates. For example, if the range of a signal $x$ is
$[-2, 2]$, then $x*x = [-2, 2]*[-2, 2] = [-4, 4]$, whereas
$x^2 = [-2, 2]^2 = [0, 4]$.
\begin{table}[htb]
\centering
\small
\caption{HCD range analysis and integral bitwidth requirement at various stages.
\label{tab:hcdbit}}
\vskip 5pt
\begin{tabularx}{0.9\linewidth}{lcc}
\toprule
Stage & Range & $\alpha$ \\
\midrule
Img & $[0,255]$ & $8$ \\
I$_x$,I$_y$ & $[-85,85]$ & $8$ \\
I$_{xy}$ & $[-85^2,85^2]$ & $14$ \\
I$_{xx}$, I$_{yy}$ & $[0, 85^2]$ & $13$ \\
S$_{xy}$ & $[-9*85^2,9*85^2]$ & $17$ \\
S$_{xx}$, S$_{yy}$ & $[0, 9*85^2]$ & $16$ \\
det & -$[-(9*85^2)^2, (9*85^2)^2]$ & $33$ \\
trace & $[0,2*9*85^2]$ & $17$ \\
harris & $[-1.16*(9*85^2)^2, (9*85^2)^2]$ & $34$ \\
\bottomrule
\end{tabularx}
\end{table}
Figure~\ref{fig:hdag} shows the DAG corresponding to the HCD benchmark.
For the sake of simplicity, instead of showing the PolyMage program,
we summarized the computations at each stage in Table~\ref{tab:hcd}.
Table~\ref{tab:hcdbit} summarizes the
ranges and worst case precision requirements of various stages in HCD
as estimated by the range analysis algorithm. It can be noted from the
Table~\ref{tab:hcdbit} that the
bitwidth requirement of $det$ and $Harris$ stages is slightly greater than
the 32 bits required for a
floating-point computation. But the power, area and the delay associated
with fixed point arithmetic at this length could still be better than the
floating-point arithmetic operations.
\fi
\iffalse
The range associated
with each pixel of input image is $[0,255]$ as it is an 8-bit image. The
required integral bitwidth $\alpha_{img}$ at this stage is 8. Stages $I_x$
and $I_y$ compute the derivative of the image along $x$ and $y$ axes
respectively. The specific 3x3 stencil operations performed by these stages
can be noted from the Table~\ref{tab:hcd}. The stencil operation here
can be expanded in the form of an expression and the range associated with
it comes out to be $[-85,85]$. Eight integral bits are required to
represent this range. Stage $I_{xy}$ performs a point-wise computation
using the expression $I_{x,y}(i,j) = I_x(i,j)*I_y(i,j)$. Based on this,
we get the range of the variable $I_{x,y}$ as $[-85^2, 85^2]$.
Our compiler deduces that the computation at
the stages $I_{xx}$ and $I_{yy}$ is a squaring operation and appropriately
infers the output signal range as $[0, 85^2]$.
Stages $S_{xx}$ , $S_{xy}$ and $S_{yy}$ perform a stencil
operation involving only summation but no division. Expanding the stencil to
an expression form gives us simple addition of neighbouring pixels. The
range we get here for stage $S_{xy}$ is $[-9*85^2,9*85^2]$, and for stages
$S_{xx}$ and $S_{yy}$ is $[0, 9*85^2]$. In a similar fashion, the range
and the associated bitwidths for the stages $det$, $trace$ and $Harris$
can be inferred. It can be noted from the Table~\ref{tab:hcdbit} that the
bitwidth requirement of $det$ and $Harris$ stages is slightly greater than
the 32 bits required for a
floating-point computation. But the power, area and the delay associated
with fixed point arithmetic at this length could still be better than the
floating-point arithmetic operations. Table~\ref{tab:hcdbit} summarizes the
ranges and worst case precision requirements of various stages in HCD.
\fi
\subsection{Bitwidth Analysis Compilation Framework}\label{sec:ca}
The interval arithmetic based range analysis algorithm proposed in the
previous section uses the fact that all the pixel signals in each stage of an
image processing DAG are homogeneous in nature and groups them to do a
combined range analysis. However, other analysis techniques such as those
based on affine arithmetic cannot be applied on the DSL level programs in the
same fashion. In this section, we show how interval and affine arithmetic based
range analysis techniques can be deployed with ease in the PolyMage compilation
framework.
Recall that the PolyMage-HLS compiler translates a DSL program into C++ code
which the Xilinx Vivado HLS compiler synthesizes into an equivalent circuit
for a target FPGA. For example, Listing~\ref{lst:sobel} depicts the C++ code
generated by the PolyMage-HLS compiler when Sobel-x filter is applied on an
input image. The generated C++ program can be run in a purely simulation
mode after compilation on any processor by providing test input images as
stimulus. It can be noted from the Listing~\ref{lst:sobel}, that the data type
of the stream, line and window buffers are parameterized by the type
{\tt\bf typ}. It can be a float or any fixed-point data type $(\alpha, \beta)$.
During the hardware synthesis or in the simulation mode, using the C++
polymorphism feature, corresponding libraries for the arithmetic operations
will be invoked based on the operand types.
Now, the parameter {\tt\bf typ} can also be set to an interval type which is
defined in a suitably chosen interval analysis library. If the generated C++
program contains a statement $x=y+z$, then depending on the type of the
variables $x$, $y$ and $z$ (like float, ia-type, aa-type etc.), appropriate
addition operation will be invoked. For example, in order
to perform affine arithmetic analysis on the Sobel-x program, using the
Yet Another Library for Affine Arithmetic (YalAA)~\cite{yalaa}, all we have to
do is to define the parameter {\tt\bf typ} appropriately as depicted in
Listing~\ref{lst:yalaa}.
\begin{table}[t]
\noindent\begin{minipage}[t]{.47\textwidth}
\begin{lstlisting}[language=C,label={lst:sobel},
breaklines=true,
basicstyle=\footnotesize,
numbers=left,
keywordstyle=\sffamily\bfseries\color{green!40!black}, commentstyle=\itshape\color{purple!40!black},
morekeywords={typ}, identifierstyle=\color{blue},frame=single
caption=Auto-generated restructured HLS code for Sobel-x.]
#include <hls_stream.h>
#include <malloc.h>
#include <cmath>
#include <arith.h>
void sobel_x(hls::stream<typ> & img,
hls::stream<typ> & sobel_x_out)
{
const int _ct0 = (2 + R);
const int _ct1 = (2 + C);
hls::stream<typ> Ix_out_stream;
hls::stream<typ> img_Iy_stream;
typ Ix_img_LBuffer[3][_ct1];
typ Ix_img_WBuffer[3][3];
typ Ix_img_Coeff[3][3];
/* Code for Sobel-x stage follows. */
}
\end{lstlisting}
\end{minipage}\hfill
\begin{minipage}[t]{.47\textwidth}
\begin{lstlisting}[language=C,label={lst:yalaa},breaklines=true,
basicstyle=\footnotesize\ttfamily,keywordstyle=\sffamily\bfseries\color{green!40!black}, commentstyle=\itshape\color{purple!40!black},
numbers=left,
morekeywords={typ}, identifierstyle=\color{blue}, frame=single,framexbottommargin=100pt
caption=Type definitions for Affine and Interval Analysis.]
// Switch for Affine analysis
#ifdef AFFINE
#include <yalaa.hpp>
typedef yalaa::aff_e_d typ;
#endif
// Switch for Interval analysis
#ifdef INTERVAL
#include <Easyval.hpp>
typedef Easyval typ;
#endif
\end{lstlisting}
\end{minipage}
\end{table}
When we run
the generated C++ program with this data type definition, the value
associated with each pixel in each stage of the pipeline DAG is its affine
signal value which contains the base signal and the coefficients for the noise
variables. From this the range of every pixel can be derived as explained
before. If the data type corresponds to interval arithmetic, then the value
associated with each pixel is an interval. Using this approach, any kind
of interval analysis technique can be deployed in the PolyMage-HLS compiler
easily without re-architecting the analysis backend.
In the next section, we show how interval arithmetic based techniques can
fare really poorly if the benchmarks contain certain kinds of computational
patterns; we use an Optical Flow algorithm as an example. We then propose our new range analysis technique using SMT solvers.
\iffalse
HAVE TO WORK ON THE NEXT PARA
In the next section, we show that the integral bitwidth requirements at
various stages could be smaller than the estimates obtained
through static analysis. This could be due to the nature of the input images
and/or certain correlation between computations on spatially proximal pixels.
Furthermore, we show how we can use simple binary search to compute the
required fractional bits via program profiling using an application-specific
error metric. In these estimations of integral and fractional bitwidth, we
can also exploit the fact that many image processing applications are
resilient to small arithmetic overflow/precision errors, to arrive at power
and area-efficient hardware designs by using practical estimates for data
range and precision.
\fi
\input{files/smt}
\subsection{Profile-Driven Analysis}
\label{sec:profile}
Profile-driven analysis can be used to accomplish two tasks. First, we can obtain
lower bounds on the bitwidth estimates, which can be compared with the estimates
obtained using static analysis. Second, depending on the application, these
estimates
can be used in the actual system design instead of the conservative
estimates obtained through static analysis techniques. However, the
bitwidth requirements estimated at each stage using profiling naturally
depend on the sample input images. Based on the analysis done by Torralba et
al. \cite{torralba2003statistics}, we hypothesize that the
images taken from a certain domain, like for example {\it nature}, has
similar properties, and hence the bitwidth estimates can be carried over to
other images drawn from the same domain.
\subsubsection{Integral Bits}
The number of integral bits required at a stage $i$ denoted as $\alpha_i$ can
be obtained by running the input PolyMage program on a sample distribution of
input images. Let $\alpha_i^s$ be the maximum number of bits required by stage
$i$ to represent a pixel from an image sample $s$. Then the average number of
bits $\alpha_i^{avgP}$ required based on a sample set $S$ is $\sum_{s\in S}
\alpha_i^s/|S|$. Similarly, the worst-case number of bits $\alpha_i^{maxP}$
required is $max_{s\in S} \alpha_i^s$. We can either use $\alpha_i^{avgP}$ or
$\alpha_i^{maxP}$ as estimates for $\alpha_i$. Even if the estimate does not
suit certain images, in many application contexts, using saturation mode
arithmetic results in satisfying the desired output quality metric. Let
$\alpha_i^{IA}$ and $\alpha^{\smtra}$ be the integral bitwidth estimates
obtained for stage $i$ through interval analysis and \smtra\ analysis
respectively. For the benchmarks we have considered, affine analysis show some
improvements in the range estimates, but it amounts to same bitwidth requirement
as with interval analysis. Hence, throughout the rest of the paper, we consider
only interval analysis.
For our experimentation, we used a subset of 200 randomly chosen images from
the Oxford Buildings dataset~\cite{oxfordimages} consisting of 5062 images.
The set of 200 images is partitioned into two equal halves: training and
test sets. The training set is used to obtain estimates of integral
bitwidths at various stages through profiling. The test set is used to
evaluate the effectiveness of the bitwidth estimates obtained for quality
and power. Figure~\ref{fig:hs} shows the average cumulative distribution of
the bitwidth required by the integral part of the pixels in stages $I_x$
and $I_{xy}$ of the HCD benchmark on the training data set.
\iffalse
The x-axis of the graphs represent the bitwidth requirement of the integral
part, and y-axis represents the percentage of pixels, averaged across test
images, that can be represented within a given bitwidth.
\fi
For example, from Figure~\ref{fig:hs}, we can infer that in stage $I_x$,
95\% of the pixels require less than 5 bits, and all pixels (100\%) can be represented
using 8 bits.
\iffalse
Since
stages $I_x$ and $I_y$ are of similar computational nature, we plot the
histogram for only $I_x$. Similarly, among $I_{xx}$, $I_{xy}$ and $I_{yy}$,
we plot for $I_{xy}$ in Figure~\ref{fig:hs2}, and from stages $S_{xx}$,
$S_{xy}$ and $S_{yy}$, we plot for $S_{xy}$ in Figure~\ref{fig:hs3}.
Figures~\ref{fig:hs4},~\ref{fig:hs5} and~\ref{fig:hs6} correspond to {\it
det}, {\it trace} and {\it Harris} stages respectively.
Table~\ref{tab:bitTableHCD},~\ref{tab:bitTableUSM_DUS} shows the bitwidth estimates obtained from
static and profile-driven analyses.
\fi
Table~\ref{tab:bitTableHCD} shows the bitwidth estimates obtained from
static and profile-driven analyses for the HCD benchmark.
\begin{table*}[h!]
\footnotesize
\caption{Comparison of integral bitwidth estimates using interval, \smtra,
and profile-guided analyses for HCD. Fractional bitwidth estimates are also provided in the last row.}
\vskip 5pt
\label{tab:bitTableHCD}
\centering
\begin{tabular}{c | c c c c c c c c c}
\toprule
& &&&&HCD&&&&\\
Stage&Img&I$_x$,I$_y$&I$_{xx}$,I$_{yy}$&I$_{xy}$&S$_{xy}$&S$_{xx}$, S$_{yy}$&det&trace&harris\\
\midrule
$\alpha^{\smtra}$ & 8&8&13&14&17&16&33&17&\bf{33}\\
$\alpha^{IA}$ & 8&8&13&14&17&16&33&17&34\\
\midrule
$\alpha^{maxP}$& 8&8&13&14&17&16&30&17&29\\
$\alpha^{avgP}$& 8&8&13&14&17&16&29&17&29\\
\midrule
$\beta$ & 8&5&4&4&3&3&1&1&1 \\
\bottomrule
\end{tabular}
\end{table*}
As can be noted from
Table~\ref{tab:bitTableHCD}, the bitwidth estimates from $\alpha^{avgP}$
and $\alpha^{maxP}$ measures are the same for all stages
except for the {\it det} stage.
The estimates from the static analysis techniques match the profile estimates
except for the {\it det}, {\it trace} and {\it harris} stages. In general, we
expect the profile estimates to be better for stages that occur deeper in
the pipeline. Unlike Optical Flow benchmark, for HCD, \smtra\ analysis
performs no better than interval analysis except for a single bit
improvement in stage {\it Harris}. Again, we note that the profile estimates
also indicate the limit to which the static analysis techniques can be
improved by using more powerful approaches. Profile information can be
easily
obtained by executing the HLS C++ program directly without the need for a
heavy weight circuit simulation.
In the next section, we propose a simple and practical greedy search
algorithm to estimate the number of fractional bits at each stage of the DAG
while respecting an application specific quality constraint.
\iffalse
\begin{figure}[hb]
\centering
\resizebox{\columnwidth}{!}{%
\harrisHist
}%
\vskip 5pt
\caption{Cumulative distribution of pixels with respect to maximum integral bitwidth length at
stages $I_x$ and $I_{xy}$ of the HCD benchmark.}
\label{fig:hs}
\end{figure}
\fi
\subsubsection{Fractional Bits ($\beta$) Analysis}\label{sec:precision}
The number of fractional bits $\beta_i$ required at a stage $i$ depends on the
application and cannot be estimated in an application independent manner
similar to the integral bitwidth analysis. Estimating the optimal number of
fractional bits at each stage for a given application metric turns out to be a
non-convex optimization problem in most cases and hence we propose a simple
heuristic search technique that requires a very small number of profile
passes.
In the profiling technique, we fix the number of integral bits required at
each stage based on static or profile-driven analysis and increase the
precision
$\beta$ uniformly across all stages. For each value $\beta$, we estimate
the application-specific error metric. For the HCD benchmark, the error
metric is the percentage of misclassified corners when compared to a design
that uses sufficiently long integral and fractional bits. We can reach an
optimal $\beta$ for a given error tolerance via binary search. Then we make
a single pass on the stages of the DAG in reverse topologically sorted
order.
At each stage $I$, we do a binary search on the number of fractional bits
required, $\beta_I$, starting from the initial estimate $\beta$ while
retaining the application specific quality requirement. The last row
of the Table~\ref{tab:bitTableHCD} shows the fractional bits estimated
at each stage of the HCD benchmark. Note that the later stages of the DAG
require fewer bits than those stages which occur earlier in the DAG.
This is due to the fact that errors in earlier stages will have a
greater impact as they get propagated to the downstream stages. Further,
our greedy algorithm is optimizing the bitwidths by considering the
stages in the reverse topologically sorted order.
\input{files/flow}
\subsection{Summary of Bitwidth Analysis Framework}
Figure~\ref{fig:flow} summarizes the proposed bitwidth analysis framework.
We can use the PolyMage-HLS compilation framework first to do a range analysis and
estimate the integral bitwidths; then use the greedy heuristic to estimate the
fractional bits required at various stages. For range analysis, we can use one of interval,
\smtra\ and profile analysis techniques. For interval analysis, the compiler generates
HLS code where the data types of the variables at various stages of the DAG are intervals.
Then the bitwidth estimates are obtained using the intervals obtained by running the HLS
code. For profile analysis, the compiler generates HLS code wherein the data types of
the variables are of fixed point type with sufficiently large integral and fractional
bitwidths. Then HLS code is run on a sample distribution of input images to arrive at
integral bitwidth estimates. For \smtra\ analysis, the compiler generates a
constraint system which is solved by an SMT solver, such as Z3, to arrive at
range estimates.
\iffalse
\begin{table*}[h!]
\footnotesize
\caption{Comparison of integral bitwidth estimates using interval, \smtra,
and profile-guided analyses for USM and DUS. Fractional bitwidth estimates are also provided in the last row.}
\vskip 5pt
\label{tab:bitTableUSM_DUS}
\centering
\begin{tabular}{c | c c c c c | c c c c c }
\toprule
&&&USM&& &&&DUS&&\\
Stage&Img&blur$_x$&blur$_y$&sharpen&mask&Img&D$_x$&D$_y$&U$_x$&U$_y$\\
\midrule
$\alpha^{Z3RA}$ &8&8&8&10&9 &8&8&8&8&8\\
$\alpha^{IA}$ &8&8&8&10&9 &8&8&8&8&8\\
\midrule
$\alpha^{maxP}$ &8&8&8&10&9 &8&8&8&8&8\\
$\alpha^{avgP}$ &8&8&8&10&9 &8&8&8&8&8\\
\midrule
$\beta$ &0&2&3&4&4 &0&3&6&8&10\\
\bottomrule
\end{tabular}
\end{table*}
\fi
\section{Conclusions}
\label{sec:conclusions}
The input, output and intermediate values generated in many image processing
applications have a limited range. Furthermore, these applications are
resilient to errors arising from factors such as limited precision
representation, inaccurate computations, and other potential noise sources. In
this work, we exploited these properties to generate power and area-efficient
hardware designs for a given image processing pipeline by using custom
fixed-point data types at various stages. We showed that domain-specific
languages facilitate the application of interval and affine arithmetic
analyses on larger benchmarks with ease. Further, we proposed a new
range analysis technique using SMT solvers, which overcomes the inherent
limitations in conventional interval/affine arithmetic techniques, when
applied to iterative algorithms. The proposed SMT solver based range
analysis technique also uses the DSL specification of the program to reduce
the number of constraints and variables in the constraint system, thereby
making it a feasible technique to adopt in practice. Then, we compared the
effectiveness of the static analysis techniques against a profile-driven
approach that automatically takes into account properties of input image
distribution and any correlation between computations on spatially proximal
pixels. In addition, the analysis revealed the limit of possible improvement
for any static analysis technique for integral bitwidth estimation.
Finally, to estimate the number of fractional bits, we used uniform
bitwidths across all the stages of the pipeline, and then used a simple
greedy search to arrive at a suitable bitwidth at each stage while
satisfying an application-specific quality criterion. Overall, the results
effectively demonstrate how information exposed through a high-level DSL
approach could be exploited in practical fixed-point data type analysis
techniques and to perform detailed impact studies on much larger image
processing pipelines than previously studied.
\iffalse
theoretical improvements we can
Our bitwidth estimation algorithms rely
In this paper, we proposed an interval arithmetic-based integral bitwidth
analysis algorithm for image processing pipelines. The algorithm is simple
and nicely fits into the data flow analysis framework used in compilers.
Then, we compared the effectiveness of the static analysis algorithm against
a profile-driven approach that automatically takes into account
properties of input image distribution and any correlation between
computations on spatially proximal pixels. For precision analysis, to
estimate fractional bits, we use uniform bitwidths across all the stages of
the pipeline, and then use a simple search to arrive at a suitable bitwidth
that satisfies an application-specific quality criterion. In conclusion,
our experimental evaluation demonstrates that by using variable bitwidth
fixed-point numbers at various stages of an image processing pipeline, get
substantial power and area savings can be obtained without a loss in desired
quality.
\fi
\iffalse In this paper, we proposed an automatic precision analysis algorithm
which in a very simple and efficient manner leverages the power and area
savings, as demonstrated on various image processing benchmarks. The algorithm
nicely fits into the data flow analysis framework used in compilers. We
developed an interval arithmetic based static algorithm for range analysis.
Then, we compared the effectiveness of the static analysis algorithm against a
profile-driven approach that automatically takes into account properties of
input image distribution and any correlation between computations on spatially
proximal pixels and showed that we were able to perform at par with static
precision using lesser area and power. For precision analysis, to estimate
fractional bits, we proposed a profiling based approach that uses uniform bit-
widths across all the stages of the pipeline, and then use a simple search to
arrive at a suitable bitwidth that satisfies an application-specific quality
criterion, which we carefully arrived at to compare the performance of our
algorithm. Finally, we demonstrate that by using the precision obtained using
our analysis, we we can get substantial power and area savings without a loss in
desired quality.
\fi
\section{Experimental Results}\label{sec:experiments}
In this section, we present a detailed area, power and throughput analysis when
variable fixed-point data types are used as against floating-point by
considering the following four benchmarks: Harris Corner Detection, Unsharp
Mask, Down and Up Sampling, and Optical Flow. Tables~\ref{tab:ofrange},
~\ref{tab:bitTableHCD},~\ref{tab:bitTableUSM} and~\ref{tab:bitTableDUS} show the integral bitwidth
estimates obtained through interval analysis ($\alpha^{IA}$), \smtra\ analysis
($\alpha^{\smtra}$) and profile analysis ($\alpha^{maxP}$ and $\alpha^{avgP}$);
and the average fractional bitwidth estimate ($\beta$) obtained through greedy
heuristic search algorithm. Table~\ref{tab:Allresults} compares the performance
of each benchmark using {\it float} data type and bitwidth estimates obtained
from different approaches. In these tables, the {\it Quality} column
corresponds to an application specific quality metric; the {\it Power}
column gives the power when the design operates at a speed specified in the
adjacent {\it Clk Period} column; {\it latency} columns provide the number
of
clock cycles required to process an HD image; the next four columns (BRAM,
DSP, FF, LUT, \%slices) summarize area usage; the {\it Min Clk Period}
column gives the maximum frequency of operation for circuit; and the next
two columns give the throughput and power consumed at the maximum frequency
of operation. Figure~\ref{fig:powersplit} gives the split of power usage by
various components of an FPGA. Unlike the Optical Flow benchmark, the integral
bitwidth estimates for the benchmarks HCD, USM and DUS using interval and
\smtra\ analysis techniques is the same. So we do not provide separate area,
power and throughput analysis for these benchmarks.
We used the Xilinx Zedboard consisting of Zynq-XC7Z020 FPGA device and
Xilinx Vivado Design Suite 2017.2 version to conduct our experiments. The
HLS design generated by our PolyMage DSL compiler is synthesized by the
Vivado HLS compiler. All characteristics are reported post Place and Route.
We ran C-RTL
co-simulations to generate switching activity (SAIF) file for reporting
detailed power consumption across the design.
\begin{table*}[t]
\footnotesize
\caption{Power, area and throughput analysis for HCD, DUS, USM and OF benchmarks using float and integral bitwidth estimates obtained using
static and profile-driven analyses. Fractional bitwidths are determined based on the greedy heuristic search approach.}
\vskip 5pt
\label{tab:Allresults}
\centering
\Ress
\end{table*}
\subsection{Harris Corner Detection}
Table~\ref{tab:bitTableHCD} summarizes the integral and fractional bitwidth estimates
obtained at each stage of the HCD benchmark through various analysis techniques.
The results in this table are commented upon in Sections~\ref{sec:profile}
and~\ref{sec:precision}. Figure~\ref{fig:uniformbeta} shows the average percentage of
pixels correctly
classified by the HCD benchmark on the test image set by varying the
fractional bits uniformly across all the stages while fixing the integral
bitwidth estimates obtained via profiling ($\alpha_i^{avgP}$). It also
contains estimates of power consumption with varying fractional bits for the
Xilinx ZED FPGA board.
It can be noted from the graph that the fractional bits do not affect the
accuracy of corner classification, and we thus get more than 99\% accuracy
even with zero fractional bits. From this graph, we infer that one can
obtain close to 100\% accuracy by using 8 fractional bits uniformly across
all the stages. We then make a backward pass on the stages of the
HCD benchmark to drop the fractional bits further without any significant loss
in accuracy and the row corresponding to $\beta$ in
Table~\ref{tab:bitTableHCD} shows the final fractional bitwidths. Due to space
constraint, we do not provide a graph such as Figure~\ref{fig:uniformbeta} for
the rest of the benchmarks. We can notice from Table~\ref{tab:Allresults} that by
using bitwidth estimates from interval analysis, we obtain 99.999\% accuracy with
a power consumption of 0.263~W. The power savings are
3.8$\times$ lower when compared with the floating-point design and 4\% more when
compared with the profile-estimate based design. The savings
on the percentage of FPGA slices used is around 6.2$\times$. From the last
3 columns of the table, we can notice that the fixed-point designs can operate
at a higher frequency achieving better throughput while consuming lesser power.
Figure~\ref{fig:powersplit} shows the detailed power analysis for floating-point and fixed-point design. It shows only the significant components of the dynamic power consumed, and in all the designs, the static power consumption is around 0.122~W.
\begin{figure}[!htb]
\centering
\begin{minipage}[t]{0.45\textwidth}
\centering
\resizebox{\linewidth}{!}{
\harrisProfPower
}
\caption{Error and power variation for the HCD benchmark by fixing the number of integral bits to profile estimated values $\alpha_i^{avgP}$ and varying $\beta$ uniformly across all the stages.}
\label{fig:uniformbeta}
\end{minipage}%
\hfill
\begin{minipage}[t]{0.45\textwidth}
\centering
\resizebox{\linewidth}{!}{
\includegraphics{./graphs/hcd_power_graph}
}
\caption{Power consumption by individual components on FPGA for HCD.\label{fig:powersplit}}
\end{minipage}
\end{figure}
\iffalse
\begin{figure}[htbp]
\centering
\resizebox{0.8\linewidth}{!}{
\harrisProfPower
\label{fig:haProf}
}
\vskip 5pt
\caption{Error and power variation for the HCD benchmark by fixing the number of
integral bits to profile estimated values $\alpha_i^{avg}$ and varying
$\beta$ uniformly across all the stages.}
\label{fig:uniformbeta}
\end{figure}
\fi
\begin{figure*}[!htb]
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{0.95\linewidth}{!}{
\usmDag
}
\caption{Pipeline DAG structure for USM benchmark.}
\label{fig:USMdag}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{\linewidth}{!}{%
\begin{tabular}{c | c c c c c }
\toprule
Stage&Img&blur$_x$&blur$_y$&sharpen&mask\\
\midrule
$\alpha^{Z3RA}$ &8&8&8&10&9 \\
$\alpha^{IA}$ &8&8&8&10&9 \\
\midrule
$\alpha^{maxP}$ &8&8&8&10&9 \\
$\alpha^{avgP}$ &8&8&8&10&9 \\
\midrule
$\beta$ &0&2&3&4&4 \\
\bottomrule
\end{tabular}
}%
\captionof{table}{Comparison of integral bitwidth estimates using interval, \smtra, and profile-guided analyses for USM. Fractional bitwidth estimates are also provided in the last row.}
\label{tab:bitTableUSM}
\end{minipage}
\end{figure*}
\subsection{Unsharp Mask (USM)}
The Unsharp Mask (USM) benchmark sharpens an input image and its computational
DAG is provided in Figure~\ref{fig:USMdag}. The input image is blurred across
x-axis and y-axis by the stencil stages {\it blurx} and
{\it blury} successively. Then it passes through
the {\it sharpen} stage, which is a point-wise computation. Finally, the
{\it masked} stage compares each pixel from the output of the {\it sharpen}
stage with a threshold value. Depending on whether the pixel value is less
than threshold, the corresponding pixel from either the original input image
or the sharpened image is chosen for output. We highlight an important
observation here: even if we make an error in computing a pixel value from
the {\it sharpen} stage, as long as it is less than the threshold, the right
output pixel is chosen. Based on this observation, we define an error metric
that is the fraction of pixels that were misclassified in the {\it masked}
stage due to variable width fixed-point representation as against
floating-point representation. We define a second quality metric that is the
root mean squared error between correctly classified pixel values and their
floating-point counterparts.
Table~\ref{tab:bitTableUSM} shows the integral and fractional bitwidths
required at various stages of the USM benchmark obtained from static (interval
and \smtra) and profile analyses. It can be noted that the estimates obtained by
the static and profile analyses are the same. Table~\ref{tab:Allresults} shows
that there is a factor of 1.6$\times$ improvement in power when compared to the
floating-point design with negligible root mean squared error and classification
error. With respect to the number of FPGA slices used, there is a factor of
2.6$\times$ improvement. Table~\ref{tab:Allresults} also shows the maximum
frequency of operation for each of the designs, the throughput at that level and
power consumption. From the last 3 columns of the table, we can infer that by
operating the fixed-point design at a higher frequency, 6\% increase in
throughput can be achieved while consuming 1.7x lower power.
\begin{figure*}[!htb]
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{0.95\linewidth}{!}{
\dusDag
}
\caption{Pipeline DAG structure for DUS benchmark.}
\label{fig:DUSdag}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{0.8\linewidth}{!}{%
\begin{tabular}{c | c c c c c }
\toprule
Stage&Img&D$_x$&D$_y$&U$_x$&U$_y$\\
\midrule
$\alpha^{Z3RA}$ &8&8&8&8&8\\
$\alpha^{IA}$ &8&8&8&8&8\\
\midrule
$\alpha^{maxP}$ &8&8&8&8&8\\
$\alpha^{avgP}$ &8&8&8&8&8\\
\midrule
$\beta$ &0&3&6&8&10\\
\bottomrule
\end{tabular}
}
\captionof{table}{Comparison of integral bitwidth estimates using interval, \smtra, and profile-guided analyses for DUS. Fractional bitwidth estimates are also provided in the last row.}
\label{tab:bitTableDUS}
\end{minipage}
\end{figure*}
\subsection{Down and Up Sampling (DUS)}
Down and Up Sampling (DUS) benchmark has a linear DAG structure as shown in Figure~\ref{fig:DUSdag}. The image is first
downsampled along the $x$-axis in stage D$_x$ and is further downsampled along the $y$-axis in stage D$_y$.
It is then upsampled again along the $x$ and $y$ axes in the stages $U_x$ and $U_y$
respectively. For the sake of conciseness, we avoid including the DUS
PolyMage code. All four stages comprise stencil computations.
The integral bitwidths estimated by both the interval and \smtra\ analyses is
equal to 8 at all the stages of DUS. We use the same set of training images as
that of HCD benchmark for estimating the integral and fractional bitwidths via
profiling. The profile estimates yielded the same integral bitwidth requirement
of 8 at all the stages. We use Peak Signal to Noise Ratio (PSNR) as a quality
metric where the reference image is obtained by using a sufficiently wide data
type. We set the required PSNR to infinity and the resulting fractional
bitwidths determined by our greedy precision analyzer is shown in the last row
of the Table~\ref{tab:bitTableDUS}. Table~\ref{tab:Allresults} shows that there
is a factor of 1.7$\times$ reduction in power using tuned fixed-point data types
when compared with using floating-point data type without loss of any accuracy.
With respect to area, there is a 4$\times$ improvement in terms of number of
slices used. Also, the fixed-point designs use no DSP blocks at all when
compared with floating-point design which uses 54 DSPs. At the peak possible
frequency of operation, fixed-point design achieves 13.6\% increase in
throughput while consuming 1.6x lesser power.
\subsection{Optical Flow (OF)}
The Optical Flow (OF) benchmark computes the velocity of individual pixels
from an image frame and its time-shifted version. Our implementation is
based on the Horn-Schunck algorithm \cite{horn1981determining} and consists
of 30 stages. The first 10 stages are pre-processing stages and the last 20
stages are obtained by repeating a set of five stages for four times. The
accuracy of motion estimation can be improved by repeating the 5-stage set
more times. Optical flow is a heavily used image processing algorithm in
many computer vision applications.
There have been many efforts in the past to implement optical flow on
FPGAs~\cite{JavierTCSVT'06,AlanISSC'10,EnZhuTCSVT'16} for power and
performance benefits.
Table~\ref{tab:ofrange} shows the estimated integral bitwidths required
at various stages of the Optical Flow benchmark. We notice that for stages
deeper in the pipeline, the difference between estimates obtained via interval
analysis and profiling are substantial. The profile estimates are obtained
from a training data set and for testing purpose, we use RubberWhale and
Dimetrodon image sequences from the Middlebury dataset~\cite{MBuryDataSet}.
Section~\ref{sec:smt} provides a detailed discussion on this and shows how
the \smtra\ analysis can overcome the inadequacies of the interval arithmetic
based analyses techniques and gives estimates which almost match profile estimates.
For computing the accuracy, we use the Average Angular Error (AAE) metric as
discussed in \cite{Fleet:1990},\cite{Otte:1994}. The reference motion
vectors are obtained by using sufficiently wide fixed-point data types at
all stages.
It can be noticed from Table~\ref{tab:Allresults} that by using bitwidth
estimates from Z3RA analysis, we obtain similar accuracy as profile-driven
analysis with a power consumption of 0.328~W. The power savings are 1.9$\times$
lower when compared with the floating-point design and 5.4\% more when compared
with the profile-estimate based design. The savings on the percentage of FPGA
slices used is around 2.5$\times$. From the last 3 columns of the table, we can
notice that the Z3RA fixed-point design can operate at a higher frequency
achieving 25\% more throughput than the floating point design while consuming
lesser power.
\iffalse
\begin{table*}[htbp]
\footnotesize
\caption{Metrics for the OF benchmark using float, and using bitwidths estimated using static analysis and profiling techniques.}
\vskip 5pt
\label{tab:OFresults}
\centering
\ofRes
\end{table*}
\begin{table*}[htbp]
\footnotesize
\caption{Metrics for USM benchmark using float, and using bitwidths estimated using static analysis and profiling techniques. }
\vskip 5pt
\label{tab:USMresults}
\centering
\usmRes
\end{table*}
\begin{table*}[htbp]
\footnotesize
\caption{Metrics for DUS using float, and using bitwidths estimated using static analysis and profiling techniques. }
\vskip 5pt
\label{tab:DUSresults}
\centering
\dusRes
\end{table*}
\fi
\iffalse
\begin{table}[tb]
\footnotesize
\centering
\caption{Comparison of bitwidth estimates using profiling technique and static analysis for the HCD benchmark.
\label{tab:hcdintegral}}
\vskip 5pt
\begin{tabularx}{0.8\linewidth}{l|c@{~~~~~~~~~~~}c@{~~~~~~~~~}c@{~~~~~~~~~}|c@{~~~~~~~~~~~}}
\toprule
Stage & $\alpha^{sa}$ & $\alpha^{max}$ & $\alpha^{avg}$ & $\beta^{avg}$ \\
\midrule
Img & 8 & 8 & 8 & 8\\
I$_x$,I$_y$ & 8 & 8 & 8 & 5\\
I$_{xx}$,I$_{yy}$ & 13 & 13 & 13 & 4\\
I$_{xy}$ & 14 & 14 & 14 & 4\\
S$_{xy}$ & 17 & 17 & 17 & 3\\
S$_{xx}$, S$_{yy}$ & 16 & 16 & 16 & 3\\
det & 33 & 30 & 29 & 1\\
trace & 17 & 17 & 17 & 1\\
harris & 34 & 29 & 29 & 1\\
\bottomrule
\end{tabularx}
\end{table}
\begin{figure*}[htbp]
\centering
\usmHist
\vskip 5pt
\caption{Cumulative distribution of pixels with respect to maximum integral bitwidth length at various stages of the USM.}
\label{fig:usm}
\end{figure*}
\begin{table}[tb]
\footnotesize
\centering
\caption{Comparison of bitwidth estimates using profiling technique and static analysis for the USM benchmark.
\label{tab:usmintegral}}
\vskip 5pt
\begin{tabularx}{0.87\linewidth}{l|c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}|c@{~~~~~}}
\toprule
Stage & $\alpha^{sa}$ & $\alpha^{max}$ & $\alpha^{avg}$ & $\beta^{avg}$ \\
\midrule
img & 8 & 8 & 8 & 0\\
blur$_x$ & 8 & 8 & 8 & 2\\
blur$_y$ & 8 & 8 & 8 & 3\\
sharpen & 10 & 10 & 10 & 4\\
mask & 9 & 9 & 9 & 4\\
\bottomrule
\end{tabularx}
\end{table}
\begin{table}[!htb]
\footnotesize
\centering
\caption{Comparison of bitwidth estimates using profiling technique and static analysis for the DUS benchmark.
\label{tab:dusintegral}}
\vskip 5pt
\begin{tabularx}{0.80\linewidth}{l|c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}|c@{~~~~~~~}}
\toprule
Stage & $\alpha^{sa}$ & $\alpha^{max}$ & $\alpha^{avg}$ & $\beta^{avg}$ \\
\midrule
img & 8 & 8 & 8 & 0 \\
D$_x$ & 8 & 8 & 8 & 3 \\
D$_y$ & 8 & 8 & 8 & 6 \\
U$_x$ & 8 & 8 & 8 & 8 \\
U$_y$ & 8 & 8 & 8 & 10 \\
\bottomrule
\end{tabularx}
\end{table}
\fi
\iffalse
\begin{figure*}[htbp]
\centering
\subfloat[Throughput]{
\includegraphics[width=0.40\linewidth]{graphs/throughput}
}
\subfloat[Power]{
\includegraphics[width=0.40\linewidth]{graphs/power}
}
\vskip 5pt
\caption{Throughput and respective power obtained with various FPGA design benchmarks\label{fig:throughput}}
\end{figure*}
\begin{figure}[t]
\centering
\resizebox{0.8\columnwidth}{!}
\dusDag
}
\vskip 5pt
\caption{Pipeline DAG structure for DUS benchmark.} \label{fig:DUSdag}
\end{figure}
\iffalse
\begin{figure}[htbp]
\centering
\resizebox{0.8\linewidth}{!}{
\harrisProfPower
\label{fig:haProf}
}
\iffalse
\subfloat[USM]{
\usmProfPower
\label{fig:usmProf}
} \\
\subfloat[DUS] {
\dusProfPower
\label{fig:dusProf}
}
\subfloat[OF]{
\ofProfPower
\label{fig:ofProf}
}
\fi
\vskip 5pt
\caption{Error and power variation for the HCD benchmark by fixing the number of
integral bits to profile estimated values $\alpha_i^{avg}$ and varying
$\beta$ uniformly across all the stages.}
\label{fig:uniformbeta}
\end{figure}
\fi
\begin{figure
\centering
\resizebox{0.8\linewidth}{!}{
\includegraphics{./graphs/hcd_power_graph}
\iffalse
\subfloat[USM]{\includegraphics{./graphs/usm_power_graph}
\label{fig:usmsplit}
}\\
\subfloat[DUS]{\includegraphics{./graphs/dus_power_graph}
\label{fig:dussplit}
}
\subfloat[OF]{\includegraphics{./graphs/of_power_graph}
\label{fig:ofsplit}
}
\fi
}
\vskip 5pt
\caption{Power consumption by individual components on FPGA for HCD.\label{fig:powersplit}}
\end{figure}
\fi
\iffalse
\begin{figure
\centering
\resizebox{0.8\linewidth}{!}{
\subfloat[HCD]{
\includegraphics{./graphs/hcd_power_graph}
}
\subfloat[USM]{\includegraphics{./graphs/usm_power_graph}
\label{fig:usmsplit}
}\\
\subfloat[DUS]{\includegraphics{./graphs/dus_power_graph}
\label{fig:dussplit}
}
\subfloat[OF]{\includegraphics{./graphs/of_power_graph}
\label{fig:ofsplit}
}
}
\vskip 5pt
\caption{Power consumption by individual components on FPGA for HCD.\label{fig:powersplit}}
\end{figure}
\fi
\section{Framework}
The polymage code is parsed into an intermediate json representation which contains complete information about the image processing pipeline represented in a DAG form. This IR is particularly useful as it provides modularity and makes addition of more analysis backends possible in the future. The json structure for point-wise and stencil computations is shown in figure ?. The whole image processing DAG can be expressed using this json structure.
From the json representation, a python code is generated which uses the Z3 python API for solving the per-stage bitwidth.
(I have a doubt on how to justify the the theory we use for Z3. I use theory of Reals. Although i have ran using Bit-Vectors as well which results in unknown solution and sometimes gives an error also for HCD particularly.. Well, as of now i am keeping the theory as an input parameter to the framework).
(Writing about the structure of our framework)
Our Z3 analysis framework is written with parameterised functions for point-wise and stencil computations which makes the code-generation from json straight forward. For each stage the parameters are extracted and the function is called with those parameters. For each stage, there is an option of breaking the correlations and make the output variables fundamental. This is done to deal with complex dags and to make the framework scalable. Figure ? shows the call structure for point-wise and stencil stages.
(Next file is running-example-smt.tex)
\section{Introduction}
Field-Programmable Gate Arrays (FPGAs) are suitable for accelerating
computations from several domains such as image processing, computer vision,
and digital signal processing. When a lower or customized precision is
desired, FPGAs are often expected to perform better than accelerators such
as GPUs with respect to performance delivered per unit of energy consumed.
When an image processing pipeline such
as Harris Corner Detection (HCD) (cf. Figure~\ref{fig:hdag} and Table~\ref{tab:hcd}),
is implemented on a CPU or a GPU, a programmer is bound to choose a pre-defined
data type such as {\tt float}, {\tt int}, or {\tt short} owing to the underlying
architectural constraints. In order to avoid arithmetic overflows, the
data types have to be chosen conservatively through over-estimation. This
leads to a wastage of memory, and hence memory bandwidth, at all levels of
the hierarchy; furthermore, additional energy is consumed both due to data
transfer and the higher precision in which the arithmetic is performed. On
the other hand, on FPGAs, it is possible to use variable length
fixed-point data types to represent the data produced and consumed at various
stages of an image processing pipeline. This saves chip area and the power
consumed by the hardware design due to the reduced precision
and internal routing logic. The other natural outcome is a better
utilization of available on-chip memory resources.
Although, FPGAs fare extremely well on the performance per watt metric,
their programmability has been a major hindrance in adoption. The reliance
on hardware description languages (HDLs) such as Verilog and VHDL makes it
extremely cumbersome for a wider programmer audience. High-Level Synthesis
(HLS) tools, which map C, C++ code into equivalent hardware designs by
generating HDL code automatically, have thus gained significant attention in
the past decade. The quality of designs generated by a HLS compiler often
depend on the analysis techniques employed. Many times programmers put in
substantial effort to drive a HLS tool to generate a hardware design of
their choice using suitable pragma annotations or code rewriting. There have
been efforts to further raise the level of abstraction --- from using
imperative languages such as C, C++ to domain-specific languages (DSLs) ---
giving rise to the term, ultra high-level synthesis; Bacon et
al.~\cite{bacon13acm} provide a comprehensive survey. DSLs not only
improve programmer productivity but also exploit the richer information
from the underlying algorithm. This facilitates compilers to generate better
code or hardware designs using relatively simple program analysis
techniques.
\subsection{DSLs for Image Processing Pipelines} An image processing
pipeline can be viewed as a directed acyclic graph (DAG) of computational
stages. Each stage transforms an
input image form into an output image form to be consumed by the subsequent
stages in the pipeline. The class of computations at each stage are simple data
parallel operations applied on all image pixels such as point-wise and stencil
computations. Figure~\ref{fig:hdag} shows the computational DAG associated with
the Harris Corner Detection (HCD) benchmark. As the name indicates, HCD is a corner
detection algorithm, commonly used in computer vision space.
Table~\ref{tab:hcd} shows the computations at each stage of the DAG.
The source code for HCD benchmark in PolyMage DSL, whose compiler infrastructure we use
in this paper,
can be obtained at the PolyMage GitHub repository~\cite{polymagebench}.
If the computations in HCD are expressed in C/C++,
then there will be a two dimensional loop associated with each stage of the DAG.
Further, these loops occur in some topologically sorted order of DAG nodes.
Thus, the rich structure in the application gets lost in the resulting C/C++
code. For example, it is hard to infer that pixels output from one stage can be
streamed to the following stage and the following stage can start computations
once it receives enough number of pixels. This observation leads to an extremely
efficient pipelined hardware architecture for the whole computational DAG and is
exploited in PolyMage-HLS compiler as in other DSL compilers for FPGAs. In this
paper, we use the fact that it is easy to infer the computations on pixels at
each stage of the DAG and further these computations are applied homogeneously on
all the pixels at the corresponding stage, to arrive at efficient range
analysis algorithms based on interval arithmetic and Satisfiability Modulo
Theory (SMT) solvers. It would be almost impossible to do
this if the design is expressed directly in Verilog/VHDL; and probably require
complex program analysis if we have to to achieve this on C/C++ programs as is
the case in HLS frameworks.
\subsection{Problem Description and Contributions}
The data at each stage of an image processing pipeline is represented using
a parametric fixed-point data type $\left(\alpha, \beta\right)$, where $\alpha$
and $\beta$
denote the number of bits used to represent integral and fractional parts. The
objective is to minimize $\alpha$ and $\beta$ at each stage while maintaining
an application specific quality metric. The optimal value of $\alpha$ at a stage
depends on the range of values produced; whereas the optimal value of $\beta$
depends on how precision impacts the quality metric.
Range analysis which is required for estimating the integral bitwidth requirement
is a well studied problem in literature. There are several works
based on variants of interval and affine arithmetic~\cite{cong09fccm, vakili13tcad,
zhang10jsip, stephenson00sigplan, lee06tcad, mahlke01tcad}.
The benchmarks considered in these works, such as FIR filter, Discrete Cosine
Transform, Polynomial Evaluation etc. are mainly from the signal processing domain
and their code size and complexity is small. These techniques are not
easily adaptable for image processing pipelines when expressed in Verilog/VHDL,
HLS C/C++ etc. due to the large number of pixel signals present at each stage
and multiple such stages in an application.
The first main contribution of our paper is an interval arithmetic based range analysis
technique in the DSL compiler which exploits the fact that the computations on all
the pixels at a given stage is homogeneous to do a combined range analysis. The
second main contribution of our paper is a range analysis framework in the DSL compiler
wherein any interval and affine arithmetic-like analysis can be incorporated with ease.
Apart from interval arithmetic based approaches, which cannot handle certain
kind of operations like divisions, techniques using powerful SMT solvers have
been proposed in the literature~\cite{ kinsman10}. However, they are applied on
extremely small benchmarks involving 2 to 3 equations. The third main
contribution of this paper is that, we propose a range analysis technique using
SAT solvers and apply it on large benchmarks such as Optical Flow involving few
tens of DAG stages not to speak of large number of pixel signals and complex
computations in the DAG structure. This is primarily possible because our
analysis is based on the DSL specification of the benchmark as against a HDL or
C/C++ specification. Further, we show that in iterative algorithms such as
Optical Flow, conservative estimates of interval analysis will have a
debilitating effect with the increase in the iterations making them unusable in
practice; and we have to resort to SMT solvers to get accurate range estimates.
The fourth main contribution of this paper is a simple greedy heuristic search
technique for precision analysis to determine the number of bits required for
representing fractional bits at each stage of the pipeline. Finally, we present
a thorough experimental study comparing the effectiveness of interval, SMT
solver and profile guided approaches with respect to power, area and speed on
large image processing benchmarks. We would like to highlight that all the
previous studies involve very small benchmarks.
We implement and evaluate our automatic bitwidth analysis approach in PolyMage
compiler infrastructure \cite{mullapudi2015asplos, chugh16pact}. With the PolyMage DSL, FPGAs are targeted by
first generating High-Level Synthesis (HLS) code after a realization of
several transformations for parallelization and reuse; the HLS code is
subsequently processed by a vendor HLS suite (Xilinx Vivado in the case of
PolyMage). There have been several recent DSL efforts that target FPGAs for image
processing; these include Darkroom~\cite{darkroom},
Rigel~\cite{rigel}, Halide~\cite{halide17hls}, HIPAcc~\cite{hipacc16fpga} and
PolyMage-HLS~\cite{chugh16pact}. While these works
have addressed several challenges in compiling DSL to FPGAs, none of them
have studied the issue of exploiting application-dependent
variable fixed-point data types for power and area savings. HiPAcc
goes to the extent of providing pragmas for specifying bitwidths of
variables, but no automatic compiler support for it.
\begin{table}[t]
\begin{minipage}[b]{0.38\linewidth}
\includegraphics[scale = 0.8]{./graphs/harris_annotated_dag}
\captionof{figure}{DAG representation of the Harris Corner Detection (HCD) algorithm.}
\label{fig:hdag}
\end{minipage}\hfill
\begin{minipage}[b]{0.58\linewidth}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r |r}
\toprule
Stage & Computation \\
\hline
$I_x$ & $\frac{1}{12}\left[ \begin{smallmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0& 1\end{smallmatrix} \right]$ \\
$I_y$ & $\frac{1}{12}\left[ \begin{smallmatrix} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2& 1\end{smallmatrix} \right]$ \\
$I_{xx}$ & $I_x\left(i,j\right) I_x\left(i,j\right)$ \\
$I_{xy}$ & $I_x\left(i,j\right) I_y\left(i,j\right)$ \\
$I_{yy}$ & $I_y\left(i,j\right) I_y\left(i,j\right)$ \\
$S_{xx}$ & $A=\left[ \begin{smallmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{smallmatrix} \right]$\\
$S_{xy}$ & $A$\\
$S_{yy}$ & $A$\\
$det$ & $S_{xx}\left(i,j\right) S_{yy}\left(i,j\right)-S_{xy}\left(i,j\right) S_{xy}\left(i,j\right)$\\
$trace$ & $S_{xx}\left(i,j\right)+S_{yy}\left(i,j\right)$\\
$Harris$ & $det\left(i,j\right) - 0.04 ~trace\left(i,j\right) trace\left(i,j\right)$\\ [1ex]
\hline
\end{tabular}
\caption{Summary of computations in HCD benchmark.}
\label{tab:hcd}
\end{minipage}
\end{table}
The rest of this paper is organized as follows. Related work is discussed in
Section~\ref{sec:related-work} and the necessary background is provided in
Section~\ref{sec:background}. Section~\ref{sec:bitwidth} presents in detail the
main contributions of this paper.
Experimental evaluation is
presented in Section~\ref{sec:experiments} and conclusions are presented in
Section~\ref{sec:conclusions}.
\section{Profile-Driven Analysis}
\label{sec:profile}
Profile-driven analysis can be used to accomplish two tasks. First, the
worst-case integral bit-width lengths at various stages obtained via static
analysis can be improved. Second, the number of fractional bits required at
each stage can be estimated using an application-specific quality metric.
However, the bit-width requirements estimated at each stage using profiling
technique naturally depends on the sample input images. Based on the
analysis done by Torralba et al. \cite{torralba2003statistics}, we
hypothesize that the images taken from a certain domain, like for example
{\it nature}, has similar properties, and hence the bit-width estimates can
be carried over to other images drawn from the same domain.
\input{files/flow}
\subsection{Integral Bits}
The number of integral bits required at a stage $i$ denoted as $\alpha_i$ can
be obtained by running the input PolyMage program on a sample distribution of
input images. Let $\alpha_i^s$ be the maximum number of bits required by stage
$i$ to represent a pixel from an image sample $s$. Then the average number of
bits $\alpha_i^{avg}$ required based on a sample set $S$ is $\sum_{s\in S}
\alpha_i^s/|S|$. Similarly, the worst-case number of bits $\alpha_i^{max}$
required is $max_{s\in S} \alpha_i^s$. We can either use $\alpha_i^{avg}$ or
$\alpha_i^{max}$ as estimates for $\alpha_i$. Even if the estimate does not suit
certain images, in many application contexts, using saturation mode arithmetic
results in satisfying the desired output quality metric. Let $\alpha_i^{sa}$ be
the integral bit-width estimation obtained for stage $i$ through static
analysis. For the benchmark programs we have considered, affine analysis show some
improvements in the range estimates, but it amounts to same bit-width requirement as
with interval analysis. Hence, throughout the rest of the paper, when
we refer to static analysis it means interval analysis only.
For our experimentation, we used a subset of 200 randomly chosen images from
the Oxford Buildings dataset~\cite{oxfordimages} consisting of 5062 images.
The set of 200 images is partitioned into two equal halves: training and
test sets. The training set is used to obtain estimates of integral
bit-widths at various stages through profiling. The test set is used to
evaluate the effectiveness of the bit-width estimates obtained for quality
and power. Figure~\ref{fig:hs} shows the average cumulative distribution of
the bit-width required by the integral part of the pixels in stages $I_x$
and $I_{xy}$ of the HCD program on the training data set.
\iffalse
The x-axis of the graphs represent the bit-width requirement of the integral
part, and y-axis represents the percentage of pixels, averaged across test
images, that can be represented within a given bit-width.
\fi
For example, from Figure~\ref{fig:hs1}, we can infer that in stage $I_x$,
95\% of the pixels require less than 5 bits, and all pixels (100\%) can be represented using 8 bits.
\iffalse
Since
stages $I_x$ and $I_y$ are of similar computational nature, we plot the
histogram for only $I_x$. Similarly, among $I_{xx}$, $I_{xy}$ and $I_{yy}$,
we plot for $I_{xy}$ in Figure~\ref{fig:hs2}, and from stages $S_{xx}$,
$S_{xy}$ and $S_{yy}$, we plot for $S_{xy}$ in Figure~\ref{fig:hs3}.
Figures~\ref{fig:hs4},~\ref{fig:hs5} and~\ref{fig:hs6} correspond to {\it
det}, {\it trace} and {\it Harris} stages respectively.
\fi
Table~\ref{tab:bitTableHCD},~\ref{tab:bitTableUSM_DUS} shows the bit-width estimates obtained from
static and profile-driven analyses.
\begin{table*}[h!]
\footnotesize
\caption{Comparison of integral bitwidth estimates using interval, \smtra,
and profile-guided analyses for HCD and DUS. Fractional bitwidth estimates are also provided.}
\label{tab:bitTableHCD}
\centering
\begin{tabularx}{1.07\linewidth}{c | c c c c c c c c c}
\toprule
& &&&&HCD&&&&\\
Stage&Img&I$_x$,I$_y$&I$_{xx}$,I$_{yy}$&I$_{xy}$&S$_{xy}$&S$_{xx}$, S$_{yy}$&det&trace&harris\\
\midrule
$Z3$ & 8&8&13&14&17&16&33&17&\bf{33}\\
$\alpha^{IA}$ & 8&8&13&14&17&16&33&17&34\\
\midrule
$\alpha^{maxP}$& 8&8&13&14&17&16&30&17&29\\
$\alpha^{avgP}$& 8&8&13&14&17&16&29&17&29\\
$\beta$ & 8&5&4&4&3&3&1&1&1 \\
\bottomrule
\end{tabularx}
\end{table*}
\begin{table*}[h!]
\footnotesize
\caption{Comparison of bit-width estimates using profiling technique and static analysis for USM and DUS.}
\vskip 5pt
\label{tab:bitTableUSM_DUS}
\centering
\begin{tabularx}{1.07\linewidth}{c | c c c c c | c c c c c }
\toprule
&&&USM&& &&&DUS&&\\
Stage&Img&blur$_x$&blur$_y$&sharpen&mask&Img&D$_x$&D$_y$&U$_x$&U$_y$\\
\midrule
$Z3$ &8&8&8&10&9 &8&8&8&8&8\\
$\alpha^{IA}$ &8&8&8&10&9 &8&8&8&8&8\\
\midrule
$\alpha^{maxP}$ &8&8&8&10&9 &8&8&8&8&8\\
$\alpha^{avgP}$ &8&8&8&10&9 &8&8&8&8&8\\
$\beta$ &0&2&3&4&4 &0&3&6&8&10\\
\bottomrule
\end{tabularx}
\end{table*}
As can be noted from
Table~\ref{tab:bitTableHCD}, the bit-width estimates from $\alpha^{avg}$
and $\alpha^{max}$ measures are the same for all stages
except for the {\it det} stage.
The estimates from the static analysis match the profile estimates except
for the {\it det}, {\it trace} and {\it harris} stages. In general, we
expect the profile estimates to be better for stages that occur deeper in
the pipeline. Note that the profile estimates also indicate the limit to
which the static analysis techniques can be improved by using more powerful
approaches. Profile information can be easily
obtained by executing the HLS C++ program directly without the need for a
heavy weight circuit simulation.
\begin{figure}[hb]
\centering
\resizebox{\columnwidth}{!}{%
\harrisHist
}%
\vskip 5pt
\caption{Cumulative distribution of pixels with respect to maximum integral bit-width length at
stages $I_x$ and $I_{xy}$ of the HCD program.}
\label{fig:hs}
\end{figure}
\subsection{Fractional Bits ($\beta$) Analysis}
The number of fractional bits $\beta_i$ required at a stage $i$ depends on the
application and cannot be estimated in an application independent way as is
the case with the integral bits analysis. Estimating the optimal number of
fractional bits at each stage for a given application metric turns out to be a
non-convex optimization problem in most cases and hence we propose a simple
heuristic search technique that requires very few profile passes.
In the profiling technique, we fix the number of integral bits required at each
stage based on static or profile-driven analysis and increase the precision
$\beta$ uniformly across all the stages. For each value $\beta$, we estimate
the application-specific error metric. For the HCD benchmark, the error
metric is the percentage of mis-classified corners when compared to a design
which uses sufficiently long integral and fractional bits. We can reach an
optimal $\beta$ for a given error tolerance via binary search. Then we make
a single pass on the stages of the DAG in reverse topologically sorted order.
At each stage $I$, we do a binary search on the number of fractional bits
required, $\beta_I$, starting from the initial estimate $\beta$ while
retaining the application specific quality requirement. Figure~\ref{fig:flow}
summarizes the proposed bitwidth analysis framework.
\iffalse
Figure~\ref{fig:uniformbeta} shows the average percentage of pixels correctly
classified by the HCD program on the test image set by varying the
fractional bits uniformly across all the stages while fixing the integral
bit-width estimates obtained via profiling ($\alpha_i^{avg}$). It also
contains estimates of power consumption with varying fractional bits for the
Xilinx ZedBoard FPGA.
It can be noted from the graph that the fractional bits do not affect the
accuracy of corner classification, and we thus get more than 99\%
accuracy even with zero fractional bits. From this graph, we infer that one
can obtain close to 100\% accuracy by using 10 fractional bits uniformly
across all the stages. We then make a backward pass on the stages of the
HCD program to drop the fractional bits further without any significant loss
in accuracy and the column corresponding to $\beta^{avg}$ in
Table~\ref{tab:bitTableHCD} shows the final fractional bitwidths.
Table~\ref{tab:HCDresults} shows the accuracy, power
and area metrics obtained when the HCD program is applied on the test data
using float, bit-widths from static and profile analysis. It can be noticed
that by using bitwidths from static analysis, we obtain 99.999\% accuracy with
a power consumption of 0.263~W. The power savings are
3.8$\times$ lower when compared with the baseline PolyMage-HLS compiler
generated code using float data type. With respect to area, the number of
slices used, the savings over using float is around 6.2$\times$. The
number of DSPs used by the fixed-point and floating-point designs 12
and 112 respectively. Similarly, the number of BRAM units used by the
fixed-point and floating-point designs are 14 and 12 respectively.
The last 3 columns of the Table~\ref{tab:HCDresults} show the maximum
frequency of operation by different designs and it can be noted that can
run at a higher frequency achieving better throughput while consuming lesser
power.
\begin{table*}[htb]
\footnotesize
\caption{Quality, post place and route power, area and throughput metrics for the HCD program using float and, bit-widths estimated using static analysis and profiling techniques.The fractional bits used are from Table~\ref{tab:bitTableHCD}.}
\vskip 5pt
\label{tab:HCDresults}
\centering
\harrisRes
\end{table*}
\fi
\section{Related Work}
\label{sec:related-work}
Besides PolyMage~\cite{mullapudi2015asplos}, Rigel~\cite{rigel},
Darkroom~\cite{darkroom}, HIPAcc~\cite{membarth16tpds}, and
Halide~\cite{kelley13pldi} are other recent domain-specific languages (DSL)
for image processing pipelines. Among them, PolyMage, Rigel, HIPAcc, and
Darkroom compilers can generate hardware designs targeting FPGAs, and none of
these currently optimize designs using bitwidth analysis.
There are several works on bitwidth estimation in digital signal processing
applications ~\cite{TEMA352, vakili13tcad, cong09fccm, lee06tcad,
zhang10jsip}. However, these techniques are not scalable and can only be applied to small circuits
like low degree polynomial multiplications, 8x8 discrete cosine transform
computation etc. which contain very few signals in the order of 10s and 100s.
Whereas the techniques proposed in this paper exploits both the image processing
domain and the PolyMage-HLS compilation framework to do interval analysis on
large image processing pipelines wherein each pixel at every stage of the pipeline
constitutes a signal. Further, there are a class of iterative algorithms such as
optical flow wherein errors in range estimation accumulate across iterations making
the analysis in essence useless. In this work, we show
how we can use SMT solvers to get accurate range estimates and thus contain errors
across iterations. Kinsman and Nicolici~\cite{kinsman10} proposed a SMT solver based
approach, however, they evaluated their approach on small signal processing
applications involving less than 10 signals. The SMT solver based approach proposed
in the current work handles large image processing applications which are iterative
in nature with potentially thousands of pixel signals being processed in each
iteration.
Usually, range analysis (integer bits) and precision (fraction bits)
analysis are performed separately. For precision analysis, there are
heuristic search~\cite{vakili13tcad, nguyen11precision} based approaches which try to minimize circuit area and power while satisfying constraints on Signal-to-Noise
ratio. The time complexity of these algorithms is usually very high and hence
impractical to use in large image processing pipelines. Whereas, the greedy heuristic
algorithm we proposed in this paper runs in linear time with respect to the number of stages
present in the image processing pipeline and is independent of the image dimensions.
Overall, ours is the first extensive study on the
application of practical range and precision analyses in image processing
applications, and their impact on power and area savings.
\iffalse
We study the effectiveness of interval
analysis in estimating integral bitwidths and compare it with the estimates
obtained via profile-driven analysis. The interval analysis algorithm nicely
fits into the data flow analysis framework used in all modern compilers. For
precision analysis to estimate fractional bits, we use uniform bitwidths
across all the stages and then use simple linear or binary search algorithms
to arrive at a suitable bitwidth meeting an application-specific quality
criteria.
\fi
Mahlke et al.~\cite{mahlke01tcad} proposed a data flow analysis based approach
for bitwidth estimation of integral variables in the PICO (Program-in Chip-out)
system for synthesizing hardware from loop nests specified in C. Along the same
lines, Gort and Anderson~\cite{anderson13range} proposed a range analysis
algorithm in the LegUp HLS tool. Their range analysis algorithm is designed
over the LLVM intermediate representation and is implemented as an LLVM
analysis pass. On the other hand, the interval arithmetic based range
analysis algorithm we will propose works at
the DSL level and furthermore, the proposed compilation framework permits
the
usage of any other range analysis algorithm nearly in a plug-and-play
manner; this can otherwise require significant effort in order to make it
into a compiler analysis pass.
The integral bitwidth analysis algorithm due to Budiu et
al.~\cite{budiu00europar} is similar to the previous work but uses a
different data flow analysis formulation.
Stephenson et al.~\cite{stephenson00sigplan} performs integer bitwidth
analysis through range propagation, again using a data flow analysis
framework.
Tong et al. \cite{tong00customfp} proposed the usage of variable
bitwidth floating-point units, which can save power for applications that
do not require the full range and precision provided by the standard
floating-point data type.
Sampson et al. \cite{sampson11enerj} proposed
EnerJ, an extension to Java that supports approximate data types and
computation. However, fixed-point data types and the associated approximate
operations are not considered in EnerJ. On the contrary, PolyMage DSL can be
enhanced by using the approximate data types as proposed by EnerJ.
Approximate computing has a rich body of
literature~\cite{mittal16approxsurvey, ramani15approx, han13approx}.
However, our context of
domain-specific automatic HLS compilation is unique. Depending on the output
quality and the application in question, our approach could either be seen
as exploiting customized precision or leveraging approximate computing.
In addition to customized precision, we can potentially use approximate
arithmetic operations \cite{kahng12adder, liu14multiplier} in the various
stages of computation.
\subsubsection{Example with Horn-Schunck Optical Flow}\label{sec:smt_running_example}
\iffalse
Comment start
\\
So, just for the terminology, I'll say that the initial input variables (stage 0) are fundamental variables and all other variables introduced in successive stages are derived variables and can be written completely in terms of stage0 vars when all correlations are preserved. Breaking correlations at a stage means that the output variables of that stage are now new fundamental vars (which i used to call dummy vars). Also, now we can also have a hybrid way of solving, where we
break the correlations at some stages, not all. And the question is how to choose those stages.
As of now, i tried various ways, and none worked for HCD.
\\
comment end
\fi
In Horn-Schunck optical flow, all stencils are 3x3. Since this optical flow solves for the correct motion vectors in an iterative manner, the DAG gets pretty huge just after 3-4 iterations and to be able to use Z3, we break the correlations before every stencil stage. We do this because otherwise every stencil stage will require at-least 9 input variables from the parent and breaking correlations at their parent significantly reduces the total number of variables and their degree. Hence, 9 fundamental variables are created at the parent of each stage.
The DAG is shown in figure \ref{fig:hsof_dag}. From this DAG, the code for SMT analysis using Z3 python API is generated as shown in listing \ref{list:of_z3}.
\ofsmtcode
\iffalse
\begin{figure}
\centering
\includegraphics[scale = 0.8]{./graphs/hs_of_dag}
\captionof{figure}{DAG representation of the Horn-Schunck Optical Flow algorithm.}
\label{fig:hsof_dag}
\end{figure}
\fi
\iffalse
\begin{figure}
\includegraphics[width=4cm]{images/HS-OF-DAG}
\caption{Horn-Schunck Optical Flow DAG}
\label{fig:hsof_dag}
\end{figure}
\fi
\section{Motivation SMT}
(Placeholder for Motivation on Z3 approach)
\\
\\
(Discussion on how the ranges are calculated is here)
\\
\\
<<<<<<< HEAD
The range(bitwidth) for each stage is solved in the topologically sorted order as per the image processing DAG. The input to any DAG is an image. We consider this as the first stage and the range associated with the variables in this stage is (0,255). This, in turn is the input to the second stage and also a constraint. Generalising, each node in the dag is a stage and parents to that node add their constraints to the solver which are used for solving the range for this stage.
The range for a stage is solved in the following manner. Depending on the nature of the stage, an expression, say $exp$, can be obtained which specifies it's operation on the input(s). The goal is to find the minimum and the maximum value this expression can take. (The expression can be non-linear etc.. The other optimisers may not give global min/max. But SMT does give a definite upper or lower bound). This is where Z3 is used. For finding the upper bound, we start with a sufficiently large negative value, say $maxVal$, and check the satisfiability of the equation $exp > maxVal$. If $there_exist$ any assignment of the variables in the $exp$ for which this is true, the solver returns $sat$ which means satisfiable. In such a case the value of $maxVal$ is reduced and this procedure is iteratively repeated until we reach a point on the real number line such that the solvers $sats$ on the left epsilon neighbourhood and $unsats$ on the right epsilon neighbourhood of that point. Epsilon is an input parameter and should be less than 1. Similarly we solve for the minimum value the expression can take. From the min and max values, the integeral bitwidth requirement for this stage is calculated as in equation ?. (I have to add the math expression here for calculating bitwidth from range).
=======
The range(bitwidth) for each stage is solved in the topologically sorted order as per the image processing DAG. The input to any DAG is an image. We consider this as the first stage and the range associated with the variables in this stage is (0,255). This, in turn is the input to the second stage and also a constraint. Generalizing, each node in the dag is a stage and parents to that node add their constraints to the solver which are used for solving the range for this stage.
The range for a stage is solved in the following manner. Depending on the nature of the stage, an expression, say $exp$, can be obtained which specifies it's operation on the input(s). The goal is to find the minimum and the maximum value this expression can take. (The expression can be non-linear etc.. The other optimizers may not give global min/max. But SMT does give a definite upper or lower bound). This is where Z3 is used. For finding the upper bound, we start with a sufficiently large negative value, say $maxVal$, and check the satisfiability of the equation $exp > maxVal$. If $there_exist$ any assignment of the variables in the $exp$ for which this is true, the solver returns $sat$ which means satisfiable. In such a case the value of $maxVal$ is reduced and this procedure is iteratively repeated until we reach a point on the real number line such that the solvers $sats$ on the left epsilon neighbourhood and $unsats$ on the right epsilon neighbourhood of that point. Epsilon is an input parameter and should be less than 1. Similarly we solve for the minimum value the expression can take. From the min and max values, the integral bit-width requirement for this stage is calculated as in equation ?. (I have to add the math expression here for calculating bitwidth from range).
>>>>>>> 4880d77b8a79889eccf4baecfae4af205af77527
The important point here is that the constraints and the expressions are stored symbolically and the information about the ancestors of a node is available to the node and this captures the correlations between the variables in the expression of the current node and this results in a tight bound. But it has it's own price. The complexity of the DAG can depend on many factors such as it's depth, degree of expressions in the stages, etc. As the complexity increases, it becomes harder and confusing
for Z3 to check the satisfiability and it severely affects the runtime and capability of Z3 to solve. For some cases, it may also return $unknown-solution$.
The input image variables are the fundamental variables and all other variables of stages are derived from them. This keeps all the correlations intact. As we move down the DAG, each derived variable becomes either higher in degree or higher in total number of unknowns. After a while z3 starts to fail. To deal with this problem, we need to keep the degree and unknowns of derived variables under control. For this we break the correlations at certain stages and introduce new fundamental
variables there. This is done by declaring completely new variables and assigning them the correct range of that stage and these flow down the dag now.
Stage 0 is the input stage and the total number of variables are equal to the input image size. To further reduce the total number of variables flowing in the system, as a first step, we calculate the minimal variables required from stage0 for full correlation capture. So, in essence, only those variables are selected at stage0 which are just enough to produce one output at the last stage. This is calculated by projecting the requirement bottom up starting from the last stage. Without loss of generality, given a 2-D matrix of size MxM, the dimension remains same if point-wise operation is applied to it. If a stencil of dimension say NxN and stride S is applied to it, then the size reduces to $floor((M-((N-1)/2))/S) \ \ X \ \ floor((M-((N-1)/2)/S)$ assuming N is odd. Using these equations, the minimum input variable matrix size can be calculated for any arbitrary DAG as after all the reductions, in whatever order, the final size should be 1. We do this compulsorily for every DAG.
For complex computer vision pipelines such as optical flow, we still have to break correlations at certain places. In section ?, this is taken as a running example to understand the code and the framework.
\\
(Next file is general-framework-smt.tex)
\subsection{Range Analysis using SMT Solvers} \label{sec:smt}
\begin{table*}[t]
\begin{minipage}[b]{0.38\linewidth}
\centering
\includegraphics[scale = 0.6]{./graphs/hs_of_dag}
\captionof{figure}{DAG representation of the Horn-Schunck Optical Flow algorithm.}
\label{fig:hsof_dag}
\end{minipage}
\hfill
\begin{minipage}[b]{0.58\linewidth}
\centering
\resizebox{\linewidth}{!}{%
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r |r}
\toprule
Stage & Computation \\
\midrule
$I_{t}$ & $img_1\left(i,j\right)-img_2\left(i,j\right)$ \\
$I_x$ & $\frac{1}{12}\left[ \begin{smallmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0& 1\end{smallmatrix} \right]$ \\
$I_y$ & $\frac{1}{12}\left[ \begin{smallmatrix} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2& 1\end{smallmatrix} \right]$ \\
$I_{xx}$ & $I_x\left(i,j\right) I_x\left(i,j\right)$ \\
$I_{yy}$ & $I_y\left(i,j\right) I_y\left(i,j\right)$ \\
$denom$ & $\alpha^2+I_{xx}\left(i,j\right)+I_{yy}\left(i,j\right)$ \\
$Common_x$ & $\frac{I_x\left(i,j\right)}{Denom\left(i,j\right)}$ \\
$Common_y$ & $\frac{I_y\left(i,j\right)}{Denom\left(i,j\right)}$ \\
$V_x^0$ & $-I_t\left(i,j\right)Common_x\left(i,j\right)$ \\
$V_y^0$ & $-I_t\left(i,j\right)Common_y\left(i,j\right)$ \\
$Av_x^0$ & $A=\frac{1}{4} \left[ \begin{smallmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{smallmatrix} \right]$\\
$Av_y^0$ & $A$\\
$Common^0$ & $I_x(i,j)Av_x^0(i,j)+ I_y(i,j)Av_y^0(i,j) + I_t(i,j) $ \\
$V_x^1$ & $Av_x^0(i,j)- Common^0(i,j)Common_x(i,j))$ \\
$V_y^1$ & $Av_y^0(i,j)-Common^0(i,j)Common_y(i,j))$ \\
\midrule
\end{tabular}
}
\caption{Summary of computations in the Optical Flow algorithm.}
\label{tab:of_comp}
\end{minipage}
\end{table*}
\iffalse
\begin{table}[htb]
\footnotesize
\centering
\caption{Comparison of bitwidth estimates using interval analysis for
the Optical Flow benchmark.
\label{tab:ofinterval}}
\vskip 5pt
\resizebox{\columnwidth}{!}
\begin{tabularx}{1.64\textwidth}{c|ccccccc|ccc|ccc|ccc|ccc}
\toprule
& & & & & & & & &\bf{Iteration 1}&& &\bf{Iteration 2}&& &\bf{Iteration 3}&& &\bf{Iteration 4} & \\
\cmidrule(lr){9-11}\cmidrule(lr){12-14}\cmidrule(lr){15-17}\cmidrule(lr){18-20}
\bf{Stage}&Img$_1$ &I$_t$&I$_x$ &I$_{xx}$ &Denom&$Common_x$ &V$_{x}^0$ &Avg$_x^0$ &Common$^0$&V$_x^1$ &Avg$_x^1$ &Common$^1$&V$_x^2$ &Avg$_x^2$ &Common$^2$&V$_x^3$ &Avg$_x^3$ &Common$^3$&V$_x^4$ \\
&Img$_2$ & &I$_y$ &I$_{yy}$ & &$Common_y$ &V$_{y}^0$ &Avg$_y^0$ & &V$_y^1$ &Avg$_y^1$ & &V$_y^2$ &Avg$_y^2$ & &V$_y^3$ &Avg$_y^3$ & &V$_y^4$ \\
\midrule
\bf{$\alpha^{sa}$}&8 &9 &8 &13 &14 &6 &14 &14 &21 &26 &26 &33 &38 &38 &45 &49 &49 &57 &61 \\
\bottomrule
\end{tabularx}}
\end{table}
\fi
\iffalse
\begin{figure}[t]
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{\linewidth}{!}{
\input{figures/bitgraph}
\label{fig:bitgraph}
}
\caption{Comparison of bitwidths obtained by SMT and Interval Analysis.}
\label{fig:bitsgraph}
\end{minipage}%
\hfill
\begin{minipage}[t]{0.49\linewidth}
\resizebox{\linewidth}{!}{
\input{figures/equation}
}
\caption{Plot for $Common_x$ stage for $I_y=60$ and $I_y=85$}
\label{fig:commonplt}
\end{minipage}
\end{figure}
\fi
Range analysis algorithms based on interval or affine arithmetic variants
have
limitations in capturing the correlations between computations
(refer Section~\ref{sec:background}). For example, consider the
Optical Flow benchmark, whose DAG and computations at
each stage of the DAG are given in the Figure~\ref{fig:hsof_dag} and
Table~\ref{tab:of_comp} respectively. Consider the point-wise stage
$Common_x$ where each pixel is computed as follows:
\[
Common_x(i,j) = \frac{I_x(i,j)}{Denom(i,j)}.
\]
The second column in Table~\ref{tab:ofrange} represents the ranges inferred at various
stages of the Optical Flow benchmark using interval analysis. The ranges obtained
using affine analysis are also very similar with no change in bitwidth estimates.
We observe that the range
at $Common_x$ stage is inferred as $[-21.25, 21.25]$ and hence requires
6 integral fixed-point bits. This range is obtained by dividing the range
of $I_x$ with the range of $Denom$, which are $[-85, 85]$ and $[4, 14454]$ respectively. However, if we symbolically expand the computation at the
stage $Common_x$, then we obtain the following formula:
\begin{eqnarray}
Common_x(i,j) = \frac{I_x(i,j)}{\alpha^2+I_x(i,j)I_x(i,j)+I_y(i,j)I_y(i,j)}.
\label{eq:eq1}
\end{eqnarray}
Now, we observe that the pixel signal $I_x(i,j)$ is present both in the
numerator and denominator. Since $\alpha=2$, the RHS in
Equation~\eqref{eq:eq1} is equivalent to $\frac{x}{x^2+a}$ for some $a\geq 4$.
Figure~\ref{fig:commonplt} shows the plot of the
function $\frac{x}{x^2+a}$ for various values of $a$.
We can analytically determine the absolute values of the maximum and minimum
of that function to be less than one. Both interval and affine
arithmetic analysis fail to arrive at this conclusion.
\input{files/range}
We address this issue using an SMT solver based range analysis approach.
The basic idea is to build a constraint system involving the variables
$I_x(i,j)$,
$I_y(i,j)$ and $Common_x(i,j)$. The constraint system consists of range
constraints on variables $I_x(x,y)$ and $I_y(i,j)$, that are inferred
through
interval analysis, and an equality constraint as specified in the
Equation~\eqref{eq:eq1}. To this base constraint system, we add a parametric
constraint $Common_x(i,j) >UB$, $UB$ being the parameter. For a given value
of $UB$, if the constraint system has no solution, then we know that the
maximum
value of $Common_x(i,j)$ is bounded by UB. We use this idea to arrive at a
tight upper bound using a binary search algorithm. The upper bound estimate
need not be too accurate as long as it does not affect the corresponding bit
width estimates. A similar approach is adopted to determine the lower
bound too.
We observe from the following recurrence relations that a bad estimate
in the bitwidth of stages $Common_x$ and $Common_y$ has a cascading effect
on the bitwidth estimates
of stages $V_x^{k}$, $V_y^{k}$, $Av_x^k$, $Av_y^{k}$ and $Common^k$ for
any $k\geq 0$:
\begin{eqnarray*}
V_x^0(i,j) & = & -I_t(i,j)Common_x(i,j) \\
V_y^0(i,j) & = & -I_t(i,j)Common_y(i,j) \\
Av_x^k(i,j) & = & \frac{1}{4}\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \circledast \begin{bmatrix} V_x^k(i-1,j-1) & V_x^k(i-1,j) & V_x^k(i-1,j+1) \\ V_x^k(i,j-1) & V_x^k(i,j) & V_x^k(i,j+1) \\ V_x^k(i+1,j-1) & V_x^k(i+1,j) & V_x^k(i+1,j+1) \end{bmatrix}\\
Av_y^k(i,j) & = & \frac{1}{4}\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \circledast \begin{bmatrix} V_y^k(i-1,j-1) & V_y^k(i-1,j) & V_y^k(i-1,j+1) \\ V_y^k(i,j-1) & V_y^k(i,j) & V_y^k(i,j+1) \\ V_y^k(i+1,j-1) & V_y^k(i+1,j) & V_y^k(i+1,j+1) \end{bmatrix}\\
Common^k(i,j) & = & I_x(i,j)Av_x^k(i,j) + I_y(i,j)Av_y^k(i,j) + I_t(i,j) \\
V_x^{k+1}(i,j) & = & Av_x^k(i,j) - Common^k(i,j)Common_x(i,j) \\
V_y^{k+1}(i,j) & = & Av_y^k(i,j) - Common^k(i,j)Common_y(i,j). \\
\end{eqnarray*}
The stages $Av_x^k$ and $Av_y^k$ are stencil stages which average the
values from the stages $V_x^{k}$ and $V_y^{k}$ respectively. Hence, any
bitwidth overestimates at the stages $V_x^{k}$ and $V_y^{k}$ will be
directly
passed down to the stages $Av_x^k$ and $Av_y^k$. These in turn will be
reflected in the bitwidth estimate of the stage $Common^k(i,j)$. Finally,
while estimating the bitwidth at the stage $V_x^{k+1}(i,j)$ , the bitwidth
estimate errors of the stages $Common^k(i,j)$ and $Common_x(i,j)$ add-up
linearly. Similar is the case for the stage $V_y^{k+1}(i,j)$.
We observe from Table~\ref{tab:ofrange} and Figure~\ref{fig:bitsgraph} as to
how bitwidth estimates explode with each stage using interval analysis,
while they are contained using SMT solver based
approach. In the next section, we provide a more detailed description of our
SMT-based range analysis algorithm called \smtra.
\begin{figure}[t]
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{\linewidth}{!}{
\input{figures/bitgraph}
\label{fig:bitgraph}
}
\caption{Comparison of bitwidths obtained by SMT and Interval Analysis.}
\label{fig:bitsgraph}
\end{minipage}%
\hfill
\begin{minipage}[t]{0.49\linewidth}
\resizebox{\linewidth}{!}{
\input{figures/equation}
}
\caption{Plot for $Common_x$ stage for $I_y=60$ and $I_y=85$}
\label{fig:commonplt}
\end{minipage}
\end{figure}
\subsubsection{\smtra \ Algorithm}
The range of a pixel signal $S_{ij}$ at a stage $S$ of the input DAG depends
on the pixel signals from the predecessor stages. Let $Dep(S_{ij})$ denote
the
pixel signals from the input stages on which $S_{ij}$ is dependent, i.e.,
\[
Dep(S_{ij}) = \{~I(k,l)~|~\textrm{$I$ is an input stage and $S_{ij}$ depends
on the $(k,l)^{th}$ pixel of $I$}. \}
\]
Then we can compute $Dep(S_{ij})$ by applying one of the following three
cases recursively:
\begin{enumerate}
\item $S$ is an input stage with no predecessors. Then $Dep(S_{ij})=\{S_{ij}\}$.
\item $S$ is a point-wise stage. Then
\[
Dep(S_{ij}) = \bigcup\limits_{P\in Predecessor(S)} Dep(P_{ij})
\]
where $Predecessor(S)$ is the set of immediate predecessors of stage $S$.
\item $S$ is a stencil stage. A stencil stage has only one predecessor
stage. Let $P$ be the predecessor stage of $S$ and
\[
\hat{P}=\{~(k,l)~|~\textrm{$S_{ij}$ depends on $P_{kl}$}\}.
\]
Then,
\[
Dep(S_{ij}) = \bigcup\limits_{(k,l) \in \hat{P}} Dep(P_{kl}).
\]
\end{enumerate}
\begin{figure}[!htb]
\centering
\begin{minipage}[t]{0.45\linewidth}
\centering
\resizebox{\linewidth}{!}{
\input{figures/grid}
}
\caption{An output pixel from the 3x3 stencil stage $S_2$ depends on a 5x5 window of pixels from the input image. This dependency is induced via a 3x3 window of pixels from stage $S_1$. }
\label{fig:grid}
\end{minipage}%
\hfill
\begin{minipage}[t]{0.45\linewidth}
\centering
\resizebox{\linewidth}{!}{
\input{./figures/plot_ha_s1}
}
\caption{Cumulative distribution of pixels with respect to maximum
integral bitwidth length at stages $I_x$ and $I_{xy}$ of the HCD
benchmark.}
\label{fig:hs}
\end{minipage}
\end{figure}
The algorithmic plan is to take pixel signal $S_{ij}$ and express its
computation using the signals from the set $Dep(S_{ij})$. The set of
equations which leads to its computation defines a constraint system.
We augment this constraint system by adding interval constraints on
input pixel signals from $Dep(S_{ij})$. In order to estimate the upper bound,
we add a constraint $S_{ij} > UB$ where $UB$ is a large enough constant,
and check if there is a solution. If there is no solution, then $UB$
is in fact an upper bound on $S_{ij}$. We continue to tighten the upper bound
using binary search until it reaches a stage where any further
improvement results in no bitwidth savings.
Although the proposed algorithmic plan is theoretically sound, in practice,
there will be an explosion in the number of variables in the constraint
system due to the presence of stencil
stages in the computational paths. For example, if an input is supplied
to a stage $S$ through a pipeline path in the DAG that consists of $k$
stencil stages such as $Av_x$ (cf. Table~\ref{tab:of_comp}), then the number
of variables in the constraint system grows quadratically, i.e.,
$|Dep(S_{ij}|=\theta(k^2)$. Even the state-of-the-art
SMT solvers may not be able to solve such large constraint systems using
reasonable computational power. Figure~\ref{fig:grid} illustrates this
scenario. Here, $I$ is the input image, stages $S_1$ and $S_2$ are two 3x3
stencil stages. A pixel in stage $S_2$ depends on 9 pixel signals from
$S_1$ which in turn leads to a dependence on 25 pixel signals from input
$I$.
We circumvent this explosion of variables in the constraint system by
limiting
the expansion of computation at a stencil stage. Towards this, we define a new
function $\widehat{Dep}(S_{ij})$ as follows.
\begin{enumerate}
\item If $S$ is an input stage with no predecessors,
$\widehat{Dep}(S_{ij})=\{S_{ij}\}$.
\item If $S$ is a point-wise stage, \[
\widehat{Dep}(S_{ij}) = \bigcup\limits_{P\in Predecessor(S)} \widehat{Dep}(P_{ij})
\]
where $Predecessor(S)$ is the set of immediate predecessors of stage $S$.
\item If $S$ is a stencil stage, \[
\widehat{Dep}(S_{ij}) = \{S_{ij}\}.
\]
\end{enumerate}
For example, in the optical flow benchmark, while computing
$\widehat{Dep}(Common_x)$, the recursion terminates with the stencil stages
$I_x$ and $I_y$. The range at stencil stages is estimated using simple
interval
analysis. In the constraint system associated with the range estimation
of the stage $Common_x$, we use the range constraints which are already derived
on the pixel signals from stages $I_x$ and $I_y$. Thus we contain the number
of variables in the constraint system from growing exponentially.
To summarize, we consider the nodes in the DAG in a topologically sorted
order. We estimate the range at a stencil stage using a simple interval
analysis. And at a point-wise stage $S$, we construct a pruned computational
DAG wherein the stage $S$ acts as a sink and the source nodes are either
input stages or stencil stages from which there exists a stencil-free path
to stage $S$. Then, the computation of a pixel signal from stage $S_{ij}$ is
expressed using pixel signals from the source and intermediate nodes. This
set of equations acts as a base constraint system to which we add the
range constraints on the source pixel signals from the set
$\widehat{Dep}(S_{ij})$. Then we search for a tight lower bound constraint,
$LB \leq S_{ij}$, and an upper bound constraint, $S_{ij} \leq UB$, using
binary search, leading to a range estimate $S_{ij}\in [LB, UB]$.
In the next section, we present a profile-driven analysis that provides
a lower bound on the bitwidth estimates, and show in the experimental
results section, that the bitwidth estimates derived from the SMT solver
based approach match the lower bounds provided by profile-driven analysis.
\iffalse
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\columnwidth}
\resizebox{\linewidth}{!}{%
\input{figures/grid}
}
\caption{Grid}
\label{fig:grid}
\end{subfigure}
\begin{subfigure}[b]{0.4\columnwidth}
\input{./figures/plot_ha_s1}
\caption{Cumulative distribution of pixels with respect to maximum
integral bitwidth at stages $I_x$ and $I_{xy}$ of the HCD benchmark.}
\label{fig:hs}
\end{subfigure}
\end{figure}
\fi
\section{Introduction}
Field-Programmable Gate Arrays (FPGAs) are suitable for accelerating
computations from several domains such as image processing, computer vision,
and digital signal processing. When a lower or customized precision is
desired, FPGAs are often expected to perform better than accelerators such
as GPUs with respect to performance delivered per unit of energy consumed.
When an image processing pipeline such
as Harris Corner Detection (HCD) (cf. Figure~\ref{fig:hdag} and Table~\ref{tab:hcd}),
is implemented on a CPU or a GPU, a programmer is bound to choose a pre-defined
data type such as {\tt float}, {\tt int}, or {\tt short} owing to the underlying
architectural constraints. In order to avoid arithmetic overflows, the
data types have to be chosen conservatively through over-estimation. This
leads to a wastage of memory, and hence memory bandwidth, at all levels of
the hierarchy; furthermore, additional energy is consumed both due to data
transfer and the higher precision in which the arithmetic is performed. On
the other hand, on FPGAs, it is possible to use variable length
fixed-point data types to represent the data produced and consumed at various
stages of an image processing pipeline. This saves chip area and the power
consumed by the hardware design due to the reduced precision
and internal routing logic. The other natural outcome is a better
utilization of available on-chip memory resources.
Although, FPGAs fare extremely well on the performance per watt metric,
their programmability has been a major hindrance in adoption. The reliance
on hardware description languages (HDLs) such as Verilog and VHDL makes it
extremely cumbersome for a wider programmer audience. High-Level Synthesis
(HLS) tools, which map C, C++ code into equivalent hardware designs by
generating HDL code automatically, have thus gained significant attention in
the past decade. The quality of designs generated by a HLS compiler often
depend on the analysis techniques employed. Many times programmers put in
substantial effort to drive a HLS tool to generate a hardware design of
their choice using suitable pragma annotations or code rewriting. There have
been efforts to further raise the level of abstraction --- from using
imperative languages such as C, C++ to domain-specific languages (DSLs) ---
giving rise to the term, ultra high-level synthesis; Bacon et
al.~\cite{bacon13acm} provide a comprehensive survey. DSLs not only
improve programmer productivity but also exploit the richer information
from the underlying algorithm. This facilitates compilers to generate better
code or hardware designs using relatively simple program analysis
techniques.
\subsection{DSLs for Image Processing Pipelines} An image processing
pipeline can be viewed as a directed acyclic graph (DAG) of computational
stages. Each stage transforms an
input image form into an output image form to be consumed by the subsequent
stages in the pipeline. The class of computations at each stage are simple data
parallel operations applied on all image pixels such as point-wise and stencil
computations. Figure~\ref{fig:hdag} shows the computational DAG associated with
the Harris Corner Detection (HCD) benchmark. As the name indicates, HCD is a corner
detection algorithm, commonly used in computer vision space.
Table~\ref{tab:hcd} shows the computations at each stage of the DAG.
The source code for HCD benchmark in PolyMage DSL, whose compiler infrastructure we use
in this paper,
can be obtained at the PolyMage GitHub repository~\cite{polymagebench}.
If the computations in HCD are expressed in C/C++,
then there will be a two dimensional loop associated with each stage of the DAG.
Further, these loops occur in some topologically sorted order of DAG nodes.
Thus, the rich structure in the application gets lost in the resulting C/C++
code. For example, it is hard to infer that pixels output from one stage can be
streamed to the following stage and the following stage can start computations
once it receives enough number of pixels. This observation leads to an extremely
efficient pipelined hardware architecture for the whole computational DAG and is
exploited in PolyMage-HLS compiler as in other DSL compilers for FPGAs. In this
paper, we use the fact that it is easy to infer the computations on pixels at
each stage of the DAG and further these computations are applied homogeneously on
all the pixels at the corresponding stage, to arrive at efficient range
analysis algorithms based on interval arithmetic and Satisfiability Modulo
Theory (SMT) solvers. It would be almost impossible to do
this if the design is expressed directly in Verilog/VHDL; and probably require
complex program analysis if we have to to achieve this on C/C++ programs as is
the case in HLS frameworks.
\subsection{Problem Description and Contributions}
The data at each stage of an image processing pipeline is represented using
a parametric fixed-point data type $\left(\alpha, \beta\right)$, where $\alpha$
and $\beta$
denote the number of bits used to represent integral and fractional parts. The
objective is to minimize $\alpha$ and $\beta$ at each stage while maintaining
an application specific quality metric. The optimal value of $\alpha$ at a stage
depends on the range of values produced; whereas the optimal value of $\beta$
depends on how precision impacts the quality metric.
Range analysis which is required for estimating the integral bitwidth requirement
is a well studied problem in literature. There are several works
based on variants of interval and affine arithmetic~\cite{cong09fccm, vakili13tcad,
zhang10jsip, stephenson00sigplan, lee06tcad, mahlke01tcad}.
The benchmarks considered in these works, such as FIR filter, Discrete Cosine
Transform, Polynomial Evaluation etc. are mainly from the signal processing domain
and their code size and complexity is small. These techniques are not
easily adaptable for image processing pipelines when expressed in Verilog/VHDL,
HLS C/C++ etc. due to the large number of pixel signals present at each stage
and multiple such stages in an application.
The first main contribution of our paper is an interval arithmetic based range analysis
technique in the DSL compiler which exploits the fact that the computations on all
the pixels at a given stage is homogeneous to do a combined range analysis. The
second main contribution of our paper is a range analysis framework in the DSL compiler
wherein any interval and affine arithmetic-like analysis can be incorporated with ease.
Apart from interval arithmetic based approaches, which cannot handle certain
kind of operations like divisions, techniques using powerful SMT solvers have
been proposed in the literature~\cite{ kinsman10}. However, they are applied on
extremely small benchmarks involving 2 to 3 equations. The third main
contribution of this paper is that, we propose a range analysis technique using
SAT solvers and apply it on large benchmarks such as Optical Flow involving few
tens of DAG stages not to speak of large number of pixel signals and complex
computations in the DAG structure. This is primarily possible because our
analysis is based on the DSL specification of the benchmark as against a HDL or
C/C++ specification. Further, we show that in iterative algorithms such as
Optical Flow, conservative estimates of interval analysis will have a
debilitating effect with the increase in the iterations making them unusable in
practice; and we have to resort to SMT solvers to get accurate range estimates.
The fourth main contribution of this paper is a simple greedy heuristic search
technique for precision analysis to determine the number of bits required for
representing fractional bits at each stage of the pipeline. Finally, we present
a thorough experimental study comparing the effectiveness of interval, SMT
solver and profile guided approaches with respect to power, area and speed on
large image processing benchmarks. We would like to highlight that all the
previous studies involve very small benchmarks.
We implement and evaluate our automatic bitwidth analysis approach in PolyMage
compiler infrastructure \cite{mullapudi2015asplos, chugh16pact}. With the PolyMage DSL, FPGAs are targeted by
first generating High-Level Synthesis (HLS) code after a realization of
several transformations for parallelization and reuse; the HLS code is
subsequently processed by a vendor HLS suite (Xilinx Vivado in the case of
PolyMage). There have been several recent DSL efforts that target FPGAs for image
processing; these include Darkroom~\cite{darkroom},
Rigel~\cite{rigel}, Halide~\cite{halide17hls}, HIPAcc~\cite{hipacc16fpga} and
PolyMage-HLS~\cite{chugh16pact}. While these works
have addressed several challenges in compiling DSL to FPGAs, none of them
have studied the issue of exploiting application-dependent
variable fixed-point data types for power and area savings. HiPAcc
goes to the extent of providing pragmas for specifying bitwidths of
variables, but no automatic compiler support for it.
\begin{table}[t]
\begin{minipage}[b]{0.38\linewidth}
\includegraphics[scale = 0.8]{./graphs/harris_annotated_dag}
\captionof{figure}{DAG representation of the Harris Corner Detection (HCD) algorithm.}
\label{fig:hdag}
\end{minipage}\hfill
\begin{minipage}[b]{0.58\linewidth}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{r |r}
\toprule
Stage & Computation \\
\hline
$I_x$ & $\frac{1}{12}\left[ \begin{smallmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0& 1\end{smallmatrix} \right]$ \\
$I_y$ & $\frac{1}{12}\left[ \begin{smallmatrix} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2& 1\end{smallmatrix} \right]$ \\
$I_{xx}$ & $I_x\left(i,j\right) I_x\left(i,j\right)$ \\
$I_{xy}$ & $I_x\left(i,j\right) I_y\left(i,j\right)$ \\
$I_{yy}$ & $I_y\left(i,j\right) I_y\left(i,j\right)$ \\
$S_{xx}$ & $A=\left[ \begin{smallmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\end{smallmatrix} \right]$\\
$S_{xy}$ & $A$\\
$S_{yy}$ & $A$\\
$det$ & $S_{xx}\left(i,j\right) S_{yy}\left(i,j\right)-S_{xy}\left(i,j\right) S_{xy}\left(i,j\right)$\\
$trace$ & $S_{xx}\left(i,j\right)+S_{yy}\left(i,j\right)$\\
$Harris$ & $det\left(i,j\right) - 0.04 ~trace\left(i,j\right) trace\left(i,j\right)$\\ [1ex]
\hline
\end{tabular}
\caption{Summary of computations in HCD benchmark.}
\label{tab:hcd}
\end{minipage}
\end{table}
The rest of this paper is organized as follows. Related work is discussed in
Section~\ref{sec:related-work} and the necessary background is provided in
Section~\ref{sec:background}. Section~\ref{sec:bitwidth} presents in detail the
main contributions of this paper.
Experimental evaluation is
presented in Section~\ref{sec:experiments} and conclusions are presented in
Section~\ref{sec:conclusions}.
\section{Related Work}
\label{sec:related-work}
Besides PolyMage~\cite{mullapudi2015asplos}, Rigel~\cite{rigel},
Darkroom~\cite{darkroom}, HIPAcc~\cite{membarth16tpds}, and
Halide~\cite{kelley13pldi} are other recent domain-specific languages (DSL)
for image processing pipelines. Among them, PolyMage, Rigel, HIPAcc, and
Darkroom compilers can generate hardware designs targeting FPGAs, and none of
these currently optimize designs using bitwidth analysis.
There are several works on bitwidth estimation in digital signal processing
applications ~\cite{TEMA352, vakili13tcad, cong09fccm, lee06tcad,
zhang10jsip}. However, these techniques are not scalable and can only be applied to small circuits
like low degree polynomial multiplications, 8x8 discrete cosine transform
computation etc. which contain very few signals in the order of 10s and 100s.
Whereas the techniques proposed in this paper exploits both the image processing
domain and the PolyMage-HLS compilation framework to do interval analysis on
large image processing pipelines wherein each pixel at every stage of the pipeline
constitutes a signal. Further, there are a class of iterative algorithms such as
optical flow wherein errors in range estimation accumulate across iterations making
the analysis in essence useless. In this work, we show
how we can use SMT solvers to get accurate range estimates and thus contain errors
across iterations. Kinsman and Nicolici~\cite{kinsman10} proposed a SMT solver based
approach, however, they evaluated their approach on small signal processing
applications involving less than 10 signals. The SMT solver based approach proposed
in the current work handles large image processing applications which are iterative
in nature with potentially thousands of pixel signals being processed in each
iteration.
Usually, range analysis (integer bits) and precision (fraction bits)
analysis are performed separately. For precision analysis, there are
heuristic search~\cite{vakili13tcad, nguyen11precision} based approaches which try to minimize circuit area and power while satisfying constraints on Signal-to-Noise
ratio. The time complexity of these algorithms is usually very high and hence
impractical to use in large image processing pipelines. Whereas, the greedy heuristic
algorithm we proposed in this paper runs in linear time with respect to the number of stages
present in the image processing pipeline and is independent of the image dimensions.
Overall, ours is the first extensive study on the
application of practical range and precision analyses in image processing
applications, and their impact on power and area savings.
\iffalse
We study the effectiveness of interval
analysis in estimating integral bitwidths and compare it with the estimates
obtained via profile-driven analysis. The interval analysis algorithm nicely
fits into the data flow analysis framework used in all modern compilers. For
precision analysis to estimate fractional bits, we use uniform bitwidths
across all the stages and then use simple linear or binary search algorithms
to arrive at a suitable bitwidth meeting an application-specific quality
criteria.
\fi
Mahlke et al.~\cite{mahlke01tcad} proposed a data flow analysis based approach
for bitwidth estimation of integral variables in the PICO (Program-in Chip-out)
system for synthesizing hardware from loop nests specified in C. Along the same
lines, Gort and Anderson~\cite{anderson13range} proposed a range analysis
algorithm in the LegUp HLS tool. Their range analysis algorithm is designed
over the LLVM intermediate representation and is implemented as an LLVM
analysis pass. On the other hand, the interval arithmetic based range
analysis algorithm we will propose works at
the DSL level and furthermore, the proposed compilation framework permits
the
usage of any other range analysis algorithm nearly in a plug-and-play
manner; this can otherwise require significant effort in order to make it
into a compiler analysis pass.
The integral bitwidth analysis algorithm due to Budiu et
al.~\cite{budiu00europar} is similar to the previous work but uses a
different data flow analysis formulation.
Stephenson et al.~\cite{stephenson00sigplan} performs integer bitwidth
analysis through range propagation, again using a data flow analysis
framework.
Tong et al. \cite{tong00customfp} proposed the usage of variable
bitwidth floating-point units, which can save power for applications that
do not require the full range and precision provided by the standard
floating-point data type.
Sampson et al. \cite{sampson11enerj} proposed
EnerJ, an extension to Java that supports approximate data types and
computation. However, fixed-point data types and the associated approximate
operations are not considered in EnerJ. On the contrary, PolyMage DSL can be
enhanced by using the approximate data types as proposed by EnerJ.
Approximate computing has a rich body of
literature~\cite{mittal16approxsurvey, ramani15approx, han13approx}.
However, our context of
domain-specific automatic HLS compilation is unique. Depending on the output
quality and the application in question, our approach could either be seen
as exploiting customized precision or leveraging approximate computing.
In addition to customized precision, we can potentially use approximate
arithmetic operations \cite{kahng12adder, liu14multiplier} in the various
stages of computation.
\section{Background}
\label{sec:background}
\begin{figure}[tb]
\centering
\overview
\vskip 5pt
\caption{PolyMage high-level compilation for FPGAs.}
\label{fig:overview}
\end{figure}
In this section, we briefly explain the architecture of the PolyMage-HLS
compilation framework; and further introduce the basics of interval and
affine arithmetic necessary to understand how range analysis techniques based
on them can be seamlessly integrated into our PolyMage-HLS compiler.
\subsection{PolyMage DSL and Compilation Framework}
In this paper, we use the PolyMage DSL and its compiler infrastructure
to implement and evaluate our automatic bitwidth analysis approach.
The PolyMage compiler infrastructure, when it was first
proposed~\cite{mullapudi2015asplos}, comprised an optimizing
source-to-source translator that generated OpenMP C++ code from an input
PolyMage DSL program. Chugh et al.~\cite{chugh16pact} developed a
backend for PolyMage targeting FPGAs by generating
HLS C++ code. The generated C++ code is translated into a hardware design
expressed in a Hardware Description Language (HDL) such as VHDL or Verilog
using High Level Synthesis compiler.
Figure~\ref{fig:overview}
shows the entire design flow using the PolyMage-HLS compiler and Xilinx Vivado
tool chain. For syntactic details and code examples, we refer the reader to
the PolyMage webpage~\cite{polymage-web}.
\subsection{Interval and Affine arithmetic}
With interval analysis, one estimates the range of an output
signal $z \leftarrow f\left(x, y\right)$ based on the range of the input signals $x$
and $y$, and the function $f$. For example, if the range of $x$ and $y$ are
$[\low{x},\high{x}]$ and $[\low{y}, \high{y}]$ respectively, and
$z\leftarrow x+y$, then the range of $z$ is
$[\low{x}+\low{y}, \high{x} +\high{y}]$. Such range estimation functions have
to be defined for different operations that are applied iteratively to
obtain the ranges of different intermediate and output signals involved in
the computation. Although interval arithmetic is simple and easy
to use in to practice, it suffers from the problem of range over-estimation.
For example, if the range of a signal $x$ is $[5, 10]$, then the interval
arithmetic estimates the range of the expression $x - x$ as $[-5, 5 ]$ whereas
the actual range is $[0, 0]$. This is due to the fact that the interval
arithmetic ignores the correlations between the operand signals if there were any.
\iffalse
Let the range of the two input signals $x$
and $y$ be $[\low{x}, \high{x}]$ and $[\low{y}, \high{y}]$ respectively. Then
range of the resulting signal $z = x~op~y$ where $op\in\{+,-,*,/, \verb|^| \}$ is
obtained by applying standard rules of interval arithmetic as follows.
\begin{enumerate}
\item {$\mathbf{z=x+y:}$} $[\low{z},\high{z}] = [\low{x}+\low{y}, \high{x}+\high{y}]$
\item {$\mathbf{z=x-y:}$} $[\low{z},\high{z}] = [\low{x}-\high{y}, \high{x}-\low{y}]$
\item {$\mathbf{z=x*y:}$} $[\low{z},\high{z}] = [t_1, t_2 ]$ where \\$t_1 = \min\left(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y}\right)$
and \\ $t_2 = \max\left(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y}\right)$.
\item $\mathbf{z=x/y:}$$[\low{z},\high{z}] = [\low{x}, \high{x}] * [1/\high{y}, 1/\low{y}]$ if $0 \not \in [\low{y}, \high{y}]$
\item $\mathbf{z = x^n}$
\begin{enumerate}
\item $n$ is odd: $[\low{z},\high{z}] = [\low{x}^n, \high{x}^n ]$
\item $n$ is even :
$\begin{aligned}[t]
[\low{z},\high{z}] & = & [\low{x}^n, \high{x}^n ] ~\textrm{if}~\low{x} \geq 0 \\
& = & [\high{x}^n, \low{x}^n ] ~\textrm{if}~\high{x} < 0\\
& = & [0, max\{\low{x}^n, \high{x}^n\} ]~\textrm{otherwise}
\end{aligned}$
\end{enumerate}
\iffalse
\begin{enumerate}
\item {\bf If} $\mathbf{\low{y} > 0}$ {\bf or} $\mathbf{\high{y}<0}$: \\$[\low{z},\high{z}] = [t_1, t_2 ]$ where \\ $t_1 = \min\left(\low{x}\mbox{\small/}\low{y}, \low{x}\mbox{\small/}\high{y}, \high{x}\mbox{\small/}\low{y}, \high{x}\mbox{\small/}\high{y}\right)$ and \\$t_2 = \max\left(\low{x}\mbox{\small/}\low{y}, \low{x}\mbox{\small/}\high{y}, \high{x}\mbox{\small/}\low{y}, \high{x}\mbox{\small/}\high{y}\right)$.
\item {\bf Otherwise:} $[\low{z}, \high{z}] = [\low{p}, \high{p}]$ where $\low{p}$ and $\high{p}$ are obtained via profiling information as explained in Section~\ref{sec:profile}.
\end{enumerate}
\fi
\end{enumerate}
Note that if an operand $x$ (or $y$) is constant, we get a tight range for
the result since $x=\low{x}=\high{x}$. In the above four cases, division
operation requires special care. If $0$ is present in the range $[\low{y},
\high{y}]$ of the denominator, then the theoretical range of $z$ can
potentially tend to infinity. Although interval arithmetic is simple and easy
to use in practice, it suffers from the problem of range over-estimation.
For example, if the range of a signal $x$ is $[5, 10]$, then the interval
arithmetic estimates the range of the expression $x - x$ as $[-5, 5 ]$ whereas
the actual range is $[0, 0]$. This is due to the fact that the interval
arithmetic ignores the correlation between the operand signals if there is
any.
\fi
With affine arithmetic analysis, a signal $x$ is
represented in an affine form as $x=x_0 + \sum_{i=1}^{n} x_i \epsilon_i$
where $\epsilon_i \in [-1, 1]$ are interpreted as independent noise signals
and their respective coefficients $x_i$'s are treated as the weights
associated with them. The interval of the signal $x$ from its affine form
can be inferred as $[x_0-r, x_0+r]$ where $r=\sum_{i=1}^{n} |x_i|$. The
addition
and subtraction operations on two input signals is defined as
$z=x\pm y = \left(x_0 \pm y_0\right) + \sum_{i=1}^{n}
\left(x_i \pm y_i\right)\epsilon_i$ and yields the resulting signal in its affine form.
The correlation between the signals
$x$ and $y$ is captured by sharing the independent noise signals $\epsilon_i,
1\leq i \leq n $ in their affine forms either partially or totally.
Now, when we perform a computation $x-x$ by considering the signal $x$ in its
affine form, the resulting range will be zero as against the over-estimated
range which we get in interval arithmetic analysis. Thus the techniques based
on affine arithmetic
arrive at better bounds when compared with interval analysis based techniques by
taking into account cancellation effects in computations involving
correlated signals.
However, note that if the operation is multiplication,
then the resulting signal contains quadratic terms and hence has to be
approximated to an affine form. A detailed discussion on affine arithmetic
analysis is beyond the scope of this paper and we recommend the reader to
Stolfi and Figueiredo~\cite{TEMA352} for the same.
\section{Bitwidth Analysis}\label{sec:bitwidth}
In this section, we present the main technical contributions of this paper which
are summarized below.
\begin{enumerate}
\item A simple interval arithmetic based range analysis algorithm
illustrating how DSLs facilitate practical and efficient program analysis
techniques when compared with C/C++ kind of languages
(Section~\ref{sec:ra}).
\item A software architecture for range analysis in DSL compilers in which variants of interval and affine arithmetic based approaches
can be easily deployed (Section~\ref{sec:ca}).
\item An SMT solver based approach for range analysis
which substantially improves the accuracy of range estimates and contains the
propagation of estimation errors across iterations. Again, such an SMT solver
based approach would have been hard to realize if not for the DSL compiler
framework (Section~\ref{sec:smt}).
\item A profile driven approach for range analysis (Section~\ref{sec:profile}).
\item A greedy heuristic search technique for precision estimation
(Section~\ref{sec:precision}).
\end{enumerate}
\subsection{Variable Width Fixed-Point Data Types}
A fixed-point data type is specified by a tuple $(\alpha, \beta)$ where
$\alpha$ and $\beta$ denote the number of bits allocated for representing
the integral and fractional parts respectively. The total bitwidth of the
data type is $\alpha + \beta$. The decimal value associated with a
fixed-point binary number $x=b_{\alpha-1}\ldots b_0 . b_{-1} \ldots
b_{-\beta}$ depends on whether it is interpreted as an unsigned integer or a
two's complement signed integer, and is given as follows:
\begin{equation*}
value(x) = \begin{cases}
\sum^{\alpha-1}_{i=-\beta} 2^i b_i & \text{unsigned} \\
-2^{\alpha-1} b_{\alpha-1} + \sum_{i=-\beta}^{\alpha-2} 2^i b_i & \text{2's
complement.}
\end{cases}
\end{equation*}
This gives us the ranges $[0, 2^{\alpha}-2^{-\beta}]$ and $[-2^{\alpha-1} ,
2^{\alpha-1}-2^{-\beta}]$ for unsigned and signed fixed-point numbers
respectively. The parameter $\alpha$ gives the range of values that can be
represented and the parameter $\beta$ indicates that the values in the range
can be represented at a resolution of $2^{-\beta}$. Hence, the range and
precision can be improved, by increasing $\alpha$ and $\beta$ respectively.
In this paper, we overload the term precision to also mean the entire data
type $(\alpha, \beta)$, and this can be disambiguated based on context.
Fixed-point data types are useful in image processing applications where the
range of values produced during computations is usually limited
and the precision requirements are less demanding when compared to many
other numerical algorithms. The data type (range and precision) requirement
at a stage depends on the input data type and the nature of local
computations carried out at that particular stage. Further, overflows during
computations can be addressed by using saturation mode arithmetic instead of
the conventional wrap around arithmetic operations performed in CPUs and GPUs.
\iffalse
This feature is
particularly useful in handling image processing pipelines using the
approximate computing paradigm.
\fi
The complexity of arithmetic operations on
fixed-point data type $(\alpha, \beta)$ is very similar to that of integer
operations on bitwidth $\alpha+\beta$.
\subsection{Range ($\alpha$) Analysis Algorithm}\label{sec:ra}
The number of integer bits required at a stage $I$ denoted as $\alpha_I$ is
a direct function of the bitwidth of the input data and the operations it
performs on them. The input data here refers to the data supplied to the
stage by its predecessor stages in the DAG. Further, the computations on
the pixel signals at each stage of DAG are identical and hence their
corresponding ranges would be the same. This information is implicitly provided
by a PolyMage DSL program and is hard to elicit from C like programs.
We use this insight to group all the pixel signals at a stage and perform a
combined range analysis using interval arithmetic. If the range of the
data produced at a stage is $[\low{x}, \high{x}]$, then the number of
integral bits $\alpha_I$ required to store the data without overflow is as
follows:
\begin{equation*}
\alpha=
\begin{cases}
\max(\ceil{\log_2(\ceil{|\low{x}|})},\ceil{\log_2(\floor{|\high{x}|}+1)})+1 & \text{if } \low{x} <0\\
\ceil{\log_2(\floor{\high{x}}+1)}, & \text{otherwise}.
\end{cases}%
\end{equation*}
The number of fractional bits
$\beta_I$ required at a stage depends on the application-specific error
tolerance or quality metric, and we propose a profile-driven estimation
technique in Section~\ref{sec:profile}.
The range analysis algorithm iterates over the stages of a DAG in a
topologically sorted order. At each stage, an equivalent expression tree
for the computations (point-wise or stencil) is built. Then the range of the
pixel signals is estimated by recursively performing interval arithmetic on the
expression tree using one of following five interval arithmetic rules.
\begin{enumerate}
\item {$\mathbf{z=x+y:}$} $[\low{z},\high{z}] = [\low{x}+\low{y}, \high{x}+\high{y}]$
\item {$\mathbf{z=x-y:}$} $[\low{z},\high{z}] = [\low{x}-\high{y}, \high{x}-\low{y}]$
\item {$\mathbf{z=x*y:}$} $[\low{z},\high{z}] = [t_1, t_2 ]$ where \\$t_1 = \min(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y})$
and \\ $t_2 = \max(\low{x}\low{y}, \low{x}\high{y}, \high{x}\low{y}, \high{x}\high{y})$.
\item $\mathbf{z=x/y:}$$[\low{z},\high{z}] = [\low{x}, \high{x}] * [1/\high{y}, 1/\low{y}]$ if $0 \not \in [\low{y}, \high{y}]$
\item $\mathbf{z = x^n}$
\begin{enumerate}
\item $n$ is odd: $[\low{z},\high{z}] = [\low{x}^n, \high{x}^n ]$
\item $n$ is even :
$\begin{aligned}[t]
[\low{z},\high{z}] & = & [\low{x}^n, \high{x}^n ] ~\textrm{if}~\low{x} \geq 0 \\
& = & [\high{x}^n, \low{x}^n ] ~\textrm{if}~\high{x} < 0\\
& = & [0, max\{\low{x}^n, \high{x}^n\} ]~\textrm{otherwise}
\end{aligned}$
\end{enumerate}
\end{enumerate}
\iffalse
. Algorithm~\ref{alg:precision}
shows the complete description of the bitwidth estimation algorithm applied
at each stage of the DAG computation. We do not show how an expression such
as $x ^ n$, where $n$ is a compile-time constant, is handled in
Algorithm~\ref{alg:precision} for the sake of simplicity. The interval
arithmetic for exponentiation operation, $[\low{x}, \high{x}]^n$ , is defined
as follows.
\begin{eqnarray*}
[\low{z},\high{z}] & = & [\low{x}^n, \high{x}^n ] ~\textrm{if}~\low{x} \geq 0 \\
& = & [\high{x}^n, \low{x}^n ] ~\textrm{if}~\high{x} < 0\\
& = & [0, max\{\low{x}^n, \high{x}^n\} ]~\textrm{otherwise}
\end{eqnarray*}
Our compiler maps simple expressions such as $x*x$ into $x^2$ as this results
in better range estimates. For example, if the range of a signal $x$ is
$[-2, 2]$, then $x*x = [-2, 2]*[-2, 2] = [-4, 4]$, whereas
$x^2 = [-2, 2]^2 = [0, 4]$.
\begin{table}[htb]
\centering
\small
\caption{HCD range analysis and integral bitwidth requirement at various stages.
\label{tab:hcdbit}}
\vskip 5pt
\begin{tabularx}{0.9\linewidth}{lcc}
\toprule
Stage & Range & $\alpha$ \\
\midrule
Img & $[0,255]$ & $8$ \\
I$_x$,I$_y$ & $[-85,85]$ & $8$ \\
I$_{xy}$ & $[-85^2,85^2]$ & $14$ \\
I$_{xx}$, I$_{yy}$ & $[0, 85^2]$ & $13$ \\
S$_{xy}$ & $[-9*85^2,9*85^2]$ & $17$ \\
S$_{xx}$, S$_{yy}$ & $[0, 9*85^2]$ & $16$ \\
det & -$[-(9*85^2)^2, (9*85^2)^2]$ & $33$ \\
trace & $[0,2*9*85^2]$ & $17$ \\
harris & $[-1.16*(9*85^2)^2, (9*85^2)^2]$ & $34$ \\
\bottomrule
\end{tabularx}
\end{table}
Figure~\ref{fig:hdag} shows the DAG corresponding to the HCD benchmark.
For the sake of simplicity, instead of showing the PolyMage program,
we summarized the computations at each stage in Table~\ref{tab:hcd}.
Table~\ref{tab:hcdbit} summarizes the
ranges and worst case precision requirements of various stages in HCD
as estimated by the range analysis algorithm. It can be noted from the
Table~\ref{tab:hcdbit} that the
bitwidth requirement of $det$ and $Harris$ stages is slightly greater than
the 32 bits required for a
floating-point computation. But the power, area and the delay associated
with fixed point arithmetic at this length could still be better than the
floating-point arithmetic operations.
\fi
\iffalse
The range associated
with each pixel of input image is $[0,255]$ as it is an 8-bit image. The
required integral bitwidth $\alpha_{img}$ at this stage is 8. Stages $I_x$
and $I_y$ compute the derivative of the image along $x$ and $y$ axes
respectively. The specific 3x3 stencil operations performed by these stages
can be noted from the Table~\ref{tab:hcd}. The stencil operation here
can be expanded in the form of an expression and the range associated with
it comes out to be $[-85,85]$. Eight integral bits are required to
represent this range. Stage $I_{xy}$ performs a point-wise computation
using the expression $I_{x,y}(i,j) = I_x(i,j)*I_y(i,j)$. Based on this,
we get the range of the variable $I_{x,y}$ as $[-85^2, 85^2]$.
Our compiler deduces that the computation at
the stages $I_{xx}$ and $I_{yy}$ is a squaring operation and appropriately
infers the output signal range as $[0, 85^2]$.
Stages $S_{xx}$ , $S_{xy}$ and $S_{yy}$ perform a stencil
operation involving only summation but no division. Expanding the stencil to
an expression form gives us simple addition of neighbouring pixels. The
range we get here for stage $S_{xy}$ is $[-9*85^2,9*85^2]$, and for stages
$S_{xx}$ and $S_{yy}$ is $[0, 9*85^2]$. In a similar fashion, the range
and the associated bitwidths for the stages $det$, $trace$ and $Harris$
can be inferred. It can be noted from the Table~\ref{tab:hcdbit} that the
bitwidth requirement of $det$ and $Harris$ stages is slightly greater than
the 32 bits required for a
floating-point computation. But the power, area and the delay associated
with fixed point arithmetic at this length could still be better than the
floating-point arithmetic operations. Table~\ref{tab:hcdbit} summarizes the
ranges and worst case precision requirements of various stages in HCD.
\fi
\subsection{Bitwidth Analysis Compilation Framework}\label{sec:ca}
The interval arithmetic based range analysis algorithm proposed in the
previous section uses the fact that all the pixel signals in each stage of an
image processing DAG are homogeneous in nature and groups them to do a
combined range analysis. However, other analysis techniques such as those
based on affine arithmetic cannot be applied on the DSL level programs in the
same fashion. In this section, we show how interval and affine arithmetic based
range analysis techniques can be deployed with ease in the PolyMage compilation
framework.
Recall that the PolyMage-HLS compiler translates a DSL program into C++ code
which the Xilinx Vivado HLS compiler synthesizes into an equivalent circuit
for a target FPGA. For example, Listing~\ref{lst:sobel} depicts the C++ code
generated by the PolyMage-HLS compiler when Sobel-x filter is applied on an
input image. The generated C++ program can be run in a purely simulation
mode after compilation on any processor by providing test input images as
stimulus. It can be noted from the Listing~\ref{lst:sobel}, that the data type
of the stream, line and window buffers are parameterized by the type
{\tt\bf typ}. It can be a float or any fixed-point data type $(\alpha, \beta)$.
During the hardware synthesis or in the simulation mode, using the C++
polymorphism feature, corresponding libraries for the arithmetic operations
will be invoked based on the operand types.
Now, the parameter {\tt\bf typ} can also be set to an interval type which is
defined in a suitably chosen interval analysis library. If the generated C++
program contains a statement $x=y+z$, then depending on the type of the
variables $x$, $y$ and $z$ (like float, ia-type, aa-type etc.), appropriate
addition operation will be invoked. For example, in order
to perform affine arithmetic analysis on the Sobel-x program, using the
Yet Another Library for Affine Arithmetic (YalAA)~\cite{yalaa}, all we have to
do is to define the parameter {\tt\bf typ} appropriately as depicted in
Listing~\ref{lst:yalaa}.
\begin{table}[t]
\noindent\begin{minipage}[t]{.47\textwidth}
\begin{lstlisting}[language=C,label={lst:sobel},
breaklines=true,
basicstyle=\footnotesize,
numbers=left,
keywordstyle=\sffamily\bfseries\color{green!40!black}, commentstyle=\itshape\color{purple!40!black},
morekeywords={typ}, identifierstyle=\color{blue},frame=single
caption=Auto-generated restructured HLS code for Sobel-x.]
#include <hls_stream.h>
#include <malloc.h>
#include <cmath>
#include <arith.h>
void sobel_x(hls::stream<typ> & img,
hls::stream<typ> & sobel_x_out)
{
const int _ct0 = (2 + R);
const int _ct1 = (2 + C);
hls::stream<typ> Ix_out_stream;
hls::stream<typ> img_Iy_stream;
typ Ix_img_LBuffer[3][_ct1];
typ Ix_img_WBuffer[3][3];
typ Ix_img_Coeff[3][3];
/* Code for Sobel-x stage follows. */
}
\end{lstlisting}
\end{minipage}\hfill
\begin{minipage}[t]{.47\textwidth}
\begin{lstlisting}[language=C,label={lst:yalaa},breaklines=true,
basicstyle=\footnotesize\ttfamily,keywordstyle=\sffamily\bfseries\color{green!40!black}, commentstyle=\itshape\color{purple!40!black},
numbers=left,
morekeywords={typ}, identifierstyle=\color{blue}, frame=single,framexbottommargin=100pt
caption=Type definitions for Affine and Interval Analysis.]
// Switch for Affine analysis
#ifdef AFFINE
#include <yalaa.hpp>
typedef yalaa::aff_e_d typ;
#endif
// Switch for Interval analysis
#ifdef INTERVAL
#include <Easyval.hpp>
typedef Easyval typ;
#endif
\end{lstlisting}
\end{minipage}
\end{table}
When we run
the generated C++ program with this data type definition, the value
associated with each pixel in each stage of the pipeline DAG is its affine
signal value which contains the base signal and the coefficients for the noise
variables. From this the range of every pixel can be derived as explained
before. If the data type corresponds to interval arithmetic, then the value
associated with each pixel is an interval. Using this approach, any kind
of interval analysis technique can be deployed in the PolyMage-HLS compiler
easily without re-architecting the analysis backend.
In the next section, we show how interval arithmetic based techniques can
fare really poorly if the benchmarks contain certain kinds of computational
patterns; we use an Optical Flow algorithm as an example. We then propose our new range analysis technique using SMT solvers.
\iffalse
HAVE TO WORK ON THE NEXT PARA
In the next section, we show that the integral bitwidth requirements at
various stages could be smaller than the estimates obtained
through static analysis. This could be due to the nature of the input images
and/or certain correlation between computations on spatially proximal pixels.
Furthermore, we show how we can use simple binary search to compute the
required fractional bits via program profiling using an application-specific
error metric. In these estimations of integral and fractional bitwidth, we
can also exploit the fact that many image processing applications are
resilient to small arithmetic overflow/precision errors, to arrive at power
and area-efficient hardware designs by using practical estimates for data
range and precision.
\fi
\input{files/smt}
\subsection{Profile-Driven Analysis}
\label{sec:profile}
Profile-driven analysis can be used to accomplish two tasks. First, we can obtain
lower bounds on the bitwidth estimates, which can be compared with the estimates
obtained using static analysis. Second, depending on the application, these
estimates
can be used in the actual system design instead of the conservative
estimates obtained through static analysis techniques. However, the
bitwidth requirements estimated at each stage using profiling naturally
depend on the sample input images. Based on the analysis done by Torralba et
al. \cite{torralba2003statistics}, we hypothesize that the
images taken from a certain domain, like for example {\it nature}, has
similar properties, and hence the bitwidth estimates can be carried over to
other images drawn from the same domain.
\subsubsection{Integral Bits}
The number of integral bits required at a stage $i$ denoted as $\alpha_i$ can
be obtained by running the input PolyMage program on a sample distribution of
input images. Let $\alpha_i^s$ be the maximum number of bits required by stage
$i$ to represent a pixel from an image sample $s$. Then the average number of
bits $\alpha_i^{avgP}$ required based on a sample set $S$ is $\sum_{s\in S}
\alpha_i^s/|S|$. Similarly, the worst-case number of bits $\alpha_i^{maxP}$
required is $max_{s\in S} \alpha_i^s$. We can either use $\alpha_i^{avgP}$ or
$\alpha_i^{maxP}$ as estimates for $\alpha_i$. Even if the estimate does not
suit certain images, in many application contexts, using saturation mode
arithmetic results in satisfying the desired output quality metric. Let
$\alpha_i^{IA}$ and $\alpha^{\smtra}$ be the integral bitwidth estimates
obtained for stage $i$ through interval analysis and \smtra\ analysis
respectively. For the benchmarks we have considered, affine analysis show some
improvements in the range estimates, but it amounts to same bitwidth requirement
as with interval analysis. Hence, throughout the rest of the paper, we consider
only interval analysis.
For our experimentation, we used a subset of 200 randomly chosen images from
the Oxford Buildings dataset~\cite{oxfordimages} consisting of 5062 images.
The set of 200 images is partitioned into two equal halves: training and
test sets. The training set is used to obtain estimates of integral
bitwidths at various stages through profiling. The test set is used to
evaluate the effectiveness of the bitwidth estimates obtained for quality
and power. Figure~\ref{fig:hs} shows the average cumulative distribution of
the bitwidth required by the integral part of the pixels in stages $I_x$
and $I_{xy}$ of the HCD benchmark on the training data set.
\iffalse
The x-axis of the graphs represent the bitwidth requirement of the integral
part, and y-axis represents the percentage of pixels, averaged across test
images, that can be represented within a given bitwidth.
\fi
For example, from Figure~\ref{fig:hs}, we can infer that in stage $I_x$,
95\% of the pixels require less than 5 bits, and all pixels (100\%) can be represented
using 8 bits.
\iffalse
Since
stages $I_x$ and $I_y$ are of similar computational nature, we plot the
histogram for only $I_x$. Similarly, among $I_{xx}$, $I_{xy}$ and $I_{yy}$,
we plot for $I_{xy}$ in Figure~\ref{fig:hs2}, and from stages $S_{xx}$,
$S_{xy}$ and $S_{yy}$, we plot for $S_{xy}$ in Figure~\ref{fig:hs3}.
Figures~\ref{fig:hs4},~\ref{fig:hs5} and~\ref{fig:hs6} correspond to {\it
det}, {\it trace} and {\it Harris} stages respectively.
Table~\ref{tab:bitTableHCD},~\ref{tab:bitTableUSM_DUS} shows the bitwidth estimates obtained from
static and profile-driven analyses.
\fi
Table~\ref{tab:bitTableHCD} shows the bitwidth estimates obtained from
static and profile-driven analyses for the HCD benchmark.
\begin{table*}[h!]
\footnotesize
\caption{Comparison of integral bitwidth estimates using interval, \smtra,
and profile-guided analyses for HCD. Fractional bitwidth estimates are also provided in the last row.}
\vskip 5pt
\label{tab:bitTableHCD}
\centering
\begin{tabular}{c | c c c c c c c c c}
\toprule
& &&&&HCD&&&&\\
Stage&Img&I$_x$,I$_y$&I$_{xx}$,I$_{yy}$&I$_{xy}$&S$_{xy}$&S$_{xx}$, S$_{yy}$&det&trace&harris\\
\midrule
$\alpha^{\smtra}$ & 8&8&13&14&17&16&33&17&\bf{33}\\
$\alpha^{IA}$ & 8&8&13&14&17&16&33&17&34\\
\midrule
$\alpha^{maxP}$& 8&8&13&14&17&16&30&17&29\\
$\alpha^{avgP}$& 8&8&13&14&17&16&29&17&29\\
\midrule
$\beta$ & 8&5&4&4&3&3&1&1&1 \\
\bottomrule
\end{tabular}
\end{table*}
As can be noted from
Table~\ref{tab:bitTableHCD}, the bitwidth estimates from $\alpha^{avgP}$
and $\alpha^{maxP}$ measures are the same for all stages
except for the {\it det} stage.
The estimates from the static analysis techniques match the profile estimates
except for the {\it det}, {\it trace} and {\it harris} stages. In general, we
expect the profile estimates to be better for stages that occur deeper in
the pipeline. Unlike Optical Flow benchmark, for HCD, \smtra\ analysis
performs no better than interval analysis except for a single bit
improvement in stage {\it Harris}. Again, we note that the profile estimates
also indicate the limit to which the static analysis techniques can be
improved by using more powerful approaches. Profile information can be
easily
obtained by executing the HLS C++ program directly without the need for a
heavy weight circuit simulation.
In the next section, we propose a simple and practical greedy search
algorithm to estimate the number of fractional bits at each stage of the DAG
while respecting an application specific quality constraint.
\iffalse
\begin{figure}[hb]
\centering
\resizebox{\columnwidth}{!}{%
\harrisHist
}%
\vskip 5pt
\caption{Cumulative distribution of pixels with respect to maximum integral bitwidth length at
stages $I_x$ and $I_{xy}$ of the HCD benchmark.}
\label{fig:hs}
\end{figure}
\fi
\subsubsection{Fractional Bits ($\beta$) Analysis}\label{sec:precision}
The number of fractional bits $\beta_i$ required at a stage $i$ depends on the
application and cannot be estimated in an application independent manner
similar to the integral bitwidth analysis. Estimating the optimal number of
fractional bits at each stage for a given application metric turns out to be a
non-convex optimization problem in most cases and hence we propose a simple
heuristic search technique that requires a very small number of profile
passes.
In the profiling technique, we fix the number of integral bits required at
each stage based on static or profile-driven analysis and increase the
precision
$\beta$ uniformly across all stages. For each value $\beta$, we estimate
the application-specific error metric. For the HCD benchmark, the error
metric is the percentage of misclassified corners when compared to a design
that uses sufficiently long integral and fractional bits. We can reach an
optimal $\beta$ for a given error tolerance via binary search. Then we make
a single pass on the stages of the DAG in reverse topologically sorted
order.
At each stage $I$, we do a binary search on the number of fractional bits
required, $\beta_I$, starting from the initial estimate $\beta$ while
retaining the application specific quality requirement. The last row
of the Table~\ref{tab:bitTableHCD} shows the fractional bits estimated
at each stage of the HCD benchmark. Note that the later stages of the DAG
require fewer bits than those stages which occur earlier in the DAG.
This is due to the fact that errors in earlier stages will have a
greater impact as they get propagated to the downstream stages. Further,
our greedy algorithm is optimizing the bitwidths by considering the
stages in the reverse topologically sorted order.
\input{files/flow}
\subsection{Summary of Bitwidth Analysis Framework}
Figure~\ref{fig:flow} summarizes the proposed bitwidth analysis framework.
We can use the PolyMage-HLS compilation framework first to do a range analysis and
estimate the integral bitwidths; then use the greedy heuristic to estimate the
fractional bits required at various stages. For range analysis, we can use one of interval,
\smtra\ and profile analysis techniques. For interval analysis, the compiler generates
HLS code where the data types of the variables at various stages of the DAG are intervals.
Then the bitwidth estimates are obtained using the intervals obtained by running the HLS
code. For profile analysis, the compiler generates HLS code wherein the data types of
the variables are of fixed point type with sufficiently large integral and fractional
bitwidths. Then HLS code is run on a sample distribution of input images to arrive at
integral bitwidth estimates. For \smtra\ analysis, the compiler generates a
constraint system which is solved by an SMT solver, such as Z3, to arrive at
range estimates.
\iffalse
\begin{table*}[h!]
\footnotesize
\caption{Comparison of integral bitwidth estimates using interval, \smtra,
and profile-guided analyses for USM and DUS. Fractional bitwidth estimates are also provided in the last row.}
\vskip 5pt
\label{tab:bitTableUSM_DUS}
\centering
\begin{tabular}{c | c c c c c | c c c c c }
\toprule
&&&USM&& &&&DUS&&\\
Stage&Img&blur$_x$&blur$_y$&sharpen&mask&Img&D$_x$&D$_y$&U$_x$&U$_y$\\
\midrule
$\alpha^{Z3RA}$ &8&8&8&10&9 &8&8&8&8&8\\
$\alpha^{IA}$ &8&8&8&10&9 &8&8&8&8&8\\
\midrule
$\alpha^{maxP}$ &8&8&8&10&9 &8&8&8&8&8\\
$\alpha^{avgP}$ &8&8&8&10&9 &8&8&8&8&8\\
\midrule
$\beta$ &0&2&3&4&4 &0&3&6&8&10\\
\bottomrule
\end{tabular}
\end{table*}
\fi
\section{Experimental Results}\label{sec:experiments}
In this section, we present a detailed area, power and throughput analysis when
variable fixed-point data types are used as against floating-point by
considering the following four benchmarks: Harris Corner Detection, Unsharp
Mask, Down and Up Sampling, and Optical Flow. Tables~\ref{tab:ofrange},
~\ref{tab:bitTableHCD},~\ref{tab:bitTableUSM} and~\ref{tab:bitTableDUS} show the integral bitwidth
estimates obtained through interval analysis ($\alpha^{IA}$), \smtra\ analysis
($\alpha^{\smtra}$) and profile analysis ($\alpha^{maxP}$ and $\alpha^{avgP}$);
and the average fractional bitwidth estimate ($\beta$) obtained through greedy
heuristic search algorithm. Table~\ref{tab:Allresults} compares the performance
of each benchmark using {\it float} data type and bitwidth estimates obtained
from different approaches. In these tables, the {\it Quality} column
corresponds to an application specific quality metric; the {\it Power}
column gives the power when the design operates at a speed specified in the
adjacent {\it Clk Period} column; {\it latency} columns provide the number
of
clock cycles required to process an HD image; the next four columns (BRAM,
DSP, FF, LUT, \%slices) summarize area usage; the {\it Min Clk Period}
column gives the maximum frequency of operation for circuit; and the next
two columns give the throughput and power consumed at the maximum frequency
of operation. Figure~\ref{fig:powersplit} gives the split of power usage by
various components of an FPGA. Unlike the Optical Flow benchmark, the integral
bitwidth estimates for the benchmarks HCD, USM and DUS using interval and
\smtra\ analysis techniques is the same. So we do not provide separate area,
power and throughput analysis for these benchmarks.
We used the Xilinx Zedboard consisting of Zynq-XC7Z020 FPGA device and
Xilinx Vivado Design Suite 2017.2 version to conduct our experiments. The
HLS design generated by our PolyMage DSL compiler is synthesized by the
Vivado HLS compiler. All characteristics are reported post Place and Route.
We ran C-RTL
co-simulations to generate switching activity (SAIF) file for reporting
detailed power consumption across the design.
\begin{table*}[t]
\footnotesize
\caption{Power, area and throughput analysis for HCD, DUS, USM and OF benchmarks using float and integral bitwidth estimates obtained using
static and profile-driven analyses. Fractional bitwidths are determined based on the greedy heuristic search approach.}
\vskip 5pt
\label{tab:Allresults}
\centering
\Ress
\end{table*}
\subsection{Harris Corner Detection}
Table~\ref{tab:bitTableHCD} summarizes the integral and fractional bitwidth estimates
obtained at each stage of the HCD benchmark through various analysis techniques.
The results in this table are commented upon in Sections~\ref{sec:profile}
and~\ref{sec:precision}. Figure~\ref{fig:uniformbeta} shows the average percentage of
pixels correctly
classified by the HCD benchmark on the test image set by varying the
fractional bits uniformly across all the stages while fixing the integral
bitwidth estimates obtained via profiling ($\alpha_i^{avgP}$). It also
contains estimates of power consumption with varying fractional bits for the
Xilinx ZED FPGA board.
It can be noted from the graph that the fractional bits do not affect the
accuracy of corner classification, and we thus get more than 99\% accuracy
even with zero fractional bits. From this graph, we infer that one can
obtain close to 100\% accuracy by using 8 fractional bits uniformly across
all the stages. We then make a backward pass on the stages of the
HCD benchmark to drop the fractional bits further without any significant loss
in accuracy and the row corresponding to $\beta$ in
Table~\ref{tab:bitTableHCD} shows the final fractional bitwidths. Due to space
constraint, we do not provide a graph such as Figure~\ref{fig:uniformbeta} for
the rest of the benchmarks. We can notice from Table~\ref{tab:Allresults} that by
using bitwidth estimates from interval analysis, we obtain 99.999\% accuracy with
a power consumption of 0.263~W. The power savings are
3.8$\times$ lower when compared with the floating-point design and 4\% more when
compared with the profile-estimate based design. The savings
on the percentage of FPGA slices used is around 6.2$\times$. From the last
3 columns of the table, we can notice that the fixed-point designs can operate
at a higher frequency achieving better throughput while consuming lesser power.
Figure~\ref{fig:powersplit} shows the detailed power analysis for floating-point and fixed-point design. It shows only the significant components of the dynamic power consumed, and in all the designs, the static power consumption is around 0.122~W.
\begin{figure}[!htb]
\centering
\begin{minipage}[t]{0.45\textwidth}
\centering
\resizebox{\linewidth}{!}{
\harrisProfPower
}
\caption{Error and power variation for the HCD benchmark by fixing the number of integral bits to profile estimated values $\alpha_i^{avgP}$ and varying $\beta$ uniformly across all the stages.}
\label{fig:uniformbeta}
\end{minipage}%
\hfill
\begin{minipage}[t]{0.45\textwidth}
\centering
\resizebox{\linewidth}{!}{
\includegraphics{./graphs/hcd_power_graph}
}
\caption{Power consumption by individual components on FPGA for HCD.\label{fig:powersplit}}
\end{minipage}
\end{figure}
\iffalse
\begin{figure}[htbp]
\centering
\resizebox{0.8\linewidth}{!}{
\harrisProfPower
\label{fig:haProf}
}
\vskip 5pt
\caption{Error and power variation for the HCD benchmark by fixing the number of
integral bits to profile estimated values $\alpha_i^{avg}$ and varying
$\beta$ uniformly across all the stages.}
\label{fig:uniformbeta}
\end{figure}
\fi
\begin{figure*}[!htb]
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{0.95\linewidth}{!}{
\usmDag
}
\caption{Pipeline DAG structure for USM benchmark.}
\label{fig:USMdag}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{\linewidth}{!}{%
\begin{tabular}{c | c c c c c }
\toprule
Stage&Img&blur$_x$&blur$_y$&sharpen&mask\\
\midrule
$\alpha^{Z3RA}$ &8&8&8&10&9 \\
$\alpha^{IA}$ &8&8&8&10&9 \\
\midrule
$\alpha^{maxP}$ &8&8&8&10&9 \\
$\alpha^{avgP}$ &8&8&8&10&9 \\
\midrule
$\beta$ &0&2&3&4&4 \\
\bottomrule
\end{tabular}
}%
\captionof{table}{Comparison of integral bitwidth estimates using interval, \smtra, and profile-guided analyses for USM. Fractional bitwidth estimates are also provided in the last row.}
\label{tab:bitTableUSM}
\end{minipage}
\end{figure*}
\subsection{Unsharp Mask (USM)}
The Unsharp Mask (USM) benchmark sharpens an input image and its computational
DAG is provided in Figure~\ref{fig:USMdag}. The input image is blurred across
x-axis and y-axis by the stencil stages {\it blurx} and
{\it blury} successively. Then it passes through
the {\it sharpen} stage, which is a point-wise computation. Finally, the
{\it masked} stage compares each pixel from the output of the {\it sharpen}
stage with a threshold value. Depending on whether the pixel value is less
than threshold, the corresponding pixel from either the original input image
or the sharpened image is chosen for output. We highlight an important
observation here: even if we make an error in computing a pixel value from
the {\it sharpen} stage, as long as it is less than the threshold, the right
output pixel is chosen. Based on this observation, we define an error metric
that is the fraction of pixels that were misclassified in the {\it masked}
stage due to variable width fixed-point representation as against
floating-point representation. We define a second quality metric that is the
root mean squared error between correctly classified pixel values and their
floating-point counterparts.
Table~\ref{tab:bitTableUSM} shows the integral and fractional bitwidths
required at various stages of the USM benchmark obtained from static (interval
and \smtra) and profile analyses. It can be noted that the estimates obtained by
the static and profile analyses are the same. Table~\ref{tab:Allresults} shows
that there is a factor of 1.6$\times$ improvement in power when compared to the
floating-point design with negligible root mean squared error and classification
error. With respect to the number of FPGA slices used, there is a factor of
2.6$\times$ improvement. Table~\ref{tab:Allresults} also shows the maximum
frequency of operation for each of the designs, the throughput at that level and
power consumption. From the last 3 columns of the table, we can infer that by
operating the fixed-point design at a higher frequency, 6\% increase in
throughput can be achieved while consuming 1.7x lower power.
\begin{figure*}[!htb]
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{0.95\linewidth}{!}{
\dusDag
}
\caption{Pipeline DAG structure for DUS benchmark.}
\label{fig:DUSdag}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
\centering
\resizebox{0.8\linewidth}{!}{%
\begin{tabular}{c | c c c c c }
\toprule
Stage&Img&D$_x$&D$_y$&U$_x$&U$_y$\\
\midrule
$\alpha^{Z3RA}$ &8&8&8&8&8\\
$\alpha^{IA}$ &8&8&8&8&8\\
\midrule
$\alpha^{maxP}$ &8&8&8&8&8\\
$\alpha^{avgP}$ &8&8&8&8&8\\
\midrule
$\beta$ &0&3&6&8&10\\
\bottomrule
\end{tabular}
}
\captionof{table}{Comparison of integral bitwidth estimates using interval, \smtra, and profile-guided analyses for DUS. Fractional bitwidth estimates are also provided in the last row.}
\label{tab:bitTableDUS}
\end{minipage}
\end{figure*}
\subsection{Down and Up Sampling (DUS)}
Down and Up Sampling (DUS) benchmark has a linear DAG structure as shown in Figure~\ref{fig:DUSdag}. The image is first
downsampled along the $x$-axis in stage D$_x$ and is further downsampled along the $y$-axis in stage D$_y$.
It is then upsampled again along the $x$ and $y$ axes in the stages $U_x$ and $U_y$
respectively. For the sake of conciseness, we avoid including the DUS
PolyMage code. All four stages comprise stencil computations.
The integral bitwidths estimated by both the interval and \smtra\ analyses is
equal to 8 at all the stages of DUS. We use the same set of training images as
that of HCD benchmark for estimating the integral and fractional bitwidths via
profiling. The profile estimates yielded the same integral bitwidth requirement
of 8 at all the stages. We use Peak Signal to Noise Ratio (PSNR) as a quality
metric where the reference image is obtained by using a sufficiently wide data
type. We set the required PSNR to infinity and the resulting fractional
bitwidths determined by our greedy precision analyzer is shown in the last row
of the Table~\ref{tab:bitTableDUS}. Table~\ref{tab:Allresults} shows that there
is a factor of 1.7$\times$ reduction in power using tuned fixed-point data types
when compared with using floating-point data type without loss of any accuracy.
With respect to area, there is a 4$\times$ improvement in terms of number of
slices used. Also, the fixed-point designs use no DSP blocks at all when
compared with floating-point design which uses 54 DSPs. At the peak possible
frequency of operation, fixed-point design achieves 13.6\% increase in
throughput while consuming 1.6x lesser power.
\subsection{Optical Flow (OF)}
The Optical Flow (OF) benchmark computes the velocity of individual pixels
from an image frame and its time-shifted version. Our implementation is
based on the Horn-Schunck algorithm \cite{horn1981determining} and consists
of 30 stages. The first 10 stages are pre-processing stages and the last 20
stages are obtained by repeating a set of five stages for four times. The
accuracy of motion estimation can be improved by repeating the 5-stage set
more times. Optical flow is a heavily used image processing algorithm in
many computer vision applications.
There have been many efforts in the past to implement optical flow on
FPGAs~\cite{JavierTCSVT'06,AlanISSC'10,EnZhuTCSVT'16} for power and
performance benefits.
Table~\ref{tab:ofrange} shows the estimated integral bitwidths required
at various stages of the Optical Flow benchmark. We notice that for stages
deeper in the pipeline, the difference between estimates obtained via interval
analysis and profiling are substantial. The profile estimates are obtained
from a training data set and for testing purpose, we use RubberWhale and
Dimetrodon image sequences from the Middlebury dataset~\cite{MBuryDataSet}.
Section~\ref{sec:smt} provides a detailed discussion on this and shows how
the \smtra\ analysis can overcome the inadequacies of the interval arithmetic
based analyses techniques and gives estimates which almost match profile estimates.
For computing the accuracy, we use the Average Angular Error (AAE) metric as
discussed in \cite{Fleet:1990},\cite{Otte:1994}. The reference motion
vectors are obtained by using sufficiently wide fixed-point data types at
all stages.
It can be noticed from Table~\ref{tab:Allresults} that by using bitwidth
estimates from Z3RA analysis, we obtain similar accuracy as profile-driven
analysis with a power consumption of 0.328~W. The power savings are 1.9$\times$
lower when compared with the floating-point design and 5.4\% more when compared
with the profile-estimate based design. The savings on the percentage of FPGA
slices used is around 2.5$\times$. From the last 3 columns of the table, we can
notice that the Z3RA fixed-point design can operate at a higher frequency
achieving 25\% more throughput than the floating point design while consuming
lesser power.
\iffalse
\begin{table*}[htbp]
\footnotesize
\caption{Metrics for the OF benchmark using float, and using bitwidths estimated using static analysis and profiling techniques.}
\vskip 5pt
\label{tab:OFresults}
\centering
\ofRes
\end{table*}
\begin{table*}[htbp]
\footnotesize
\caption{Metrics for USM benchmark using float, and using bitwidths estimated using static analysis and profiling techniques. }
\vskip 5pt
\label{tab:USMresults}
\centering
\usmRes
\end{table*}
\begin{table*}[htbp]
\footnotesize
\caption{Metrics for DUS using float, and using bitwidths estimated using static analysis and profiling techniques. }
\vskip 5pt
\label{tab:DUSresults}
\centering
\dusRes
\end{table*}
\fi
\iffalse
\begin{table}[tb]
\footnotesize
\centering
\caption{Comparison of bitwidth estimates using profiling technique and static analysis for the HCD benchmark.
\label{tab:hcdintegral}}
\vskip 5pt
\begin{tabularx}{0.8\linewidth}{l|c@{~~~~~~~~~~~}c@{~~~~~~~~~}c@{~~~~~~~~~}|c@{~~~~~~~~~~~}}
\toprule
Stage & $\alpha^{sa}$ & $\alpha^{max}$ & $\alpha^{avg}$ & $\beta^{avg}$ \\
\midrule
Img & 8 & 8 & 8 & 8\\
I$_x$,I$_y$ & 8 & 8 & 8 & 5\\
I$_{xx}$,I$_{yy}$ & 13 & 13 & 13 & 4\\
I$_{xy}$ & 14 & 14 & 14 & 4\\
S$_{xy}$ & 17 & 17 & 17 & 3\\
S$_{xx}$, S$_{yy}$ & 16 & 16 & 16 & 3\\
det & 33 & 30 & 29 & 1\\
trace & 17 & 17 & 17 & 1\\
harris & 34 & 29 & 29 & 1\\
\bottomrule
\end{tabularx}
\end{table}
\begin{figure*}[htbp]
\centering
\usmHist
\vskip 5pt
\caption{Cumulative distribution of pixels with respect to maximum integral bitwidth length at various stages of the USM.}
\label{fig:usm}
\end{figure*}
\begin{table}[tb]
\footnotesize
\centering
\caption{Comparison of bitwidth estimates using profiling technique and static analysis for the USM benchmark.
\label{tab:usmintegral}}
\vskip 5pt
\begin{tabularx}{0.87\linewidth}{l|c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}|c@{~~~~~}}
\toprule
Stage & $\alpha^{sa}$ & $\alpha^{max}$ & $\alpha^{avg}$ & $\beta^{avg}$ \\
\midrule
img & 8 & 8 & 8 & 0\\
blur$_x$ & 8 & 8 & 8 & 2\\
blur$_y$ & 8 & 8 & 8 & 3\\
sharpen & 10 & 10 & 10 & 4\\
mask & 9 & 9 & 9 & 4\\
\bottomrule
\end{tabularx}
\end{table}
\begin{table}[!htb]
\footnotesize
\centering
\caption{Comparison of bitwidth estimates using profiling technique and static analysis for the DUS benchmark.
\label{tab:dusintegral}}
\vskip 5pt
\begin{tabularx}{0.80\linewidth}{l|c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}c@{~~~~~~~~~~~}|c@{~~~~~~~}}
\toprule
Stage & $\alpha^{sa}$ & $\alpha^{max}$ & $\alpha^{avg}$ & $\beta^{avg}$ \\
\midrule
img & 8 & 8 & 8 & 0 \\
D$_x$ & 8 & 8 & 8 & 3 \\
D$_y$ & 8 & 8 & 8 & 6 \\
U$_x$ & 8 & 8 & 8 & 8 \\
U$_y$ & 8 & 8 & 8 & 10 \\
\bottomrule
\end{tabularx}
\end{table}
\fi
\iffalse
\begin{figure*}[htbp]
\centering
\subfloat[Throughput]{
\includegraphics[width=0.40\linewidth]{graphs/throughput}
}
\subfloat[Power]{
\includegraphics[width=0.40\linewidth]{graphs/power}
}
\vskip 5pt
\caption{Throughput and respective power obtained with various FPGA design benchmarks\label{fig:throughput}}
\end{figure*}
\begin{figure}[t]
\centering
\resizebox{0.8\columnwidth}{!}
\dusDag
}
\vskip 5pt
\caption{Pipeline DAG structure for DUS benchmark.} \label{fig:DUSdag}
\end{figure}
\iffalse
\begin{figure}[htbp]
\centering
\resizebox{0.8\linewidth}{!}{
\harrisProfPower
\label{fig:haProf}
}
\iffalse
\subfloat[USM]{
\usmProfPower
\label{fig:usmProf}
} \\
\subfloat[DUS] {
\dusProfPower
\label{fig:dusProf}
}
\subfloat[OF]{
\ofProfPower
\label{fig:ofProf}
}
\fi
\vskip 5pt
\caption{Error and power variation for the HCD benchmark by fixing the number of
integral bits to profile estimated values $\alpha_i^{avg}$ and varying
$\beta$ uniformly across all the stages.}
\label{fig:uniformbeta}
\end{figure}
\fi
\begin{figure
\centering
\resizebox{0.8\linewidth}{!}{
\includegraphics{./graphs/hcd_power_graph}
\iffalse
\subfloat[USM]{\includegraphics{./graphs/usm_power_graph}
\label{fig:usmsplit}
}\\
\subfloat[DUS]{\includegraphics{./graphs/dus_power_graph}
\label{fig:dussplit}
}
\subfloat[OF]{\includegraphics{./graphs/of_power_graph}
\label{fig:ofsplit}
}
\fi
}
\vskip 5pt
\caption{Power consumption by individual components on FPGA for HCD.\label{fig:powersplit}}
\end{figure}
\fi
\iffalse
\begin{figure
\centering
\resizebox{0.8\linewidth}{!}{
\subfloat[HCD]{
\includegraphics{./graphs/hcd_power_graph}
}
\subfloat[USM]{\includegraphics{./graphs/usm_power_graph}
\label{fig:usmsplit}
}\\
\subfloat[DUS]{\includegraphics{./graphs/dus_power_graph}
\label{fig:dussplit}
}
\subfloat[OF]{\includegraphics{./graphs/of_power_graph}
\label{fig:ofsplit}
}
}
\vskip 5pt
\caption{Power consumption by individual components on FPGA for HCD.\label{fig:powersplit}}
\end{figure}
\fi
\section{Conclusions}
\label{sec:conclusions}
The input, output and intermediate values generated in many image processing
applications have a limited range. Furthermore, these applications are
resilient to errors arising from factors such as limited precision
representation, inaccurate computations, and other potential noise sources. In
this work, we exploited these properties to generate power and area-efficient
hardware designs for a given image processing pipeline by using custom
fixed-point data types at various stages. We showed that domain-specific
languages facilitate the application of interval and affine arithmetic
analyses on larger benchmarks with ease. Further, we proposed a new
range analysis technique using SMT solvers, which overcomes the inherent
limitations in conventional interval/affine arithmetic techniques, when
applied to iterative algorithms. The proposed SMT solver based range
analysis technique also uses the DSL specification of the program to reduce
the number of constraints and variables in the constraint system, thereby
making it a feasible technique to adopt in practice. Then, we compared the
effectiveness of the static analysis techniques against a profile-driven
approach that automatically takes into account properties of input image
distribution and any correlation between computations on spatially proximal
pixels. In addition, the analysis revealed the limit of possible improvement
for any static analysis technique for integral bitwidth estimation.
Finally, to estimate the number of fractional bits, we used uniform
bitwidths across all the stages of the pipeline, and then used a simple
greedy search to arrive at a suitable bitwidth at each stage while
satisfying an application-specific quality criterion. Overall, the results
effectively demonstrate how information exposed through a high-level DSL
approach could be exploited in practical fixed-point data type analysis
techniques and to perform detailed impact studies on much larger image
processing pipelines than previously studied.
\iffalse
theoretical improvements we can
Our bitwidth estimation algorithms rely
In this paper, we proposed an interval arithmetic-based integral bitwidth
analysis algorithm for image processing pipelines. The algorithm is simple
and nicely fits into the data flow analysis framework used in compilers.
Then, we compared the effectiveness of the static analysis algorithm against
a profile-driven approach that automatically takes into account
properties of input image distribution and any correlation between
computations on spatially proximal pixels. For precision analysis, to
estimate fractional bits, we use uniform bitwidths across all the stages of
the pipeline, and then use a simple search to arrive at a suitable bitwidth
that satisfies an application-specific quality criterion. In conclusion,
our experimental evaluation demonstrates that by using variable bitwidth
fixed-point numbers at various stages of an image processing pipeline, get
substantial power and area savings can be obtained without a loss in desired
quality.
\fi
\iffalse In this paper, we proposed an automatic precision analysis algorithm
which in a very simple and efficient manner leverages the power and area
savings, as demonstrated on various image processing benchmarks. The algorithm
nicely fits into the data flow analysis framework used in compilers. We
developed an interval arithmetic based static algorithm for range analysis.
Then, we compared the effectiveness of the static analysis algorithm against a
profile-driven approach that automatically takes into account properties of
input image distribution and any correlation between computations on spatially
proximal pixels and showed that we were able to perform at par with static
precision using lesser area and power. For precision analysis, to estimate
fractional bits, we proposed a profiling based approach that uses uniform bit-
widths across all the stages of the pipeline, and then use a simple search to
arrive at a suitable bitwidth that satisfies an application-specific quality
criterion, which we carefully arrived at to compare the performance of our
algorithm. Finally, we demonstrate that by using the precision obtained using
our analysis, we we can get substantial power and area savings without a loss in
desired quality.
\fi
\section{Acknowledgments}
\label{sec:acknowledgments}
We would like to gratefully acknowledge the Science and Engineering Research Board (SERB), Government of India for funded this research work in part through a grant under its EMR program (EMR/2016/008015).
\bibliographystyle{ACM-Reference-Format}
|
{
"timestamp": "2018-12-20T02:03:19",
"yymm": "1803",
"arxiv_id": "1803.02660",
"language": "en",
"url": "https://arxiv.org/abs/1803.02660"
}
|
\section{Introduction}
Recent years have seen tremendous experimental advances in the nascent
field of strongly-coupled light-matter systems~\cite{ebbesen2016, sukharev2017}. In particular, new experimental advances have been demonstrated in polaritonic
chemistry~\cite{george2015,hiura2018,thomas2019}, solid-state physics~\cite{riek2015}, biological
systems~\cite{coles2017}, nanoplasmonics~\cite{chikkaraddy2016,benz2016}, two-dimensional materials~\cite{kleemann2017, bisht2019} or optical waveguides~\cite{mirhosseini2018}, among others.
In this so-called strong-coupling regime, as a result of mixing matter and photon degrees-of-freedom~\cite{ruggenthaler2017b,flick2018b}, novel effects emerge such as changes in chemical pathways~\cite{galego2015,galego2016,kowalewski2016} ground-state electroluminescence~\cite{cirio2016}, cavity-controlled chemistry for molecular ensembles~\cite{herrera2016, galego2017}, or optomechanical coupling in optical cavities~\cite{roelli2015}, new topological phases of matter~\cite{shin2018}, superradiance~\cite{mazza2019} or superconductivity~\cite{sentef2018}.
Due to the inherent complexity of such coupled fermion-boson problems described in general by quantum electrodynamics (QED), the theoretical treatment is usually drastically simplified. One common approximation is to restrict the description of the system to simplified effective models that heavily rely on input parameters. Current state of the art in the theoretical description of strong light-matter coupling very often employs a few-level approximation. This approximation leading to the Rabi or Jaynes-Cummings model~\cite{braak2011, xie2016} in the single-emitter case, or the Dicke model~\cite{garraway2011} in the many-emitter case, is however often not sufficient~\cite{rabl2018,schaefer2018}, in particular when observables besides the energy are of interest~\cite{schaefer2018}, such as in experimental setups involving the modification of chemical reactivity~\cite{ebbesen2016}.
Alternatively, in linear spectroscopy, the current theoretical description is built on the \textit{semi-classical} approximation~\cite{gilbert2010}. Herein, the many-particle electronic system is treated quantum mechanically and the electromagnetic field appears as an external perturbation. As an external perturbation, the electromagnetic field probes the quantum system, but is not a dynamical variable of the complete system ({see also supplemental material ~\ref{sec:semi-classics}}). Since in the strong-coupling regime light and matter {must be} on the same level, a semi-classical approximation is not adequate and the feedback between light and matter has to be considered.
{It is, however, long known that the radiative lifetimes are finite.} {Furthermore, experimentally excited-state properties are usually inferred from (de)excitations of the photon field, which is in stark contrast to the usual semi-classical theoretical description based solely on the electronic subsystem.}
In free-space, {this mismatch can be circumvented since excited-state properties such as} radiative lifetimes of atoms and molecules can be calculated {perturbatively} using the theory of Wigner-Weisskopf~\cite{weisskopf1930} employing the Markov approximation. {However, this perturbative treatment of the coupling of light and matter becomes insufficient in the case that strong light-matter coupling is achieved, e.g., due to many emitters or due to reducing the mode volume of a cavity. In such cases} the Markov approximation breaks down and the Wigner-Weisskopf theory is not applicable anymore~\cite{buzek1999}. Additionally it is not straightforward how to extend the original formulation of Wigner-Weisskopf to many electronic levels and hence {to an ab-initio treatment of electronic} systems.
As a consequence, the current literature shows a large gap for situations, where light and matter is strongly coupled and observables such as excited-state densities, radiative lifetimes, or electron-photon correlated observables of interest. {A good example is} the control of the radiative lifetimes of single molecules~\cite{lettow2007,wang2017} by changing the environment. In such cases the properties of the many-body system are changed, e.g., the excitation energies and lifetimes are {strongly} modified. This happens because certain modes of the photon vacuum field are enhanced which can lead to a strong coupling of light with matter. {Alternatively, increasing the number of particles leads to an enhancement of the coupling due to the self-consistent back-reaction of matter onto the photon field and vice versa. It is important to realize that such changes are non-perturbative for the photon field as well as for the matter subsystem and hence need a self-consistent { implementation}. This fact is most pronounced in the appearance of polaritonic states and their influence on chemical and physical properties of matter~\cite{ebbesen2016,ruggenthaler2017b}.}
In this paper, we close this gap by presenting a practical and general framework that subsumes electronic-structure theory, nanoplasmonics, and quantum optics. We present a {new description that challenges our conception of light and matter as distinct entities~\cite{ruggenthaler2017a}} {and that expresses the excited states as modifications of the photon field}. {We do so by introducing a linear-response formalism {for coupled matter-photon systems}. This {formalism} leads naturally to modifications of Maxwell's equations and the ability to calculate radiative lifetimes in arbitrary photon environments, including free-space, high-Q optical cavity or nanoplasmonic structures.} {We make this approach practical by introducing a linear-response framework for quantum-electrodynamical density-functional theory (QEDFT)~\cite{ruggenthaler2017b,tokatly2013,ruggenthaler2014,ruggenthaler2015,flick2018b}. This development is specifically timely since QEDFT has now been successfully applied to real systems in equilibrium~\cite{flick2017c} -- which demonstrates the feasibility of ab-initio strong-coupling calculations -- yet an accurate and efficient approach to excited states within QEDFT has been missing. This work therefore furthermore closes a gap within the QEDFT framework.}
\section{Light-matter interaction in the long wavelength limit}
\label{sec:theory}
Our fundamental description of how the charged constituents of atoms, molecules and solid-state systems, i.e., electrons and positively charged nuclei, interact is based on QED~\cite{ryder1996, craig1998, spohn2004, ruggenthaler2017b}, thus the interaction is mediated via the exchange of photons. Adopting the Coulomb gauge for the photon field allows us to single out the longitudinal interaction among the particles which gives rise to the well-known Coulomb interaction and leaves the photon field purely transversal. Assuming then that the kinetic energies of the nuclei and electrons are relatively small, allows us to take the non-relativistic limit for the matter subsystem of the coupled photon-matter Hamiltonian, which gives rise to the so-called Pauli-Fierz Hamiltonian~\cite{spohn2004, ruggenthaler2014, ruggenthaler2017b} of non-relativistic QED. In a next step one then usually assumes that the combined matter-photon system is in its ground state such that the transversal charge currents are small and that the coupling to the (transversal) photon field is very weak. Besides the Coulomb interaction it is then only the physical mass of the charged constituents (bare plus electromagnetic mass~\cite{spohn2004}) that is a reminder of the photon field in the usual many-body Schr\"odinger Hamiltonian. In this work, however, we will not disregard the transversal photon field{, which makes the presented framework much more versatile and applicable to situation outside of standard quantum mechanics (see also appendix~\ref{app:lifetimes}).}
\subsection{Novel Spectroscopy from quantum description of light-matter interaction}
In the following, we consider cases, in which the semi-classical approximation breaks down, as outlined in the introduction. From the Pauli-Fierz Hamiltonian, we make the long-wavelength or dipole approximation in the length-gauge~\cite{rokaj2017} since the wavelength of the photon modes are usually much larger than the extend of the electronic subsystem which leads (in SI units) to~\cite{tokatly2013,ruggenthaler2014,pellegrini2015}~\footnote{In principle, QEDFT can be formulated for each level of theory of QED as presented in Ref.~\cite{ruggenthaler2014}. As a consequence, our formalism can be extended to more general formulations, including full minimal coupling, beyond the dipole approximation.}
\begin{align}
\label{eqn:h-dipole-2}
\hat{H}(t)&={\hat{H}_e}+\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha}+\omega^2_{\alpha}\left(\hat{q}_{\alpha}-\frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}} \cdot \textbf{R} \right)^2\right]+\frac{j_{\alpha}(t)}{\omega_\alpha}\hat{q}_\alpha,
\end{align}
{where $\hat{H}_e$ is the standard many-body electronic Hamiltonian}~\cite{szabo1989}. We further restrict ourselves to arbitrarily many but a finite number $M$ of modes $\alpha\equiv (\textbf{k},s)$ with $s$ being the two transversal polarization directions that are perpendicular to the direction of propagation $\textbf{k}$. The frequency $\omega_\alpha$ and polarization ${\boldsymbol \epsilon_\alpha}$ that enter in ${\boldsymbol \lambda_\alpha} = \boldsymbol \epsilon_\alpha \lambda_\alpha$ with $\lambda_\alpha = S_\textbf{k} (\textbf{r})/\sqrt{\epsilon_0}$ and mode function $S_\textbf{k} (\textbf{r})$ define these electromagnetic modes. $S_\textbf{k} (\textbf{r})$ is normalized, has the unit $1/\sqrt{\text{V}}$ with the volume $V$ and we choose a reference point $\textbf{r}_0$ where we have placed the matter subsystem to determine the fundamental coupling strength~\footnote{All results presented in this paper are independent of $\textbf{r}_0$.}. These photon modes couple via the displacement coordinate $\hat{q}_{\alpha} = \sqrt{\frac{\hbar}{2\omega_{\alpha}}}(\hat{a}_{\alpha} + \hat{a}_{\alpha}^{\dagger})$, where $\hat{q}_\alpha$ is given in terms of photon annihilation $\hat{a}_{\alpha}$ and creation $\hat{a}_{\alpha}^{\dagger}$ operators, to the total dipole moment $\textbf{R} = \sum_{i=1}^{N} e\textbf{r}_i$~\footnote{{Throughout this paper, we use the implicit definition $e=-|e|$.}}. The $\hat{q}_{\alpha}$ {appears in the contribution of mode $\alpha$ to the displacement field} $\hat{\vec{D}}_{\alpha} = \epsilon_0 \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \hat{q}_{\alpha}$~\cite{rokaj2017}. Further, the conjugate momentum of the displacement coordinate is given by $\hat{p}_{\alpha} = -i\sqrt{\frac{\hbar\omega_{\alpha}}{2}}(\hat{a}_{\alpha} - \hat{a}_{\alpha}^{\dagger})$. Besides a time-dependent external potential $v(\textbf{r},t)$, we also have an external perturbation $j_{\alpha}(t)$ that acts directly on the mode $\alpha$ of the photon subsystem. Here $j_{\alpha}(t)$ is connected to a classical external charge current $\textbf{J}(\textbf{r},t)$ that acts as a source for the inhomogeneous Maxwell's equation.
Formally, however, due to the length-gauge transformations, the $j_{\alpha}(t)$ corresponds to the time-derivative of this (mode-resolved) classical external charge current~\cite{tokatly2013,ruggenthaler2014} {(see also appendix~\ref{app:Maxwell})}. Physically the static part $j_{\alpha,0}$ merely polarizes the vacuum of the photon field and leads to a static electric field~\cite{dimitrov2017, ruggenthaler2015}. The time-dependent part $\delta j_{\alpha}(t)$ then generates real photons in the mode $\alpha$. This term is also known as a source term in quantum field theory~\cite{ryder1996}, where it generates the particles (here the photons) that are studied. From this perspective it becomes obvious that instead of using $\delta j_{\alpha}(t)$ one could equivalently slightly change the initial state of the fully coupled system by adding incoming photons that then scatter off the coupled light-matter ground state~\cite{spohn2004}.
\subsection{Linear Response in the Length Gauge}%
With the Hamiltonian of Eq.~(\ref{eqn:h-dipole-2}) in length gauge we can then in principle solve the corresponding time-dependent Schr\"odinger equation (TDSE) for a given initial state of the coupled matter-photon system $\Psi_0(\vec{r}_1 \sigma_1,...,\vec{r}_N \sigma_N, q_{1},...,q_M)$
\begin{align}
\label{eqn:TDSE}
\i \hbar \frac{\partial}{\partial t} \Psi(\vec{r}_1 \sigma_1,...,t) = \hat{H}(t)\Psi(\vec{r}_1 \sigma_1,...,t),
\end{align}
where $\sigma$ correspond to the spin degrees-of-freedom. However, instead of trying to solve for the infeasible time-dependent many-body wave function, we restrict ourselves to weak perturbations $\delta v(\vec{r},t)$ and $\delta j_{\alpha}(t)$ and assume that our system is in the ground state of the coupled matter-photon system initial time. In this case, first-order time-dependent perturbation theory can be used to approximate the dynamics of the coupled matter-photon system ({for details see supplemental material ~\ref{app:linresp}}). This framework gives us access to linear spectroscopy, e.g., the absorption spectrum of a molecule. Traditionally, if we made a decoupling of light and matter, i.e., we assumed $\Psi_0(\vec{r}_1 \sigma_1,...,\vec{r}_N \sigma_N, q_{1},...,q_M) \simeq \psi_0(\vec{r}_1 \sigma_1,...,\vec{r}_N \sigma_N) \otimes \varphi_0(q_{1},...,q_M)$, we would only consider the matter subsystem $\psi$ (the photonic part $\varphi$ would be completely disregarded). Physically, we would investigate the classical dipole field that the electrons induced due to a classical external perturbation $\delta v(\vec{r},t)$. To determine this induced dipole field we would only consider the linear response of the density operator $\hat{n}(\vec{r}) = \sum_{i=1}^{N}\delta(\vec{r} - \vec{r}_i)$ which would be given by the usual density-density response function in terms of the electronic wave function $\psi_0$ only~\footnote{In the following, we suppress the spin component of the wave function and focus exclusively on the spatial and mode dependence, i.e., $\Psi(\vec{r}_1 ,...,\vec{r}_N , q_{1},...,q_M;t)$.}
In this work however, since we do not assume the decoupling of light and matter, the full density-density response is taken with respect to the combined ground-state wave function $\Psi_0$ and is consequently different to the traditional density-density response. Further, since we can also perturb the photon field in the cavity by $\delta j_{\alpha}(t)$ which will subsequently induce density fluctuations, the density response $\delta n$ gets a further contribution leading to
\begin{align}
\delta n(\textbf{r}t) = &\int dt' \int d\textbf{r}' \chi^n_n(\textbf{r}t,\textbf{r}'t') \delta v(\textbf{r}'t')\label{eq:delta_n1}\\
&+ \sum_{\alpha=1}^{M} \int dt' \chi^n_{q_\alpha}(\textbf{r}t,t') \delta j_\alpha(t')\nonumber.
\end{align}
Here the response function $\chi^n_n(\textbf{r}t,\textbf{r}'t')$ corresponds to the density-density response but with respect to the coupled light-matter ground state and $\chi^n_{q_\alpha}(\textbf{r}t,t')$ corresponds to the response induced by changing the photon field. In the standard linear-response formulation, due to the decoupling ansatz, changes in the transversal photon field would not induce any changes in the electronic subsystem. Since obviously we now have a cross-talk between light and matter, we accordingly have also a genuine linear-response of the quantized light field
\begin{align}
\delta q_\alpha(t) = &\int dt' \int d\textbf{r}' \chi^{q_\alpha}_{n}(t,\textbf{r}'t') \delta v(\textbf{r}'t')\label{eq:delta_q1}
\\
&+ \sum_{\alpha'=1}^{M}\int dt' \chi^{q_\alpha}_{q_{\alpha'}}(t,t')\delta j_{\alpha'}(t'),\nonumber
\end{align}
where $\chi^{q_\alpha}_{n}(t,\textbf{r}'t')$ is the full response of the photons due to perturbing the electronic degrees, and $\chi^{q_\alpha}_{q_\alpha'}(t,t')$ is the photon-photon response function. The response function $\chi^{q_\alpha}_{n}(t,\textbf{r}t')$ is in general not trivially connected to $\chi_{q_\alpha}^{n}(\textbf{r}t,t')$, due to the different time-ordering of $t$ and $t'$.
The entire linear-response in non-relativistic QED for the density and photon coordinate can also be written in matrix form~\cite{hoffmann2016}. In this form we clearly see that the density response of the coupled matter-photon system depends on whether we use a classical field $\delta v(\vec{r},t)$, photons, which are created by $\delta j_{\alpha}(t)$, or combinations thereof for the perturbation. Furthermore, we can also decide to not consider the classical response of the coupled matter-photon system due to $\delta n(\vec{r},t)$, but rather directly monitor the quantized modes of the photon field $\delta q_{\alpha}(t)$. This response yet again depends on whether we choose to use a classical field $\delta v(\vec{r},t)$ that induces photons in mode $\alpha$ or whether we directly generate those photons by an external current $\delta j_{\alpha}(t)$. And we also see that the different modes are coupled, i.e., that photons interact. Similarly as charged particles interact via coupling to photons, also photons interact via coupling to the charged particles.
Keeping the coupling to the photon field explicitly therefore, on the one hand, changes the standard spectroscopic observables, and on the other hand also allows for many more spectroscopic observables than in the standard matter-only theory.
\subsection{Maxwell-Kohn-Sham linear-response theory}%
\begin{figure*}
\centerline{\includegraphics[width=\textwidth]{tddft-qedft.pdf}}
\caption{Schematics of the {Maxwell} KS approach contrasted with schematics of the usual semi-classical {KS} theory. While in the semi-classical approach the {KS} orbitals are used as fixed input into the {mode-resolved} inhomogeneous Maxwell's equation in vacuum {through the total dipole $\textbf{R}(t) = \int d \textbf{r} \, {e}\vec{r} \, \sum_{i} |\varphi_i(\vec{r},t)|^2 $ (see also { appendix}~\ref{app:Maxwell})}, in the {Maxwell KS} framework the induced field acts back on the orbitals, which leads to an extra {self-consistency} cycle.}
\label{fig:change-maxwell}
\end{figure*}
The problem of this general framework in practice is that already in the simplified matter-only theory we usually cannot determine the exact response functions of a many-body system. The reason is that the many-body wave functions, which we use to define the response functions, are difficult, if not impossible to determine beyond simple model systems. So in practice we need a different approach that avoids the many-body wave functions. Several approaches exist that employ reduced quantities instead of wave functions~\cite{fetter2003, stefanucci2013, bonitz1998}. The workhorse of these many-body methods is DFT and its time-dependent formulation TDDFT~\cite{dreizler2012, engel2011, ullrich2011}. Both theories have been extended to general coupled matter-photon systems within the framework of QED~\cite{ruggenthaler2017b, tokatly2013, ruggenthaler2014, ruggenthaler2015, flick2018a}.
QEDFT allows us to solve instead of the TDSE equivalently a non-linear fluid equation for the charge density $n(\vec{r},t)$ coupled non-linearly to the mode-resolved inhomogeneous Maxwell's equation~\cite{ruggenthaler2011, tokatly2013, ruggenthaler2014, ruggenthaler2015}. While these equations are in principle easy to handle numerically, we do not know the forms of all the different terms explicitly in terms of the basic variables of QEDFT, i.e. $(n(\vec{r},t), q_{\alpha}(t))$. To find accurate approximations one then employs the Kohn-Sham (KS) scheme, where we model the unknown terms by a numerically easy to handle auxiliary system in terms of wave functions. The simplest approach is to use non-interacting fermions and bosons which lead to a similar set of equations, which are however uncoupled. Enforcing that both give the same density and displacement field dynamics gives rise to mean-field exchange-correlation (Mxc) potentials and currents~\cite{flick2015, ruggenthaler2015b, dimitrov2017}. Formally this Mxc potential and current is defined as the difference of the potential/current that generate a prescribed internal pair in the auxiliary non-interacting and uncoupled system $(v_{\rm s}([n], \vec{r},t), j_{\alpha}^{\rm s}([q_{\alpha}],t))$ and the potential/current that generates the same pair in the physical system defined by Eq.~(\ref{eqn:h-dipole-2}) which we denote by $(v([n, q_{\alpha}], \vec{r},t),j_{\alpha}([n, q_{\alpha}],t))$, i.e.,
\begin{align}
v_{\rm Mxc}([n, q_{\alpha}], \vec{r},t) &= v_{\rm s}([n], \vec{r},t) - v([n, q_{\alpha}], \vec{r},t),
\\
j_{\alpha, {\rm M}}([n],t)& = j_{\alpha}^{\rm s}([q_{\alpha}],t) - j_{\alpha}([n,q_{\alpha}],t)
\label{m_current}\\
&= - \omega_{\alpha}^2 \boldsymbol{\lambda}_{\alpha}\cdot \textbf{R}(t). \nonumber
\end{align}
In the time-dependent case we only have a mean-field contribution to the Mxc current~\cite{tokatly2013, ruggenthaler2015} where the total dipole moment is written as $\textbf{R}(t) = \int d \textbf{r} \, {e}\vec{r} \, n(\vec{r},t) $. Further, we have ignored the so-called initial-state dependence because we assume (for notational simplicity and without loss of generality) in the following that we always start from a ground state~\cite{maitra2002, ruggenthaler2015b} of the matter-photon coupled system. In this way we can recast the coupled Maxwell-quantum-fluid equations in terms of coupled non-linear Maxwell-KS equations for auxiliary electronic orbitals, which sum to the total density $\sum_{i} |\varphi_i(\vec{r},t)|^2 = n(\vec{r},t)$, and the displacement fields $q_{\alpha}(t)$, i.e.,
\begin{align}
{i \hbar }\frac{\partial}{\partial t}\varphi_i(\textbf{r},t)=&\left[-\frac{\hbar^2}{2m_e}\vec{\boldsymbol\nabla}^2+ v_{\rm KS}([v, n, q_{\alpha}],\textbf{r},t)\right] \varphi_i(\textbf{r},t) ,
\label{KS-matter}
\\
\left(\frac{\partial^{2}}{\partial t^{2}} + \omega_{\alpha}^{2}\right) & q_{\alpha}(t) = -\frac{j_{\alpha}(t)}{\omega_{\alpha}} + \omega_{\alpha}\boldsymbol{\lambda}_{\alpha}\cdot \textbf{R}(t). \label{Max0}
\end{align}
Here we use the self-consistent KS potential $v_{\rm KS}([v, n, q_{\alpha}],\textbf{r},t) = v(\vec{r},t) + v_{\rm Mxc}([n, q_{\alpha}], \vec{r},t)$ that needs to depend on the fixed physical potential $v(\vec{r},t)$~\cite{ruggenthaler2015b}, and instead of the full bosonic KS equation for the modes $\alpha$ we just provide the Heisenberg equation for the displacement field. Although the auxiliary bosonic wave functions might be useful for further approximations it is only $q_{\alpha}(t)$ that is physically relevant and thus we get away with merely coupled classical harmonic oscillators, i.e., the mode resolved inhomogeneous Maxwell's equation. {To highlight the extra self-consistency due to coupling between light and matter we contrast the traditional electron-only KS theory with the Maxwell KS theory in Fig.~\ref{fig:change-maxwell}.} It is then useful to divide the Mxc potential into the usual Hartree-exchange-correlation (Hxc) potential that we know from electronic TDDFT and a correction term that we call photon-exchange-correlation {potential} (pxc), i.e.,
\begin{align*}
v_{\rm Mxc}([n, q_{\alpha}], \vec{r},t) = v_{\rm Hxc}([n], \vec{r},t) + v_{\rm pxc}([n, q_{\alpha}], \vec{r},t).
\end{align*}
Clearly, the correction term $v_{\rm pxc}$ will vanish if we take the coupling $|\boldsymbol{\lambda}_{\alpha}|$ to zero and recover the purely electronic case. Since by construction the Maxwell KS system reproduces the exact dynamics, we also recover the exact linear-response of the interacting coupled system ({see also supplemental material ~\ref{app:linresp1}}). We can express this with the help of the Mxc kernels defined by the functional derivatives of the Mxc quantities
\begin{align*}
f^{n}_\text{Mxc}(\textbf{r}t,\textbf{r}'t') &= \frac{\delta v_\text{Mxc}(\textbf{r}t)}{\delta n(\textbf{r}'t')}, \quad
f^{q_\alpha}_\text{Mxc}(\textbf{r}t,t') = \frac{\delta v_\text{Mxc}(\textbf{r}t)}{\delta q_\alpha(t')}, \\
g^{n_\alpha}_\text{M}(t,\textbf{r}'t') &= \frac{\delta j_{\alpha,\text{M}}(t)}{\delta n(\textbf{r}'t')}, \quad
g^{q_{\alpha'}}_\text{M}(t,t') = \frac{\delta j_{\alpha,\text{M}}(t)}{\delta q_{\alpha'}(t')} \equiv 0.
\end{align*}
and use the corresponding definitions for the Hxc kernel (that only for the variation with respect to $n$ has a non-zero contribution) and the pxc kernels. We note that using Eq.~(\ref{m_current}) we explicitly find
\begin{align}
g^{n_{\alpha}}_\text{M}(t-t',\textbf{r}) = -\delta (t-t') \, \omega_{\alpha}^2 \boldsymbol{\lambda}_{\alpha}\cdot {e}\textbf{r}.
\label{eq:gnm}
\end{align}
and $g^{q_{\alpha'}}_\text{M}(t,t')$ vanishes, since $j_{\alpha,M}$ in Eq.~(\ref{m_current}) has no functional dependency on $q_\alpha$.
Via these kernels we find with $\chi^{n}_{n, {\rm s}}(\vec{r}t,\vec{r}' t')$ and $\chi^{q_{\alpha}}_{q_{\alpha'}, {\rm s}}(t,t')$, where $\chi^{q_{\alpha}}_{q_{\alpha'}, {\rm s}}(t,t') \equiv 0$ for $\alpha \neq \alpha'$, the uncoupled and non-interacting response functions that
\begin{widetext}
\begin{align}
\label{eq:chi_nn}
\chi^n_n(\textbf{r}t,\textbf{r}'t') &=\chi^n_{n, {\rm s}}(\textbf{r}t,\textbf{r}'t') +\iint \text{d}\textbf{x} \text{d}\tau \chi^n_{n, {\rm s}}(\textbf{r}t,\textbf{x}\tau)\left(\iint \text{d}\tau'\text{d}\textbf{y}f^n_\text{Mxc}{(\textbf{x}\tau,\textbf{y}\tau')}\chi^n_n{(\textbf{y}\tau',\textbf{r}'t')}\right.
\\
&\left.\quad + \sum_\alpha \int \text{d}\tau'f^{q_\alpha}_\text{Mxc}{ (\textbf{x}\tau,\tau')}\chi^{q_\alpha}_n{(\tau',\textbf{r}'t')}\right), \nonumber
\\
\label{eq:chi_qn}
\chi_{q_{\alpha'}}^{q_{\alpha}}(t,t') &= \chi_{q_{\alpha',s}}^{q_{\alpha}}(t,t') + \sum_{\beta}\iiint d\tau d\tau'd\textbf{x} \; \chi_{q_{\beta,s}}^{q_{\alpha}}(t,\tau) g_{M}^{n_{\beta}}(\tau,\textbf{x}\tau') \chi_{q_{\alpha'}}^{n}(\textbf{x}\tau',t') ,
\end{align}
\end{widetext}
and accordingly for the mixed matter-photon response functions
\begin{widetext}
\begin{align}
\label{eq:chi_nq}
\chi_{q_{\alpha}}^{n}(\textbf{r}t,t') &= \iint d\tau d\textbf{x} \; \chi_{n,s}^{n}(\textbf{r}t,\textbf{x}\tau) \left(\iint d\tau' d\textbf{y}f_{\text{Mxc}}^{n}(\textbf{x}\tau,\textbf{y}\tau') \chi_{q_{\alpha}}^{n}(\textbf{y}\tau',t') + \sum_{\alpha'}\int d\tau' f_{\text{Mxc}}^{q_{\alpha'}}(\textbf{x}\tau,\tau') \chi_{q_{\alpha}}^{q_{\alpha'}}(\tau',t')\right) ,\\
\label{eq:chi_qq}
\chi_{n}^{q_{\alpha}}(t,\textbf{r}'t') &= \sum_{\beta}\iiint d\tau d\tau' d\textbf{y} \; \chi_{q_{\beta,s}}^{q_{\alpha}}(t,\tau) g_{M}^{n_{\beta}}(\tau,\textbf{y}\tau') \chi_{n}^{n}(\textbf{y}\tau',\textbf{r}'t') .
\end{align}
\end{widetext}
Here we employed the formal connection between response functions and functional derivatives $\chi^{n}_{n}(\vec{r} t,\vec{r}' t') = \delta n(\vec{r},t)/ \delta v(\vec{r}',t')$ as well as $\chi^{q_{\alpha}}_{q_{\alpha'}}(t,t') = \delta q_{\alpha}(t)/ \delta j_{\alpha'}(t')$ and accordingly for the auxiliary system. The Mxc kernels correct the unphysical responses of the auxiliary system to match the linear response of the interacting and coupled problem. So in practice, instead of the full wave function, what we need are approximations to the unknown Mxc kernels. Later we will provide such approximations, show how accurate they perform for a model system and then apply them to real systems. If we decouple light and matter, i.e., $\Psi_0 \simeq \psi_{0} \otimes \varphi_{0}$, and disregard the photon part $\varphi_0$ (as is usually done in many-body physics), we recover the response function of Eq.~(\ref{eq:chi_nn}) with $f^{q_{\alpha}}_{\text{Mxc}} \equiv 0$, and $f^{n}_{\text{Mxc}}\rightarrow f^{n}_{\text{Hxc}}$. The response function, which is calculated with the bare matter initial state $\psi_0$, then obeys the usual Dyson-type equation relating the noninteracting and interacting response in TDDFT~\cite{petersilka1996, casida1996} with $v_{\rm Mxc}([n, q_{\alpha}],\vec{r},t) \rightarrow v_{\rm Hxc}([n],\vec{r},t)$.
\subsection{{Excited states as properties of the photon field}}%
~\label{sec:Observables}
\begin{figure*}
\centerline{\includegraphics[width=\textwidth]{maxwell.pdf}}
\caption{
{Schematics that contrasts the usual Maxwell's equation (left) with the fully self-consistent Maxwell's equation (right). Top: The induced transversal electric field $\textbf{E}_{\perp}$ as a consequence of the induced polarization $\textbf{P}_{\perp}$, which can be equivalently expressed in terms of the auxiliary displacement field $\textbf{D}_{\perp}$. Left: mode-resolved non-self-consistent Maxwell's equation with no backreaction. The external charge current $\textbf{j}_{\alpha}$ induces the external electric field in $\textbf{E}_{\alpha}^{\textrm{tot}}=\textbf{E}_{\alpha}+\textbf{E}_{\alpha}^{\textrm{ext}}$ which acts as an external perturbation through the dipole. Since the constituents of $\tilde{\chi}^n_n$ expressed in TDDFT are purely electronic, the induced field does not couple back to the Maxwell field. Right: self-consistent Maxwell's equation in which $\textbf{j}_{\alpha}$ induces the internal field $q_{\alpha}(t)$ through the electron-photon correlated dipole which has an explicit dependence as seen in the QEDFT form of $\chi_{q_{\alpha}}^{n}$. The self-consistency of the induced field through the dipole introduces nonlinearities in the coupled system thus changes the Maxwell field at the level of linear-response.} }
\label{fig:maxwell}
\end{figure*}
Following the above discussion, the usual response functions will change and novel response functions are introduced if we keep the matter-photon coupling explicitly. {This leads to many exciting consequences. Firstly, we get the completely self-consistent response of the system including all screening, retardation and other effects that become important when either the matter subsystem is becoming large~\cite{ehrenreich1966, mochan1985, maki1991,luppi2010} or when strong-coupling situations are considered. Since light and matter influence each other non-perturbatively the usual simplified approximations that only treat one part of the system accurately become unreliable~\cite{schaefer2018,rabl2018} (see also discussion in Sec.~\ref{sec:beyond}). Secondly, due to the matter-mediated photon-photon interactions (see appendix~\ref{app:Maxwell} and Fig.~\ref{fig:maxwell}) the usual Maxwell's equations are changed. A very interesting consequence is that in contrast to a purely classical theory we can theoretically distinguish whether a system is perturbed by a free current (that in turn would generate a classical electromagnetic field) or by a free electromagnetic field, e.g., a classical laser pulse. Thirdly, we rectify fundamental failings of standard quantum mechanics, such as the prediction of infinitely-lived excited states. The inclusion of the photon modes introduces the missing photon bath that leads to finite lifetimes (see appendix~\ref{app:lifetimes} and Sec.~\ref{sec:num-lifetime}). In connection to this it becomes important that we suddenly have access to a wealth of new observables that describe the photon field. Most importantly this implies the possibility to completely change our perspective of excited states of atoms and molecules. Indeed, in line with the experimental situation where changes in the photon field give us information on the excited states, we can view excited-state properties as arising from quantum modifications of the Maxwell's equations in matter}
\begin{align*}
\left(\frac{\partial^{2}}{\partial t^{2}} + \omega_{\alpha}^{2}\right)\delta q_{\alpha}(t) = -\frac{\delta j_{\alpha}(t)}{\omega_{\alpha}} + \omega_{\alpha}\boldsymbol{\lambda}_{\alpha}\cdot\int d\textbf{r} \; {e}\textbf{r} \delta n(\textbf{r},t).
\end{align*}
The response of the density is then found with help of the response functions Eqs.~(\ref{eq:chi_nn})-(\ref{eq:chi_qq}). In the usual case of an external classical field $\delta v(\vec{r},t)$ and $\delta j_\alpha(t) = 0$ we then find the induced field by (suppressing detailed dependencies with $\int d\textbf{r}\rightarrow \int $ and $\int d\textbf{r} \sum_\alpha \rightarrow\SumInt $)
\begin{widetext}
\begin{align}
\left(\frac{\partial^{2}}{\partial t^{2}} + \omega_{\alpha}^{2}\right)\delta q_{\alpha}(t) = \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \int {e}\vec{r} \; \chi^n_{n, {\rm s}} \delta v + \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \int {e}\vec{r} \; \chi^n_{n, {\rm s}} f^{n}_\text{Mxc}\chi^{n}_{n}\delta v + \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \SumInt {e}\vec{r} \; \chi^n_{n, {\rm s}} f^{q_{\alpha'}}_\text{Mxc} \delta q_{\alpha'} . \label{ v(r,t)}
\end{align}
\end{widetext}
Here the first term on the right-hand side corresponds to the non-interacting matter-response. However, due to the electron-electron interaction we need to take into account also the self-polarization of interacting matter (second term). Finally, the third term describes the matter-mediated photon-photon response. {The excited states of the coupled light-matter system are in this description changes in the photon field. That this perspective is actually quite natural becomes apparent if one considers the nature of the emerging resonances for a real system (see Fig.~\ref{fig:azulene-lifetime}). These resonances are mainly photonic in nature, as they describe the emission/absorption of photons (see appendix~\ref{app:lifetimes}).} Let us consider {now in more detail} what the terms on the right-hand side {of the modified Maxwell's equations} mean physically. First of all, in a matter-only theory the self-consistent solution of the Maxwell's equations together with the response of the bare matter-system would correspond approximately to the first two terms on the right-hand side {(see appendix~\ref{app:Maxwell})}. The photon-photon interaction would not be captured in such an approximate approach. Secondly, to highlight the physical content of the different terms we can make the mean-field contributions due to
\begin{align}
v_{M}(\textbf{r}t) &= \sum_{\alpha} \left(\int d\textbf{r}' \boldsymbol{\lambda}_{\alpha} \cdot {e}\textbf{r}'n(\textbf{r}'t) - \omega_{\alpha} q_{\alpha}(t) \right)\boldsymbol{\lambda}_{\alpha} \cdot {e}\textbf{r} \label{eq:vm}
\\
&\quad + \int \d \textbf{r}'\frac{e^2 n(\vec{r}'t)}{4 \pi \epsilon_0 |\textbf{r} - \textbf{r}'|} \nonumber
\end{align}
explicit
\begin{widetext}
\begin{align*}
\left(\frac{\partial^{2}}{\partial t^{2}} + \omega_{\alpha}^{2}\right)\delta q_{\alpha}(t) &= \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \int {e}\vec{r} \; \chi^n_{n, {\rm s}} \delta v + \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \int {e}\vec{r} \; \chi^n_{n, {\rm s}} \left[ \frac{e^2}{4 \pi \epsilon_0 |\textbf{r}' - \textbf{r}''| } + \sum_{\alpha'} \left(\boldsymbol{\lambda}_{\alpha'} \cdot {e}\textbf{r}''\right)\boldsymbol{\lambda}_{\alpha'} \cdot {e}\textbf{r}' \right]\chi^{n}_{n}\delta v \nonumber
\\
& - \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \SumInt {e}\vec{r} \; \chi^n_{n, {\rm s}} \left(\omega_{\alpha'} \boldsymbol{\lambda}_{\alpha'} \cdot {e}\vec{r}' \right)\delta q_{\alpha'} + \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \int {e}\vec{r} \; \chi^n_{n, {\rm s}} f^{n}_\text{xc}\chi^{n}_{n}\delta v \nonumber\\
&+ \omega_{\alpha} \boldsymbol{\lambda}_{\alpha} \cdot \SumInt {e}\vec{r} \; \chi^n_{n, {\rm s}} f^{q_{\alpha'}}_\text{xc}\delta q_{\alpha'} .
\end{align*}
\end{widetext}
The second term on the right-hand side then corresponds to the random-phase approximation (RPA) to the instantaneous matter-matter polarization. Here a new term that corresponds to the dipole self-energy induced by the coupling to the photons arises. The third term on the right hand side is the RPA approximation to the dipole-dipole mediated photon interaction. {To give these terms further physical meaning note that in the usual perturbative derivation of the van-der-Waals interaction~\cite{craig1998} the first two terms would cancel and leave the photonic dipole-dipole interaction that gives rise to the $R^{-6}$ for small distances and the $R^{-7}$ for larger distances.} The rest are exchange-correlation (xc) contributions that arise due to more complicated interactions among the electrons and photons. The last term effectively describe photon-photon interactions mediated by matter. {In addition, we want to highlight that xc contributions are directly responsible for multi-photon effects, such as two-photon or three-photon processes (see Fig.~\ref{mixed}).} If we only keep the mean-field contributions of the coupled problem, we will denote the resulting approximation in the following as photon RPA (pRPA) to distinguish it from the bare RPA of only the Coulomb interaction. We see how the Maxwell's equations in matter change for bound charges, i.e., fields due to the polarization of matter, only. A new term, the photon-photon interaction, appears. For free charges, i.e., due to an external charge current $\delta j_{\alpha}(t)$, we see similar changes. Clearly, if we would not have a coupling to matter, then there would be no induced density change and we just find the vacuum Maxwell's equations coupled to an external current for the electric field. In other terms, the displacement field trivially corresponds to the electric field {(see appendix~\ref{app:Maxwell})}.
\section{Examples for the coupled matter-photon response}
~\label{sec:rabi}
\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{Diag1.png}}
\caption{Two-level system {(with excitation $\omega_0$)} coupled to one mode of the radiation field {(with frequency $\omega_c$)}. The matter subsystem is driven by an external classical field $v(t)$ and the photon mode is driven by an external classical current $j(t)$ {and both subsystems are coupled with a coupling strength $\lambda$}.}
\label{Fig:TwoLevel}
\end{figure}
\begin{figure*}
\begin{minipage}[c]{0.32\textwidth}
\includegraphics[width=2.2in,height=2.2in]{fig1_new}
\end{minipage}%
\begin{minipage}[c]{0.32\textwidth}
\includegraphics[width=2.2in,height=2.2in]{fig2_new}
\end{minipage}%
\begin{minipage}[c]{0.32\textwidth}
\includegraphics[width=2.2in,height=2.2in]{fig3_new}
\end{minipage}\\
\begin{minipage}[c]{0.39\textwidth}
\includegraphics[width=2.5in,height=2.2in]{absorption_spectrum}
\end{minipage}%
\begin{minipage}[c]{0.32\textwidth}
\includegraphics[width=2.5in,height=2.2in]{absorption_spectrum_w0=2_wc=1}
\end{minipage
\hspace{2em}\begin{minipage}[c]{0.22\textwidth}
\includegraphics[width=0.5in,height=2.0in]{Legends}
\end{minipage}
\caption{Linear-response spectra for the extended Rabi model (dotted-red) compared to the pRPA (dashed-blue) and RWA (full-orange) approximations and for different coupling strengths $\lambda$. (a.) Absorption spectra due to matter-matter response, (b.) spectra due to photon-photon response, (c.) spectra due to matter-photon or photon-matter response. {(d.) The case for $\lambda=0.7$ shows all excitations that arise in strong coupling. {(a.) through (d.) describes resonant coupling.} In (e.) the field is half-way detuned from atomic resonance, i.e., $\omega_{0}=2$ and $\omega_{c}=1$ with strength and energies shifted to frequencies favoring 2-photon processes. The insets in (d.) and (e.) zoom into the frequency axis showing many-photon process.}} \label{mixed}
\end{figure*}
{In this section, we discuss the new perspective enabled by the linear response formalism of QEDFT in more detail for a simple {and illustrative} model system. {We discuss} a slight generalization of the Rabi model~\cite{rabi1936,rabi1937}, which is the standard model of quantum optics. The Rabi model describes a single electron on two lattice sites{/energy levels} interacting with a single photon mode. We schematically depict the system in Fig~\ref{Fig:TwoLevel} and present all further details of this system in appendix~\ref{sec:app:rabi}.}
{First,} let us analyze the optical spectra for such a system and scrutinize the different approximations to the Mxc kernels. {We will compare the numerical exact results, with the mean-field (pRPA) and the rotating-wave approximation (RWA).} In Fig.~\ref{mixed}~(a), (b) and (c) we see how the optical spectra {of the resonantly coupled system (i.e. $\delta = \omega_{0} - \omega_{c} = 0$)} change for an increasing electron-photon coupling strength $\lambda$. Already for small coupling, the splitting of the electronic state into an upper and lower polariton becomes apparent. Approximately these states are given in terms of the RWA as $|+, 0\rangle$ and $|-,0\rangle$. The difference in energy between the lower and upper polariton is called the Rabi splitting $\Omega_R$ and is used to indicate the strength of the matter-photon coupling. In molecular experiments values of up to $\Omega_R/\omega_c \simeq 0.25$ have been measured~\cite{shalabney2015,george2016}. {Up to $\lambda = 0.1$ the different spectra for the exact (dotted-red), the pRPA (dashed-blue) as well as the RWA (full-orange) are in close agreement before they start to differ.} Already the mean-field treatment is enough to recover the quantized matter-photon responses, even for the coupled matter-photon spectra in Fig~\ref{mixed}~c. Consequently the pRPA seems a reasonable approximation for linear-response spectra even for relatively strong coupling situations. Only upon increasing the coupling strength further and thus going into the ultra-strong coupling regime, {the} discrepancies becomes large. For ultra-strong coupling (for $\lambda = 0.3$ the Rabi splitting is already of the order of $0.5{\omega_c}$) the approximations do not recover the exact results. Increasing further leads then to not only a disagreement in transition frequencies but also the weights of the transitions become increasingly different.
Besides a simple check for the approximations to the Mxc kernels, the extended Rabi model also allows us to get some understanding of the novel response functions $\chi^{\sigma_x}_q$, $\chi_{\sigma_x}^q$ and $\chi^{q}_q${, where $\sigma_{x}$ is the expectation-value of the corresponding Pauli matrix and describes the density/occupation changes between the two sites/energy levels}.
{This means,} we consider mixed spectroscopic observables where we perturb one subsystem and then consider the response in the other. We analogously employ $ \chi_{\sigma_{x}}^{q}(\omega)$ and $\chi_{q}^{\sigma_{x}}(\omega)$, respectively, to determine a ``mixed polarizability'' {(see supplemental material ~\ref{app:linresp3})}. If we plot this mixed spectrum (see Fig.~\ref{mixed}~(c) displayed in dotted-red for the numerically exact case), we find that we have positive and negative peaks. Indeed, this highlights that excitations due to external perturbations can be exchanged between subsystems, i.e., energy absorbed in the electronic subsystem can excite the photonic subsystem and vice versa. The oscillator strength of the photonic spectrum (based on $\chi^{q}_q$) in Fig.~\ref{mixed}~(b) provides us with a measure of how strong the displacement field (and with this also the electric field) reacts to an external classical charge current with frequency $\omega$. Similarly, the mixed spectrum (based on $\chi^{\sigma_x}_q$ or $\chi_{\sigma_x}^q$) in Fig.~\ref{mixed}~(c) provides us with information of how strong one subsystem of the coupled system reacts upon perturbing the other one. The oscillator strength here is not necessarily positive. What is absorbed by one subsystem can be transferred to the other.
{In Fig.~\ref{mixed} (d) and (e), we show specifically the absorption spectra of the Rabi model for ultra-strong coupling, i.e., $\lambda = 0.7$. In this regime, three new peaks arise for the exact case accounting for high-lying excited states with non-vanishing dipole moments due to the strong electron-photon coupling. The new absorption peaks in Fig.~\ref{mixed}~(d), also shown in the inset, describes the resonant coupling case which the RWA and pRPA fail to capture in strong coupling, since processes beyond one-photon are involved. Similarly, Fig.~\ref{mixed}~(e) depicts the case were the field is half-detuned from the electronic resonance indicating a two-photon process. Clearly in ultra-strong coupling the absorption peaks are merely shifted close to the bare frequencies of the individual subsystems, but remain dressed by the photon field as new peaks arise due to the coupling.} {The pRPA and RWA capture the first of the two peaks around $\omega=2$, which is also the frequency of the atom, but fail to capture higher lying non-vanishing contributions to the spectra. These higher-lying peaks correspond to multi-photon processes.} {With more accurate approximation for the xc potential results closer to the exact ones can be obtained.} We note at this point that the peaks in Fig.~\ref{mixed} are artificially broadened and in reality correspond to sharp transitions due to excited states with infinite lifetimes. How to get lifetimes {quantitatively} will be discussed in the next section.
\section{Coupled matter-photon response: real systems}
\label{sec:general}
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth]{azulene4.png}}
\caption{Schematic of absorption spectroscopy in optical cavities: Benzene (C$_{6}$H$_6$) molecule and ${\boldsymbol\lambda_\alpha}$ denotes the polarization direction of the photon field.}
\label{fig:azulene}
\end{figure}
{In this section, we apply the introduced formalism in pRPA approximation to real systems. We make the linear-response formulation practical by reformulating the problem as an eigenvalue equation in the frequency-domain. For electron-only problems this formulation is known as the \textit{Casida equation}~\cite{casida1996}. We refer the reader to appendix~\ref{sec:app:casida} for a derivation of our extension of the Casida equation, which includes transverse photon fields.}
For the following discussion, we consider benzene molecules in an optical cavity. In Fig.~\ref{fig:azulene} we schematically depict the experimental setup for a photoabsorption experiment under strong light-matter coupling {for a single molecule}. First we study the prototypical cavity QED setup where {a} molecule is strongly coupled to a single cavity mode of a high-Q cavity. In the second setup, we lift the restriction of only one mode and instead couple the benzene molecule to many modes that sample the electromagnetic vacuum field without enhancing the coupling to a specific mode by hand. In the third setup, we study the behavior of two molecules in an optical cavity, as well as a dissipative situation, where only {a few} modes {are} strongly coupled, embedded in a quasi-continuum of modes. {In the last example, we analyze the strong coupling of a single molecule to a continuum of modes. We find a transition from Lorentzian lineshape to a Fano lineshape~\cite{Ott2013} for increasing electron-photon coupling strength.} {These different setups provide us with the first ab-initio calculation for the spectrum of a real molecule in a high-Q cavity, the first ab-initio determination of intrinsic lifetimes and the first ab-initio calculation of the non-perturbative interplay between electronic structure, lifetime and strong-coupling. The {two} last situations need a self-consistent treatment of photons and matter alike and cannot be captured by any available electronic-structure or quantum-optical method.} All of those examples highlight the novel possibilities and perspectives that the QEDFT framework provides.
\subsection{Strong light-matter coupling}
The first results we discuss are a set of calculations, where a benzene molecule is strongly coupled to a single photon mode in an optical high-Q cavity. {We have implemented the linear-response pseudo-eigenvalue equation of Eq.~(\ref{eq:casida-deltav1}) into the real-space code OCTOPUS~\cite{marques2003,andrade2014} and details of the numerical parameters {are given in appendix~}\ref{sec:numerics}}~\footnote{{The routines used to perform all calculations in this work will be made publicly available. They can be easily transported to any other first principles code that has the matter linear-response equations implemented to make them ready to describe the complete QED response, i.e. joint matter-photon response, as described in this work.}}.
In the first calculation, we include a single cavity mode in resonance to the $\Pi$-$\Pi^*$ transition of the benzene molecule~\cite{yabana1999,marques2003}, i.e., $\omega_\alpha=6.88$~eV. For the light-matter coupling strength $\lambda_\alpha= |\boldsymbol\lambda_\alpha|$, we choose five different values, i.e. $\lambda_\alpha = (0, 2.77, 5.55, 8.32, 11.09)$ eV$^{1/2}$/nm that correspond to a transition from the weak to the strong-coupling limit and the cavity mode is assumed to be polarized along the x-direction.
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth]{figure_benzene.pdf}}
\caption{Absorption spectra for the benzene molecule in free space (black) and under strong light-matter coupling in an optical cavity to ultra-strong coupling (blue). The value for $\lambda_\alpha$ is given in units of $[$eV$^{1/2}$/nm$]$.}
\label{fig:azulene-results}
\end{figure}
In Fig.~\ref{fig:azulene-results}, we show the absorption spectra for these different values of $\lambda_\alpha$. We start by discussing the $\lambda_\alpha=0$ case that is shown in black. This spectrum corresponds to a calculation of the benzene molecule in free space and the spectrum is within the numerical capabilities identical to Ref.~\cite{marques2003}~\footnote{The spectrum in Ref.~\cite{marques2003} has been obtained using an explicit time-propagation with finite time. In the limit of zero broadening and including all unoccupied states, we would find identical spectra with very long propagated spectra.}. {We stress that here the broadening of the peaks is only done artificially since the photon bath is not included in the calculation. In the examples of Sec.~\ref{sec:num-lifetime} and \ref{sec:novel} we include many modes and hence sample the photon bath non-perturbatively.} We tune the electron-photon coupling strength $\lambda_\alpha$ in Fig.~\ref{fig:azulene-results}. We find for increasing coupling strength a Rabi splitting of the $\Pi$-$\Pi^*$ peak into two polaritonic branches. The lower polaritonic branch has higher intensity, compared to the upper polaritonic peak. Numerical values for the excitation energy $E_I$, the transition dipole moment $x_I$ and the oscillator strength $f_I$ are given in Tab.~\ref{tab:casida} {in the appendix}. This demonstrates that ab-initio theory is able to describe excited-state properties of strong light-matter coupling situations and captures the hybrid character of the combined matter-photon states. Thus predictive theoretical first-principle calculations for excited-states properties of real systems strongly coupled to the quantized electromagnetic field are now available. This will allow unprecedented insights into coupled light-matter systems, {since we have access to many observables that are not (or not well~\cite{schaefer2018}) captured by quantum-optical models}.
\subsection{Lifetimes of excitations from first principles}
\label{sec:num-lifetime}
Next we consider how {to} obtain lifetimes from QEDFT linear-response theory. In this example, we explicitly couple the benzene molecule to a wide range of photon modes similar as in the spontaneous emission calculation of Ref.~\cite{flick2017}. While in Ref.~\cite{flick2017}, {the system was simulated} with 200 photon modes, we choose here now 80.000 photon modes. The energies of the sampled photon modes cover densely a range from $0.19$~meV, for the smallest energy up to 30.51~eV for the largest one with a spacing of $\Delta \omega = 0.38$~meV. However, we do not sample the full three-dimensional mode space together with the two polarization possibilities per mode but rather consider a one-dimensional slice in mode space. This one-dimensional sampling of mode frequencies will change the actual three-dimensional lifetimes, but for demonstrating the possibilities of obtaining lifetimes this is sufficient~\footnote{A detailed analysis of real lifetimes would besides a proper sampling of the mode space also include considerations with respect to the bare mass of the particles.}. The sampling of the photon modes corresponds to the modes of a quasi-one dimensional cavity. We choose a cavity of length $L_x$~\cite{flick2017} in $x$-direction with a finite width in the other two directions that are much more confined. Thus we employ $\omega_\alpha = \alpha c\pi /L_x$ and $\boldsymbol\lambda_\alpha = \sqrt{\frac{2}{\hbar\epsilon_0 L_xL_y L_z}} \text{sin}( \omega_\alpha/c \, x_0)\textbf{e}_x$, where $x_0 = L_x/2$ is the position of the molecule in $x$-direction. While we have a sine mode function in the $x$-direction, we assume a constant mode function in the other directions. For this example, we choose a cavity of length $L_x= 3250\mu \text{m}$ in $x$-direction, $L_y=10.58 \AA$ in $y$-direction and $L_z = 2.65\AA$ in $z$-direction.\\
\begin{figure}[ht]
\centerline{\includegraphics[width=0.5\textwidth]{many_modes.pdf}}
\caption{{First principles} lifetime calculation of the electronic excitation spectrum of the benzene molecule in an quasi one-dimensional cavity: (a) Full spectrum of the benzene molecule, (b) zoom to the $\Pi-\Pi^*$ transition{, where the black arrow indicates the full width at half maximum (FWHM) $\Delta E$}, (c) zoom to a peak contributing to the $\sigma-\sigma^+$ transition. The gray spectrum is obtained by Wigner-Weisskopf theory~\cite{weisskopf1930}. The dotted spectral data points correspond to many coupled electron-photon excitation energies which together comprise the natural lineshape of the excitation. Blue color refers to a more photonic nature of the excitations, vs. red color to a more electronic nature.}
\label{fig:azulene-lifetime}
\end{figure}
The results of this calculation are shown in Fig.~\ref{fig:azulene-lifetime}. In Fig.~\ref{fig:azulene-lifetime}~(a) we show the full spectrum. The electron-photon absorption function that has been obtained by coupling the benzene molecule to the quasi one-dimensional cavity with 80.000 cavity modes is plotted in blue. Since we have sampled the photon part densely, we do not need to artificially broaden the peaks anymore. Formulated differently, we can directly plot the oscillator strength and the excitation energies of our resulting eigenvalue equation and do not need anymore to employ the Lorentzian broadening. In Fig.~\ref{fig:azulene-lifetime} from blue (more photonic) to red (more electronic) for the electron-photon absorption spectrum we plot the different contributions of each pole in the response function. {These results confirm our intuition that resonances are mainly photonic in nature and that a Maxwell's perspective of excited states is quite natural.} In (b) we zoom to the $\Pi$-$\Pi^*$ transition. Due to quasi one-dimensional nature of the quantization volume, we find a broadening of the peak that is larger than it is for the case of a three-dimensional cavity due to the sampling of the electromagnetic vacuum. This is similar to changing the vacuum of the electromagnetic field. Accordingly the lifetimes of the electronic states are shorter if the electromagnetic field is confined to one dimension and we will discuss this in the next section.
\subsection{Connection to {standard} Wigner-Weisskopf theory}
\label{sec:beyond}
{{If the coupling between light and matter is very weak and neither subsystem gets appreciably modified due to the other, in contrast to the previous strong light-matter coupling case,} the radiative lifetimes of atoms and molecules can be calculated using {the perturbative} Wigner-Weisskopf theory~\cite{weisskopf1930} in single excitation approximation, as well as under the assumption of the Markov approximation. {These approximations are justified in the usual free-space case, where the results of Wigner and Weisskopf reproduce the prior results of Einstein based on the ad-hoc $\textit A$ and $\textit B$ coefficients.} {However it does not include the treatment of ensembles of molecules that effectively enhance the matter-photon coupling strength, as shown below.
}{Under the assumption of Wigner-Weisskopf theory, the radiative decay rate is given by}
\begin{align}
\Gamma_{3D} = \frac{\omega_0^3|\textbf{d}|^2}{3 \pi \epsilon_0\hbar c^3}.
\label{eq:gamma-ww3}
\end{align}
{For a one-dimensional cavity in x-dimension the results change to~\cite{buzek1999}}
\begin{align}
\Gamma_{1D} = \frac{\omega_0|\textbf{d}|^2}{L_yL_z \epsilon_0\hbar c}
\label{eq:gamma-ww}
\end{align}
{For comparison, we show in Fig.~\ref{fig:azulene-lifetime} in grey the peaks that are predicted by Wigner-Weisskopf theory. Since our sampling is very dense, we find for both peaks shown in the bottom a good agreement with Eq.~\ref{eq:gamma-ww}.\\
In fact, if we take the continuum limit for the photon modes, we recover {in our framework} the lifetimes predicted by Wigner-Weisskopf theory including the diverging energy shifts~\cite{milonni1976}, i.e. the Lamb shift. Due to the Lamb shift, our resulting peaks are slightly shifted, due to the divergencies. These divergencies can be handled by renormalization theory.} The lifetimes can now be obtained the following way: We measure the full width at half maximum (FWHM), indicated by the black arrow in (b). In this case, we find $\Delta E_\text{FWHM} = 0.0204$~eV and the corresponding lifetime $\tau_{\Pi-\Pi^*}$ follows by $\tau_{\Pi-\Pi^*}=\hbar/\Delta E_\text{FWHM} = 32.27$~fs. {Using the Wigner-Weisskopf formula from Eq.~\ref{eq:gamma-ww}, and the dipole moments and energies from the LDA calculation without a photon field, we find a lifetime of $32.21$~fs. As a side remark, the same transition using Eq.~\ref{eq:gamma-ww3} has a free-space lifetime of $0.89$~ns, roughly in the range of the 2p-1s lifetime of the Hydrogen atom of $1.6$~ns.}\\
In Fig.~\ref{fig:azulene-lifetime}~c we finally show the ab-initio peak of the $\sigma-\sigma^+$ transition. We find a narrow ab-initio peak {that is not as well sampled as the $\Pi-\Pi^*$. We note in passing that we find a ionization energy of $9.30$~eV using $\Delta$-SCF in the benzene molecule with the LDA exchange-correlation functional. We note in passing that we find a ionization energy of $9.30$~eV using $\Delta$-SCF in the benzene molecule with the LDA exchange-correlation functional. In our simulation, coupling to peaks higher than the ionization energy are broadened by continuum (box) states.}
\subsection{Beyond the single molecule limit and dissipation in QEDFT}
\label{sec:novel}
\begin{figure}[ht]
\centerline{\includegraphics[width=0.5\textwidth]{3.pdf}}
\caption{{(a) Two molecules of benzene strongly coupled to 80.000 cavity modes of an one-dimensional cavity. The further apart the molecules are, the closer the peak gets to the single molecule peak. Also we notice the doubled peak broadening (shorter lifetime). The gray spectrum is obtained by Wigner-Weisskopf theory~\cite{weisskopf1930}. (b) We show the Rabi splitting in a situation of a single strongly coupled mode with 80.000 cavity modes (green), and three strongly coupled modes with 80.000 cavity modes (blue). The red lines correspond to the same setup as in (a). The dashed lines refer to the frequency of the cavity modes. The peaks become broadened due to the interaction with the continuum.}}
\label{fig:azulene-two}
\end{figure}
{In contrast to the free-space result, where weak coupling as well as the assumption of a dilute gas of molecules are implied, in the case of single-molecule strong coupling~\cite{chikkaraddy2016} or when nearby molecules {or an ensemble of interacting molecules} modify the vacuum, the usual perturbative theories break down. Changes in the electronic and the photonic subsystem become self-consistent and the usual distinction of light and matter becomes less clear. In such situations the linear-response formulation of QEDFT as well as the Maxwell's perspective of excited-state properties becomes most powerful. Consider, for instance, two benzene molecules weakly coupled to a one-dimensional continuum of photon modes. If the molecules are far apart we just find the usual Wigner-Weisskopf result. But if we bring the molecules closer (see Fig.~\ref{fig:azulene-two} (a)), we see that the combined resonance shifts and the combined linewidth becomes broader, implying a shortened lifetime. In Fig.~\ref{fig:azulene-two} (b), we consider the case of single-molecule strong coupling, where {a few} out of the 80.000 modes ha{ve} an enhanced coupling strength.} {In red, we show the spectrum where the molecule is coupled to the continuum, as is also shown in Fig.~\ref{fig:azulene-lifetime}. We then introduce a single strongly coupled mode at the $\Pi-\Pi^*$ transition energy and the resulting spectra is shown in green. We note that in the figure, the cavity frequencies are plotted in dashed lines. The single mode introduces the expected Rabi splitting into the upper and lower polariton} {and the peaks of the upper and lower polariton become broadened due to the interaction with the continuum. Interestingly, we find a different line broadening for the lower and the upper polaritonic peak, since only the sum of both has to be conserved. The smaller broadening for these two lower polaritonic states implies that the radiative lifetime of the lower and upper polaritonic state is longer than the lifetime of the excitation in weakly-coupled free-space.} {In blue, we show the spectra, where we have introduced three strongly coupled modes in addition to the cavity 80.000 modes of the continuum. We tune the two additional cavity modes in resonance to the lower and upper polariton peak of the green plot. We find additional peak splitting, but also a shifting of peak positions at 7.8~eV.}
{In the last numerical example, we study the strong coupling to the continuum for the case of a single molecule. The results are shown in Fig.~\ref{fig:azulene-fano}. Here, we effectively enhance the light-matter coupling strength by reducing the volume of the cavity along the $y$ and $z$ direction. For comparison, we show in red the setup that is also shown in Fig.~\ref{fig:azulene-lifetime}, where the excitations have Lorentzian lineshape consistent with Wigner-Weisskopf theory as discussed in the previous section. By gradually reducing the dimensions along the $y$ and $z$ direction, we find drastic changes in the lineshape of the excitations. These changes lead to the transition of the lineshape from a Lorentzian to a Fano lineshape, as becomes clearly visible for $L_xL_z=0.28\AA$.}
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth]{4.pdf}}
\caption{{Ab-intio lifetime calculation of the electronic excitation spectrum of the benzene molecule in an one-dimensional cavity along $x$-direction with different length in $L_y$ and $L_z$ direction. The red spectra refer to the same setup as in Fig.~\ref{fig:azulene-lifetime}. Effectively the electron-photon strength increases with smaller $L_y$ and $L_z$ length leading to a transition from a Lorentzian lineshape to a Fano lineshape.}}
\label{fig:azulene-fano}
\end{figure}
As a summary, {we have presented in this section, that lineshapes, as well as lifetimes can be inferred directly from first principle calculations. In case of Lorentzian lineshape, we find that the width of the} {calculated peaks (no need to introduce any artificial broadening as commonly done)} correspond to the lifetimes. These calculations demonstrate that ab-initio theory is able to capture the true nature of excitations, i.e., resonances with finite intrinsic lifetimes, without the need of an artificial bath or post-processing. This allows a new perspective { of well-known results. Furthermore, we find} that the excitations measured in absorption/emission experiments are mainly photonic in nature, and it is only the peak position that is dominated by the matter constituents. This is of course very physical, since what we see is the absorption/emission of a photon, not of the matter constituents. Further, since we describe the photon vacuum on the same theoretical footing as the matter subsystem, we have full control over the photon field making it straightforward to simulate very intricate changes, e.g., changing the character of a specific mode out of basically arbitrarily many, and investigating its influence on excited-states properties such as the radiative lifetime. This allows predictive first-principle calculations for intricate experimental situations similar to the ones encountered in Ref.~\cite{lettow2007,wang2017}.
\section{Summary and Outlook}
\label{sec:summary}
In this work we have introduced linear-response theory for non-relativistic quantum-electrodynamics in the long wavelength limit. Compared to the conventional matter-only response approaches, we have highlighted how in the coupled matter-photon case the usual response functions change, how novel photon-photon and matter-photon response functions are introduced, {how these novel response functions provide a photonic perspective on excited state properties, how the results lead to changes in the usual Maxwell's equation in matter} and how we can efficiently calculate {all} these response functions in the framework of QEDFT. By investigating a simple model system, we have shown how the spectrum of the matter subsystem is changed upon coupling to the photon field. Further we have demonstrated the range of validity of a simple yet reliable approximation to the in general unknown mean-field exchange-correlation kernels. Using this approximation we have presented the first ab-initio calculations of the spectrum of real system{s} (benzene {molecules}) coupled to the modes of the quantized electromagnetic field. In one example we have calculated the change upon strong coupling to a single mode of a high-Q cavity, which leads to a large Rabi splitting. In the second example we have calculated from first principles the natural linewidths of benzene coupled to a specific sampling of the vacuum field. {In the last examples, we demonstrated the abilities to calculate many-molecule systems, as well as dissipative strong-coupling situations, {as well as strong coupling to the continuum, where we find a transition from Lorentzian lineshape to Fano lineshape,} {where the usual (perturbative) approaches to light-matter coupling fail}.}
These results demonstrate the versatility and possibilities of QEDFT, where light and matter are treated on equal quantized footing. In the context of strong light-matter coupling, e.g., in polaritonic chemistry, the presented linear-response formulation allows now to determine polaritonically modified spectra from first principles. Together with ab-initio ground-state calculations~\cite{flick2017c} QEDFT now provides a workable first-principle description to analyze and predict photon-dressed chemistry and material sciences. In particular, our novel approach provides {a unique} practical computational scheme to compute photon-dressed excited-state potential-energy surfaces and non-adiabatic coupling elements that are required for ab-initio calculations in the emerging field of polaritonic chemistry. Further, in the context of standard ab-initio theory, the linear-response formulation of QEDFT now allows the calculation of intrinsic lifetimes {and provides access to quantum-optical observables. Specifically, due to the non-perturbative nature of the approach, quantum-optical problems where the self-consistent feedback between light and matter has to be taken into account, e.g., that many molecules change the photon vacuum and hence the Markov approximation breaks down, become feasible}. For optical physics, the presented linear-response framework presents an interesting opportunity to study the modifications of the Maxwell's equations in matter from first principles.
Finally we want to highlight that although the QEDFT linear-response framework is new, its similarity to the usual matter-only linear-response formulation in terms of an pseudo-eigenvalue problem makes it very easy to include in already existing first-principle codes. This, together with the above discussed novel possibilities in different fields of physics, shows that there are many interesting cases that can be studied with the presented method.
\section{Acknowledgements}
We would like to thank Christian Sch\"afer and Norah Hoffmann for insightful discussions, and Sebastian Ohlmann for the help with the efficient massive parallel implementation. JF acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) under Contract No. FL 997/1-1 and all of us acknowledge financial support from the European Research Council (ERC-2015-AdG-694097).
|
{
"timestamp": "2019-06-19T02:03:36",
"yymm": "1803",
"arxiv_id": "1803.02519",
"language": "en",
"url": "https://arxiv.org/abs/1803.02519"
}
|
\section{introduction}
\label{sec01}
With the great progress and good performance of the Belle, BaBar and LHCb
experiments, many three-body nonleptonic $B$ meson weak decay channels are
accessible and have been measured \cite{pdg}. Recently, based on the $3\,fb^{-1}$
$pp$ collision data recorded by the LHCb detector, the three-body nonleptonic
decay $B_{s}^{0}$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$
was investigated with the requirements on the ${\pi}^{+}{\pi}^{-}$ invariant mass
in the range 400 MeV $<$ $m({\pi}{\pi})$ $<$ 1600 MeV, then an analysis of the
$m({\pi}{\pi})$ spectrum including the $S$-, $P$-, and $D$-wave amplitudes was
further performed to study the possible resonant contributions \cite{prd95.012006}.
Some prominent maxima in the $m({\pi}{\pi})$ spectrum are observed around the
${\rho}(770)$, $f_{0}(980)$, $f_{2}(1270)$ and $f_{0}(1500)$ resonant regions.
One of the formal public announcement of the LHCb Collaboration is that \cite{prd95.012006}
the three-body sequential rare decay $B_{s}^{0}$ ${\to}$ ${\phi}f_{0}(980)$
${\to}$ ${\phi}{\pi}^{+}{\pi}^{-}$ was first observed with a statistical
significance of $8\,{\sigma}$ and the branching fraction of
\begin{equation}
{\cal B}(B_{s}^{0}\,{\to}\,{\phi}\,f_{0}(980)\,{\to}\,{\phi}\,{\pi}^{+}\,{\pi}^{-})
\, =\, (1.12{\pm}0.16^{+0.09}_{-0.08}{\pm}0.11){\times}10^{-6}
\label{eq:br-exp-01},
\end{equation}
where the errors are statistical, systematic and from the normalization,
respectively.
Although it is still a controversial issue whether the isospin-singlet
particle $f_{0}(980)$ should be regarded as the conventional $q\bar{q}$
meson, or the exotic tetraquark $q\bar{q}q\bar{q}$ state, or the
meson-meson $K\bar{K}$ molecule, it is usually suggested that the
unflavored scalar $f_{0}(980)$ meson has a substantial $s\bar{s}$ component,
and decays dominantly into the ${\pi}{\pi}$ final states \cite{pdg}.
Therefore, on the one hand, the $B_{s}^{0}$ ${\to}$ ${\phi}f_{0}(980)$
${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay is interesting and helpful
to explore the compositive structure of the $f_{0}(980)$;
on the other hand, the importance of the $B_{s}^{0}$ ${\to}$
${\phi}f_{0}(980)$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay is obvious,
{\em i.e.}, this decay is induced by the flavor-changing-neutral-current
(FCNC) $\bar{b}$ ${\to}$ $\bar{s}s\bar{s}$ process at the elementary
particle level within the Standard model (SM), which is absolutely
forbidden at the tree level by the Cabibbo-Kobayashi-Maskawa (CKM)
quark-mixing mechanism within SM but sensitive to the new physics
effects beyond SM.
Along with the experimental advances, the theoretical research on the
three-body nonleptonic $B$ weak decay is really necessary.
Although there exist some attractive QCD-inspired phenomenological
methods to deal with the two-body nonleptonic $B$ decays, such as
the perturbative QCD (PQCD) approach \cite{plb348.597,prd52.3958,prd55.5577,
prd56.1615,plb504.6,prd63.054008,prd63.074006,prd63.074009,epjc23.275},
the QCD factorization (QCDF) approach \cite{prl83.1914,npb591.313,
npb606.245,plb488.46,plb509.263,prd64.014036,npb774.64,prd77.074013},
and so on, the theoretical description of the three-body nonleptonic
$B$ decays is still in the early stage of modeling.
This is not surprising because that the more hadrons participated,
the more intricate the interferences among different contributions
(such as the possible resonances and final state interactions) will
certainly become. Moreover, for the three-body hadronic decays,
the kinematical configurations will vary from region to region in
the Dalitz plot, and in principle correspond to different dynamical
components and theoretical treatments with special scales.
The resonant contributions are entirely engulfed by the blurry
background clouds, so any phenomenological parametrization and
interpretations of the resonant structures are process- and
model-dependent.
The effective separations between the perturbative and nonperturbative
contributions to the three-body nonleptonic $B$ decays will be much
more complicated, which is by no means trivial.
However at the same time, the phase space distributions make the
theoretical calculation of three-body nonleptonic $B$ meson decays
to be very meaningful for exploring some fresh and potentially important
information, such as the natures and effects of possible resonances,
the energy dependence of observables, the local $CP$ asymmetry
distributions in the Dalitz plot, and so on.
In the past years, there were plenty of theoretical studies of the
three-body nonleptonic $B$ meson decays, such as
Refs.\cite{prd39.3346,prd44.1454,prd52.6356,prd68.015004,plb564.90,
prd72.094031,prd75.014002,plb726.337,plb727.136,plb728.206,prd89.074043,
prd84.034040,prd84.034041,prd84.056002,prd85.016010,prl90.061802,
prd74.051301,ijmpa29.1450011,plb728.579,prd91.014029}
based on SU(3) relations,
Refs.\cite{prd52.5354,prd60.054029,plb447.313,plb539.67,prd69.114020,
prd70.034033,prd65.094004,prd66.054015,ijmpa23.3229,prd72.094003,prd76.094006,
prd88.114014,prd89.074025,prd89.094007,prd94.094015,plb665.30}
based on both heavy quark effective theory and chiral perturbation
theory,
Refs.\cite{epjc31.215,jhep1602.009,prd65.034003,prd66.054004,prd67.034012,
prd70.034032,epjc33.s253,epja50.122,cjp93.339,prd89.095026,prd95.036013,
prd96.113003,prd62.036001,prd62.114011,prl86.216,prd87.076007}
with factorization approach,
Refs.\cite{jpg31.199,prd83.014002,prd90.034014,ahep2014.785648,jhep1710.117,
npb899.247,prd93.116008,aplb42.2013,plb622.207,prd74.114009,prd79.094005,
prd81.094033,plb699.102,plb737.201,ps89.095301,ahep2014.451613}
with the QCDF approach,
Refs.\cite{plb561.258,prd70.054006,npa930.117,prd89.074031,prd91.094024,
epjc76.675,plb763.29,cpc41.083105,npb923.54,npb924.745,prd95.056008,
prd96.036014,prd96.093011,epjc77.199,prd94.034040,epjc77.518}
with the PQCD approach.
Both the PQCD and QCDF approaches have been widely employed in the two-body
nonleptonic $B$ meson decays in recent years.
In Ref.\cite{plb561.258}, Chen and Li attempted to generalize the PQCD
approach to the three-body nonleptonic $B^{+}$ ${\to}$ $K^{+}{\pi}^{+}{\pi}^{-}$
decay for the particular configuration topologies where the kinematics is very
similar to a two-body decay.
In this paper, we shall follow the method of Ref.\cite{plb561.258} to investigate
the $B_{s}^{0}$ ${\to}$ ${\phi}f_{0}(980)$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$
decay with the PQCD approach.
The overall layout of this paper is as follows.
The theoretical framework and the amplitudes for the $B_{s}^{0}$ ${\to}$
${\phi}f_{0}(980)$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay are
elaborated in section \ref{sec02}.
The numerical results and discussion are presented in Section \ref{sec03}.
The last section is a short summary.
\section{theoretical framework}
\label{sec02}
\subsection{The effective Hamiltonian}
\label{sec0201}
The nonleptonic weak decays of the $B$ mesons involve three fundamental scales,
including the weak interaction scale $M_{W}$, the $b$ quark mass scale $m_{b}$,
and the QCD characteristic scale ${\Lambda}_{\rm QCD}$, which are strongly
ordered: $M_{W}$ ${\gg}$ $m_{b}$ ${\gg}$ ${\Lambda}_{\rm QCD}$.
To deal with the multi-scale problems, one usually has to resort to the
effective theory approximation.
Using the operator product expansion and the renormalization group (RG) equation,
the low energy effective Hamiltonian for the FCNC $B_{s}^{0}$ ${\to}$
${\phi}f_{0}(980)$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay within
the SM can be written as \cite{rmp68.1125}:
\begin{equation}
{\cal H}_{\rm eff}\, =\,
\frac{G_{F}}{\sqrt{2}}\, \Big(V_{ub}^{\ast}\,V_{us}+V_{cb}^{\ast}\,V_{cs}\Big)\,
\sum_{i=3}^{10}\,C_{i}({\mu})\,Q_{i}({\mu})\, +\, {\rm h.c.}
\label{eq:Heff},
\end{equation}
where the Fermi coupling constant $G_{F}$ ${\simeq}$ $1.166{\times}10^{-5}$
${\rm GeV}^{-2}$ \cite{pdg}. $V_{ub}^{\ast}\,V_{us}$ and $V_{cb}^{\ast}\,V_{cs}$
are the CKM factors. The scale ${\mu}$ separates the effective Hamiltonian into
two distinct parts: the Wilson coefficients $C_{i}$ and the local four-quark
operators $Q_{i}$.
The expressions of the operators $Q_{i}$ are written as:
\begin{eqnarray}
Q_{3} &=& \sum_{q}\,
(\bar b_{\alpha}\,s_{\alpha})_{V-A}\,
(\bar q_{\beta}\,q_{\beta})_{V-A}, \,
\qquad \quad \
Q_{4}\, =\, \sum_{q}\,
(\bar b_{\alpha}\,s_{\beta})_{V-A}\,
(\bar q_{\beta}\,q_{\alpha})_{V-A},
\label{q3-q4} \\
Q_{5} &=& \sum_{q}\,
(\bar b_{\alpha}\,s_{\alpha})_{V-A}\,
(\bar q_{\beta}\,q_{\beta})_{V+A}, \,
\qquad \quad \
Q_{6}\, =\, \sum_{q}\,
(\bar b_{\alpha}\,s_{\beta})_{V-A}\,
(\bar q_{\beta}\,q_{\alpha})_{V+A},
\label{q5-q6} \\
Q_{7} &=& \frac{3}{2}\,\sum_{q}\,Q_{q}\,
(\bar b_{\alpha}\,s_{\alpha})_{V-A}\,
(\bar q_{\beta}\,q_{\beta})_{V+A},
\quad
Q_{8}\, =\, \frac{3}{2}\,\sum_{q}\,Q_{q}\,
(\bar b_{\alpha}\,s_{\beta})_{V-A}\,
(\bar q_{\beta}\,q_{\alpha})_{V+A},
\label{q7-q8} \\
Q_{9} &=& \frac{3}{2}\,\sum_{q}\,Q_{q}\,
(\bar b_{\alpha}\,s_{\alpha})_{V-A}\,
(\bar q_{\beta}\,q_{\beta})_{V-A},
\quad
Q_{10}\, =\, \frac{3}{2}\,\sum_{q}\,Q_{q}\,
(\bar b_{\alpha}\,s_{\beta})_{V-A}\,
(\bar q_{\beta}\,q_{\alpha})_{V-A},
\label{q9-q10}
\end{eqnarray}
where $Q_{3,...,6}$ and $Q_{7,...,10}$ are called as the QCD and
electroweak penguin operators, respectively.
${\alpha}$ and ${\beta}$ are color indices.
$q$ denotes all the active quark at the scale of ${\cal{O}}(m_{b})$,
{\em i.e.}, $q$ $=$ $u$, $d$, $s$, $c$, $b$.
$Q_{q}$ is the electric charge of quark $q$ in the unit of
${\vert}e{\vert}$.
The operators $Q_{i}$ govern the dynamics of the $B$ meson weak decay.
The coupling strength of the effective interactions among four quarks
of the operators $Q_{i}$ is proportionate to the Wilson coefficients
$C_{i}$. The physical contributions from the scale higher than ${\mu}$
are summarized in the Wilson coefficients $C_{i}$, while
the physical contributions from the scale lower than ${\mu}$ are
incorporated into the hadronic matrix elements (HMEs) where the operators
$Q_{i}$ are sandwiched between the initial and final hadron states.
The Wilson coefficients $C_{i}$ are process independent and computable
order by order with the RG improved pertrubative theory as long as the
scale ${\mu}$ is not too small. The expressions of the Wilson coefficients
$C_{i}$ including the next-to-leading order corrections can be found in
Ref.\cite{rmp68.1125}.
The HMEs describe the transition from the quarks of the operators $Q_{i}$
to the participating hadrons.
The operators $Q_{i}$ comprise of four quarks at the local interaction
point, the initial and final states are hadronic states. The transition
between the quarks and hadrons necessarily involves the hadronization
and other rescattering effects.
Due to the low-energy long-distance QCD effects and the entanglement
of nonperturbative and perturbative contributions, the main obstacles
of the calculation of the nonleptonic $B$ decays is how to properly
evaluate the HMEs of the local four-quark operators.
\subsection{Hadronic matrix element}
\label{sec0202}
As for the two-body nonleptonic $B$ decays with both the PQCD and QCDF approaches,
the HMEs are usually written as the convolution of the universal wave functions
(WFs) or distribution amplitudes (DAs) reflecting the nonperturbative contributions
with the scattering amplitudes containing perturbative contributions, based on
the factorization theorem for exclusive processes \cite{plb87.359,prl43.545,
prd22.2157,plb91.239,plb90.159}. Similarly, the HMEs for the three-body nonleptonic
$B$ decays could generally be written as:
\begin{eqnarray}
{\langle}h_{1}h_{2}h_{3}{\vert}Q_{i}{\vert}B{\rangle} &{\sim}&
{\int}dk_{1}\,dk_{2}\,dk_{3}\,dk\,
{\Phi}_{h_{1}}(k_{1})\,{\Phi}_{h_{2}}(k_{2})\,{\Phi}_{h_{3}}(k_{3})\,
{\Phi}_{B}(k)\,{\cal T}(k_{1},k_{2},k_{3},k)
\label{hme-01}, \\ \text{or} &{\sim}&
{\int}dx_{1}\,dx_{2}\,dx_{3}\,dx\,
{\phi}_{h_{1}}(x_{1})\,{\phi}_{h_{2}}(x_{2})\,{\phi}_{h_{3}}(x_{3})\,
{\phi}_{B}(x)\,\widetilde{\cal T}(x_{1},x_{2},x_{3},x)
\label{hme-02},
\end{eqnarray}
where ${\Phi}_{h_{i}}(k_{i})$ and ${\phi}_{h_{i}}(x_{i})$ are the
WFs and DAs for the $h_{i}$ hadron, respectively; $k_{i}$ ($x_{i}$)
is the momentum (the longitudinal momentum fraction) of the valence
quark; ${\cal T}$ and $\widetilde{\cal T}$ are the scattering kernels.
It is assumed that the nonperturbative contributions are contained
within the WFs and DAs. The DAs are universal.
The DAs either extracted from experimental data or obtained from
nonperturbative means could be employed for other processes
involving the same hadron.
The scattering kernels, ${\cal T}$ and $\widetilde{\cal T}$, could be
computed systematically in an expansion in the strong coupling
${\alpha}_{s}$ and the power $1/m_{b}$ with the perturbation theory.
As analyzed in Refs.\cite{npb899.247}, the Dalitz plot for the three-body
nonleptonice $B$ decays could be divided into different regions with
distinct kinematic and dynamic properties.
In the center region of the Dalitz plot, all three final hadrons have a
large energy and none of them moves collinearly to the others.
This kinematical configurations have two hard gluons and the perturbative
calculation of the scattering kernels seems to be applicable.
Unfortunately, as analyzed in Ref.\cite{plb561.258}, the scattering kernels
of this region contain two virtual gluons at the lowest order with the PQCD
approach, which is not practical due to a huge number of Feynman diagrams.
In addition, the amplitudes for this region are power suppressed with
respect to the amplitude at the edges. At the edges of the Dalitz plot,
two hadrons move collinearly or back-to-back, so the three-body decay
could be approximately regarded as the quasi-two-body decay \cite{npb899.247}.
The $B_{s}$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay observed by the LHCb
Collaboration \cite{prd95.012006} with the ${\pi}^{+}{\pi}^{-}$ invariant mass
less than 1.6 GeV is the case, where the possible ${\pi}^{+}{\pi}^{-}$ resonant
states show up.
The three-body $B_{s}$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay could be
approximated as the quasi two-body $B_{s}$ ${\to}$ ${\phi}\,({\pi}^{+}{\pi}^{-})$
decay. It seems reasonable to assume that the two-pion pair originates
from a quark-antiquark state and postulate the validity of factorization
for this quasi two-body $B_{s}$ decay.
In this paper, we will follow Ref.\cite{plb561.258} as a hypothesis,
and write the HMEs for the three-body nonleptonic $B_{s}$ ${\to}$
${\phi}\,{\pi}^{+}{\pi}^{-}$ decay as follow.
\begin{equation}
{\langle}{\phi}\,{\pi}^{+}{\pi}^{-}{\vert}Q_{i}{\vert}B_{s}{\rangle}\, {\sim}\,
{\int}dx\,dy\,dz\,
{\phi}_{B}(x)\,{\phi}_{\phi}(y)\,{\phi}_{{\pi}{\pi}}(z)\,
\widetilde{\cal T}(x,y,z)
\label{hme-quasi},
\end{equation}
where one new input, the ${\pi}^{+}{\pi}^{-}$ pair DA ${\phi}_{{\pi}{\pi}}$
parameterizing both the resonant and nonresonant contributions, is introduced
in order to factorize the HMEs for the three-body decay.
It is possible to combine the ${\pi}^{+}{\pi}^{-}$ pair with the $S$, $P$, $D$
waves. The $S$-, $P$- and $D$-wave transition matrix elements between the two-pion
pair and the vacuum are proportional to the time-like scale, vector, and tensor
form factors, respectively.
It is clear that for the sequential $B_{s}^{0}$ ${\to}$ ${\phi}f_{0}(980)$
${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay in question, only the $S$ wave
contribution from the scalar $f_{0}(980)$ meson needs to be considered.
\subsection{Kinematic variable}
\label{sec0203}
It is convenient to describe the kinematical variables in terms of the light
cone coordinates. The relations between the four-dimensional space-time
coordinates ($x^{0}$, $x^{1}$, $x^{2}$, $x^{3}$) $=$ ($t$, $x$, $y$, $z$)
and the light-cone coordinates ($x^{+}$, $x^{-}$, $x_{\perp}$)
are defined as $x^{\pm}$ $=$ $(x^{0}{\pm}x^{3})/\sqrt{2}$ and
$x_{\perp}$ $=$ ($x^{1}$, $x^{2}$).
The scalar product of two vectors is given by $a{\cdot}b$ $=$
$a_{\mu}b^{\mu}$ $=$ $a^{+}b^{-}$ $+$ $a^{-}b^{+}$
$-$ $a_{\perp}{\cdot}b_{\perp}$.
$n_{+}^{\mu}$ $=$ $(1,0,0)$ and $n_{-}^{\mu}$ $=$ $(0,1,0)$
are the plus and minus null vectors, respectively \cite{prd96.036010}.
The momenta of the participating mesons in the rest frame of the $B_{s}$
meson are defined as follow.
\begin{equation}
p_{B_{s}}\, =\, \frac{m_{B_{s}}}{\sqrt{2}}(1,1,0)
\label{kine-bs},
\end{equation}
\begin{equation}
p_{\phi}\, =\, (p_{\phi}^{+},p_{\phi}^{-},0)
\label{kine-phi},
\end{equation}
\begin{equation}
{\epsilon}_{\phi}^{\parallel}\, =\, \frac{1}{m_{\phi}}(p_{\phi}^{+},-p_{\phi}^{-},0)
\label{kine-epsilon},
\end{equation}
\begin{equation}
p_{2{\pi}}\, =\, q\, =\, (q^{-},q^{+},0)
\label{kine-2pi},
\end{equation}
\begin{equation}
p_{{\pi}^{+}}\, =\, (\bar{\zeta}q^{-},{\zeta}q^{+},+\sqrt{{\zeta}\bar{\zeta}}w)
\label{kine-pip},
\end{equation}
\begin{equation}
p_{{\pi}^{-}}\, =\, ({\zeta}q^{-},\bar{\zeta}q^{+},-\sqrt{{\zeta}\bar{\zeta}}w)
\label{kine-pim},
\end{equation}
\begin{equation}
p_{\phi}^{\pm}\, =\, (E_{\phi}\,{\pm}\,p_{\rm cm})/{\sqrt{2}}
\label{kine-p1-pm},
\end{equation}
\begin{equation}
q^{\pm}\, =\, (E_{w}\,{\pm}\,p_{\rm cm})/{\sqrt{2}}
\label{kine-q-pm},
\end{equation}
\begin{equation}
E_{\phi}\, =\, (m_{B_{s}}^2+m_{\phi}^2-w^2)/(2\,m_{B_{s}})
\label{kine-E1},
\end{equation}
\begin{equation}
E_{w}\, =\, (m_{B_{s}}^2-m_{\phi}^2+w^2)/(2\,m_{B_{s}})
\label{kine-Ew},
\end{equation}
\begin{equation}
p_{\rm cm}\, =\, \frac{\sqrt{[m_{B_{s}}^2-(m_{\phi}+w)^2][m_{B_{s}}^2-(m_{\phi}-w)^2]}}{2\,m_{B_{s}}}
\label{kine-pcm},
\end{equation}
\begin{equation}
q^2\, =\, (p_{B_{s}}-p_{\phi})^2\, =\, (p_{{\pi}^{+}}+p_{{\pi}^{-}})^2\, =\,w^2
\label{kine-q2},
\end{equation}
where ${\epsilon}_{\phi}^{\parallel}$ is the longitudinal polarization
vector of the ${\phi}$ meson. $\bar{\zeta}$ $=$ $1$ $-$ ${\zeta}$.
The variable ${\zeta}$ ($\bar{\zeta}$) is the ${\pi}^{+}$ (${\pi}^{-}$)
meson momentum fraction of the ${\pi}^{+}{\pi}^{-}$ meson pair with
the invariant mass $w$ $=$ $m({\pi}{\pi})$.
The momenta of the spectator quark of the $B_{s}$ meson and the valence quarks
of the final states are defined as $p$, $k$ and $l$ (see Fig.\ref{fey} for detail)
with the longitudinal momentum fraction of $x$, $y$, $z$ and the transverse momentum
of $p_{T}$, $k_{T}$, $l_{T}$, respectively,
\begin{equation}
p\, =\, (xp_{B_{s}}^{+},xp_{B_{s}}^{-},p_{T})
\label{kine-k},
\end{equation}
\begin{equation}
k\, =\, (yp_{\phi}^{+},yp_{\phi}^{-},k_{T})
\label{kine-k1},
\end{equation}
\begin{equation}
l\, =\, (zq^{-},zq^{+},l_{T})
\label{kine-l}.
\end{equation}
\subsection{The distribution amplitudes}
\label{sec0204}
Within the pQCD framework, the WFs and/or DAs are the essential input parameters.
Following the notations in Refs.\cite{jhep0703.069,prd65.014007,prd92.074028,
plb751.171,plb752.322}, the WFs of the $B_{s}$ meson and the longitudinally
polarized ${\phi}$ meson are defined as:
\begin{equation}
{\langle}0{\vert}\bar{b}_{i}(0)s_{j}(z){\vert}B_{s}(p){\rangle}\, =\,
-\frac{i\,f_{B_{s}}}{4}{\int}d^{4}k\,e^{-ik{\cdot}z}\, \Big\{
\Big[ \!\!\not{p}\,{\Phi}_{B}^{a}(k) + m_{B_{s}} {\Phi}_{B}^{p}(k) \Big]
{\gamma}_{5} \Big\}_{ji}
\label{wf-bs},
\end{equation}
\begin{equation}
{\langle}{\phi}(p,{\epsilon}^{\parallel}){\vert}s_{i}(z)\bar{s}_{j}(0){\vert}0{\rangle}\,=\,
\frac{1}{4} {\int}_{0}^{1}dk\,e^{ik{\cdot}z}\,\Big\{ \!\!\not{\epsilon}^{\parallel}
m_{\phi}{\Phi}_{\phi}^{v}(k)+ \!\!\not{\epsilon}^{\parallel}\!\!\!\not{p}\,
{\Phi}_{\phi}^{t}(k)-m_{\phi}{\Phi}_{\phi}^{s}(k) \Big\}_{ji}
\label{wf-phi},
\end{equation}
where $f_{B_{s}}$ $=$ $227.2{\pm}3.4$ MeV \cite{pdg} is the decay
constant of the $B_{s}$ meson.
The WFs of ${\Phi}_{B}^{a}$ and ${\Phi}_{\phi}^{v}$ are twist-2,
while the WFs of ${\Phi}_{B}^{P}$ and ${\Phi}_{\phi}^{t,s}$ are twist-3.
By integrating out the transverse momentum from the wave functions,
one can obtain the corresponding DAs.
In our calculation, the expressions of the $B_{s}$ DAs are
\cite{prd92.074028,plb751.171,plb752.322}:
\begin{equation}
{\phi}_{B}^{a}(x) \,=\, {\cal N}_{a}\,x\,\bar{x}\,
{\exp}\Big\{-\frac{1}{8\,{\omega}_{B}} \Big(
\frac{m_{s}^{2}}{x}+\frac{m_{b}^{2}}{\bar{x}} \Big) \Big\}
\label{DA-B-a},
\end{equation}
\begin{equation}
{\phi}_{B}^{p}(x) \,=\, {\cal N}_{p}\,
{\exp}\Big\{-\frac{1}{8\,{\omega}_{B}} \Big(
\frac{m_{s}^{2}}{x}+\frac{m_{b}^{2}}{\bar{x}} \Big) \Big\}
\label{DA-B-p},
\end{equation}
where $x$ and $\bar{x}$ $=$ $1$ $-$ $x$ are the longitudinal momentum
fractions of light and heavy quarks, respectively; $m_{b}$ $=$
$4.78{\pm}0.06$ GeV \cite{pdg} and $m_{s}$ $=$ ${\simeq}$ 0.51 GeV \cite{book.kamal}
are the mass of the $b$ and $s$ quarks. The parameter ${\omega}_{B}$
determines the average transverse momentum of partons, and ${\omega}_{B}$
${\simeq}$ $m_{i}{\alpha}_{s}$. The parameters ${\cal N}_{a}$ and
${\cal N}_{p}$ are the normalization coefficients,
\begin{equation}
{\int}_{0}^{1}dx\,{\phi}_{B}^{a, p}(x)\, =\, 1
\label{DA-B-coe}.
\end{equation}
One distinguish feature of the above DAs is the exponential functions,
which strongly suppress the contribution from the end point of $x$,
$\bar{x}$ ${\to}$ $0$ and naturally provide the effective truncation
for the end point and soft contributions. In addition, the exponential
factors are proportional to the ratio of the parton mass squared $m^{2}_{i}$
to the momentum fraction $x_{i}$. Hence, the above DAs are generally
consistent with the ansatz that the momentum fractions are shared among
the valence quarks according to the quark mass, {\em i.e.}, the light $s$ quark
carries relatively less momentum fraction in the heavy-light $B_{s}$ meson.
The expressions of the two-particle DAs of the ${\phi}$ meson are
\cite{jhep0703.069,prd65.014007}:
\begin{equation}
{\phi}_{\phi}^{v}(x) \, =\, 6\,f_{\phi}\,x\,\bar{x}\,
\Big\{1+a_{2}^{\phi}C_{2}^{3/2}({\xi})+{\cdots}\Big\}
\label{phi-v},
\end{equation}
\begin{equation}
{\phi}_{\phi}^{t}(x) \, =\, 3\,f_{\phi}^{T}\,{\xi}^2
\label{phi-t},
\end{equation}
\begin{equation}
{\phi}_{\phi}^{s}(x) \, =\, 3\,f_{\phi}^{T}\,{\xi}
\label{phi-s},
\end{equation}
where ${\xi}$ $=$ $x$ $-$ $\bar{x}$; $f_{\phi}$ $=$ $(215{\pm}5)$ MeV
and $f_{\phi}^{T}$ $=$ $(186{\pm}9)$ MeV \cite{jhep0703.069} are the
longitudinal and transverse decay constants for the ${\phi}$ meson.
$C_{2}^{3/2}({\xi})$ is the Gegenbauer polynomial. The nonperturbative
parameter $a_{2}^{\phi}$ $=$ $0.18{\pm}0.08$ \cite{jhep0703.069} is
the Gegenbauer moment.
The $S$-wave two-pion WFs have been defined in Ref.\cite{npb555.231,plb467.263}
\begin{equation}
{\Phi}_{{\pi}^{+}{\pi}^{-}}\, =\,
\frac{1}{4} \Big\{ \!\!\not{q}\,{\phi}_{-}(z,{\zeta},w^{2})
+{\omega}\,{\phi}_{s}(z,{\zeta},w^{2})
-{\omega}\,(\!\not{n}_{+}\!\not{n}_{-}-1)\,{\phi}_{+}(z,{\zeta},w^{2}) \Big\}
\label{wf-2pi-01},
\end{equation}
where the variable $z$ gives the momentum fraction of the quark.
The variables ${\zeta}$ and $w^{2}$ concern the hadronic system but
not the partons. The asymptotic expressions of the two-pion DAs are
the variable ${\zeta}$ independent \cite{npb555.231,plb467.263,prd91.094024,epjc76.675},
\begin{equation}
{\phi}_{-}(z,{\zeta},w^{2})
\, =\, 18\,F_{s}(w^{2})\,a_{{\pi}{\pi}}\,z\,\bar{z}\,(\bar{z}-z)
\, =\, {\phi}_{-}
\label{da-2pi-m},
\end{equation}
\begin{equation}
{\phi}_{s}(z,{\zeta},w^{2})\, =\, F_{s}(w^{2}) \, =\, {\phi}_{s}
\label{da-2pi-s},
\end{equation}
\begin{equation}
{\phi}_{+}(z,{\zeta},w^{2})\, =\, F_{s}(w^{2})\,(\bar{z}-z)\, =\, {\phi}_{+}
\label{da-2pi-p},
\end{equation}
where $F_{s}(w^{2})$ is the time-like scalar form factor, and the
parameter $a_{{\pi}{\pi}}$ $=$ $0.2{\pm}0.2$ \cite{prd91.094024}.
Clearly, the DAs of ${\phi}_{-}$ and ${\phi}_{+}$ are antisymmetric
under the interchange $z$ ${\leftrightarrow}$ $\bar{z}$.
The $F_{s}(w^2)$ involves the strong interaction between the $S$-wave
resonance and two-pion, as well as elastic rescattering of pion pair.
Because the mass of the $f_{0}(980)$ meson is near the $K\overline{K}$
threshold, the form factor $F_{s}(w^{2})$ for the $S$-wave $f_{0}(980)$
resonance is usually parameterized with the Flatt\'{e} model
\cite{plb63.228,prd86.052006,prd87.052001,prd89.092006}.
\begin{equation}
F_{s}(w^{2})\, =\,
\frac{m_{f_{0}(980)}^2}
{m_{f_{0}(980)}^2-w^{2}-im_{f_{0}(980)}
(g_{{\pi}{\pi}}{\rho}_{{\pi}{\pi}}+
g_{KK}{\rho}_{KK})}
\label{ff-f980},
\end{equation}
where in our calculation, the mass of the $f_{0}(980)$ meson is fixed to
the value used by the LHCb Collaboration in the amplitude analysis for
the $B_{s}$ ${\to}$ ${\phi}\,{\pi}^{+}{\pi}^{-}$ decay, {\em i.e.},
$m_{f_{0}(980)}$ $=$ 0.98 GeV \cite{prd95.012006}.
The parameters of $g_{{\pi}{\pi}}$ and $g_{KK}$ are the $f_{0}(980)$
couplings to the ${\pi}{\pi}$ and $K\overline{K}$ states, respectively.
Their values are fitted by the LHCb Collaboration through the $B_{s}$ ${\to}$
$J/{\psi}{\pi}^{+}{\pi}^{-}$ decay, and $g_{{\pi}{\pi}}$ $=$ $167{\pm}7$ MeV
and $g_{KK}$ $=$ $(3.47{\pm}0.12)g_{{\pi}{\pi}}$ \cite{prd89.092006}.
The expressions of the phase space factors are written as
\cite{prd86.052006,prd87.052001,prd89.092006}:
\begin{equation}
{\rho}_{{\pi}{\pi}}\, =\,
\frac{2}{3}\,\sqrt{ 1-\frac{ 4m^{2}_{{\pi}^{\pm}} }{ w^{2} } }
+\frac{1}{3}\,\sqrt{ 1-\frac{ 4m^{2}_{{\pi}^{0}} }{ w^{2} } }
\label{rho-pipi},
\end{equation}
\begin{equation}
{\rho}_{KK}\, =\,
\frac{1}{2}\,\sqrt{ 1-\frac{ 4m^{2}_{K^{\pm}} }{ w^{2} } }
+\frac{1}{2}\,\sqrt{ 1-\frac{ 4m^{2}_{K^{0}} }{ w^{2} } }
\label{rho-kk}.
\end{equation}
\begin{figure}[h]
\includegraphics[width=0.95\textwidth,bb=90 340 520 760]{fey.pdf}
\caption{Feynman diagrams for the $B_{s}$ ${\to}$ ${\phi}\,f_{0}(980)$
${\to}$ ${\phi}\,\pi^{+}\pi^{-}$ decay with the PQCD approach.}
\label{fey}
\end{figure}
\subsection{Deacy amplitude}
\label{sec0205}
The Feynman diagrams for the $B_{s}$ ${\to}$ ${\phi}\,f_{0}(980)$ ${\to}$
${\phi}\,\pi^{+}\pi^{-}$ decay within the pQCD framework are show in
Fig.\ref{fey}, including (1) the ${\phi}$ meson emission while the $B_{s}$
meson transition into the two-pion pair through the $f_{0}(980)$ resonance
in Fig.\ref{fey}(a-d), (2) the two-pion pair emission while the $B_{s}$ meson
transition into the ${\phi}$ meson in Fig.\ref{fey}(e-h), (3) the $B_{s}$
annihilation in Fig.\ref{fey}(i-p). In addition, the diagrams in the first
(last) two columns are called the (non)factorizable topologies.
In general, the amplitudes of the factorizable topologies have the relatively
simple structures. For the factorizable topologies of Fig.\ref{fey}(a,b),
the ${\phi}$ meson can be isolated from the $B_{s}{\pi}{\pi}$ system,
so the amplitudes can be written as the product of the decay constant $f_{\phi}$
and the $B_{s}$ ${\to}$ ${\pi}{\pi}$ transition form factors.
Similarly, for the factorizable topologies of Fig.\ref{fey}(e,f),
the two-pion pair can be isolated from the $B_{s}{\phi}$ system, and the
transition matrix elements between the vacuum and the two-pion pair can
be expressed as the time-like scalar form factor $F_{s}(w^{2})$.
So the HMEs of the local operators can be written as the $B_{s}$ ${\to}$
${\phi}$ transition form factors multiplied by the form factor $F_{s}(w^{2})$.
Likewise, for the factorizable topologies of Fig.\ref{fey}(i,j) and (m,n),
the $B_{s}$ meson can be isolated from the final states, so the amplitudes
can be written as the product of the decay constant $f_{B_{s}}$ and
the time-like form factors for the transition between the ${\phi}$ meson
and the two-pion pair.
The amplitudes of the nonfactorizable topologies involve the DAs of all
participating mesons.
Using the PQCD formula in Eq.(\ref{hme-quasi}) for the quasi two-body decay,
the amplitude for the $B_{s}$ ${\to}$ ${\phi}\,f_{0}(980)$ ${\to}$
${\phi}\,\pi^{+}\pi^{-}$ decay is written as follow.
\begin{eqnarray}
{\cal A} &=& \, \Big(V_{ub}^{\ast}\,V_{us}+V_{cb}^{\ast}\,V_{cs}\Big)\,
\Big\{ {\cal A}_{ef}^{LL}[a_{3} - \frac{1}{2} a_{9}]
+ {\cal A}_{ef}^{LR}[a_{5} - \frac{1}{2} a_{7}]
+ {\cal A}_{ef}^{SP}[a_{6} - \frac{1}{2} a_{8}]
\nonumber \\ & & \quad
+ {\cal A}_{nf}^{LL}[C_{4} - \frac{1}{2} C_{10}]
+ {\cal A}_{nf}^{LR}[C_{6} - \frac{1}{2} C_{8} ]
+ {\cal A}_{nf}^{SP}[C_{5} - \frac{1}{2} C_{7} ] \Big\}
\label{amp},
\end{eqnarray}
\begin{equation}
{\cal A}_{ef}^{\rho}\, =\, {\cal A}_{a}^{\rho}+{\cal A}_{b}^{\rho}
+ {\cal A}_{e}^{\rho}+{\cal A}_{f}^{\rho}
+ {\cal A}_{i}^{\rho}+{\cal A}_{j}^{\rho}
+ {\cal A}_{m}^{\rho}+{\cal A}_{n}^{\rho},
\quad \text{for } {\rho}\,=\, LL,LP,SP
\label{a-ef}
\end{equation}
\begin{equation}
{\cal A}_{nf}^{\rho}\, =\, {\cal A}_{c}^{\rho}+{\cal A}_{d}^{\rho}
+ {\cal A}_{g}^{\rho}+{\cal A}_{h}^{\rho}
+ {\cal A}_{k}^{\rho}+{\cal A}_{l}^{\rho}
+ {\cal A}_{o}^{\rho}+{\cal A}_{p}^{\rho},
\quad \text{for } {\rho}\,=\, LL,LP,SP
\label{a-nf}
\end{equation}
\begin{equation}
a_{i}\, =\, \Bigg\{ \begin{array}{l}
C_{i}+C_{i+1}/N_{c} \quad \text{for odd }i; \\
C_{i}+C_{i-1}/N_{c} \quad \text{for even }i,
\end{array}
\label{eq:ai}
\end{equation}
where the Wilson coefficients $C_{i}$ are looked as the function variables
of the amplitudes of ${\cal A}_{ef}^{\rho}$, ${\cal A}_{nf}^{\rho}$ and
${\cal A}_{\sigma}^{\rho}$, and $N_{c}$ $=$ $3$ is the color number.
${\cal A}_{ef}^{\rho}$ (${\cal A}_{ff}^{\rho}$) is the sum of the amplitudes
for the (non)factorizable topologies. The superscript ${\rho}$ of the
amplitude building block ${\cal A}_{\sigma}^{\rho}$ refers to the three
possible Dirac structures ${\Gamma}_{1}{\otimes}{\Gamma}_{2}$ of the operators
$(\bar{q}_{1}q_{2})_{{\Gamma}_{1}}(\bar{q}_{3}q_{4})_{{\Gamma}_{2}}$,
namely ${\rho}$ $=$ $LL$ for $(V-A){\otimes}(V-A)$, ${\rho}$ $=$ $LR$
for $(V-A){\otimes}(V+A)$ and ${\rho}$ $=$ $SP$ for $-2(S-P){\otimes}(S+P)$.
The subscript ${\sigma}$ of ${\cal A}_{\sigma}^{\rho}$ (${\sigma}$ $=$ $a$
$b$, ${\cdots}$, $p$) corresponds to the sub-diagram indices of Fig.\ref{fey}.
${\cal A}_{ef}^{\rho}$, ${\cal A}_{ff}^{\rho}$ and ${\cal A}_{\sigma}^{\rho}$
are the functions of the Wilson coefficient $C_{i}$.
The analytical expressions of the amplitude building blocks ${\cal A}_{\sigma}^{\rho}$
are listed in Appendix \ref{block} in detail.
\section{numerical results and discussion}
\label{sec03}
The differential branching ratio for the sequential $B_{s}$ ${\to}$
${\phi}\,f_{0}(980)$ ${\to}$ ${\phi}\,\pi^{+}\pi^{-}$ decay is \cite{pdg}:
\begin{equation}
\frac{d{\cal B}}{d w}\, =\,
\frac{ {\tau}_{B_{s}}\,p_{\pi}^{\ast}\,p_{\rm cm}}
{4\,(2{\pi})^{3}\, m_{B_{s}}^{2}}
{\vert}{\cal A}{\vert}^{2}
\label{eq:br},
\end{equation}
where ${\tau}_{B_{s}}$ $=$ $(1.510{\pm}0.005)$ ps is the lifetime of
the $B_{s}$ meson \cite{pdg}. The kinematic variable $p_{\pi}^{\ast}$
is the pion momentum in the rest frame of the two-pion pair,
\begin{equation}
p_{\pi}^{\ast}\, =\, \frac{1}{2}\,\sqrt{w^{2}-4m_{{\pi}^{\pm}}^{2}}
\label{kine-p-pi}.
\end{equation}
In our calculation, besides the aforementioned parameters, other related
parameters, such as the mass of the mesons and quarks, will take their
values given in Ref.\cite{pdg}. And if it is not specified explicitly,
their central values will be fixed as the default inputs.
Our numerical result of the branching ratio is
\begin{equation}
{\cal B}(B_{s}{\to}{\phi}\,f_{0}(980){\to}{\phi}\,\pi^{+}\pi^{-})\, =\,
\Big[1.31_{-0.31}^{+0.40}(a_{{\pi}{\pi}}){}_{-0.16}^{+0.19}(m_{b})
{}_{-0.09}^{+0.10}(\text{CKM})\Big]{\times}10^{-6}
\label{num-br},
\end{equation}
where the uncertainties come from the parameter $a_{{\pi}{\pi}}$ of
DA in Eq.(\ref{da-2pi-m}), the $b$ quark mass $m_{b}$, the CKM factors
$V_{ub}^{\ast}V_{us}$ and $V_{cb}^{\ast}V_{cs}$, respectively.
It is clear that the result in Eq.(\ref{num-br}) agrees with the LHCb
measurement in Eq.(\ref{eq:br-exp-01}) within uncertainties.
\begin{figure}[h]
\includegraphics[width=0.7\textwidth,bb=15 50 540 530]{dw.pdf}
\caption{The distributions from some resonances versus the invariant mass
of the ${\pi}^{+}{\pi}^{-}$ pair for the $B_{s}$ ${\to}$ ${\phi}\,f_{0}(980)$
${\to}$ ${\phi}\,\pi^{+}\pi^{-}$ decay, where the solid (green) line
is our result with the PQCD approach, the dot-dashed (blue) line is
the $f_{0}(980)$ meson contribution given by the LHCb Collaboration
in Fig.7(d) of Ref.\cite{prd95.012006}, and a full explanation of
other lines can be found in Ref.\cite{prd95.012006}.}
\label{fig:f980}
\end{figure}
To illustrate the $S$-wave $f_{0}(980)$ contribution to the decay in question,
and to compare our result with the experimental measurement, the dependence
of the calibrated differential branching ratio $d{\cal B}/dw$ on the pion-pair
invariant mass $w$ $=$ $m({\pi}^{+}{\pi}^{-})$ is shown in Fig.\ref{fig:f980}.
It is seen that the result with the PQCD approach is generally consistent
with the shape line of the $f_{0}(980)$ meson fitted by the LHCb Collaboration
\cite{prd95.012006}.
In addition, the contributions from different topologies to the $B_{s}$ ${\to}$
${\phi}\,f_{0}(980)$ ${\to}$ ${\phi}\,\pi^{+}\pi^{-}$ decay are investigated.
It shows that
(1) the main contributions come from the factorizable ${\phi}$
emission topologies of Fig.\ref{fey}(a,b).
(2) Due to the renormalization conditions of the two-pion DAs, only the
amplitudes corresponding to the Dirac current structure of
${\Gamma}_{1}{\otimes}{\Gamma}_{2}$ $=$ $-2(S-P){\otimes}(S+P)$ have
nonzero contributions [see Eq.(\ref{amp-e-lr})-Eq.(\ref{amp-f-sp})] for
the factorizable two-pion emission topologies of Fig.\ref{fey}(e,f).
(3) Because of the opposite sign of the quark propagators between the
factorizable annihilation topologies of Fig.\ref{fey}(i) [Fig.\ref{fey}(j)]
and Fig.\ref{fey}(n) [Fig.\ref{fey}(m)], the interference cancelation
mechanism results in the relatively small total contributions from the
factorizable annihilation topologies.
(4) For each type diagrams, such as the ${\phi}$ emission diagrams
in Fig.\ref{fey}(a-d) or the two-pion emission diagrams in Fig.\ref{fey}(e-h),
the nonfactorizable contributions are small relative to the factorizable
contributions because of the $1/N_{c}$ suppression.
(5) The relative magnitudes of decay amplitudes basically correspond with
the power estimations in Ref.\cite{plb561.258}, {\em i.e.},
\begin{equation}
\sum\limits_{i=a,b,c,d \atop {\alpha}=LL,LR,SP}\!\!\!\!\!\! {\cal A}_{i}^{\alpha}
\, :\!\!\!\!\!\!\!
\sum\limits_{j=e,f,g,h \atop {\beta}=LL,LR,SP}\!\!\!\!\!\! {\cal A}_{j}^{\beta}
\, :\!\!\!\!\!\!\!
\sum\limits_{k=i,{\cdots},p \atop {\rho}=LL,LR,SP}\!\!\!\!\!\! {\cal A}_{k}^{\rho}
\, =\, 1\, :\, \frac{w}{m_{B_{s}}}\, :\, \frac{{\Lambda}_{\rm QCD}}{m_{B_{s}}}
\label{eq:power}.
\end{equation}
It should be pointed out that the $B_{s}$ ${\to}$ ${\phi}\,\pi^{+}\pi^{-}$ decay
could be approximately handled as the quasi two-body sequential $B_{s}$ ${\to}$
${\phi}\,f_{0}(980)$ ${\to}$ ${\phi}\,\pi^{+}\pi^{-}$ decay at the edges
of the Dalitz plot, and there are still many factors that can affect the
theoretical result. For example, the contributions from the center regions of
the Dalitz plot and the nonresonant contributions to the $B_{s}$ ${\to}$
${\phi}\,\pi^{+}\pi^{-}$ decay are not considered in this paper.
It has shown in Refs.\cite{prd88.114014,prd89.094007,prd94.094015} that the
nonresonant contributions are important and deserve much attention, which is
beyond the scope of this paper.
\section{summary}
\label{sec04}
The rare cascade $B_{s}$ ${\to}$ ${\phi}f_{0}(980)$ ${\to}$
${\phi}\,{\pi}^{+}{\pi}^{-}$ decay is induced by the FCNC
$b$ ${\to}$ $\bar{s}s\bar{s}$ process within SM, where the
isoscalar $f_{0}(980)$ meson has a substantial $s\bar{s}$
component. Given the two-pion pair with small invariant mass
GeV comes from the $S$-wave resonant $f_{0}(980)$ state,
the three-body $B_{s}$ ${\to}$ ${\phi}f_{0}(980)$ ${\to}$
${\phi}\,{\pi}^{+}{\pi}^{-}$ decay can be approximated
as the quasi two-body decay.
By introducing the nonperturbative two-pion DAs to describe
the two-pion system, and parameterizing the scalar form factor
for the $f_{0}(980)$ resonance with the Flatt\'{e} model, the
$B_{s}$ ${\to}$ ${\phi}f_{0}(980)$ ${\to}$
${\phi}\,{\pi}^{+}{\pi}^{-}$ decay is studied with the PQCD
approach. It is found that with appropriate parameters, the
theoretical result of the branching ratio in the mass range
400 MeV $<$ $m({\pi}^{+}{\pi}^{-})$ $<$ 1600 MeV is in agreement
with the recent LHCb data \cite{prd95.012006} within uncertainties.
\section*{Acknowledgments}
The work is supported by the National Natural Science Foundation
of China (Grant Nos. 11705047, U1632109, 11547014 and 11475055).
\begin{appendix}
\section{The amplitude building blocks for the $B_{s}$ ${\to}$
${\phi}\,f_{0}(980)$ ${\to}$ ${\phi}\,\pi^{+}\pi^{-}$ decay}
\label{block}
\begin{equation}
{\cal F}\, =\, -i\,{\pi}\,C_{F}\,f_{B_{s}}
\label{amp-f},
\end{equation}
\begin{eqnarray}
{\cal A}_{a}^{LL}[C_{i}] &=& 2\, {\cal F}\, f_{\phi}\,
{\int}^{1}_{0} dx\,dz {\int}_{0}^{\infty} db\,db_{f}\, b\,b_{f}\,
{\alpha}_{s}(t_{a})\, H_{ef}({\alpha}_{a},{\beta}_{a},b,b_{f})\,
E_{\phi}(t_{a})\,C_{i}(t_{a})
\nonumber \\ &{\times}& m_{B_{s}}
\Big\{ {\phi}_{B}^{a}(x)\, m_{B_{s}}
\Big[ {\phi}_{-}\, m_{B_{s}} p_{\rm cm}\, (1-z+z\,r_{\phi}^{2})
+ {\phi}_{s}\, r_{b}\, w\, p_{\rm cm}
\nonumber \\ &+&
{\phi}_{+}\, r_{b}\, w\, E_{\phi} \Big]
- 2\, {\phi}_{B}^{p}(x)\, \Big[ {\phi}_{+}\, w\,
( m_{B_{s}} E_{\phi} + z\, w^{2} - z\, E_{w}\, m_{B_{s}} )
\nonumber \\ &+&
{\phi}_{-}\, r_{b}\, m_{B_{s}}^{2} p_{\rm cm}
+ {\phi}_{s}\, m_{B_{s}} w\, p_{\rm cm}\, (1-z)
\Big] \Big\}
\label{amp-a-ll},
\end{eqnarray}
\begin{equation}
{\cal A}_{a}^{LR}[C_{i}]\, =\, {\cal A}_{a}^{LL}[C_{i}]
\label{amp-a-lr},
\end{equation}
\begin{equation}
{\cal A}_{a}^{SP}[C_{i}]\, =\, 0
\label{amp-a-sp},
\end{equation}
\begin{eqnarray}
{\cal A}_{b}^{LL}[C_{i}] &=& 2\, {\cal F}\, f_{\phi}\,
{\int}^{1}_{0} dx\,dz {\int}_{0}^{\infty} db\,db_{f}\, b\,b_{f}\,
{\alpha}_{s}(t_{b})\, H_{ef}({\alpha}_{a},{\beta}_{b},b_{f},b)\,
E_{\phi}(t_{b})\,C_{i}(t_{b})
\nonumber \\ &{\times}& m_{B_{s}} p_{\rm cm}\,
\Big\{ {\phi}_{B}^{a}(x)\, {\phi}_{-}\, ( \bar{x}\,w^2+x\,m_{\phi}^2 )
- {\phi}_{B}^{p}(x)\,{\phi}_{s}\,2\,m_{B_{s}} w\, \bar{x} \Big\}
\label{amp-b-ll},
\end{eqnarray}
\begin{equation}
{\cal A}_{b}^{LR}[C_{i}]\, =\, {\cal A}_{b}^{LL}[C_{i}]
\label{amp-b-lr},
\end{equation}
\begin{equation}
{\cal A}_{b}^{SP}[C_{i}]\, =\, 0
\label{amp-b-sp},
\end{equation}
\begin{eqnarray}
{\cal A}_{c}^{LL}[C_{i}] &=&
\frac{{\cal F}}{N_{c}} {\int}^{1}_{0} dx\,dy\,dz
{\int}_{0}^{\infty} db\, db_{\phi}\, db_{f}\, b_{\phi}\, b_{f}\,
{\alpha}_{s}(t_{c})\, H_{en}({\alpha}_{a},{\beta}_{c},b_{\phi},b,b_{f})\,
E_{n}(t_{c})\,C_{i}(t_{c})
\nonumber \\ &{\times}& {\phi}_{\phi}^{v}(y)\,
\Big\{ 4\,m_{B_{s}}^{2} p_{\rm cm}\,{\phi}_{B}^{a}(x)\,
{\phi}_{-}\, (y\,E_{\phi}+z\,E_{w}-x\,m_{B_{s}})
+\Big[ {\phi}_{s}\,m_{B_{s}} p_{\rm cm}\,(x-z)
\nonumber \\ &+&
{\phi}_{+}\, (x\,m_{B_{s}} E_{\phi}-y\,m_{\phi}^{2}-z\,m_{B_{s}} E_{w}+z\,w^2)
\Big] 2\,m_{B_{s}} w\, {\phi}_{B}^{p}(x)\, \Big\}
\label{amp-c-ll},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{c}^{LR}[C_{i}] &=&
\frac{{\cal F}}{N_{c}} {\int}^{1}_{0} dx\,dy\,dz
{\int}_{0}^{\infty} db\, db_{\phi}\, db_{f}\, b_{\phi}\, b_{f}\,
{\alpha}_{s}(t_{c})\, H_{en}({\alpha}_{a},{\beta}_{c},b_{\phi},b,b_{f})\,
E_{n}(t_{c})\,C_{i}(t_{c})
\nonumber \\ &{\times}& 2\,m_{B_{s}} {\phi}_{\phi}^{v}(y)\,
\Big\{ 2\,p_{\rm cm}\,{\phi}_{B}^{a}(x)\,
{\phi}_{-}\, (y\,p_{\rm cm}^2+y\,E_{w}\,E_{\phi}
+z\,w^2-x\,E_{w}\,m_{B_{s}})
\nonumber \\ &+& w\, {\phi}_{B}^{p}(x)
\Big[ {\phi}_{s}\,m_{B_{s}} p_{\rm cm}\,(x-z)
- {\phi}_{+}\, (x\,m_{B_{s}} E_{\phi}-y\,m_{\phi}^{2}
- z\,m_{B_{s}} E_{w}+z\,w^2) \Big] \Big\},\qquad
\label{amp-c-lr}
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{c}^{SP}[C_{i}] &=&
\frac{-2\,{\cal F}}{N_{c}} {\int}^{1}_{0} dx\,dy\,dz
{\int}_{0}^{\infty} db\, db_{\phi}\, db_{f}\, b_{\phi}\, b_{f}\,
{\alpha}_{s}(t_{c})\, H_{en}({\alpha}_{a},{\beta}_{c},b_{\phi},b,b_{f})\,
E_{n}(t_{c})\,C_{i}(t_{c})
\nonumber \\ &{\times}&
m_{B_{s}} m_{\phi}\, \Big\{ {\phi}_{\phi}^{t}(y)\,
\Big[ {\phi}_{B}^{a}(x)\,w\, \{ {\phi}_{s}\,p_{\rm cm}\,(y-z)
- {\phi}_{+}\,(y\,E_{\phi}+z\,E_{w}-x\,m_{B_{s}}) \}
\nonumber \\ &-&
{\phi}_{B}^{p}(x)\,{\phi}_{-}\,m_{B_{s}} p_{\rm cm}\,(x-y)\Big]
+ {\phi}_{\phi}^{s}(y)\,\Big[ {\phi}_{B}^{a}(x)\,w\, \{
{\phi}_{s}\,(y\,E_{\phi}+z\,E_{w}-x\,m_{B_{s}})
\nonumber \\ &+&
{\phi}_{+}\,p_{\rm cm}\,(z-y) \} - {\phi}_{B}^{p}(x)\,{\phi}_{-}\,
(x\,m_{B_{s}} E_{w}-y\,E_{\phi}\,E_{w}-y\,p_{\rm cm}^2 -z\,w^2)
\Big] \Big\}
\label{amp-c-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{d}^{LL}[C_{i}] &=&
\frac{{\cal F}}{N_{c}} {\int}^{1}_{0} dx\,dy\,dz
{\int}_{0}^{\infty} db\, db_{\phi}\, db_{f}\, b_{\phi}\, b_{f}\,
{\alpha}_{s}(t_{d})\, H_{en}({\alpha}_{a},{\beta}_{d},b_{\phi},b,b_{f})\,
E_{n}(t_{d})\,C_{i}(t_{d})
\nonumber \\ &{\times}& 2\,m_{B_{s}}\, {\phi}_{\phi}^{v}(y)\,
\Big\{ {\phi}_{B}^{a}(x)\,{\phi}_{-}\,2\, p_{\rm cm}\,
\Big[ x\,m_{B_{s}} E_{w}-\bar{y}\,(m_{B_{s}} E_{\phi}-m_{\phi}^2) -z\,w^2 \Big]
\nonumber \\ &+& {\phi}_{B}^{p}(x)\,w
\Big[ {\phi}_{s}\,m_{B_{s}} p_{\rm cm}\, (z-x)
+ {\phi}_{+}\, (x\,m_{B_{s}} E_{\phi} -\bar{y}\,m_{\phi}^{2}
- z\,m_{B_{s}} E_{w}+z\,w^2 ) \Big] \Big\},\qquad
\label{amp-d-ll}
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{d}^{LR}[C_{i}] &=&
\frac{{\cal F}}{N_{c}} {\int}^{1}_{0} dx\,dy\,dz
{\int}_{0}^{\infty} db\, db_{\phi}\, db_{f}\, b_{\phi}\, b_{f}\,
{\alpha}_{s}(t_{d})\, H_{en}({\alpha}_{a},{\beta}_{d},b_{\phi},b,b_{f})\,
E_{n}(t_{d})\,C_{i}(t_{d})
\nonumber \\ &{\times}& 2\,m_{B_{s}}\, {\phi}_{\phi}^{v}(y)\, \Big\{
{\phi}_{B}^{a}(x)\,{\phi}_{-}\,2\, m_{B_{s}} p_{\rm cm}\,
\Big[ x\,m_{B_{s}}-\bar{y}\,E_{\phi}-z\,E_{w} \Big]
\nonumber \\ &+& {\phi}_{B}^{p}(x)\,w
\Big[ {\phi}_{s}\,m_{B_{s}} p_{\rm cm}\,(z-x)
- {\phi}_{+}\,(x\,m_{B_{s}}E_{\phi}-\bar{y}\,m_{\phi}^{2}
- z\,m_{B_{s}} E_{w}+z\,w^2 ) \Big] \Big\},\qquad
\label{amp-d-lr}
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{d}^{SP}[C_{i}] &=&
\frac{-2\,{\cal F}}{N_{c}} {\int}^{1}_{0} dx\,dy\,dz
{\int}_{0}^{\infty} db\, db_{\phi}\, db_{f}\, b_{\phi}\, b_{f}\,
{\alpha}_{s}(t_{d})\, H_{en}({\alpha}_{a},{\beta}_{d},b_{\phi},b,b_{f})\,
E_{n}(t_{d})\,C_{i}(t_{d})
\nonumber \\ &{\times}& m_{B_{s}} m_{\phi}\, \Big\{
{\phi}_{\phi}^{t}(y)\,\Big[ {\phi}_{B}^{a}(x)\,w\,
\{ {\phi}_{s}\,p_{\rm cm}\,(\bar{y}-z)
+ {\phi}_{+}\,( y\,E_{w}-E_{\phi}+x\,m_{B_{s}}-z\,E_{w} ) \}
\nonumber \\ &-&
{\phi}_{B}^{p}(x)\,{\phi}_{-}\,m_{B_{s}} p_{\rm cm}\,(x-\bar{y}) \Big]
+ {\phi}_{\phi}^{s}(y)\,\Big[ {\phi}_{B}^{a}(x)\,w\, \{ {\phi}_{s}\,
(x\,m_{B_{s}}-z\,E_{w}+y\,E_{\phi}-E_{\phi})
\nonumber \\ &+&
{\phi}_{+}\,p_{\rm cm}(\bar{y}-z) \}
+{\phi}_{B}^{p}(x)\,{\phi}_{-}\,( x\,m_{B_{s}}E_{w}
-\bar{y}\,E_{\phi}\,E_{w} -\bar{y}\,p_{\rm cm}^2-z\,w^2)
\Big] \Big\}
\label{amp-d-sp},
\end{eqnarray}
\begin{equation}
{\cal A}_{e}^{LL}[C_{i}]\, =\, {\cal A}_{e}^{LR}[C_{i}]\, =\, 0
\label{amp-e-lr},
\end{equation}
\begin{eqnarray}
{\cal A}_{e}^{SP}[C_{i}] &=& -4\, m_{B_{s}}\, {\cal F}\,
{\int}^{1}_{0} dx\,dy {\int}_{0}^{\infty} db\, db_{\phi}\, b\, b_{\phi}\,
{\alpha}_{s}(t_{e})\, H_{ef}({\alpha}_{e},{\beta}_{e},b,b_{\phi})\,
E_{f}(t_{e})\,C_{i}(t_{e})
\nonumber \\ &{\times}& {\phi}_{s}\,w\,\Big\{ {\phi}_{B}^{a}(x)
\Big[ {\phi}_{\phi}^{s}(y)\, m_{\phi}\, (m_{B_{s}}-y\,E_{\phi})
- {\phi}_{\phi}^{v}(y)\,m_{b}\,p_{\rm cm}
- {\phi}_{\phi}^{t}(y)\,m_{\phi}\, p_{\rm cm}\,y \Big]
\nonumber \\ &+& 2\,
{\phi}_{B}^{p}(x)\, \Big[ {\phi}_{\phi}^{v}(y)\, m_{B_{s}} p_{\rm cm}
-{\phi}_{\phi}^{s}(y)\,m_{\phi}\,m_{b}\ \Big] \Big\}
\label{amp-e-sp},
\end{eqnarray}
\begin{equation}
{\cal A}_{f}^{LL}[C_{i}]\, =\, {\cal A}_{f}^{LR}[C_{i}]\, =\, 0
\label{amp-f-lr},
\end{equation}
\begin{eqnarray}
{\cal A}_{f}^{SP}[C_{i}] &=& -4\, m_{B_{s}} {\cal F}\,
{\int}^{1}_{0} dx\,dy {\int}_{0}^{\infty} db\, db_{\phi}\, b\, b_{\phi}\,
{\alpha}_{s}(t_{f})\, H_{ef}({\alpha}_{e},{\beta}_{f},b_{\phi},b)\,
E_{f}(t_{f})\,C_{i}(t_{f})
\nonumber \\ &{\times}& {\phi}_{s}\,w\,\Big\{
{\phi}_{B}^{a}(x)\,{\phi}_{\phi}^{s}(y)\,2\,m_{\phi}\,(E_{\phi}-x\,m_{B_{s}})
-{\phi}_{B}^{p}(x)\,{\phi}_{\phi}^{v}(y)\, m_{B_{s}} p_{\rm cm}\,x \Big\}
\label{amp-f-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{g}^{LL}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{g})\, H_{en}({\alpha}_{e},\, {\beta}_{g},\, b_{f},\, b_{\phi},\, b)\, E_{n}(t_{g})\,C_{i}(t_{g})\nonumber\\
&& \times \,
2\,m_{B_{s}}^2\,{\phi}_{-}\,\Big\{{\phi}_{B}^{a}(x)\,{\phi}_{\phi}^{v}(y)\,2\,p_{\rm cm}
\,(y\,E_{\phi}+z\,E_{w}-x\,m_{B_{s}})\nonumber\\
&&+\,{\phi}_{B}^{p}(x)\,r\,\bigg[{\phi}_{\phi}^{s}(y)\,
\big[E_{w}\,(y\,E_{\phi}-x\,m_{B_{s}})+y\,p_{\rm cm}^2+z\,w^2 \big]\nonumber\\
&&
-\,{\phi}_{\phi}^{t}(y)\,m_{B_{s}}\,p_{\rm cm}(y-x)
\bigg]\Big\}
\label{amp-g-ll},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{g}^{LR}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{g})\, H_{en}({\alpha}_{e},\, {\beta}_{g},\, b_{f},\, b_{\phi},\, b)\, E_{n}(t_{g})\,C_{i}(t_{g})\nonumber\\
&& \times \,
2\,m_{B_{s}}^2\,{\phi}_{-}\,\Big\{{\phi}_{B}^{a}(x)\,{\phi}_{\phi}^{v}(y)\,2\,E_{w}\,p_{\rm cm}
\,(z-x)\nonumber\\
&&+\,{\phi}_{B}^{p}(x)\,r\,\bigg[{\phi}_{\phi}^{s}(y)\,
\big[E_{w}\,(y\,E_{\phi}-x\,m_{B_{s}})+y\,p_{\rm cm}^2+z\,w^2\big]\nonumber\\
&&+\,{\phi}_{\phi}^{t}(y)\,m_{B_{s}}\,p_{\rm cm}(y-x)\bigg]\Big\}
\label{amp-g-lr},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{g}^{SP}[C_{i}] &=& \frac{(-2){\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{g})\, H_{en}({\alpha}_{e},\, {\beta}_{g},\, b_{f},\, b_{\phi},\, b)\, E_{n}(t_{g})\,C_{i}(t_{g})\nonumber\\
&& \times \,m_{B_{s}}\,w\,
\Big\{r\,m_{B_{s}}\,{\phi}_{B}^{a}(x)\,\bigg[({\phi}_{\phi}^{t}(y)\,{\phi}_{s}\,-
{\phi}_{\phi}^{s}(y)\,{\phi}_{+})\,p_{\rm cm}\,
(y-z)\nonumber\\
&&+ \,({\phi}_{\phi}^{t}(y)\,{\phi}_{+}\,-
{\phi}_{\phi}^{s}(y)\,{\phi}_{s}) (y\,E_{\phi}-x\,m_{B_{s}}+z\,E_{w})\bigg]\nonumber\\
&&+\,{\phi}_{B}^{p}(x)\,{\phi}_{\phi}^{v}(y)\,\Big[{\phi}_{s}\,m_{B_{s}}\,p_{\rm cm}\,(z-x)\nonumber\\
&&+\,{\phi}_{+}\,(x\,m_{B_{s}}\,E_{\phi}-y\,m_{\phi}^2+z\,w^2
-z\,E_{w}\,m_{B_{s}})\Big]\Big\}
\label{amp-g-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{h}^{LL}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{h})\, H_{en}({\alpha}_{e},\, {\beta}_{h},\, b_{f},\, b_{\phi},\, b)\, E_{n}(t_{h})\,C_{i}(t_{h})\nonumber\\
&& \times \,2\,m_{B_{s}}^2\,
{\phi}_{-}\,\Big\{{\phi}_{B}^{a}(x)\,{\phi}_{\phi}^{v}(y)\,2\,E_{w}\,p_{\rm cm}
\,(x+z-1)\nonumber\\
&& +\,{\phi}_{B}^{p}(x)\,r\,\bigg[{\phi}_{\phi}^{s}(y)
\big[E_{w}(x\,m_{B_{s}}-y\,E_{\phi})-y\,p_{\rm cm}^2+(z-1)\,w^2\big]\nonumber\\
&&
-\,{\phi}_{\phi}^{t}(y)\,m_{B_{s}}\,p_{\rm cm}(y-x)\bigg]\Big\}
\label{amp-h-ll},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{h}^{LR}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{h})\, H_{en}({\alpha}_{e},\, {\beta}_{h},\, b_{f},\, b_{\phi},\, b)\, E_{n}(t_{h})\,C_{i}(t_{h})\nonumber\\
&& \times \,2\,m_{B_{s}}^2\,
{\phi}_{-}\,\Big\{{\phi}_{B}^{a}(x)\,{\phi}_{\phi}^{v}(y)\,2\,p_{\rm cm}
\,\Big[x\,m_{B_{s}}-y\,E_{\phi}+(z-1)\,E_{w}\Big]\nonumber\\
&& +\,{\phi}_{B}^{p}(x)\,r\,\bigg[{\phi}_{\phi}^{t}(y)\,m_{B_{s}}\,p_{\rm cm}(y-x)\nonumber\\
&& +\,{\phi}_{\phi}^{s}(y)\,
\big[E_{w}(x\,m_{B_{s}}-y\,E_{\phi})-y\,p_{\rm cm}^2+(z-1)\,w^2\big]\bigg]\Big\}
\label{amp-h-lr},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{h}^{SP}[C_{i}] &=& \frac{(-2){\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{h})\, H_{en}({\alpha}_{e},\, {\beta}_{h},\, b_{f},\, b_{\phi},\, b)\, E_{n}(t_{h})\,C_{i}(t_{h})\nonumber\\
&& \times \,m_{B_{s}}\,w\,
\Big\{r\,m_{B_{s}}{\phi}_{B}^{a}(x)\,\bigg[({\phi}_{\phi}^{t}(y)\,{\phi}_{s}\,+\,{\phi}_{\phi}^{s}(y)\,{\phi}_{+})
\,p_{\rm cm}\,(1-y-z) \nonumber\\
&&+({\phi}_{\phi}^{t}(y)\,{\phi}_{+}\,+\,{\phi}_{\phi}^{s}(y)\,{\phi}_{s})
\,(y\,E_{\phi}-x\,m_{B_{s}}-z\,E_{w}+E_{w})\bigg]\nonumber\\
&& +\,{\phi}_{B}^{p}(x)\,{\phi}_{\phi}^{v}(y)\,\Big[{\phi}_{s}\,m_{B_{s}}\,p_{\rm cm}\,(x+z-1)\nonumber\\
&&+\,{\phi}_{+}\,[x\,m_{B_{s}}\,E_{\phi}-y\,m_{\phi}^2+(w^2
-m_{B_{s}}\,E_{w})(1-z)]\Big]\Big\}
\label{amp-h-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{i}^{LL}[C_{i}] &=& {\cal F} {\int}^{1}_{0} dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{i})\, H_{af}({\alpha}_{i},\, {\beta}_{i},\, b_{\phi},\, b_{f})\, E_{B}(t_{i})\,C_{i}(t_{i})\nonumber\\
&&\times\,2\,m_{B_{s}}^{2}\Big[ {\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,m_{B_{s}}\,p_{\rm cm}\,(z\,r^2-r^2-z)\nonumber\\
&&-\,2\,r\,w\,{\phi}_{\phi}^{s}(y)\,[{\phi}_{s}\,(z\,E_{w}+E_{\phi})-\,{\phi}_{+}\,p_{\rm cm}\,(1-z)]
\Big]
\label{amp-i-ll},
\end{eqnarray}
\begin{equation}
{\cal A}_{i}^{LR} = {\cal A}_{i}^{LL}
\label{amp-i-lr},
\end{equation}
\begin{eqnarray}
{\cal A}_{i}^{SP}[C_{i}] &=& (-2){\cal F} {\int}^{1}_{0} dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, H_{af}({\alpha}_{i},\, {\beta}_{i},\, b_{\phi},\, b_{f})\, E_{B}(t_{i})\,C_{i}(t_{i})\nonumber\\
&& \times\,2\,m_{B_{s}}\Big[w\,{\phi}_{\phi}^{v}(y)\,[{\phi}_{s}
z\,m_{B_{s}}\,p_{\rm cm}\,+
\,{\phi}_{+}\,(z\,w^2-m_{\phi}^2-z\,m_{B_{s}}\,E_{w})]\nonumber\\
&&
+\,{\phi}_{\phi}^{s}(y)\,{\phi}_{-}\,2\,r\,m_{B_{s}}\,(p_{\rm cm}^2+z\,w^2+E_{\phi}\,E_{w})
\Big]
\label{amp-i-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{j}^{LL}[C_{i}] &=& {\cal F} {\int}^{1}_{0} dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{j})\, H_{af}({\alpha}_{i},\, {\beta}_{j},\, b_{f},\, b_{\phi})\, E_{B}(t_{j})\,C_{i}(t_{j})\nonumber\\
&& \times\,2\,m_{B_{s}}\, \Big\{{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,p_{\rm cm}
\Big[(1-y)(m_{B_{s}}\,E_{\phi}+p_{\rm cm}^2+E_{\phi}\,E_{w})+w^2\Big] \nonumber\\
&&
-\,2\,r\,w\,m_{B_{s}}\,{\phi}_{s}\,\bigg[{\phi}_{\phi}^{t}(y)\,y\,p_{\rm cm}
-\,{\phi}_{\phi}^{s}(y)\,[E_{w}-E_{\phi}\,(y-1)]\bigg]\Big\}
\label{amp-j-ll},
\end{eqnarray}
\begin{equation}
{\cal A}_{j}^{LR}[C_{i}] = {\cal A}_{j}^{LL}[C_{i}]
\label{amp-j-lr},
\end{equation}
\begin{eqnarray}
{\cal A}_{j}^{SP}[C_{i}] &=& (-2){\cal F} {\int}^{1}_{0} dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{j})\, H_{af}({\alpha}_{i},\, {\beta}_{j},\, b_{f},\, b_{\phi})\, E_{B}(t_{j})\,C_{i}(t_{j})\nonumber\\
&& \times\,2\,m_{B_{s}}^2\,\Big\{r\,{\phi}_{-}\,\bigg[{\phi}_{\phi}^{t}(y)
(y-1)\,m_{B_{s}}\,p_{\rm cm}
- {\phi}_{\phi}^{s}(y)\,[(y-1)\,p_{\rm cm}^2+(y-1)\,E_{\phi}\,E_{w}-w^2]
\nonumber\\
&& +\,{\phi}_{s}\,{\phi}_{\phi}^{v}(y)2\,w\,p_{\rm cm}\Big\}
\label{amp-j-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{k}^{LL}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{k})\, H_{an}({\alpha}_{i},\, {\beta}_{k},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{k})\,C_{i}(t_{k})\nonumber\\
&& \times \,2\,m_{B_{s}}^{2}
\Big\{{\phi}_{B}^{a}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,2\,E_{w}\,p_{\rm cm}\,
(x+z-1)\nonumber\\
&& +\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))\,r\,w\,p_{\rm cm}\,(y+z-1)\nonumber\\
&&
+\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))\,r\,w\,[(x-1)\,m_{B_{s}}-(y-1)\,E_{\phi}+z\,E_{w}]
\Big]\nonumber\\
&&+\,r_{b}\,m_{B_{s}}\,{\phi}_{B}^{p}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,p_{\rm cm}
+\,{\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,2\,r\,w
\Big]\Big\}
\label{amp-k-ll},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{k}^{LR}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{k})\, H_{an}({\alpha}_{i},\, {\beta}_{k},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{k})\,C_{i}(t_{k})\nonumber\\
&& \times \,2\,m_{B_{s}}\,
\Big\{{\phi}_{B}^{a}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,2\,p_{\rm cm}\,
(x\,m_{B_{s}}\,E_{w}-y\,E_{w}\,E_{\phi}-y\,p_{\rm cm}^2-m_{\phi}^2+z\,w^2)\nonumber\\
&& +\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))
\,r\,w\,m_{B_{s}}\,p_{\rm cm}\,(1-y-z)\nonumber\\
&&
+\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))\,r\,w\,m_{B_{s}}
\,[(x-1)\,m_{B_{s}}-(y-1)\,E_{\phi}+z\,E_{w}]\nonumber\\
&&+\,r_{b}\,m_{B_{s}}^{2}
{\phi}_{B}^{p}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,p_{\rm cm}
+\,{\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,2\,r\,w
\Big]\Big\}
\label{amp-k-lr},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{k}^{SP}[C_{i}] &=& \frac{(-2){\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{k})\, H_{an}({\alpha}_{i},\, {\beta}_{k},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{k})\,C_{i}(t_{k})\nonumber\\
&& \times \,m_{B_{s}}
\Big\{{\phi}_{B}^{a}(x)\,
\bigg[-r\,m_{B_{s}}^{2}\,{\phi}_{-}\,({\phi}_{\phi}^{t}(y)\,r_{b}\,p_{\rm cm}\,+\,{\phi}_{\phi}^{s}(y)\,E_{w})\nonumber\\
&&+\,r_{b}\,w\,m_{B_{s}}\,{\phi}_{\phi}^{v}(y)\,(
{\phi}_{s}\,p_{\rm cm}+\,{\phi}_{+}\,E_{\phi})\big]\nonumber\\
&& +\,
{\phi}_{B}^{p}(x)\bigg[r\,m_{B_{s}}\,{\phi}_{-}\,\big[
{\phi}_{\phi}^{t}(y)\,m_{B_{s}}\,p_{\rm cm}\,(x-y)\nonumber\\
&&-\,{\phi}_{\phi}^{s}(y)\,\big((1-x)\,m_{B_{s}}\,E_{w}-(1-y)\,E_{\phi}\,E_{w}-(1-y)\,p_{\rm cm}^2-z\,w^2\big)\big]
\nonumber\\
&&-\,w\,{\phi}_{\phi}^{v}(y)\big[
{\phi}_{s}\,m_{B_{s}}\,p_{\rm cm}\,(x+z-1)\,\nonumber\\
&&-\,
{\phi}_{+}\,\big((y-1)\,m_{\phi}^2
+(1-x)\,m_{B_{s}}\,E_{\phi}+z\,(w^2-m_{B_{s}}\,E_{w})\big)\big]\bigg]\Big\}
\label{amp-k-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{l}^{LL}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{l})\, H_{an}({\alpha}_{i},\, {\beta}_{l},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{l})\,C_{i}(t_{l})\nonumber\\
&& \times \,2\,m_{B_{s}}
{\phi}_{B}^{a}(x)\,\Big\{{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,2\,p_{\rm cm}\,
\Big[x\,m_{B_{s}}\,E_{w}-z\,w^2+(y-1)(m_{B_{s}}\,E_{\phi}-m_{\phi}^{2})\Big]\nonumber\\
&& +\,r\,w\,m_{B_{s}}\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))p_{\rm cm}\,(y+z-1)\nonumber\\
&&
+\,r\,w\,m_{B_{s}}\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))
\,[x\,m_{B_{s}}+(y-1)\,E_{\phi}-z\,E_{w}]
\Big\}
\label{amp-l-ll},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{l}^{LR}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{l})\, H_{an}({\alpha}_{i},\, {\beta}_{l},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{l})\,C_{i}(t_{l})\nonumber\\
&& \times \,2\,m_{B_{s}}^2
{\phi}_{B}^{a}(x)\,\Big\{{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,2\,E_{w}\,p_{\rm cm}\,
(x-z)\nonumber\\
&& +\,r\,w\,p_{\rm cm}\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))\,(1-y-z)\nonumber\\
&&
+\,r\,w\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))\,[x\,m_{B_{s}}+(y-1)\,E_{\phi}-z\,E_{w}]\Big\} \label{amp-l-lr},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{l}^{SP}[C_{i}] &=& \frac{(-2){\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{l})\, H_{an}({\alpha}_{i},\, {\beta}_{l},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{l})\,C_{i}(t_{l})\nonumber\\
&& \times \,
{\phi}_{B}^{p}(x)\Big\{r\,m_{B_{s}}^2
{\phi}_{-}\,\bigg[{\phi}_{\phi}^{t}(y)\,m_{B_{s}}\,p_{\rm cm}\,(y+x-1)\nonumber\\
&&-\,{\phi}_{\phi}^{s}(y)\,[(1-y)\,E_{\phi}\,E_{w}-x\,m_{B_{s}}\,E_{w}-(y-1)\,p_{\rm cm}^2+z\,w^2]\bigg]
\nonumber\\
&&-\,w\,m_{B_{s}}\,{\phi}_{\phi}^{v}(y)\,\bigg[
{\phi}_{s}\,m_{B_{s}}\,p_{\rm cm}\,(x-z)\nonumber\\
&&+\,{\phi}_{+}\big[(y-1)\,m_{\phi}^2
+x\,m_{B_{s}}\,E_{\phi}+z\,(w^2-E_{w}\,m_{B_{s}})\Big]\bigg]\Big\}
\label{amp-l-lr},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{m}^{LL}[C_{i}] &=& -2\,{\cal F} dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{m})\, H_{af}({\alpha}_{m},\, {\beta}_{m},\, b_{f},\, b_{\phi})\, E_{B}(t_{m})\,C_{i}(t_{m})\nonumber\\
&& \times\,
\Big\{m_{B_{s}}\,p_{\rm cm}\,{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\Big[
y\,(E_{\phi}\,m_{B_{s}}+p_{\rm cm}^2+E_{\phi}\,E_{w})+w^2\Big]\nonumber\\
&& +2\,r\,w\,m_{B_{s}}^2\,{\phi}_{s}\,\Big[{\phi}_{\phi}^{t}(y)\,\,p_{\rm cm}\,(1-y)\,+\,
{\phi}_{\phi}^{s}(y)\,(y\,E_{\phi}+E_{w})\Big]
\Big\}
\label{amp-m-ll},
\end{eqnarray}
\begin{equation}
{\cal A}_{m}^{LR}[C_{i}] = {\cal A}_{m}^{LL}[C_{i}]
\label{amp-m-lr},
\end{equation}
\begin{eqnarray}
{\cal A}_{m}^{SP}[C_{i}] &=& 4{\cal F} {\int}^{1}_{0} dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{m})\, H_{af}({\alpha}_{m},\, {\beta}_{m},\, b_{f},\, b_{\phi})\, E_{B}(t_{m})\,C_{i}(t_{m})\nonumber\\
&&\times\,m_{B_{s}}^{2} \Big\{
r\,{\phi}_{-}\,\bigg[{\phi}_{\phi}^{t}(y)\,y\,m_{B_{s}}\,p_{\rm cm}\,+\,{\phi}_{\phi}^{s}(y)\,
(y\,p_{\rm cm}^2+y\,E_{\phi}\,E_{w}+w^2)\bigg]
\nonumber \\
&&
+\,{\phi}_{s}\,{\phi}_{\phi}^{v}(y)\,2\,w\,p_{\rm cm}
\Big\}
\label{amp-m-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{n}^{LL}[C_{i}] &=& {\cal F} {\int}^{1}_{0} dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{n})\, H_{af}({\alpha}_{m},\, {\beta}_{n},\, b_{f},\, b_{\phi})\, E_{B}(t_{n})\,C_{i}(t_{n})\nonumber\\
&& \times\,2\,m_{B_{s}}^{2}
\Big\{{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,m_{B_{s}}\,p_{\rm cm}(z\,r^2-z+1)\nonumber\\
&& +\,2\,r\,w\,{\phi}_{\phi}^{s}(y)\,\bigg[{\phi}_{s}\,[E_{\phi}-(z-1)\,E_{w}] +\,{\phi}_{+}\,z\,p_{\rm cm}\bigg]
\Big\}
\label{amp-n-ll},
\end{eqnarray}
\begin{equation}
{\cal A}_{n}^{LR}[C_{i}] = {\cal A}_{n}^{LL}[C_{i}]
\label{amp-n-lr},
\end{equation}
\begin{eqnarray}
{\cal A}_{n}^{SP}[C_{i}] &=& (-2){\cal F} {\int}^{1}_{0}dy\,dz {\int}_{0}^{\infty} b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{n})\, H_{af}({\alpha}_{m},\, {\beta}_{n},\, b_{f},\, b_{\phi})\, E_{B}(t_{n})\,C_{i}(t_{n})\nonumber\\
&&\times\,2\,m_{B_{s}} \Big\{{\phi}_{-}\,{\phi}_{\phi}^{s}(y)\,2\,r\,m_{B_{s}}\,[(z-1)\,w^2-p_{\rm cm}^2-E_{\phi}\,E_{w}]\nonumber\\
&&
-\,w\,{\phi}_{\phi}^{v}(y)\,\bigg[{\phi}_{s}\,(1-z)\,m_{B_{s}}\,p_{\rm cm}\,-\,{\phi}_{+}\,[(1-z)(w^2-E_{w}\,m_{B_{s}})-m_{\phi}^2
]\bigg]
\Big\}
\label{amp-n-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{o}^{LL}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{o})\, H_{an}({\alpha}_{m},\, {\beta}_{o},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{o})\,C_{i}(t_{o})\nonumber\\
&& \times \,2\,m_{B_{s}}
\Big\{{\phi}_{B}^{a}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,2\,p_{\rm cm}\,
\big[E_{w}\big((x-1)m_{B_{s}}+y\,E_{\phi}\big)+y\,p_{\rm cm}^2-(z-1)\,w^2\big]\nonumber\\
&&+\,r\,w\,m_{B_{s}}\,p_{\rm cm}\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))\,\,(1-y-z)\nonumber\\
&&
+\,r\,w\,m_{B_{s}}\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))\,[(x-1)\,m_{B_{s}}+y\,E_{\phi}-(z-1)\,E_{w}]
\nonumber\\
&&+\,r_{b}\,m_{B_{s}}^2\,{\phi}_{B}^{p}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,p_{\rm cm}\,+\,{\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,2\,r\,w
\Big]\Big\}
\label{amp-o-ll},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{o}^{LR}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{o})\, H_{an}({\alpha}_{m},\, {\beta}_{o},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{o})\,C_{i}(t_{o})\nonumber\\
&& \times \,2\,m_{B_{s}}^2
\Big\{{\phi}_{B}^{a}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,2\,E_{w}\,p_{\rm cm}\,
(x-z)\nonumber\\
&& +\,r\,w\,p_{\rm cm}\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))\,(y+z-1)\nonumber\\
&&
+\,r\,w\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))\,[(x-1)\,m_{B_{s}}+y\,E_{\phi}-(z-1)\,E_{w}]
\Big]\nonumber\\
&&+\,r_{b}\,m_{B_{s}}\,{\phi}_{B}^{p}(x)\,\Big[{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,p_{\rm cm}\,+\,{\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,2\,r\,w
\Big]\Big\}
\label{amp-o-lr},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{o}^{SP}[C_{i}] &=& \frac{(-2){\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{o})\, H_{an}({\alpha}_{m},\, {\beta}_{o},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{o})\,C_{i}(t_{o})\nonumber\\
&& \times \,m_{B_{s}}\,
\Big\{{\phi}_{B}^{a}(x)\,
\bigg[r\,r_{b}\,m_{B_{s}}^{2}\,{\phi}_{-}\,({\phi}_{\phi}^{t}(y)\,p_{\rm cm}\,-\,{\phi}_{\phi}^{s}(y)\,E_{w})
\nonumber\\
&&+\,r_{b}\,w\,m_{B_{s}}\,{\phi}_{\phi}^{v}(y)\,({\phi}_{s}\,p_{\rm cm}\,-\,
{\phi}_{+}\,E_{\phi})\bigg]\nonumber\\
&& +\,
{\phi}_{B}^{p}(x)\Big[r\,m_{B_{s}}\,{\phi}_{-}\,[\,{\phi}_{\phi}^{t}(y)\,m_{B_{s}}\,p_{\rm cm}\,(1-y-x)\nonumber\\
&&\,-\,
{\phi}_{\phi}^{s}(y)\,
\big((z-1)\,w^2-y\,p_{\rm cm}^2+(1-x)\,m_{B_{s}}\,E_{w}-y\,E_{\phi}\,E_{w}\big)]
\nonumber\\
&&-\,w\,
{\phi}_{\phi}^{v}(y)\,[{\phi}_{s}\,m_{B_{s}}\,p_{\rm cm}\,(x-z)\nonumber\\
&&+\,
{\phi}_{+}\,\big((1-y)\,m_{\phi}^2
-x\,m_{B_{s}}\,E_{\phi}+z\,(m_{B_{s}}\,E_{w}-w^2)\big)]\Big]\Big\}
\label{amp-o-sp},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{p}^{LL}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{p})\, H_{an}({\alpha}_{m},\, {\beta}_{p},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{p})\,C_{i}(t_{p})\nonumber\\
&& \times \,2\,m_{B_{s}}^2\,
{\phi}_{B}^{a}(x)\,\Big\{2\,E_{w}\,p_{\rm cm}\,{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,
(x+z-1)\nonumber\\
&&+\,r\,w\,p_{\rm cm}\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))\,(1-y-z)\nonumber\\
&&+\,r\,w\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))
\,[x\,m_{B_{s}}-y\,E_{\phi}+(z-1)\,E_{w}]
\Big\}
\label{amp-p-ll},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{p}^{LR}[C_{i}] &=& \frac{{\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{p})\, H_{an}({\alpha}_{m},\, {\beta}_{p},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{p})\,C_{i}(t_{p})\nonumber\\
&& \times \,2\,m_{B_{s}}\,
{\phi}_{B}^{a}(x)\,\Big\{2\,p_{\rm cm}\,{\phi}_{-}\,{\phi}_{\phi}^{v}(y)\,
\big[E_{w}(x\,m_{B_{s}}-E_{\phi}\,y)-y\,p_{\rm cm}^2+(z-1)w^2\big]\nonumber\\
&& +\,r\,w\,p_{\rm cm}\,m_{B_{s}}\,({\phi}_{s}\,{\phi}_{\phi}^{t}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{s}(y))\,(y+z-1)\nonumber\\
&&+\,r\,w\,m_{B_{s}}\,({\phi}_{s}\,{\phi}_{\phi}^{s}(y)\,+\,{\phi}_{+}\,{\phi}_{\phi}^{t}(y))
\,[x\,m_{B_{s}}-y\,E_{\phi}+(z-1)\,E_{w}]\Big\}
\label{amp-p-lr},
\end{eqnarray}
\begin{eqnarray}
{\cal A}_{p}^{SP}[C_{i}] &=& \frac{(-2){\cal F}}{N_{C}} {\int}^{1}_{0} dx\,dy\,dz {\int}_{0}^{\infty} b db \, b_{\phi}db_{\phi}\, b_{f} db_{f}\, {\alpha}_{s}(t_{p})\, H_{an}({\alpha}_{m},\, {\beta}_{p},\, b,\, b_{\phi},\, b_{f})\, E_{n}(t_{p})\,C_{i}(t_{p})\nonumber\\
&& \times \,m_{B_{s}}\,
{\phi}_{B}^{p}(x)\Big\{r\,m_{B_{s}}\,
{\phi}_{-}\,\bigg[m_{B_{s}}\,p_{\rm cm}\,{\phi}_{\phi}^{t}(y)\,(y-x)\nonumber\\
&&-\,{\phi}_{\phi}^{s}(y)\,
[y\,E_{\phi}\,E_{w}-x\,m_{B_{s}}\,E_{w}+y\,p_{\rm cm}^2-(z-1)\,w^2]\bigg]
\nonumber\\
&&-\,w\,
{\phi}_{\phi}^{v}(y)\,\bigg[m_{B_{s}}\,p_{\rm cm}\,{\phi}_{s}\,(x+z-1)\nonumber\\
&&+\,
{\phi}_{+}\,\big[y\,m_{\phi}^2
-x\,m_{B_{s}}\,E_{\phi}+(1-z)(m_{B_{s}}\,E_{w}-w^2)\big]\bigg]\Big\}
\label{amp-p-sp},
\end{eqnarray}
where the color number $N_{c}$ $=$ $3$ and the color factor $C_{F}$ $=$ $4/3$.
The superscript ${\rho}$ of the amplitude building block ${\cal A}_{\sigma}^{\rho}$
refers to the three possible Dirac structures ${\Gamma}_{1}{\otimes}{\Gamma}_{2}$
of the operators $(\bar{q}_{1}q_{2})_{{\Gamma}_{1}}(\bar{q}_{3}q_{4})_{{\Gamma}_{2}}$,
namely ${\rho}$ $=$ $LL$ for $(V-A){\otimes}(V-A)$, ${\rho}$ $=$ $LR$
for $(V-A){\otimes}(V+A)$ and ${\rho}$ $=$ $SP$ for $-2(S-P){\otimes}(S+P)$.
The subscript ${\sigma}$ of ${\cal A}_{\sigma}^{\rho}$ (${\sigma}$ $=$ $a$
$b$, ${\cdots}$, $p$) corresponds to the sub-diagram indices of Fig.\ref{fey}.
The variables of $b$, $b_{\phi}$, $b_{f}$ are the conjugate variables of
the transverse momentum $p_{T}$, $k_{T}$, $l_{T}$, respectively.
The function $H_{i}$ and Sudakov factor $E_{i}$ are defined as
\begin{eqnarray}
H_{ef}({\alpha},{\beta},b_{i},b_{j}) &=&
K_{0}(b_{i}\sqrt{-{\alpha}}) \Big\{
{\theta}(b_{i}-b_{j})\,K_{0}(b_{i}\sqrt{-{\beta}})\,I_{0}(b_{j}\sqrt{-{\beta}})
+ (b_{i}{\leftrightarrow}b_{j}) \Big\}
\label{hef}, \\
H_{en}({\alpha},{\beta},b_{i},b_{j},b_{k}) &=&
\Big\{ {\theta}(-{\beta})\,K_{0}(b_{i}\sqrt{-{\beta}})
+ \frac{\pi}{2}{\theta}({\beta}) \Big[
i\,J_{0}(b_{i}\sqrt{\beta})-Y_{0}(b_{i}\sqrt{\beta}) \Big] \Big\}
\nonumber \\ &{\times}&
\Big\{ {\theta}(b_{i}-b_{j})\,K_{0}(b_{i}\sqrt{-{\alpha}})\,I_{0}(b_{j}\sqrt{-{\alpha}})
+ (b_{i}{\leftrightarrow}b_{j}) \Big\} {\delta}(b_{j}-b_{k})
\label{hen}, \\
H_{af}({\alpha},{\beta},b_{i},b_{j}) &=&
\Big\{ {\theta}(b_{i}-b_{j}) \Big[ i\,J_{0}(b_{i}\sqrt{\beta})
- Y_{0}(b_{i}\sqrt{\beta}) \Big] J_{0}(b_{j}\sqrt{\beta})
+ (b_{i}{\leftrightarrow}b_{j}) \Big\}
\nonumber \\ &{\times}&
\frac{{\pi}^{2}}{4} \Big\{ i\,J_{0}(b_{i}\sqrt{\alpha})
- Y_{0}(b_{i}\sqrt{\alpha}) \Big\}
\label{haf}, \\
H_{an}({\alpha},{\beta},b_{i},b_{j},b_{k}) &=&
\frac{\pi}{2} \Big\{ {\theta}(-{\beta})K_{0}(b_{i}\sqrt{-{\beta}})
+\frac{\pi}{2} {\theta}({\beta}) \Big[ i\,J_{0}(b_{i}\sqrt{\beta})
-Y_{0}(b_{i}\sqrt{\beta}) \Big] \Big\}
\nonumber \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! {\times}\,
\Big\{ {\theta}(b_{i}-b_{j}) \Big[ i\,J_{0}(b_{i}\sqrt{\alpha})
- Y_{0}(b_{i}\sqrt{\alpha})\Big] J_{0}(b_{j}\sqrt{\alpha})
+ (b_{i}{\leftrightarrow}b_{j}) \Big\} {\delta}(b_{j}-b_{k})
\label{han},
\end{eqnarray}
\begin{eqnarray}
E_{\phi}(t) &=& {\exp} \Big\{-S_{B_{s}}(t)-S_{f_{0}}(t)\Big\}
\label{e-phi}, \\
E_{f}(t) &=& {\exp}\Big\{-S_{B_{s}}(t)-S_{\phi}(t)\Big\}
\label{e-f}, \\
E_{B}(t) &=& {\exp}\Big\{-S_{f_{0}}(t)-S_{\phi}(t)\Big\}
\label{e-b}, \\
E_{n}(t) &=& {\exp}\Big\{-S_{B_{s}}(t)-S_{f_{0}}(t)-S_{\phi}(t)\Big\}
\label{e-n},
\end{eqnarray}
\begin{eqnarray}
S_{B_{s}}(t) &=& s(x,b,p_{B_{s}}^{+})
+2{\int}_{1/b}^{t}\,\frac{d{\mu}}{\mu}\,{\gamma}_{q}
\label{s-b}, \\
S_{\phi}(t) &=& s(y,b_{\phi},p_{\phi}^{+})
+s(\bar{y},b_{\phi},p_{\phi}^{+})
+2{\int}_{1/b_{\phi}}^{t}\,\frac{d{\mu}}{\mu}\,{\gamma}_{q}
\label{s-phi}, \\
S_{f_{0}}(t) &=& s(z,b_{f},q^{+})+s(\bar{z},b_{f},q^{+})
+2{\int}_{1/b_{f}}^{t}\,\frac{d{\mu}}{\mu}\,{\gamma}_{q}
\label{s-f},
\end{eqnarray}
where the subscripts $i$ $=$ $ef$, $en$, $af$, $an$ of the function $H_{i}$
correspond to the factorizable emission topologies, the nonfactorizable e
mission topologies, the factorizable annihilation topologies, and the
nonfactorizable annihilation topologies, respectively.
$I_{0}$, $J_{0}$, $K_{0}$ and $Y_{0}$ are the Bessel functions.
The expression of $s(x,b,Q)$ can be found in of Ref.\cite{prd52.3958}.
${\gamma}_{q}$ $=$ $-{\alpha}_{s}/{\pi}$ is the quark anomalous dimension.
The parameters of ${\alpha}_{i}$ and ${\beta}_{i}$ are the virtualities of
gluons and quarks. The subscript $i$ of ${\alpha}_{i}$, ${\beta}_{i}$, $t_{i}$
corresponds to the indices of Fig.\ref{fey}. The explicit definitions of the
virtualities and typical scale $t_{i}$ are given as follows.
\begin{eqnarray}
{\alpha}_{a} &=& x^2\,m_{B_{s}}^2+z^2\,w^2-2\,x\,z\,m_{B_{s}}\,E_{w}
\label{alpha-a}, \\
{\alpha}_{e} &=& x^2\,m_{B_{s}}^2+y^2\,m_{\phi}^2-2\,x\,y\,m_{B_{s}}\,E_{\phi}
\label{alpha-e}, \\
{\alpha}_{i} &=& \bar{y}^2\,m_{\phi}^2+z^2\,w^2+\bar{y}\,z\,(m_{B_{s}}^2-m_{\phi}^2-w^2)
\label{alpha-i}, \\
{\alpha}_{m} &=& y^2\,m_{\phi}^2+\bar{z}^2\,w^2+ y\,\bar{z}(m_{B_{s}}^2-m_{\phi}^2-w^2)
\label{alpha-m}, \\
{\beta}_{a} &=& (1-r_{b}^2)\,m_{B_{s}}^2+z^2\,w^2-2\,z\,m_{B_{s}}\,E_{w}
\label{beta-a}, \\
{\beta}_{b} &=& w^2+x^2\,m_{B_{s}}^2-2\,x\,m_{B_{s}}\,E_{w}
\label{beta-b}, \\
{\beta}_{c} &=& (x-y)\,(x-z)\,m_{B_{s}}^{2}
\nonumber \\ &+& (y-z)\,(y-x)\,m_{\phi}^{2}
\nonumber \\ &+& (z-x)\,(z-y)\,w^{2}
\label{beta-c}, \\
{\beta}_{d} &=& \left.{\beta}_{c}\right\vert_{y{\to}\bar{y}}
\label{beta-d}, \\
{\beta}_{e} &=& (1-r_{b}^2)\,m_{B_{s}}^2+y^2\,m_{\phi}^{2}-2\,y\,m_{B_{s}}\,E_{\phi}
\label{beta-e}, \\
{\beta}_{f} &=& m_{\phi}^2+x^2\,m_{B_{s}}^2-2\,x\,m_{B_{s}}\,E_{\phi}
\label{beta-f}, \\
{\beta}_{g} &=& {\beta}_{c}
\label{beta-g}, \\
{\beta}_{h} &=& \left.{\beta}_{g}\right\vert_{z{\to}\bar{z}}
\label{beta-h}, \\
{\beta}_{i} &=& m_{\phi}^{2}+z^2\,w^2+z\,(m_{B_{s}}^2-m_{\phi}^2-w^2)
\label{beta-i}, \\
{\beta}_{j} &=& \bar{y}^2\,m_{\phi}^2+w^2+\bar{y}\,(m_{B_{s}}^2-m_{\phi}^2-w^2)
\label{beta-j}, \\
{\beta}_{k} &=& \left.{\beta}_{c}\right\vert^{x{\to}\bar{x}}_{y{\to}\bar{y}}
-m_{b}^2
\label{beta-k}, \\
{\beta}_{l} &=& \left.{\beta}_{c}\right\vert_{y{\to}\bar{y}}
\label{beta-l}, \\
{\beta}_{m} &=& w^2+y^2\,m_{\phi}^2+y\,(m_{B_{s}}^2-m_{\phi}^2-w^2)
\label{beta-l}, \\
{\beta}_{n} &=& \bar{z}^2\,w^2+m_{\phi}^2+\bar{z}\,(m_{B_{s}}^2-m_{\phi}^2-w^2)
\label{beta-n}, \\
{\beta}_{o} &=& \left.{\beta}_{c}\right\vert^{x{\to}\bar{x}}_{z{\to}\bar{z}}
-m_{b}^2
\label{beta-o}, \\
{\beta}_{p} &=& \left.{\beta}_{c}\right\vert_{z{\to}\bar{z}}
\label{beta-p},
\end{eqnarray}
\begin{eqnarray}
t_{a,b} &=& {\max}\{\sqrt{-{\alpha}_{a}},\sqrt{{\vert}{\beta}_{a,b}{\vert}},1/b,1/b_{f}\}
\label{t-ab}, \\
t_{c,d} &=& {\max}\{\sqrt{-{\alpha}_{a}},\sqrt{{\vert}{\beta}_{c,d}{\vert}},1/b,1/b_{\phi}\}
\label{t-cd}, \\
t_{e,f} &=& {\max}\{\sqrt{-{\alpha}_{e}},\sqrt{{\vert}{\beta}_{e,f}{\vert}},1/b,1/b_{\phi}\}
\label{t-ef}, \\
t_{g,h} &=& {\max}\{\sqrt{-{\alpha}_{e}},\sqrt{{\vert}{\beta}_{g,h}{\vert}},1/b,1/b_{f}\}
\label{t-gh}, \\
t_{i,j} &=& {\max}\{\sqrt{{\alpha}_{i}},\sqrt{{\vert}{\beta}_{i,j}{\vert}},1/b_{\phi},1/b_{f}\}
\label{t-ij}, \\
t_{k,l} &=& {\max}\{\sqrt{{\alpha}_{i}},\sqrt{{\vert}{\beta}_{k,l}{\vert}},1/b,1/b_{f}\}
\label{t-kl}, \\
t_{m,n} &=& {\max}\{\sqrt{{\alpha}_{m}},\sqrt{{\vert}{\beta}_{m,n}{\vert}},1/b_{\phi},1/b_{f}\}
\label{t-mn}, \\
t_{o,p} &=& {\max}\{\sqrt{{\alpha}_{m}},\sqrt{{\vert}{\beta}_{o,p}{\vert}},1/b,1/b_{f}\}
\label{t-op}.
\end{eqnarray}
\end{appendix}
|
{
"timestamp": "2018-03-08T02:08:42",
"yymm": "1803",
"arxiv_id": "1803.02656",
"language": "en",
"url": "https://arxiv.org/abs/1803.02656"
}
|
\section{Introduction}
Since Jaynes \cite{Jaynes1}, the entropy maximization has been a widely used tool in many different fields benefiting from Shannon entropy. Although the initial aim of Jaynes was to derive the equilibrium distribution associated with the Shannon entropy subject to the linear constraints, recent progress in the generalized entropies such as Tsallis \cite{Tsallis1988} or R\'enyi \cite{Renyi} entropies, to mention but a few, also benefited from the very same entropy maximization procedure with various applications \cite{Bagci1,Rotundo,Van1,Van2,Wong,reis,
Portesi,Bagci2,Campisi,Oik2007,high3,third,Rob1,esc3}.
However, we have recently shown that the entropy maximization with linear constraints does not yield a distribution which can be cast into the appropriate form so as to include the partition function when it is used for the generalized entropies \cite{Th}. In other words, the distributions are not of the form $p_i=f^{-1}(\beta \varepsilon_i)/\sum_k f^{-1}(\beta \varepsilon_k)$ ($\beta$ being the Lagrange multiplier of the internal energy constraint and $\varepsilon_i$ is the energy of the $i$th micro state) so that the the denominator (i.e. normalization term) could not be identified as the partition function. The sole possibility for such a distribution has been found to be the one associated with the Shannon entropy.
In this work, we do not interest ourselves with the explicit form of the distribution. Instead, in its all generality, we consider the entropy maximization with linear constraints as Jaynes previously did \cite{Jaynes1} and show that the only admissible entropy expression is the Shannon (or Boltzmann-Gibbs) entropy. Therefore, we point out that the entropy maximization construed {\`a} la Jaynes is suitable only for the Shannon entropy and thereby excludes the use of any generalized entropies.
\section{Maximization procedure revisited}\label{SecII}
The entropy functional with linear constraints reads
\begin{eqnarray}\label{eq02}
L(\{p\},\alpha,\beta,U)=S(\{p\})-\alpha\sqbr{\sum_{i=1}^{n} p_i-1} - \beta\sqbr{\sum_{i=1}^{n} p_i\varepsilon_i-U}\,,
\end{eqnarray}
where $S$ denotes the entropy measure and $U$ is the internal energy. As usual, $p_i$ is the probability of occurrence for the $i$th micro state and $(\alpha,\beta)$ are the respective Lagrange multipliers. Considering the maximization functional in Eq. (\ref{eq02}) and using the definition $\partial S(\{p\})/\partial p_i=:f(p_i)$, the maximization procedure yields the following $n+3$ equations \cite{Karabulut}
\begin{subequations}\label{eq03}
\begin{eqnarray}\label{eq03a}
f(p_i)&=&\alpha + \beta \varepsilon_i=x_{i}\,,\\
\label{eq03b}
1&=&\sum_{i=1}^{n} p_i\,,\\
\label{eq03c}
U&=&\sum_{i=1}^{n} p_i\varepsilon_i\,,\\
\label{eq03d}
\beta &=& \frac{\partial S}{\partial U}\,.
\end{eqnarray}
\end{subequations}
Taking the derivative of Eq. (\ref{eq03b}) with respect to $\beta$, we have
\begin{eqnarray}\label{alphaRelation}
0&=& \sum_{i=1}^{n}\frac{\partial p_i}{\partial \beta}
=\sum_{i=1}^{n}\frac{\partial f^{-1}(\alpha+\beta\varepsilon_i)}{\partial \beta}
=\sum_{i=1}^{n}\frac{\partial f^{-1}(\alpha+\beta\varepsilon_i)}{\partial (\alpha+\beta\varepsilon_i)}\frac{\partial(\alpha+\beta\varepsilon_i)}{\partial\beta}\,.
\end{eqnarray}
Introducing the normalized $P_i$ as
\begin{eqnarray}\label{BigP}
P_i= \rbr{\sum_{k=1}^{n}\frac{\partial f^{-1}(x_k)}{\partial x_k}}^{-1}\frac{\partial f^{-1}(x_i)}{\partial x_i}\,,
\end{eqnarray}
Eq. (\ref{alphaRelation}) yields
\begin{eqnarray}\label{delalpha}
\frac{\partial \alpha}{\partial \beta} &=& - \sum_{i=1}^{n} P_i\varepsilon_i=-\widetilde{U}\,.
\end{eqnarray}
The quantity $\widetilde{U}$ is related to $U$ as (combine Eqs. (\ref{eq03})- (\ref{delalpha}))
\begin{eqnarray}
\label{eqNew01}
\widetilde{U}&=& U - \sum_{i=1}^{n}p_i\frac{\partial f(p_i)}{\partial\beta}\,.
\end{eqnarray}
Similarly to Eq. (\ref{alphaRelation}), since $P_i$ satisfies the normalization condition, we have
\begin{eqnarray}\label{delbeta}
0=\sum_{i=1}^{n} \frac{\partial P_i}{\partial \beta}= \sum_{i=1}^{n} \frac{\partial P_i}{\partial x_i}\left(\varepsilon_i - \widetilde{U}\right)\quad\Rightarrow\quad
\widetilde{U}=\sum_{i=1}^{n}\frac{\frac{\partial P_i}{\partial x_i}}{\sum_{k=1}^{n}\frac{\partial P_k}{\partial x_k}}\varepsilon_i\,.
\end{eqnarray}
Comparing Eqs. (\ref{delalpha}) and (\ref{delbeta}) we read
\begin{eqnarray}\label{DefY_i}
\sum_{i=1}^{n}Y_i\varepsilon_i=0\,,\qquad
Y_i:= P_i - \frac{\frac{\partial P_i}{\partial x_i}}{\sum_{k=1}^{n}\frac{\partial P_k}{\partial x_k}}\,.
\end{eqnarray}
The validity of this equation presents us with two cases we inspect below:\\
\noindent(i.) The first possibility, assuming $Y_i\neq0$, is that the total sum can be equal to zero. Then, applying the $m$th derivative with respect to $\beta$ yields
\begin{eqnarray}\label{eqA3}
\sum_{i=1}^{n}\frac{\partial^m Y_i}{\partial \beta^m} \varepsilon_i=0
\end{eqnarray}
This is a $m\times n$ homogeneous system of the form $A_{ij}X_i=0$ ($i=1,\ldots,n$, $j=1,\ldots,m$) to be solved with $A_{ij}\equiv \frac{\partial^{j} Y_j}{\partial \beta^{j}}$ and $X_i\equiv\varepsilon_i$. Then, we know from linear algebra that the former system has either the zero solution, i.e., $X_i=0$, or a set of infinite solutions with $A_{ij}=A_{i\ell}$. The zero solution is apparently not an option. Thus, we have infinite solutions yielding $\frac{\partial^{j}}{\partial \beta^{j}}Y_i = \frac{\partial^{\ell}}{\partial\beta^{\ell}}Y_i\;\Rightarrow\; Y_i=ce^{\beta}$. Summing over all $i$'s and using the normalization condition we see that the former relation is only possible when $c=0\;\Rightarrow\;Y_i=0$, which is a contradiction to our initial assumption.\\
\noindent(ii.) The second and only possibility that is left is
\begin{eqnarray}\label{eq10}
Y_i=0\,.
\end{eqnarray}
Then, substituting the definition of $Y_i$ in Eq. (\ref{DefY_i}) into the former equality, we have
\begin{eqnarray}\label{important}
\frac{\partial}{\partial x_i} \ln(P_i) =
\sum_{k=1}^{n} \frac{\partial P_k}{\partial x_k}
\end{eqnarray}
Since the l.h.s. and r.h.s. have an open and a closed $i$ dependence (or equivalently, the former depends and the latter does not depend on $i$), respectively, the only option satisfying this relation is $\ln(P_i)\sim x_i$ so that the derivative eliminates the $i$-dependence. Thus, the only option is that the measure $P_i$ has to be the inverse logarithmic function, i.e.,
\begin{eqnarray}\label{ffun}
P_i= \exp\rbr{\displaystyle\pm\frac{x_i}{k}}\,,
\end{eqnarray}
where $k$ is merely a constant. By virtue of Eq. (\ref{ffun}), we read in Eq. (\ref{BigP})
\begin{eqnarray}\label{ffun2}
\sum_{k=1}^{n}\frac{\partial f^{-1}(x_k)}{\partial x_k} = \exp\left(\mp\frac{x_i}{k}\right)\frac{\partial f^{-1}(x_i)}{\partial x_i}\,.
\end{eqnarray}
Then, a similar discussion to Eq. (\ref{important}) uniquely yields $P_i=f^{-1}(x_i)=p_i$, hence
\begin{eqnarray}\label{ffun3}
f^{-1}(x_i)= \exp\rbr{\displaystyle\pm\frac{x_i}{k}} \qquad\Leftrightarrow\qquad
f(p_i)=\pm k\ln(p_i)\,.
\end{eqnarray}
To reiterate, the MaxEnt procedure with linear constraints leads to two distinct, at first glance, probability distribution sets, $\{p_i\}$ and $\{P_i\}$, respectively. The former is used in the maximization procedure itself and the latter was deduced from the normalization condition of $p_i$. However, the normalization of $P_i$ in turn shows that these two distribution sets are actually one and the same, $P_i=p_i\;\Rightarrow\; U=\widetilde{U}$, exhibiting an exponential behavior with respect to the energy values $\varepsilon_i$.
\section{Determining the Entropy Uniquely}
We now show how considerations in the previous section uniquely leads to the Shannon-Boltzmann-Gibbs entropy. Integrating Eq. (\ref{eq03d}) with respect to $U$, we have
\begin{eqnarray}\label{entropy01}
S=\beta U - \int U\mathrm{d}\beta+C_1\,,
\end{eqnarray}
where $C_1$ is the integration constant and does not depend on $\beta$. Using the mean value constraint in Eq. (\ref{eq03c}) the former equation can be written as
\begin{eqnarray}\label{entropy02}
S=\sum_{i=1}^{n} p_i(\beta \varepsilon_i) -\int U\mathrm{d}\beta +C_1\,,
\end{eqnarray}
Taking into account Eqs. (\ref{eq03a}) and (\ref{ffun}) and then Eqs. (\ref{delalpha}) and (\ref{eq10}), Eq. (\ref{entropy02}) can be written as
\begin{eqnarray}\label{entropy03}
S=\pm k \sum_{i=1}^{n}p_i\ln(p_i) + C\,.
\end{eqnarray}
This is the most general structure of the entropy $S$ satisfying the MaxEnt procedure with linear constraints.
The term $C$ includes all additive constants. The sign in Eq. (\ref{entropy03}) depends on whether the entropy $S$ is to be maximized or minimized (negative or positive sign, respectively).
For $k=1$ this is identified with the Shannon entropy and for $k=k_\text{\tiny{B}}$ with the Boltzmann-Gibbs entropy within the information theory and statistical thermodynamics, respectively.
\section{Conclusions}
Since the seminal work of Jaynes \cite{Jaynes1}, entropy maximization procedure has been utilized in the literature. However, in the recent decades, this procedure has been used for various entropy definitions such as Tsallis \cite{Tsallis1988} or R{\'e}nyi entropies \cite{Renyi}, although Jaynes originally used only the Shannon entropy (or Boltzman-Gibbs entropy which differs from Shannon entropy by a multiplicative constant) with linear constraints.
Instead of specifying a particular entropy measure right from the beginning, we have considered a very general treatment of the entropy maximization in this work and shown that the only entropy measure compatible with the entropy maximization {\`a} la Jaynes is the Shannon entropy if the linear constraints are to be used. In this sense, the procedure devised by Jaynes is strictly devised for the Shannon entropy. As a matter of fact, this has exactly been the point of the well-known Shore-Johnson axioms \cite{Shore}, too. However, we note that we have not used a joint probability composition rule in above derivation thereby rendering our calculations in essence different from the approach of the Shore-Johnson axioms \cite{notelast}.
When we consider for example the R{\'e}nyi entropy (or Tsallis entropy for that matter) in virtue of Eq. (\ref{eqNew01}), one obtains $0 = (1-q) \beta \frac{\partial \widetilde{U}}{\partial \beta}$. This relation either forces us to use Shannon entropy i.e. setting $q=1$ or assuming $\frac{\partial \widetilde{U}}{\partial \beta} =\frac{\partial U}{\partial \beta}= 0$, which leads to a contradiction since $\frac{\partial p_i}{\partial\beta}\neq0$, as can be seen in Eq. (\ref{eq03a}). Therefore, the use of entropy maximization with linear constraints should not be extended to the uses of the deformed entropies. However, note that our work is limited to the linear constraints i.e. linear averaging schemes so that other averaging schemes is beyond the scope of present treatment.
\begin{acknowledgments}
T.O. acknowledges the state-targeted program ``Center of Excellence for Fundamental and Applied Physics" (BR05236454) by the Ministry of Education and Science of the Republic of Kazakhstan and ORAU grant entitled ``Casimir light as a probe of vacuum fluctuation simplification" (090118FD5350).
\end{acknowledgments}
|
{
"timestamp": "2018-05-01T02:03:17",
"yymm": "1803",
"arxiv_id": "1803.02556",
"language": "en",
"url": "https://arxiv.org/abs/1803.02556"
}
|
\section{Introduction}
In recent years there have been many reports of experimental~\cite{Han2017,Palacci2013, Kumar2014} and simulated~\cite{Fily2012, Redner2013, Cates2015, Liebchen2016b, Nguyen2014b, Yeo2015b} particle systems with purely repulsive interactions that are always homogeneous at equilibrium but undergo phase separation when driven out of equilibrium. Understanding how non-equilibrium driving modifies interfacial fluctuations in these cases - and material properties in general - is an important and open question. For instance, surface fluctuations play a central role in micro-scale applications~\cite{Sackmann2014}, and understanding how to control them can contribute to our ability to exploit the engineering promise of nonequilibrium particle systems~\cite{Whitesides2002, Grunwald2016a, Grzybowski2017}.
A few examples of nonequilibrium phase separation are motility-induced phase separation (MIPS), undergone by Brownian particles when they are given the ability to self-propel~\cite{Fily2012, Redner2013, Bialke2013, Cates2015, Stenhammar2015, Whitelam2018}, lane or stripe formation of charged particles in an electric field~\cite{Dzubiella2002, Wysocki2009, Vissers2011, Vissers2011a, Klymko2016} and shaken granular matter~\cite{Mullin2000, Pooley2004}, and the separation of particles with rotational dynamics based on phase synchronization~\cite{Liebchen2017} or chirality~\cite{Nguyen2014b, Han2017, Yeo2016}.
Recently we reported phase separation of this last kind and stable, system-spanning interfaces in simulations of a liquid of 2-dimensional disks with repulsive interactions where half of the particles are driven by a time dependent field so that they orbit in phase~\cite{delJunco2018}. This model was inspired by a recent experimental study in which magnetic particles are driven by a rotating magnetic field and undergo phase separation~\cite{Han2017}. Here, we combine simulations with an analysis based on capillary wave theory (CWT)~\cite{Bedeaux1985} to study the effect of the time-dependent forces on the interfacial properties of the liquid with repulsive interactions, and of a closely related liquid with attractive interactions. To distinguish these systems, we will refer to the repulsive model studied in Ref.~\citenum{delJunco2018} as the Weeks-Chandler-Andersen (WCA) model, and to the new attractive model as the Lennard-Jones (LJ) model.
The main result of CWT predicts that the power spectrum of height fluctuations of an interface parallel to a prescribed horizontal axis satisfies $\langle |h(k)|^2\rangle \propto 1/ (\sigma k^2)$, where $k$ denotes the wavevector, $h(k)$ denotes the Fourier transform of height fluctuations, and $\sigma$ is the surface tension. This $1/k^2$ scaling - also known as capillary scaling - is found in systems ranging from the 2D Ising model to water~\cite{Fisher1982, Sides1999,Schwartz1990a}. CWT has also been used to study interfaces in non-equilibrium liquids and extract effective surface tensions~\cite{Bialke2015, Paliwal2017, Derks2006, Patch2018, Lee2017}.
It has been shown that phase separation in the WCA model belongs to the Ising universality class~\cite{Han2017}, which would lead us to expect capillary scaling of the interface modes~\cite{Fisher1982}. This expectation is brought in to question by our first finding, which is that the time-dependent driving forces result in persistent particle currents along the interface of the WCA model (Fig.~\ref{fig:velocityprof}). These can affect the statistics of interface fluctuations. For instance, the currents present in a non-equilibrium Ising model with an applied electric field can cause the scaling to decrease to $1/k^{0.67}$\cite{Leung1993}.
The fluctuations of active interfaces have been studied recently in systems of the MIPS type~\cite{Bialke2015, Paliwal2017, Lee2017, Patch2018} where there can be local tangential flows~\cite{Patch2018} but not system-spanning currents at the interface. In our system, and in others with rotational dynamics, we observe system spanning currents qualitatively similar to those in the non-equilibrium driven Ising model~\cite{Leung1993}. The presence of these currents makes it important to examine the full spectrum of capillary fluctuations. This examination will allow us to assess whether the system obeys capillary scaling and for what range of wavenumbers, to check the convergence of interface statistics, and to accurately measure the surface tension.
In the WCA model, we find that the scaling of interface fluctuations depends on the amplitude of the driving forces. For one amplitude that we studied, we find close to $1/k^2$ scaling, while for all others we find that $\langle |h(k)|^2\rangle$ is inversely correlated with $k$, but decreases less rapidly than predicted by CWT (Fig.~\ref{fig:hk_WCA}). The effect of the driving forces on the stability of interfaces in the WCA model is non-monotonic, because they cause the system to phase separate at low amplitudes but to become mixed again at large amplitudes~\cite{delJunco2018}. Moreover, since the system is mixed at equilibrium, there is no reference value for the surface tension in the absence of driving. For these reasons, the WCA model is not ideal to systematically investigate the effect of driving forces on surface tension.
For this purpose, we introduce the LJ model, which is phase-separated with a well-defined surface tension at equilibrium~\cite{Paliwal2017}. We find that the LJ model exhibits capillary fluctuations over a wide range of wavevectors $k$ even in the presence of driving. Over an order of magnitude in the driving forces, the effect of driving in the LJ model is a linear increase in the surface tension ($\sigma$). We discuss two ways that the driving forces can increase the force imbalance at the interface, thereby causing the observed increase in $\sigma$: first, by inducing a restoring force on the interface that is proportional to the curvature, and second, by changing the density of the liquid and gas phases of LJ particles. We show that both of these effects can contribute to the increase in the surface tension, but a full account of the linear trend remains an open problem that these equilibrium-like arguments are insufficient to address.
\section{Methods}
\subsection{Models and Simulation Details}
We studied interfaces in two models of driven liquids: one in which the particles have repulsive interactions only, which does not phase separate at equilibrium, and a second with attractive interactions between driven particles and repulsive interactions between undriven particles, which phase separates and possesses stable interfaces at equilibrium.
Both models consist of 2-dimensional disks whose positions evolve in time according to driven Brownian dynamics:
\begin{equation}
{\bf \dot r}_i(t) = D_0\beta\left({\bf F}_{ c,i}(t) + {\bf F}_{ d}(t)\right)+\boldsymbol\eta_i (t).
\label{eq:xEOM}
\end{equation}
Here, $D_0$ is the diffusion constant of a single particle and $\boldsymbol\eta_i (t) = (\eta_{i,x}(t),\eta_{i,y}(t))$ are Gaussian-distributed random variables with $\left<\boldsymbol\eta_i (t)\right>=0$ and $\left<\eta_{i,\mu}(t)\eta_{j,\nu}(t')\right> = 2D_0\delta_{i,j}\delta_{\mu,\nu}\delta(t-t')$. $D_0$ is related to the friction coefficient $\gamma$ by $D_0 = k_BT/\gamma$. In all of our simulations and calculations, we set $\beta=(k_BT)^{-1}=1$. The length scale of the system is set by the particle diameter, $r_0$, and the time scale is set by $t_0 = D_0/r_0^2$.
In the WCA model, previously described in Ref.~\citenum{delJunco2018} and motivated by Ref.~\citenum{Han2017}, ${\bf F}_{ c,i}$ is the (purely repulsive) conservative force on particle $i$ due to the Weeks-Chandler-Andersen interaction potential~\cite{WCA1971}:
\begin{equation}
u(r_{ij})=\left\{ \begin{matrix} 4\epsilon_{WCA} \left[ \left( \frac{r_0}{r_{ij}} \right)^{12}-\left( \frac{r_0}{r_{ij}}\right)^{6}\right]+\epsilon_{WCA}, & r \leq 2^{1/6}r_0 \\
0, & r>2^{1/6}r_0 \end{matrix} \right .
\label{WCA}
\end{equation}
We set $\epsilon_{WCA}=1$. In addition to the conservative forces, half of the particles are driven by an external force acting on the center of mass of the particle whose direction changes with a period $\tau$ according to:
\begin{align}
&{\bf F}_{ d}=A\sin\theta \hat e_x+A\cos\theta\hat e_y \label{eq:FexEq} \\
&\theta=2\pi t/\tau. \label{eq:theta}
\end{align}
We characterize the driving forces in terms of the P\'eclet number ($Pe$,) a dimensionless measure of the ratio of advective to diffusive velocity in the system that we define here as $Pe = \frac{A/\gamma}{D_0/r_0}$~\cite{delJunco2018}. For a driven particle, the effect of ${\bf F}_d$ is to cause the particle to orbit in a circle of radius $D_0\beta Pe\tau/(2\pi)$. For the other half of the particles, ${\bf F}_{d}=0$.
The second model consists of a mixture of driven LJ particles and undriven WCA particles. We refer to it as the LJ model. The WCA particles move according to Eq.\,\ref{eq:xEOM} and \ref{WCA} with ${\bf F}_{ d}=0$. The LJ particles move according to Eq.\,\ref{eq:xEOM} with ${\bf F}_{ c,i}$ due to the truncated LJ potential~\cite{Jones1924}:
\begin{equation}
u(r_{ij})=\left\{ \begin{matrix} 4\epsilon_{LJ} \left[ \left( \frac{r_0}{r_{ij}} \right)^{12}-\left( \frac{r_0}{r_{ij}}\right)^{6}\right] , & r \leq 2.5r_0 \\
0, & r>2.5r_0 \end{matrix} \right.
\label{LJ}
\end{equation}
with $\epsilon_{LJ} = 2.25$, and with ${\bf F}_{ d}$ given by Eqs.\,\ref{eq:FexEq} and \ref{eq:theta}.
Molecular dynamics simulations of both models were performed using a custom Brownian dynamics integrator in LAMMPS~\cite{Plimpton1995}. Results reported here are for square simulation boxes with sides of length $L = 100r_0$ unless otherwise indicated and periodic boundary conditions. We initiated the simulations by placing a slab $50r_0$ wide of driven particles in the middle of the box, spanning the system in the $y$-direction, so that there were two interfaces of length $L$ along the $y$-direction.
We characterized the phase diagram of the WCA model at a number density $\rho=N/L^2=0.5$ and chose the parameters of the driving force accordingly. At $\tau = 0.1$, the system phase separates in to regions of driven and undriven particles when $Pe \approx 50$ and becomes mixed again at large values of $Pe > 150$, so we chose to simulate interfaces at $\tau = 0.1, \rho = 0.5$, and $Pe = 60, 80, 100$ and 120. In Fig.~\ref{fig:snapshots} we show a snapshot of the system with $Pe = 100$ in the steady state.
For the LJ model we chose the initial density of the slab of LJ particles, $\rho_{LJ} = 0.85$, such that they would exhibit liquid-vapor coexistence in the absence of driving forces~\cite{Smit1991}, and we chose a density of passive WCA particles so that the total density of the system was $0.5$. At equilibrium this results in a liquid phase of LJ particles with a density of $\sim0.72$ in coexistence with a gas of LJ and WCA particles (Fig.~\ref{fig:snapshots}). We fixed $\tau = 0.1$ and varied $Pe$ from 0 to 80. Examples of steady-state configurations of both models are shown in Fig.~\ref{fig:snapshots}.
\begin{figure}
\center
\includegraphics[width=\linewidth]{snapshots.pdf}
\caption{{\bf Two models of driven liquids exhibit stable, system-spanning interfaces.} Snapshots of (a) the WCA model with $Pe = 100$, of the LJ model (b) at equilibrium, and (c) with $Pe = 40$ show the slab geometry used in our simulations. Active particles are colored red, and passive particles are colored blue. In the driven cases, a gap is visible at the right interface, which is particularly noticeable in the WCA system. The gap switches from one interface to the other with a period $\tau$.}
\label{fig:snapshots}
\end{figure}
The expected relaxation time of the longest-wavelength interface mode was approximated as $\tau_r = L^2 / D$, where $L$ is the length of the interface and $D$ is the diffusion constant of the WCA model in the absence of driving~\cite{delJunco2018}. In the WCA system, we ran each simulation for 10$\tau_r$, discarded the first $\tau_r$ of the trajectory and performed the CWT analysis on the remaining 9$\tau_r$. In the LJ system, we ran each simulation for 20$\tau_r$, discarded the first 10$\tau_r$ of the trajectory and performed the CWT analysis on the remaining 10$\tau_r$.The center of mass was adjusted in the simulation at intervals of $t_0$ to compensate for drift, and as an extra precaution we also subtracted any center-of-mass motion before analyzing the trajectories.
The box dimension perpendicular to the interfaces, $L_x$, was wide enough that the interfaces were stable along the $y$-direction, and that the width of the interfaces was unrestricted.
\subsection{Interface Current, Density, and Work}
Driven liquids with rotational dynamics can exhibit currents along boundaries and interfaces\cite{Nguyen2014b, VanZuiden2016}. Because of the slab geometry of the present system, any currents would have to be in the $y$-direction. To calculate the particle current, the simulation box was divided in to slices of width $r_0$. For all particles in a given slice of the box at time $t + t_0$, the displacement $\Delta y = y(t + t_0) - y(t)$ was calculated. Although there is no velocity in Brownian equations of motion, we report $v_y = \Delta y /t_0$ as an analog of the velocity. The average $v_y$ as a function of $x$ was then calculated by averaging over all of the particles in the slice between $x$ and $x+r_0$ over an interval $\tau_r = 20000t_0$ after the system has reached a steady state. The average $v_y$ in the bulk phase of driven particles was subtracted.
The density profile of driven particles was measured by dividing in to slices of width $r_0$ and calculating $\rho(x) = N/(L\times r_0)$ in each slice of the box at intervals of $t_0$, where $N$ is the number of particles located in the slice between $x$ and $x+r_0$. The average density profile was obtained by averaging $\rho(x)$ over an interval $\tau_r = 20000t_0$ in the steady state.
We define the work done on the system by the driving forces as~\cite{delJunco2018}
\begin{equation}
\label{eq:work}
\langle \dot{w} \rangle=-\sum_{i=1}^{N} \frac{1}{\tau}\int_0^\tau \dfrac{\langle {\bf F}_{{ c,i}}(t)\rangle\cdot {\bf F}_{{ d,i}}(t)}{\gamma}dt
\end{equation}
where ${\bf F}_{{ c}}$ and ${\bf F}_{{ d}}$ are defined in Eqs.~\ref{eq:xEOM}-\ref{LJ}. This definition of work quantifies the energy input to the system as the driving forces push particles in to one another at each timestep; this energy is subsequently dissipated to the bath as heat. To measure the work in simulations, at each timestep we summed ${\bf F}_{c, i} \cdot {\bf F}_d \Delta t /\gamma$ over all driven particles. This quantity was summed over intervals of $\tau$ and divided by $\tau$ to get $\dot w$. The averaged $\langle \dot w\rangle$ and errors shown in Fig.~\ref{fig:work} are the average and standard deviation of $\dot w$ over 300 periods of $\tau$ after the system has reached a steady state.
\subsection{Capillary Wave Theory and Analysis}
Our analysis of interfacial fluctuations is motivated by CWT~\cite{Bedeaux1985}. For a flat interface of length $L$, CWT posits that fluctuations in the height of the interface are described by the effective Hamiltonian:
\begin{equation}
H = \frac{ \sigma }{ 2 } \int_L dx \left| \frac{ dh }{ dx } \right| ^2
\label{eq:CWTHamiltonian}
\end{equation}
where $h(x)$ is the height of a 1D interface. Using Parseval's identity to take the Fourier transform yields a quadratic Hamiltonian in Fourier space, so we can apply equipartition theorem and obtain an expression for the average height fluctuations of the interface~\cite{Bedeaux1985}:
\begin{equation}
\langle |h(k)|^2\rangle = \frac{k_BT}{L\sigma k^2}.
\label{eq:CWT}
\end{equation}
Here $k$ is a scalar since we are considering straight, 1D interfaces in this work, but Eqs.~\ref{eq:CWTHamiltonian} and \ref{eq:CWT} are easily generalized to higher dimensions~\cite{Bedeaux1985}. In equilibrium, the $\sigma$ appearing in Eq.~\ref{eq:CWT} should match the surface tension obtained by any other means~\cite{MTOC}. Out of equilibrium that may or may not be the case~\cite{Bialke2015, Patch2018} - nonetheless, if we find that the height fluctuations of the interface scale as $1/k^2$, we can use Eq.~\ref{eq:CWT} to extract $\sigma$ which we may call an effective surface tension~\cite{Bialke2015, Paliwal2017, Derks2006}. We note that in systems where capillary scaling is not obeyed, deviations from $1/k^2$ scaling have been connected to the violation of fluctuation-dissipation theorem - in other words, height fluctuations can still provide insight in to how energy input affects correlations in the system~\cite{Zia1991b}.
To clearly define the location of the interface, we performed a coarse-graining of snapshots of the system at intervals of $t_0$ by dividing the simulation box up in to a grid with cells $2r_0\times 2r_0$ in dimension, yielding a lattice of dimensions $n \times n$ with $n = L/2$. We assigned a value of 1 to a grid site if it contained at least one driven particle, and a value of 0 otherwise. For the subsequent analysis we only considered one of the two interfaces. We used an image processing algorithm on each frame to extract two contiguous clusters of grid sites, one with value 1 and the other with value 0, separated by an interface. The interface height at $j = y/2$ is the number of sites with value 1 in column $j$. To obtain $|h(k)|^2$, we took the discrete Fourier transform of $h'(j) = h(j) - \langle h(j) \rangle$. We averaged over all of the snapshots in an interval $\tau_r$ to obtain $\langle |h(k)|^2\rangle$, and checked that the statistics did not change systematically between segments of $\tau_r$. We took the segments to be statistically independent, and we averaged over them to get a second average $\langle \langle |h(k)|^2\rangle\rangle$ - this double average is the value reported in Figs.~\ref{fig:hk_WCA} and ~\ref{fig:LJ}. The error was estimated as the standard deviation of $\langle |h(k)|^2\rangle$ between the analyzed segments. The code used for the analysis is available upon request.
In the WCA system where the scaling of $\langle |h(k)|^2\rangle$ was not $1/k^2$, we extracted the scaling exponent by fitting the linear part of a log-log plot of $\langle |h(k)|^2\rangle L$ vs $1/k^2$, judged by eye from the data in Fig.~\ref{fig:hk_WCA}. Where applicable, the surface tension was extracted by fitting $\langle |h(k)|^2\rangle$ according to Eq.~\ref{eq:CWT} over a range from $k_{min}$ to $k_{max}$, where $k_{max}$ was defined as the largest value of $k$ for which $\langle |h(k)|^2\rangle$ was greater than the coarse-graining length of the system and $k_{min}$ was defined as the the smallest value of $k$ for which $1/k^2$ was a good fit to $\langle |h(k)|^2\rangle$, judged by eye from the data in Fig.~\ref{fig:LJ}.
\section{Results}
\subsection{Phase Separation}
First we briefly recapitulate the mechanism of phase separation in the WCA model, which was explored in more detail in Refs.~\citenum{delJunco2018} and \citenum{Han2017}. In Ref.~\citenum{delJunco2018}, we found that the driving forces do work (as defined in Eq.~\ref{eq:work}) on the system by inducing collisions between particles. These collisions result in an increased diffusion coefficient which scales roughly proportional to $w$, the amount of work done per period of driving, which in turn scales as $Pe^2$.
Because the driven particles are always in phase, in a region with only driven particles or only undriven particles, the nonequilibrium forces do not induce any collisions. The work done and therefore the diffusion coefficient thus depend on the local composition, and particles diffuse faster out of regions with mixed configurations than back in to them. If the gradient of the diffusion with respect to composition is sufficiently high, this results in phase separation of driven and undriven particles.
This mechanism is similar to what has been proposed for systems that undergo laning (separation of two types of particles moving in opposite directions in to lanes parallel to their velocity vectors)~\cite{Klymko2016} and stripe formation (separation in to stripes perpendicular to the direction of periodic forcing)~\cite{Mullin2000, Pooley2004, Wysocki2009}. In both cases, the differential mobility of the particles in the presence of the other particle type leads to separation.
We note that this mechanism of phase separation depends on a high degree of synchronization between the displacement vectors of the driven particles of each type - in our case, the driving force on all driven particles is the same (Eq.~\ref{eq:FexEq}), so that all the driven particles are in phase. If the directions of the driven particles are not correlated, for instance if we assigned random phases to each driven particle, phase separation of the kind seen here would not occur. Instead, for sufficiently high $Pe$ and slowly changing particle direction, we would expect motility-induced phase separation~\cite{Liebchen2017, Cates2015}.
Similar to Refs.~\citenum{Mullin2000, Pooley2004, Wysocki2009, Vissers2011a}, in the WCA system there is a gap at one of the interfaces between the red and blue particles (Fig.~\ref{fig:snapshots}). The location of the gap switches periodically from one interface to the other. This is because the red (driven) particles effectively occupy a larger volume than the blue particles and push them out of the way when the driving force pushes red and blue particles in to one another. When the force changes directions, the red particles move en masse away from the blue particles, but diffusion is not fast enough for the blue particles to fill the space left by the retreating red particles, so a gap opens up.
\subsection{Currents Along the Interface}
Measuring the $y$-direction displacement of particles in the WCA model reveals that there are particle currents along the interface. In Fig.~\ref{fig:velocityprof} we show that the direction of the flow is chiral - by which we mean that it moves in only one direction along the interface as determined by the direction of orbit of the driven particles - and that its maximum value is roughly linear in $Pe$. This feature distinguishes interfaces in this system from ones previously studied in MIPS-type systems with WCA~\cite{Bialke2015, Lee2017, Patch2018} or LJ~\cite{Paliwal2017} interactions, where no flows exist in the steady state due to the random orientation of the active forces.
\begin{figure}
\includegraphics[width=\linewidth]{vprof.pdf}
\caption{{\bf There is a net particle current along the interface in the driven WCA system.} $v_y$, defined in Methods, quantifies the current along the interface in the $y$-direction. The maximum value of $v_y$ scales roughly linearly with $Pe$. Since the average position of the interface varies between simulations, the curves have been shifted in the $x$-direction to facilitate comparison.}
\label{fig:velocityprof}
\end{figure}
\subsection{Scaling of Interface Fluctuations and Surface Tension}
Based on studies of driven lattice gases~\cite{Leung1988b, Leung1993} we might expect currents parallel to the interface in the WCA model to cause deviations of the interface height fluctuations from capillary scaling. Indeed, most of the parameters that we studied do not exhibit capillary scaling. However, for $Pe = 120$, the spectrum of interface fluctuations has an exponent close to -2, but only over roughly an order of magnitude of wavenumbers (Fig.~\ref{fig:hk_WCA}). For this value of $Pe$ we calculated an effective surface tension of $\sigma/k_BT = 0.9$. For lower values of $Pe$, $\langle |h(k)|^2\rangle$ decreases less rapidly with $k$ than predicted by CWT. The scaling exponents extracted from fits are shown in Fig.~\ref{fig:hk_WCA}; however we emphasize that these should not be interpreted as analytical exponents resulting from some underlying physics. We note that the system undergoes a reentrant mixing transition as the value of $Pe$ is increased~\cite{delJunco2018}. In particular, the point $Pe=120$ is close to the rentrant transition. Due to this, we were unable to systematically probe the effects of increasing the driving force amplitude on the interfacial fluctuations.
At all values of $Pe$, fluctuations for the smallest ($k < 0.4$ in Fig.~\ref{fig:hk_WCA}) and largest ($k > 2$) wavevectors do not follow the same trend as the rest of the data. At large wavevectors $\langle |h(k)|^2\rangle$ flattens out as a result of the lower limit on fluctuations set by our coarse-graining of the system. To test whether the flattening at small wavevectors was a real feature or an artefact of the finite simulation time, we simulated a trajectory with $Pe = 120$ and $L = 200r_0$ for 8 times longer than the $L = 100r_0$ simulations. There, $1/k^2$ scaling persists to larger wavelengths, suggesting that the fall-off is indeed due to the simulation time.
The results in Fig.~\ref{fig:hk_WCA}, as well as previous results on interfaces in active systems~\cite{Paliwal2017}, suggest that driving can change the effective surface tension and modify the statistics of interfaces in nonequilibrium liquids. Studying these effects in a systematic way is complicated in the WCA model by the fact that the driving has a non-monotonic effect on the interface statistics over a relatively narrow range of values of $Pe$, but a linear effect on the magnitude of the particle flow along the interface. In addition, since this system cannot phase separate in the absence of driving, there is no reference equilibrium interface to compare the driven interfaces to. To address this issue, we use the LJ model, which exhibits liquid-vapor coexistence at equilibrium. Interfaces in LJ liquids have been well-studied and are known to exhibit capillary scaling~\cite{Sides1999, MTOC}, so the LJ model provides a clear reference point that is lacking in the WCA model, and moreover, we can study the effect of driving forces starting well below $Pe = 60$.
\begin{figure}
\includegraphics[width=\linewidth]{hk-WCA-w-fits.pdf}
\caption{{\bf The WCA model exhibits capillary scaling for ${\mathbf{Pe = 120}}$.} Scaling of interface modes ($\langle |h(k)|^2\rangle$) multiplied by interface length ($L$) in the WCA model as a function of $k$, for interfaces of length $100r_0$ (solid colored lines). The legend indicates values of $Pe$ and of the scaling exponent $\alpha$, as in $\langle |h(k)|^2\rangle \propto k^{-\alpha}$, obtained by fitting over the region indicated by dashed black lines, in the format ($Pe: \alpha$). The fluctuations for $Pe = 60, 120$ are larger than for $Pe = 80, 100$. In this system, $Pe = 60$ is close to the point where the system first phase separates, while $Pe = 120$ is close to the point where the system becomes mixed again. $Pe = 80$ and 100 are further inside the bulk of the phase separated region of the phase diagram. For $Pe = 120$, the scaling of fluctuations is close to the $1/k^2$ signature of capillary wave theory over roughly an order of magnitude in $k$, so for this case we also fit a line (solid black line) $\propto 1/k^2$ to calculate an effective surface tension $\sigma/k_BT = 0.9$. The error bars are negligibly small except for at $k < 0.3$.}
\label{fig:hk_WCA}
\end{figure}
We first verified that the LJ model produced the expected behavior at equilibrium. We show in Fig.~\ref{fig:LJ} that at $Pe = 0$, the LJ system exhibits capillary fluctuations with a value of the surface tension that is in reasonable agreement with literature values~\cite{Santra2009}. The range of capillary scaling in $k$ is again limited from above by the coarse-graining length and from below by the simulation time. We then measured the effect of driving the system with $Pe$ ranging from $5 - 80$. We find that the surface tension increases linearly over the whole range of $Pe$ (Fig.~\ref{fig:LJ}). Based on our own previous work~\cite{delJunco2018}, which shows that driving can stabilize interfaces in this system, and on other studies of surface tension in driven systems~\cite{Paliwal2017}, we expected an increase in surface tension. However, those results do not indicate that the increase would be linear and persist over the entire range of $Pe$ investigated here, which is an order of magnitude larger than in Ref.~\citenum{Paliwal2017}. In the following section, we present phenomenological arguments and simulation data that partially account for this observation.
\begin{figure}
\centering
\includegraphics[width = \linewidth]{LJ.pdf}
\caption{{\bf Driving increases the surface tension linearly and modifies the scaling of interface fluctuations of LJ particles.} (a) Scaling of interface modes ($\langle |h(k)|^2\rangle$) multiplied by interface length $L$ in the LJ model as a function of $k$, for interfaces of length $100r_0$. The curves for $Pe = 0 - 70$ have been offset to make it easier to see that the range of $k$ for which $\langle |h(k)|^2\rangle$ scales as $1/k^2$ is largest close to equilibrium and becomes smaller as $Pe$ increases; the $Pe = 80$ curve is not offset to show that the magnitude of fluctuations is comparable to the WCA system. Black lines are $\propto 1/k^2$ and show the range of the fits used to extract $\sigma$; in this range, the error bars are very small. (b) Surface tension (measured from the fits of $\langle |h(k)|^2\rangle$) as a function of $Pe$, with a fit showing the linear correlation between $\sigma$ and $Pe$.}
\label{fig:LJ}
\end{figure}
\subsection{Origin of the Increase in Surface Tension}
Despite our heuristic understanding (summarized in the first part of the Results) of how the driving forces in our model cause phase separation and therefore how they can create interfaces (in the WCA model) or stabilize them (in the LJ model) by increasing the surface tension, it is not clear why the increase should be linear in $Pe$. Surface tension arises due to an imbalance in the forces on particles near to the interface. We now consider two ways that time-dependent driving forces of the kind studied here can magnify this force imbalance, and whether these can explain the observed doubling of the surface tension (Fig.~\ref{fig:LJ}).
First, we propose that the driving forces can cause the undriven WCA particles to exert a restoring force on regions of the interface with high curvature. To see why, consider a section of the interface like the one shown in Fig.~\ref{fig:schematic}. All LJ particles at the interface experience a force ${\mathbf F}_d\propto Pe$ that pushes them in to undriven WCA particles. In the linear response regime, WCA particles will push back with a conservative force also proportional to $Pe$~\cite{delJunco2018}. A driven particle at the interface will therefore feel a downward force proportional to $Pe$ and to the number of undriven particles in its neighborhood~\footnote{Although the force exerted by the driven particle is not always pointed straight in to the undriven phase as illustrated in Fig.~\ref{fig:schematic} - it rotates according to Eq.~\ref{eq:FexEq} - when the driven particles are moving away from the undriven particles they exert no force on them, since the driven-undriven particle interactions are purely repulsive. This results in the gaps that we observe at the interface in the WCA system. To a first approximation, we therefore assume that the most important contribution to ${\mathbf F}_d$ points out normal to the interface and restoring force ${\mathbf F}_d$ points back down.}.
As we illustrate in Fig.~\ref{fig:schematic}, if the driven particle is at a point with negative curvature, it is surrounded by more undriven particles than if it is at a point with positive curvature. Thus, the excess downward force on the interface is proportional to the curvature: $\langle F_c\rangle_{int}\propto Pe\nabla^2 h$. Combining this argument with the CWT Hamiltonian in Eq.~\ref{eq:CWTHamiltonian} we can write down a phenomenological equation of motion for $h(x)$:
\begin{equation}
\frac{\delta h}{\delta t} = \frac{\sigma}{2} \nabla^2 h(x) + Pe \nabla^2 h(x) + \eta(x, t),
\label{eq:CWTEOM}
\end{equation}
where $\eta$ is a white noise with statistics $\left<\eta(x, t)\right>=0$ and $\left<\eta(x, t)\eta(x',t')\right> = 2k_BT\delta (x - x') \delta(t-t')$. We immediately see that this will result in an apparent surface tension $\propto Pe$.
For this picture to correctly explain our observations, $\langle {\bf F}_c\rangle$ must scale with $Pe$, which implies that the work done on the system at the interface by the driving forces should scale as $Pe^2$, since the work is proportional to ${\bf F}_c\cdot {\bf F}_d$ (Eq.~\ref{eq:work}). Motivated by earlier work on this system in which we found that work in a region of mixed driven and undriven particles scales as $Pe^2$, we hypothesized that this could also be the case at the interface. To check whether this is indeed the case, we measured the work in the system according to Eq.~\ref{eq:work}. Work can only be done where there are driven and undriven particles in contact, so although we measured the work in the whole system, the small number of LJ particles in the WCA bulk and vice-versa (Fig.~\ref{fig:snapshots}) ensures that the interfacial region provides the important contribution to the total work. Contrary to our hypothesis, we show in Fig.~\ref{fig:work} that in the LJ system the work is only quadratic in $Pe$ for $Pe < 15$, and then follows a linear trend up to $Pe = 80$.
This means that $\langle F_c\rangle_{int}\propto Pe\nabla^2 h$ can only partially explain the linear scaling of $\sigma$ with $Pe$.
\begin{figure}
\includegraphics[width = 0.6\linewidth]{schematic.pdf}
\caption{{\bf WCA particles exert a force proportional to $\mathbf{Pe\nabla^2 h}$ on LJ particles near the interface.} At the moment of the snapshot, all red LJ particles are pushing up on the blue WCA particles with a force ${\bf F}_d\propto Pe$. In the box labeled 1, where the curvature is positive, LJ particles experience an opposing conservative force from 1 WCA particle. In the box labeled 2, where the curvature is positive, LJ particles experience an opposing conservative force from 3 WCA particles. On average, this leads to a force on the interface $\propto Pe\nabla^2 h$.}
\label{fig:schematic}
\end{figure}
\begin{figure}
\includegraphics[width = \linewidth]{work.pdf}
\caption{{\bf The rate of work done on the system by driving forces scales linearly with $Pe$ for $Pe > 15$.} In the inset we show that for values of $Pe \leq 15$ the work scales as $Pe^2$, in agreement with the results of Ref.~\citenum{delJunco2018}. Error bars are smaller than the points except for at $Pe = 80$.}
\label{fig:work}
\end{figure}
Another way that the driving forces can modify the force imbalance is by increasing the density of the LJ liquid phase, so that the imbalance in attractive forces is magnified. We measured the density of the driven LJ particles as a function of position to see if there was a significant change. Indeed, as $Pe$ is increased the density of LJ particles in the liquid phase increases, and the density of LJ particles in the gas phase decreases. To quantify the change we fit the density of the left interface to a hyperbolic tangent function of the form:
\begin{equation}
\rho(x) = C\tanh(x - x_0) + b.
\label{eq:tanh}
\end{equation}
where $C, b$ and $x_0$ are fitting parameters. Assuming this form for the density, the force imbalance on a particle located at the interface is proportional to $C$, so $C$ should predict the increase in surface tension due to the change in density. In Fig.~\ref{fig:density}, we show that $C$ increases roughly linearly with $Pe$. However, the change in $C$ is only on the order of 15\% and cannot explain the full increase in the surface tension that we observed. The driving forces must therefore have effects on the interface in addition to a force proportional to $Pe\nabla^2 h$ and an increase in density; what these effects might be remains an open question. Importantly, in both of these arguments we ignored the time dependence of the driving forces. The time dependence is what causes currents at the interface, which are expected to affect fluctuations~\cite{Leung1993}. We therefore expect that it will be necessary to take in to account time-dependent affects such as coupling between interface modes to account for our results.
\begin{figure}
\includegraphics[width = \linewidth]{C.pdf}
\caption{{\bf The density gradient near the interface scales linearly with $\mathbf{Pe}$.} The slope of the density near the interface, given by $C$ as defined in Eq.~\ref{eq:tanh}, as a function of $Pe$. Error bars are the standard deviation of the values of $C$ obtained from fitting density profiles of four independent segments of the simulation of length $\tau_r$. (Inset) An example of the fits of the density of LJ particles to Eq.~\ref{eq:tanh} for $Pe = 0, 40, 80$ shows the liquid density increasing and the gas density decreasing with increasing $Pe$.}
\label{fig:density}
\end{figure}
\section{Conclusions}
In this simulation study we presented results regarding the surface tension and statistics of interfacial fluctuations in two closely related systems of driven particles: one where all particles have repulsive WCA interactions and half are driven, and a second where the driven particles have attractive LJ interactions. The WCA system is phase separated for a range of P\'eclet numbers from approximately $Pe = 50-150$. Over this range the interfaces exhibit chiral particle currents parallel to the interface whose velocity is proportional to $Pe$. At one value of $Pe$ near to the reentrant phase transition, height fluctuations of the interface exhibit the $1/k^2$ scaling that is a signature of capillary wave theory. For other values of $Pe$, the spectra of height fluctuations are inversely proportional to $k$ but less steep than $1/k^2$. In the system with LJ interactions, stable interfaces with capillary scaling already exist at equilibrium. Upon driving, we found that capillary scaling persists and that surface tension increases linearly over two orders of magnitude in $Pe$ - from small values in the linear response regime to well above the value of $Pe$ required for phase separation in the WCA system.
The driving force in our system can be reproduced in an experiment using rotating magnetic fields~\cite{Han2017}. Our findings therefore suggest a way of controlling the surface tension of assemblies of particles from a distance, without the need to change any properties of the particles. However, although we discussed two possible explanations for the excess force imbalance at the interface that causes the increase in the surface tension with $Pe$ (a force proportional to the curvature of the interface induced by the driving forces, and the increased density gradient of the LJ particles), neither captures the doubling in the magnitude of the surface tension that we observed. Our work thus poses the challenge of fully explaining how the system channels the energy input at the smallest possible length scale in to modes at the interface that span the largest length scale of the system, which we expect will require a theory taking in to account the genuinely non-equilibrium nature of the steady state. This understanding will be necessary to fully control the surface tension of experimental particle systems.
\section{Acknowledgements}
Thanks to Glen Hocky and Bodhi Vani for helpful comments on this draft. This work was partially supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation under award number DMR-1420709. CdJ and SV also acknowledge support from the Sloan Fellowship and the University of Chicago.
|
{
"timestamp": "2019-01-24T02:03:06",
"yymm": "1803",
"arxiv_id": "1803.02678",
"language": "en",
"url": "https://arxiv.org/abs/1803.02678"
}
|
\section{Introduction}
Multilayer neural network can approximate any smooth function. For example, the transient of a state variables gyro stabilizer or control by time. The main advantage of a neural network is not requiring of a complete mathematical model of gyro stabilizer. An example of these case is using MEMS gyroscopes for creating gyro stabilizer. But, general algorithm of the synthesis of neural network not formulated yet. Such of algorithms is the goal for scientists. This paper is describing a development of this algorithm for the class of dynamic systems.
\section{Problem definition}
Dynamic a channel of uniaxial gyro stabilizer can be described by the following system of nonlinear differential equations \cite{IEEEhowto:Lysov}:
\begin{dmath}
\begin{array}{c}
A_1\ddot{\alpha_1} -\frac{(J_{xp} -J_{yp})} {2}\ddot{{\alpha }}_2 \cos \alpha_{2} \sin 2\alpha_{2}+\\+h\dot{{\alpha }}_1 + (J_{ze} -J_{xp})\dot{\alpha_1}\,\dot{\alpha_2} =M_{1} +M_{\hbox{cont.}1}; \\
A_2 \ddot{{\alpha}}_2-\frac{(J_{\hbox{ye}} -J_{\hbox{xe}} )} {2} \ddot{{\alpha }}_1 \cos \alpha_{2} \sin 2\alpha
_{2} + \\+h \dot{{\alpha }}_2 + (J_{xp} -J_{ye})\dot{\alpha_3}\,\dot{\alpha_1} =M_{2} + M_{\hbox{cont.}2} ; \\
A_3 \ddot{{\alpha }}_3 -J_{\hbox{ze}}
\ddot{{\alpha }}_1 \sin \alpha_{3} +\\+ h_3 \dot{{\alpha }}_3 + (J_{zp} -J_{yi})\dot{\alpha_2}\,\dot{\alpha_3}=M_{3} +M_{\hbox{cont.}3} ;\\
A_1 =J_{\hbox{ye}} +J_{\hbox{уi}} \cos^2\alpha_{2} +J_{\hbox{zi}} \sin ^2\alpha_{2} + \\ + J_{\hbox{yн}} \cos^2\alpha_{2}\cos^2\alpha_{3} +J_{\hbox{xp}}
\cos^2\alpha_{2} \sin^2\alpha_{3} ;\\
A_2 =J_{\hbox{xi}} +J_{\hbox{xp}} \cos^2\alpha_{3} +J_{\hbox{yp}} \sin^2\alpha
_{3} ;\\
A_3 = J_{\hbox{zp}};\\
\end{array}
\label{eq:platf}
\end{dmath}
where $H$- Kinetic moment gyro unit, $J_{xp},J_{yp},J_{zp}$ - inertia moment of platform, $J_{xi},J_{yi},J_{zi}$ - inertia moment of internal frame, $J_{xe},J_{ye},J_{ze}$ - inertia moment of external frame, $h$ - damping factor, $\alpha_1,\alpha_2,\alpha_3$- angle pumping platform, internal frame and external frame.
The nonlinear system \ref{eq:platf} append equation of inertial measurement unit:
\begin{equation}
\begin{array}{c}
\textbf{u}_g = \textbf{f}_g(\dot{\alpha}_1,\dot{\alpha}_2,\dot{\alpha}_3), \\
\textbf{u}_a = \textbf{f}_a(\gamma_1,\gamma_2,\gamma_3),
\end{array}
\label{eq:imu_g}
\end{equation}
where $u_g$ - the signal from gyroscope, $u_a$ - signal of accelerometr. $\gamma_i$ - angle between platform and horizone plane. The model of IMU \ref{eq:imu_g} have a some features:
\begin{itemize}
\item the model have a dynamic properties;
\item the model is nonlinear;
\item the model is incomplete.
\end{itemize}
The problem of generating a control torque for compensation of the external torque by mesuared of signal of gyro($\textbf{u}_g$). Control is expected to form the law gyro stabilizer $M_{con.}=g(\alpha_1,\alpha_2,\alpha_3)$. Where $M_{con.}$ is nonlinear dynamic link that is in a feedback loop. There some types of feedback controllers:
\begin{itemize}
\item correcting unit;
\item observer with regulator;
\item neural network.
\end{itemize}
We consider case, when the feedback loop contains a neural network.
\begin{figure}
\begin{tikzpicture}[matrix=true]
\newcommand{\Gitter}[4]{
\draw[very thin,color=gray] (#1,#3) grid (#2,#4);
\node at (#1,#3) {(#1,#3)};
\node at (#2,#4) {(#2,#4)};
\node at (0,0) {(0,0)};
}
\node at (2,7) (xix) {$M$};
\node at (4,7) (Sum1x) [summe] {$\Sigma$};
\node at (6.75,7) (Intx) [object] {{Gyroplatform}};
\draw [>->] [ultra thick] (xix) -- (Sum1x.west);
\draw [->] [ultra thick] (Sum1x.east) -- (Intx.west);
\node at (10.25,7.35) (yxx) {\begin{small}$\textbf{y}= [\textbf{u}_{\hbox{g}},\textbf{u}_{\hbox{a}}]^T$\end{small}};
\node at (9.05,4.25) (Bp) [object] {{MU}};
\node at (6.75,4.25) (K2) [object] {{NN}};
\node at (3.0,5) (yxx) {{$\textbf{M}_{\hbox{cont.}}$}};
\draw[ultra thick] [->] (Intx.east) -- (10,7) |- (Bp);
\draw [->] [thick] (Bp) -- (K2);
\draw [ultra thick] (4.55,4.0) -- (4.55,4.5);
\draw [->] (K2)-- (4.55,4.25);
\draw [->] [ultra thick] (4.55,4.25) -- (4,4.25) -- (Sum1x);
\end{tikzpicture}
\caption{The structure of the control system. Neural network as regulator.}
\end{figure}
\begin{figure}
\begin{tikzpicture}[matrix=true]
\node at (2,7) (xix) {$M$};
\node at (4,7) (Sum1x) [summe] {$\Sigma$};
\node at (6.75,7) (Intx) [object] {{Gyroplatform}};
\draw [>->] [ultra thick] (xix) -- (Sum1x.west);
\draw [->] [ultra thick] (Sum1x.east) -- (Intx.west);
\node at (10.25,7.35) (yxx) {\begin{small}$\textbf{y}= [\textbf{u}_{\hbox{g}},\textbf{u}_{\hbox{a}}]^T$\end{small}};
\node at (6.55,4.65) (Ip) {$\hat{\textbf{x}(t)}$};
\node at (9.25,4.25) (Bp) [object] {{ MU}};
\node at (5.55,4.25) (Reg) [object] {{ P}};
\node at (7.75,4.25) (K2) [object] {{NN}};
\node at (3.0,5) (yxx) {{$\textbf{M}_{\hbox{cont.}}$}};
\draw[ultra thick] [->] (Intx.east) -- (10,7) |- (Bp);
\draw [->] [thick] (Bp) -- (K2);
\draw [->] [thick] (K2)-- (Reg);
\draw [->] [ultra thick] (Reg) -- (4,4.25) -- (Sum1x);
\end{tikzpicture}
\caption{The structure of the control system. Neural network as observer.}
\end{figure}
Two schemes of control are proposed. In the first scheme gyro signal put into the memory unit. The memory unit is generated a vector containing the current value and the previous several values of the vector $\hat{x}$. This vector $\hat{x}$ is input for neural network \cite{IEEEhowto:yan}.
The neural network connected to a motor. The motor creates a moment, which balance external moment (Figure 1). In the second case, neural network estimate of the state vector, which is connected to the regulator, the signal from regulator is connected to the motors of stabilization (Figure 2).
In the case when gyro stabilizer has a several channels of stabilization feedback loop consists of several of parallel neural networks. This allow reduce the load on each neural network.
\section{Optimizing algorithm of synthesis}
We consider the "classic" algorithm for the synthesis of neural networks, which consists of five stages. In the first stage, formalization of the problem. The unknown function is determined that the neural network during its work will be interpolated, the number of input and output variables. Next the step is selecting structure of the neural network: definition of topology and network settings, types activation functions. After, creation of the training sample is following, which should reflect all the possible modes. Next step is a choice of algorithm training parameters and train neural network. And the final stage is verification of the trained neural network on the test sample. When a result of checking is positive neural network is considered trained and may be used in the work.
Describe the algorithm for the synthesis of the control device consists of a neural network and regulator. So that the system (1) will be defined as:
\begin{equation}
u=-Px,
\end{equation}
where $x$- State vector, $P$ - regulator.
\subsection{The formalization of the problem}
We proposed formulation of the problem a neural network works as an observer. The input of the neural network is vector of the measured signal and several previous values of it, and the output of the neural network - estimation of the state vector.
For base topology is selected multilayer neural network ("multilayer perceptron"). Mathematical model of the network is described by the equation \cite{IEEEhowto:Ossowski}:
\begin{eqnarray}
\begin{array}{l}
\widehat{\textbf{x}}(u_{\hbox{g.}})=f^{(n)} ...(f^{(2)}(\textbf{w}^{(2)}(f^{(1)}(^{(1)} [u_{\hbox{g.}}(k)\,u_{\hbox{g.}} (k-1)\,\cdots \\
\phantom{aa}\cdots\,u_{\hbox{g.}}(k-m) ] ^{T}
+\textbf{b}^{(1)} )+\textbf{b}^{(2)})...\textbf{b}^{(n)}_{i,0}),
\end{array}
\label{eq:MnogosloyDynamic}
\end{eqnarray}
where $w^{(j)}$ - Weighting matrix $j$-th layer of the neural network, $b^{(j)}$- bias vector of $j$-th layer of the neural network, $f^{(j)}$ - activation function $j$-th layer of the neural network, $u_{\hbox{g.}}(k)$ -current value of the measured signal,$\widehat{\textbf{x}}$ - An output vector of the neural network, $k$- depth of memory.
Tables \ref{tab:max_angel_logsig},\ref{tab:max_angel_sig} are provided the result of simulation gyro stabilizer. Many of neural networks were synthesized during experiments for detecting the relationship between the parameters of the neural network and the features of the transient process in the stabilization of the platform. The tables show not all network can work as observer. Sometimes the transient process is unstable($\inf$ in tables). Stable transient process can be find when correct inequality:
\begin{equation}
m<k
\label{eq:ineq}
\end{equation}
where $m$- number neutral in hidden layer.
The sub optimal, in case minimization of angle plumping, estimate when numbers neural in hidden layer is approximate equal to order of system (1)-(2). So, inequality \ref{eq:ineq} can be appended:
\begin{equation}
m \approx order(System)
\label{eq:ineq}
\end{equation}
Neural network with nonlinear activation, like “tansig” or “logsig” function in hidden layer and linear in output layer are preferred. These features also work when the neural network is used for observing a linear dynamic system.
\subsection{Creating a training sample}
The training set is prepared with a special algorithm. The main aim is all system state variables are observable. Next, a closed system is formed by including a feedback loop controller by state. In some works are recommended use a harmonic signal with increasing frequency to the input of gyroscope stabilizer. But most prefer is use random normalized input signal or a harmonic signal at a fixed frequency. These results were obtained on the basis of numerical modeling.
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}
[
declare function={unipdf(\x,\xl)= (\x>\xl)?1:0.005*rand);}
]
\begin{axis}[
minor tick num=3,
axis y line=center,
axis x line=middle,
enlargelimits = true,
xmin = 0,xmax=7,
ymin=-50,ymax =30000,
ylabel={$N$, epochs}, xlabel={f, Hz},
no markers,
y tick label style={/pgf/number format/.cd,%
scaled y ticks = false,
set thousands separator={},
fixed},
x tick label style={/pgf/number format/.cd,%
scaled x ticks = false,
set decimal separator={,},
fixed} ]
\addplot [domain=0:6, very thick, blue, samples=100,smooth]{25000*unipdf(x,4.2)*(1+0.005*rand) };
\end{axis}
\end{tikzpicture}
\caption{The relationship between the number of iterations and the frequency of the disturbance. The cutoff frequncy of gyro stabilizer 4 Hz.}
\end{center}
\label{pic:Epochs}
\end{figure}
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}
[
declare function={unipdf(\x,\xl)= (\x>\xl)?1:0;}
]
\begin{axis}[
minor tick num=3,
axis y line=center,
axis x line=middle,
enlargelimits = true,
xmin = 0,xmax=7,
ymin=-15,ymax =90,
ylabel={$N$, epochs}, xlabel={f, Hz},
no markers,
y tick label style={/pgf/number format/.cd,%
scaled y ticks = false,
set thousands separator={},
fixed},
x tick label style={/pgf/number format/.cd,%
scaled x ticks = false,
set decimal separator={,},
fixed} ]
\addplot [domain=0:6,smooth,very thick, blue] table {
0.5 79
0.6 75
0.8 68
0.9 60
1.4 38
2.4 24
3.4 20
4.2 14
6.1 9
};
\end{axis}
\end{tikzpicture}
\caption{The relationship between the the maximum angle of the pumping and the frequency of the disturbance. The cutoff frequncy of gyro stabilizer 4 Hz.}
\end{center}
\label{pic:freq}
\end{figure}
For determining the optimum frequency of the input harmonic signal numeric experiment was conducted.The input to the reference model supplied harmonic signal with a fixed frequency, after which the obtained sample was trained the neural network. The number of epochs required to train the neural network, and maximum angle leveling platforms comprising a feedback loop neural network was measured in the experiments.
The results of numeric experiments are shown in Figures \ref{pic:freq} and \label{pic:Epochs}. The figure shows that increasing the frequency of the input harmonic signal decreases the maximum angle pumping platform, but after a certain frequency is a sharp increase in the number of periods required for training. The frequency, then a sharp increasing the number of periods, was close to the cutoff frequency of the reference model. I Thus, the optimum in terms of the ratio of the time of training and the maximum angle of pumping, is situated at the cutoff frequency.
\subsection{Selecting learning algorithm }
The goal of training the neural network is changing the weight coefficients, which the minimization of functional \cite{IEEEhowto:Haykin}:
\begin{equation}
E(w)=\frac{1}{2\,N}\Sigma \left(x(t)-\widehat{x}(t,\textbf{w}) \right)^2=
\frac{1}{2\,N}\Sigma\varepsilon^2,
\label{eq:crit}
\end{equation}
where $x(t)$ - the training sample, $\widehat{x}(t,\textbf{w})$ - the output of neural network, $N$- the number of training samples.
\begin{itemize}
\item Gradient method;
\item Hewton method;
\item Levenberg-Marquardt method \cite{IEEEhowto:Lera},\cite{IEEEhowto:Ngia}.
\end{itemize}
The learning of neural network consist of several steps. At the begin training set is shuffle. After that minimization of \label{eq:crit} is doing. These two steps repeat until \label{eq:crit} achieving preset value, or number of loop iteration not be a huge.
As shown by mathematical modeling, the most efficient is the Levenberg-Marquardt algorithm.
\begin{figure}[ht!]
\begin{center}
\begin{tikzpicture}
\begin{axis}[
minor tick num=3,
axis y line=center,
axis x line=middle,
ymode=log,
xmode = log,
enlargelimits = true,
ymax = 1e5,
no markers,
y tick label style={/pgf/number format/.cd,%
scaled y ticks = false,
set thousands separator={},
fixed},
x tick label style={/pgf/number format/.cd,%
scaled x ticks = false,
precision = 1,
set decimal separator={.},
fixed}
]
\addplot[color=red,samples=500,mark=*,very thick] table[y=y1,x=x] {learn.dat};
\addlegendentry{Levenrg-Marquardt alorithm};
\addplot[color=blue,samples=500,mark=*,very thick] table[y=y2,x=x] {learn.dat};
\addlegendentry{Gradient algorithm};
\addplot[color=black,samples=500,mark=*,very thick] table[y=y3,x=x] {learn.dat};
\addlegendentry{Newton algorithm};
\end{axis}
\end{tikzpicture}
\caption{The learning rate}
\end{center}
\label{pic:Learn}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\plotffind{0}{{t, sec.}}{1}{{$\alpha_1$, arcmin}}{alpha_tansig_1_1_2.dat}
\end{tikzpicture}
\caption{Angle plumping when NN(tansig, 1,2)}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tikzpicture}
\plotffind{0}{{t, sec.}}{1}{{$\alpha_1$, arcmin}}{alpha_logsig_1_3_7.dat}
\end{tikzpicture}
\caption{Angle plumping when NN(logsig, 3,7)}
\end{figure}
\subsection{Verification of a neural network }
The neural network operates in a feedback loop of a dynamic system, so that traditional methods verification of the neural network are not applicable. In this regard, the only method of verification is a simulation of a closed system. In this case modeling gyro stabilizer However, in some cases, simulations can take time comparable to the time of training, and even exceed it. In addition to mathematical modeling, it is proposed to check the performance of Neural network at the stand.
\section{Conclusion}
The article was considered a neural network algorithm for controlling a uniaxial gyro stabilizer. The optimal parameters of neural network based observer are determing. The optimum frequency of the harmonic signal input od ideal model for the formation of a training sample. The results can be used in the synthesis of control devices built using the device of neural networks for tracking systems.
|
{
"timestamp": "2018-03-13T01:21:46",
"yymm": "1803",
"arxiv_id": "1803.02738",
"language": "en",
"url": "https://arxiv.org/abs/1803.02738"
}
|
\section{Introduction}\label{sec:introduction}
It is common believed that the usual parton distributions (PDFs) can only give the longitudinal information of a hadron target in the deep inelastic scattering (DIS) processes, while the generalized parton distributions (GPDs) have the promising ability to shade light on the transverse information, which gives rise to the idea of ``quark/gluon imaging" of hadrons~\cite{Marukyan2015}. Moreover, the impact parameter distributions (IPDs),
obtained by the Fourier transform of GPDs with respect to the transverse momentum transfer, may show some
information about the transverse impact space position of partons~\cite{Burkardt:2002hr}. This impact parameter representation is useful in processes such as high-energy scattering and hard processes~\cite{Diehl2003}. It is also argued that, in position space, IPDs play a similar role to the charge distributions, and are, thus,
very promising for understanding the hadron internal structures. \\
As we know, $G_C(Q^2)$ is the form factor of the conserved local current, and is thus independent of the renormalization scale $\mu$. It can be obtained through the sum rules from GPDs, which by definition are probed in hard processes~\cite{Diehl2003}. In the case of Fourier transforms of GPDs, Burkardt pointed out that, when $\xi=0$, the Fourier transforms of GPDs have the interpretation of a density of partons with longitudinal momentum fraction $x$, localized at ${\bf b}_\perp$ relative to the transverse center in the impact parameter space, which is allowed by the Heisenberg uncertainty principle~\cite{Burkardt:2000za,Diehl:2002he}. Due to the significance of the form factors in the impact parameter space, many theoretical works have been devoted to study the IPDs of pions, kaons and nucleons~\cite{Miller2009,Miller:2007uy,Miller:2010nz,Miller:2009qu,Miller:2010tz,
Carmignotto:2014rqa,Nam:2011yw,Kumar:2015yta,Diehl:2002he,Dalley2003Jul21,Broniowski2003}. \\
It should be mentioned that our recent work~\cite{Sun:2017gtz} gave a discussion of the $\rho$ meson unpolarized GPDs in momentum space with a Light-Cone Constituent Quark Model (LCCQM).
The form factors and some other low-energy observables of the $\rho$ meson were calculated and our numerical results agreed with the previous publications and some experimental data~\cite{Krutov:2018mbu}.
In the literature, the constituent quark model is also used to describe the form factors of pions, nucleons, deuterons, etc.~\cite{Frederico:2009fk,Sun:2016ncc,Dahiya:2017fmp}.
Moreover, the contributions from the valence and non-valence regimes to the form factors and generalized parton distributions were discussed and analyzed in detail.
In addition, the reduced matrix elements, which are the moments of the DIS structure functions, were also estimated and the obtained values were compatible with the available lattice calculation at the same scale ratio~\cite{Best1997}.
In general, our numerical results for the unpolarized GPDs~\cite{Sun:2017gtz} were reasonable and satisfying. Therefore, in this work, we extend the phenomenological model to study the IPDs of the $\rho$ meson and to calculate the impact parameter dependent PDFs of $q(x,{\bf b}_\perp)$ and
$q({\bf b}_\perp)$ and the form factors of $q^{C,M,Q}(x,{\bf b}_\perp)$ and $q^{C,M,Q}({\bf b}_\perp)$. \\
The paper is organized as follows. In Section~\ref{Impact_parameter_dependent_PDF}, the framework of the impact
parameter dependent PDFs is presented. In Section~\ref{sec:Wave_packets}, we discuss the wave packets and the
cutoff for the numerical calculation. The definitions of the impact parameter dependent FFs are
given in Section~\ref{sec:impact_parameter_space_FFs}. Our numerical results for the PDFs and FFs in the impact
parameter space are shown in Section~\ref{sec:Results}, and Section~\ref{sec:summary} gives a short
summary and conclusion.\\
\section{Impact parameter dependent PDFs}
\label{Impact_parameter_dependent_PDF}
When considering the nucleon GPDs without helicity flip, Burkardt~\cite{Burkardt2003} identifies the Fourier transform of its GPD $H_q(x,\xi=0,-{\bf \Delta}_\perp^2)$ w.r.t. $-{\bf \Delta}_\perp^2$ as a distribution of partons in the transverse plane, i.e., the probability of finding a quark with longitudinal momentum fraction $x$ and at transverse impact space position ${\bf b}_\perp$.
The impact parameter dependent PDF for a nucleon (a spin-1/2 target), given by Ref.~\cite{Burkardt2003}, reads
\end{multicols}
\eq
q_N(x,{\bf b}_\perp)
&=& \left|{\cal N}\right|^2
\int \frac{d^2{\bf p}_\perp}{(2\pi)^2}
\int \frac{d^2{\bf p}_\perp^\prime}{(2\pi)^2}\times
\langle p^+, {\bf p}^\prime_\perp, \lambda | \left[
\int\frac{dz^-}{4\pi} \bar{q}(-\frac{z^-}{2}, {\bf b}_\perp) \gamma^+ q(\frac{z^-}{2}, {\bf b}_\perp)
e^{-\imath x p^+ z^-}
\right] | p^+, {\bf p}_\perp, \lambda \rangle \nonumber\\
&=& \left|{\cal N}\right|^2
\int \frac{d^2{\bf p}_\perp}{(2\pi)^2}
\int \frac{d^2{\bf p}_\perp^\prime}{(2\pi)^2}
H_q(x,\xi=0,-\left({\bf p}_\perp-{\bf p}_\perp^\prime \right)^2)
e^{i{{\bf b}_\perp} \cdot ({\bf p}_\perp-{\bf p}_\perp^\prime)}
\nonumber\\
&=&
\int \frac{d^2{\bf \Delta}_\perp}{(2\pi)^2}
H_q(x,0,-{\bf \Delta}_\perp^2)
e^{-i{\bf b_\perp} \cdot {\bf \Delta}_\perp}
\nonumber \\
&=&
\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi} J_0 (b \Delta_\perp) H_q(x,0,-\Delta_\perp^2) \nonumber \\
&=& q_N(x,{b}),
\label{eq:result1}
\en
\begin{multicols}{2}
where the normalization factor ${\cal N}$ satisfies $\left|{\cal N}\right|^2\int\frac{d{\bf p_\perp}}{(2\pi)^2}=1$,
and $\Delta_\perp=|{\bf \Delta}_\perp|=\sqrt{\Delta_x+\Delta_y}$ and $b=|{\bf b}_\perp|=\sqrt{b_x+b_y}$. Cylindrical symmetry is applied in the last but one step and $J_0$ is the Bessel function of the first kind $J_\nu(z)$ with $\nu=0$. The parton distribution depends on transverse impact space position ${\bf b}_\perp$ only through its norm $b$ being the consequence of the longitudinal polarization.
In the third step the integral turns to the total and transverse momentum transfer, i.e., ${d^2{\bf p}_\perp}{d^2{\bf p}_\perp^\prime}={d^2{\bf \Delta}_\perp}{d^2{\bf P}_\perp}$, with ${\bf \Delta}_\perp={\bf p}_\perp^\prime-{\bf p}_\perp$ and $\bf P_\perp= (\bf p^\prime_\perp + \bf p_\perp)/2$, and using the fact that GPD $H$ is independent of total transverse momentum $\bf P_\perp$.
Ignoring the helicity flip, the spin projection $\lambda$ can be dropped. In the forward limit, namely $\xi=0$, we have $t=(p'-p)^2=-{\bf \Delta}_\perp^2$. \\
Note that Hoodbhoy~\cite{Hoodbhoy:1988am} has already pointed out the DIS structure function
$F_1$, $F_2$, $g_1$, and $g_2$ of spin-1 targets can be precisely measured in the same way as that
of spin-1/2 targets. Analogous to the fact that the structure function $F_1$ connects to GPD $H_q$ for spin-1/2 targets, we simply assume $F_1$ connects to the GPD $H_1^q$ for spin-1 targets as well.
As shown by Eqs.~(37$\sim$39) in Ref.~\cite{Sun:2017gtz}, the isospin combination implies that
\eq
\int_{-1}^1 dx \; H_i^{u} (x,\xi,t) = \int_{-1}^1 dx \; H_i^{I=1} (x,\xi,t) \ .
\en
Hereafter we omit the label of quark flavor $u$ and isospin $I=1$ for simplicity.
Due to the similar roles of $H_q$ and $H_1$, we introduce the impact parameter dependent PDF for spin-1 targets (for the $u$ quark),
\eq
q(x,b)
&=&
\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi} J_0 (b \Delta_\perp) H_1(x,0,-\Delta_\perp^2) \ ,
\label{eq:result2}
\en
One can further define the total parton distribution in the impact parameter space as
\eq
\hspace{-10mm} q(b)
&=&\int_0^1 dx \; q(x,b) \ .
\label{eq:result3}
\en
\par\noindent
Notice that $\int d^2{\bf b}_\perp \; q(x,{ b})=H_1(x,0,0)$, which is equal to the usual PDF $q(x)$ in the forward limit $t=\Delta^2\rightarrow0$. Therefore, $q(x,b)$, the Fourier transform of the GPD $H_1(x,\xi=0,-\Delta_\perp^2)$ w.r.t. $- \Delta_\perp^2$, can be identified, in analogy to the nucleon case, with the probability of finding a quark
with longitudinal momentum fraction $x$ and transverse impact space position ${\bf b}_\perp$ in the $\rho$ meson. \\
It should be emphasized that in Ref.~\cite{Burkardt:2002hr}, the nucleon impact parameter dependent PDF $q_N$ was proved to satisfy the positive constraints for the so-called ``good" quark field. In our model calculation, the phenomenological vertexes (see Eq.~(24) in Ref.~\cite{Sun:2017gtz}) involve the loop momentum ($k$), and the form of the vertexes is fixed according to the constraints from isospin symmetry. Our sophisticated model cannot simply reproduce the procedure of Ref.~\cite{Burkardt:2002hr} to fold the correlation function into a norm of a quantity (see Eq.~(23) of Ref.~\cite{Burkardt:2002hr}). Therefore, the positive constraint for $q(x,b)$ with a realistic model calculation needs to be proven further.\\
\section{Wave packets}
\label{sec:Wave_packets}
The Fourier transform of a plane wave is not well defined, thus, one can start with the wave packets instead of the plane wave. In the non-relativistic limit, the Fourier transform of the charge form factor $G_C(Q^2)$ can
be interpreted as the charge distribution in the transverse direction. In other words, as long as the wave packets peak sharply at some point in position space, by taking the non-relativistic limit, the Fourier transform of the charge distribution equals the form factor. By the way, a Gaussian weighting factor was also adopted in a recent lattice QCD calculation~\cite{Chen:2017lnm}, in order to suppress the unphysical oscillatory behaviour. The oscillation
is due to the finite lattice size and nucleon momentum. The result in the small Bjorken $x$($<0.3$) region is changed
by weighting. In Ref.~\cite{Pire:2002ut}, the Gaussian ansatz is also applied to shape the hadron when calculating generalized distribution amplitudes of the pion pair production process.
\\
Moreover, as pointed out by Burkardt~\cite{Burkardt:2000za,Burkardt:2002hr}, the interpretation of the Fourier transform of the form factor as the charge distribution may receive relativistic corrections in the rest frame. However, such a problem may disappear in either Breit frame or infinite momentum frame (IMF). In the relativistic case, the transform receives relativistic corrections when the wave packet is localized with a size smaller than the Compton wavelength
of the system. In the IMF, the relativistic correction can be managed to be very small, and therefore, the wave packet
does not change the interpretation, as long as the wave packets are set slowly varying w.r.t. $\bf \Delta_\perp$. To be specific, the width of the wave packets must be much larger than a typical QCD scale $\Lambda_{QCD}$ ($\sim0.23~\gev$). For a Gaussian form wave packet, one gets $\sigma\ll 1/\Lambda_{QCD} \sim 3/M$, with $M$ being the $\rho$ meson mass. \\
On the other hand, as Diehl~\cite{Diehl:2002he} has discussed, a real hadron is an extended object and is smeared out by an amount $\sigma$. From the experimental viewpoint, there is a largest measured value $|t|_{\text{max}}$ and thus there is the accuracy of the measurement $\sigma\sim(|t|_{\text{max}})^{-1/2}$. According to the observations and to the limit of the effect from unmeasured values of $t$, a Gaussian form wave packet can also be reasonably introduced. Thus we have
\eq
&&\int \frac{d^2{\bf p}_\perp dp^+}{(2\pi)^2 p^+} p^+\delta(p^+-p^+_0) G({\bf p}_\perp, \frac{1}{\sigma^2}) | p,\lambda \rangle
\nonumber \\
&&\sim \;
\int \frac{d^2{\bf p}_\perp}{(2\pi)^2} \text{exp}\left( -\frac{{\bf p}^2_\perp \sigma^2}{2} \right)| p^+, {\bf p}_\perp, \lambda \rangle \ ,
\label{eq:wp}
\en
where $G({\bf p}_\perp, 1/\sigma^2)=\text{exp}( -{\bf p}^2_\perp \sigma^2/2 )$ and
the mixed state is modified to be
\eq \label{eq:state}
| p^+, {\bf b}_\perp, \lambda \rangle_\sigma
&=&
{\cal N}_{\sigma} \int \frac{d^2{\bf p}_\perp}{(2\pi)^2}
e^{-\imath {\bf b}_\perp \cdot {\bf p}} G({\bf p}_\perp, \frac{1}{\sigma^2})
| p^+, {\bf p}_\perp, \lambda \rangle
\nonumber \\
&\stackrel{\sigma\rightarrow0}{=} &| p^+, {\bf b}_\perp, \lambda \rangle
\ ,
\en
where the normalization factor ${\cal N}_{\sigma}$ satisfies $\left|{\cal N}_{\sigma}\right|^2\int\frac{d{\bf p_\perp}}{(2\pi)^2}=1$ and $\lim_{\sigma\rightarrow0} {\cal N}_{\sigma}={\cal N} $. Note that our normalization of states is different from that in Ref.~\cite{Diehl:2002he}.
This action will add two Gaussian functions in the expression, $G({\bf p}_\perp, \frac{1}{\sigma^2})$ and $G({\bf p}^\prime_\perp, \frac{1}{\sigma^2})$, into the definition of $q(x,{b})$
(see eq. (2)). We can still change variables to remove the dependence of ${\bf P}_\perp$, which leaves only one $G({\bf \Delta}_\perp, \frac{1}{\sigma^2})$.
Consequently, the definition of the impact parameter dependent PDF is modified to be
\begin{eqnarray}
q_{\sigma}( x,{ b} )
&=&
\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi}
J_0 (b \Delta_\perp) G({\bf \Delta}_\perp,\frac{2}{\sigma^2})
H_1(x,0,-\Delta_\perp^2) \nonumber \\
&=&
\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi}
J_0 (b \Delta_\perp) e^{-{{\Delta}_\perp^2}{\sigma^2}/4}
H_1(x,0,-\Delta_\perp^2) \ , \nonumber \\
\label{fb}
\end{eqnarray}
and
\begin{eqnarray}
q_{\sigma}({b} )
&=&
\int_0^1 dx \; q_{\sigma}( x,{ b} ) \ .
\label{fb2}
\end{eqnarray}
Reference~\cite{Diehl:2002he} also argued that in order to give a well-defined (positive, or without sign flip) longitudinal momentum $p^3$, $|{\bf p}_\perp|\ll p^+$ is required. However, as one can see in Eq.~(\ref{eq:wp}), ${\bf p}_\perp$ and $p^+$ are separated in the wave packet and thus this requirement actually does not affect the result of the integrals. This can also be seen from the property of GPDs. In the forward limit, $H(x,0,-{\Delta}_\perp^2)$ is not affected by this requirement either. Moreover, Ref.~\cite{Brodsky:1997de} emphasized that since the longitudinal momentum is $p^+$ in the front form, one needs not to go to infinite momentum along the moving direction, and not to impose the constraint on the $p^3$ component either. \\
According to the above discussions, the relation $\sigma\sim(|t|_{\text{max}})^{-1/2}$ inspires us to introduce a
cutoff ($\Delta_0$) of the momentum transfer in the integral as well
\begin{eqnarray}
\hspace{-5mm} q( x,{b},\Delta_0 )
&=&
\int_0^{\Delta_0} \frac{\Delta_\perp d
\Delta_\perp}{2\pi}
J_0 (b \Delta_\perp) H_1(x,0,-{\Delta}_\perp^2) \ ,
\label{fxbcut}
\end{eqnarray}
and
\begin{eqnarray}
\hspace{-10mm} q({b},\Delta_0 )
&=&
\int_0^1 dx \; q( x,{b},\Delta_0 ) \ .
\label{fbcut}
\end{eqnarray}
This assumption is supported by a comparison between the results of the integrals with a wave packet, $q_\sigma(b)$ (width $\sigma\sim1/\Delta_0$) and the one with a cutoff $q(b,\Delta_0)$. This will be shown in Section~\ref{sec:Results}. \\
\section{Impact parameter dependent FFs}
\label{sec:impact_parameter_space_FFs}
We emphasize that the unpolarized impact parameter dependent PDFs are proposed to describe the transverse distribution of unpolarized partons in an unpolarized target. As shown in previous sections, the IPDs can be obtained through Fourier transform of the unpolarized GPD $H_1$. We notice that the conventional charge,
magnetic dipole and quadrupole FFs are the integrals of the linear combination of $H_i$. This motivates us to explore the possibility of obtain the IPDs with respect to the three FFs. The sum rules relating to the GPDs and the
FFs $G_i$ are~\cite{Berger2001}
\eq\label{eq:sumrule}
\int_{-1}^{1} dx H_i (x,\xi,t) &=& G_i(t) \quad (i=1,2,3) \ , \nonumber \\
\int_{-1}^{1} dx H_i (x,\xi,t) &=& 0 \quad (i=4,5) \ ,
\en
where $G_i^q$ are the FFs in the decomposition of the local current.
The FFs $G_{C,M,Q}$ can be expressed in terms of $G_{1,2,3}$ as~\cite{Choi2004}
\begin{eqnarray}
G_C(t)&=&G_1(t) + \frac{2}{3}{\eta} G_Q(t) \ , \ \nonumber \\
G_M(t)&=&G_2(t) \ , \ \nonumber \\
G_Q(t)&=&G_1(t) - G_2(t) + (1+\eta)G_3(t)\ , \
\label{eq:Gcmq}
\end{eqnarray}
where $\eta=-t/4M^2$. Together with Eq.~(\ref{eq:sumrule}), one can obtain $G_{C,M,Q}$ directly from GPDs $H_{1,2,3}$. This allows us to bypass the well-known ambiguity of the angular condition~\cite{Melo1997}.
With the above two equations, one can get the relations
\end{multicols}
\eq
G_C(t)&=&\int_{-1}^{1} dx \Big [H_1(x,\xi,t) + \frac{2}{3}{\eta} \left[H_1(x,\xi,t) - H_2(x,\xi,t) + (1+\eta)H_3(x,\xi,t) \right] \Big ]\ , \ \nonumber \\
G_M(t)&=&\int_{-1}^{1} dx H_2(x,\xi,t) \ , \ \nonumber \\
G_Q(t)&=&\int_{-1}^{1} dx \Big [H_1(x,\xi,t) - H_2(x,\xi,t) + (1+\eta)H_3(x,\xi,t)\Big ]\ . \
\en
\begin{multicols}{2}
Notice that by taking $\xi=0$ and $\eta=-t/4M^2={\Delta}_\perp^2/4M^2$,
one can get quantities similar to the integrands in Eq.~(\ref{eq:result1}).
We have the impact parameter dependent FFs
\end{multicols}
\eq
\label{eq:ipFFsGC}
q^C_\sigma(x,{b})
&=&
\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi}
J_0 (b \Delta_\perp)
e^{-{{\Delta}_\perp^2}{\sigma^2}/4}
\nonumber \\ &&\times
\Bigg[ H_1(x,0,-{\Delta}_\perp^2) + \frac{2}{3}{\frac{{\Delta}_\perp^2}{4M^2}} \Big[H_1(x,0,-{\Delta}_\perp^2) - H_2(x,0,-{ \Delta}_\perp^2)+ (1+{\frac{{\Delta}_\perp^2}{4M^2}})H_3(x,0,-{\Delta}_\perp^2) \Big] \Bigg] \ , \\\
\label{eq:ipFFsGM}
q^M_\sigma(x,{b})
&=&
\frac{1}{G_M(0)}\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi}
J_0 (b \Delta_\perp)
e^{-{{\Delta}_\perp^2}{\sigma^2}/4}
H_2(x,0,-{ \Delta}_\perp^2) \ , \ \\
q^Q_\sigma(x,{b})
&=&
\frac{1}{G_Q(0)}\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi}
J_0 (b \Delta_\perp)
e^{-{{\Delta}_\perp^2}{\sigma^2}/4} \nonumber \\ &&\times
\left[H_1(x,0,-{ \Delta}_\perp^2) - H_2(x,0,-{ \Delta}_\perp^2) + (1+{\frac{{ \Delta}_\perp^2}{4M^2}})H_3(x,0,-{ \Delta}_\perp^2) \right] \ , \
\label{eq:ipFFsGQ}
\en
\begin{multicols}{2}
and
\begin{eqnarray}
\hspace{-10mm} q^{C,M,Q}_{\sigma}({ b} )
=
\int_0^1 dx \; q^{C,M,Q}_{\sigma}( x,{ b} ) \ .
\end{eqnarray}
Comparing the impact parameter dependent FFs, Eq.~(\ref{eq:ipFFsGC}), with the impact parameter dependent PDFs, Eq.~(\ref{fb}), we introduce the ``difference" quantities
\end{multicols}
\eq
q^{QC}_\sigma(x,{b})
&=&
\int_0^{\infty} \frac{\Delta_\perp d
\Delta_\perp}{2\pi}
J_0 (b \Delta_\perp)
e^{-{{\Delta}_\perp^2}{\sigma^2}/4} \nonumber \\ &&\times
\left(\frac{2}{3}{\frac{{\Delta}_\perp^2}{4M^2}}\right)
\Bigg[H_1(x,0,-{\Delta}_\perp^2) - H_2(x,0,-{\Delta}_\perp^2)
+ (1+{\frac{{\Delta}_\perp^2}{4M^2}})H_3(x,0,-{\Delta}_\perp^2) \Bigg] \ , \\
q^{QC}_{\sigma}({ b} )
&=& \int_0^1 dx \; q^{QC}_{\sigma}( x,{b} ) \ ,
\en
\begin{multicols}{2}
which receive the contribution from the quadrupole moment.
The ``difference" quantities satisfy
\eq
\label{eq:sumCQC}
q^{QC}_{\sigma}( x,{ b} ) &=& q^{C}_{\sigma}( x,{b} ) - q_{\sigma}( x,{ b} ), \; \nonumber \\
q^{QC}_{\sigma}( { b} ) &=& q^{C}_{\sigma}( { b} ) - q_{\sigma}( { b} ).
\en
It is clear that the impact parameter dependent PDFs relate to the impact parameter dependent FFs and
\eq
\label{eq:sumCMQ}
\int_0^1 dx \int_{-\infty}^{\infty} d^2{\bf b} \; q^{C,M,Q}_\sigma(x,{ b})=1 \; \ .
\en
Thus, it is possible to interpret $q^{C}_\sigma$, $q^{M}_\sigma$ and $q^{Q}_\sigma$ as the percentage of the
contributions to the charge (normalized to 1), magnetic dipole $\mu_{\rho}$ and quadrupole moment $Q_\rho$ respectively,
from the parton with the longitudinal momentum fraction $x$ and transverse impact space position ${\bf b}_\perp$.
\\
\section{Results}\label{sec:Results}
In our previous work~\cite{Sun:2017gtz} with a light-cone constituent quark model, we took the two model parameters of the constituent mass $m=0.403~\gev$ and regulator mass $m_R=1.61~\gev$, and we calculated the GPDs of the $\rho$ meson.
In our LCCQM, we introduced an effective Lagrangian for the $\rho-q\bar q$ interaction with a phenomenological vertex $\Gamma^u$ and a Bethe-Salpeter amplitude. By integrating the minus component of the quark momentum $k^-$ analytically and rest of the components numerically, we obtained the GPDs and FFs of the $\rho$ meson.
In this work, we simply extend the calculation to the impact parameter dependent PDFs $q(b)$ and impact parameter dependent FFs $q^{C,M,Q}_\sigma(b)$.
Figure~\ref{fig:ipPDF} gives the $q(b)$ with a wave packet, $q_\sigma(b)$, and with a cutoff on the momentum transfer, $q(b,\Delta_0)$, respectively.
The comparison shows that the cutoff ($\Delta_0$) has a similar effect as the wave packet with width $\sigma\sim1/\Delta_0$.
Of course, we expect that the prediction of the constituent quark model is reasonable only in the region of $|t|^{1/2}\leq 2~\gev$ and when the momentum transfer becomes larger the constituent quark model fails.
The width of the wave packet is also constrained by the uncertainty principle: to have a valid probability interpretation of the initial and finial states, the position dispersion ($\sim\sigma$) cannot be smaller than the Compton wavelength. In the later content, our numerical results in Fig.~\ref{fig:ipPDFCQC:C} agree with this point of view. \\
\end{multicols}
\begin{figure}
\centering
{\hskip -1.5cm}
\subfigure[~$q_\sigma(b)$ with packet width $\sigma=1/2~\gev^{-1}$, $1~\gev^{-1}$, and $2~\gev^{-1}$.]{\label{fig:ipPDF:s}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-H1-Qb-Sigma-212-c3.eps}
\end{minipage}
}
{\hskip 1cm}
\subfigure[~$q(b,\Delta_0)$ with cutoff $\Delta_0=1/2~\gev$, $1~\gev$ and $2~\gev$.]{\label{fig:ipPDF:c}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-H1-Qb-Cut-212-c3.eps}
\end{minipage}
}
\caption{\label{fig:ipPDF}{ \small The impact parameter dependent PDF $q(b)$ with (a) a wave packet and (b) a cutoff on the momentum transfer.
}
\end{figure}
\begin{multicols}{2}
Figure~\ref{fig:ipPDFc} gives the contour plots of the impact parameter dependent PDF $q_{\sigma}(b)$ with $\sigma=1~\gev^{-1}$ and $2~\gev^{-1}$.
Since we choose the polarization in the $z$ direction, the parton distribution is invariant under rotation around the $z$ direction.
We see that as $\sigma$ becomes smaller, the wave functions of the initial and final states get closer to a plane wave, and the parton distribution also becomes more transversely localized in the position space, as shown in Fig.~\ref{fig:ipPDF} and Fig.~\ref{fig:ipPDFc}. \\
\end{multicols}
\begin{figure}
\centering
{\hskip -1.5cm}
\subfigure[~$q_\sigma(b)$ (fm${}^{-2}$) with packet width $\sigma=1~\gev^{-1}$.]{\label{fig:ipPDFc:s2}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-H1-Qb-Sigma-1c-c3.eps}
\end{minipage}
}
{\hskip 1cm}
\subfigure[~$q_\sigma(b)$ (fm${}^{-2}$) with packet width $\sigma=2~\gev^{-1}$.]{\label{fig:ipPDFc:s4}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-H1-Qb-Sigma-2c-c3.eps}
\end{minipage}
}
\caption{\label{fig:ipPDFc}{\small Contour plots of the impact parameter dependent PDF $q(b)$ with a wave packet.
}
\end{figure}
\begin{multicols}{2}
Figures~\ref{fig:ipPDFCQC} and \ref{fig:ipPDFMQ} give the impact parameter dependent FFs $q^{C,M,Q}_\sigma(b)$ and $q^{QC}_\sigma(b)$ with $\sigma=1/2~\gev^{-1},\; 1~\gev^{-1},\; 2~\gev^{-1}$ respectively.
Figure~\ref{fig:ipPDFMQ} shows that, as the wave packet becomes more sharply localized ($\sigma$ decreases), the contributions are concentrated more in the small ${\bf b}_\perp$ region for both the magnetic dipole $\mu_{\rho}$ and quadrupole moment $Q_\rho$.
For the impact parameter charge density, Fig.~\ref{fig:ipPDFCQC:C}, the distributions with $\sigma$ less than about $1~\gev^{-1}$ become obscure due to the oscillation.
As we argued before, the $\rho$ meson is an extended object and its Compton wavelength is $1/m_{\rho}=1.3 \gev^{-1}$.
The position dispersion $\langle\Delta x\rangle=\sigma$ in the case of the Gaussian wave packet. The uncertainty principle ($\langle\Delta x\rangle\langle\Delta p\rangle \ge 1/2$ in natural units) gives the constraint that, to maintain the probability interpretation of the states, the position dispersion $\langle\Delta x\rangle$ should not be smaller than the Compton wavelength.
Otherwise, localizing a wave packet to less than its Compton wavelength in size will in general induce various relativistic corrections~\cite{Burkardt:2000za}.
With the help of Figs.~\ref{fig:ipPDF} and \ref{fig:ipPDFCQC:QC}, and Eq.~\ref{eq:sumCQC}, the oscillation in $q^{C}_\sigma(b)$ can be explained as the behaviour of $q^{QC}_\sigma(b)$ which is related to the quadrupole moment.
From the experimental aspect, since the $\rho$ meson quadrupole moment is small, this phenomenon is hard to determine. \\
\end{multicols}
\begin{figure}
\centering
{\hskip -1.5cm}
\subfigure[~$q_\sigma^C(b)$ with packet width $\sigma=1/2~\gev^{-1}$, $1~\gev^{-1}$, and $2~\gev^{-1}$.]{\label{fig:ipPDFCQC:C}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-Gc-Qb-Sigma-212-c3.eps}
\end{minipage}
}
{\hskip 1cm}
\subfigure[~$q_\sigma^{QC}(b)$ with packet width $\sigma=1/2~\gev^{-1}$, $1~\gev^{-1}$, and $2~\gev^{-1}$.]{\label{fig:ipPDFCQC:QC}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-Gqc-Qb-Sigma-212-c3.eps}
\end{minipage}
}
\caption{\label{fig:ipPDFCQC}{\small The impact parameter dependent FFs $q^{C}_\sigma(b)$ and $q^{QC}_\sigma(b)$ with $\sigma=1/2~\gev^{-1}$, $1~\gev^{-1}$ and $2~\gev^{-1}$.
}
\end{figure}
\begin{figure}
\centering
{\hskip -1.5cm}
\subfigure[~$q_\sigma^M(b)$ with packet width $\sigma=1/2~\gev^{-1}$, $1~\gev^{-1}$, and $2~\gev^{-1}$.]{\label{fig:ipPDFMQ:M}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-Gm-Qb-Sigma-212-c3.eps}
\end{minipage}
}
{\hskip 1cm}
\subfigure[~$q_\sigma^Q(b)$ with packet width $\sigma=1/2~\gev^{-1}$, $1~\gev^{-1}$, and $2~\gev^{-1}$.]{\label{fig:ipPDFMQ:Q}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-Gq-Qb-Sigma-212-c3.eps}
\end{minipage}
}
\caption{\label{fig:ipPDFMQ}{\small The impact parameter dependent FFs $q^{M,Q}_\sigma(b)$ with $\sigma=1/2~\gev^{-1}$, $1~\gev^{-1}$, and $2~\gev^{-1}$.
}
\end{figure}
\begin{multicols}{2}
Figures~\ref{fig:ipPDF1CQCx} and \ref{fig:ipPDFMQx} show the numerical result of $q_\sigma(x,b)$ and $q^{C,M,Q,QC}_\sigma(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10,\;3/10 \; \text{and} \; 1/2$ respectively. When $x~\le~1/10$, $q^{C}_\sigma(x,b)$ has negative values as $b<0.4$~fm (see Fig.~\ref{fig:ipPDF1CQCx:C}), due to the oscillation of $q^{QC}_\sigma(x,b)$ (see Fig.~\ref{fig:ipPDF1CQCx:QC}).
In the small $x$ region (like $x~<~1/10$ in our case), it is believed that the contribution of the gluon GPDs becomes more important, which is beyond the scope of the present model. The symmetry around $x\sim1/2$ of the parton distributions, implied by the isospin symmetry, is not satisfied well due to this reason.
In addition, we found, from Fig.~\ref{fig:ipPDFMQx}, that the distributions approximately remain the same in $q^Q_\sigma(x,{b})$ when $1/10~\le~x~\le~3/10$.
\end{multicols}
\begin{figure}
\centering
{\hskip -1.5cm}
\subfigure[~$q_\sigma(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10$, $3/10$, and $1/2$.]{\label{fig:ipPDF1CQCx:1}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-H1-Qxb-Sigam-1-c3.eps}
\end{minipage}
}
{\hskip 1cm}
\subfigure[~$q_\sigma^C(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10$, $3/10$ and $1/2$.]{\label{fig:ipPDF1CQCx:C}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-HC-Qxb-Sigam-1-c3.eps}
\end{minipage}
}
\\
{\hskip -1.5cm}
\subfigure[~$q_\sigma^{QC}(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10$, $3/10$ and $1/2$.]{\label{fig:ipPDF1CQCx:QC}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-HQC-Qxb-Sigam-1-c3.eps}
\end{minipage}
}
\caption{\label{fig:ipPDF1CQCx}{\small The impact parameter dependent PDFs $q_\sigma(x,b)$ and FFs $q^{C,QC}_\sigma(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10$, $3/10$ and $1/2$.
}
\end{figure}
\begin{figure}
\centering
{\hskip -1.5cm}
\subfigure[~$q_\sigma^M(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10$, $3/10$ and $1/2$.]{\label{fig:ipPDFMQx:M}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-HM-Qxb-Sigam-1-c3.eps}
\end{minipage}
}
{\hskip 1cm}
\subfigure[~$q_\sigma^Q(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10$, $3/10$ and $1/2$.]{\label{fig:ipPDFMQx:Q}
\begin{minipage}[b]{0.4\textwidth}
\includegraphics[width=1\textwidth]{IPS-HQ-Qxb-Sigam-1-c3.eps}
\end{minipage}
}
\caption{\label{fig:ipPDFMQx}{\small The impact parameter dependent FFs $q^{M,Q,QC}_\sigma(x,b)$ with $\sigma=1~\gev^{-1}$ and $x=1/10$, $3/10$ and $1/2$.
}
\end{figure}
\begin{multicols}{2}
\section{Summary and conclusions}
\label{sec:summary}
In this work, analogous to the definition of the pion and nucleon impact parameter dependent PDFs, we introduce the $\rho$ meson impact parameter dependent PDFs ($q(x,{b})$ and $q({b})$) and impact parameter dependent FFs ($q^{C,M,Q}(x,{b})$ and $q^{C,M,Q}({b})$).
By employing the LCCQ, as we have done previously, we carried out the numerical calculation of those quantities for the first time.
We believe that $q^{C,M,Q}(x, {b})$ may be interpreted as the percentages of the contributions to the charge (normalized to 1), magnetic dipole $\mu_\rho$, and quadrupole moment $Q_\rho$, respectively, from a parton with a longitudinal momentum fraction $x$ and a transverse impact space position ${\bf b}_\perp$.
Considering the facts that the $\rho$ meson is an extended object and there exists a largest measured value of momentum transfer in realistic measurements, a Gaussian form wave packet is employed in our numerical calculation.
Our numerical results show that the wave packet approach plays a similar effect to the cutoff in the integral, which is due to the validity of the constituent quark model.
Our numerical results for impact parameter charge distributions also show that the width of the Gaussian wave packet should be larger than the Compton wavelength.
We expect that this approach is needed in a phenomenological model calculation in order to remove the possible negative values of the impact parameter charge distributions $q^C_{\sigma}(x,{b})$, which cannot be understood by the density interpretation.
\section*{Acknowledgements}
We would like to thank Stanley J. Brodsky, M. V. Polyakov, and Haiqing Zhou for their encouragement and constructive discussions.
This work is supported by the National Natural Science Foundation of China under Grant No. 11475192, by the fund provided to the Sino-German CRC 110 ``Symmetries and the Emergence of Structure in QCD" project by the NSFC under Grant No.11621131001, and the Key Research Program of Frontier Sciences, CAS, Grant No. Y7292610K1.
\end{multicols}
\vspace{2mm}
\centerline{\rule{80mm}{0.1pt}}
\vspace{2mm}
\begin{multicols}{2}
|
{
"timestamp": "2018-05-16T02:09:17",
"yymm": "1803",
"arxiv_id": "1803.02521",
"language": "en",
"url": "https://arxiv.org/abs/1803.02521"
}
|
\section*{Introduction}
The dual concepts of coverings and packings are well studied in
graph theory. Coverings of graphs with balls of radius one and
packings of vertices with pairwise distances at least
two are the well-known concepts of domination and independence respectively.
Typically we are interested in minimum (cost) coverings and maximum
packings. Natural questions to ask are for what graph do these dual
problems have equal (integer) values, and in the case they are not equal, can we
bound the difference between the two values? The second question is
the focus of this paper.
The particular covering problem we study is broadcast domination.
Let $G=(V,E)$ be a graph.
Define the \emph{ball of radius $r$ around $v$} by
$N_r(v) = \{ u : d(u,v) \leq r \}$. A \emph{dominating broadcast} of $G$
is a collection of balls $N_{r_1}(v_1), N_{r_2}(v_2), \dots, N_{r_t}(v_t)$
(each $r_i > 0$) such that $\bigcup_{i=1}^t N_{r_i}(v_i) = V$.
Alternatively, a dominating broadcast is a function $f: V \to \mathbb{N}$
such that for any vertex $u \in V$, there is a vertex $v \in V$
with $f(v)$ positive and $\mathrm{dist}(u,v) \leq f(v)$. (The ball around
$v$ with radius $f(v)$ belongs to the covering.)
The \emph{cost} of a dominating broadcast $f$ is
$\sum_{v \in V} f(v)$ and the minimum cost of a dominating broadcast in $G$,
its \emph{broadcast number}, is denoted by $\ensuremath{\gamma_b}(G)$.\footnote{
One may consider the cost to be any function of the powers
(for example the sum of the squares), see e.g.~\cite{HeggernesLokshtanov2006}.
We shall stick to the classical convention of linear cost.
}
When broadcast
domination is formulated as an integer linear program, its dual
problem is \emph{multipacking}~\cite{Brewster2013,Teshima2012}. A
multipacking in a graph $G$ is a subset $P$ of its vertices such that
for any positive integer $r$ and any vertex $v$ in $V$, the ball of
radius $r$ centered at $v$ contains at most $r$ vertices of $P$. The
maximum size of a multipacking of $G$, its \emph{multipacking number},
is denoted by $\ensuremath{\mathrm{mp}}(G)$.
Broadcast domination was introduced by Erwin~\cite{Erwin2001,
Erwin2004} in his doctoral thesis in 2001. Multipacking was then
defined in Teshima's Master's Thesis~\cite{Teshima2012} in 2012, see
also~\cite{Brewster2013}
(and~\cite{Brewster2017,hartnell_2014,Yang2015} for subsequent
studies). As we have already mentioned, this work fits into the
general study of coverings and packings, which has a rich history
in Graph Theory: Cornu\'ejols
wrote a monograph on the topic~\cite{Cornuejols2001}.
In early work, Meir and Moon~\cite{MeirMoon1975} studied
various coverings and packings in trees, providing several inequalities relating
the size of a minimum covering and a maximum packing. Giving such inequalities
connecting the parameters $\ensuremath{\gamma_b}$ and $\ensuremath{\mathrm{mp}}$ is the focus of our work.
Since broadcast domination and multipacking are dual problems,
we know that for any graph $G$,
\begin{equation*}
\ensuremath{\mathrm{mp}}(G) \leq \ensuremath{\gamma_b}(G).
\end{equation*}
This bound is tight, in particular for strongly chordal graphs,
see~\cite{Farber84,Lubiw87,Teshima2012}.
(In a recent companion work we prove equality
for grids~\cite{Beaudou2018}.)
A natural question comes to mind. How far
apart can these two parameters be? Hartnell and
Mynhardt~\cite{hartnell_2014} gave a family of graphs $(G_k)_{k \in
\mathbb{N}}$ for which the difference between both parameters is
$k$. In other words, the difference can be arbitrarily
large. Nonetheless, they proved that for any graph $G$ with
$\ensuremath{\mathrm{mp}}(G)\geq 2$,
\begin{equation*}
\ensuremath{\gamma_b}(G) \leq 3 \ensuremath{\mathrm{mp}}(G) - 2
\end{equation*}
and asked~\cite[Section~5]{hartnell_2014} whether the factor~$3$ can
be improved. Answering their question in the affirmative, our main
result is the following.
\begin{theorem}
\label{thm:bounding}
Let $G$ be a graph. Then,
\begin{equation*}
\ensuremath{\gamma_b}(G) \leq 2 \ensuremath{\mathrm{mp}}(G) + 3.
\end{equation*}
\end{theorem}
Moreover, we conjecture that the additive constant in the bound of
Theorem~\ref{thm:bounding} can be removed.
\begin{conjecture}
\label{conj:fac2}
For any graph $G$, $\ensuremath{\gamma_b}(G) \leq 2 \ensuremath{\mathrm{mp}}(G)$.
\end{conjecture}
In Section~\ref{sec:bound}, we prove Theorem~\ref{thm:bounding}. In
Section~\ref{sec:discussion}, we show that Conjecture~\ref{conj:fac2}
holds for all graphs with multipacking number at most~$4$. We conclude the paper with some discussions in Section~\ref{sec:remarks}.
\section{Proof of Theorem~\ref{thm:bounding}}
\label{sec:bound}
We want to bound the broadcast number of a graph by a function of its
multipacking number. We first state a key counting result which is
used throughout the remainder of this paper.
For any two relative integers $a$ and $b$ such that $a \leq b$,
$\llbracket a, b\rrbracket$ denotes the set $\mathbb{Z} \cap [a,b]$.
\begin{lemma}
\label{lem:path}
Let $G$ be a graph, $k$ be a positive integer and
$(u_0,\ldots,u_{3k})$ be an isometric path in $G$. Let \mbox{$P=\{u_{3i} | i \in \llbracket
0,k \rrbracket \}$} be the
set of every third vertex on this path. Then, for any positive integer $r$ and any ball
$B$ of radius $r$ in~$G$,
\begin{equation*}
|B \cap P| \leq \left\lceil \frac{2r+1}{3} \right\rceil.
\end{equation*}
\end{lemma}
\begin{proof}
Let $B$ be a ball of radius $r$ in $G$, then any two vertices in $B$
are at distance at most $2r$. Since the path $(u_0,\ldots,u_{3k})$
is isometric the intersection of the path and $B$ is included in a
subpath of length $2r$. This subpath contains at most $2r+1$
vertices and only one third of those vertices can be in $P$.
\end{proof}
Any positive integer $r$ is greater than or equal to $\left\lceil
\frac{2r+1}{3} \right\rceil$. Thus, Lemma~\ref{lem:path} ensures that
$P$ is a valid multipacking of size $k+1$. We have the following
(see also~\cite{dun_al_2006}):
\begin{proposition}
For any graph $G$,
\begin{equation*}
\ensuremath{\mathrm{mp}}(G) \geq \left\lceil\frac{\text{diam}(G)+1}{3}\right\rceil.
\end{equation*}
\end{proposition}
Building on this idea, we have the following result.
\begin{theorem}
\label{thm:main}
Given a graph $G$ and two positive integers $k$ and $k'$ such that
\mbox{$k' \leq k$}, if there are four vertices $x,y,u$ and $v$ in $G$ such
that
\begin{equation*} d_G(x,u) = d_G(x,v) = 3k \text{, } d_G(u,v) = 6k \text{ and }d_G(x,y) = 3k
+ 3k',
\end{equation*}
then
\begin{equation*}
\ensuremath{\mathrm{mp}}(G) \geq 2k + k'.
\end{equation*}
\end{theorem}
\begin{proof}
Let $(u_{-3k},\ldots,u_0,\ldots,u_{3k})$ be the vertices of an
isometric path from $u$ to $v$ going through $x$. Note that $u =
u_{-3k}$, $x = u_0$ and $v = u_{3k}$. We shall select every third
vertex of this isometric path and let $P_1$ be the set $\{u_{3i}
| i \in \llbracket -k, k \rrbracket\}$.
We thus have already selected $2k+1$ vertices. In order to complete
our goal, we need $k'-1$ additional vertices. Let $(x_0,\ldots,x_{3k + 3k'})$
be the vertices of an isometric path from $x$ to $y$. Note that $x =
x_0$ and $y = x_{3k+3k'}$. We shall select every third vertex on
this isometric path starting at $x_{3k+6}$. Formally, we let $P_2$
be the set $\{ x_{3k+3(i+2)} | i \in \llbracket 0,k'-2
\rrbracket\}$. Finally, we let $P$ be the union of $P_1$ and
$P_2$. An illustration of this is displayed in
Figure~\ref{fig:firstscheme}.
\begin{figure}[ht]
\scriptsize
\begin{center}
\begin{tikzpicture}
\node[vertex] (u) at (-3,0) {};
\node[vertex] (v) at (3,0) {};
\node[vertex] (x) at (0,0) {};
\node[vertex] (y) at (0,-5) {};
\node[vertex] (x3k6) at (0,-3.4) {};
\node[unselected] (x3k) at (0,-2.6) {};
\node[unselected] (x3k3) at (0,-3) {};
\node[vertex] (um3) at (-.4,0) {};
\node[vertex] (u3) at (.4,0) {};
\node[unselected] (x3) at (0,-.4) {};
\draw (u) -- (v);
\draw (x) -- (x3) -- (x3k) -- (x3k3) -- (y);
\node[above] at ($(x) + (0,.2)$) {$x = x_0 = u_0$};
\node[above] at ($(u) + (0,.2)$) {$u = u_{-3k}$};
\node[above] at ($(v) + (0,.2)$) {$v = u_{3k}$};
\node[below] at ($(y) + (0,-.2)$) {$y = x_{3k+3k'}$};
\node[below right] at ($(x3) + (0,0)$) {$x_3$};
\node[right] at (x3k) {$x_{3k}$};
\node[right] at ($(x3k6)+(.1,0)$) {$x_{3k+6}$};
\node[right] at (x3k3) {$x_{3k+3}$};
\node[left] at (-.1,-4.2) {$P_2$};
\node[below] at (-1.5,-0.2) {$P_1$};
\draw (-3.1,.1) rectangle (3.1,-.1);
\draw (-.1,-3.3) rectangle (.1,-5.1);
\end{tikzpicture}
\end{center}
\caption{Building of $P$.}
\label{fig:firstscheme}
\end{figure}
Since every vertex of $P_2$ is at distance at least $3k + 6$ from
$x$, while every vertex of $P_1$ is at distance at most $3k$ from
$x$, we infer that $P_1$ and $P_2$ are disjoint. Thus $|P| =
2k+k'$. We shall now prove that $P$ is a valid multipacking.
Let $r$ be an integer between 1 and $|P| - 1$, and let $B$ be a ball
of radius $r$ in $G$ (we do not care about the center of the
ball). If this ball $B$ intersects only $P_1$ or only $P_2$, then we
know by Lemma~\ref{lem:path} that it cannot contain more than $r$
vertices of $P$. We may then consider that the ball $B$ intersects
both $P_1$ and $P_2$. Let $l$ denote the greatest integer $i$ such
that $x_{3k+3(i+2)}$ is in $B$ and in $P_2$. Let us name this vertex
$z$. From this, we may say that
\begin{equation}
\label{eq:P2}
|B \cap P_2| \leq l + 1
\end{equation}
Before ending this preamble, we state an easy
inequality. For every integer $n$,
\begin{equation}
\label{eq:mod3}
\left\lceil\frac{n}{3}\right\rceil \leq \frac{n}{3} + \frac{2}{3}
\end{equation}
We now split the remainder of the proof into two cases.
\paragraph{Case 1: $3(l+2) \leq r$.}
In this case, we just use Lemma~\ref{lem:path} for
$P_1$. We have
\begin{equation*}
|B \cap P_1 | \leq \left\lceil \frac{2r + 1}{3} \right\rceil,
\end{equation*}
and by Inequality~\eqref{eq:mod3}, this quantity is bounded above by
$\frac{2r+1}{3} + \frac{2}{3}$. We obtain with
Inequality~\eqref{eq:P2},
\begin{flalign*}
&&|B \cap P| & \leq l+1 + \frac{2r+1}{3} + \frac{2}{3}&&\\
&& & \leq l+2 + \frac{2r}{3}&&\\
&& & \leq \frac{r}{3} + \frac{2r}{3} &\text{(by our case hypothesis)}&\\
&& & \leq r.&&
\end{flalign*}
Therefore, the ball $B$ contains at most $r$ vertices of $P$, as required.
\paragraph{Case 2: $3(l+2) > r$.}
Here we need some more insight. Recall that $l + 2 $
cannot exceed $k'$ and that $k' \leq k$. Thus
\begin{align*}
r & < 3(l+2) \\
& < 2k' + l +2\\
& < 2k + l + 2,
\end{align*}
and since $r$ is an integer, we get
\begin{equation}
\label{eq:nice}
r \leq 2k + l + 1.
\end{equation}
We also note that any vertex $u_i$ for $|i| \leq 3k + 3(l+2) -
(2r+1)$ is at distance at least $2r+1$ from $z$. By the triangle
inequality $d(z,u_i) \geq d(z,x)-d(u_i,x)$, where $d(z,x)=3k + 3(l+2)$,
and $d(u_i,x) = |i|$. Since the ball $B$ has
radius $r$, no such vertex can be in $B$. Since we
assumed that $B$ intersects $P_1$, not all the vertices of
the $uv$-path are excluded from $B$. This means that
\begin{equation}
\label{eq:nonzero}
3k > 3k + 3(l+2) - (2r+1).
\end{equation}
We partition the vertices of $P_1$ into three sets: $U_L, U_M, U_R$.
The vertex $u_i$ belongs to: $U_L$ if $i < -3k - 3(l+2) + 2(r+1)$;
$U_M$ if $|i| \leq 3k + 3(l+2) - (2r+1)$; and $U_R$ if
$i > 3k + 3(l+2) - (2r+1)$. See Figure~\ref{fig:case21}.
The distance from $u = u_{-3k}$ to the first vertex (smallest positive index)
in $U_R$ is then $6k + 3(l+2) - (2r+1) + 1$. We
compare this distance with $2r+1$.
\subparagraph{Case 2.1: $6k + 3(l+2) - (2r+1) + 1 \geq 2r+1$.}
We match $U_L$ with $U_R$ so that each pair is at distance at
least $2r+1$ (match $u_{-3k}$ with the first vertex in $U_R$ and so on,
as pictured in Figure~\ref{fig:case21}). Therefore the ball $B$ contains
at most one vertex of each matched pair. In other words, $B$ contains
at most $\lceil |U_R|/3 \rceil$ vertices from $U_L \cup U_R$, and so
\begin{equation*}
|B \cap P_1| \leq \left\lceil \frac{3k - (3k + 3(l+2) -
2r) + 1}{3} \right\rceil.
\end{equation*}
By using Inequality~\eqref{eq:P2} again,
\begin{align*}
|B \cap P| & \leq l+1 + \left\lceil \frac{2r+1}{3} \right\rceil - (l+2)\\
& \leq r.
\end{align*}
Therefore, the ball $B$ contains at most $r$ vertices of $P$, as required.
\begin{figure}[ht]
\begin{center}
\scriptsize
\subfigure[Case 2.1.]{
\label{fig:case21}
\begin{tikzpicture}[xscale=1.6]
\node[vertex] (u) at (-3,0) {};
\node[vertex] (v) at (3,0) {};
\node[vertex] (x) at (0,0) {};
\node[vertex] (um31) at (-2.4,0) {};
\node[vertex] (u31) at (2.4,0) {};
\draw (u) -- (v);
\node[above] at ($(x) + (0,.2)$) {$u_0$};
\node[left] at ($(u) + (-.2,0)$) {$u_{-3k}$};
\node[right] at ($(v) + (.2,0)$) {$u_{3k}$};
\node[below] at ($(-.5,-1.0) + (0,-.2)$) {$6k+3(l+2)-(2r+1)$};
\node[below] at ($(-.95,-.5) + (0,-.2)$) {$2r+1$};
\node[below] at ($(-2.65,0) + (0,-.2)$) {$U_L$};
\node[below] at ($(2.65,0) + (0,-.2)$) {$U_R$};
\node[below] at ($(0,0) + (0,-.2)$) {$U_M$};
\draw (-2.1,.1) rectangle (2.1,-.1);
\draw (-3.1,.1) rectangle (-2.2,-.1);
\draw (2.2,.1) rectangle (3.1,-.1);
\draw[dashed] (u) to[bend left] (u31);
\draw[dashed] (um31) to[bend left] (v);
\draw[arrows=<->] (-3,-.7) -- (1.1,-.7);
\draw[arrows=<->] (-3,-1.2) -- (2.1,-1.2);
\end{tikzpicture}
} \quad
\subfigure[Case 2.2.]{
\label{fig:case22}
\begin{tikzpicture}[xscale=1.6]
\node[vertex] (u) at (-3,0) {};
\node[vertex] (v) at (3,0) {};
\node[vertex] (x) at (0,0) {};
\node[vertex] (um31) at (-2.4,0) {};
\node[vertex] (u31) at (2.4,0) {};
\draw (u) -- (v);
\node[above] at ($(x) + (0,.2)$) {$u_0$};
\node[left] at ($(u) + (-.2,0)$) {$u_{-3k}$};
\node[right] at ($(v) + (.2,0)$) {$u_{3k}$};
\node[below] at ($(-1,-.5) + (0,-.2)$) {$6k+3(l+2)-(2r+1)$};
\node[below] at ($(-.4,-1) + (0,-.2)$) {$2r+1$};
\node[below] at ($(-2.65,0) + (0,-.2)$) {$U'_L$};
\node[below] at ($(2.65,0) + (0,-.2)$) {$U'_R$};
\node[below] at ($(-1.6,0) + (0,-.2)$) {$U''_L$};
\node[below] at ($(1.6,0) + (0,-.2)$) {$U''_R$};
\node[below] at ($(0,0) + (0,-.2)$) {$U_M$};
\draw (-1.1,.1) rectangle (1.1,-.1);
\draw (-2.1,.1) rectangle (-1.2,-.1);
\draw (1.2,.1) rectangle (2.1,-.1);
\draw (-3.1,.1) rectangle (-2.2,-.1);
\draw (2.2,.1) rectangle (3.1,-.1);
\draw[dashed] (u) to[bend left] (u31);
\draw[dashed] (um31) to[bend left] (v);
\draw[arrows=<->] (-3,-.7) -- (1.1,-.7);
\draw[arrows=<->] (-3,-1.2) -- (2.2,-1.2);
\end{tikzpicture}
}
\end{center}
\caption{Illustrations for Case 2.}
\end{figure}
\subparagraph{Case 2.2: $6k + 3(l+2) - (2r+1) + 1 < 2r+1$.}
We partition each of $U_L$ and $U_R$ as shown
in Figure~\ref{fig:case22}. The vertices that are
distance at least $2r+1$ from a vertex in $U_L \cup U_R$
are the sets $U'_L$ and $U'_R$,
and those that are close to all other vertices are
$U''_L$ and $U''_R$.
We can match pairs of vertices $U'_L \cup U'_R$.
This allows us to say that the extremities of $P_1$ will
contribute at most $\left\lceil \frac{6k - (2r+1) + 1}{3}
\right\rceil$ which equals $2k + \lceil\frac{-2r}{3}\rceil$. Using
again Inequality~\eqref{eq:mod3}, this is bounded above by $2k -
\frac{2r}{3} + \frac{2}{3}$.
For any integer $i$ between $6k + 3(l+2) - (2r+1) + 1$ and $2r$,
vertices $u_{-i}$ and $u_{i}$ belong to $U''_L$ and $U''_R$ respectively.
Such vertices may be in $B$. Since $P_1$ contains every third vertex on these two
subpaths, this amounts to at most
\begin{equation*}
2 \left\lceil\frac{2r - 6k - 3(l+2) + (2r+1)}{3}\right\rceil
\end{equation*}
such vertices. This quantity is equal to
\begin{equation*}
2\left\lceil \frac{4r+1}{3} \right\rceil -4k - 2(l+2),
\end{equation*}
which in turn, using Inequality~\eqref{eq:mod3} is bounded above by
\begin{equation*}
\frac{8r}{3} + 2 -4k -2(l+2).
\end{equation*}
By putting everything together, we derive that
\begin{flalign*}
&& |B \cap P| & \leq (l+1) + \left(2k - \frac{2r}{3} + \frac{2}{3}\right) + \left(\frac{8r}{3} +2 -4k - 2(l+2)\right) &&\\
&&& \leq 2r - 2k - l - \frac{1}{3}.&&
\end{flalign*}
But since $|B \cap P|$ is an integer, we may rewrite this last
inequality as
\begin{flalign*}&& |B \cap P| & \leq r + (r - 2k - l - 1) &&\\
&&& \leq r. &\text{(by
Inequality~\eqref{eq:nice})}&
\end{flalign*}
Thus, $|B \cap P|$ cannot exceed $r$ and the ball $B$ contains at most $r$ vertices of $P$, as required. This concludes the proof of Theorem~\ref{thm:main}.
\end{proof}
Theorem~\ref{thm:main} allows us to give a lower bound on the size of
a maximum multipacking in a graph in terms of its diameter and radius.
\begin{corollary}\label{coro:diam-rad}
For any graph $G$ of diameter $d$ and radius r,
\begin{equation*}
\ensuremath{\mathrm{mp}}(G) \geq \frac{d}{6} + \frac{r}{3} - \frac{3}{2}.
\end{equation*}
\end{corollary}
\begin{proof}
We just pick the integer $k$ such that $d$ can be expressed as
$6k + \alpha$ where $\alpha$ is in $\llbracket 0,5 \rrbracket$ and
the integer $k'$ such that $r$ can be expressed as $3k +
3k'+\beta$ where $\beta$ is in $\llbracket 0,2\rrbracket$.
We must have two vertices at distance $6k$ in $G$. On a shortest path of length $6k$, the middle
vertex has some vertex at distance $3k+3k'$. We can then apply
Theorem~\ref{thm:main}.
\begin{align*}
\ensuremath{\mathrm{mp}}(G) &\geq 2k + k'\\
& \geq \frac{1}{3}(d - \alpha) + \frac{1}{3} \left(r - \beta - \frac{1}{2}(d - \alpha)\right)\\
& \geq \frac{d}{6} + \frac{r}{3} - \frac{9}{6}.\qedhere
\end{align*}
\end{proof}
We can now finalize the proof of our main theorem.
\begin{proof}[Proof of Theorem~\ref{thm:bounding}]
Since the diameter of a graph is always greater than or equal to its
radius, we conclude from Corollary~\ref{coro:diam-rad} that
$$
\frac{\text{rad}(G)-3}{2} \leq \ensuremath{\mathrm{mp}}(G) \leq \ensuremath{\gamma_b}(G) \leq \text{rad}(G).
$$
Hence, for any graph $G$,
\begin{equation*}
\ensuremath{\gamma_b}(G) \leq 2 \ensuremath{\mathrm{mp}}(G) + 3,
\end{equation*}
proving Theorem~\ref{thm:bounding}.
\end{proof}
Note that in our proof, we chose the length of the long path to be a
multiple of~$6$ for the reading to be smooth. We think that the same
ideas implemented with more care would work for multiples of~$3$.
This might slightly improve the additive constant in our bound, but we
believe that it would not be enough to prove
Conjecture~\ref{conj:fac2} (while adding too much complexity to the
proof).
\section{Proving Conjecture~\ref{conj:fac2} when $\ensuremath{\mathrm{mp}}(G)\leq 4$}\label{sec:discussion}
The following collection of results shows that
Conjecture~\ref{conj:fac2} holds for graphs whose multipacking number
is at most~$4$.
\begin{lemma}\label{lemma-distances}
Let $G$ be a graph and $P$ a subset of vertices of $G$. If, for every
subset $U$ of at least two vertices of $P$, there exist two vertices
of $U$ that are at distance at least $2|U|-1$, then $P$ is a
multipacking of $G$.
\end{lemma}
\begin{proof}
We prove the contrapositive. Let $G$ be a graph and $P$ a subset of
its vertices which is not a multipacking. Then there is a ball $B$
of radius $r$ which contains $r+1$ vertices of $P$.
Let $U$ be the set $B \cap P$, then $U$ has size at least
$r+1$. Moreover, any two vertices in $U$ are at distance at most
$2r$ which is stricly smaller than $2|U|-1$.
\end{proof}
\begin{proposition}\label{prop:mp=3}
Let $G$ be a graph. If $\ensuremath{\mathrm{mp}}(G)=3$, then $\ensuremath{\gamma_b}(G)\leq 6$.
\end{proposition}
\begin{proof}
We prove the contrapositive again. Let $G$ be a graph with
broadcast number at least 7. Then, the eccentricity of any vertex is
at least 7 (otherwise we could cover the whole graph by broadcasting
with power 6 from a single vertex).
Let $x$ be any vertex of $G$. There must be a vertex $y$ at distance
7 from $x$. Let $u$ be any vertex at distance 3 from $x$ and on a
shortest path from $x$ to $y$. Then $u$ is at distance 4 from
$y$. But $u$ has also eccentricity at least 7. So there is a vertex
$v$ at distance 7 from $u$. By the triangle inequality, $v$ is at
distance at least 4 from $x$ and at least 3 from $y$. Therefore the
set $\{u,v,x,y\}$ satisfies the condition of
Lemma~\ref{lemma-distances} and the multipacking number of $G$ is at
least 4 (and so it is not equal to 3).
\end{proof}
The following proposition improves Theorem~\ref{thm:bounding} for
graphs $G$ with $\ensuremath{\mathrm{mp}}(G) \leq 6$ and shows that
Conjecture~\ref{conj:fac2} holds when $\ensuremath{\mathrm{mp}}(G) = 4$.
\begin{proposition}\label{prop:mp=4}
Let $G$ be a graph. If $\ensuremath{\mathrm{mp}}(G)\geq 4$, then $\ensuremath{\gamma_b}(G)\leq 3\ensuremath{\mathrm{mp}}(G)-4$.
\end{proposition}
\begin{proof}
For a contradiction, let $G$ be a counterexample, that is a graph
with multipacking number $p$ at least 4 while $\ensuremath{\gamma_b}(G)\geq
3p-3$. Then, the eccentricity of any vertex of $G$ is at least
$3p-3$ (otherwise we could broadcast at distance $3p-4$ from a
single vertex). Let $x$ be a vertex of $G$ and let $V_i$ denote the
set of vertices at distance exactly~$i$ of $x$. By our previous
remark, $V_{3p-3}$ is non-empty. Let $y$ be a vertex in $V_{3p-3}$
and consider a shortest path $P_{xy}$ from $x$ to $y$ in $G$. Let
$v_0=x$, and for $1\leq i\leq p-1$, let $v_i$ be the vertex on
$P_{xy}$ belonging to $V_{3i}$ (thus $v_{p-1}=y$).
Now, since $\ensuremath{\gamma_b}(G)\geq 3p-3$, there must be a vertex $u$ at distance
at least $3p-3$ of $v_{p-2}$ (otherwise we could broadcast from that
single vertex). Note that the triangle inequality ensures that the
distance between $u$ and $v_i$ is at least $3+3i$ for $i$ between
$0$ and $p-2$. The distance from $u$ to $v_{p-1}$ is at least $3p-6$
which is at least 6 since $p$ is at least 4. Consider the set
$P=\{u,v_0,\ldots, v_{p-1}\}$. We claim that $P$ is a multipacking
of $G$ of size $p+1$, which is a contradiction.
Let $B$ be a ball of radius $r$. Since $P_{xy}$ is an isometric
path, Lemma~\ref{lem:path} ensures us that $B$ contains at most
\begin{equation*}
\left\lceil \frac{2r+1}{3} \right\rceil
\end{equation*}
vertices from $P \cap P_{xy}$ which is smaller than $r$. When $B$
does not include $u$, the ball is satisfied. For balls that contain
vertex $u$, the maximum size of $P \cap B$ is
\begin{equation*}
\left\lceil \frac{2r+1}{3} \right\rceil + 1.
\end{equation*}
Whenever $r$ is 4 or more, this quantity does not exceed $r$. So
every ball with radius $4$ or more is satisfied. We still need to
check balls of radius 1,2, and 3 which contain $u$.
\begin{itemize}
\item Balls of radius 1 are easy to check since every vertex of $P_{xy}$
is at distance at least 3 from $u$.
\item For balls of radius 2, it is enough to check that there is only one
vertex at distance 4 or less from $u$ in $P \cap P_{xy}$.
\item For balls of radius 3, there is only one way to select $u$ and three
vertices in $P \cap P_{xy}$ within distance 6 from $u$. We should
take $v_0, v_1$ and $v_{p-1}$. But since $v_0$ and $v_{p-1}$ are at
distance $3p-3$ from each other, they cannot appear simultaneously in
a ball of radius 3 (since $p$ is at least 4, $3p-3$ is at least 9).
\end{itemize}
Therefore $P$ is a multipacking of size $p+1$, which is a
contradiction.
\end{proof}
\begin{corollary}
Let $G$ be a graph. If $\ensuremath{\mathrm{mp}}(G)\leq 4$, then $\ensuremath{\gamma_b}(G)\leq 2\ensuremath{\mathrm{mp}}(G)$.
\end{corollary}
\begin{proof}
When $\ensuremath{\mathrm{mp}}(G)\leq 2$, this is shown in~\cite{hartnell_2014}. The case $\ensuremath{\mathrm{mp}}(G)=3$ is implied by Proposition~\ref{prop:mp=3}, and the case $\ensuremath{\mathrm{mp}}(G)=4$ follows from Proposition~\ref{prop:mp=4}.
\end{proof}
\section{Concluding remarks}\label{sec:remarks}
We conclude the paper with some remarks.
\subsection{The optimality of Conjecture~\ref{conj:fac2}}
We know a
few examples of connected graphs $G$ which achieve the conjectured
bound, that is, $\ensuremath{\gamma_b}(G)=2\ensuremath{\mathrm{mp}}(G)$. For example, one can easily check
that $C_4$ and $C_5$ have multipacking number~$1$ and broadcast
number~$2$. In Figure~\ref{fig:twoFour}, we depict three examples
having multipacking number~$2$ and broadcast number~$4$. By making
disjoint unions of these graphs, we can build further extremal graphs
with arbitrary multipacking number. However, if we only consider
connected graphs, we do not even know an example with multipacking
number~$3$ and broadcast number~$6$. Hartnell and
Mynhardt~\cite{hartnell_2014} constructed an infinite family of
connected graphs $G$ with $\ensuremath{\gamma_b}(G)=\tfrac{4}{3}\ensuremath{\mathrm{mp}}(G)$, but we do not
know any construction with a higher ratio. Are there arbitrarily large
connected graphs that reach the bound of Conjecture~\ref{conj:fac2}?
\begin{figure}[!ht]
\begin{center}
\scalebox{1.0}{\begin{tikzpicture}[join=bevel,inner sep=0.5mm,scale=0.7]
\node[vertex](0) at (0:1) {};
\node[vertex](1) at (60:1) {};
\node[vertex](2) at (120:1) {};
\node[vertex](3) at (180:1) {};
\node[vertex](4) at (240:1) {};
\node[vertex](5) at (300:1) {};
\node[vertex](a) at (0:2) {};
\node[vertex](b) at (60:2) {};
\node[vertex](c) at (120:2) {};
\node[vertex](d) at (180:2) {};
\node[vertex](e) at (240:2) {};
\node[vertex](f) at (300:2) {};
\draw[-] (0)--(1)--(2)--(3)--(4)--(5)--(0)
(a)--(b)--(c)--(d)--(e)--(f)--(a)
(0)--(a) (2)--(c) (4)--(e);
\node at (270:2.5) {(a)};
\begin{scope}[xshift=6cm]
\node[vertex](0) at (0:1) {};
\node[vertex](1) at (60:1) {};
\node[vertex](2) at (120:1) {};
\node[vertex](3) at (180:1) {};
\node[vertex](4) at (240:1) {};
\node[vertex](5) at (300:1) {};
\node[vertex](a) at (0:2) {};
\node[vertex](b) at (60:2) {};
\node[vertex](c) at (120:2) {};
\node[vertex](d) at (180:2) {};
\node[vertex](e) at (240:2) {};
\node[vertex](f) at (300:2) {};
\node[vertex](x) at (0:1.5) {};
\node[vertex](y) at (180:1.5) {};
\draw[-] (0)--(1)--(2)--(3)--(4)--(5)--(0)
(a)--(b)--(c)--(d)--(e)--(f)--(a)
(1)--(c) (2)--(b) (4)--(f) (5)--(e) (0)--(x)--(a) (3)--(y)--(d);
\node at (270:2.5) {(b)};
\end{scope}
\begin{scope}[xshift=12cm,rotate=-22.5]
\foreach \i in {0,1,2,3,4,5,6,7}
{
\node[vertex](x\i) at (45*\i:1) {};
\node[vertex](y\i) at (45*\i:2) {};
}
\draw[-] (x0)--(x1)--(x2)--(x3)--(x4)--(x5)--(x6)--(x7)--(x0)
(y0)--(y1)--(y2)--(y3)--(y4)--(y5)--(y6)--(y7)--(y0)
(x0)--(y1) (y0)--(x1) (x2)--(y3) (y2)--(x3)
(x4)--(y5) (y4)--(x5) (x6)--(y7) (x7)--(y6);
\node at (292.6:2.5) {(c)};
\end{scope}
\end{tikzpicture}}
\end{center}
\caption{\label{fig:twoFour} Graphs with multipacking number $2$ and broadcast number $4$.
Graph (b) comes from L.~Teshima's Master Thesis~\cite{Teshima2012} and (c) was found by C.~R.~Dougherty (private communication).}
\end{figure}
\subsection{An approximation algorithm}
The computational complexity of broadcast domination has been
extensively studied, see for
example~\cite{Dabney2009,HeggernesLokshtanov2006} and references
of~\cite{Brewster2013,Teshima2012,Yang2015}. It is particularly
interesting to note that, unlike most other natural covering problems,
broadcast domination is solvable in polynomial (sextic)
time~\cite{HeggernesLokshtanov2006}. It is not known whether this is
also the case for multipacking, but a cubic-time algorithm exists for
strongly chordal graphs~\cite{Brewster2017,Yang2015}, as well as a
linear-time algorithm for
trees~\cite{Brewster2013,Brewster2017,Yang2015}. We note that our
proof of Theorem~\ref{thm:bounding}, being constructive, implies the
existence of a $(2+o(1))$-factor approximation algorithm for the
multipacking problem.
\begin{corollary}
There is a polynomial-time algorithm that, given a graph $G$,
constructs a multipacking of $G$ of size at least
$\frac{\ensuremath{\mathrm{mp}}(G)-3}{2}$.
\end{corollary}
\begin{proof}
To construct the multipacking, one first needs to compute the radius
$r$ and diameter $d$ of the graph $G$. Then, as described in the
proof of Corollary~\ref{coro:diam-rad}, we compute $\alpha$ and $k$,
and find the four vertices $x$, $y$, $u$, $v$ and the two isometric
paths $P_1$ and $P_2$ described in Theorem~\ref{thm:main}. Finally,
we proceed as in the proof of Theorem~\ref{thm:main}, that is, we
essentially select every third vertex of these two paths to obtain
the multipacking $P$. All distances and paths can be computed in
polynomial time using classic methods. By
Corollary~\ref{coro:diam-rad}, $P$ has size at least
$\frac{\text{rad}(G)-3}{2}$. Since $\ensuremath{\mathrm{mp}}(G)\leq \text{rad}(G)$, the
approximation factor follows.
\end{proof}
|
{
"timestamp": "2019-05-29T02:14:01",
"yymm": "1803",
"arxiv_id": "1803.02550",
"language": "en",
"url": "https://arxiv.org/abs/1803.02550"
}
|
\section{Introduction}
Traveltime tomography is an important class of inverse problems which appear in various applications such as global seismology~\cite{kennett1991traveltimes,inoue1990whole,bishop1985tomographic,clarke20013d,minkoff1996computationally}, ocean acoustic tomography~\cite{munk1979ocean, munk2009ocean,collins1994inverse,jensen2011computational}, ultrasound tomography~\cite{hormati2010robust,schomberg1978improved,jin2006thermoacoustic} in biomedical imaging and so on. It determines the internal velocity of the medium by measuring the wave traveltime between points on the boundary.
Theoretically, the traveltime tomography is very closely related to boundary rigidity and lens rigidity problems in differential geometry~\cite{croke1991rigidity,frigyik2008x,stefanov2004stability,stefanov2005boundary,stefanov2008boundary,guillarmou2017lens,kurylev2010rigidity,pestov2004characterization,stefanov2012geodesic,pestov2005two}. The boundary rigidity problem is to determine the metric of compact Riemannian manifold up to a diffeomorphism from first arrival time information, and the traveltime is the length of geodesic, which is also called ray in geometric optics context, connecting two points on boundary. The lens rigidity problem utilizes multiple arrival times information to determine the Riemann metric. The multiple arrival times are encoded in scattering relation which consists of incoming and outgoing points and directions as well as the traveltime.
For boundary rigidity, the uniqueness of reconstruction (up to an action of a diffeomorphism) is known for simple metrics (see~\cite{croke1991rigidity,mukhometov1981problem,michel1981rigidite,stefanov2005boundary,stefanov2005recent} and references therein) and many other cases~\cite{gromov1983filling,besson1995entropies,croke1990rigidity,lassas2003semiglobal,sharafutdinov1994integral}. A compact Riemannian manifold $(M,\partial M, g)$ is \emph{simple} if the boundary $\partial M$ is strictly convex with respect to its metric $g$ and there are no conjugate points along any geodesic. Moreover, for simple manifolds, the knowledge of scattering relation does not provide more information than boundary distance function. See~\cite{stefanov2007local,stefanov2008boundary,stefanov2008microlocal,guillarmou2017lens,stefanov2008integral} and references therein for recent progress on lens rigidity for non-simple manifolds. Numerically, there are many numerical algorithms motivated by the theoretical progress in boundary rigidity and lens rigidity problems,
see~\cite{chung2011adaptive,chung2008phase,chung2007new,leung2006adjoint,leung2007transmission,li2013fast,li2014level,glowinski2015penalization} for algorithmic developments.
When there are strong scattering effects or impenetrable obstacles inside the medium, then the geodesics could be broken. In~\cite{kurylev2010rigidity}, Kurylev, Lassas and Ulhmann established a uniqueness result for reconstructing Riemannian metric from the broken scattering relation. For reflective obstacles, we consider an incident ray jointed with its corresponding reflected ray as a broken geodesic by imposing reflection condition at the joint point. When there is only one strictly convex obstacle inside the manifold, then under certain conditions such as simple manifolds of dimension $\ge 2$ with real analytic metric~\cite{krishnan2009support} or manifolds of dimension $\ge 3$ with convex hypersurface foliation~\cite{uhlmann2016inverse}, the Riemannian metric outside the obstacle can be uniquely recovered from all nonbroken rays by Helgason support theorem~\cite{helgason2011radon}. However, all of the proofs of uniqueness are not constructive, and another difficulty in practice would be how to efficiently distinguish the broken and nonbroken scattering relation in measurements. In~\cite{chung2011adaptive}, Chung, Qian, Ulhmann and Zhao proposed a numerical reconstruction algorithm which is able to distinguish nonbroken and broken rays by measuring mismatch in scattering relation data during each iteration, if a broken ray is falsely predicted as nonbroken one, then there could be an $O(1)$ mismatch in the data.
In~\cite{chung2007new,chung2008phase,chung2011adaptive}, the authors have developed a phase-space approach for transmission and reflection traveltime tomography for acoustic and elastic media by using the Stefanov-Ulhmann identity formulated in~\cite{stefanov1998rigidity}. The method is advantageous over traditional methods in inverse kinematic problems~\cite{romanov1987inverse,bishop1985tomographic,sei1994gradient,sei1995convergent,washbourne2002crosswell}, because it uses multiple arrival times systematically and has the potential to handle anisotropic metrics as well, while these traditional methods can only recover isotropic metrics by utilizing first arrival times.
However, for the challenging case where both the medium and the scatterers are unknown, adaptive phase space method developed in \cite{chung2011adaptive} only uses non-broken ray to recover the medium outside the convex hull of the scatterers. In this work, we combine the adaptive phase space method, which is an optimization based iterative method, with a direct imaging method using selected broken-once ray data. This will give us the possibility to recover non-convex part of the boundary of the unknown scatterers. We also make several improvements that include preprocessing of the scattering relation data to classify data corresponding to non-broken rays, broken-once rays, and the others respectively and improvements in efficiency and robustness for the adaptive phase method.
Although Stefanov-Ulhmann identity can be used as feedback from computed metric to the exact metric, however, the identity itself is nonlinear. The linearization of Stefanov-Ulhmann identity will require the two metrics to be close enough, therefore the initial guess is critical for stable reconstruction. In our method, we first consider those geodesics with short traveltimes, by taking Taylor expansion for such geodesics, we can obtain Dirichlet and Neumann data of the metric on boundary, then we can extrapolate the initial guess of metric from these boundary data and the initial guess should be quite close to the exact solution near boundary. For the construction, our method also follows the layer stripping idea, but quite different from \cite{chung2011adaptive}, which selects the rays according to smallness in mismatch and could end up with some long rays which may deviate from layer stripping process. In our method, we introduce an auxiliary fidelity function to guide the layer stripping process. It can seen from our numerical experiments in Section~\ref{sec:num} that the iteration number can be reduced and the reconstruction process is very stable. For the reflection traveltime tomography, the method in~\cite{chung2011adaptive} will take more iterations and more time due to its trial and error strategy in distinguishing broken and nonbroken rays, while our method first preprocess the data and directly detect non-broken rays from the scattering relation by scanning discontinuities in derivatives. The non-broken rays can immediately be used to reconstruct the metric outside the convex hull of obstacles by Helgason support theorem~\cite{helgason2011radon}. Furthermore, when the obstacles are not large and not too concave or the metric does not vary too much near obstacle, then our method can be used to capture non-convex shape of the obstacles by tracking those rays which hit the obstacle in normal direction. Such rays will reverse their trace back to their initial location after reflection and provide a direct and stable way of locating points on boundaries of obstacles by tracing the ray to half of the traveltime (see the numerical experiments in Section~\ref{sec:num}).
The paper is organized as follows: we introduce the mathematical formulation for reflection traveltime tomography and broken geodesics in Section~\ref{sec:2}. Then we describe our numerical algorithm and the hybrid method in Section~\ref{sec:3}. Test results of our method for different setups are presented in Section~\ref{sec:num}.
\section{Mathematical formulation for reflection traveltime tomography}\label{sec:2}
\subsection{Broken scattering relation}
Let $(M,g)$ be a compact Riemann manifold with boundary dimension of $d$, and denote $S(M)$ its unit tangent bundle. The scattering relation or lens relation~\cite{kurylev2010rigidity} is
\begin{equation}
\begin{aligned}
\mathcal{L} = \{((\bx,\boldsymbol \xi), (\by, \boldsymbol \zeta), t)\in S(M)\times S(M)\times \mathbb{R}_{+}\cup \{0\}: \bx,\by \in \partial M, \\(\gamma_{\bx,\boldsymbol \xi}(t), \dot{\gamma}_{\bx,\boldsymbol \xi}(t)) = (\by,\boldsymbol \zeta)\text{ for some }t\ge 0\},
\end{aligned}
\end{equation}
where $\gamma_{\bx,\boldsymbol \xi}$ is the geodesic of $(M,g)$ starts from $\bx$ with direction $\boldsymbol \xi$ at $t=0$.
As defined in~\cite{kurylev2010rigidity}, a broken-once geodesic is a path $\alpha = \alpha_{\bx, \boldsymbol \xi, \bz,\boldsymbol \eta}(t)$ where $\bz = \gamma_{\bx, \boldsymbol \xi}(s) \in M$ for some $s\ge 0$, $\boldsymbol \eta\in S_{\bz}(M)$, and
\begin{equation}
\alpha_{\bx,\boldsymbol \xi, \bz, \boldsymbol \eta}(t) = \begin{cases}
\gamma_{\bx,\boldsymbol \xi}(t),& \text{ for } t < s,\\
\gamma_{\bz,\boldsymbol \eta}(t- s), & \text{ for } t\ge s.
\end{cases}
\end{equation}
The entering and exiting points of broken geodesics define the broken scattering relation~\cite{kurylev2010rigidity}
\begin{equation}
\begin{aligned}
\mathcal{R} = \{((\bx,\boldsymbol \xi), (\by, \boldsymbol \zeta), t)\in S(M)\times S(M)\times \mathbb{R}_{+}\cup\{0\}: (\bx,\boldsymbol \xi)\in \Gamma_{+}, (\by,\boldsymbol \zeta)\in \Gamma_{-}, \\t = l(\alpha_{\bx,\boldsymbol \xi, \bz,\boldsymbol \eta}),\text{ and } (\alpha_{\bx,\boldsymbol \xi, \bz,\boldsymbol \eta}(t), \dot{\alpha}_{\bx,\boldsymbol \xi, \bz,\boldsymbol \eta}(t)) = (\by,\boldsymbol \zeta) \text{ for some } (\bz,\boldsymbol \eta)\in S(M)\},
\end{aligned}
\end{equation}
where $ l(\alpha_{\bx,\boldsymbol \xi, \by,\boldsymbol \eta}) \in \mathbb{R}_{+}\cup\{\infty\}$ denotes the smallest $l>0$ that $\alpha_{\bx,\boldsymbol \xi, \by,\boldsymbol \eta} \in\partial M$. Let ${\boldsymbol \nu}$ be the interior unit normal vector of $\partial M$ and we define the following incoming and outgoing subbundles:
\begin{equation}
\begin{aligned}
&\Gamma_{+} = \{(\bx, \boldsymbol \xi)\in S(M): \bx\in \partial M, \langle \boldsymbol \xi, {\boldsymbol \nu}\rangle_g > 0\},\\
&\Gamma_{-} = \{(\bx, \boldsymbol \xi)\in S(M): \bx\in \partial M, \langle \boldsymbol \xi, {\boldsymbol \nu}\rangle_g < 0\}.
\end{aligned}
\end{equation}
Note that the scattering relation does not contain any information about the point $\bz$ or its corresponding direction $\boldsymbol \eta$ where the broken ray $\alpha_{\bx,\boldsymbol \xi, \bz,\boldsymbol \eta}$ changes its direction~\cite{kurylev2010rigidity}.
\subsection{Mathematical formulation}
Let $\Omega\subseteq \mathbb{R}^d$ be a compact domain and let $(g_{ij})$ be a Riemann metric on it. We define the Hamiltonian $H_g$ by
\begin{equation}
H_g(\bx,\boldsymbol \xi) = \frac{1}{2}\left(\sum_{1\le i, j\le d}g^{ij}(\bx)\xi_i\xi_j - 1\right),
\end{equation}
where $(g^{ij}) =(g_{ij})^{-1}$. Let $X^{(0)} = (\bx^{(0)}, \boldsymbol \xi^{(0)})$ be the initial condition belonging to the inflow set:
\begin{equation}
\label{eq:inflow}
\mathcal{S}^{-} = \{(\bx,\boldsymbol \xi) \,|\, \bx\in\partial\Omega, H_g(\bx,\boldsymbol \xi) = 0,\, \sum_{1\le i,j\le d}g^{ij}\xi_i\nu_j < 0\},
\end{equation}
where ${\boldsymbol \nu}(x)$ is the unit outward normal vector at $\bx\in\partial\Omega$ and $\nu_j(\bx)$ is the $j$-th component of ${\boldsymbol \nu}(\bx)$. The geodesic $X_g(s, X^{(0)})$ satisfies following Hamiltonian system:
\begin{equation}\label{eq:hamiltonian}
\frac{d\bx}{ds} = \partial_{\boldsymbol \xi}H_{g},\quad \frac{d\boldsymbol \xi}{ds} = -\partial_{\bx}H_{g},
\end{equation}
with initial condition $(\bx(0),\boldsymbol \xi(0)) = X^{(0)}$. Then the solution $X_g(s, X^{(0)}) = (\bx(s),\boldsymbol \xi(s))$ defines a geodesic (or a ray) in phase space, where $\bx(s)$ is the projection onto physical space $\Omega$, $\boldsymbol \xi(s)$ is the cotangent vector at $\bx(s)$, and $s$ denotes the traveltime.
In the following sections, we consider the case in which there are obstacles inside the domain $\Omega$. Rays are broken and reflected at the boundary when they hit the obstacles. Then the Hamiltonian system~\eqref{eq:hamiltonian} needs to impose a jump condition at the reflection point. The jump condition for the case when there is only one obstacle strictly lying in $\Omega$ has been derived in~\cite{chung2011adaptive}. Let $\Gamma$ be the interface where rays are reflected, for each reflected ray, there is a unique time $s^{\ast} > 0$ that $\bx(s^{\ast})\in\Gamma$ hits the interface at an incoming direction $\boldsymbol \xi_{\texttt{in}}:=\boldsymbol \xi(s^{\ast})$. The ray will be reflected to an outgoing direction $\boldsymbol \xi_{\texttt{out}}:=(I - 2\bn\bn^T)\boldsymbol \xi_{\texttt{in}}$, where $\bn$ is unit outward normal vector at $\bx(s^{\ast})$ of the obstacle. The Hamiltonian system for a broken-once ray will be
\begin{equation}
\begin{aligned}
&\frac{d\bx}{ds} = \partial_{\boldsymbol \xi}H_{g},\quad \frac{d\boldsymbol \xi}{ds} = -\partial_{\bx}H_{g},\quad 0<s\le s^{\ast}, \quad (\bx(0), \boldsymbol \xi(0)) = X^{(0)}, \\
&\frac{d\bx}{ds} = \partial_{\boldsymbol \xi}H_{g},\quad \frac{d\boldsymbol \xi}{ds} = -\partial_{\bx}H_{g},\quad s>s^{\ast},\quad (\bx(s^{\ast}),\boldsymbol \xi(s^{\ast})) = (\bx(s^{\ast}),\boldsymbol \xi_{\texttt{out}}).
\end{aligned}
\end{equation}
To derive Stefanov-Ulhmann identity, we need the following Jacobian matrix with respect to the initial condition
\begin{equation}
J_g(s, X^{(0)}) := \frac{\partial X_g}{\partial X^{(0)}}(s, X^{(0)}) = \begin{pmatrix}
\frac{\partial \bx}{\partial \bx(0)} & \frac{\partial\bx}{\partial \boldsymbol \xi(0)}\\
\frac{ \partial \boldsymbol \xi}{\partial \bx(0)} & \frac{\partial\boldsymbol \xi}{\partial\boldsymbol \xi(0)}
\end{pmatrix}.
\end{equation}
Let
\begin{equation}
M = \begin{pmatrix}
H_{\boldsymbol \xi,\bx} & H_{\boldsymbol \xi,\boldsymbol \xi}\\
-H_{\bx,\bx} & -H_{\bx,\boldsymbol \xi}
\end{pmatrix},
\end{equation}
then $J_g(s, X^{(0)})$ satisfies system
\begin{equation}\label{eq:jacobian}
\begin{aligned}
&\frac{dJ}{ds} = MJ,\quad J(0) = I \quad \text{ for } 0 < s < s^{\ast},\\
&\frac{dJ}{ds} = MJ,\quad J(s^{\ast}) = B\quad \text{ for } s>s^{\ast}.
\end{aligned}
\end{equation}
where
\begin{equation}\label{eq:jump-condition}
\begin{aligned}
B = \begin{pmatrix}
J(s^{\ast})_{11} & J(s^{\ast})_{12}\\
\partial_{\boldsymbol \xi}\boldsymbol \xi_{\texttt{out}} J(s^{\ast})_{21} + \partial_{\bx} \boldsymbol \xi_{\texttt{out}}J(s^{\ast})_{11} & \partial_{\boldsymbol \xi}\boldsymbol \xi_{\texttt{out}} J(s^{\ast})_{22} + \partial_{\bx}\boldsymbol \xi_{\texttt{out}}J(s^{\ast})_{12}
\end{pmatrix}.
\end{aligned}
\end{equation}
Similar to~\cite{chung2011adaptive}, we consider function
\begin{equation}
F(s) = X_{g_2}(t-s, X_{g_1}(s, X^{(0)})),
\end{equation}
where $t = t_{g_1}$ and
\begin{equation}
\int_0^{t} F'(s) ds = X_{g_1}(t,X^{(0)}) - X_{g_2}(t, X^{(0)}).
\end{equation}
The left-hand side is
\begin{equation}
\int_0^{t} F'(s) ds = \int_0^t J_{g_2}(t-s, X_{g_1}(s,X^{(0)})) \times (V_{g_1} - V_{g_2})(X_{g_1}(s, X^{(0)})) ds,
\end{equation}
where $V_g = \left(\partial_{\boldsymbol \xi}H_g, -\partial_{\bx}H_g \right)$. Linearize above integral's right-hand side at metric $g_2$, then we approximately have
\begin{equation}\label{eq:linearized-su}
\begin{aligned}
X_{g_1}(t,X^{(0)}) - &X_{g_2}(t, X^{(0)}) \simeq \\&\int_0^t J_{g_2}(t-s, X_{g_2}(s,X^{(0)})) \times \partial_g V(g_1-g_2, g_2,X_{g_2}(s, X^{(0)})) ds.
\end{aligned}
\end{equation}
For simplicity, we only consider isotropic metric in following sections. Let $X=(\bx,\boldsymbol \xi)$, then
\begin{equation}
g_{ij} = c^{-2}\delta_{ij},\quad \partial_g V(\lambda, g, X) = (2c\lambda \boldsymbol \xi, -(\lambda \nabla c + c\nabla \lambda )|\boldsymbol \xi|^2).
\end{equation}
\section{A hybrid method for reconstruction}\label{sec:3}
\subsection{Stabilized adaptive phase space method}\label{sec:stab-adaptive}
In~\cite{chung2011adaptive}, the authors have introduced the adaptive phase space method. The numerical method is an iterative algorithm based on linearized Stefanov-Uhlmann identity~\eqref{eq:linearized-su}, and the algorithm automatically follows layer-stripping process by choosing those rays with small mismatches on exiting phase measurements. However, using \emph{only} mismatch information could deviate from layer-stripping since small mismatches do not guarantee small errors on paths. Hence we propose a stabilized iterative method to overcome the ``false picking".
For simplicity, we first introduce the method for the medium without interior obstacle. The metric $g$ is discretized over an underlying Eulerian grid in the physical domain $\Omega$. The linearized Stefanov-Ulhmann identity~\eqref{eq:linearized-su} is discretized along the ray for each initial condition $X^{(0)}$ in phase space, the Jacobian matrix along the ray is computed by~\eqref{eq:jacobian}, the value of metric $g$ on non-grid points are linearly interpolated from neighborhood grid values. Therefore each integral equation along the ray $X_g(s,X^{(0)})$ represents a linear equation for neighboring grid values.
Let $X_k^{(0)}, k = 1,2,\dots, m$, be initial coordinates in phase space of those measurements $X_g(t_k, X_k^{(0)})$, where $t_k$ is the traveltime of corresponding ray starts from $X_k^{(0)}$. We then iteratively construct a sequence of metric $g_n$ as follows.
First, we need to construct a good initial guess $g_0$ for the linearized problem, which is important for convergence of the metric.
For short geodesic $X_g(s, X^{(0)}) = (\bx(s), \boldsymbol \xi(s)), 0\le s \ll 1$, with assumption on differentiability and local analyticity of $g$, we can easily deduce that
\begin{equation}
\begin{aligned}
\bx(s) &= \bx^{(0)} + s H_{g,\boldsymbol \xi} + \frac{s^2}{2} (H^{(0)}_{g,\boldsymbol \xi,\bx}H^{(0)}_{g,\boldsymbol \xi} - H^{(0)}_{g,\boldsymbol \xi,\boldsymbol \xi}H^{(0)}_{g,\bx}) + O(s^3),\\
\boldsymbol \xi(s) &= \boldsymbol \xi^{(0)} - sH^{(0)}_{g,\bx} - \frac{s^2}{2}(H^{(0)}_{g,\bx,\bx} H^{(0)}_{g,\boldsymbol \xi} - H^{(0)}_{g,\bx,\boldsymbol \xi}H^{(0)}_{g,\bx}) + O(s^3),
\end{aligned}
\end{equation}
where $H^{(0)}_g = H_g(X^{(0)})$. In case of isotropic metric, $g^{ij} = c^2(\bx)\delta_{ij}$, choose $X^{(0)} = (\bx^{(0)}, c^{-1}(\bx^{(0)})\bv)$ for a unit direction $\bv$ that is closest to tangential direction, and we have the following Talyor expansions
\begin{equation}
\begin{aligned}
\bx(s) &= \bx^{(0)} + s c \bv + \frac {s^2}{2}c (2\bv(\nabla c)^T \bv - \nabla c) + O(s^3),\\
\boldsymbol \xi(s) &= \boldsymbol \xi^{(0)} - s \frac{\nabla c}{c} - \frac{s^2}{2}((\nabla c\nabla c^T + c\nabla^2 c) c\bv - \nabla c \bv^T \frac{\nabla c}{c}) + O(s^3).
\end{aligned}
\end{equation}
By ignoring the $O(s^3)$ term of the first approximation above at a boundary point $\bx^{0}\in\partial\Omega$, we can solve $\nabla c(\bx_0)$ from the following linear equation
\begin{equation}
\bx(s) = \bx^{(0)} + s c \bv + \frac {s^2}{2}c (2\bv(\nabla c)^T \bv - \nabla c).
\end{equation}
After solving $\nabla c$ for all boundary points, we can construct a smooth initial guess $g_0^{ij} = c_0^{2}(\bx)\delta_{ij}$ by solving following minimization problem
\begin{equation}
c_0 = \argmin_{\tilde{c}} \frac{1}{2}\int_{\partial\Omega} \left(|\nabla c - \nabla \tilde{c}|^2 + |c - \tilde{c}|^2 \right)ds + \frac{\gamma}{2}\int_{\Omega} |\nabla \tilde{c}|^2 dx,
\end{equation}
where $\gamma$ is a small regularization parameter.
Define measurement mismatch $d^n_k$ of $k$-th ray as
\begin{equation}
d_k^n = X_g(t_k, X_k^{(0)}) - X_{g_n}(t_k, X_k^{(0)}).
\end{equation}
By linearized Stefanov-Ulhmann identity, we define linear operator $F_k^n$ along the $k$-th ray
\begin{equation}
\label{eq:SU}
F_k^n \tilde{g} = J_{g_n}(t, X_k^{(0)})\int_0^t J_{g_n}^{-1}(s, X_k^{(0)})\partial_g V_{g_n}(\tilde{g}, X_{g_n}(s, X_k^{(0)})) ds.
\end{equation}
The linear operators $F_k^n$ are matrices of size $4\times N$, where $N$ is the number of unknowns in physical domain $D$, $d^n_k$ are vectors of size $4\times 1$. For each $g^n$, we define block matrix $A^n$ of size $m \times N$, with each block of size $4\times 1$ that
\begin{equation}\label{eq:linear-eq}
A^n = \begin{bmatrix}
F^n_1 \\ F^n_2 \\ \vdots \\ F_m^n
\end{bmatrix}, \mbox{ and } A^n g \simeq b^n:=\begin{bmatrix}
d_1^n \\ d_2^n \\\vdots \\ d_m^n
\end{bmatrix}.
\end{equation}
From a geometrical viewpoint, if the matrix $F_k^n$'s $l$-th column is nonzero, then it means the $k$-th ray passes nearby the $l$-th grid, the total number of nonzero columns in $F_k^n$ roughly represents the length of this ray. During each iteration, suppose the $k$-th ray's mismatch is negligible, then it is highly possible that the matrix $F_k^n$ is close to the correct one. In other words, the velocity field at those grid points used in the computation of the linear operator $F_k^n$ in \eqref{eq:SU} along the $k$-th ray is likely to be accurate. In order to characterize this property, we define a fidelity function $0 \le p^n(\bx)\le 1$ for all grid points $ \bx\in \Omega$ at $n$-th iteration, which approximately represents the confidence of the current value of $g$ at $\bx$ and initially $p^0(\bx)\equiv 0$ over $\Omega$.
We also define a residual function for $F_k^n$ of matrix $A^n$,
\begin{equation}
\texttt{res}(F_k^n) = \texttt{nnz}(F_k^n) - \sum_{i:F_k^n(:,i)\neq 0} p(\bx_i), \quad
\end{equation}
where $\texttt{nnz}(F_k^n)$ is the number of nonzero columns in $F_k^n$ and $\bx_i, i=1, 2, \ldots, N$ are the grid points.
Geometrically, $\texttt{res}(F_k^n)$ approximately reveals effective length of the part of the $k$-th ray along which the velocity field is unknown. Or simply an indicator of the accuracy of the linearized Stefanov-Ulhmann identity \eqref{eq:linearized-su} along the $k$-th ray. At $n$-th iteration, we use $S_n$ to represent a subset of row blocks, which corresponds to all indices $k$ that $\texttt{res}(F_k^n)$ are under some effective length threshold $r_{\min}$,
\begin{equation}
S_n = \{k\in \{1, 2, \dots, m\} |\, \texttt{res}(F_k^n) \le r_{\min}\}.
\end{equation}
We notice that the rays belonging to $S_n$ will provide more stable reconstruction than other rays do.
At $n$-th iteration, we construct a perturbation $\tilde{g}$ by minimizing functional
\begin{equation}\label{eq:regularize}
\mathcal{H}(\tilde{g}) = \frac{1}{2} \|A^n(S_n, :) \tilde{g} - b(S_n)\|^2 + \frac{\beta}{2}\|\nabla \tilde{g}\|^2.
\end{equation}
where $\beta$ is a regularization parameter. We then update $g^{n+1}$ by
\begin{equation}
g^{n+1} = g^n + \tilde{g},
\end{equation}
until the relative error of measurement is below certain tolerance level. The fidelity function $p^{n+1}$ is updated by the following steps.
\begin{enumerate}
\item For each $k\in S_n$, we calculate the residual error along the $k$-th ray $$e_k = \|A^n(k, :)\tilde{g} - b(k)\|.$$
\item If $e_k$ is under some threshold $\tau$, we update the fidelity for grid points involved in the linear operator $F^n_k$ by
\begin{equation}
p^{n+1}(\bx_i) = \max(p^n(\bx_i), 1 - \alpha e_k),\quad\text{ when } A^n(k,i)\neq \mathbf{0}_{4\times 1}.
\end{equation}
where the threshold $\tau$ is chosen to be small and $\alpha$ is a parameter to control the fidelity decay. Here we have made assumption that if an effectively ``short'' ray has smaller mismatch with measurement, then metric $g$'s value along the ray has higher fidelity.
\end{enumerate}
This adaptive method follows a layer-stripping process by using shorter rays near-boundary first and stripping them by updating fidelity function, which corresponds to the foliation process introduced in~\cite{uhlmann2016inverse}. The layers are implicitly charaterized by $p^n$ at each iteration, and only depend on the measurements/data. We can see from the numerical experiments in Section~\ref{sec:num} that this method not only improves stability and efficiency over the previous adaptive phase space method introduced in \cite{chung2011adaptive}, but also achieves a better accuracy on reconstructed metric.
\subsection{Interior reflection detection}\label{sec:ref-detect}
When there is a reflective obstacle lying inside the medium, and neither the obstacle nor the metric is known, then the reconstruction of both can be very challenging. In~\cite{chung2011adaptive}, the authors used the adaptive phase space method to distinguish most of the unbroken rays. These rays can help to recover the convex hull (under the same metric) of the unknown obstacle and the medium outside the convex hull. However, for concave part of the boundary of the obstacle, one also needs to use those broken rays. In the following we propose a direct imaging method to find points on the boundary of the obstacle using broken-once rays and the metric $g$ recovered from the adaptive phase method based on non-broken rays as described in the previous section.
Let's consider a simple case in this scenario. Suppose the physical domain $\Omega$ is \emph{convex} and interior reflector $D\subseteq \Omega$ have $C^2$ boundaries, which are homotopic equivalent to $S^{d}$.
We call a point $\by\in\partial D$ a \emph{radiative} boundary point if a ray starts from $\by$ with direction along the outward normal $\bn(\by)$ of $\partial D$ only intersects with $\partial\Omega$. Denote the set of all radiative boundary points by $R$, assume $R$ is nonempty, then consider a continuous mapping $L_g:R\to \partial\Omega$ that
\begin{equation}
L_g(\by)= P_{\bx}X_g(t, (\by, c^{-1}\bn(\by))),
\end{equation}
where $t$ is the traveltime such that $X_g(t, (\by, c^{-1}\bn(\by))) \in \Gamma_{-}$ and $P_{\bx}$ is projection mapping from phase space to physical space. Let the range of $L_g$ be $T_g$, then for each $\bx^{(0)}\in T_g$, there exists a direction $\boldsymbol \xi^{(0)}$ and travel time $s^{\ast}(\bx^{(0)},\boldsymbol \xi^{(0)}) $ such that $ \bx(s^{\ast})\in \partial D, \boldsymbol \xi_{\texttt{out}} = -\boldsymbol \xi(s^{\ast})$, which means the ray hits the obstacle in the inward normal direction. Such reflected rays will go back to its initial physical location and have exact opposite directions. Though numerically such rays are rare, but they are very stable and can be used to parametrize the obstacle implicitly, because the reflection occurs exactly at halfway of a broken-once ray, see Figure~\ref{fig:orthorays}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.45]{orthorays.png}
\caption{The broken rays hit the obstacle in (nearly) normal direction.}
\label{fig:orthorays}
\end{figure}
We remark that once such rays exist, then an obstacle will be detected because otherwise these non-broken rays (or geodesics) are not unique along the their initial phases.
Numerically, let $X^{(0)}_k, k=1,2,\dots,m$ be initial coordinates in phase space as previous section, $t_k$ is the traveltime of corresponding ray $(\bx_k(s), \boldsymbol \xi_k(s))$ starts from $X^{(0)}_k=(\bx_k^{(0)}, \boldsymbol \xi_k^{(0)})$. We use $T$ to represent a subset of index $\{1,2,\dots, m\}$ of rays satisfying
\begin{equation}
\label{eq:broken}
T = \Big\{k\,\Big|\,\|\bx_k^{(0)} - \bx_k(t_k)\|_{L^2} + \|\boldsymbol \xi_k^{(0)} + \boldsymbol \xi_k(t_k)\|_{L^2} < \epsilon\Big\}.
\end{equation}
from the given data, i.e., the scattering relations for certain numerical tolerance $\epsilon>0$. These rays are considered to be hitting the reflector in the normal direction. And the reflection happens at middle point $s^{\ast}_k = t_k/2$. Comparing to other broken rays, these are more predictable and easier to use to locate the boundary due to the knowledge of reflection time and angle.
\subsection{Non-broken rays detection}\label{sec:non-broken-detect}
The adaptive phase space method~\cite{chung2011adaptive} has shown its potential to distinguish most of the non-broken rays during the layer-stripping process, if a broken ray or a non-broken ray is falsely predicted, then it is likely to produce an $O(1)$ mismatch in scattering relation due to the jump condition~\eqref{eq:jump-condition}. But this method would have difficulties in critical cases, e.g. distinguishing near-tangent broken and non-broken rays.
For each physical location $\bx^{(0)}$ on the boundary of the domain, $\partial\Omega$, we define the set of directions for broken-rays (including tangential rays) by
\begin{equation}
\mathcal{B}(\bx^{(0)}) = \{ \boldsymbol \xi~|~ (\bx^{(0)},\boldsymbol \xi)\in \mathcal{S}^-, X_g(t, \bx^{(0)}, \boldsymbol \xi) \cap \partial D \neq \emptyset \text{ for some } t > 0\}.
\end{equation}
Let $\mathcal{C}(\bx^{(0)})$ be the smallest simply connected set containing $\mathcal{B}(\bx^{(0)})$, then we only have to find out the directions $\boldsymbol \xi\in \partial \mathcal{C}(\bx^{(0)})$, since the rays with directions outside $\mathcal{C}(\bx^{(0)})$ are all non-broken. In order to be able to detect the set $\partial \mathcal{C}^{(0)}$ with both medium and object unknown, we need to make a few further assumptions on the metric and obstacle. For simplicity, we focus our study on the case where the medium is isotropic, i.e., $H(\bx, \boldsymbol \xi)=\frac{1}{2}(c^2(\bx)\|\boldsymbol \xi\|^2-1)$, although our results can be extended to the more general anisotropic case.
\begin{enumerate}
\item $c(\bx)$ is a $C^3$ function and non-trapping.\label{as:1}
\item Both boundaries $\partial \Omega$ and $\partial D$ are $C^2$. Without loss of generality, we assume the boundaries $\partial\Omega$ and $\partial D$ are represented by $G(\bx) = 0$ and $F(\bx) = 0$ respectively, where $G$ and $F$ are $C^2$ functions. \label{as:2}
\item If a broken ray $X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)}) = (\bx(s), \boldsymbol \xi(s))$ is tangential to $\partial D$ and its exiting phase is $(\by, \boldsymbol \zeta) = (\bx(t), \boldsymbol \xi(t))$, where $t$ is traveltime, then there is a constant $\delta >0 $ that $|\boldsymbol \zeta \cdot \bn(\by)| > \delta$, where $\bn(\by)$ is the outward unit normal vector of $\partial\Omega$ at $\by$. Physically speaking, this means for those the rays that are tangential to the obstacle's boundary, they exit in non-tangential directions. \label{as:3}
\item If a broken ray $X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)}) = (\bx(s), \boldsymbol \xi(s))$ is tangential to $\partial D$ at $\bp = \bx(t_{\bp})\in\partial D$,
\begin{equation}\label{eq:as4}
\bn_{\bp}\cdot \frac{\partial \boldsymbol \xi}{\partial s}\Big|_{t_{\bp}} +\left( \frac{\partial \bx}{\partial s}\nabla\frac{\nabla F}{\|\nabla F\| }\boldsymbol \xi\right)\Big|_{t_{\bp}}\neq 0,
\end{equation}
where $\bn_{\bp}$ is outward unit normal at $\bp$. This term stands the interaction between the geometry of the obstacle and the medium. When this term is nonzero, reflection at the obstacle boundary would impose a significant change on an impinging ray direction compared to the change of direction due to the medium variation. In the isotropic case, this requirement is equivalent to
\begin{equation}
-\bn_{\bp}\cdot \frac{\nabla c(\bp)}{c(\bp)} + \mathbf k} \newcommand{\bl}{\mathbf l\cdot \frac{\textrm{Hess}(F)}{\|\nabla F\|}\Big|_{\bp}\mathbf k} \newcommand{\bl}{\mathbf l \neq 0,
\end{equation}
where $\mathbf k} \newcommand{\bl}{\mathbf l = \frac{\boldsymbol \xi}{\|\boldsymbol \xi\|}$. The second term is bounded below by the minimal principal curvature $\lambda_{\min}$, therefore, if we assume that
\begin{equation}
\frac{\|\nabla c(\bp)\|}{c(\bp)} < \lambda_{\min},
\end{equation}
then~\eqref{eq:as4} is satisfied. Simply speaking, if the obstacle boundary's minimal principal curvature is not small, or the speed $c$ varies little, then reflection's is strong enough to be observed.
In addition, we require $\bn_{\bp}$ not parallel to $\boldsymbol \xi^{(0)}$ and $\bn_{\bp}\cdot \frac{\partial\bx}{\partial\boldsymbol \xi^{(0)}}\Big|_{t_{\bp}} \neq{\mathbf 0}$. This condition guarantees one to find a ray starting at $\bx^{(0)}$ with a direction in a small neighborhood of $\boldsymbol \xi^{(0)}$ whose reflection by the obstacle can be observed in the scattering relation (see Lemma \ref{lm:4} and \ref{lm:6}).
\label{as:4}
\end{enumerate}
By the differentiability theorem of initial value problems, we can easily show the following lemma.
\begin{lemma}\label{LM:CONT}
If metric $g_{ij}=c^{-2}\delta_{ij}$ satisfies that $c(\bx)$ is $C^{k}, k\ge 3$ in $\Omega$, then the Hamiltonian system's solution $X_g$ is $C^{k-1}$ before and after hitting the obstacle.
\end{lemma}
\begin{lemma}
For any $\boldsymbol \xi\in \partial \mathcal{C}(\bx^{(0)})$, $X_g(\cdot, \bx^{(0)}, \boldsymbol \xi)$ is tangential to $\partial D$.
\end{lemma}
\begin{proof}
We prove by contradiction. If $X_g(s, \bx^{(0)}, \boldsymbol \xi)$ impinges on the obstacle $D$ with non-tangential direction, then there is an open neighborhood $U$ at $\boldsymbol \xi$ such that $\forall \boldsymbol \xi_b\in U$, $X_g(s, \bx^{(0)}, \boldsymbol \xi_b)$ still intersects with $D$, which contradicts with the assumption of $\mathcal{C}(\bx^{(0)})$ contains $\mathcal{B}(\bx^{(0)})$. On the other hand, if $X_g(s, \bx^{(0)}, \boldsymbol \xi)$ is non-broken, then there is also an open neighborhood $V$ at $\boldsymbol \xi$ such that $\forall \boldsymbol \xi_n\in V$, $X_g(s, \bx^{(0)}, \boldsymbol \xi_n)$ are non-broken, which contradicts the smallness assumption of $\mathcal{C}(\bx^{(0)})$.
\end{proof}
\begin{lemma}\label{lm:3}
If a broken ray $X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)})= (\bx(s), \boldsymbol \xi(s))$ is tangential to $\partial D$ and $\boldsymbol \xi^{(0)}\in \partial\mathcal{C}(\bx^{(0)})$. $U$ is an open neighborhood of $(\bx^{(0)}, \boldsymbol \xi^{(0)})$, if the phase $(\bx_n^{(0)}, \boldsymbol \xi_n^{(0)})\in U$ and the ray $X_g(s, \bx_n^{(0)}, \boldsymbol \xi_n^{(0)})= (\bx_n(s), \boldsymbol \xi_n(s))$ is non-broken with $\|(\bx^{(0)}, \boldsymbol \xi^{(0)}) - (\bx_n^{(0)}, \boldsymbol \xi_n^{(0)})\|\ll 1$, denote the traveltimes of $X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)})$ and $X_g(s, \bx_n^{(0)}, \boldsymbol \xi_n^{(0)})$ are $t$ and $t_n$ respectively, then
\begin{equation}
t_n - t = -\bu \cdot (\bx^{(0)}_n - \bx^{(0)}) -\bv\cdot (\boldsymbol \xi^{(0)}_n - \boldsymbol \xi^{(0)})+ o(\|\bx^{(0)}_n - \bx^{(0)}\|) + o(\|\boldsymbol \xi^{(0)}_n - \boldsymbol \xi^{(0)}\|).
\end{equation}
where $\bu$ and $\bv$ are defined as
\begin{equation}
\begin{aligned}
&\bu = \left(\nabla G\cdot \pdr{\bx}{s}\Big|_{t}\right)^{-1}\left(\nabla G\cdot \pdr{\bx}{\bx{(0)}}\Big|_{t} \right),\\
&\bv = \left(\nabla G\cdot \pdr{\bx}{s}\Big|_{t}\right)^{-1}\left(\nabla G\cdot \pdr{\bx}{\boldsymbol \xi{(0)}}\Big|_{t} \right).
\end{aligned}
\end{equation}
\end{lemma}
\begin{proof}
The existence of such a non-broken ray in $U$ is directly from previous lemma. At the exiting locations, $G(\bx(t)) = 0$ and $G(\bx_n(t_n)) = 0$. Since $G$ is twice differentiable, we have the following expansion,
\begin{equation}\nonumber
\begin{aligned}
G(\bx_n(t_n)) =&~ G(\bx(t)) + \nabla G\cdot \pdr{\bx}{s}\Big|_{t }(t_n- t) \\&+ \nabla G\cdot \pdr{\bx}{\bx^{(0)}}\Big|_{t } (\bx^{(0)}_n - \bx^{(0)}) + \nabla G\cdot \pdr{\bx}{\boldsymbol \xi^{(0)}}\Big|_{t} (\boldsymbol \xi^{(0)}_n - \boldsymbol \xi^{(0)}) \\
&+ O((t_n - t)^2) + o(\|\bx^{(0)}_n- \bx^{(0)}\|) + o(\|\boldsymbol \xi^{(0)}_n - \boldsymbol \xi^{(0)}\|).
\end{aligned}
\end{equation}
According to the assumption~\ref{as:3}, there exists a constant $\eta > 0$ such that $ \Big|\nabla G\cdot \pdr{\bx}{s}\Big|_{t}\Big| > \eta$. Therefore we can find vectors $\bu$ and $\bv$ that
\begin{equation}
t_n - t = -\bu \cdot (\bx^{(0)}_n - \bx^{(0)}) -\bv\cdot (\boldsymbol \xi^{(0)}_n - \boldsymbol \xi^{(0)})+ o(\bx^{(0)}_n - \bx^{(0)}) + o(\boldsymbol \xi^{(0)}_n - \boldsymbol \xi^{(0)}),
\end{equation}
where
\begin{equation}
\begin{aligned}
&\bu = \left(\nabla G\cdot \pdr{\bx}{s}\Big|_{t}\right)^{-1}\left(\nabla G\cdot \pdr{\bx}{\bx^{(0)}}\Big|_{t} \right),\\
&\bv = \left(\nabla G\cdot \pdr{\bx}{s}\Big|_{t}\right)^{-1}\left(\nabla G\cdot \pdr{\bx}{\boldsymbol \xi^{(0)}}\Big|_{t} \right).
\end{aligned}
\end{equation}
\end{proof}
\begin{lemma}\label{lm:4}
Suppose $\boldsymbol \xi\in \bbS^{d-1}$ is a fixed vector and $\bv\neq {\mathbf 0}$ is not parallel to $\boldsymbol \xi$. For any open set $B\subset \bbS^{d-1}$
$\sup_{\boldsymbol \zeta\in B}\|\boldsymbol \xi - \boldsymbol \zeta\| <\varepsilon \ll 1$,
there exists $\hat{\boldsymbol \zeta}\in B$ such that
\begin{equation}
|\bv\cdot (\boldsymbol \xi - \hat{\boldsymbol \zeta})| = O(\|\boldsymbol \xi - \hat{\boldsymbol \zeta}\|).
\end{equation}
\end{lemma}
\begin{proof}
Without loss of generality, we assume $\|\bv\| = 1$. Let $r=\boldsymbol \xi\cdot\bv, |r|<1$. Since $B$ is an open set in $\bbS^{d-1}$, there exists $\hat{\boldsymbol \zeta} \in B$, $\hat{\boldsymbol \zeta} \neq \boldsymbol \xi$ such that $|(\boldsymbol \xi - \hat{\boldsymbol \zeta})\cdot (r\boldsymbol \xi - \bv)| \ge \hat{\gamma}t\sqrt{1-r^2}$ for some $\hat{\gamma}>0$, where $t=\|\boldsymbol \xi-\hat{\boldsymbol \zeta}\|<\epsilon$. Since $|(\boldsymbol \xi - \hat{\boldsymbol \zeta})\cdot \boldsymbol \xi|=O(t^2)$, we have $|\bv\cdot (\boldsymbol \xi - \hat{\boldsymbol \zeta})| = O(\|\boldsymbol \xi - \hat{\boldsymbol \zeta}\|)$.
\end{proof}
\begin{lemma}\label{lm:5}
For any initial phase $X^{(0)} = (\bx^{(0)}, \boldsymbol \xi^{(0)})$, suppose the ray $$X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)}) = (\bx(s ), \boldsymbol \xi(s)),\quad \bx(0)=\bx^{(0)},\quad \boldsymbol \xi(0)=\boldsymbol \xi^{(0)}$$ is non-broken on $s\in (0, T)$. If vector $\bn$ satisfies
\begin{equation}
\frac{\partial \bx(t)}{\partial \boldsymbol \xi{(0)}} \bn = {\mathbf 0},\quad \frac{\partial \boldsymbol \xi(t)}{\partial \boldsymbol \xi{(0)}} \bn = {\mathbf 0}, \quad \forall t\in(0,T)
\end{equation}
then $\bn = {\mathbf 0}$.
\end{lemma}
\begin{proof}
Since the determinant of the Jacobian matrix $J = \begin{pmatrix}
\frac{\partial \bx}{\partial \bx{(0)}} & \frac{\partial \bx}{\partial \boldsymbol \xi{(0)}}\\
\frac{\partial \boldsymbol \xi}{\partial \bx{(0)}} & \frac{\partial \boldsymbol \xi}{\partial \boldsymbol \xi{(0)}}
\end{pmatrix}$ is 1, if the vector $\bn\neq {\mathbf 0}$, then
\begin{equation}
J\begin{pmatrix}
{\mathbf 0}\\\bn
\end{pmatrix} = {\mathbf 0}
\end{equation}
which contradicts to the non-degeneracy of $J$.
\end{proof}
\begin{lemma}\label{lm:6}
From a fixed point $\bx^{(0)}\in\partial\Omega$, we denote the ray with initial direction $\boldsymbol \zeta$ as $X_g(s, \bx^{(0)}, \boldsymbol \zeta) = (\bx(s, \boldsymbol \zeta), \boldsymbol \xi(s, \boldsymbol \zeta))$ and let $T(\boldsymbol \zeta)$ be the traveltime.
Suppose $\boldsymbol \xi^{(0)}\in \partial \mathcal{C}(\bx^{(0)})$,
then either $\partial_{\boldsymbol \zeta} T$ or $\partial_{\boldsymbol \zeta}\bx$ or $\partial_{\boldsymbol \zeta}\boldsymbol \xi$ is discontinuous at $\boldsymbol \xi^{(0)}$.
\end{lemma}
\begin{proof}
Consider a small open neighborhood $V\subset \mathcal{S}^-$ of $\boldsymbol \xi^{(0)}$, then the following sets
\begin{equation}
\begin{aligned}
&N = \{ \boldsymbol \zeta \in V~|~ X_g(\cdot, \bx^{(0)},\boldsymbol \zeta) \text{ is nonbroken}\},\\
&B = \{ \boldsymbol \zeta\in V ~|~ X_g(\cdot, \bx^{(0)}, \boldsymbol \zeta) \text{ is broken}\}.
\end{aligned}
\end{equation}
are both nonempty sets.
Then by Lemma~\ref{lm:3}, for any $\epsilon > 0$, we can select $\boldsymbol \xi^{(0)}_n\in N$ that $\| \boldsymbol \xi^{(0)}_n - \boldsymbol \xi^{(0)}\| < \epsilon$ and
\begin{equation}
\begin{aligned}
|T(\boldsymbol \xi^{(0)}) - T(\boldsymbol \xi^{(0)}_n)| = O(\|\boldsymbol \xi^{(0)} - \boldsymbol \xi^{(0)}_n\|).
\end{aligned}
\end{equation}
and the differences between the exiting locations and phases are also of order $O(\|\boldsymbol \xi^{(0)} - \boldsymbol \xi^{(0)}_n\|)$.
\begin{figure}[!htb]
\begin{center}
\includegraphics[scale=0.20]{example.png}
\caption{The illustration of the tangential ray and broken ray, both rays start from the same physical locations but different directions. The tangential ray intersects with the obstacle at $\bp$ and the other broken ray intersects with the obstacle at $\mathbf q} \newcommand{\br}{\mathbf r$.}
\label{fig:example}
\end{center}
\end{figure}
On the other hand, suppose the broken ray $X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)})$ is tangential to $\partial D$ at point $\bp = \bx(t_{\bp}, \boldsymbol \xi^{(0)})$, let
$$\bz = \nabla F(\bp)\cdot \frac{\partial \bx}{\partial \boldsymbol \zeta}\Big|_{t_{\bp},\boldsymbol \xi^{(0)}},$$
then by the assumption~\ref{as:4}, $\bz \neq {\mathbf 0}$ and not parallel to $\boldsymbol \xi^{(0)}$, using the Lemma~\ref{lm:4}, For the same $\epsilon$, we can select $\boldsymbol \xi_b^{(0)}\neq \boldsymbol \xi^{(0)}$ with $\boldsymbol \xi_b^{(0)}\in B$, $\|\boldsymbol \xi_b^{(0)} - \boldsymbol \xi^{(0)}\|<\epsilon$ and
\begin{equation}\label{eq:nonzero}
\bz \cdot (\boldsymbol \xi_b^{(0)} - \boldsymbol \xi^{(0)}) = O(\|\boldsymbol \xi_b^{(0)} - \boldsymbol \xi^{(0)}\|).
\end{equation}
Then the ray with initial phase $X_b^{(0)} = (\bx^{(0)}, \boldsymbol \xi^{(0)}_b)$ is broken.
Note that the tangent ray $X_g(\cdot, \bx^{(0)}, \boldsymbol \xi^{(0)})$ satisfies
\begin{equation}\label{eq:tangent}
\begin{aligned}
\nabla F(\bp)\cdot \frac{\partial \bx}{\partial s}\Big|_{t_{\bp}, \boldsymbol \xi^{(0)}} = 0,\\
F(\bp) = 0.
\end{aligned}
\end{equation}
Assume the broken ray $X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)}_b)$ intersects $\partial D$ at $\mathbf q} \newcommand{\br}{\mathbf r = \bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}_b)$ as illustrated in Figure~\ref{fig:example}, then take Taylor expansion at $(t_{\bp}, \boldsymbol \xi^{(0)})$,
\begin{equation}
\begin{aligned}
0 = F( \bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}_b)) =& F(\bx(t_{\bp}, \boldsymbol \xi^{(0)})) \\&+ \nabla F(\bp)\cdot \frac{\partial \bx}{\partial s}\Big|_{(t_{\bp},\boldsymbol \xi^{(0)})}(t_{\mathbf q} \newcommand{\br}{\mathbf r} - t_{\bp})+ \nabla F(\bp)\cdot \frac{\partial \bx}{\partial \boldsymbol \zeta}\Big|_{(t_{\bp},\boldsymbol \xi^{(0)})}(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \\
& + o(\|\boldsymbol \xi^{(0)}_b - \boldsymbol \xi^{(0)}\|) + O((t_{\mathbf q} \newcommand{\br}{\mathbf r} - t_{\bp})^2).
\end{aligned}
\end{equation}
The first two terms on the right hand side are zero according~\eqref{eq:tangent}. By using~\eqref{eq:nonzero} $$\nabla F(\bp)\cdot \frac{\partial \bx}{\partial \boldsymbol \zeta}\Big|_{(t_{\bp},\boldsymbol \xi^{(0)})}(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) = \bz\cdot (\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) = O(\|\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}\|),$$ we then conclude
\begin{equation}
|t_{\bp} - t_{\mathbf q} \newcommand{\br}{\mathbf r}|= O\left(\sqrt{\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|}\right).
\end{equation}
After reflection at time $t_{\mathbf q} \newcommand{\br}{\mathbf r}$, the direction of the broken ray $X_g(s, \bx^{(0)}, \boldsymbol \xi^{(0)}_b)$ has been reflected to $$\boldsymbol \xi_{b,\textrm{out}} = (I - 2\bn_{\mathbf q} \newcommand{\br}{\mathbf r}\otimes \bn_{\mathbf q} \newcommand{\br}{\mathbf r})\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}^{-}, \boldsymbol \xi^{(0)}_b),$$
where $\bn_{\mathbf q} \newcommand{\br}{\mathbf r} = \frac{\nabla F(\mathbf q} \newcommand{\br}{\mathbf r)}{\|\nabla F(\mathbf q} \newcommand{\br}{\mathbf r)\|}$ is the outward unit normal vector at $\mathbf q} \newcommand{\br}{\mathbf r$. Then by taking Taylor expansion at $(t_{\bp}, \boldsymbol \xi^{(0)})$, the difference between the directions of the two rays $X_g(t_{\mathbf q} \newcommand{\br}{\mathbf r}^+,\bx^{(0)}, \boldsymbol \xi^{(0)})$ and $X_g(t_{\mathbf q} \newcommand{\br}{\mathbf r}^+,\bx^{(0)}, \boldsymbol \xi^{(0)}_b)$ is
\begin{equation}
\begin{aligned}
\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \boldsymbol \xi_{b,\textrm{out}}&= 2 \bn_{\bp} \left(\bn_{\bp}\cdot \frac{\partial \boldsymbol \xi}{\partial s}\Big|_{t_{\bp},\boldsymbol \xi^{(0)}} +\left( \frac{\partial \bx}{\partial s}\nabla\frac{\nabla F}{\|\nabla F\| }\boldsymbol \xi\right)\Big|_{t_{\bp},\boldsymbol \xi^{(0)}}\right) (t_{\mathbf q} \newcommand{\br}{\mathbf r} - t_{\bp}) \\&\quad + O((t_{\mathbf q} \newcommand{\br}{\mathbf r} - t_{\bp})^2).
\end{aligned}
\end{equation}
where $\bn_{\bp} = \frac{\nabla F(\bp)}{\|\nabla F(\bp)\|}$ is the outward normal vector at $\bp$. According to assumption~\ref{as:4}, the first term on right hand side does not vanish, therefore
\begin{equation}\label{eq:k1}
\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \boldsymbol \xi_{b,\textrm{out}} = \bn_{\bp} \cdot O(t_{\mathbf q} \newcommand{\br}{\mathbf r} - t_{\bp}) = \bn_{p}\cdot O\left(\sqrt{\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|}\right).
\end{equation}
Regarding the phases at time $t_{\mathbf q} \newcommand{\br}{\mathbf r}^{+}$ as initial phases, then we can define the rest of the tangential ray as
\begin{equation}
X_g(s, \bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}), \boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)})) = (\by(s), {\boldsymbol\theta}(s)), \quad \by(0)=\bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}),\quad {\boldsymbol\theta}(0)=\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}),
\end{equation}
and denote the traveltime of this partial ray as $t_{\by}$. For the other ray, we define the rest of the broken ray as
\begin{equation}
X_g(s, \mathbf q} \newcommand{\br}{\mathbf r, \boldsymbol \xi_{b,\textrm{out}}) = (\bw(s), \boldsymbol \eta(s)), \quad \bw(0) = \mathbf q} \newcommand{\br}{\mathbf r,\quad \boldsymbol \eta(0) = \boldsymbol \xi_{b, \textrm{out}},
\end{equation}
and similarly, the traveltime is denoted as $t_{\bw}$.
Then use the Lemma~\ref{lm:3} and~\eqref{eq:k1}
\begin{equation}
\begin{aligned}
T(\boldsymbol \xi^{(0)}) - T(\boldsymbol \xi^{(0)}_b) &= (t_{\mathbf q} \newcommand{\br}{\mathbf r} + t_{\by}) - (t_{\mathbf q} \newcommand{\br}{\mathbf r} + t_{\bw})\\&= -\bu \cdot (\by(0) - \bw(0)) -\bv \cdot ({\boldsymbol\theta}(0) - \boldsymbol \eta(0)) \\&\quad+ o(\|\by(0) - \bw(0)\|)+ o(\|{\boldsymbol\theta}(0) - \boldsymbol \eta(0)\|) \\
&= -(\bv\cdot \bn_{p}) \cdot O\left(\sqrt{\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|}\right) + O(\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|),
\end{aligned}
\end{equation}
where $\bu$ and $\bv$ are defined as following,
\begin{equation}
\begin{aligned}
\bu = \left(\nabla G\cdot \frac{\partial \by}{\partial s}\Big|_{t_{\by}}\right)^{-1} \left(\nabla G\cdot \frac{\partial {\by}}{\partial \by(0)}\Big|_{t_{\by}}\right),\\
\bv = \left(\nabla G\cdot \frac{\partial \by}{\partial s}\Big|_{t_{\by}}\right)^{-1} \left(\nabla G\cdot \frac{\partial {\by}}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\right).
\end{aligned}
\end{equation}
We consider following three cases,
\begin{enumerate}
\item If $\bv\cdot \bn_p\neq 0$, then $\partial_{\zeta}T$ is discontinuous at $\boldsymbol \xi^{(0)}$. Otherwise, $T(\boldsymbol \xi^{(0)}) - T(\boldsymbol \xi^{(0)}_b) =t_{\by} - t_{\bw}= O(\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|)$ and we consider the next case.
\item Use~\eqref{eq:k1} and $\|\bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \mathbf q} \newcommand{\br}{\mathbf r\|=O(\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|)$, the difference between exiting physical locations is
\begin{equation
\begin{aligned}
\by(t_{\by}) - \bw(t_{\bw}) &= \frac{\partial \by}{\partial s}\Big|_{t_{\by}} (t_{\by} - t_{\bw}) \\&\quad+\frac{\partial \by}{\partial \by(0)}\Big|_{t_{\by}}(\bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \mathbf q} \newcommand{\br}{\mathbf r) +\frac{\partial \by}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}(\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \boldsymbol \xi_{b,\textrm{out}})\\&\quad+O ((t_{\by} - t_{\bw})^2)+O(\|\bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)})-\mathbf q} \newcommand{\br}{\mathbf r\|^2) + O(\|(\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \boldsymbol \xi_{b,\textrm{out}}\|^2)\\
&= \left( \frac{\partial \by}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\cdot\bn_p \right)\cdot O\left(\sqrt{\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|}\right) + O(\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|).
\end{aligned}
\end{equation}
If $\frac{\partial \by}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\cdot\bn_p $ is nonzero vector, then the derivative $\partial_{\boldsymbol \zeta}\bx$ will suffer from a discontinuity at $\by(t_{\by})$, otherwise if $\frac{\partial \by}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\cdot\bn_p = {\mathbf 0}$, we consider the next case.
\item Similarly, the difference between exiting directions is
\begin{equation}
\begin{aligned}
{\boldsymbol\theta}(t_{\by}) - \boldsymbol \eta(t_{\bw}) &= \frac{\partial {\boldsymbol\theta}}{\partial s}\Big|_{t_{\by}}(t_{\by} - t_{\bw})\\&\quad +\frac{\partial {\boldsymbol\theta}}{\partial \by(0)}\Big|_{t_{\by}}(\bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \mathbf q} \newcommand{\br}{\mathbf r) + \frac{\partial {\boldsymbol\theta}}{\partial{\boldsymbol\theta}(0)}\Big|_{t_{\by}}(\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \boldsymbol \xi_{b,\textrm{out}})\\&\quad + O ((t_{\by} - t_{\bw})^2)+O(\|\bx(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)})-\mathbf q} \newcommand{\br}{\mathbf r\|^2) + O(\|(\boldsymbol \xi(t_{\mathbf q} \newcommand{\br}{\mathbf r}, \boldsymbol \xi^{(0)}) - \boldsymbol \xi_{b,\textrm{out}}\|^2)\\
&= \left( \frac{\partial {\boldsymbol\theta}}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\cdot\bn_p \right)\cdot O\left(\sqrt{\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|}\right) + O(\|(\boldsymbol \xi^{(0)}_{b} - \boldsymbol \xi^{(0)}) \|).
\end{aligned}
\end{equation}
If $ \frac{\partial \by}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\cdot\bn_p$ is nonzero vector, then the derivative $\partial_{\boldsymbol \zeta}\boldsymbol \xi$ will suffer from a discontinuity at ${\boldsymbol\theta}(t_{\by})$. Otherwise we will have following equations
\begin{equation}
\begin{aligned}
\frac{\partial \by}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\cdot\bn_p &= 0, \\
\frac{\partial {\boldsymbol\theta}}{\partial {\boldsymbol\theta}(0)}\Big|_{t_{\by}}\cdot\bn_p &= 0,
\end{aligned}
\end{equation}
by Lemma~\ref{lm:5}, we must have $\bn_{\bp}={\mathbf 0}$, which is a contradiction. Therefore either $\partial_{\boldsymbol \zeta}T$ or $\partial_{\boldsymbol \zeta}\bx$ or $\partial_{\boldsymbol \zeta}\boldsymbol \xi$ must have a discontinuity at $\boldsymbol \xi^{(0)}$.
\end{enumerate}
\end{proof}
With these assumptions, we can directly detect non-broken rays from the measurements by scanning the traveltimes, exiting directions and exiting locations for jumps in the derivatives with respect to initial directions, see Figure~\ref{fig:jumps}. And using these non-broken rays enables us to recover the metric outside of the convex hull of the obstacle~\cite{krishnan2009support,ilmavirta2014broken}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.14]{15t.png}
\includegraphics[scale=0.14]{30t.png}\\
\includegraphics[scale=0.14]{15a.png}
\includegraphics[scale=0.14]{30a.png}\\
\includegraphics[scale=0.14]{15l.png}
\includegraphics[scale=0.14]{30l.png}
\caption{The plots of traveltimes, exiting directions, exiting locations from numerical example~\ref{num:5}. Left: From top to bottom, the plots are the traveltimes, exiting directions, exiting locations corresponding to the initial phases with varying directions at the \nth{15} boundary point, the shadowed spots are placed at the detected jumps near \nth{52} and \nth{195} rays respectively. Right: From top to bottom, the plots are the traveltimes, exiting directions, exiting locations corresponding to the initial phases with varying directions at the \nth{30} boundary point, the shadowed spots are placed at the detected jumps near \nth{98} and \nth{175} rays respectively.}
\label{fig:jumps}
\end{figure}
\subsection{Reconstruction of metric and obstacle}
In this section, we present the hybrid method for reconstructing both the metric and included obstacles. First we find out if there exists an obstacle inside the medium by checking if the set $T$ defined in \eqref{eq:broken} is empty.
If no obstacle is detected, then we can use the improved adaptive phase space method in Section~\ref{sec:stab-adaptive} to recover the metric efficiently and stably with the layer-stripping strategy.
Once an obstacle is detected, then we distinguish the broken rays and non-broken ones by scanning the exiting traveltimes, locations and directions in scattering relation explained in Section~\ref{sec:non-broken-detect} and use the improved adaptive phase space method in Section~\ref{sec:stab-adaptive} on non-broken rays to recover the metric. In this case, our layer-stripping reconstruction strategy will be able to recover the metric starting from the boundary and continuing inward all the way to the convex hull of the obstacle. Since our adaptive phase space method is based on optimization formulation with a regularization \eqref{eq:regularize} for the metric at all grid points in the domain, the numerically reconstructed metric in the whole domain can be viewed as a good approximation of the true metric outside the convex hull of the obstacle plus a harmonic extension to the interior of the convex hull.
If the obstacle is convex (under the metric), one can reconstruct both the metric and the obstacle using non-broken rays as we can see from the numerical experiments in Section~\ref{sec:num}. Since non-broken rays only contain information outside the convex hull of the obstacle, it is impossible to reconstruct a concave obstacle with only non-broken rays.
On the other hand, since our adaptive phase method using non-broken rays reconstructs the metric on the whole domain, then by tracing back the rays in $T$ (defined in \eqref{eq:broken}) to half of the traveltime, we will get the approximated reflection points of such rays, if the true metric varies slowly inside the convex hull. This gives us a direct imaging method for the boundary of the obstacle, see the numerical experiments in Section~\ref{sec:num}.
For the cases that the metric has \emph{large} variations inside the convex hull, our method then can not recover the obstacle and metric inside the convex hull without using other broken rays. One possible way is to introduce an representation of the obstacle's boundary and iteratively reconstruct the metric inside the convex hull as well as morph the boundary simultaneously to minimize the mismatch, e.g., using the result from our hybrid method as an initial guess. However, this will be a daunting task due to the highly non-convex and coupled optimization problem.
Finally, we briefly discuss the computational cost of our hybrid method in 2D. Suppose we have $N_s$ sources and each source probes $N_a$ directions, then there are $N_sN_a$ scattering relation measurements. For obstacle detection, it will take $O(N_s N_a)$ complexity at the worst case. For non-broken rays detection, it will take $O(N_s N_a)$ complexity due to linear scan. At each iteration of the stabilized adaptive phase space method, we have to solve the Hamiltonian system for $X_g(s, X^{(0)})$ in~\eqref{eq:hamiltonian} and Jacobian matrix~\eqref{eq:jacobian}, which has the worst complexity as $O(T_{\max}N_s N_a)$, where $T_{\max}$ is length of the longest geodesic. These solutions are then used to calculate mismatch and formulate the linearized Stefanov-Ulhmann identity~\eqref{eq:linearized-su} over the Eulerian grid in~\eqref{eq:linear-eq} with the worst complexity $O(T_{\max}N_sN_a)$. Then we use the standard multifrontal solver \texttt{umfpack} to solve the perturbation for the minimization problem~\eqref{eq:regularize}, the complexity is $O(n^3)$ in general, where $n$ is the number of unknowns. Therefore the total time complexity is $O(K( T_{\max}N_sN_a + n^3))$, where $K$ is the number of iterations.
\input{numerical}
\section{Conclusion}
In this work, we proposed a hybrid phase space method for traveltime tomography which includes both an unknown medium and unknown scatterer. The underlying medium outside the convex hull of the scatterer is reconstructed by a optimization based iterative method. The newly developed method is more stable than the previous adaptive phase method proposed in \cite{chung2011adaptive} due to the introduction of an auxiliary fidelity function to guide the layer stripping process and a direct detection of all non-broken rays. To image the boundary of the scatterer, we use a direct imaging method that can locate points on the boundary of the scatterer by selecting those broken-once rays that hit the scatterer almost normally and tracing back those
rays to half traveltime in the reconstructed medium.
\section*{Acknowledgement}
H. Zhao is partially supported by NSF grant DMS-1418422. Both authors would like to thank ICERM 2017 Fall program on Mathematical and Computational Challenges in Radar and Seismic Reconstruction, where this project was started. The authors also would like to thank Kui Ren for valuable discussions.
\bibliographystyle{unsrtnat}
\section{Numerical experiments}\label{sec:num}
All numerical experiments are implemented in \texttt{Julia} and performed on a dual-core laptop of $2.7\texttt{GHz}$ CPU and $16\texttt{GByte}$ memory. Source code is hosted on \href{https://github.com/lowrank/ray}{https://github.com/lowrank/ray}.
We take the physical domain $\Omega$ as unit disk for all examples. The discretization of metric $g$ over a uniform grid is parametrized by $\texttt{Q4}$ element. If a ray passes through a grid, then it will involve $12$ surrounding grid values, under such situation we can set rank threshold $r_{\min} = 12$. For other parameters, our selections are conservative, we take $\tau = 5\%$ and $\alpha = 10$ for fidelity function updating, and regularization parameter $\beta = 0.5$ (see Section \ref{sec:stab-adaptive}), the numerical tolerance $\epsilon=0.5\%$ for obstacle detection (see Section \ref{sec:ref-detect}). We keep them fixed for all of the examples.
\subsection{Scenario 1: no obstacle}
In this scenario, we experiment our improved adaptive phase space method described in Section \ref{sec:stab-adaptive} on simple cases without interior obstacle.
\subsubsection{Example 1}
The exact solution is $$c(x,y) = 1+0.3\sin(\pi x)\sin(\pi y).$$ The grid's resolution is $h = 1/15$. We put $50$ equispaced sources and each source probes $100$ uniformly distributed directions. The method converges to a solution with relative $L^2$ error $2.41\times10^{-3}$ at $11$th iteration. We plot the numerical and the exact solutions in Figure~\ref{fig:ex1}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.35]{ex1-tr.png}
\includegraphics[scale=0.35]{ex1-rec.png}
\includegraphics[scale=0.35]{ex1-err.png}
\caption{Left: the exact solution. Middle: the numerical solution at $11$th iteration. Right: the error between numerical solution and exact solution.}
\label{fig:ex1}
\end{figure}
\subsubsection{Example 2}
The exact solution is $$c(x,y) = 1+0.3\sin(1.5\pi x)\sin(1.5\pi y).$$
The grid's resolution is $h = 1/25$. We put $100$ equispaced sources and each source probes $100$ uniformly distributed directions. The method converges to a solution with relative $L^2$ error $3.07\times 10^{-3}$ at $22$th iteration. We plot the numerical and the exact solutions in Figure~\ref{fig:ex2}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.35]{ex2-tr.png}
\includegraphics[scale=0.35]{ex2-rec.png}
\includegraphics[scale=0.35]{ex2-err.png}
\caption{Left: the exact solution. Middle: the numerical solution at $22$th iteration. Right: the error between numerical solution and exact solution.}
\label{fig:ex2}
\end{figure}
We also plot the auxiliary fidelity function at three different iterations to illustrate the layer stripping process in Figure~\ref{fig:ex2-fidelity}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.38]{2-6}
\includegraphics[scale=0.38]{2-11}
\includegraphics[scale=0.38]{2-17}
\caption{The fidelity function $p^n$ in different iterations. From left to right: $6$th, $11$th, $17$th iteration.}
\label{fig:ex2-fidelity}
\end{figure}
\subsection{Scenario 2: convex unknown obstacle}
In this scenario, we experiment our hybrid method for imaging an unknown convex obstacle inside an unknown metric.
\subsubsection{Example 3}\label{sec:ex3}
We consider the obstacle as a circle at center
$$x^2 + y^2 = \frac{1}{16},$$
and the exact solution is given by
$$c(x,y) = 1+0.4\sin\left(\pi \sqrt{(x-0.5)^2 + (y-0.2)^2}\right) + 0.4 \sin\left(\pi \sqrt{(x+0.4)^2 + (y+0.3)^2}\right).$$
The grid's resolution is $1/15$. We put $50$ equispaced sources and each source probes $300$ uniformly distributed directions. We first detect the obstacle by checking the scattering relation as in Section~\ref{sec:ref-detect}, and then distinguish the non-broken rays from the broken rays as in Section~\ref{sec:non-broken-detect}, see Figure~\ref{fig:ex3-ortho}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.35]{ex3-ortho}
\includegraphics[scale=0.35]{ex3-nonbroken}
\caption{Left: The broken rays in $T$, which hit the obstacle in (nearly) normal direction in Example 3. Right: Detected tangent rays in Example 3.}
\label{fig:ex3-ortho}
\end{figure}
And then we use all the non-broken rays to reconstruct the metric by the stabilized adaptive phase space method, the method converges to a solution with relative $L^2$ error $4.93\times 10^{-3}$ at $9$th iteration. We plot the numerical and exact solutions in Figure~\ref{fig:ex3}.
\begin{figure}[!htb]
\includegraphics[scale=0.35]{ex3-tr}
\includegraphics[scale=0.35]{ex3-rec}
\includegraphics[scale=0.35]{ex3-err}
\caption{Left: exact solution. Middle: numerical solution at $9$th iteration. Right: error between numerical solution and exact solution.}
\label{fig:ex3}
\end{figure}
From the experiment, we can see that the metric outside of the obstacle has been recovered well. Then the obstacle's convex hull can be approximated by the envelope of all the tangent rays computed through the recovered metric, see Figure~\ref{fig:ex3-convex-hull-rec}. After the reconstruction of the convex hull of the obstacle and the metric outside the convex hull, we trace the rays in collection $T$ (defined in \eqref{eq:broken}) to half of the traveltime to get the reflection points on the boundary, see also in Figure~\ref{fig:ex3-convex-hull-rec}. We can see that the computed reflection points are quite close to the boundary.
\begin{figure}
\centering
\includegraphics[scale=0.35]{ex3-nonbroken-rec}
\includegraphics[scale=0.35]{ex3-ob-rec}
\caption{Left: The tangent rays computed from recovered metric in Example 3. Right: Traced the rays in collection $T$ to half traveltime, the dashed blue circle at center is the exact boundary.}
\label{fig:ex3-convex-hull-rec}
\end{figure}
\subsection{Scenario 3: non-convex unknown obstacle}
In this scenario, we will use our hybrid method to recover a non-convex unknown interior obstacle and the underlying metric.
\subsubsection{Example 4}
In this example, we consider an easier case. The obstacle's boundary is parameterized in polar coordinate $(r, \theta)$ as
$$r(\theta) = 0.25 + 0.05\sin(3\theta),$$
which is a \emph{slightly} concave shape, the exact solution is given by
$$c(x,y) = 1+0.4\sin\left(\pi \sqrt{(x-0.5)^2 + (y-0.2)^2}\right) + 0.4 \sin\left(\pi \sqrt{(x+0.4)^2 + (y+0.3)^2}\right).$$
The grid's resolution is $1/15$. We put $50$ equispaced sources and each source probes $300$ uniformly distributed directions. From the scattering relation, we can directly extract the rays that hit the obstacle in almost normal direction, and also distinguish the non-broken rays by detecting the jumps in scattering relation. We plot those rays in Figure~\ref{fig:ex4-shape}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.35]{ex4-ortho}
\includegraphics[scale=0.35]{ex4-nonbroken}
\caption{Left: The rays that hit the obstacle in (nearly) normal direction in Example 4. Right: The tangent rays detected from scattering relation.}
\label{fig:ex4-shape}
\end{figure}
Then we follow the method in Section~\ref{sec:non-broken-detect} to distinguish the non-broken rays and broken rays. Then we use all the non-broken rays to recover the metric outside the obstacle by the stabilized adaptive phase space method. The method converges to a solution with relative $L^2$ error $5.58\times 10^{-3}$ at $9$th iteration. We plot the numerical and exact solutions in Figure~\ref{fig:ex4}.
\begin{figure}[!htb]
\includegraphics[scale=0.35]{ex4-tr}
\includegraphics[scale=0.35]{ex4-rec}
\includegraphics[scale=0.35]{ex4-err}
\caption{Left: exact solution. Middle: numerical solution at $9$th iteration. Right: error between the numerical solution and the exact solution.}
\label{fig:ex4}
\end{figure}
Since the error of metric is small outside the obstacle, then the obstacle's convex hull can be approximated well by all the tangent rays computed through the recovered metric, see Figure~\ref{fig:ex4-nonbroken-rec}. By tracing back the rays in $T$, we approximately obtain the reflection points on the obstacle, also see Figure~\ref{fig:ex4-nonbroken-rec}. However, since no information is available inside the convex hull, the error of reflection points can be large in general.
\begin{figure}
\centering
\includegraphics[scale=0.35]{ex4-nonbroken-rec}
\includegraphics[scale=0.35]{ex4-ob-rec}
\caption{Left: The tangent rays computed from recovered metric in Example 4. Right: The traced rays in collection $T$ to half of traveltime. The blue dashed line is the exact boundary of obstacle.}
\label{fig:ex4-nonbroken-rec}
\end{figure}
\subsubsection{Example 5}\label{num:5}
In this example, we take a more challenging obstacle. The obstacle's boundary is parameterized in polar coordinate $(r, \theta)$ as
$$r(\theta) = 0.4 + 0.2\sin(3\theta),$$
which is a \emph{more} concave shape than previous example,
and the exact solution is again given by
$$c(x,y) = 1+0.4\sin\left(\pi \sqrt{(x-0.5)^2 + (y-0.2)^2}\right) + 0.4 \sin\left(\pi \sqrt{(x+0.4)^2 + (y+0.3)^2}\right).$$
The grid's resolution is $1/15$. We put $50$ equispaced sources and each source probes $300$ uniformly distributed directions. From the scattering relation, we can directly extract the rays that hit the obstacle in almost normal direction, and also distinguish the non-broken rays by detecting the jumps in scattering relation. We plot such rays in Figure~\ref{fig:ex5-rays}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.35]{ex5-ortho}
\includegraphics[scale=0.35]{ex5-nonbroken}
\caption{Left: The rays that hit the obstacle in (nearly) normal direction in Example 5. Right: The tangent rays detected from scattering relation.}
\label{fig:ex5-rays}
\end{figure}
Then we use the stabilized adaptive phase space method in Section~\ref{sec:stab-adaptive} on all the non-broken rays to recover the metric as much as possible. The method converges to a solution with relative $L^2$ error of $2.90\times 10^{-2}$ at $9$th iteration. We plot the numerical and exact solutions in Figure~\ref{fig:ex5}.
\begin{figure}[!htb]
\includegraphics[scale=0.35]{ex5-tr}
\includegraphics[scale=0.35]{ex5-rec}
\includegraphics[scale=0.35]{ex5-err}
\caption{Left: exact solution. Middle: numerical solution at $9$th iteration. Right: error between the numerical solution and the exact solution.}
\label{fig:ex5}
\end{figure}
After having recovered the metric from the non-broken rays' scattering relation, we can approximate the convex hull of the obstacle by the recovered non-broken rays, see Figure~\ref{fig:ex5-nonbroken-rec}. And by tracing the rays in collection $T$, we can approximately obtain the reflection points on the boundary of the obstacle, also see Figure~\ref{fig:ex5-nonbroken-rec}.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.35]{ex5-nonbroken-rec}
\includegraphics[scale=0.35]{ex5-ob-rec}
\caption{Left: The tangent rays computed from recovered metric in Example 5. Right: The traced rays in collection $T$ to half of traveltime. The blue dashed line is the exact boundary of obstacle.}
\label{fig:ex5-nonbroken-rec}
\end{figure}
|
{
"timestamp": "2018-03-08T02:03:33",
"yymm": "1803",
"arxiv_id": "1803.02501",
"language": "en",
"url": "https://arxiv.org/abs/1803.02501"
}
|
\section{Introduction}\label{s:intro}
It is now well established that planet formation processes are robust,
and proceed around stars of a wide range of masses. At the higher mass
end, planets have been discovered around evolved stars with masses up to
three times the Sun's
\citep[e.g.][]{2005A&A...437L..31S,2007ApJ...665..785J,2015A&A...574A.116R}. At
the lower mass end the results have been equally impressive, with
planets discovered around objects ten times less massive than the Sun,
and whose luminosity is a thousand times weaker
\citep[e.g.][]{2016Natur.533..221G,2016Natur.536..437A}. This wide mass
range provides a unique way to study planet formation processes, and has
shown that while the occurence rate of giant planets increases towards
higher mass stars
\citep{2007ApJ...670..833J,2010PASP..122..905J,2015A&A...574A.116R}, the
converse is true for the frequency of Earth to Neptune-mass planets
\citep{2015ApJ...798..112M}.
In tandem with these searches, observations that seek to detect the
building blocks of these planets have also been conducted. These mid and
far-infrared (IR) surveys detect `debris disks', the collections of
small dust particles that are seen to orbit other stars (the `dust'
comprises various constituents, such as silicates, ice, and organic
compounds). Since their discovery in the 1980's, a growing body of
evidence has shown that they can be interpreted as circumstellar disks
made up of bodies ranging from
$\sim$$\mu$m to many km in size; while the observations only detect
$\mu$m to mm-size particles, the lifetime of these particles is commonly
shorter than the age of the host star, leading to the conclusion that
they must be replenished through the collisional destruction of a mass
reservoir of larger planetesimals
\citep[e.g.][]{1993prpl.conf.1253B}. For main-sequence stars this
paradigm is generally accepted, so in terms of the dust having an origin
in collisions between larger bodies, debris disks can be genuinely
thought of as analogues of the Solar System's Asteroid and Kuiper
belts. A key unknown is how the planetesimals acquire high enough
relative velocities for their collisions to be destructive; while it is
possible that planets excite these velocities
\citep{2009MNRAS.399.1403M}, it may be a natural outcome upon emergence
from the gas rich phase of evolution, or the planetesimals may `stir'
themselves \citep[e.g.][]{2004AJ....127..513K}, in which case planets
are not necessarily needed in order for debris disks to exist.
However, it is well known that the Solar System planets play an
important role in sculpting the Asteroid and Kuiper belts, two examples
being the presence of the Kirkwood gaps and the capture of Pluto into
2:3 mean motion resonance by Neptune \citep{1993Natur.365..819M}. In
attempts to make analogous link in other planetary systems, hypotheses
that connect the properties of the disks and planets have been
developed, and vary in complexity. The most basic is that some systems
are simply `better' at forming large bodies (whether those bodies be
planetesimals or planets), and more detailed models suggest that the
outcomes depend on whether planetary instabilities occurred
\citep{2011A&A...530A..62R}. As with planets, merely detecting these
belts is challenging, so quantifying the connection between the planets
and disks in these systems is typically limited to searching for
correlations between their basic properties \citep[such as disk
brightness,
e.g.][]{2009ApJ...700L..73K,2009ApJ...705.1226B,2012MNRAS.424.1206W,2014A&A...565A..15M,2015AJ....149...86W,2015ApJ...801..143M}. Ultimately,
these searches yielded a significant correlation between the presence of
radial velocity planets and the brightness of debris disks around
Sun-like stars \citep{2014prpl.conf..521M}. This trend is unfortunately
not strong, so while splitting the sample to look for trends among
sub-samples (e.g. as a function of planet mass) yields tentative trends
\citep[e.g.][]{2012MNRAS.424.1206W} it also lowers the significance.
Thus, while there is evidence that some Sun-like stars are indeed better
at forming disks and planets than other, the origin of this correlation
remains unclear.
In the case of low-mass stars the challenge of finding connections
between the planet and disk populations is even greater; for disks at
the typical radial distances of a few tens of astronomical units, the
low stellar luminosities do not heat the dust to temperatures greater
than about 50K. While the Stefan-Boltzmann law therefore limits the
luminosity of these disks, the low temperatures further hinder detection
because discoveries must be made at far infrared and millimeter
wavelengths
\citep[e.g.][]{2006A&A...460..733L,2012A&A...548A..86L}. Thus, it is not
particularly surprising that efforts to discover debris disks around
late-type stars at mid-infrared wavelengths have often been unsuccessful
\citep[e.g.][]{2007ApJ...667..527G,2012A&A...548A.105A}. Further, the
sensitivity of surveys is normally such that the non-detections are not
sufficiently constraining to rule out disks that have similar properties
to those that are known to orbit Sun-like stars
\citep{2007ApJ...667..527G,2014A&A...565A..58M}.
In this paper, we present far infrared \emph{Herschel}\footnote{Herschel
is an ESA space observatory with science instruments provided by
European-led Principal Investigator consortia and with important
participation from NASA} \citep{2010A&A...518L...1P} observations that
aim to detect Kuiper belt analogues around a sample of 21 nearby late K
and M-type stars that host planets discovered by the radial velocity
technique. The primary aim is to search for a correlation between the
presence of planets and the brightness of disks, and secondary aims are
to detect new disks that may be amenable to further detailed
investigation, and to obtain more sensitive observations than were
possible with larger surveys. We present the sample and observations in
section \ref{s:obs}, discuss the results in section \ref{s:disc}, and
summarise and conclude in section \ref{s:conc}.
\section{Sample and Observations}\label{s:obs}
\begin{table}
\begin{center}
\caption{PACS observations of 16 targets taken as part of our
programme (OT2\_gbryden\_2). OD is the Herschel Observing Day, and
Reps is the number of repeats of a standard PACS mini scan-map
used to reach the desired sensitivity.}\label{tab:obs}
\begin{tabular}{llll}
\hline
Name & ObsIDs & OD & Reps \\
\hline
GJ 176 & 1342250278/279 & 1202 & 6\\
GJ 179 & 1342250276/277 & 1202 & 6\\
GJ 317 & 1342253029/030 & 1245 & 6\\
GJ 3634 & 1342257175/176 & 1310 & 6\\
GJ 370 & 1342256997/998 & 1308 & 6\\
GJ 433 & 1342257567/568 & 1316 & 6\\
GJ 1148 & 1342247393/394 & 1138 & 6\\
GJ 436 & 1342247389/390 & 1138 & 6\\
GJ 9425 & 1342249877/878 & 1194 & 6\\
GJ 9482 & 1342248728/729 & 1170 & 6\\
HIP 79431 & 1342262219/220 & 1355 & 6\\
GJ 649 & 1342252819/820 & 1244 & 6\\
GJ 1214 & 1342252011/012 & 1237 & 6\\
GJ 674 & 1342252841/842 & 1244 & 6\\
GJ 676 A & 1342243794/795 & 1058 & 6\\
GJ 849 & 1342246764/765 & 1121 & 6\\
\end{tabular}
\end{center}
\end{table}
Our sample comprises nearly all low-mass planet-host stars within
20pc. Most stars are M spectral type, but we include three that are late
K types (GJ~370, GJ~9425 and GJ~9482). Not all systems in the final
sample were known to host planets at the time the observations were
proposed (2011 September), but some in which planets were subsequently
discovered were observed by the volume-limited DEBRIS Key Programme
\citep{2010A&A...518L.135M}. The final sample has 21 stars, 16 of which
were observed by \emph{Herschel} in this programme, and which are listed
in Table \ref{tab:obs}. Five more targets, GJ~15~A, GJ~581, GJ~687,
GJ~842, and GJ~876, were observed by the DEBRIS survey so are also
included in our sample \citep[see][for results for
GJ~581]{2012A&A...548A..86L}.
The sample does not include the planet host Proxima Centauri
\citep{2016Natur.536..437A}, as it was not observed by
\emph{Herschel}. While it has been suggested to host excess emission
arising from a debris disk \citep{2017arXiv171100578A}, these
observations use the Atacama Large Millimeter Array (ALMA) and this
system is therefore not easily integrated into our sample. Two of our
targets are possible wide binaries; GJ~15~A (NLTT~919) is a common
proper motion pair at a projected separation of 35\arcsec\ with NLTT~923
\citep{2004ApJS..150..455G}, and GJ~676~A has a wide common proper
motion companion (GJ~676~B) at a projected separation of 50\arcsec\
\citep{1994RMxAA..28...43P}. We do not expect the planetary systems to
be affected seriously by these companions, so retain them in our sample.
\begin{figure*}
\begin{center}
\hspace{-0.4cm} \includegraphics[width=0.275\textwidth]{figs/im100-0.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im100-1.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im100-2.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im100-3.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im100-4.eps} \\
\vspace{-0.95cm}
\hspace{-0.4cm} \includegraphics[width=0.275\textwidth]{figs/im160-0.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im160-1.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im160-2.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im160-3.eps}
\hspace{-1.8cm} \includegraphics[width=0.275\textwidth]{figs/im160-4.eps}
\caption{\emph{Herschel} images of the two targets found here to
host debris disks (GJ~433 and GJ~649, in the left two columns),
and the three targets for which excess emission near the star was
seen, but which was assumed not to be associated with the star in
question (right three columns). In each panel the black cross
marks the estimated stellar position at the time of
observation. Each image is centered either on the star, or in the
case of GJ~649 and GJ~3634 between the two visible source
detections. White contours are at 2, 4, and 6 times the 1$\sigma$
noise level in each image. The disk around GJ~649 appears to be
marginally resolved; see Figure \ref{fig:im2}.}\label{fig:im}
\end{center}
\end{figure*}
The targets were observed using the Photodetector Array Camera and
Spectrometer \citep[PACS,][]{2010A&A...518L...2P}, using the so-called
`mini-scan map' mode. A series of ten parallel scans with a separation
of 4\arcsec\ are taken to make a single map, which is repeated six times
to build up the signal. One such sequence coresponds to a single
observation ID number, or ObsID. The observatory is then rotated by
40$^\circ$, and the sequence repeated, to provide some robustness to
striping artefacts and low-frequency noise. The total integration time
for each source is 56 minutes. For our observations the noise level at
100$\mu$m was typically 1mJy, while observations carried out by DEBRIS
(integration time of 15 minutes) had fewer repeats and a noise level
nearer 2mJy. The images used in the analysis are the standard `level
2.5' observatory products obtained from the Herschel Science
Archive,\footnote{\href{http://archives.esac.esa.int/hsa/whsa/}{http://archives.esac.esa.int/hsa/whsa/}}
which combine the two observing sequences (ObsIDs) into a single image.
Photometry $F_{\rm obs}$ for each source was extracted using point
spread function (PSF) fitting. Observations of the calibration star
$\gamma$ Dra, again level 2.5 observatory products, were used as PSFs,
which were rotated to a position angle appropriate for each
observation. The fitting was done at 100 and 160$\mu$m simultaneously,
so the four free parameters in each fit were a position common to both
wavelengths, and two fluxes (i.e. $F_{100}$ and
$F_{160}$). Uncertainties $\sigma_{100}$ and $\sigma_{160}$ were
estimated by measuring the flux in apertures at hundreds of random
locations near the center of the images; this method was found to be
more reliable and provide more realistic flux distributions than
attempting to fit PSFs at random locations. The apertures were chosen to
be those optimal for source extraction (5 and 8\arcsec~for 100 and
160$\mu$m respectively, derived using calibration observations). In the
case of GJ~649 there is evidence that the source (i.e. disk) is
marginally resolved (see Figure \ref{fig:im}), so the flux for this
source at 100$\mu$m is measured using an aperture radius of 10\arcsec,
and the uncertainty estimated as above but with 10\arcsec\
apertures. The results of the source extraction are summarised in Table
\ref{tab:fluxes}, and the results for a few problematic sources are
described in more detail below.
To assess whether each star shows the infrared excess that is indicative
of a debris disk requires an estimate of the flux density expected from
the stellar photosphere $F_\star$ at the PACS wavelengths. These
estimates are made by fitting stellar photosphere models to optical and
near-IR photometry. The method has been described elsewhere, and for
example has been used for the DEBRIS survey and shown to provide
photospheric fluxes that are sufficiently precise that the detection of
excesses is limited by the \emph{Herschel} photometry, not the
photosphere models \citep[i.e.
$\sigma_{\rm obs} >
\sigma_\star$,][]{2012MNRAS.426.2115K,2012MNRAS.421.2264K}. While
photospheric models for late-type stars are less precise than for
earlier types (e.g. because of uncertain molecular opacity), the flux of
many of our target stars is predicted to be near our noise level and the
models are not a limiting factor. The photospheric predictions at 100
and 160$\mu$m are given in Table~\ref{tab:fluxes}.
The significance of any excess is then given in each PACS bandpass by
$\chi = (F_{\rm obs} - F_\star) / \sqrt{\sigma_{\rm obs}^2 +
\sigma_\star^2}$, where $\chi>3$ is taken to be a significant
excess. To summarise the observational results; in addition to the disk
known to orbit GJ~581, we find two new systems that show strong evidence
for infrared excesses: GJ~433 and GJ~649, whose images are shown in
Figure \ref{fig:im}.
\begin{table*}
\caption{The 21 stars in our sample, comprising 16 stars observed in programme
OT2\_gbryden\_2, and five stars observed in programme
KPOT\_bmatthew\_1 (DEBRIS): GJ~15, GJ~581 \citep[multiple observations,
see][]{2012A&A...548A..86L}, GJ~687, GJ~832, and GJ~876. We have not
reported flux densities for the two strongly confused sources,
HIP~79431 and GJ~674.}\label{tab:fluxes}
\begin{tabular}{lrlrrrrrrrrrl}
\hline
GJ & HIP no. & SpTy & Dist &
$F_{\star,100}$ & $F_{\rm 100}$ & $\sigma_{\rm 100}$ & $\chi_{\rm 100}$ &
$F_{\star,160}$ & $F_{\rm 160}$ & $\sigma_{\rm 160}$ & $\chi_{\rm 160}$ &
Notes \\
& & & (pc) &
(mJy) & (mJy) & (mJy) & &
(mJy) & (mJy) & (mJy) & & \\
\hline
GJ 15 A & 1475 & M2V & 3.6 & 15.3 & 14.9 & 2.2 & -0.2 & 5.9 & 13.2 & 3.0 & 2.4 & Photosphere at 100$\mu$m \\
GJ 176 & 21932 & M2.5V & 9.4 & 4.1 & 3.6 & 1.6 & -0.3 & 1.6 & -3.6 & 6.4 & -0.8 & No detection \\
GJ 179 & 22627 & M2V & 12.4 & 1.3 & -1.6 & 1.2 & -2.5 & 0.5 & -4.7 & 2.8 & -1.9 & No detection \\
GJ 317 & - & M3.5V & 15.3 & 1.1 & 3.2 & 1.1 & 1.9 & 0.4 & 4.9 & 2.2 & 2.0 & No detection \\
GJ 370 & 48331 & K6Vk: & 11.3 & 5.7 & 6.9 & 0.9 & 1.3 & 2.2 & -5.2 & 3.2 & -2.3 & Photosphere at 100$\mu$m \\
GJ 3634 & - & M2.5 & 19.8 & 0.7 & -0.9 & 1.2 & -1.3 & 0.3 & 3.0 & 2.8 & 1.0 & Detection at 6\arcsec~SW \\
\textbf{GJ 433} & 56528 & M2V & 9.1 & 3.9 & 11.9 & 1.3 & 6.2 & 1.5 & 13.9 & 4.3 & 2.9 & Excess detection \\
GJ 1148 & 57050 & M4.0Ve & 11.1 & 1.5 & 1.4 & 1.0 & -0.1 & 0.6 & -2.1 & 3.3 & -0.8 & No detection \\
GJ 436 & 57087 & M3V & 9.7 & 2.4 & 3.4 & 1.1 & 0.9 & 0.9 & 4.3 & 2.3 & 1.5 & No detection \\
GJ 9425 & 63833 & K9Vk: & 15.9 & 3.1 & -0.7 & 2.1 & -1.8 & 1.2 & -15.1 & 7.6 & -2.1 & No detection \\
GJ 9482 & 70849 & K7Vk & 23.6 & 1.0 & 1.6 & 1.4 & 0.5 & 0.4 & -4.1 & 3.4 & -1.3 & No detection \\
GJ 581 & 74995 & M3V & 6.3 & 3.8 & 21.8 & 1.5 & 11.8 & 1.5 & 22.4 & 5.0 & 4.2 & Excess {\citet{2012A&A...548A..86L}} \\
- & 79431 & M3V & 14.4 & 1.7 & - & - & - & 0.6 & - & - & - & Extended detection at 5\arcsec~N \\
\textbf{GJ 649} & 83043 & M2V & 10.4 & 3.6 & 22.6 & 2.4 & 7.9 & 1.4 & 16.3 & 5.2 & 2.9 & Excess detection, extended? \\
GJ 1214 & - & M4.5V & 14.6 & 0.3 & 0.9 & 1.1 & 0.6 & 0.1 & -0.1 & 2.3 & -0.1 & No detection, source 10\arcsec~W \\
GJ 674 & 85523 & M3V & 4.5 & 8.1 & - & - & - & 3.1 & - & - & - & Extended, high background \\
GJ 676 A & 85647 & M0V & 15.9 & 2.6 & 0.9 & 1.1 & -1.6 & 1.0 & 0.5 & 2.2 & -0.2 & No detection, source 10\arcsec~SW \\
GJ 687 & 86162 & M3.0V & 4.5 & 10.1 & 6.1 & 1.6 & -2.5 & 3.9 & 0.2 & 3.4 & -1.1 & No detection \\
GJ 832 & 106440 & M2/3V & 5.0 & 10.4 & 12.5 & 1.6 & 1.3 & 4.0 & 1.2 & 3.5 & -0.8 & Photosphere at 100$\mu$m \\
GJ 849 & 109388 & M3.5V & 8.8 & 4.3 & 4.2 & 1.2 & -0.1 & 1.7 & 3.6 & 1.8 & 1.1 & No detection \\
GJ 876 & 113020 & M3.5V & 4.7 & 8.1 & 6.5 & 1.6 & -1.0 & 3.2 & 6.5 & 3.5 & 0.9 & Photosphere at 100$\mu$m \\
\hline
\end{tabular}
\end{table*}
Several other targets were also found to have emission at or near the
source position, but in these cases we do not believe the emission to be
associated with the star in question. These are shown in Figure
\ref{fig:im}.
\begin{itemize}
\item GJ~3634: A bright ($\sim$14mJy) source is seen 6\arcsec~SW of the
expected position of GJ~3634. This offset is larger than expected
given the $\sim$2\arcsec~1$\sigma$ pointing accuracy of
\emph{Herschel}\footnote{\href{http://herschel.esac.esa.int/Docs/Herschel/html/ch02s04.html}{http://herschel.esac.esa.int/Docs/Herschel/html/ch02s04.html}}
and our small sample size. By comparing the positions of several other
sources detected in the 100 $\mu$m PACS image with the (optical) DSS2
plates\footnote{\href{https://archive.stsci.edu/dss/}{https://archive.stsci.edu/dss/}}
we found that three were almost perfectly coincident. Thus, we
conclude that the 6\arcsec~offset seen is real, and that the PACS
detection near GJ~3634 is not associated with this star.
\item HIP~79431: Extended structure is seen to the North of the stellar
position, but the peak is 5\arcsec~away. Only one low S/N source was
seen to be common between the PACS and DSS2 images, with perfect
coincidence. The background as seen in IRAS and WISE images is complex
and variable. We conclude that the large offset and high background
mean that the detected source is unlikely to be associated with
HIP~79431.
\item GJ~674: The background level around GJ~674 is significantly above
zero. At 100 $\mu$m the flux in the image peaks at the position of
GJ~674, but if a point source with the photospheric flux of GJ~674 is
subtracted the background becomes uniform. Thus, we conclude that the
image shows emission from the star GJ~674 superimposed on a
non-negligible background, and that there is no evidence for excess
emission from the star itself.
\end{itemize}
\section{Discussion}\label{s:disc}
Our survey finds two new excess detections, around the stars GJ~433 and
GJ~649. We first consider these detections as part of our sample, and
then take a closer look at the architecture of these two systems in more
detail.
\subsection{Planet - disk correlation}\label{ss:corr}
One of our goals was to test for a correlation between the brightness of
debris disks around low-mass stars and the presence of planets. That is,
all stars may host debris disks, but we can only detect those above a
given dust level, so we cannot test for a correlation between the
`existence' of planets and disks. The same is true for planet detection
of course, so we are in fact testing for a correlation between disks
above a given brightness threshold and planets above a given semi-major
axis vs. mass threshold (acknowledging that the star-to-star sensitivity
also varies). These thresholds are discussed below.
A significant correlation has been seen among Sun-like stars that host
radial velocity planets \citep{2014prpl.conf..521M}, and tentative
evidence that this trend is stronger for stars that host low-mass
planets was found among a small sample of nearby stars
\citep{2012MNRAS.424.1206W,2014A&A...565A..15M}. No clear trends were
seen in the volume-limited DEBRIS FGK-type sample considered by
\citet{2015ApJ...801..143M}, illustrating the tentative nature of the
latter trend, and that samples that do not specifically target
planet-host stars suffer from small numbers of planet hosts that limit
the power to discover trends.
Here, our sample comprises 21 planet-hosting low-mass stars that were
observed in search of IR excesses by \emph{Herschel}, for which three
were found to host disks. Thus, our detection rate is 14\%, but clearly
suffers from a small number of detections. As a control sample, we
consider the volume-limited DEBRIS M-type sample, which comprises 89
nearby stars \citep{2010MNRAS.403.1089P}. Of these, two were discovered
to host debris disks; the planet host GJ~581 \citep{2012A&A...548A..86L}
and the third star in the very wide Fomalhaut triple system, Fomalhaut~C
\citep{2014MNRAS.438L..96K}. We remove GJ~581 and the four other
planet-host stars from this sample, leaving 84 stars with one disk
detection, a rate of 1.2\%.
A Fisher's exact test to determine whether these two populations could
arise from the same underlying distribution yields a $p$-value of 0.025,
thus showing reasonable evidence that the planet-host stars have a
tendency to have more detectable (i.e. brighter) debris disks. The
Fomalhaut system is known to be relatively young, at 440Myr
\citep{2012ApJ...754L..20M}; if we were to assume that all of the planet
host systems are older than this and exclude Fomalhaut~C from the
control sample the $p$-value decreases to 0.01. However, we cannot be
sure that the planet-host stars are all older than the Fomalhaut system,
since for example GJ~674 may also be a relatively young system
\citep{2007A&A...474..293B}.
\begin{figure*}
\begin{center}
\hspace{-0.5cm} \includegraphics[width=0.48\textwidth]{figs/det-lims-mskarps.eps}
\hspace{-0.cm} \includegraphics[width=0.48\textwidth]{figs/det-lims-debris.eps}
\caption{Detection space for our sample (\emph{left panel}) and the
control sample (\emph{right panel}). Contours show the number of
stars for which a disk of a given fractional luminosity and
temperature could have been detected. The upper and lower red
contours show where disks around all, and one, systems could have
been detected. The intermediate curves are for 75, 50, and 25\% of
systems. The difference in sensitivity between our sample and the
DEBRIS control sample is a factor of a few.}\label{fig:detlims}
\end{center}
\end{figure*}
Thus, we find suggestive evidence that debris disks are more easily
detected around M-type stars that also host planets. A further
consideration however is whether the observations are biased towards
detections for the planet-host sample. This might be expected given that
our noise level is about half that of the DEBRIS observations of the
control sample, but might also be balanced by the fact that all DEBRIS
M-type stars are within 10pc, and thus on average closer than our
planet-host stars.
The relative sensitivities for the two samples is shown in Figure
\ref{fig:detlims}, where the grey scale shows the number of systems for
which disks at a given temperature and above a certain fractional
luminosity ($f = L_{\rm disk}/L_\star$) could have been detected. The
lowest red contour shows the maximum sensitivity (disks that could have
been detected around only one star), the highest shows the level above
which disks could have been detected around all stars, and the
intermediate contours show where disks could have been detected around
25, 50, and 75\% of systems. By comparing the red contours it can be
seen that our observations could typically detect disks that are a
factor of two to three lower in fractional luminosity than those
observed by DEBRIS (as expected from observations that are 2-3 times
deeper). While the three disks around planet-host stars could have been
detected around 75\% of our sample, they could only have been detected
around about 30\% of the DEBRIS sample. Thus, the evidence for any
correlation between planets and debris disk brightness is weaker than
suggested by the $p$-value above.
The significance of the $p$-value may be further reduced by future
radial velocity observations, because an implicit assumption is that the
stars in the control sample do not host planets in a similar parameter
space range as those around our planet-host sample. This is unlikely to
be true because not all systems in our control sample will have been
observed in search of planets, and our control sample is best termed
`stars with no known planets'. If any of the systems in the control
sample that do not host disks were in fact found to host planets, the
significance of our result would decrease further. If however
Fomalhaut~C were found to host a planet (and a search may be well
motivated by our results), the significance would increase.
As noted earlier, it is not yet known whether M-type stars host a disk
population that is the same or different to those that orbit Sun-like
stars, and a major problem is that obtaining comparably sensitive
observations is challenging. This sensitivity difference can be seen by
comparing the contours in the right panel of Figure \ref{fig:detlims}
with those in Figure 4 of Sibthorpe et al. (2017, MNRAS in press), which
shows the sensitivity for FGK-type stars observed as part of the DEBRIS
survey (and for which an FGK-type disk detection rate of 17\% was
obtained). The 50\% contour for our survey is at best about
$f = 5 \times 10^{-6}$, an order of magnitude better than achieved by
DEBRIS for M-type stars. In comparison, our survey is about midway
between the two in terms of sensitivity. Therefore, with the caveats
that the number of detections is small, and that the results could be
biased by a planet-disk correlation, the fact that we have here obtained
a disk detection rate similar to that seen for Sun-like stars suggests
that in surveys of equal sensitivity in fractional luminosity the disk
detection rate among Sun-like and M-type stars should be approximately
the same.
\subsection{A marginally resolved disk around GJ 649}\label{ss:649}
\begin{figure*}
\begin{center}
\hspace{-0.5cm} \includegraphics[width=0.48\textwidth]{figs/CD-31-9113.eps}
\hspace{0.5cm}
\hspace{-0.5cm} \includegraphics[width=0.48\textwidth]{figs/BD+25-3173.eps}
\caption{Flux distributions showing the disk detections for GJ~433
(left panel) and GJ~649 (right panel). Solid lines show the star
(blue), disk (red), and total (black) models. Black dots and
triangles show measured photometry and upper limits. The best fit
disk temperatures are 30 and 50K, though the large uncertainties
in the 160$\mu$m measurements make these very
uncertain.}\label{fig:seds}
\end{center}
\end{figure*}
GJ~649 (HIP 83043, BD+25~3173, LHS 3257) was reported to host a planet
with a minimum mass similar to Saturn's, in an eccentric 598 day (1.1au)
orbit \citep{2010PASP..122..149J}. The age of the star is uncertain,
though it was classed as a member of the `old disk' (as opposed to the
young disk or halo) based on kinematics \citep{1992ApJS...82..351L}, and
noted to be among the 20\% most chromospherically active early M-type
stars \citep{2010PASP..122..149J}. Using constraints from the disk
temperature and \emph{Herschel} images we can therefore build a picture
of the system's architecture.
The flux density distribution for GJ~649 is shown in Figure
\ref{fig:seds}. The excess flux above the photosphere is modelled using
a modified blackbody function, where the disk spectrum is divided by
$\lambda/210\mu{\rm m}$ beyond 210$\mu$m. This steeper long-wavelength
spectral slope approximates the poor efficiency of dust emission at
wavelengths longer than the grain size, though in this case is not
constrained and included simply in order to make the extrapolations to
millimeter wavelengths more realistic. The main point to take away from
this figure is that the dust thermal emission is very cold, so could not
have been detected in the WISE observations at 22$\mu$m. The best-fit
disk temperature is $50$K with $f = 7 \times 10^{-5}$, but is uncertain
because the 160$\mu$m observation is not formally a 3$\sigma$ detection
of the disk (i.e. Table \ref{tab:fluxes} shows that $\chi_{160}$ for
GJ~649 is 2.9). The non-detection of an excess at 22$\mu$m means that
the temperature cannot be significantly more than 100K.
\begin{figure}
\begin{center}
\hspace{-0.cm} \includegraphics[width=0.45\textwidth]{figs/im100-1-starsub.eps}
\caption{\emph{Herschel} image of GJ~649 after subtracting point
sources near the location of GJ~649 (at the white +) and at the
bright peak to the SE (at the black +, see Figure
\ref{fig:im}). The low level residual structure around GJ~649
provides circumstantial, though not conclusive, evidence, that the
disk is resolved. The asymmetry in the residuals suggests that the
disk position angle is near to North, and that the disk is closer
to edge-on than face-on. White contours are at 1, 2, and 3 times
the 1$\sigma$ noise level. The center of the image is
approximately midway between the plus symbols.}\label{fig:im2}
\end{center}
\end{figure}
Given a stellar luminosity of 0.044$L_\odot$ the best fit temperature of
50K corresponds to a radial distance of 6au if the disk material behaves
as a blackbody, while a temperature of 100K yields a distance of about
2au. Given that most debris disks are comprised of dust small enough to
have super-blackbody temperatures, the disk around GJ~649 would be
expected to be larger than blackbody estimates, by a factor of several
at least
\citep[e.g.][]{2012ApJ...745..147R,2013MNRAS.428.1263B,2014ApJ...792...65P,2016ApJ...831...97M}.
This factor was found to be 6-20 for GJ~581 \citep{2012A&A...548A..86L},
with the large uncertainty arising because the disk radius depends on
the square of the temperature. At a distance of 10.4pc
\citep{2016A&A...595A...4L} the GJ~649 disk may therefore have an
angular diameter large enough to be resolved. This extent may be
confirmed by the \emph{Herschel} images, which at 100$\mu$m show some
extended residual emission after PSF subtraction (see Figure
\ref{fig:im2}). The fact that these residuals are extended in a
non-axisymmetric pattern suggests that the disk may be nearer to edge-on
than face-on, as might be expected given in the case of a planet
detection with the radial velocity technique. Given that most of the
residual contours are only 1$\sigma$ however, we consider that these
residuals provide circumstantial evidence that the disk is resolved, in
which case the disk diameter would be similar to the PACS beam size of
6\arcsec. We therefore conclude that the disk radius could lie in the
range 2-50au, but is more likely to be a few tens of au.
The system layout is shown in Figure \ref{fig:sys}, where the planet
GJ~649~b is indicated by the dot, and the error bar indicates the range
of radii covered by the eccentric orbit. The solid line shows limits
estimated based on the radial velocity residuals once the best-fit
planet orbit is subtracted,\footnote{The inner part of this limit can be
derived using Kepler's laws and the residual noise in the RV data once
the planet(s) have been subtracted, but the steeper outer part where
the orbital period is longer than the span of observations was
empirically estimated from full simulations of radial velocity
sensitivity \citep[e.g.][]{2015MNRAS.449.3121K}} indicating that
planets more massive than Saturn that orbit beyond about 5au would not
have been detected. The range of estimated disk locations is shown by
the hatched region, where we have taken the marginally resolved image to
indicate that the disk has a radius between 10-30au. The basic
conclusion is that while the separation between the planet and disk is
probably large, it is possible that this gap is occupied by one or more
undetected planets. A further conclusion is that lower mass planets at
smaller radii could have been detected, though the sensitivity is a
factor of two poorer than for the other systems discussed below.
\subsection{An unresolved disk around GJ~433}\label{ss:433}
GJ~433 (HIP~56528, LHS~2429) was reported to host a low-mass planet
GJ~433~b ($M \sin i = 5.8 M_\oplus$) on a 7.4 day period at 0.058 au
\citep{2013A&A...553A...8D}. They detected an additional significant
signal with a much longer period of 10 years (3.6au), but based on the
variation of activity indices on a similar timescale
\citep{2011A&A...534A..30G}, concluded that a magnetic cycle of the star
was a more likely origin. The same signals were recovered by
\citet{2014MNRAS.441.1545T}, who considered the second signal to be a
candidate planet. Given the uncertain nature of the outer planet we do
not include it here. The age of GJ~433 is uncertain, but the dynamical,
x-ray, and Ca~II emission properties show that the star is not young
\citep{2013A&A...553A...8D}.
As above we can constrain the disk location relative to the planet's,
but in the case of GJ~433 there is no clear evidence that the disk is
resolved with \emph{Herschel}. The best fit disk temperature is $30$K
(see Figure \ref{fig:seds}, but again the temperature is poorly
constrained by a weak detection at 160$\mu$m, and could be as warm as
100K. The fractional luminosity is also poorly constrained, but is
approximately $2.5 \times 10^{-5}$. For the stellar luminosity of
0.033$L_\odot$ a disk temperature range from 100 to 30K yields a
blackbody radius range of about 1 to 16au, or about 0.2 to 3.5\arcsec\
diameter at the 9.1pc distance of the system. As for GJ~649, the disk
structure as seen at 100$\mu$m can constrain the disk extent to less
than the PACS beam size, but as with GJ~649 only limits the disk radius
to less than about 30au, and does not constrain the inclination or
position angle.
The system layout is shown in Figure \ref{fig:sys}. While the
observational limits on the disk radius are poor, a radius of 1au would
make GJ~433 host to an unusually small disk \citep{2007ApJ...658..569W},
so it seems most likely that the disk extent is similar to that expected
for GJ~649. If this is indeed the case, there is again space for
undetected planets in the region between the known planet and the disk.
\subsection{Summary of system architectures}\label{ss:arch}
\begin{figure}
\begin{center}
\hspace{-0.5cm} \includegraphics[width=0.48\textwidth]{figs/sys.eps}
\caption{Mass semi-major axis diagrams showing the GJ~433, GJ~581,
and GJ~649 planets (dots), the approximate RV sensitivity (lines),
and the possible range of disk locations (hatched regions, showing
the disk extent in the case of GJ~581). GJ~581~e lies below the
sensitivity curve because the RV amplitude (1.7 m s$^{-1}$) is
smaller than the RMS (2.12 m s$^{-1}$) reported by
\citet{2014Sci...345..440R}. In each case, with the possible
exception of GJ~433, there remains room in the detection space for
sizeable planets that reside between the known planets and the
disk, but that could not have been detected with the current RV
observations.}\label{fig:sys}
\end{center}
\end{figure}
Figure \ref{fig:sys} summarises the architecture of the planet-host
systems in our sample, and includes the multi-planet system GJ~581. The
number of planets residing in this system is contentious, and stellar
activity has been proposed as the cause of some of the periodic signals
seen; here we show the three planets proposed by
\citet{2014Sci...345..440R}, and the hatched disk region shows the
extent of the disk derived by \citet{2012A&A...548A..86L}. As with
GJ~433 and GJ~649, there is space for undetected planets in the
intervening region.
Given the lack of strong evidence for any correlation between the
presence of planets and debris disk brightness, we should not
necessarily expect clear trends when looking at plots such as Figure
\ref{fig:sys}. We might however note trends that are glossed over by a
simple disk brightness metric, such as tendencies for systems to show
particular architectures or scales. Again noting that a disk as small as
1au around GJ~433 would be very unusual, the radii of the disks is
consistent with being a few tens of au. However, this size is also
inferred for the disk that orbits Fomalhaut~C
\citep{2014MNRAS.438L..96K}, so there is no evidence that this
preference is related to the presence of planets. Indeed, this radius
range is also preferred for disks around FGK-stars, independent of
whether planets are known (Sibthorpe et al. 2017).
There is no obvious link between the disks and the layout of the planets
that orbit closer in, but in each case there remains room in the
detection space for sizeable planets that reside between the known
planets and the disk, but that could not have been detected with the
current RV observations. In this regard the M-type planet + disk systems
appear to be analogues of Sun-like planet + disk systems such as
HD~20794, HD~38858, and 61 Vir
\citep{2012MNRAS.424.1206W,2015MNRAS.449.3121K}. This similarity may
however simply reflect that detecting long period planets takes time,
and that small debris disks grind down to undetectable levels more
rapidly than large ones, and that these biases are present regardless of
the mass of the host star. That is, there may be differences in the
architectures of planetary systems across different spectral types, but
that this difference is in the type or existence of planets that reside
near 10au. For further discussion of planet formation scenarios, we
refer the reader to \citet{2012MNRAS.424.1206W},
\citet{2015MNRAS.449.3121K}, and \citet{2017MNRAS.469.3518M}.
The very cool disk temperatures shown in Figure \ref{fig:seds} make it
clear that progress in our understanding of these disks, and the links
with the planets, can only be made by far infrared and millimeter-wave
observations. The present observations are hindered by the low spatial
resolution of \emph{Herschel}, which means that we are constrained to
estimating disk locations. With no far infrared missions on the near
horizon, and an expectation of sub-mJy disk flux densities, observations
with the Atacama Large Millimeter Array (ALMA) are the main avenue for
progress. These will be challenging, but necessary to obtain further
discoveries, and in cases such as GJ~433, GJ~581, and GJ~649 could
provide higher resolution images that instead of yielding disk location
estimates, will allow the discussion of disk structure.
\section{Conclusions}\label{s:conc}
This paper presents the results of a \emph{Herschel} survey of 21 nearby
late-type stars that host planets discovered by the radial velocity
technique. These observations were obtained with the aim of discovering
new debris disks in these systems, and in search of any correlation
between planet presence and disk brightness.
We report the discovery of two previously undetected disks, residing at
a few tens of au around the stars GJ~433 and GJ~649. The disk around
GJ~649 appears marginally resolved and more consistent with being viewed
edge-on. Despite uncertainty in their radii these disks orbit well
beyond the known planets, and it is possible that other as-yet
undetected planets reside in the intervening regions. The layout of
these systems therefore appears similar to star + disk systems around
Sun-like stars such as HD~20794, HD~38858, and 61~Vir. Estimating the
ages of M-type stars is challenging, but neither star shows evidence of
youth, so there is no evidence that the ages of these stars are special
compared to the rest of the sample.
Including the previously known disk around GJ~581, our sample comprises
three planet hosts with disks, a detection rate of 14\%. While this rate
is higher than for a control sample of M-type stars without reported
planets observed by the DEBRIS survey (1 out of 84 stars), the
difference is only significant at 98\% confidence. This evidence is
further shown to be optimistic, because the observations of the
planet-host sample were somewhat more sensitive to debris disks than
those in the control sample, and because not all systems in the control
sample have been searched for planets (or reported not to have planets
above some detection threshold).
Though this survey represents an improvement over previous surveys of
M-tye stars, the fractional luminosity sensitivity achieved remains
about a factor of three poorer than similar surveys of Sun-like
stars. Nevertheless, the fact that we find disks around 14\% of M-type
stars, in comparison to 17\% of Sun-like stars, provides circumstantial
evidence that there is no difference in their disk populations.
\section{Acknowledgements}
We thank the referee for a useful report. GMK is supported by the Royal
Society as a Royal Society University Research Fellow. This work was
supported by the European Union through ERC grant number 279973 (GMK \&
MCW).
The Digitized Sky Survey was produced at the Space Telescope Science
Institute under U.S. Government grant NAG W-2166. The images of these
surveys are based on photographic data obtained using the Oschin Schmidt
Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates
were processed into the present compressed digital form with the
permission of these institutions.
|
{
"timestamp": "2018-03-09T02:00:16",
"yymm": "1803",
"arxiv_id": "1803.02832",
"language": "en",
"url": "https://arxiv.org/abs/1803.02832"
}
|
\section{Introduction}
Owing to the deep potential well of the Galactic Center (GC),
all aspects of the environment are found at extreme conditions compared to what is seen in the Galactic spiral arms:
the dense nuclear star cluster at whose center lies a $4 \times 10^6$ M$_\odot$ black hole
at a distance of {approximately 8} kpc ({Reid et al. 2014; Boehle et al. 2016,} Bland-Hawthorn \& Gerhard 2016),
massive dense molecular clouds (residing in the so-called the Central Molecular Zone, CMZ, Morris \& Serabyn 1996),
and a radiation field including both synchrotron radiation (Yusef-Zadeh, Hewitt, \& Cotton 2004)
and substantial X-rays (e.g., Ponti et al. 2015).
Figure 1 shows the prominent features of the GC that will be discussed in this paper:
blue, green, and red indicate the 21 \micron\ emission observed by the {\it MidCourse Space Experiment}
({\it MSX}, Price et al. 2001) and
the 70 and 500 \micron\ emission from {\it Herschel} (Molinari et al. 2011).
The 21 and 70 \micron\ emission shows the warm dust heated by nearby massive stars;
it consequently delineates the star-forming regions Sgr B and Sgr C and the dust heated by the massive star clusters
known as the Quintuplet and Arches Clusters as well as the nucleus of the Galaxy itself, Sgr A
(Morris \& Serabyn 1996).
The far-infrared emission from the cold dust shows the locations of the dense molecular clouds that
have very few indicators of active star formation, even though their overall masses and densities
seem to be adequately high.
These clouds have the appearance of a lopsided ring of gas and dust
circling the center (Molinari et al. 2011).
\begin{figure*}
\includegraphics[width=173mm]{fig1.eps}
\caption{Three-color image of the Galactic Center.
Blue is the 21\micron\ Band E {\it MSX} image from Price et al. (2001),
green is the 70 \micron\ image from
Hi-GAL (Molinari et al. 2011) taken with the Photodetector Array Camera and Spectrometer
(PACS, Poglitsch et al. 2010) on {\it Herschel Space Observatory} (Pilbratt et al. 2010),
and red is the 500 \micron\ image from Hi-GAL taken with the
Spectral and Photometric Imaging Receiver (SPIRE, Griffin et al. 2010).
}
\end{figure*}
Longmore et al. (2013) showed that if one describes the positions of the molecular clouds
at positive Galactic longitudes as lying on the Earth-side of the cloud orbit,
the amount of star formation in the clouds increases as a function of time since the clouds
passed the pericenter of the Galaxy's exact center, Sgr A.
They suggested that passing the pericenter compresses the clouds, thereby triggering stars to form.
Kruijssen, Dale, \& Longmore (2015) then showed from the radial velocities of the clouds
that the gas orbit could not be closed; instead the cloud motions can be better portrayed
as streams of gas in orbit about Sgr A.
Modeling the orbits, they estimated times since pericenter passage of
0.30 Myr for the Brick, 0.74 Myr for Sgr B2, and 3.58 Myr for Sgr C (Figure~1).
The Arches and Quintuplet Clusters formed from gas clouds that passed the pericenter
at even earlier times.
Krumholz \& Kruijssen (2015) and Krumholz, Kruijssen, \& Crocker (2017)
expanded the discussion of gas streams in the GC to include the effects of the Milky Way's Bar.
Gas flows in along the Bar and is highly turbulent within the inner Lindblad resonance
at $\sim 1000$~kpc from the center owing to the high shear.
However, the turbulence dissipates at $\sim 100$~pc (the edge of the CMZ)
where the rotation curve becomes close to solid-body,
thereby allowing the massive molecular clouds to form with resulting star formation.
This processs is episodic because feedback from winds from the stars and supernovae
blow much of the gas out, greatly reducing the star formation rate.
Krumholz et al. (2017) suggest that the GC is currently in a low star-formation state,
the last major star formation having occurred $\sim 8$~Myr ago.
This explains the apparent low star-formation rate of the GC (Kruijssen et al. 2014).
This is an exciting model because of the wealth of important observations that it explains.
However, there are still uncertainties with the details of this model.
The model posits that the positions in the foreground of Sgr A at positive longitudes
are all very young, $< 1$~Myr,
with Sgr B1 (already on the back loop of the orbit) somewhat but not greatly older than Sgr B2
(1.5 Myr vs. 0.7 Myr, Barnes et al. 2017)
and Sgr C closing in on its second pass by the pericenter (Sgr A).
However, the appearance of Sgr B1 is that of an \ion{H}{2} region in the process of dispersal,
indicating a much older age than that of Sgr B2,
even though both it and Sgr C contain substantial numbers of young stellar object (YSO) candidates
(An et al. 2011; Lu et al. 2017).
There are additional suggestions for the causes of the low star-formation rate in the GC:
Federrath et al. (2016) suggest that solenoidal driving of the turbulence owing to the high shear in the GC
compared to the compressive turbulence in spiral arm star forming regions is the cause
of the low star formation rate (see also Kruijssen et al. 2014).
High levels of turbulence in the GC is the usual suggestion for the high gas temperature
(line widths much larger than thermal)
in molecular clouds compared to the dust temperature (e.g., Immer et al. 2016).
In addition, recent higher resolution observations indicate that
the temperature and density conditions in the GC gas clouds
are not as simple as the previous theories assumed:
Kauffmann et al. (2017a) observe that although the line widths are large when integrated
over whole clouds, the line widths for individual clumps within a cloud can be of normal width
(compared to spiral arm molecular clouds).
Moreover, Kauffmann et al. (2017b) find that the densities in the clumps do not increase fast enough
towards their centers to be able to initiate star formation
and that there is no clear trend of increased and then decreased star formation
as predicted by the Kruijssen et al. (2015) model.
They suggest that initial conditions may be as important as position on the orbital path since pericenter
and that data on more clouds are needed to test the model.
Further insight can be ascertained from mid-infrared (MIR) spectroscopy.
In typical observing programs, sources are first identified with imaging
on either the {\it Infrared Space Observatory}'s (ISO) ISOGAL survey (Schuller et al. 2006)
or {\it Spitzer Space Telescope}'s (Werner et al. 2004) Infrared Array Camera (IRAC, Fazio et al. 2004);
then spectra are taken with the Infrared Spectrograph (IRS, Houck et al. 2004).
The IRS has four modules: short-low, SL, with resolution $R \sim 100$ and range 5.2 -- 14.5 \micron,
long-low, LL, with $R \sim 100$ and spectral range 14.0 -- 37 \micron,
short-high, SH, with $R \sim 600$ and range 9.9 -- 19.6 \micron, and
long-high, LH, with $R \sim 600$ and range 19 -- 36 \micron.
In spectra from 5 -- 20 \micron\ one can identify young stellar objects (YSOs)
from the presence of ice features (e.g., Simpson et al. 2012)
and organic molecules evaporated from ice (e.g., An et al. 2009)
in absorption in their dusty envelopes.
From the ionic abundances of a number of elements of differing ionization potential ($IP$),
one can determine the ionization structure of a region, get probable locations of
the exciting stars, and, if the cluster of stars that produce the energetic photons is large enough to
include a representative sample of the initial mass function, get an estimate of the age of the cluster.
A number of {\it Spitzer} programs observed targets in the GC with the IRS.
Immer et al. (2012) found 14 YSOs from IRS spectra of 68 ISOGAL sources, based on the shapes
of their spectral energy distributions (SEDs) and the presence or lack of
polycyclic aromatic hydrocarbon (PAH) features or atomic fine structure lines characteristic of \ion{H}{2} regions.
Out of 107 candidates, An et al. (2011) found 16 YSOs and 19 possible YSOs from the presence of
methanol mixed with the CO$_2$ ice in their deep 15 \micron\ absorption features.
Note that these same candidate YSOs also have deep ice features at 5.8 and 6.8 \micron,
features that are also characteristic of YSOs (e.g., Simpson et al. 2012 and references therein).
An, Ram\'irez, \& Sellgren (2013) used the SH and LH background spectra from the same data set
to test whether the GC is ionized by flux from an active galactic nucleus (AGN);
by comparing their flux ratios to the external galaxy fluxes measured by Dale et al. (2009),
they concluded that there is no significant AGN activity in the GC.
In Simpson et al. (2007) we analyzed SH and LH spectra of the Arched Filaments, Arches Cluster,
regions near the Quintuplet Cluster, and the Radio Arc Bubble.
In it we showed that the Arches Cluster is the source of the ionizing photons of the Arched Filaments,
and the Quintuplet Cluster is likewise the source of the ionizing photons of the Bubble.
The Quintuplet Cluster also ionizes the rim of the Bubble at its lowest Galactic latitudes,
from which we inferred that the G0.10$-$0.08 molecular cloud is totally unrelated along the line of sight.
From measurements of the Fe/Ne ratio we concluded that
strong shocks have contributed to destroying grains in the Bubble,
thereby increasing the abundance of gas-phase iron compared to abundances
in the Arched Filaments and the Bubble rim.
Simpson et al. (2007) also found that
that even though the SED of the photons ionizing the observed gas
could be described as having the effective temperature, $T_{\rm eff}$, of $\sim 36,000$~K
(late O, Sternberg et al. 2003),
essentially all the observed positions contained detectable [\ion{O}{4}] 25.9 \micron\ lines.
This is surprising because
O$^{3+}$ has an $IP = 54.9$ eV and this ionization stage is only observed in \ion{H}{2} regions
ionized by the most massive O stars, and then only by stars with low metallicity,
which the GC is not
(O$^{3+}$ is sometimes detected in the \ion{H}{2} regions of low metallicity dwarf galaxies, Lutz et al. 1998).
In fact, An et al. (2013) showed that the [\ion{O}{4}] line is observed at many positions in the GC;
nhey used its presence compared to its immediate (and partially blended) neighbor [\ion{Fe}{2}] 26.0 \micron\
to distinguish starbursts from LINERS (low-ionization nuclear emission-line regions) or Seyferts
by their relative excitation.
In this paper I address why there is such a dichotomy between
the generally observed, very low excitation gas interspersed with quite high excitation gas.
The {\it Spitzer} Heritage Archive contains a great deal more data of the same high quality.
Although both the SH and LH modules have compact apertures, measuring barely more
than one spatial resolution element in both directions,
the SL and LL modules are long slit, with slit lengths $57''$ and $168''$, respectively.
Moreover, both SL and LL have two grating orders each, displaced by $22''$ or $24''$ on the sky,
such that when one is observing some source in one order, the other order is taking data
a slit length away.
The result is that although the papers referenced above analyze the point source targets of each program,
a substantial amount of the GC was actually covered when one includes
the full slit lengths of the SL and LL modules and both orders, the order
planned for the astronomical observing request (AOR) and the `other' order,
observed but not normally analyzed.
No one to this time has analyzed any of that data!
This is the first of several papers in which we analyze the IRS spectra of all four modules from programs
0018 (PI: J. Houck), 3121 (PI: K. Kraemer), 3189 (PI: F. Schuller),
3295 (PI: J. Simpson), 3616 (PI: J. Chiar), and 40230 (PI: S. Ram\'irez).
Subsequent papers will discuss individual objects such as Sgr A and Sgr B1 (J. Simpson et al., in preparation),
and exceptionally-highly excited sources that could be shocks (e.g., Allen et al. 2008)
or candidate planetary nebulae (D. An et al., in preparation).
Section 2 describes the data and its analysis,
Section 3 discusses the results, including the observation that [\ion{O}{4}] line emission is
prevalent over the whole GC,
Section 4 presents \ion{H}{2} region models computed with the code Cloudy {(Ferland et al. 2017)}
and compares them to the observed line ratios to estimate the parameters of the
exciting star clusters,
and Section 5 presents the summary and conclusions.
\section{Observations}
\input tab1.tex
The observations were taken with the {\it Spitzer} IRS in Cycles 1 and 4 and were
downloaded from the {\it Spitzer} Heritage Archive\footnote{sha.ipac.caltech.edu/applications/Spitzer/SHA/}.
The spectra from programs 3295 (line fluxes published in Simpson et al. 2007) and 0018
(mentioned in Simpson et al. but published in this paper)
were reduced and calibrated with the {\it Spitzer} S13.2 pipeline;
all the other data were reduced and calibrated with the final version of the pipeline, S18.18.
SH and LH spectra are found in programs 0018, 3295, and 40230, and
SL and LL spectra are found in programs 3121, 3189, 3616, and 40230.
The data reduction subsequent to the pipeline for the SH and LH spectra from program 40230
follows that described by Simpson et al. (2007) for program 3295 (and unpublished 0018):
the basic calibrated data (bcds) for each telescope pointing
(all program 40230 spectra were taken in staring mode)
were median combined and cleaned of rogue and exceptionally noisy pixels
(rejecting all spectra where the bcds show `jailbars').
Background subtraction was not performed because no background spectra at large enough distances
from the GC were ever taken for any of these programs and the GC line-of-sight itself
has at least some emission at all locations, as will be discussed later.
{Sample SH and LH spectra are shown in figure 2 of Simpson et al. (2007).}
CUBISM (Smith et al. 2007a) was used to extract the spectra of {both} the low resolution SL and LL
{bcds and the high resolution SH and LH bcds}.
CUBISM was written to analyze the spectral maps of galaxies in the
SIRTF Nearby Galaxies Survey (SINGS) program (Kennicutt et al. 2003);
the software takes the bcds of the map and
produces a three-dimensional cube of x, y, and wavelength,
where x and y refer to coordinates parallel and perpendicular to the
{various slits.}
The GC programs observed in both mapping (multiple adjacent slit positions on the sky)
and staring (single slit placed on a requested target) modes.
{ For the low resolution modules,}
after cleaning the bad pixels,
CUBISM was used to extract spectra from both modes at multiple positions along the slit,
both for the order that had the requested target and also for the `other' order a slit length
(plus $22''$ or $24''$) distant.
For staring mode (or for mapping mode where the multiple slit positions did not touch),
the x-y slit in the CUBISM GUI appears as a line of about 32 pixels long and 2 pixels wide,
where the pixel size is $1.8''$ for SL and $5.1''$ for LL; the physical slit widths
and spatial resolution are approximately 2 pixels for all modules.
The extraction boxes were 3 pixels by 2 pixels for the single slits
(or sometimes 3 by 3 pixels for the small maps of program 3189)
spaced by 2 pixels along the slit, thus producing typically 15 spectra per slit per order.
Care was taken to have the extraction boxes of the two SL or LL orders overlap on the sky
so that the spectra could be joined into single 5.3 -- 14.5 \micron\ or 14 -- 37 \micron\ spectra.
However, since the SL and LL slits are close to orthogonal on the sky,
spectra from the two modules could not be joined except at the target positions.
See figure 2 of An et al. (2011) for the layout of the slits as used by program 40230.
Not all SL pointings could produce usable spectra because the IRS peak-up cameras
are on the same module and saturation in the peak-up arrays produced uncalibratable spectra.
{For the high resolution modules, the CUBISM extraction boxes were
5 pixels by 2 pixels for SH and 3 pixels by 2 pixels for LH.}
Line and continuum intensities were obtained from all spectra by fitting Gaussian profiles
and a sloping continuum.
The lines measured are given in Table 1.
The uncertainties for the line fluxes were estimated from the rms deviation of the data
from the fit.
Because there were hundreds of SH and LH spectra and thousands of SL and LL spectra,
this fitting had to be done with an automated line fitting program without hand checking,
using a fixed set of wavelengths for each line such that there would be no interference
with another line or PAH feature.
{A tool in the Spectroscopic Modeling Analysis and Reduction Tool
(SMART, Higdon et al. 2004) can also be used to fit Gaussian line profiles to spectra,
both single lines and partially blended lines with multiple Gaussians.}
Because the weak H 7--6 lines are crucial to the estimation of abundances with respect to hydrogen
and because the baseline for this line
can be affected by the close, usually stronger H$_2$ S(2) line,
they were measured by hand with SMART in the SH spectra.
Additionally, those lines that are known to be blends were measured by hand with SMART:
[\ion{O}{4}] 25.9 and [\ion{Fe}{2}] 26.0 \micron, [\ion{P}{3}] 17.89 \micron\ and [\ion{Fe}{2}] 17.94 \micron,
and, rarely, [\ion{Ne}{5}] 14.32 and [\ion{Cl}{2}] 14.37 \micron, all in the high resolution spectra.
The first two blends for program 3295 are plotted in figure 10 of Simpson et al. (2007).
Tests of measuring the same line with both the automated line fitting program and with SMART
show a difference in intensities of only a few percent for lines with signal/noise $> 3$
and less than $2\%$ for bright lines,
the difference being due to the flexible choice of baseline wavelengths with SMART
versus the uniform baseline wavelengths with the automatic measuring program.
{The weak H 7-6 recombination line exhibits a larger difference in intensities with a standard deviation
(SD) of $\sim 6\%$.}
However, whereas hand measurement with SMART could always identify those cases where
no line could be fit, that was not the case for the automated Gaussian fitting program,
which always found some sort of fit, good or bad.
Consequently, its bad fits were identified as having
negative fluxes, too big or too small line widths, or extreme radial velocities
and were removed from the data set.
{The SH and LH spectra from program 40230 were also extracted with SMART.
This presents a good test of the reliability of spectral extraction with the various extraction software programs.
The spectra are dissimilar enough that the estimated line intensities differ by
typically 4 or 5\% (SD) for the brighter lines and up to 14\% (SD) for the weaker lines, such as the H 7-6
recombination line at 12.37 \micron.
These systematic uncertainties are included in quadrature with the measured rms uncertainties
from the Gaussian fits for the computation of abundances in the next section.
For the sake of consistency, almost all line intensities in this paper
are from spectra extracted with CUBISM.
The exceptions are those from the previously described SH and LH spectra from programs 0018 and 3295,
the blended line pair [P III] 17.89 and [Fe II] 17.94 \micron\ (e.g., Simpson et al. 2007),
and the quite faint [S I] 25.25 \micron\ line, which line is often obscured by the bad fringing in the LH spectra
that can be removed by SMART but not by CUBISM (the absolute intensities of this line are equal to the
intensities measured from the SMART spectra times a scale factor computed from the
other lines measured from CUBISM-extracted spectra).
}
\section{Results}
\subsection{Line Intensities}
\begin{figure*}
\includegraphics[width=184mm]{fig2_abc.eps}
\caption{Logarithms of the intensities of the observed lines, grayscale from {\it MSX} Band E.
The color scale is given by the bar on the right.
Crosses mark the positions of the SH or LH apertures from Cycle 1 (programs 0018 and 3295)
and diamonds mark the positions of the SH or LH apertures from Cycle 4 (program 40230).
The locations of the LL or SL apertures are drawn as extracted by CUBISM.
(a) The intensity of the [\ion{S}{3}] 19 \micron\ line.
(b) The intensity of the [\ion{S}{3}] 33 \micron\ line.
The positions that have the [\ion{S}{3}] 19 \micron\ line overlapping with the 33 \micron\ line
show the slit locations from the observing plan AORs;
the positions that do not overlap are the other order on the LL arrays.
(c) The intensity of the [\ion{Si}{2}] 34 \micron\ line.
}
\end{figure*}
\setcounter{figure}{1}
\begin{figure*}
\includegraphics[width=184mm]{fig2_def.eps}
\caption{{\it Continued.}
(d) The intensity of the [\ion{Ne}{2}] 12.8 \micron\ line.
(e) The intensity of the [\ion{Ne}{3}] 15.6 \micron\ line.
(f) The intensity of the [\ion{O}{4}] 26 \micron\ line.
Positions colored white are the locations of LH observations where the [\ion{O}{4}] 26 \micron\ line was not detected.
Positions colored dark blue are where the signal/noise for the [\ion{O}{4}] 26 \micron\ line is $< 2$.
Note that for those positions with low S/N or no measurements,
the $2\sigma$ uncertainties, which are approximately proportional to the continuum intensity,
are usually similar to the fluxes of the nearby positions with good detections.
The positions colored magenta at the top of the color scale in panels (e) and (f) may have intensities
much larger than $9.65 \times 10^{-17}$ or $5.6 \times 10^{-18}$ W m$^{-2}$ arcsec$^{-2}$, respectively ---
these are the locations of the highly excited positions in Tables 8 and 9.
}
\end{figure*}
\setcounter{figure}{1}
\begin{figure*}
\includegraphics[width=184mm]{fig2_ghi.eps}
\caption{{\it Continued.}
(g) The intensity of the H$_2$ S(0) 28.2 \micron\ line.
(h) The intensity of the H$_2$ S(1) 17.0 \micron\ line.
(i) The intensity of the H$_2$ S(2) 12.28 \micron\ line.
}
\end{figure*}
The observed line intensities are given in Tables 2 -- 7
for modules SL2 (second order), SL1 (first order), SH, LL2 (second order), LL1 (first order), and LH,
respectively, and are plotted in Figure 2.
These tables contain the intensities of the common \ion{H}{2} region lines plus [\ion{O}{4}] 26 \micron,
which is detected widely in the GC although it is not an \ion{H}{2} region line.
The apertures used in the extraction of the SL or LL spectra are given in the tables.
{
The apertures for the CUBISM-extracted spectra for SH and LH are $\sim 51$
and $\sim 119$ arcsec$^2$, respectively,
and the apertures for the programs 0018 and 3295 spectra that were extracted with SMART
by Simpson et al. (2007)
are assumed to be the IRS's nominal 53.11 and 247.5 arcsec$^2$
for SH and LH, respectively (Houck et al. 2004).
The latter SH and LH line intensities were also corrected for slitloss (telescope and instrument diffraction),
necessary because SMART assumed the spectra being extracted are from point sources and the GC emission
is extended (the default for CUBISM is to assume extended emission).
}
The units of the intensities from CUBISM (MJy sr$^{-1}$)
were adjusted to be W m$^{-2}$ s$^{-1}$ arcsec$^{-2}$ from W m$^{-2}$ s$^{-1}$ sr$^{-1}$ for the figures
and W m$^{-2}$ s$^{-1}$ sr$^{-1}$ for the online machine readable tables.
So that the final line intensities could be compared, the SH and LH fluxes were also converted
to W m$^{-2}$ s$^{-1}$ arcsec$^{-2}$ or W m$^{-2}$ s$^{-1}$ sr$^{-1}$.
In addition to the relatively smooth emission from the inner Galaxy,
there are intriguing serendipitous discoveries in this archived GC data set
of locations that are so highly excited that the [\ion{Ne}{5}] lines ($IP = 97 - 126$ eV, Table~1) are strong;
these locations are listed in Table~8.
These locations also have very strong [\ion{Ne}{3}] 15.6 \micron\ and [\ion{O}{4}] 26 \micron\ if
the relevant module was observed (in many cases, the module order with the [\ion{Ne}{5}] line
was the `other' module order from the LL order in the planned AOR and so the observed spectrum
contains only one order of the nominal LL wavelength range).
These highly excited lines require substantial amounts of ionizing photons with much higher energy
than are found in OB stars;
such photons are often emitted by the white dwarf stars that ionize
planetary nebulae or are emitted by active galactic nuclei (e.g., Feuchtgruber et al. 1997),
or are found in the high-energy shocks of supernova remnants (e.g., Sankrit et al. 2014).
These positions will be discussed further in future papers.
Probably in the same classes of exciting sources are found an additional number of positions
that have very strong [\ion{O}{4}] 26 \micron\ but no detected [\ion{Ne}{5}] at either wavelength.
These positions are listed in Table~9.
The references in Tables 8 and 9 are for the radio or Paschen~$\alpha$ sources
found at approximately the same locations on the sky.
A few positions observed with the LH module have detected [\ion{S}{1}] 25.25 \micron.
Because sulfur in the interstellar medium (ISM) is either singly ionized (the $IP$ for S$^+$ is 10.4 eV)
or found in molecules such as SO,
it is generally thought that this line from neutral sulfur is shock-excited (e.g., Hollenbach \& McKee 1989).
This line is seen in two orders in the LH module and must be detected in both orders
to be considered an acceptable measurement.
These positions and line intensities are given in Table~10.
Finally, a few positions seen in order 2 of SL have deep ice absorption features at 6.0 and 6.8 \micron\
(e.g., Boogert et al. 2015).
These candidate YSOs are in addition to those described by Immer et al. (2012) and An et al. (2011)
and are listed in Table 11.
It is seen that the \ion{H}{2} regions marked in Figure 1
are all detected in the ionized lines of Figure 2,
although the intensities of the lines in Sgr B2 are substantially lower
than the line intensities of the other \ion{H}{2} regions.
This is due to the large extinction towards Sgr B2, which has been well known for a long time.
The other \ion{H}{2} regions appear prominently, particularly the Arched Filaments,
the small \ion{H}{2} regions at the base of the Arched Filaments (Zhao et al. 1993; Cotera et al. 2000;
{ Dong et al. 2017}),
the `Sickle' next to the Quintuplet Cluster, the diffuse gas of Sgr A East and
the Bubble Rim, Sgr B2, and Sgr C (see the 20 cm radio survey of Yusef-Zadeh et al. 2004
for images taken at similar resolutions to the {\it Spitzer} spectra).
Note that the [\ion{Si}{2}] 34 \micron\ line displays a pattern much more similar to that of the \ion{H}{2} region lines,
such as [\ion{S}{3}] 33 \micron,
than it does to the spatial appearance of the cold molecular clouds that appear in red in Figure~1.
Although the [\ion{Si}{2}] 34 \micron\ line is often treated as a PDR line (e.g., Kaufman et al. 2006),
in the GC it is at least as much an \ion{H}{2} region line, and so will be treated as such in the following sections.
{
Regarding the high-excitation lines,
although there appears to be a good correlation of the emission line intensities (Figure~2)
with star-forming regions (partly due to the selection bias of the {\it Spitzer} observing programs),
there does not appear to be any correlation of the most highly-excited gas
with any of the numerous non-thermal filaments detected by Yusef-Zadeh et al. (2004)
nor with the 6.4 keV X-ray line produced by low-ionization iron.
The latter has been reviewed by Koyama (2018), who concluded that this time-variable line is
consistent with fluorescence in molecular-cloud iron from an input X-ray flare from Sgr A*.
This line has also been attributed to cosmic-ray ionization by Yusef-Zadeh et al. (2007)
because of its correlation with non-thermal filaments and molecular clouds.
Relativistic cosmic-ray electrons, however, although the source of the non-thermal emission,
would produce an ionization equilibrium more radical than what is seen here in the GC,
with the higher ionization-potential [\ion{Ne}{5}] lines stronger than the [\ion{O}{4}] line,
assuming that there is no contribution to either line
from the ordinary low-excitation \ion{H}{2} regions of the GC.
In fact, the exceptionally highly-excited lines (Tables 8 and 9) occur in decidedly thermal regions,
because they all have counterparts as either thermal radio sources or even Paschen alpha sources
(references in Tables 8 and 9) where such observations with adequate sensitivity have been performed.
I suggest these are either shocks, or for the compact or symmetric sources, candidate planetary nebulae
(D. An et al., in preparation).
}
\input tab2.tex
\input tab3.tex
\input tab4.tex
\input tab5.tex
\input tab6.tex
\input tab7.tex
\input tab8.tex
\clearpage
\input tab9.tex
\input tab10.tex
\input tab11.tex
\subsection{Line Ratios}
\begin{figure*}
\includegraphics[width=184mm]{fig3.eps}
\caption{Ratios of some of the observed lines, grayscale from {\it MSX} Band E.
The color scale is given by the bar on the right.
Crosses mark the positions of the SH or LH apertures from Cycle 1 (programs 0018 and 3295)
and diamonds mark the positions of the SH or LH apertures from Cycle 4 (program 40230).
The locations of the LL apertures are drawn as extracted by CUBISM.
(a) The ratio of the [\ion{Ne}{3}] 15.6 \micron\ line divided by the [\ion{S}{3}] 18.7 \micron\ line.
(b) The ratio of the [\ion{S}{3}] 33 \micron\ line divided by the [\ion{Si}{2}] 34 \micron\ line.
(c) The ratio of the [\ion{O}{4}] 26 \micron\ line divided by the [\ion{Ne}{3}] 15.6 \micron\ line.
{ The boxes marked `1', `2', `3', and `4' circumscribe the regions known as the GC Filaments,
the Quintuplet Region, Sgr B1, and Sgr C that are discussed in Sections 3.4 and 4.}
}
\end{figure*}
Some of the interesting line ratios are plotted in Figure 3.
Although all three line ratios are indicators of excitation (see the required $IP$ in Table~1),
it is clear that different parts of the GC are affected differently
by those processes that affect the excitation: the exciting source SED
and the dilution of the radiation field.
In Figure 3a the [\ion{Ne}{3}] 15.6/[\ion{S}{3}] 18.7 \micron\ line ratio indicates
the relative numbers of high energy photons ($> 41$~eV) needed for doubly ionized neon,
and hence the effective temperatures ($T_{\rm eff}$) of the stars whose SEDs ionize the \ion{H}{2} regions.
In Figure 3b the [\ion{S}{3}] 33/[\ion{Si}{2}] 34 \micron\ line ratio indicates
the dilution of the radiation field, since the singly and doubly ionized excitation structures
of both silicon and sulfur are quite similar (see Section 4):
both elements are at least singly ionized in the non-molecular interstellar gas
but the $> 16$ or $> 23$ eV photons of a nearby hot star are required to photoionize
the silicon or sulfur, respectively, in its local \ion{H}{2} region.
Thus positions colored red or orange in Figure~3b have their exciting stars relatively close by,
whereas positions colored blue or green have very dilute radiation fields,
indicating that their exciting stars are at some distance.
Finally, in Figure 3c the [\ion{O}{4}] 26/[\ion{Ne}{3}] 15.6 \micron\ line ratio indicates
the presence of photons with energies $> 54.9$~eV, as are necessary to produce O$^{3+}$.
In Figure~3c the isolated measurements with exceptionally large ratios, colored red,
are the candidate planetary nebulae or shocks of Tables 8 and 9;
however, the majority of observed [\ion{O}{4}] 26 \micron\ lines, colored green through orange,
are scattered along the Galactic plane with sources of higher ratios at all Galactic longitudes
(note that the colors represent the logs of the line ratios for panel Figure~3c, unlike
panels 3a and 3b, where the line ratios have a linear scale).
\subsection{Extinction}
\begin{figure*}
\includegraphics[width=184mm]{fig4.eps}
\caption{Optical depths at 9.6 \micron\ computed by two different methods, grayscale from {\it MSX} Band E.
The color scale is given by the bar on the right.
Crosses mark the positions of the SH or LH apertures from Cycle 1 (programs 0018 and 3295),
{ squares mark the positions of the SL or LH apertures from Cycle 4 (program 40230), and }
the locations of the SL or LL apertures are drawn as extracted by CUBISM.
(a) The optical depth at 9.6 \micron\ estimated by fitting the SL spectrum with PAHFIT, see text.
(b) Lower limits to the 9.6 \micron\ extinction computed from the ratios
of the [\ion{S}{3}] 19 \micron\ lines divided by the [\ion{S}{3}] 33 \micron\ lines.
The plotted extinction is that required to make the ratio equal to at least 0.508,
the minimum value for $T_e = 6000$~K.
(c) Final estimated extinction from the combination of the two methods, see text.
The values of $\tau_{9.6 \micron}$ in this plot are found in Tables 2 -- 7.
}
\end{figure*}
Although the extinction to the GC has been studied extensively at near-infrared (NIR) wavelengths
(e.g., { Nishiyama et al. 2008}; Schultheis et al. 2009),
the results are always referenced to visible or K-band extinction, $A_V$ or $A_K$, respectively.
{ For longer wavelengths, Lutz et al. (1996), Nishiyama et al. (2009), and Fritz et al. (2011)
showed that the steep extinction law seen in the NIR flattens substantially between 3 and 8 \micron\
using either hydrogen recombination line ratios measured with ISO or Spitzer IRAC photometry
(see also Indebetouw et al. 2005 for other Galactic plane sources).}
For { the longer} MIR wavelengths where almost all the measured lines have wavelengths longer than 10 \micron,
the GC extinction should be referenced to the optical depth, $\tau_{9.6 \micron}$,
of the deepest part of the 10 \micron\ silicate feature,
since it has long been known that the ratio of $\tau_{9.6 \micron}$/$A_V$
is a function of position in the Galaxy, and in particular, it is quite different
in the GC itself (Roche \& Aitken 1985; An et al. 2013).
\begin{figure}
\includegraphics[width=60mm,angle=90,origin=c]{fig5.ps}
\caption{Example of a spectrum where the optical depth at 9.6 \micron\ is estimated
by fitting the SL spectrum with PAHFIT (Smith et al. 2007b) as modified by Simpson et al. (2012).
The fitted function consists of a template (Simpson et al. 2012)
for the 6.2, 7.7, 8.6, 11.2, and 12.5 \micron\ PAH features (gray),
various blackbodies representing the stellar (magenta) and large grain contributions (red),
additional PAH features (blue),
atomic and molecular (H$_2$) emission lines (purple), all multiplied by
the GC extinction law of Chiar \& Tielens (2006).
The sum of all components is plotted in green; this often obscures the strongest features,
particularly the purple [\ion{Ar}{2}] 6.98 \micron\ and [\ion{Ne}{2}] 12.8 \micron\ lines.
The spectrum extracted with CUBISM consists of the black boxes with error bars.
}
\end{figure}
The extinction, as described by $\tau_{9.6 \micron}$, can be computed in two independent ways:
1) The 9.6 \micron\ silicate feature can be modeled and the low-resolution spectra can be fit with
a combination of this deep absorption feature, plus various black bodies to represent
the warm dust in the line of sight and the Rayleigh-Jeans tail of the stellar emission,
and individual broad emission features representing the PAH features that are ubiquitous in the ISM.
A program to fit this combination of features, PAHFIT, was written by Smith et al. (2007b)
and modified by Simpson et al. (2012) to use an unextincted template for the PAH features
instead of the multitudinous individual features.
Because there are fewer degrees of freedom, this produces more reliable results for
the sole absorption feature, the 9.6 \micron\ silicate feature.
Figure~4a shows the values of $\tau_{9.6}$ for the short-low spectra, which cover the wavelength range
5.2 -- 14.4 \micron.
The GC extinction law of Chiar \& Tielens (2006) was used.
Figure 5 shows an example of a high signal/noise (S/N) fit with PAHFIT from the Sickle region.
PAHFIT, as described { by} Smith et al. (2007b), is supposed to fit
the full 5 -- 37 \micron\ low-resolution spectrum;
however, there are very few positions that have both SL and LL spectra, and these are
mostly YSOs, which have additional dust extinction from their dusty envelopes and so
are not representative of the interstellar extinction towards the GC.
Tests were made fitting a few of those positions with both the combination SL--LL wavelength range
and the SL range as shown here --- the results are not significantly different.
2) Lower limits on $\tau_{9.6}$ can be computed from the ratio of the [\ion{S}{3}] 18.7/33.5 \micron\ lines
since the extinction at wavelengths longer than 10 \micron\ is due to the same silicate dust
grains (Chiar \& Tielens 2006 and references therein).
The ratio at the lowest density is determined from the atomic data
and the gas electron temperature, $T_e$ (e.g., Simpson et al. 2007).
At higher densities, the ratio is higher because of collisional de-excitation.
Figure~4b shows the minimum value of $\tau_{9.6}$ needed to make the [\ion{S}{3}] 18.7/33.5 \micron\ line ratio
equal to the minimum ratio, 0.508, predicted for $T_e = 6000$ K
(effective collision strengths from Grieve et al. 2014, but see the comments in Rubin et al. 2016).
Note that since the deepest part of the 9.6 \micron\ silicate absorption feature is close to zero,
the contribution from the foreground Zodiacal emission is not negligible.
This was estimated from the four spectra with the smallest integrated flux -- all of these are
from Program 3121 (PI: K. Kraemer) on the spectra of infrared dark clouds.
These sources are all much fainter than any of the rest of our spectra.
Estimates of the intensity of the Zodiacal emission plus telescope at 9.6 \micron\ computed
by the {\it Spitzer} software SPOT program
were not used because these estimates are
higher than the observed minima in these GC spectra.
The spectra from program 3121 also show low luminosity PAH emission in addition to flux at 9.6 \micron.
This PAH emission, along with some of the continuum, is probably foreground emission
along the line of sight through the Galactic plane to the GC.
The estimated foreground emission, Zodiacal plus line of sight, was subtracted from
the SL spectra when fitting the spectra with PAHFIT.
The final estimated extinction is a combination of both methods \#1 and \#2.
Since the optical depths estimated by method \#2 are only lower limits,
I increased these values to those estimated by method \#1 for nearby positions.
This was done by making a 6 arcsec grid covering the entire GC and
then interpolating or extrapolating the observed values of $\tau_{9.6}$ for the grid pixels
using the Interactive Data Language (IDL) function, GRIDDATA, and the method of `Nearest Neighbor'.
Two grids were made, for method \#1 and for method \#2.
The final extinction grid is the maximum of the method \#2 grid and the method \#1 grid.
Clearly the validity of the results depends on the closeness of each pixel to a location
where there were either LL observations of both [\ion{s}{3}] lines or PAHFIT computations,
and thus I plot only the regions with observations in Figure~4c.
It is interesting to note that the extinction is fairly uniform across the GC with a value of
$\tau_{9.6} \sim 3$, with the regions with exceptionally higher extinction also being the regions
with known dense molecular clouds: the `Brick' at G0.26+0.0, Sgr B2 at G0.7$-$0.0, and
the Galactic Center dust ridge described by Immer et al. (2012).
The low extinction regions are mostly at the higher Galactic latitudes,
where the foreground gas at moderate distances over the Galactic plane
contributes much more to the line of sight than the gas at the larger distances above
the Galactic plane of the GC.
The values of the estimated extinction are given in Tables 2 -- 7.
It should be noted here that the extinction values for those positions in
the extreme excitation sources of Tables 8 -- 10 may be erroneous ---
these sources are typically very compact and are not detectable in continuum images.
Thus extinction values estimated from the continuum (method \#1)
or extinction values estimated from that of their local neighboring positions (method \#2)
may not be applicable.
At least some sources may be foreground and actually have much lower optical depths.
Such sources may be indicated by strong observed [\ion{S}{4}] 10.5 \micron\ line intensities,
which lines are normally extremely weak or not detectable in regions
with deep 10 \micron\ silicate absorption.
{ An et al. (2013) also made extinction estimates from the SH spectra of program 40230
by measuring the depths of the 9.6 \micron\ silicate feature from the ratios of the observed intensities
at approximately 10.24 and 13.9 \micron.
These estimates are shown in their figure 4, top.
The estimates of the depth of the 9.6 \micron\ silicate feature in this paper
for those positions on the sky that have both usable SL spectra and SH spectra
are in most cases somewhat smaller (the median of the ratio equals 66\%)
except in the regions of high extinction in Sgr B2, where the extinction estimated in this paper
is somewhat higher.
At least part of the difference is probably due to the fact that neither wavelength region
used by An et al. (2013) is free of PAH emission, with the 13.9 \micron\ wavelength region
in the PAH template (figure 4 of Simpson et al. 2012) that is used in the PAHFIT computation
having a larger PAH contribution
than the 10.24 \micron\ wavelength region.
Thus after subtraction of the PAH emission, the 9.6 \micron\ silicate feature is not as deep,
producing a smaller computation of the optical depth by PAHFIT.}
\subsection{Electron Densities and Elemental Abundances}
\input tab12.tex
\begin{figure*}
\includegraphics[width=184mm]{fig6.eps}
\caption{Electron densities estimated from the [\ion{S}{3}] 18.7/[\ion{S}{3}] 33 \micron\ line ratios, grayscale from {\it MSX} Band E.
The color scale is given by the bar on the right.
Crosses mark the positions of the SH or LH apertures from Cycle 1 (programs 3295);
Square boxes mark the positions of SH/LH apertures from programs 3189 and 40230,
{ and the LL apertures are drawn as extracted with CUBISM}.
}
\end{figure*}
Electron densities, $N_e$, were estimated from the extinction-corrected and co-located
[\ion{S}{3}] 19/33 \micron\ line intensity ratios
using the effective collision strengths referenced in Table~1.
Estimated foreground intensities were first subtracted to compensate for the integrated intensities
along the lines of sight to the GC;
this foreground was estimated from the minimum observed intensities
(3.5 for [\ion{Ne}{2}] 12.8 \micron, 0.2 for [\ion{Ne}{3}] 15.6 \micron,
0.6 and 3.0 for [\ion{S}{3}] 18.7 \micron, 1.5 and 10.0 for [\ion{S}{3}] 33 \micron,
and 8.0 and 19.0 for [\ion{Si}{2}] 34 \micron, where all estimates
should be multiplied by $10^{-19}$ W m$^{-2}$ arcsec$^{-2}$ and
pairs of numbers are for positive and negative Galactic longitudes, respectively).
The densities estimated from the { observed line ratios}
are plotted in Figure 6,
along with the densities from Simpson et al. (2007).
The low resolution lines are preferred for the density computation because
there are far more observed positions per \ion{H}{2} region
and because there is no need for correction for differences in aperture size
for lines at 19 versus 33 \micron.
Elemental abundances with respect to hydrogen were estimated from the high-resolution line measurements.
In order to produce sufficiently small errors in the ratios,
only measurements of the hydrogen line intensities with S/N$ > 4$ were used.
The measured densities and abundance ratios were then averaged over regions
that I call the GC Filaments, the Quintuplet Cluster region, Sgr B1, and Sgr C;
each region should be localized in space and ionized by the stars local to that region.
{ Outlines of these regions are plotted in Figure~3c.}
A separate average for the Arched Filaments observed by Simpson et al. (2007)
was also computed
since the references for the effective collision strengths in Table~1
have changed substantially since that paper.
These average densities and ionic abundance ratios are listed in Table~12.
Note that there is very little overlap between the positions measured by Simpson et al. (2007)
and the positions in the GC Filaments seen in Figure~2f, which are heavily weighted to
the small \ion{H}{2} regions at the base of the Arched Filaments.
Approximately half of the SL positions with observed [\ion{Ne}{2}] have detectable [\ion{Ar}{2}] 6.98 \micron\ lines.
For these positions, the average [\ion{Ar}{2}] 6.98/[\ion{Ne}{2}] 12.8 \micron\ line ratio equals 0.515.
The positions in the Radio Arc Bubble are excluded from this average
because they have substantial Ne$^{++}$ (Simpson et al. 2007)
and probably also substantial Ar$^{++}$,
making this ratio too dissimilar to that of the rest of the GC.
For $T_e = 6000$~K, this intensity ratio corresponds to an Ar$^{+}$/Ne$^+$ abundance ratio of { 0.032}.
The measured abundances are all higher than those of the Orion Nebula
[(Ne$^+$ + Ne$^{++}$)/H$^+ = 1.02 \pm 0.03 \times 10^{-4}$ and
(S$^{++}$ + S$^{3+}$)/H$^+ = 9.6 \pm 0.3 \times { 10^{-6}}$, Rubin et al. 2016];
this is expected considering the known abundance gradients in the Galaxy
(e.g., Mart\'in-Hern\'andez et al. 2002; Rudolph et al. 2006).
Since it is essential, in a comparison of abundances, that all abundances
are computed using the same atomic physics (e.g., Table~1),
it is beyond the scope of this paper to recompute the Galactic abundance gradients
using these new results.
\section{Discussion}
\subsection{Comparison with Models}
By comparing the line flux ratios to ratios produced by \ion{H}{2} region models,
one can gain insight into the gas chemical composition
and the nature of the sources ionizing the gas.
Such models need to cover a range of gas densities and geometries,
gas compositions, ionizing source SEDs, etc.
At one extreme, one can employ the million model grid of
Morisset, Delgado-Inglada, \& Flores-Fajardo (2015) with its access via a data-base query language,
and at the other extreme, one can compute detailed models specific to each source,
such as the models of the Galactic \ion{H}{2} regions W43 and G333.6$-$0.2 of Simpson et al. (2004).
In general, large grids are useful to get a first guess of the range of model parameters,
which are then refined with detailed modeling.
Because of the extreme nature of the GC and because it has already been seen that
there are difficulties in reproducing the observed combination
of both low and high excitation gas (e.g., Simpson et al. 2007),
the pre-computed models from the extant databases are not adequate to represent
the observations in the previous section.
I here expand the model parameter space from that of Simpson et al. (2007)
to demonstrate that X-rays in addition to the SEDs of OB stars
are required to fit the observed line intensities.
I have modeled the ionization structure of the GC gas
with a relatively coarse grid of model \ion{H}{2} regions
computed with the code { Cloudy 17.00 (Ferland et al. 2017)}.
Because of the low gas density in the GC, as seen in the previous section,
the models all have low hydrogen density ($N_{\rm H} = N_p = 100$~cm$^{-3}$)
but high photon input, to correspond to the massive clusters ionizing the gas.
The parameters that are varied to make a grid of models
consist of the shapes of the ionizing SEDs, which determine the amounts of
the more highly excited states of oxygen and neon,
and the dilution of the radiation field that so strongly affects
the low-ionization states of silicon and sulfur.
The radiation field intensity is usually described by
the ionization parameter, $U$, which is defined as the ratio of
the photon density divided by $N_e$ at the outer edge
of the fully ionized gas sphere of radius, $R_S$ (e.g., Mathis 2000).
The low values of $U$ needed to produce the low observed [\ion{S}{3}] 33/[\ion{Si}{2}] 34 \micron\ ratio (Figure~3b)
are found only for \ion{H}{2} regions at extremely large distances from their exciting stars
(thus producing very dilute radiation fields) or
for \ion{H}{2} regions with very low average densities, much lower than the densities
estimated from the [\ion{S}{3}] line ratios.
Here I define the filling factor $f$ as the fraction of the volume with clumps
of proton density $N_p$ such that the average proton density equals $fN_p$.
With this definition, $R_S$ and $U$ are given by
\begin{equation}
R_S = {\left( {\frac{3 N_{\rm LyC}}{4 \pi f N_{\rm H}^2 (N_e/N_p) \alpha_{\rm B} F_{\rm He}}} \right)}^{\frac{1}{3}}
\end{equation}
and
\begin{equation}
U = \frac{N_{\rm LyC}}{4 \pi R_S^2 c} \frac{1}{N_{\rm H}(N_e/N_p)} \approx
\left[ \frac{N_{\rm LyC}}{36 \pi c^3} f^2 \alpha_{\rm B}^2 F_{\rm He}^2 \frac{N_p}{N_e}N_{\rm H} \right]^{\frac{1}{3}},
\end{equation}
where
$F_{\rm He} = 1/(1+f_i <{\rm He}^+/({\rm H}^+ + {\rm He}^+>)$ (Rubin 1968 as written by Simpson \& Rubin 1990),
$f_i$ is the fraction of helium recombination photons to excited states that are energetic enough
to ionize hydrogen ($f_i \sim 0.65$),
$N_{\rm LyC}$ is the number of photons emitted per second in the Lyman continuum
that can ionize hydrogen,
$N_e$ is the electron density, $c$ is the speed of light,
and $\alpha_{\rm B}$ is the total recombination rate to the second level of hydrogen.
For this grid I use SEDs from the compilation of SEDs with Solar abundances from Starburst99,
most recently described by Leitherer et al. (2014), with individual
O-star atmosphere models from Leitherer et al. (2010).
{ These input SEDs were computed using the Starburst99 default parameters for an instantaneous burst of star formation.}
In Starburst99, an initial mass function for a massive stellar cluster is assumed,
the cluster stars of varying masses are evolved following the evolutionary tracks for each star,
and a composite SED for the whole cluster is calculated using the stellar types that the stars have for the given cluster age.
In Cloudy, the SED is input by calling for it by atmospheric model type, and
for Starburst99, by model age, expressed as the logarithm of the age.
The ages used for this grid are $10^{6.0}$, $10^{6.2}$, $10^{6.3}$, $10^{6.4}$, $10^{6.5}$, $10^{6.6}$, $10^{6.65}$,
and $10^{6.7}$ years; these ages produced line intensity ratios
that bracket the observed ratios.
The 2014 version of Starburst99 used the `Geneva' evolutionary tracks of Ekstr\"om et al. (2012)
for stellar models with both zero rotation and rotation with velocities of 40\%\
of the break-up velocity.
The effect of the higher rotational velocity is to produce SEDs that are bluer and more luminous
due to more mixing in the stellar core.
The result is that the cluster SED for a given age has more high energy photons
than the SED for zero rotation, or conversely, if two SEDs must be essentially identical
to produce an \ion{H}{2} region model that best fits the observations,
the age of the Starburst99 SED with rotation is significantly older than the age
of the Starburst99 model without rotation (Leitherer et al. 2014).
I will show that such models with rotation require ages that are almost certainly too old
for fitting the spectra of the diffuse gas of the GC.
In Simpson et al. (2007) we attempted to model the gas of the Radio Arc Bubble
with \ion{H}{2} region SEDs consisting of multiple component blackbodies, with the higher temperature
blackbodies having temperatures of either $10^5$ or $10^6$~K.
None of these models was satisfactory.
In retrospect, as a result of computing the models in this paper, I conclude that the reason
is surely the use of blackbodies to represent the stellar spectra ---
such SEDs all have too much flux in the 40 -- 54 eV range compared to the 54 -- 77 eV range
and so produce too much Ne$^{++}$ compared to O$^{+3}$.
The models here use stellar SEDs from Starburst99 plus blackbodies with temperatures
of either $10^6$ or $10^{6.5}$~K and blackbody luminosities ranging from $10^{37}$ to $10^{39}$ erg s$^{-1}$
in increments of $10^{0.5}$;
these models adequately cover the range of line ratios observed by Simpson et al. (2007).
The SEDs used in this paper are shown in Figure 7.
As mentioned above, the SEDs for the Starburst99 model with age $= 10^{6.7}$~yrs and
rotation 40\% of breakup have much more flux than SEDs of the same age but zero rotation.
Blackbodies were chosen only as a way of adding a smooth, well-defined component to the SED
between 54 and $\sim 100$ eV;
other spectral shapes may be more appropriate, particularly a non-thermal shape
if the actual X-ray emission is optically thin.
\begin{figure*}
\includegraphics[width=165mm]{fig7.eps}
\caption{Spectral energy distributions from Starburst99 for various ages with
additional fluxes from blackbodies
with the given luminosities (labels in erg s$^{-1}$) and temperatures (in K).
The numbers of photons emitted by the central source, $Ph_\nu$, are multiplied
by the frequency, $\nu$.
All the Starburst99 models drawn with colored lines have zero rotation velocity;
the black line is the Starburst99 model with rotation 0.4 times the break-up velocity
and log age = 6.7.
Note that the SED with $T_{\rm BB} = 10^{6.0}$~K and $L_{\rm BB} = 10^{39.0}$~erg s$^{-1}$
has substantially more flux in the important energy range of 41--100 eV than
the SED with $T_{\rm BB} = 10^{6.5}$~K and $L_{\rm BB} = 10^{39.0}$~erg s$^{-1}$.
}
\end{figure*}
The abundances used in the models are my best estimate of the abundances
of the atomic gas component of the GC.
{
With respect to hydrogen, these are, for He, C, N, O, Ne, Si, S, Ar, and Fe:
0.095, 5.13e-4, 1.16e-4, 6.84e-4, 1.74e-4, 2.40e-5, 1.90e-5, 6.20e-6, and 2.6e-6, respectively.
The abundances of Ne/H, (S$^{++}$ + S$^{3+}$)/H$^+$, and Si$^+$/H$^+$ are the averages of the abundances in Table~12.
Ionization correction factors (icf) were computed for Si$^+$/H$^+$ (0.56)
and Ar$^+$/Ne$^+$ (0.90) from the fully ionized regions of the models computed herein.
For S/H there is a 15.6\% addition to the (S$^{++}$ + S$^{3+}$)/H$^+$ ratio for S$^+$ (Rubin et al. 2016).
Since both Ne/H and S/H are factors of 1.71 times the abundances used by Rubin et al. (2016)
to describe the Orion Nebula,
I multiplied their Orion Nebula abundances for C, N, O, and Fe by this same factor
to get the GC abundances in this paper (omitting the effects of the likely larger N/H abundance gradient,
e.g., Rudolph et al. 2006).
}
The other elements have the abundances used for the Cloudy \ion{H}{2} region mix,
which are essentially those of the Orion Nebula and so are probably of too low abundance for the GC.
However, the elements in this mix all have very low abundances compared to the elements
listed above and so contribute very little to the cooling of the \ion{H}{2} region gas.
The other chief parameters for the grid are the electron and photon densities.
Because the observed line ratios indicate that the electron density is never high
(except in a few of the high excitation sources of Tables 8 and 9),
densities of $N_p = 100$~cm$^{-3}$ and cluster photon luminosities
$N_{\rm LyC} = 10^{50}$ photons s$^{-1}$ were used in the models.
The average gas densities were varied by using a range of filling factors $f$ of 1.0, 0.31623, 0.1,
0.031623, 0.01, 0.0031623, and 0.001.
The photon densities at the inner edges of the \ion{H}{2} regions were varied by
using inner \ion{H}{2} region radii, $R_{\rm inner}$, of 1.0, 3.1623, and 10 pc;
a few models were also computed with inner radii of 31.623 and 100 pc but
these models have such low $U$ that their predicted line ratios are outside the observed range
($\log U \lesssim -3.5$).
Given the constant density and filling factor with distance from the exciting star cluster,
the integrated line fluxes should scale with the ionizing luminosities $N_{\rm LyC}$,
and adjustments can be made for changes in $N_e$ keeping $U$ constant
according to equations 1 and 2.
This is not exact, as changing the density changes the cooling rates, and hence $T_e$,
thus affecting $R_S$ and $U$ through the $\sim T^{-0.8}$ temperature dependence of $\alpha_{\rm B}$.
MIR line emissivities are relatively insensitive to $T_e$ (proportional to $\sim T_e^{-0.3}$)
and the predicted line ratios of the MIR forbidden lines are quite insensitive.
Changing the input $R_{\rm inner}$ does not significantly change $U$ or $R_S$ until
$R_{\rm inner}$ is quite a bit larger than 10 pc.
\begin{figure*}
\includegraphics[width=165mm]{fig8.eps}
\caption{
Ionization structures for different elements for models with Starburst99 SEDs for zero rotation and age 6.6 Myr.
The models are {\tt starburst0\_66\_0001} and {\tt starburst0\_66\_6570} and have inner radii of 1.0 pc, Str\"omgren radii $\sim 14$~pc, and filling factor equal to 0.1.
The different elements and models with or without X-rays are indicated by color, as shown in the figure.
The bottom panel shows the ionization of oxygen and neon and the top panel shows the ionization of silicon and sulfur.
For all elements, the neutral ionization stage is plotted with a solid line, singly ionized is dotted, doubly ionized is dashed, triply ionized is dash-dot, quadruply ionized is long dashes, and five times ionized is plotted with a solid line again.
}
\end{figure*}
Examples of the ionization structures of models with and without additional X-rays are given
in Figure 8.
In particular, the locations of gas with doubly ionized silicon and sulfur are seen to be
quite similar and unlike either oxygen or neon.
For this reason the easily observed ratio of doubly ionized sulfur to singly ionized silicon
([\ion{S}{3}] 33 \micron/[\ion{Si}{2}] 34 \micron)
is an excellent indicator of the local ionization parameter $U$.
\begin{figure*}
\includegraphics[width=160mm]{fig9.eps}
\caption{
Plots of the ratios of the lines observed with {\it Spitzer} IRS (Section 3) computed from all the models
in the grid of \ion{H}{2} regions.
The points marked with asterisks include ionizing fluxes by blackbodies with $T = 10^{6.5}$~K
and the points marked with plus signs include ionizing fluxes by blackbodies with $T = 10^{6.0}$~K.
The points marked with black diamonds in panels (c) and (d) employ the Starburst99 SED
for rotation 0.4 times the break-up velocity and log age = 6.7.
The colors of the data points in panels (a) and (b) are functions of the log of the ionization parameter $U$
and the colors in panels (c) and (d) are functions of the numbers of stellar photons with energies
between 54.4 and 280 eV, here written as $Q_X$.
Such photons from GC sources are too soft to be detectable by {\it Chandra}
but have the right energy range to ionize the higher-excitation ionization states reported here.
It is seen that $Q_X$ has an especially strong influence on the [\ion{O}{4}] 26/[\ion{S}{3}] 33 \micron\ line ratios
and that $U$ is the main parameter influencing the [\ion{S}{3}] 33/[\ion{Si}{2}] 34 \micron\ line ratios,
as expected.
}
\end{figure*}
The line ratios predicted by this grid of \ion{H}{2} region models, seen in Figure~9,
can be compared to the observed ratios in the GC.
Here I subdivide the area of the GC to analyze specific \ion{H}{2} regions of interest:
the regions ionized by the Quintuplet or the Arches Clusters, Sgr B1, and Sgr C.
Obviously, Sgr B2 would be of great interest; however, the extinction is so large
towards Sgr B2 that the {\it Spitzer} IRS data set includes only a few sources (mostly candidate YSOs)
and thus the region was not well enough observed by the IRS (Figure~2).
The models and data for the four regions are plotted in Figures 10 to 13,
and summaries of the best-fitting models are given in Table 13.
For each region, the line intensities divided by the [\ion{S}{3}] intensity
observed in the same {\it Spitzer} IRS module and corrected for extinction
were averaged, with the low-resolution modules and high-resolution modules
computed separately, since there are far more low-resolution spectra
but only the high-resolution spectra include measurable [\ion{O}{4}] 26 \micron\ line intensities
(not including the shocked regions of Tables 8 and 9).
Then one by one the log of these intensity ratios with respect to a sulfur line was compared to the logs
of the same ratios of each model, and the sum of the squares of the differences (`$chisq$') was computed:
\begin{equation}
chisq = \sum_{i} \left[ \log (\frac{x_i}{y_i}) - \log (\frac{x^\prime_i}{y^\prime_i}) \right]^2 ,
\end{equation}
where $x_i$ is one of the observed [\ion{Ne}{3}] 15, [\ion{Si}{2}] 34, or [\ion{O}{4}] 26 \micron\ line intensities,
$y_i$ is the [\ion{S}{3}] line flux observed in the same IRS module,
and $x^\prime_i$ and $y^\prime_i$ are the same lines as computed by one of the Cloudy models.
The model parameters were not iterated, and since the spacing of the parameters of the grid is
not small (0.5 dex), the models with minimum $chisq$ give only an indication
of what is needed to produce a better fit.
The results of three models from each X-ray SED group are given in Table 13 --
although there are differences in $chisq$, these models are probably equally good fits
and simply show the ranges of acceptable parameters.
\input tab13.tex
Table 13 contains the numbers of ionizing photons, $N_{\rm LyC}$, for each of the \ion{H}{2} regions.
These were estimated from the single-dish radio continuum
measurements, $S_\nu$, of Altenhoff et al. (1978), Downes et al. (1980), Reifenstein et al. (1970), and
Wilson et al. (1970) at $\sim 5$~GHz
using the relation of Rubin (1968) as written by Simpson \& Rubin (1990) for an assumed $T_e = 6000$~K
(Simpson et al. 2007).
It is especially important to use single-dish radio telescope measurements for these estimates
because the total numbers of ionizing photons are needed for computing scale factors for the models,
which were all computed for $10^{50}$ photons s$^{-1}$,
and interferometers lose flux owing to their lack of dishes with almost zero spacing.
The estimates of the photon numbers for the Quintuplet Cluster region and the Arched Filaments
are particularly uncertain because of the large contribution to the radio fluxes from
the non-thermal Radio Arc (e.g., Yusef-Zadeh \& Morris 1987).
However, the values in Table~13 must be underestimates since neither cluster
is embedded in its natal molecular cloud, thereby allowing a sizable fraction of ionizing photons
to escape the region, and the $N_{\rm LyC}$ vs. $S_\nu$ relation assumes that
the \ion{H}{2} region producing the radio continuum is ionization-bounded in all directions (Rubin 1968).
In fact, Figer et al. (1999, 2002) estimated ionizing fluxes for both the Quintuplet and Arches clusters
of close to or more than $10^{51}$ photons s$^{-1}$ from the numbers of OB and Wolf-Rayet (WR) stars.
In summary, to compare the X-ray fluxes from the best fit models to X-ray observations of the GC,
one should multiply the X-ray $L_{\rm BB}$ by the observed $N_{\rm LyC}$ in Table~13
and divide by the $N_{\rm LyC}$ used in the models, which was $N_{\rm LyC} = 10^{50}$ photons s$^{-1}$.
\subsection{Arched Filaments}
\begin{figure}
\includegraphics[width=84mm]{fig10.eps}
\caption{
Plots of the ratios of the lines observed with {\it Spitzer} IRS (Section 3) computed from all the models
with zero rotation
in the grid of \ion{H}{2} regions and the ratios observed in the Arched Filaments.
The points marked with asterisks include ionizing fluxes by blackbodies with $T = 10^{6.5}$~K
and the points marked with plus signs include ionizing fluxes by blackbodies with $T = 10^{6.0}$~K.
The colors of the points representing the models in both panels are functions of
the ages of the zero rotation models from Starburst99: purple is log age = 6.0,
magenta is 6.2, red is 6.3, red-orange is 6.4, yellow-orange is 6.5,
yellow is 6.6, green is 6.65, and blue is 6.7, where the age is in yrs.
For the locations of the points representing the models computed with Starburst99 SEDs
with rotation velocity 0.4 times the break-up velocity and log age = 6.7, see Figure~9.
In the lower panel the small black crosses are ratios of lines observed with the LL module
and in both panels the black dots are the ratios of lines observed with the SH and LH modules
in program 40230.
The black X's show the ratios of lines observed with the SH and LH modules
in program 3295 (Simpson et al. 2007) { or program 0018}.
The large black crosses plot the averages of the observed ratios and the error bars their
standard deviations, with the thicker error bars from the high resolution modules.
}
\end{figure}
The line ratios from the Arched Filament positions used in this section are the combination
of the Arched Filament positions of Simpson et al. (2007)
and a subset of the GC Filaments, described in Table~12, with Galactic longitudes between 0.10 and 0.25,
and are plotted in Figure~10.
The Arches Cluster is assumed to be the source of the ionizing photons for the Filaments
along Galactic longitude 0.15 -- the high [\ion{S}{3}] 33/[\ion{Si}{2}] 34 \micron\ ratio in this region (Figure~3b)
shows that the exciting stars are near by, providing the undiluted radiation field
(high ionization parameter) needed to produce this ratio.
The average model ages are $10^{6.6}$ to $10^{6.7}$ yr with average $chisq$ = { 0.07 and 0.05
for the models with X-ray $T_{\rm BB} = 10^{6.5}$~K and $L_{\rm BB} = 10^{38.0}$ erg s$^{-1}$
or $L_{\rm BB} = 10^{38.5}$ erg s$^{-1}$
or X-ray $T_{\rm BB} = 10^{6.0}$~K and $L_{\rm BB} = 10^{37.0}$ or $10^{37.5}$ erg s$^{-1}$, respectively.}
The fitted SEDs all have ages in the range of $4 - 5 \times 10^6$ yrs;
a substantially younger age does not produce any reasonable fit.
On the other hand, going back to Figure~9, notice that the black diamonds that mark the Starburst99 SEDs
for rotation of 0.4 times the break-up velocity all have
much higher [\ion{Ne}{3}] 15/[\ion{S}{3}] 19 \micron\ line ratios,
even though they all also have ages $\sim 5 \times 10^6$ yrs.
To get models using the Starburst99 SEDs for the 0.4 times break-up sequence
that also match the observed [\ion{Ne}{3}] 15/[\ion{S}{3}] 19 \micron\ line ratios for the Arched Filaments,
one would need substantially older cluster ages than $5 \times 10^6$ yrs.
Such long ages are in conflict with { ages estimated by other means, for example, the
$3.5 \pm 0.7$ Myr for the Arches Cluster (Schneider et al. 2014).}
Moreover, long ages are not reasonable considering the short orbital time for clusters in the GC
and the subsequent loss of stars due to tidal interactions (e.g., Portegies Zwart et al. 2002;
{ Habibi et al. 2014}).
For these reasons, that the Starburst99 SEDs computed for stellar models with rotation cannot produce reasonable fits, they are not discussed further in this paper.
The GC filaments of { Table 12} include the small \ion{H}{2} regions
at the base of the Filaments near Galactic longitude 0.05.
These \ion{H}{2} regions also have hot stars (Cotera et al. 1999)
and so should be analyzed separately from the Arched Filaments.
Moreover, any models should use individual stellar SEDs instead of the composite cluster SED
of Starburst99. Such models will be discussed in a later paper.
{ However, if the data for only the small \ion{H}{2} regions
are compared to the models, the best fits have similar ages to those of the Arched Filaments
in Table 13 but require slightly more X-rays, possibly owing to their location closer to Sgr A*,
and a lower $U$.
The similarity in inferred ages to those of the Arches Cluster stars
could be a possible indication that these stars were not formed independently from
the Arches Cluster but were originally in the Cluster and have been stripped away
by the tidal field of the GC (e.g., Habibi et al. 2014).
}
Note that there { is only one value } plotted in the top panel of Figure 10 ---
the reason is that the [\ion{O}{4}] 26 \micron\ line is required to have a signal/noise ratio of at least 2
(the observed S/N for the { plotted point is 2.5}).
Although all positions in the Arched Filaments appear to have detectable [\ion{O}{4}] 26 \micron\ lines,
the other positions are in the high-continuum filaments,
with the result that the noise from uncorrectable rogue pixels is also high, leading to poor S/N
in the measurement of the faint [\ion{O}{4}] 26 \micron\ line (Simpson et al. 2007).
Thus the estimate of the X-ray luminosity for the Arched Filaments is the least certain of
all four \ion{H}{2} region estimates.
Erickson et al. (1991), Colgan et al. (1996), and Cotera et al. (2005)
found that the excitation of the Filaments is consistent with the exciting source being
located at some distance from the Filaments, such as the location of the Arches Cluster.
Simpson et al. (2007) also found that that the excitation of the `W' Filament is
much lower than that of the `E2' Filament, again in accord with the supposition
that the Filaments are ionized by the Arches Cluster.
Unfortunately, the new IRS fluxes reported in this paper do not include
much additional coverage of the E2 Filament and none of the W Filament beyond
that described by Simpson et al. (2007), as seen in Figure 2.
The most likely configuration of the Filaments with respect to the Arches Cluster
is that the Filaments are the lit-up edges of molecular clouds at
distances of $\sim 10 - 20$~pc from the Arches Cluster (Lang et al. 2001; Yasuda et al. 2009).
On the other hand, Hankins et al. (2017) mapped the Arched Filaments in several MIR bands with
the Faint Object InfraRed CAmera for the {\it SOFIA} Telescope (FORCAST).
They found that if they assumed standard size interstellar dust grains ($\sim 0.1$~\micron),
the heating source of the dust would have to be much closer to the Filaments
than the distances observed for the Arches Cluster on the sky.
Either the heating sources would have to be more localized in the Filaments
or the dust grains would have to be significantly smaller, $\sim 0.01$~\micron\
if the heating source is required to be the Arches Cluster.
{ They suggest that the presence of} such small grains would indicate substantial dust processing in the GC.
{ Small grains could result from processes such as shattering by supernovae (e.g., Andersen et al. 2011)
or shattering by high grain velocities due to the turbulence in the GC molecular clouds (Hankins et al. 2017; Hirashita \& Yan 2009).}
\subsection{Quintuplet Cluster Region}
\begin{figure}
\includegraphics[width=84mm]{fig11.eps}
\caption{
Plots of the ratios of the lines observed with {\it Spitzer} IRS (Section 3) computed from all the models
with zero rotation
in the grid of \ion{H}{2} regions and the ratios observed in the Quintuplet Cluster region.
The colored points marked with asterisks include ionizing fluxes by blackbodies with $T = 10^{6.5}$~K
and the colored points marked with plus signs include ionizing fluxes by blackbodies with $T = 10^{6.0}$~K.
The colors of the points representing the models and the black symbols in both panels,
representing the line ratios observed in the Quintuplet Cluster region,
are described in Figure 10.
}
\end{figure}
The region surrounding the Quintuplet Cluster (G0.1659$-$0.0656),
$0.11 < l < 0.25$ and $-0.07 < b < -0.01$ in Galactic coordinates ($l,b$),
was chosen to specifically avoid the Radio Arc Bubble, which is known to
contain shocks (Simpson et al. 2007).
Previous observations (Rodr\'{\i}guez-Fern\'andez et al. 2001; Simpson et al. 2007)
have demonstrated that the gas excitation decreases uniformly with distance from the Quintuplet Cluster,
from which I infer that this massive cluster is the ionizing source.
Modeling the Quintuplet Cluster region (Figure~11), I find that
the average Starburst99 model age is $10^{6.4}$ yr for the models with $T_{\rm BB} = 10^{6.5}$ erg s$^{-1}$
and $10^{6.5}$ yr for the models with $T_{\rm BB} = 10^{6.0}$ erg s$^{-1}$,
significantly younger than the average Starburst99 model age of { $10^{6.66}$} yr
needed for the Arches Cluster.
The Quintuplet Cluster, like the Arches Cluster, contains numerous WR stars,
indicating that there has been considerable evolution away from the main sequence
for its most massive stars.
This change in the individual stellar SEDs owing to such evolution is included
in the composite SED of Starburst99 (Leitherer et al. 2014).
Schneider et al. (2014) modeled the change in the stellar mass function
for these two clusters taking into account that most massive stars are found in binary systems
(see, e.g., Sana et al. 2014)
and exchange mass, thereby changing their spectral types and the apparent mass function of the cluster.
By fitting the observed stellar mass functions to their population synthesis models,
they find { an age for the Quintuplet Cluster of $4.8 \pm 1.1$ Myr, older than their $3.5$ Myr age of the Arches Cluster}.
Although the uncertainties are sizable, the turn-off point from the main sequence
is well determined in both clusters; thus they find there is no uncertainty
in their determination that the Quintuplet Cluster is older than the Arches Cluster.
Another method of determining ages of stellar clusters is by isochrone fitting;
using this method Liermann et al. (2014 and references therein) estimate an age of $3.0 \pm 0.5$ Myr
for the OB stars in the Quintuplet Cluster.
This should not be surprising, according to Schneider et al. (2014),
since the most massive stars in the Quintuplet Cluster are effectively rejuvenated
by receiving mass transferred from their binary companions.
However, I find from the {\it Spitzer} IRS spectra of the [\ion{Ne}{3}]/[\ion{S}{3}] ratios
(Figures 10 and 11) that the Quintuplet Cluster
{ Region is ionized by higher energy photons than the Arched Filaments. }
This { conclusion} is not new, if one looks back at previous observations of the [\ion{O}{3}], [\ion{S}{3}], and [\ion{Si}{2}] lines
taken from the Kuiper Airborne Observatory (KAO, Colgan et al. 1996; Simpson et al. 1997)
and the [\ion{O}{3}] 88 and [\ion{N}{2}] 122 \micron\ lines measured by Yasuda et al. (2009) with {\it AKARI}.
In the KAO spectra the O$^{++}$/S$^{++}$ ratios were measured to be higher in the Sickle than in the Arched Filaments,
indicating ionization by higher temperature stellar SEDs in the former region.
{ Given the assumption that the two clusters
are the sources of the photons ionizing their respective regions,
one would normally conclude, therefore, that the Quintuplet Cluster
must contain hotter stars than the Arches Cluster.
It is important to note that both }
clusters are massive enough that they should not be affected by small numbers of stars
producing uneven initial mass functions.
{ On the other hand, if the Quintuplet Cluster is indeed older than the Arches Cluster
(e.g., Schneider et al. 2014),
then there are additional high energy photons in the region of the Quintuplet Cluster
that are contributing to the ionization of the Sickle,
photons that are not included in the composite SEDs that are produced by Starburst99.}
One possibility is that the stars of the Quintuplet Cluster, unlike the Arches Cluster,
rotate with a high velocity, thereby making the models computed with Starburst99 SEDs
with rotation 0.4 times the breakup velocity applicable.
Another possibility is that the actual SED includes energetic photons from some
transient event, such as a supernova or a mass-loss episode from
the Quintuplet Cluster's Pistol Star.
(I note, however, that the Pistol Nebula is not as highly ionized as the rest of the Sickle
--- Simpson et al. 1997 --- thereby making the latter suggestion less likely.)
In fact, Ponti et al. (2015) suggest that the Radio Arc Bubble is the result of multiple
supernova explosions from Quintuplet Cluster stars, occurring over a range of time
as the cluster moved from lower to higher Galactic longitudes (Stolte et al. 2014),
and Heard \& Warwick (2013) suggest that there is a supernova remnant at G0.13-0.12.
\subsection{Sgr B1}
\begin{figure}
\includegraphics[width=84mm]{fig12.eps}
\caption{
Plots of the ratios of the lines observed with {\it Spitzer} IRS (Section 3) computed from all the models
with zero rotation
in the grid of \ion{H}{2} regions and the ratios observed in Sgr B1.
The points marked with asterisks include ionizing fluxes by blackbodies with $T = 10^{6.5}$~K
and the points marked with plus signs include ionizing fluxes by blackbodies with $T = 10^{6.0}$~K.
The colors of the points representing the models and the black symbols in both panels,
representing the line ratios observed in Sgr B1,
are described in Figure 10.
}
\end{figure}
In Galactic coordinates, Sgr B1 is defined to be located in the region $0.45 < l < 0.60$ and $-0.11 < b < -0.01$.
The line ratios observed in Sgr B1 are plotted in Figure~12
and the best fit models are tabulated in Table~13.
Sgr B1 appears to have two components, `East' and `West',
where the West component has relatively less absorption from both \ion{H}{1} (Lang et al. 2010)
and formaldehyde (Mehringer et al. 1995) than the East component.
The measurements of the MIR dust extinction also show that the eastern part of the source
has higher $\tau_{9.6 \micron}$ than the western part of Sgr B1 (Figure~4).
Although Sgr B2 is thought to be extremely young because of the multiple compact components
in its radio thermal continuum emission
(e.g., De Pree et al. 2015 and references therein)
{ and its numerous hot molecular cores (e.g., Vogel et al. 1987; Etxaluze et al. 2013; Schmiedeke 2016)},
this is not true for Sgr B1.
The radio continuum of Sgr B1 mostly appears to consist of extended ridges and shell-like structures
(Mehringer et al. 1992).
The measurement of the density of Sgr B1 equal to { $\sim 290$} cm$^{-3}$ (Table 12)
is in accord with this finding that the gas has no { truly} compact components.
Such morphology in an \ion{H}{2} region can arise only as the result of strong stellar winds dispersing the gas.
{ Such strong winds might originate in stars like the O4-6I and three WR WN7--8 stars
found in Sgr B1 (Mauerhan et al. 2010),
where the O star is located in a cavity in the dust at G0.52$-$0.046 (Ponti et al. 2015)
and one of the WR stars is in a possible bubble
(although on the other hand,
these stars might actually be unrelated to Sgr B1 and instead be some of the stars
that were stripped from the Arches or Quintuplet Clusters and have drifted many pc along the
plane of the Galaxy, Habibi et al. 2014).
However, there might be another indicator of significant age in that}
Nobukawa et al. (2008) and Ponti et al. (2015) suggest that there is a supernova remnant
at G0.42$-$0.04 in Sgr B1 (an area not covered by the IRS),
which they detect in X-ray observations with {\it Suzaku} and {\it XMM-Newton}, respectively.
The models of Table 13 require Starburst99 SEDs with ages of $10^{6.65}$ to $10^{6.7}$ yrs.
However, in contrast to the above suggestions of substantial age for Sgr B1,
I note that it does contain YSOs (e.g., An et al. 2011) and
OH and H$_2$O masers (Mehringer et al. 1993),
both indicators that star formation is currently occurring.
{ To summarize, the status of Sgr B1 still presents a major puzzle to the theory of star formation
in the orbital streams around the gravitational center of the Galaxy at Sgr A
proposed by Kruijssen et al. (2015).
In this theory molecular clouds are compressed and stars form as a result of close passage to Sgr A*
with Sgr B2 following Sgr B1 around an extreme of the orbit as viewed from the Earth,
Sgr B2 still on the front side of the orbit relative to the Earth but Sgr B1 already on the back side.
From their orbital positions, Sgr B2 would have an age of $\sim 0.7$ Myr and Sgr B1 would have an age
of $\sim 1.5$ Myr (Barnes et al. 2017).
The presence of YSOs in Sgr B1 would be compatible with this age although the $\sim 5$ Myr age
inferred from the best fitting models along with the dispersed appearance of the ionized gas in Figure 1
indicate a significantly larger age.
However, the location of Sgr B1 is not exactly on this orbital path
as seen in figure 4 of Barnes et al. (2017),
and there is substantially more extinction towards the supposedly closer Sgr B2 than Sgr B1.
This extra optical depth is seen in the extinction map of Figure~4
and in both formaldehyde absorption by Mehringer et al. (1995)
and in neutral hydrogen absorption by Lang et al. (2010).
For the age difference between Sgr B2 and Sgr B1 to be as large as the 4 Myr that
the combination of orbital kinematics and the ages inferred from the models suggest,
Sgr B1 would have to be much further along in its orbit and there would have to
be substantial line-of-sight separation between the two \ion{H}{2} regions.
On the other hand, the velocity structure of both Sgr B2 and Sgr B1 seems to indicate
that they are physically connected,
with both \ion{H}{2} regions consisting of multiple sources at different distances along the line of sight
interspersed with dense molecular cloud material
(Mehringer et al. 1995; Lang et al. 2010).
Clearly, this region has no simple description, and given the measured velocities,
there does not appear to be any good evidence
that Sgr B1 lies on the back side of the orbiting molecular cloud streams.
}
Sgr B1 will be discussed in more detail in a later paper (J. Simpson et al., in preparation).
\subsection{Sgr C}
\begin{figure}
\includegraphics[width=84mm]{fig13.eps}
\caption{
Plots of the ratios of the lines observed with {\it Spitzer} IRS (Section 3) computed from all the models
with zero rotation
in the grid of \ion{H}{2} regions and the ratios observed in Sgr C.
The points marked with asterisks include ionizing fluxes by blackbodies with $T = 10^{6.5}$~K
and the points marked with plus signs include ionizing fluxes by blackbodies with $T = 10^{6.0}$~K.
The colors of the points representing the models and the black symbols in both panels,
representing the line ratios observed in Sgr C,
are described in Figure 10.
}
\end{figure}
Sgr C, G359.43$-$0.08,
is the least luminous of the bright \ion{H}{2} regions discussed in this paper.
In addition to the thermal continuum of the \ion{H}{2} region, there appears
to be an associated non-thermal filament, G359.45$-$0.06, reminiscent of the radio arc
immediately adjacent and partially superposed on the Sickle and Arched Filaments
(Liszt 1985; Liszt \& Spiker 1995).
Tsuru et al. (2009) suggested that a nearby X-ray source, G359.41$-$0.12, is a supernova remnant
with an outflow in the direction of Sgr C,
although Ponti et al. (2015) find additional structures with {\it XMM-Newton}
and suggest that it is part of the Galactic Center Lobe.
Chuard et al. (2017) also find additional components and suggest that G359.41$-$0.12
may include multiple supernova remnants.
The line ratios observed in Sgr C are plotted in Figure~13
and the best fit models are tabulated in Table~13.
An age of { $\sim 10^{6.6}$} yrs is estimated for Sgr C ---
this is not unreasonable given its position at negative Galactic longitudes
in the models described in the Introduction.
{ However, there is also evidence of ongoing star formation in the presence
of a YSO with an `extended green object' outflow (Kendrew et al. 2013)
and the possible YSOs detected by An et al. (2011).
Thus this source appears to be similar to Sgr B1 with signs of both current star formation
along with indications of significantly evolved stars. }
\subsection{Other Sources}
In summary, there appears to be extensive low-excitation gas in the GC,
with a few regions of high-excitation gas that are confined to either known \ion{H}{2} regions
or the high-excitation sources of Tables 8 and 9.
There are two \ion{H}{2} regions seen in Figure~3b that are not otherwise discussed in this paper:
G0.00+0.20 and G359.25$-$0.25.
The former is probably the foreground \ion{H}{2} region S17 (Langer et al. 2017)
and the latter may also be foreground of the GC, given its relatively low $\tau_{9.6 \micron}$.
\subsection{\ion{H}{2} Regions and the Warm Ionized Medium of Galaxies}
Extragalactic astronomers are often urged to study the Milky Way's GC because
its relative nearness allows one to distinguish different phenomena that
are confused in the effectively large apertures used on external galaxies.
Figures 2 and 3 show that the GC has individual \ion{H}{2} regions/star-forming regions
widely spaced and surrounded by regions of lower density gas.
The locations of the very low-excitation gas are seen in Figure~3b,
where the [\ion{S}{3}] 33/[\ion{Si}{2}] 34 \micron\ ratio is
proportional to the ratio of S$^{++}$/Si$^+$,
which is the equivalent of the S$^{++}$/S$^+$ ratio (see Figure~8).
This low ratio is indicative of a low ionization parameter or a very dilute radiation field.
Such dilute radiation fields occur when ionizing photons escape from density-bounded \ion{H}{2} regions,
sometimes traveling great distances to high galactic latitudes (see Haffner et al. 2009 for a review).
The gas ionized by this dilute radiation field is known as the warm ionized medium (WIM).
Many galaxies have over half of their H$\alpha$ emission originating in the WIM (Oey et al. 2007).
Usually, visible observations of the WIM largely compare the lines of H$\alpha$,
[\ion{Ne}{2}], and [\ion{S}{2}] to [\ion{O}{3}], although
another possibility is the [\ion{Ne}{3}] 3869/[\ion{O}{2}] 3727 \AA\ ratio
for low-extinction galaxies (Levesque \& Richardson 2014).
To study the sources of the photons ionizing high-galactic latitude gas,
Pellegrini et al. (2012) devised a technique they call `ionization-parameter mapping',
wherein they form images of extra-galactic \ion{H}{2} regions in the [\ion{S}{2}]/[\ion{O}{3}] line ratios.
They notice that sources with noticeable [\ion{S}{2}]/[\ion{O}{3}] halos must be optically thick
in the Lyman continuum, such that no ionizing photons escape outside of the \ion{H}{2} region.
In contrast, \ion{H}{2} regions with no noticeable [\ion{S}{2}] must be optically thin (i.e., density bounded),
and ionizing photons do escape.
Blister \ion{H}{2} regions are optically thick (ionization-bounded) on one side and density-bounded on the other.
Similarly, the escape of ionizing radiation in starburst galaxies was studied by Zastrow et al. (2013),
who find, from their observations of [\ion{S}{3}], [\ion{S}{2}], and H$\alpha$, that
the star formation history and galaxy morphology are both important, with the radiation
mostly escaping through ionization cones produced by feedback from stellar winds and supernovae.
Although these visible wavelength lines cannot be detected in the GC, owing to the GC's extinction,
the two bright [\ion{S}{3}] 33 and [\ion{Si}{2}] 34 \micron\ lines can be used to test
the same questions regarding the WIM in the Milky Way's GC.
In Figure~3b it is seen that the [\ion{S}{3}] 33/[\ion{Si}{2}] 34 \micron\ line ratio
is high only in the region of the Arched Filaments and in Sgr B1 and B2,
but the ratio is substantially lower in the regions to the west of Sgr B and
all around Sgr C.
Even though the mapping is incomplete to the east of Sgr B,
it is not to the west, where the two \ion{H}{2} regions
must be ionization bounded, possibly by the thick molecular clouds
of the Dust Ridge, seen in Figure~1.
On the other hand,
I have already noted that the numbers of ionizing photons estimated from the radio luminosities
are much smaller than the numbers of photons estimated by counting stars
for the Arches and Quintuplet Clusters.
Since these clusters are well separated from their natal molecular clouds (e.g., Stolte et al. 2014),
a significant fraction of their ionizing photons probably do escape to the surrounding
ISM above and below the Galactic plane.
\subsection{X-rays in the Galactic Center}
The high-excitation [\ion{O}{4}] 26 \micron\ line was observed
in a number of starburst galaxies by Lutz et al. (1998) using {\it ISO}
and Dale et al. (2009) using {\it Spitzer}'s IRS.
Lutz et al. (1998) discussed several possible sources of the $> 54$~eV photons needed to ionize O$^{++}$ to O$^{3+}$,
including weak AGNs, low-metallicity OB stars and WR stars,
and ionizing shocks in starburst outflows.
They concluded the last is the most likely source of the O$^{3+}$.
In addition, Schaerer \& Stasi\'nska (1999) pointed out that the [\ion{O}{4}] 26 \micron\ line
could be easily formed by photoionization in `Wolf-Rayet galaxies'; however, their examples
are all low metallicity, which the GC is not.
Dale et al. (2009) and An et al. (2013) compared the [\ion{O}{4}] 26 \micron\ lines
to other lines observed with the IRS and concluded
that strong [\ion{O}{4}] 26 \micron\ lines are most commonly found in AGN.
In particular, An et al. (2013) observed that the line ratios found in the GC
are not compatible with those found in LINER galaxies or AGN;
they preferred the starburst designation for the GC.
However, there is still the problem that the high excitation [\ion{O}{4}] lines
are not seen in any of the giant \ion{H}{2} regions that are not low metallicity;
there must be some way to ionize the gas in addition to the shocks in starburst outflows.
Correcting for the larger numbers of ionizing photons of Table~13,
I estimate from the best-fit models that
the X-ray luminosities for the four regions are
{ $10^{38.9}$, $10^{38.7}$}, $10^{38.7}$, and $10^{38.7}$ erg s$^{-1}$ for
the Quintuplet Cluster region, the Arched Filaments, Sgr B1, and Sgr C, respectively.
This distribution is surprisingly flat across the GC --- I conclude that the source of
the 54 -- 77 eV photons ionizing O$^{3+}$ is not the Sgr A* black hole.
The models, of course, do not have exactly the same $N_{\rm Lyc}$ and density as the estimates
in Tables 13 and 12, respectively; however, if models are run with values of $N_{\rm Lyc}$ and $N_p$
from the observations and $R_{\rm inner}$ (for $R_S$) and filling factor $f$ adjusted using Equation (2)
to produce the same value of $U$,
the new models produce almost the same forbidden line ratios and almost the same $chisq$
if the X-ray luminosity is scaled by the same factor as $N_{\rm Lyc}$.
After adjusting for the actual numbers of ionizing photons, then,
the best-fitting models for the Arched Filaments, Sgr B1, and Sgr C
all have ages of $10^{6.6}$ to $10^{6.7}$ yrs and X-ray luminosities of $\sim 10^{38.7}$
{ to $\sim 10^{38.9}$} erg s$^{-1}$.
For producing the observed triply ionized oxygen, only the energy range of $\sim 55 - 100$ eV is important.
Integrating the $10^{6.6}$ yr SED with X-ray blackbody $T_{\rm BB} = 10^{6.5}$~K and
luminosity $10^{38.5}$ erg s$^{-1}$,
I find that the integrated luminosity for this energy range is $5.7 \times 10^{35}$ erg s$^{-1}$.
These extreme ultraviolet (EUV) photons are completely absorbed by hydrogen gas in the intervening ISM
and cannot be detected from Earth.
In the models they have been represented by the Rayleigh-Jeans tail of the black-body function
of the added X-rays (Figure 7),
but there is no reliable way to extrapolate this SED to energies observable with {\it Chandra}
or some other X-ray telescope;
bremsstrahlung SEDs, for example, could have a much lower integrated luminosity
than the blackbody SEDs of our models.
In fact, a thermal emission model could be tweaked to fit the observed 3 -- 10 keV {\it Suzaku} observations
of the Arches Cluster (Tsujimoto et al. 2007).
The source of the diffuse X-ray emission seen in the GC in both soft
and hard X-ray bands has no conclusive identification
(e.g., Ponti et al. 2015).
For example, Yusef-Zadeh et al. (2002) suggested that low-energy cosmic ray electrons
interact with the gas, thereby producing the diffuse bremsstrahlung X-ray emission as well as
fluorescent line emission,
whereas Muno et al. (2004) and Park et al. (2004) suggest the energy source
of the softer component of the emission is shocks from supernovae or the winds of O and WR stars.
On the other hand, as spatial resolution and sensitivity due to deep integrations improved,
more and more of the apparently diffuse hard X-ray emission has been resolved into point sources:
Muno et al. (2009) found 9017 point sources with {\it Chandra} in the 2 -- 8 keV range
and Hong et al. (2016) found 77 in the harder 3 -- 79 keV X-ray range with {\it NuSTAR}.
Most of these point sources are thought to be accreting white dwarfs (e.g., cataclysmic variables, CV,
Muno et al. 2006; see also the review of Mukai 2017) or binaries with active coronae.
Further observations of CVs with {\it Chandra}, {\it Swift}, and {\it Suzaku}
indicate that they could indeed make up a significant contribution
to the diffuse hard X-ray emission from the Galactic Ridge (e.g., Revnivtsev et al. 2009; Reis et al. 2013)
but not to the central GC itself (Nobukawa et al. 2016).
Not all of the {\it Chandra} point sources are CVs, however.
Thanks to the high spatial resolution of {\it Chandra}, NIR counterparts have been identified
for some of these point sources, and some of these have been determined
to be O supergiants or WR stars by NIR spectroscopy (Mauerhan et al. 2010).
I conclude that the EUV photons that triply ionize oxygen
to produce the observed [\ion{O}{4}] 26 \micron\ line
are probably the low-energy tails of the X-ray spectrum produced by the known sources in the GC:
accreting white dwarfs, hot O and WR stars with winds, and supernova remnants.
This would explain why the [\ion{O}{4}] line intensities are distributed widely across the GC ---
they are correlated with the mass distribution of the inner parts of the Galaxy
and not with the locations of the star forming regions such as Sgr B and Sgr C.
\section{Summary and Conclusions}
Spectra taken of the Galactic Center region with all four modules of the {\it Spitzer} IRS were
downloaded from the {\it Spitzer} Heritage Archive and analyzed with
SMART { and/or} CUBISM.
From these spectra, intensities were measured over a large fraction of the GC of the lines
H$_2$ S(0), S(1), S(2), and S(7), \ion{H}{1} 7--6 12.37 \micron,
[\ion{O}{4}] 26 \micron, [\ion{Ne}{2}] 12.8 \micron, [\ion{Ne}{3}] 15.6 \micron, [\ion{Si}{2}] 34 \micron,
[\ion{S}{1}] 25.3 \micron, [\ion{S}{3}] 18.7 and 33 \micron, [\ion{S}{4}] 10.5 \micron, [\ion{Cl}{2}] 14.4 \micron,
[\ion{Ar}{2}] 6.98 \micron, [\ion{Fe}{2}] 26 \micron, and [\ion{Fe}{3}] 23 \micron.
Locally strong intensities were measured for the lines
[\ion{Fe}{2}] 5.34 \micron\ and [\ion{S}{4}] 10.5 \micron\
but for most spectra only upper limits could be obtained for these lines
(in particular, the gas with strong [\ion{S}{4}] 10.5 \micron\ intensities may be foreground to the GC).
Extinction was estimated from the ratios of the [\ion{S}{3}] 18.7 and 33 \micron\ lines
and from the depths of the 9.6 \micron\ silicate absorption features
seen in the short-low module spectra.
In general, it is seen that the [\ion{Ne}{3}] 15.6/[\ion{S}{3}] 18.7 \micron\ line ratio
is fairly low, indicating ionization by late O stars.
From this one can infer either a dearth of the highest mass main-sequence stars
and/or substantial evolution away from the main sequence.
All together, after rejection of those spectra affected by saturation,
intensities measured from { 47,469} spectra are published in this paper (Tables 2 -- 7).
Serendipitously, seventeen of the locations on the sky, almost all very compact, show high excitation
characteristic of planetary nebulae or high-velocity shocks (Tables 8 and 9).
In addition to significantly stronger [\ion{O}{4}] 26 \micron\ line emission compared to
their surroundings, these regions
sometimes exhibit emission in the [\ion{Ne}{5}] 14.3 and 24.3 \micron\ lines.
{ I suggest that the high excitation sources that have compact radio counterparts
be investigated as candidate planetary nebulae;
the shocked sources may be indicators of outflows from so-far undetected, isolated massive hot stars
that have been modeled as escapees from the Arches or Quintuplet Clusters by Habibi et al. (2014).}
At the other extreme, emission from low-velocity shocks,
characterized by emission in the [\ion{S}{1}] 25.3 \micron\ line,
was detected in { 24} locations on the sky (Table 10).
Twenty-three candidate YSOs or sources behind dense molecular clouds
with ice absorption features at 6.0 and 6.8 \micron\ were also detected (Table 11).
Electron densities and ionic abundances were estimated from the line intensities
for the GC \ion{H}{2} regions usually named Sgr B1, Sgr C, the Arched Filaments, and
the Sickle, here called the Quintuplet Region to include somewhat more area on the sky.
{ The average densities ranged from 270 to 310 cm$^{-3}$.}
I conclude that the hot stars in these \ion{H}{2} regions are not
ionizing gas from local molecular clouds but that the natal molecular { clouds have} already been dispersed
by the strong winds from O and WR stars, and possibly a supernova in Sgr B1.
{ Other possibilities include the idea that
the ionized gas consists of the lit-up edges of dense molecular clouds,
particularly in those regions close to Sgr A (e.g., Lang et al. 2001).}
The computed ionic abundances range from
{ 1.55 -- $1.97 \times 10^{-4}$ for the (Ne$^+$ + Ne$^{++}$)/H$^+$ ratio
and 0.84 -- $2.09 \times 10^{-5}$ for the (S$^{++}$ + S$^{3+}$)/H$^+$ ratio.}
Although the former ratio is a good approximation to the Ne/H ratio,
there being very little neutral or triply ionized neon in \ion{H}{2} regions,
the latter ratio needs some sort of correction for singly and quadruply ionized sulfur.
S$^+$ is especially prevalent in regions with very low ionization parameters
and is probably the reason why the measurement for Sgr C diverges so much
from that of the other \ion{H}{2} regions (Table 12, Figure~3b).
The Ar$^+$/Ne$^+$ abundance ratio was measured to be { $\sim 0.032$};
since the median of the ratio of the ionization fractions of Ar$^+$ and Ne$^+$
is measured to be 0.90 for the computed models,
the inferred Ar/Ne abundance ratio is $\sim 10$\% larger, or { $\sim 0.036$}.
I conclude from these measurements that the Galactic Center,
{ with Ne/H, S/H, and Ar/H ratios of $\sim 1.7 \times 10^{-4}$, $\sim 1.9 \times 10^{-5}$, and $\sim 6.2 \times 10^{-6}$, respectively,}
has { abundances somewhat higher than Solar abundances}
(Ekstr\"om et al. 2012 assumed $1.29 \times 10^{-4}$ for Ne/H, which, { like Ar/H}, cannot be measured in the Sun,
and Scott et al. 2015 inferred $1.35 \times 10^{-5}$ for S/H
from their analysis of Solar spectra).
The average measurements of Si$^+$/H$^+$ range from { 1.17 to $1.46 \times 10^{-5}$};
{ after correcting for higher ionization stages,
the average value of gas-phase Si/H equals $2.4 \times 10^{-5}$,
whereas the Solar value of Si/H equals $3.3 \times 10^{-5}$ (Scott et al. 2015).
Thus, even though the estimated gas-phase abundance of Si/H is still below that of the Solar Si/H ratio,
it is }
much more than what is often assumed for the abundances used in modeling \ion{H}{2} regions or PDRs.
Because the observed [\ion{Ne}{3}] 15.6/[\ion{S}{3}] 18.7 \micron\ line ratios are
very sensitive to the input SEDs of the ionizing stars
and the [\ion{S}{3}] 33/[\ion{Si}{2}] 34 \micron\ line ratios are
very sensitive to the ionization parameters,
I performed tests of both by modeling the \ion{H}{2} region line ratios with Cloudy
{ (Ferland et al. 2017)}.
This is a well known procedure for estimating the effective temperatures
of individual stars exciting small \ion{H}{2} regions and/or testing the SEDs produced
by various stellar atmosphere codes (e.g., Simpson et al. 2004; Rubin et al. 2008).
Here for the first time I estimate the ages of the four GC massive star clusters
by employing the composite spectra suitable for the clusters
from Starburst99 (Leitherer et al. 2014).
There are two significant results:
(1) { The ages estimated for the clusters, $4.6$ Myr for Sgr B1, $4.5$ Myr for the Arches Cluster,
$3.0$ Myr for the Quintuplet Cluster, and $3.8$ Myr for Sgr C,
do not correspond to the formation sequence predicted by recent models
of the gas streams in the GC by Kruijssen et al. (2015).
In the orbiting gas stream model, the positions of Sgr B1 and Sgr C past pericenter approach to Sgr A*
correspond to ages of
$\sim 1.5$ and $\sim 3.6$ Myr, respectively (Barnes et al. 2017; Kruijssen et al. 2015);
the molecular clouds at the locations of the Arches and Quintuplet Clusters
are intermediate (Kruijssen et al. 2015).
The two clusters
must have formed a full orbit earlier than their current positions would suggest
(Kruijssen et al. 2015; Habibi et al. 2014).
Considering the overlap in models of different ages seen in Figures 10 -- 13
and the spread in acceptable models in Table~13,
I estimate that there is an uncertainty of $\sim 0.5$ Myr in the ages estimated for the clusters
through the fitting of these models (that is, one step in either direction in this fairly coarse model grid).
With these uncertainties, the only real outlier among the four sources is Sgr B1.
Here the 4.6 Myr age is not in particular disagreement with its dispersed appearance,
but it is in disagreement with the velocity measurements that indicate that Sgr B1 is
part of the same system with the known much-younger Sgr B2 (e.g., Lang et al. 2010; Mehringer et al. 1992)
or if not, that it is on the back stream past Sgr B2 but still with a positional age of only 1.5 Myr
(Barnes et al. 2017).
I suggest that the velocity agreement is accidental and that Sgr B1 is, in fact,
of a larger age and lies on a continuation of the stream from Sgr C past Sgr A to higher Galactic longitudes
on the front side of the orbit with its lesser extinction.
Another problem concerns the respective ages for the Arches and Quintuplet Clusters,
estimated by Schneider et al. (2014) to be $3.5 \pm 0.7$ and $4.8 \pm 1.1$ Myr, respectively,
in contrast to the ages estimated from the model fits.
Comparing the [\ion{Ne}{3}] 15/[\ion{S}{3}] 19 \micron\ ratios for the Arched Filament region
and the Quintuplet Cluster region seen in Figures 10 and 11,
the excitation is without doubt substantially higher in the gas excited by the Quintuplet Cluster compared
to the gas excited by the Arches Cluster.
If this excitation is caused by stellar photons, the only possible conclusion is that the Quintuplet Cluster
contains hotter stars than the Arches Cluster, and in models of the spectral evolution of the SEDs
produced by massive clusters such as predicted by Starburst99 (e.g., Leitherer et al. 2014),
hotter stars mean the cluster is younger.
However, if the Quintuplet Cluster really is older,
as seems to be convincingly shown by Schneider et al. (2014),
its higher excitation would have to be produced by some means other than stellar photons.
Shocks from stellar winds or a possible supernova remnant are the most likely possibilities.
Thus I conclude that the current cluster models as applied to these \ion{H}{2} regions are inadequate.
Consequently, because our detailed studies of the Galaxy should be informing us
on the conditions in the rest of the Universe,
I suggest that such additional sources of excitation should be considered
by modelers of extragalactic \ion{H}{2} regions in starburst galaxies.
}
(2) It is not possible to predict {\it all} the observed line ratios from the SEDs
from the 2014 version of Starburst99 ---
additional photons with energies $\gtrsim 55$ eV are needed to produce
the wide-spread detected [\ion{O}{4}] 26 \micron\ line.
The 55 -- 100 eV photons required for the models to fit the data
are probably the low energy tails of the X-rays observed by {\it Chandra},
{\it XMM-Newton}, {\it Suzaku}, etc.
These X-rays could have several possible origins:
the low-mass X-ray sources of the old stars populating most of the mass of the GC,
X-rays from the energetic shocks of supernova remnants,
or the Starburst99 model SEDs inadequately representing the interactions of the stellar winds
of the multiple massive stars found in the massive clusters of their models.
Probably there are contributions from all these sources of high energy photons in the GC.
Future observers may well want to use the observations discussed in this paper
to plan additional observations of GC sources.
In addition to that found at numerous ground-based observatories,
there will be MIR spectroscopic capability out to 28.5 \micron\ with
the Mid-Infrared Instrument (MIRI) on NASA's {\it James Webb Space Telescope} ({\it JWST}),
to be launched in { 2020}.
For longer wavelengths it will be necessary to use the instruments on
the Stratospheric Observatory for Infrared Astronomy ({\it SOFIA}),
such as for additional observations of the [\ion{S}{3}] 33 and [\ion{Si}{2}] 34 \micron\ lines,
which are very bright at many locations.
Above all, what is needed is a more complete identification
of the OB and WR stars that are not in clusters but
that contribute the diffuse ionizing photons that are ubiquitous in the GC.
\acknowledgments
This work is based on observations made with the {\it Spitzer
Space Telescope}, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with NASA.
The IRS was a collaborative venture between Cornell University
and Ball Aerospace Corporation
funded by NASA through the Jet Propulsion Laboratory and Ames Research Center.
SMART was developed by the IRS Team at Cornell University and is available
through the Spitzer Science Center at Caltech.
This research has made use of the NASA/IPAC Infrared Science Archive, which is operated
by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration.
I acknowledge {\it SOFIA} grant 04\_0113 for the payment of the page charges.
I thank Angela Cotera for reading the manuscript
{ and the referee for the thoughtful and detailed comments that improved the presentation in this paper}.
\vspace{5mm}
\facility{Spitzer(IRS)}
\software{Cloudy \citep{cloudy17},
CUBISM \citep{cubism},
PAHFIT \citep{pahfit}
}
|
{
"timestamp": "2019-01-23T02:26:57",
"yymm": "1803",
"arxiv_id": "1803.02806",
"language": "en",
"url": "https://arxiv.org/abs/1803.02806"
}
|
\section{Introduction}
Learning meaningful representations of data is an important step for models to understand the world \cite{Bengio2013}. Recently, the Generative Adversarial Network (GAN) \cite{Goodfellow2014} has been proposed as a method that can learn characteristics of data distributions without the need for labels. GANs traditionally consist of a generator $G$, which generates data from randomly sampled vectors $Z$, and a discriminator $D$, which tries to distinguish generated data from real data $x$. During training, the generator learns to generate realistic data samples $G(Z)$, while the discriminator becomes better at distinguishing between the generated and the real data $x$. As a result, both the generator and the discriminator learn characteristics about the underlying data distribution without the need for any labels \cite{Radford2015}.
One desirable characteristic of learned representations is disentanglement \cite{Bengio2013}, which means that different parts of the representation encode different factors of the data-generating distribution. This makes representations more interpretable, easier to modify, and is a useful property for many tasks such as classification, clustering, or image captioning.
To achieve this, Chen et al. \cite{Chen2016} introduced a GAN variant in which the generator's input is split into two parts $z$ and $c$. Here, $z$ encodes unstructured noise while $c$ encodes meaningful, data-generating factors. Through enforcing high mutual information between $c$ and and the generated images $G(z, c)$ the generator is trained using the inputs $c$ as meaningful encodings for certain image characteristics. For example, a ten-dimensional categorical code for $c$ could represent the ten different digit classes in the MNIST data set. Since no labels are provided the generator has to learn by itself which image characteristics can be represented through $c$.
One drawback of this model is that the only way to perform inference, i.e.\ map real data samples into a (disentangled) representation, is to use the discriminator. However, there is no guarantee that the discriminator learns good representations of the data in general, as it is trained to discriminate between real and generated data and may therefore focus only on features that are helpful for discriminating these two, but are not necessarily descriptive of the data distribution in general \cite{Donahue2017}.
Zhang et al. \cite{Zhang2017} tried to enforce disentangled representations in order to improve the controllability of the generator. The latent representation is split up into two parts encoding meaningful information and unknown factors of variation. Two additional inference networks are introduced to enforce the disentanglement between the two parts of the latent representation. While this setup yields a better controllability over the generative process it depends on labeled samples for its training objective and can not discover unknown data-generating factors, but only encodes known factors of variation (obtained through labels) in its disentangled representation.
Donahue et al. \cite{Donahue2017} and Dumoulin et al. \cite{Dumoulin2017} introduced an extension which includes an encoder $E$ that learns the encodings of real data samples. The discriminator gets as input both the data sample $x$ (either real or generated) and the according representation (either $Z$ or $E(x)$) and has to classify them as either coming from the generator or the encoder. The generator and the encoder try to fool the discriminator into misclassifying the samples. As a result, the encoder $E$ learns to approximate the inverse of the generator $G$ and can be used to map real data samples into representations for other applications. However, in these approaches the representations follow a simple prior, e.g.\ a Gaussian or uniform distribution, and do not exhibit any disentangled properties.
Our model, the Bidirectional-InfoGAN, integrates some of these approaches by extending traditional GANs with an encoder that learns disentangled representations in an unsupervised setting. After training, the encoder can map data points to meaningful, disentangled representations which can potentially be used for different tasks such as classification, clustering, or image captioning. Compared to the InfoGAN \cite{Chen2016} we introduce an encoder to mitigate the problems of using a discriminator for both the adversarial loss and the inference task. Unlike the Structured GAN \cite{Zhang2017} our training procedure is completely unsupervised, can detect unknown data-generating factors, and only introduces one additional inference network (the encoder). In contrast to the Bidirectional GAN \cite{Donahue2017, Dumoulin2017} we replace the simple prior on the latent representation with a distribution that is amenable to disentangled representations and introduce an additional loss for the encoder and the generator to achieve disentangled representations.
On the MNIST, CelebA \cite{Liu2015}, and SVHN \cite{Netzer2011} data sets we show that the encoder does learn interpretable representations which encode meaningful properties of the data distribution. Using these we can sample images that exhibit certain characteristics, e.g. digit identity and specific stroke widths for the MNIST data set, or different hair colors and clothing accessories in the CelebA data set.
\begin{figure}
\centering
\input{binfog-tikz.tex}
\vspace{-0.5cm}
\caption{High-level overview of the Bidirectional-InfoGAN. The generator $G$ generates images from the vector $(z, c)$ and tries to fool the discriminator into classifying them as real. The encoder $E$ encodes images into a representation and tries to fool the discriminator $D$ into misclassifying them as fake if its input is a real image while trying to approximate $P(c\vert x)$ if its input is a generated image.}
\vspace{-0.3cm}
\label{fig:model_architecture}
\end{figure}
\section{Methodology}
Our model, shown in Fig.~\ref{fig:model_architecture}, consists of a generator $G$, a discriminator $D$, and an encoder $E$, which are implemented as neural networks. The input vector $Z$ that is given to the generator $G$ is made up of two parts $Z = (z, c)$. Here, $z$ is sampled from a uniform distribution, $z\sim U(-1, 1)$, and is used to represent unstructured noise in the images. On the other hand, $c$ is the part of the representation that encodes meaningful information in a disentangled manner and is made up of both categorical values $c_{\text{cat}}$ and continuous values $c_{\text{cont}}$. $G$ takes $Z$ as input and transforms it into an image $x$, i.e.\ $G: Z\rightarrow x$.
$E$ is a convolutional network that gets as input either real or fake images and encodes them into a latent representation $E: x\rightarrow Z$. $D$ gets as input an image $x$ and the corresponding representation $Z$ concatenated along the channel axis. It then tries to classify the pair as coming either from the generator $G$ or the encoder $E$, i.e.\ $D: Z\times x\rightarrow \{0,1\}$, while both $G$ and $E$ try to fool the discriminator into misclassifying its input. As a result the original GAN minimax game \cite{Goodfellow2014} is extended and becomes:
\[\underset{G, E}{\text{min}}\ \underset{D}{\text{max}}\ V(D, G, E) = \mathbb{E}_{x\sim P_{\text{data}}}[logD(x, E(x))] + \mathbb{E}_{Z\sim P_Z}[log(1-D(G(Z), Z))],\]
where $V(D, G, E)$ is the adversarial cost as depicted in Fig.~\ref{fig:model_architecture}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.8\textwidth}
\begin{footnotesize}
\def\linewidth{\linewidth}
\def0.5{0.5}
\import{}{mnist_disc.ps_tex}
\end{footnotesize}
\vspace{-0.6cm}
\caption{}
\label{fig:mnist:discrete}
\end{subfigure}
\begin{subfigure}[b]{0.8\textwidth}
\begin{footnotesize}
\def\linewidth{\linewidth}
\def0.5{0.5}
\import{}{mnist_cont.ps_tex}
\end{footnotesize}
\vspace{-0.4cm}
\caption{}
\label{fig:mnist:continuous}
\end{subfigure}
\caption{Images sampled from the MNIST test set. (a) Each row represents one value of the ten-dimensional code $c_1$, which encodes different digits despite never seeing labels during the training process. (b) Images with maximum and minimum values for $c_2$ and $c_3$ for each categorical value from $c_1$.}
\vspace{-0.3cm}
\label{fig:mnist}
\end{figure}
In order to force the generator to use the information provided in $c$ we maximize the mutual information $I$ between $c$ and $G(z,c)$. Maximizing the mutual information directly is hard, as it requires the posterior $P(c\vert x)$ and we therefore follow the approach by Chen et al. \cite{Chen2016} and define an auxiliary distribution $E(c\vert x)$ to approximate $P(c\vert x)$. We then maximize the lower bound $L_I(G, E) = \mathbb{E}_{c\sim P(c), z\sim P(z), x\sim G(z,c)}[log\ E(c\vert x)]+H(c) \leq I(c;G(z,c))$, where $L_I(G, E)$ is the mutual information depicted in Fig.~\ref{fig:model_architecture}. For simplicity reasons we fix the distribution over $c$ and, therefore, the entropy term $H(c)$ is treated as a constant. In our case $E$ is the encoder network which gets images generated by $G$ as input and is trained to approximate the unknown posterior $P(c\vert x)$. For categorical $c_{\text{cat}}$ we use the softmax nonlinearity to represent $E(c_{\text{cat}}\vert x)$ while we treat the posterior $E(c_{\text{cont}}\vert x)$ of continuous $c_{\text{cont}}$ as a factored Gaussian. Given this structure, the minimax game for the Bidirectional-InfoGAN (BInfoGAN) is then
\[\underset{G, E}{\text{min}}\ \underset{D}{\text{max}}\ V_\text{BInfoGAN}(D, G, E) = V(D, G, E) - \lambda L_I(G, E)\]
where $\lambda$ determines the strength of the impact of the mutual information criterion $L_I$ and is set to $1.0$ in all our experiments.
\section{Experiments}
We perform experiments on the MNIST, the CelebA \cite{Liu2015}, and the SVHN \cite{Netzer2011} data set. While the final performance of the model is likely influenced by choosing the ``optimal'' characteristics for $c$ this is usually not possible, since we do not know all data-generating factors beforehand. When choosing the characteristics and dimensionality of the disentangled vector $c$ we therefore mostly stick with the values previously chosen by Chen et al. \cite{Chen2016}.
For further information on the network architectures and more examples of the learned characteristics on the different data sets see our Git: \url{https://github.com/tohinz/Bidirectional-InfoGAN}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.8\textwidth}
\begin{footnotesize}
\def\linewidth{\linewidth}
\def0.5{0.5}
\import{}{celeba_discrete.ps_tex}
\end{footnotesize}
\vspace{-0.6cm}
\caption{}
\label{fig:celeba}
\end{subfigure}
\begin{subfigure}[b]{0.8\textwidth}
\begin{footnotesize}
\def\linewidth{\linewidth}
\def0.5{0.5}
\import{}{svhn_discrete.ps_tex}
\end{footnotesize}
\vspace{-0.6cm}
\caption{}
\label{fig:svhn}
\end{subfigure}
\caption{Images sampled from the (a) CelebA and (b) SVHN test sets. Each row shows images sampled according to one specific categorical variable $c_{\text{cat}}$ which represents a learned characteristic.}
\vspace{-0.3cm}
\label{fig:celeba:svhn}
\end{figure}
On the MNIST data set we model the latent code $c$ with one categorical variable $c_1\sim Cat(K=10, p=0.1)$ and two continuous variables $c_2, c_3\sim U(-1,1)$. During the optimization process and without the use of any labels the encoder learns to use $c_1$ to encode different digit classes, while $c_2$ and $c_3$ encode stroke width and digit rotation. Fig.~\ref{fig:mnist:discrete} shows images randomly sampled from the test set according to the ten different categorical values. We can see that the encoder has learned to reliably assign a different categorical value for different digits. Indeed, by manually matching the different categories in $c_1$ to a digit type, we achieve a test set accuracy of 96.61\% ($\pm 0.32\%$, averaged over 10 independent runs) without ever using labels during the training, compared to Chen et al. \cite{Chen2016} (unsupervised) with an accuracy of 95\%, and Zhang et al. \cite{Zhang2017} (semi-supervised, 20 labels) with an accuracy of 96\%. Fig.~\ref{fig:mnist:continuous} shows images sampled from the test set for different values of $c_2$ and $c_3$. We see that we can use the encodings from $E$ to now sample for digits with certain characteristics such as stroke width and rotation, even though this information was not explicitly provided during training.
On the CelebA data set the latent code is modeled with four categorical codes $c_1, c_2, c_3, c_4\sim Cat(K=10, p=0.1)$ and four continuous variables $c_5, c_6, c_7, c_8\sim U(-1,1)$. Again, the encoder learns to associate certain image characteristics with specific codes in $c$. This includes characteristics such as the presence of glasses, hair color, and background color and is visualized in Fig.~\ref{fig:celeba}.
On the SVHN data set we use the same network architecture and latent code representations as for the CelebA data set. Again, the encoder learns interpretable, disentangled representations encoding characteristics such as image background, contrast and digit type. See Fig.~\ref{fig:svhn} for examples sampled from the SVHN test set. These results indicate that the Bidirectional-InfoGAN is indeed capable of mapping data points into disentangled representations that encode meaningful characteristics in a completely unsupervised manner.
\section{Conclusion}
We showed that an encoder coupled with a generator in a Generative Adversarial Network can learn disentangled representations of the data without the need for any explicit labels. Using the encoder network we maximize the mutual information between certain parts of the generator's input and the images that are generated from it. Through this the generator learns to associate certain image characteristics with specific parts of its input. Additionally, the adversarial cost from the discriminator forces both the generator to generate realistic looking images and the encoder to approximate the inverse of the generator, leading to disentangled representations that can be used for inference.
The learned characteristics are often meaningful and humanly interpretable, and can potentially help with other tasks such as classification and clustering. Additionally, our method can be used as a pre-training step on unlabeled data sets, where it can lead to better representations for the final task. However, currently we have no influence over which characteristics are learned in the unsupervised setting which means that the model can also learn characteristics or features that are meaningless or not interpretable by humans. In the future, this can be mitigated by combining our approach with semi-supervised approaches, in which we can supply a limited amount of labels for the characteristics we are interested in to exert more control over which data-generating factors are learned while still being able to discover ``new'' generating factors which do not have to be known or specified beforehand.
\begin{footnotesize}
\bibliographystyle{unsrt}
|
{
"timestamp": "2018-03-08T02:07:50",
"yymm": "1803",
"arxiv_id": "1803.02627",
"language": "en",
"url": "https://arxiv.org/abs/1803.02627"
}
|
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